awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Measure.Regular
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∞)
⊢ ∀ (n : ℕ), MeasurableSet (s n)

Tactic:
have hm : ∀ n, MeasurableSet (s n) := fun n => (h' n).measurableSet
-- Module: Mathlib.MeasureTheory.Measure.Regular
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 → Set α),
    (∀ (n : ℕ), MeasurableSet (A' n)) ∧ (∀ (n : ℕ), A' n ⊆ s n) ∧ Pairwise (Disjoint on A') ∧ A = ⋃ (n : ℕ), A' n

Tactic:
⟨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, ?_⟩

Premises used: disjointed, inf_le_right, HasSubset.Subset.trans, Inter.inter, And.intro, Exists.intro, MeasurableSet.disjointed, disjointed_subset, disjoint_disjointed, Disjoint.mono, Set.inter_subset_right, Pairwise.mono, MeasurableSet.inter
-- Module: Mathlib.MeasureTheory.Measure.Regular
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
rw₃ rw₄ : A = A
⊢ A = ⋃ (n : ℕ), (fun (n : ℕ) ↦ A ∩ disjointed s n) n

Tactic:
rw [← inter_iUnion, iUnion_disjointed, univ_subset_iff.mp h'', inter_univ]

Premises used: Set.univ_subset_iff, Set.inter_univ, iUnion_disjointed, Iff.mp, Set.inter_iUnion
-- Module: Mathlib.MeasureTheory.Measure.Regular
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)
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
⊢ ∃ U ⊇ A, 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]

Premises used: rfl
-- Module: Mathlib.MeasureTheory.Measure.Regular
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∞)
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
⊢ ∃ 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⟩

Premises used: Iff.mpr, ENNReal.exists_pos_sum_of_countable', tsub_pos_iff_lt, LT.lt.ne', Nat
-- Module: Mathlib.MeasureTheory.Measure.Regular
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∞)
⊢ 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

Premises used: lt_tsub_iff_right, add_comm
-- Module: Mathlib.MeasureTheory.Measure.Regular
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
⊢ ∀ (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)
-- Module: Mathlib.MeasureTheory.Measure.Regular
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∞)
rw : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (A n ∩ s n) : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ Ne (α := ℝ≥0∞) ((μ.restrict (s n) : (a : Set α) → ℝ≥0∞) (A n) : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
rw [H₁]
-- Module: Mathlib.MeasureTheory.Measure.Regular
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 : (a : Set α) → ℝ≥0∞) (A n ∩ s n) : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
((measure_mono (inter_subset_left.trans (subset_iUnion A n))).trans_lt HA).ne

Premises used: HasSubset.Subset.trans, Set.inter_subset_left, Set.subset_iUnion, MeasureTheory.measure_mono, LE.le.trans_lt, LT.lt.ne
-- Module: Mathlib.MeasureTheory.Measure.Regular
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 : (a : Set α) → ℝ≥0∞) (A n ∩ s n) : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
exact ((measure_mono (inter_subset_left.trans (subset_iUnion A n))).trans_lt HA).ne

Premises used: HasSubset.Subset.trans, Set.inter_subset_left, Set.subset_iUnion, MeasureTheory.measure_mono, LE.le.trans_lt, LT.lt.ne
-- Module: Mathlib.MeasureTheory.Measure.Regular
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∞)

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
-- Module: Mathlib.MeasureTheory.Measure.Regular
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∞)
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)
⊢ ∃ 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⟩

Premises used: Set.exists_isOpen_lt_add, LT.lt.ne'
-- Module: Mathlib.MeasureTheory.Measure.Regular
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)
⊢ (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

Premises used: Set.inter_eq_self_of_subset_left
-- Module: Mathlib.MeasureTheory.Measure.Regular
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 : (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)
⊢ ∃ 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:
⟨U ∩ s n, subset_inter hAU (hAs _), hUo.inter (h' n), hU⟩

Premises used: Inter.inter, Set.subset_inter, And.intro, IsOpen.inter, Exists.intro
-- Module: Mathlib.MeasureTheory.Measure.Regular
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 : (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)
⊢ ∃ 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:
exact ⟨U ∩ s n, subset_inter hAU (hAs _), hUo.inter (h' n), hU⟩
-- Module: Mathlib.MeasureTheory.Measure.Regular
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)

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⟩
-- Module: Mathlib.MeasureTheory.Measure.Regular
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)
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
⊢ ∃ 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
-- Module: Mathlib.MeasureTheory.Measure.Regular
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

Tactic:
⟨⋃ n, U n, iUnion_mono hAU, isOpen_iUnion hUo, ?_⟩

Premises used: isOpen_iUnion, And.intro, Exists.intro, Set.iUnion_mono, Set.iUnion
-- Module: Mathlib.MeasureTheory.Measure.Regular
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)
refine : (μ : (a : Set α) → ℝ≥0∞) (⋃ (n : ℕ), U n) < r
⊢ ∃ U ⊇ ⋃ (n : ℕ), A n, IsOpen[inst_1] U ∧ (μ : (a : Set α) → ℝ≥0∞) U < r

Tactic:
refine ⟨⋃ n, U n, iUnion_mono hAU, isOpen_iUnion hUo, ?_⟩