awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
d : Set α → ℝ :=
  fun (s : Set α) ↦
    (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ) - (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ)
c : Set ℝ := d '' {s : Set α | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hν : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
to_nnreal_μ : ∀ (s : Set α), (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (μ : (a : Set α) → ℝ≥0∞) s
to_nnreal_ν : ∀ (s : Set α), (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (ν : (a : Set α) → ℝ≥0∞) s
d_split : ∀ (s t : Set α) (ht : MeasurableSet t), d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α) (hm : Monotone s), Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α) (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n),
    Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ (s : Set α), MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun (n m : ℕ) ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ} (hab : a ≤ b) (hcd : c ≤ d), f a d ⊆ f b c
f_succ : ∀ (n m : ℕ) (hnm : n ≤ m), f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ) (h : m ≤ n), γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ sᶜ
rw : d s + d t = d ((s ∪ t) \ t) + d ((s ∪ t) ∩ t)
rw₁ : d s + d t = d (s \ t) + d ((s ∪ t) ∩ t)
⊢ d s + d t = d (s ∪ t)
have rw₂ : d s + d t = d (s \ t) + d t
Tactic: rw [d_split (s ∪ t) _ ht, union_diff_right, union_inter_cancel_right,
              (subset_compl_iff_disjoint_left.1 hts).sdiff_eq_left]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
d : Set α → ℝ :=
  fun (s : Set α) ↦
    (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ) - (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ)
c : Set ℝ := d '' {s : Set α | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hν : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
to_nnreal_μ : ∀ (s : Set α), (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (μ : (a : Set α) → ℝ≥0∞) s
to_nnreal_ν : ∀ (s : Set α), (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (ν : (a : Set α) → ℝ≥0∞) s
d_split : ∀ (s t : Set α) (ht : MeasurableSet t), d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α) (hm : Monotone s), Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α) (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n),
    Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ (s : Set α), MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun (n m : ℕ) ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ} (hab : a ≤ b) (hcd : c ≤ d), f a d ⊆ f b c
f_succ : ∀ (n m : ℕ) (hnm : n ≤ m), f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ) (h : m ≤ n), γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ sᶜ
rw : d s + d t = d ((s ∪ t) \ t) + d ((s ∪ t) ∩ t)
rw₁ : d s + d t = d (s \ t) + d ((s ∪ t) ∩ t)
rw₂ : d s + d t = d (s \ t) + d t
⊢ d s + d t = d (s ∪ t)
have rw₃ : d s + d t = d s + d t
Tactic: rw [d_split (s ∪ t) _ ht, union_diff_right, union_inter_cancel_right,
              (subset_compl_iff_disjoint_left.1 hts).sdiff_eq_left]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
d : Set α → ℝ :=
  fun (s : Set α) ↦
    (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ) - (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ)
c : Set ℝ := d '' {s : Set α | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hν : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
to_nnreal_μ : ∀ (s : Set α), (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (μ : (a : Set α) → ℝ≥0∞) s
to_nnreal_ν : ∀ (s : Set α), (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (ν : (a : Set α) → ℝ≥0∞) s
d_split : ∀ (s t : Set α) (ht : MeasurableSet t), d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α) (hm : Monotone s), Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α) (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n),
    Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ (s : Set α), MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun (n m : ℕ) ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ} (hab : a ≤ b) (hcd : c ≤ d), f a d ⊆ f b c
f_succ : ∀ (n m : ℕ) (hnm : n ≤ m), f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ) (h : m ≤ n), γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ sᶜ
⊢ d (s ∪ t) ≤ γ + 0
have rw : d (s ∪ t) ≤ γ
Tactic: rw [add_zero]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (hypothesis)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
d : Set α → ℝ :=
  fun (s : Set α) ↦
    (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ) - (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ)
c : Set ℝ := d '' {s : Set α | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hν : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
to_nnreal_μ : ∀ (s : Set α), (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (μ : (a : Set α) → ℝ≥0∞) s
to_nnreal_ν : ∀ (s : Set α), (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (ν : (a : Set α) → ℝ≥0∞) s
d_split : ∀ (s t : Set α) (ht : MeasurableSet t), d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α) (hm : Monotone s), Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α) (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n),
    Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this : ∀ (n : ℕ), ∃ (s : Set α), MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun (n m : ℕ) ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ} (hab : a ≤ b) (hcd : c ≤ d), f a d ⊆ f b c
f_succ : ∀ (n m : ℕ) (hnm : n ≤ m), f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ) (h : m ≤ n), γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ sᶜ
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
have this₁ : d t ≤ 0
Tactic: have : d t ≤ 0 :=
      (add_le_add_iff_left γ).1 <|
        calc
          γ + d t ≤ d s + d t := by gcongr
          _ = d (s ∪ t) := by
            rw [d_split (s ∪ t) _ ht, union_diff_right, union_inter_cancel_right,
              (subset_compl_iff_disjoint_left.1 hts).sdiff_eq_left]
          _ ≤ γ + 0 := by rw [add_zero]; exact d_le_γ _ (hs.union ht)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
d : Set α → ℝ :=
  fun (s : Set α) ↦
    (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ) - (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ)
c : Set ℝ := d '' {s : Set α | MeasurableSet s}
γ : ℝ := sSup c
hμ : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
hν : ∀ (s : Set α), Ne (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) (∞ : ℝ≥0∞)
to_nnreal_μ : ∀ (s : Set α), (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (μ : (a : Set α) → ℝ≥0∞) s
to_nnreal_ν : ∀ (s : Set α), (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) s)) : ℝ≥0∞) = (ν : (a : Set α) → ℝ≥0∞) s
d_split : ∀ (s t : Set α) (ht : MeasurableSet t), d s = d (s \ t) + d (s ∩ t)
d_Union : ∀ (s : ℕ → Set α) (hm : Monotone s), Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))
d_Inter :
  ∀ (s : ℕ → Set α) (hs : ∀ (n : ℕ), MeasurableSet (s n)) (hm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n),
    Tendsto (fun (n : ℕ) ↦ d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))
bdd_c : BddAbove c
c_nonempty : c.Nonempty
d_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ
this_1 : ∀ (n : ℕ), ∃ (s : Set α), MeasurableSet s ∧ γ - (1 / 2) ^ n < d s
e : ℕ → Set α
he : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)
he₁ : ∀ (n : ℕ), MeasurableSet (e n)
he₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)
f : ℕ → ℕ → Set α := fun (n m : ℕ) ↦ (Finset.Ico n (m + 1)).inf e
hf : ∀ (n m : ℕ), MeasurableSet (f n m)
f_subset_f : ∀ {a b c d : ℕ} (hab : a ≤ b) (hcd : c ≤ d), f a d ⊆ f b c
f_succ : ∀ (n m : ℕ) (hnm : n ≤ m), f n (m + 1) = f n m ∩ e (m + 1)
le_d_f : ∀ (n m : ℕ) (h : m ≤ n), γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)
s : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n
γ_le_d_s : γ ≤ d s
hs : MeasurableSet s
t : Set α
ht : MeasurableSet t
hts : t ⊆ sᶜ
this : d t ≤ 0
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
have refine_2 : (↑(ENNReal.toNNReal ((μ : (a : Set α) → ℝ≥0∞) t)) : ℝ) ≤ (↑(ENNReal.toNNReal ((ν : (a : Set α) → ℝ≥0∞) t)) : ℝ)
Tactic: rw [← to_nnreal_μ, ← to_nnreal_ν, ENNReal.coe_le_coe, ← NNReal.coe_le_coe]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
⊢ ∃ (s : Set α), IsHahnDecomposition μ ν s
have obtain : ∀ (s : Set α) (hs : MeasurableSet s)
  (h₁ :
    ∀ (t : Set α),
      MeasurableSet t →
        t ⊆ s → LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞))
  (h₂ :
    ∀ (t : Set α),
      MeasurableSet t →
        t ⊆ sᶜ → LE.le (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)),
  ∃ (s : Set α), IsHahnDecomposition μ ν s
Tactic: obtain ⟨s, hs, h₁, h₂⟩ := hahn_decomposition ν μ
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
s : Set α
hs : MeasurableSet s
h₁ :
  ∀ (t : Set α),
    MeasurableSet t → t ⊆ s → LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
h₂ :
  ∀ (t : Set α),
    MeasurableSet t → t ⊆ sᶜ → LE.le (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
⊢ ∃ (s : Set α), IsHahnDecomposition μ ν s
have refine_1 : μ.restrict s ≤ ν.restrict s
Tactic: refine ⟨s, hs, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
s : Set α
hs : MeasurableSet s
h₁ :
  ∀ (t : Set α),
    MeasurableSet t → t ⊆ s → LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
h₂ :
  ∀ (t : Set α),
    MeasurableSet t → t ⊆ sᶜ → LE.le (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
refine_1 : μ.restrict s ≤ ν.restrict s
⊢ ∃ (s : Set α), IsHahnDecomposition μ ν s
have refine_2 : ν.restrict sᶜ ≤ μ.restrict sᶜ
Tactic: refine ⟨s, hs, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
s : Set α
hs : MeasurableSet s
h₁ :
  ∀ (t : Set α),
    MeasurableSet t → t ⊆ s → LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
h₂ :
  ∀ (t : Set α),
    MeasurableSet t → t ⊆ sᶜ → LE.le (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
⊢ μ.restrict s ≤ ν.restrict s
have refine_1 : ∀ (s_1 : Set α),
  MeasurableSet s_1 →
    LE.le (α := ℝ≥0∞) ((μ.restrict s : (a : Set α) → ℝ≥0∞) s_1 : ℝ≥0∞) ((ν.restrict s : (a : Set α) → ℝ≥0∞) s_1 : ℝ≥0∞)
Tactic: rw [Measure.le_iff]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.Hahn (goal)
Goal:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
s : Set α
hs : MeasurableSet s
h₁ :
  ∀ (t : Set α),
    MeasurableSet t → t ⊆ s → LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
h₂ :
  ∀ (t : Set α),
    MeasurableSet t → t ⊆ sᶜ → LE.le (α := ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞)
⊢ ν.restrict sᶜ ≤ μ.restrict sᶜ
have refine_2 : ∀ (s_1 : Set α),
  MeasurableSet s_1 →
    LE.le (α := ℝ≥0∞) ((ν.restrict sᶜ : (a : Set α) → ℝ≥0∞) s_1 : ℝ≥0∞)
      ((μ.restrict sᶜ : (a : Set α) → ℝ≥0∞) s_1 : ℝ≥0∞)
Tactic: rw [Measure.le_iff]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv (goal)
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsProbabilityMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (ε := ℝ) (fun (x : 𝓧) ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Ici 0) x
⊢ ContinuousWithinAt f (Ici 0) x
have inl : ∀ (hx : Membership.mem (γ := Set ℝ) (Ici 0) 0), ContinuousWithinAt f (Ici 0) 0
Tactic: rcases eq_or_lt_of_le (α := ℝ) (hx : 0 ≤ x) with rfl | hx_pos
Cleared hypotheses: x, hx
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv (goal)
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsProbabilityMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (ε := ℝ) (fun (x : 𝓧) ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Ici 0) x
inl : ∀ (hx : Membership.mem (γ := Set ℝ) (Ici 0) 0), ContinuousWithinAt f (Ici 0) 0
⊢ ContinuousWithinAt f (Ici 0) x
have inr : ∀ (hx_pos : 0 < x), ContinuousWithinAt f (Ici 0) x
Tactic: rcases eq_or_lt_of_le (α := ℝ) (hx : 0 ≤ x) with rfl | hx_pos
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv (hypothesis)
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsProbabilityMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (ε := ℝ) (fun (x : 𝓧) ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Ici 0) x
hx_pos : 0 < x
⊢ ContinuousWithinAt f (Ici 0) x
have h : Membership.mem (γ := Set ℝ) (interior (Ici 0)) x → ContinuousWithinAt f (interior (Ici 0)) x
Tactic: have h := hf_cvx.continuousOn_interior x
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv (hypothesis)
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsProbabilityMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (ε := ℝ) (fun (x : 𝓧) ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Ici 0) x
hx_pos : 0 < x
h : Membership.mem (γ := Set ℝ) (interior (Ici 0)) x → ContinuousWithinAt f (interior (Ici 0)) x
⊢ ContinuousWithinAt f (Ici 0) x
have h₁ : 0 < x → ContinuousWithinAt f (Ioi 0) x
Tactic: simp only [nonempty_Iio, interior_Ici', mem_Ioi] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv (hypothesis)
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsProbabilityMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (ε := ℝ) (fun (x : 𝓧) ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
x : ℝ
hx : Membership.mem (γ := Set ℝ) (Ici 0) x
hx_pos : 0 < x
h : 0 < x → ContinuousWithinAt f (Ioi 0) x
⊢ ContinuousWithinAt f (Ici 0) x
have h₁ : 0 < x → ContinuousAt f x
Tactic: rw [continuousWithinAt_iff_continuousAt (Ioi_mem_nhds hx_pos)] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv (hypothesis)
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsProbabilityMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (ε := ℝ) (fun (x : 𝓧) ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
⊢ f (μ.real univ) ≤ ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
have hf_cont' : ContinuousOn f (Ici 0)
Tactic: have hf_cont' : ContinuousOn f (Ici 0) := by
    intro x hx
    rcases eq_or_lt_of_le (α := ℝ) (hx : 0 ≤ x) with rfl | hx_pos
    · exact hf_cont
    · have h := hf_cvx.continuousOn_interior x
      simp only [nonempty_Iio, interior_Ici', mem_Ioi] at h
      rw [continuousWithinAt_iff_continuousAt (Ioi_mem_nhds hx_pos)] at h
      exact (h hx_pos).continuousWithinAt
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv (goal)
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsProbabilityMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (ε := ℝ) (fun (x : 𝓧) ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hf_cont' : ContinuousOn f (Ici 0)
⊢ f (μ.real univ) = f (∫ (x : 𝓧), (μ.rnDeriv ν x).toReal ∂ν)
have rw : f (μ.real univ) = f (μ.real univ)
Tactic: rw [Measure.integral_toReal_rnDeriv hμν]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv (goal)
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsProbabilityMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (ε := ℝ) (fun (x : 𝓧) ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hf_cont' : ContinuousOn f (Ici 0)
⊢ f (∫ (x : 𝓧), (μ.rnDeriv ν x).toReal ∂ν) ≤ ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
have rw : f (⨍ (x : 𝓧), (μ.rnDeriv ν x).toReal ∂ν) ≤ ⨍ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
Tactic: rw [← average_eq_integral, ← average_eq_integral]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv (goal)
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsProbabilityMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (ε := ℝ) (fun (x : 𝓧) ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hf_cont' : ContinuousOn f (Ici 0)
⊢ f (μ.real univ) ≤ ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
have calc : f (μ.real univ) = f (∫ (x : 𝓧), (μ.rnDeriv ν x).toReal ∂ν)
Tactic: calc f (μ.real univ)
    = f (∫ x, (μ.rnDeriv ν x).toReal ∂ν) := by rw [Measure.integral_toReal_rnDeriv hμν]
  _ ≤ ∫ x, f (μ.rnDeriv ν x).toReal ∂ν := by
    rw [← average_eq_integral, ← average_eq_integral]
    exact ConvexOn.map_average_le hf_cvx hf_cont' isClosed_Ici (by simp)
      Measure.integrable_toReal_rnDeriv hf_int
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv (goal)
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsProbabilityMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (ε := ℝ) (fun (x : 𝓧) ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hf_cont' : ContinuousOn f (Ici 0)
calc : f (μ.real univ) = f (∫ (x : 𝓧), (μ.rnDeriv ν x).toReal ∂ν)
⊢ f (μ.real univ) ≤ ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
have calc₁ : f (∫ (x : 𝓧), (μ.rnDeriv ν x).toReal ∂ν) ≤ ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
Tactic: calc f (μ.real univ)
    = f (∫ x, (μ.rnDeriv ν x).toReal ∂ν) := by rw [Measure.integral_toReal_rnDeriv hμν]
  _ ≤ ∫ x, f (μ.rnDeriv ν x).toReal ∂ν := by
    rw [← average_eq_integral, ← average_eq_integral]
    exact ConvexOn.map_average_le hf_cvx hf_cont' isClosed_Ici (by simp)
      Measure.integrable_toReal_rnDeriv hf_int
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv (goal)
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (ε := ℝ) (fun (x : 𝓧) ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
⊢ ν.real univ * f (μ.real univ / ν.real univ) ≤ ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
have pos : ∀ (hν : ν = 0), ν.real univ * f (μ.real univ / ν.real univ) ≤ ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
Tactic: by_cases hν : ν = 0
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv (goal)
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (ε := ℝ) (fun (x : 𝓧) ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
pos : ∀ (hν : ν = 0), ν.real univ * f (μ.real univ / ν.real univ) ≤ ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
⊢ ν.real univ * f (μ.real univ / ν.real univ) ≤ ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
have neg : ∀ (hν : ¬ν = 0), ν.real univ * f (μ.real univ / ν.real univ) ≤ ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
Tactic: by_cases hν : ν = 0
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv (hypothesis)
Goal:
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst : IsFiniteMeasure μ
inst_1 : IsFiniteMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (ε := ℝ) (fun (x : 𝓧) ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hν : ¬ν = 0
⊢ ν.real univ * f (μ.real univ / ν.real univ) ≤ ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
have this : NeZero ν
Tactic: have : NeZero ν := ⟨hν⟩