awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
β : Type u_2
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
mβ : MeasurableSpace β
inst_2 : TopologicalSpace β
μ : Measure β
inst_3 : μ.OuterRegular
f : α → β
f_cont : Continuous[inst_1, inst_2] f
f_me : MeasurableEmbedding f
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > (μ : (a : Set β) → ℝ≥0∞) (f '' A)
U : Set β
hUA : U ⊇ f '' A
Uopen : IsOpen[inst_2] U
hμU : (μ : (a : Set β) → ℝ≥0∞) U < r
⊢ ((comap f μ : Measure α) : (a : Set α) → ℝ≥0∞) (f ⁻¹' U) < r
have rw : (μ : (a : Set β) → ℝ≥0∞) (f '' f ⁻¹' U) < r
Tactic: rw [f_me.comap_apply]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
μ : Measure α
inst_2 : μ.OuterRegular
x : ℝ≥0∞
hx : x ≠ ∞
⊢ (x • μ).OuterRegular
have inl : ∀ (hx : Ne (α := ℝ≥0∞) 0 ∞), (HSMul.hSMul (α := ℝ≥0∞) 0 μ).OuterRegular
Tactic: rcases eq_or_ne x 0 with (rfl | h0)
Cleared hypotheses: x, hx
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
μ : Measure α
inst_2 : μ.OuterRegular
x : ℝ≥0∞
hx : x ≠ ∞
inl : ∀ (hx : Ne (α := ℝ≥0∞) 0 ∞), (HSMul.hSMul (α := ℝ≥0∞) 0 μ).OuterRegular
⊢ (x • μ).OuterRegular
have inr : ∀ (h0 : x ≠ 0), (x • μ).OuterRegular
Tactic: rcases eq_or_ne x 0 with (rfl | h0)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
μ : Measure α
inst_2 : μ.OuterRegular
hx : Ne (α := ℝ≥0∞) 0 ∞
⊢ (HSMul.hSMul (α := ℝ≥0∞) 0 μ).OuterRegular
have inl : OuterRegular.{u_1} (α := α) 0
Tactic: rw [zero_smul]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
μ : Measure α
inst_2 : μ.OuterRegular
x : ℝ≥0∞
hx : x ≠ ∞
h0 : x ≠ 0
⊢ (x • μ).OuterRegular
have inr : ∀ (A : Set α),
  MeasurableSet A →
    ∀ (r : ℝ≥0∞) (hr : r > (x • μ : (a : Set α) → ℝ≥0∞) A),
      ∃ U ⊇ A, IsOpen[inst_1] U ∧ (x • μ : (a : Set α) → ℝ≥0∞) U < r
Tactic: refine ⟨fun A _ r hr => ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (hypothesis)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
μ : Measure α
inst_2 : μ.OuterRegular
x : ℝ≥0∞
hx : x ≠ ∞
h0 : x ≠ 0
A : Set α
h : MeasurableSet A
r : ℝ≥0∞
hr : r > (x • μ : (a : Set α) → ℝ≥0∞) A
⊢ ∃ U ⊇ A, IsOpen[inst_1] U ∧ (x • μ : (a : Set α) → ℝ≥0∞) U < r
have hr₁ : r > x * ⨅ (U : Set α), ⨅ (_ : A ⊆ U), ⨅ (_ : IsOpen[inst_1] U), (μ : (a : Set α) → ℝ≥0∞) U
Tactic: rw [smul_apply, A.measure_eq_iInf_isOpen, smul_eq_mul] at hr
Cleared hypotheses: hr
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
inst_2 : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst_1] (s n)
h'' : Subset.{u_1} (α := Set α) univ (⋃ (n : ℕ), s n)
⊢ μ.OuterRegular
have refine : ∀ (A : Set α) (hA : MeasurableSet A) (r : ℝ≥0∞) (hr : r > (μ : (a : Set α) → ℝ≥0∞) A),
  ∃ U ⊇ A, IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < r
Tactic: refine ⟨fun A hA r hr => ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (hypothesis)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
inst_2 : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst_1] (s n)
h'' : Subset.{u_1} (α := Set α) univ (⋃ (n : ℕ), s n)
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > (μ : (a : Set α) → ℝ≥0∞) A
⊢ ∃ U ⊇ A, IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < r
have HA : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞) (∞ : ℝ≥0∞)
Tactic: have HA : μ A < ∞ := lt_of_lt_of_le hr le_top
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (hypothesis)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
inst_2 : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst_1] (s n)
h'' : Subset.{u_1} (α := Set α) univ (⋃ (n : ℕ), s n)
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > (μ : (a : Set α) → ℝ≥0∞) A
HA : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ ∃ U ⊇ A, IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < r
have hm : ∀ (n : ℕ), MeasurableSet (s n)
Tactic: have hm : ∀ n, MeasurableSet (s n) := fun n => (h' n).measurableSet
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
inst_2 : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst_1] (s n)
h'' : Subset.{u_1} (α := Set α) univ (⋃ (n : ℕ), s n)
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > (μ : (a : Set α) → ℝ≥0∞) A
HA : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞) (∞ : ℝ≥0∞)
hm : ∀ (n : ℕ), MeasurableSet (s n)
⊢ A = ⋃ (n : ℕ), (fun (n : ℕ) ↦ A ∩ disjointed s n) n
have rw : A = A ∩ ⋃ (i : ℕ), disjointed s i
Tactic: rw [← inter_iUnion, iUnion_disjointed, univ_subset_iff.mp h'', inter_univ]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
inst_2 : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst_1] (s n)
h'' : Subset.{u_1} (α := Set α) univ (⋃ (n : ℕ), s n)
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > (μ : (a : Set α) → ℝ≥0∞) A
HA : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞) (∞ : ℝ≥0∞)
hm : ∀ (n : ℕ), MeasurableSet (s n)
rw : A = A ∩ ⋃ (i : ℕ), disjointed s i
⊢ A = ⋃ (n : ℕ), (fun (n : ℕ) ↦ A ∩ disjointed s n) n
have rw₁ : A = A ∩ ⋃ (i : ℕ), s i
Tactic: rw [← inter_iUnion, iUnion_disjointed, univ_subset_iff.mp h'', inter_univ]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
inst_2 : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst_1] (s n)
h'' : Subset.{u_1} (α := Set α) univ (⋃ (n : ℕ), s n)
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > (μ : (a : Set α) → ℝ≥0∞) A
HA : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞) (∞ : ℝ≥0∞)
hm : ∀ (n : ℕ), MeasurableSet (s n)
rw : A = A ∩ ⋃ (i : ℕ), disjointed s i
rw₁ : A = A ∩ ⋃ (i : ℕ), s i
⊢ A = ⋃ (n : ℕ), (fun (n : ℕ) ↦ A ∩ disjointed s n) n
have rw₂ : A = A ∩ univ
Tactic: rw [← inter_iUnion, iUnion_disjointed, univ_subset_iff.mp h'', inter_univ]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
inst_2 : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst_1] (s n)
h'' : Subset.{u_1} (α := Set α) univ (⋃ (n : ℕ), s n)
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > (μ : (a : Set α) → ℝ≥0∞) A
HA : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞) (∞ : ℝ≥0∞)
hm : ∀ (n : ℕ), MeasurableSet (s n)
rw : A = A ∩ ⋃ (i : ℕ), disjointed s i
rw₁ : A = A ∩ ⋃ (i : ℕ), s i
rw₂ : A = A ∩ univ
⊢ A = ⋃ (n : ℕ), (fun (n : ℕ) ↦ A ∩ disjointed s n) n
have rw₃ : A = A
Tactic: rw [← inter_iUnion, iUnion_disjointed, univ_subset_iff.mp h'', inter_univ]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
inst_2 : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst_1] (s n)
h'' : Subset.{u_1} (α := Set α) univ (⋃ (n : ℕ), s n)
A : Set α
hA : MeasurableSet A
r : ℝ≥0∞
hr : r > (μ : (a : Set α) → ℝ≥0∞) A
HA : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) A : ℝ≥0∞) (∞ : ℝ≥0∞)
hm : ∀ (n : ℕ), MeasurableSet (s n)
⊢ ∃ U ⊇ A, IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < r
have obtain : ∀ (A : ℕ → Set α) (hAm : ∀ (n : ℕ), MeasurableSet (A n)) (hAs : ∀ (n : ℕ), A n ⊆ s n) (hAd : Pairwise (Disjoint on A))
  (hA : MeasurableSet.{u_1} (α := α) (⋃ (n : ℕ), A n)) (hr : r > (μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n))
  (HA : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∞ : ℝ≥0∞)),
  ∃ U ⊇ ⋃ (n : ℕ), A n, IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < r
Tactic: obtain ⟨A, hAm, hAs, hAd, rfl⟩ :
    ∃ A' : ℕ → Set α,
      (∀ n, MeasurableSet (A' n)) ∧
        (∀ n, A' n ⊆ s n) ∧ Pairwise (Disjoint on A') ∧ A = ⋃ n, A' n := by
    refine
      ⟨fun n => A ∩ disjointed s n, fun n => hA.inter (MeasurableSet.disjointed hm _), fun n =>
        inter_subset_right.trans (disjointed_subset _ _),
        (disjoint_disjointed s).mono fun k l hkl => hkl.mono inf_le_right inf_le_right, ?_⟩
    rw [← inter_iUnion, iUnion_disjointed, univ_subset_iff.mp h'', inter_univ]
Cleared hypotheses: A, hA, hr, HA
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
inst_2 : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst_1] (s n)
h'' : Subset.{u_1} (α := Set α) univ (⋃ (n : ℕ), s n)
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet.{u_1} (α := α) (⋃ (n : ℕ), A n)
hr : r > (μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n)
HA : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ ∃ U ⊇ ⋃ (n : ℕ), A n, IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < r
have rcases : ∀ (δ : ℕ → ℝ≥0∞) (δ0 : ∀ (i : ℕ), 0 < δ i)
  (hδε : ∑' (i : ℕ), δ i < HSub.hSub (β := ℝ≥0∞) r ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞)),
  ∃ U ⊇ ⋃ (n : ℕ), A n, IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < r
Tactic: rcases ENNReal.exists_pos_sum_of_countable' (tsub_pos_iff_lt.2 hr).ne' ℕ with ⟨δ, δ0, hδε⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (hypothesis)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
inst_2 : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst_1] (s n)
h'' : Subset.{u_1} (α := Set α) univ (⋃ (n : ℕ), s n)
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet.{u_1} (α := α) (⋃ (n : ℕ), A n)
hr : r > (μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n)
HA : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∞ : ℝ≥0∞)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : ∑' (i : ℕ), δ i < HSub.hSub (β := ℝ≥0∞) r ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞)
⊢ ∃ U ⊇ ⋃ (n : ℕ), A n, IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < r
have hδε₁ : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∑' (i : ℕ), δ i : ℝ≥0∞) < r
Tactic: rw [lt_tsub_iff_right, add_comm] at hδε
Cleared hypotheses: hδε
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (hypothesis)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
inst_2 : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst_1] (s n)
h'' : Subset.{u_1} (α := Set α) univ (⋃ (n : ℕ), s n)
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet.{u_1} (α := α) (⋃ (n : ℕ), A n)
hr : r > (μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n)
HA : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∞ : ℝ≥0∞)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∑' (i : ℕ), δ i : ℝ≥0∞) < r
n : ℕ
⊢ ∃ U ⊇ A n,
    IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (A n) : ℝ≥0∞) (δ n)
have H₁ : ∀ (t : Set α),
  Eq (α := ℝ≥0∞) ((μ.restrict (s n) : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ s n) : ℝ≥0∞)
Tactic: have H₁ : ∀ t, μ.restrict (s n) t = μ (t ∩ s n) := fun t => restrict_apply' (hm n)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
inst_2 : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst_1] (s n)
h'' : Subset.{u_1} (α := Set α) univ (⋃ (n : ℕ), s n)
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet.{u_1} (α := α) (⋃ (n : ℕ), A n)
hr : r > (μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n)
HA : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∞ : ℝ≥0∞)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∑' (i : ℕ), δ i : ℝ≥0∞) < r
n : ℕ
H₁ :
  ∀ (t : Set α),
    Eq (α := ℝ≥0∞) ((μ.restrict (s n) : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ s n) : ℝ≥0∞)
⊢ Ne (α := ℝ≥0∞) ((μ.restrict (s n) : (a : Set α) → ℝ≥0∞) (A n) : ℝ≥0∞) (∞ : ℝ≥0∞)
have rw : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (A n ∩ s n) : ℝ≥0∞) (∞ : ℝ≥0∞)
Tactic: rw [H₁]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (hypothesis)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
inst_2 : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst_1] (s n)
h'' : Subset.{u_1} (α := Set α) univ (⋃ (n : ℕ), s n)
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet.{u_1} (α := α) (⋃ (n : ℕ), A n)
hr : r > (μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n)
HA : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∞ : ℝ≥0∞)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∑' (i : ℕ), δ i : ℝ≥0∞) < r
n : ℕ
H₁ :
  ∀ (t : Set α),
    Eq (α := ℝ≥0∞) ((μ.restrict (s n) : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ s n) : ℝ≥0∞)
⊢ ∃ U ⊇ A n,
    IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (A n) : ℝ≥0∞) (δ n)
have Ht : Ne (α := ℝ≥0∞) ((μ.restrict (s n) : (a : Set α) → ℝ≥0∞) (A n) : ℝ≥0∞) (∞ : ℝ≥0∞)
Tactic: have Ht : μ.restrict (s n) (A n) ≠ ∞ := by
      rw [H₁]
      exact ((measure_mono (inter_subset_left.trans (subset_iUnion A n))).trans_lt HA).ne
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
inst_2 : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst_1] (s n)
h'' : Subset.{u_1} (α := Set α) univ (⋃ (n : ℕ), s n)
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet.{u_1} (α := α) (⋃ (n : ℕ), A n)
hr : r > (μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n)
HA : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∞ : ℝ≥0∞)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∑' (i : ℕ), δ i : ℝ≥0∞) < r
n : ℕ
H₁ :
  ∀ (t : Set α),
    Eq (α := ℝ≥0∞) ((μ.restrict (s n) : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ s n) : ℝ≥0∞)
Ht : Ne (α := ℝ≥0∞) ((μ.restrict (s n) : (a : Set α) → ℝ≥0∞) (A n) : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ ∃ U ⊇ A n,
    IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (A n) : ℝ≥0∞) (δ n)
have rcases : ∀ (U : Set α) (hAU : U ⊇ A n) (hUo : IsOpen[inst_1] U)
  (hU :
    (μ.restrict (s n) : (a : Set α) → ℝ≥0∞) U <
      HAdd.hAdd (α := ℝ≥0∞) ((μ.restrict (s n) : (a : Set α) → ℝ≥0∞) (A n) : ℝ≥0∞) (δ n)),
  ∃ U ⊇ A n,
    IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (A n) : ℝ≥0∞) (δ n)
Tactic: rcases (A n).exists_isOpen_lt_add Ht (δ0 n).ne' with ⟨U, hAU, hUo, hU⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (hypothesis)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
inst_2 : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst_1] (s n)
h'' : Subset.{u_1} (α := Set α) univ (⋃ (n : ℕ), s n)
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet.{u_1} (α := α) (⋃ (n : ℕ), A n)
hr : r > (μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n)
HA : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∞ : ℝ≥0∞)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∑' (i : ℕ), δ i : ℝ≥0∞) < r
n : ℕ
H₁ :
  ∀ (t : Set α),
    Eq (α := ℝ≥0∞) ((μ.restrict (s n) : (a : Set α) → ℝ≥0∞) t : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (t ∩ s n) : ℝ≥0∞)
Ht : Ne (α := ℝ≥0∞) ((μ.restrict (s n) : (a : Set α) → ℝ≥0∞) (A n) : ℝ≥0∞) (∞ : ℝ≥0∞)
U : Set α
hAU : U ⊇ A n
hUo : IsOpen[inst_1] U
hU :
  (μ.restrict (s n) : (a : Set α) → ℝ≥0∞) U <
    HAdd.hAdd (α := ℝ≥0∞) ((μ.restrict (s n) : (a : Set α) → ℝ≥0∞) (A n) : ℝ≥0∞) (δ n)
⊢ ∃ U ⊇ A n,
    IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (A n) : ℝ≥0∞) (δ n)
have hU₁ : (μ : (a : Set α) → ℝ≥0∞) (U ∩ s n) < HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (A n) : ℝ≥0∞) (δ n)
Tactic: rw [H₁, H₁, inter_eq_self_of_subset_left (hAs _)] at hU
Cleared hypotheses: hU
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (hypothesis)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
inst_2 : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst_1] (s n)
h'' : Subset.{u_1} (α := Set α) univ (⋃ (n : ℕ), s n)
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet.{u_1} (α := α) (⋃ (n : ℕ), A n)
hr : r > (μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n)
HA : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∞ : ℝ≥0∞)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∑' (i : ℕ), δ i : ℝ≥0∞) < r
⊢ ∃ U ⊇ ⋃ (n : ℕ), A n, IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < r
have this : ∀ (n : ℕ),
  ∃ U ⊇ A n,
    IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (A n) : ℝ≥0∞) (δ n)
Tactic: have : ∀ n, ∃ U ⊇ A n, IsOpen U ∧ μ U < μ (A n) + δ n := by
    intro n
    have H₁ : ∀ t, μ.restrict (s n) t = μ (t ∩ s n) := fun t => restrict_apply' (hm n)
    have Ht : μ.restrict (s n) (A n) ≠ ∞ := by
      rw [H₁]
      exact ((measure_mono (inter_subset_left.trans (subset_iUnion A n))).trans_lt HA).ne
    rcases (A n).exists_isOpen_lt_add Ht (δ0 n).ne' with ⟨U, hAU, hUo, hU⟩
    rw [H₁, H₁, inter_eq_self_of_subset_left (hAs _)] at hU
    exact ⟨U ∩ s n, subset_inter hAU (hAs _), hUo.inter (h' n), hU⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
inst_2 : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst_1] (s n)
h'' : Subset.{u_1} (α := Set α) univ (⋃ (n : ℕ), s n)
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet.{u_1} (α := α) (⋃ (n : ℕ), A n)
hr : r > (μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n)
HA : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∞ : ℝ≥0∞)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∑' (i : ℕ), δ i : ℝ≥0∞) < r
this :
  ∀ (n : ℕ),
    ∃ U ⊇ A n,
      IsOpen[inst_1] U ∧
        (μ : (a : Set α) → ℝ≥0∞) U < HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (A n) : ℝ≥0∞) (δ n)
⊢ ∃ U ⊇ ⋃ (n : ℕ), A n, IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < r
have choose : ∀ (U : (n : ℕ) → Set α) (hAU : ∀ (n : ℕ), U n ⊇ A n) (hUo : ∀ (n : ℕ), IsOpen[inst_1] (U n))
  (hU :
    ∀ (n : ℕ), (μ : (a : Set α) → ℝ≥0∞) (U n) < HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (A n) : ℝ≥0∞) (δ n)),
  ∃ U ⊇ ⋃ (n : ℕ), A n, IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < r
Tactic: choose U hAU hUo hU using this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
inst_1 : TopologicalSpace α
inst_2 : OpensMeasurableSpace α
μ : Measure α
s : ℕ → Set α
h : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular
h' : ∀ (n : ℕ), IsOpen[inst_1] (s n)
h'' : Subset.{u_1} (α := Set α) univ (⋃ (n : ℕ), s n)
r : ℝ≥0∞
hm : ∀ (n : ℕ), MeasurableSet (s n)
A : ℕ → Set α
hAm : ∀ (n : ℕ), MeasurableSet (A n)
hAs : ∀ (n : ℕ), A n ⊆ s n
hAd : Pairwise (Disjoint on A)
hA : MeasurableSet.{u_1} (α := α) (⋃ (n : ℕ), A n)
hr : r > (μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n)
HA : LT.lt (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∞ : ℝ≥0∞)
δ : ℕ → ℝ≥0∞
δ0 : ∀ (i : ℕ), 0 < δ i
hδε : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), A n) : ℝ≥0∞) (∑' (i : ℕ), δ i : ℝ≥0∞) < r
U : (n : ℕ) → Set α
hAU : ∀ (n : ℕ), U n ⊇ A n
hUo : ∀ (n : ℕ), IsOpen[inst_1] (U n)
hU : ∀ (n : ℕ), (μ : (a : Set α) → ℝ≥0∞) (U n) < HAdd.hAdd (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (A n) : ℝ≥0∞) (δ n)
⊢ ∃ U ⊇ ⋃ (n : ℕ), A n, IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < r
have refine : (μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), U n) < r
Tactic: refine ⟨⋃ n, U n, iUnion_mono hAU, isOpen_iUnion hUo, ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
μ : Measure α
inst_1 : TopologicalSpace α
inst_2 : R1Space α
inst_3 : μ.OuterRegular
k : Set α
hk : IsCompact k
⊢ Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closure[inst_1] k) : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) k : ℝ≥0∞)
have apply : LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closure[inst_1] k) : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) k : ℝ≥0∞)
Tactic: apply le_antisymm ?_ (measure_mono subset_closure)
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
μ : Measure α
inst_1 : TopologicalSpace α
inst_2 : R1Space α
inst_3 : μ.OuterRegular
k : Set α
hk : IsCompact k
⊢ LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closure[inst_1] k) : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) k : ℝ≥0∞)
have simp : ∀ (i : Set α) (i_1 : k ⊆ i) (i_2 : IsOpen[inst_1] i),
  LE.le (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (closure[inst_1] k) : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) i : ℝ≥0∞)
Tactic: simp only [measure_eq_iInf_isOpen k, le_iInf_iff]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Regular (goal)
Goal:
α : Type u_1
inst : MeasurableSpace α
μ : Measure α
inst_1 : TopologicalSpace α
ν : Measure α
inst_2 : μ.OuterRegular
inst_3 : ν.OuterRegular
hμν :
  ∀ (U : Set α),
    IsOpen[inst_1] U → Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) U : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) U : ℝ≥0∞)
⊢ μ = ν
have h : ∀ (s : Set α) (ms : MeasurableSet s),
  Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞) ((ν : (a : Set α) → ℝ≥0∞) s : ℝ≥0∞)
Tactic: ext s ms