awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
x : α
hx : Membership.mem (γ := Set α) {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} x
pos : ∀ (h : x ∈ s), x ∈ toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)
⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)
have neg : ∀ (h : x ∉ s), x ∈ toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)
Tactic: by_cases h : x ∈ s
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
x : α
hx : Membership.mem (γ := Set α) {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} x
h : x ∈ s
⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)
have pos : x ∈ toMeasurable (ρ + μ) (u (spanningSetsIndex (ρ + μ) x))
Tactic: refine Or.inr (mem_iUnion.2 ⟨spanningSetsIndex (ρ + μ) x, ?_⟩)
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
x : α
hx : Membership.mem (γ := Set α) {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} x
⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n)
have pos : ∀ (h : x ∈ s), x ∈ toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n)
Tactic: by_cases h : x ∈ s
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
x : α
hx : Membership.mem (γ := Set α) {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} x
pos : ∀ (h : x ∈ s), x ∈ toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n)
⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n)
have neg : ∀ (h : x ∉ s), x ∈ toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n)
Tactic: by_cases h : x ∈ s
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
x : α
hx : Membership.mem (γ := Set α) {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} x
h : x ∈ s
⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n)
have pos : x ∈ toMeasurable (ρ + μ) (w (spanningSetsIndex (ρ + μ) x))
Tactic: refine Or.inr (mem_iUnion.2 ⟨spanningSetsIndex (ρ + μ) x, ?_⟩)
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
H :
  ∀ (m n : ℕ),
    Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)
      (0 : ℝ≥0∞)
⊢ (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩
      (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n)) ⊆
    toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
have simp : (⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩ toMeasurable μ sᶜ ⊆
    toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) ∧
  toMeasurable μ sᶜ ∩ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n) ⊆
      toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) ∧
    (⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n) ⊆
      toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
Tactic: simp only [inter_union_distrib_left, union_inter_distrib_right, true_and,
        subset_union_left, union_subset_iff, inter_self]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
H :
  ∀ (m n : ℕ),
    Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)
      (0 : ℝ≥0∞)
⊢ (⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩ toMeasurable μ sᶜ ⊆
      toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) ∧
    toMeasurable μ sᶜ ∩ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n) ⊆
        toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) ∧
      (⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n) ⊆
        toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
have refine_1 : (⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩ toMeasurable μ sᶜ ⊆
  toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
Tactic: refine ⟨?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
H :
  ∀ (m n : ℕ),
    Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)
      (0 : ℝ≥0∞)
refine_1 :
  (⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩ toMeasurable μ sᶜ ⊆
    toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
⊢ (⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩ toMeasurable μ sᶜ ⊆
      toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) ∧
    toMeasurable μ sᶜ ∩ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n) ⊆
        toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) ∧
      (⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n) ⊆
        toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
have refine_2 : toMeasurable μ sᶜ ∩ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n) ⊆
  toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
Tactic: refine ⟨?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
H :
  ∀ (m n : ℕ),
    Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)
      (0 : ℝ≥0∞)
refine_1 :
  (⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩ toMeasurable μ sᶜ ⊆
    toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
refine_2 :
  toMeasurable μ sᶜ ∩ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n) ⊆
    toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
⊢ (⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩ toMeasurable μ sᶜ ⊆
      toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) ∧
    toMeasurable μ sᶜ ∩ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n) ⊆
        toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) ∧
      (⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n) ⊆
        toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
have refine_3 : (⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n) ⊆
  toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
Tactic: refine ⟨?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
H :
  ∀ (m n : ℕ),
    Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)
      (0 : ℝ≥0∞)
⊢ (⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n) ⊆
    toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
have refine_3 : ⋃ (i : ℕ), ⋃ (i_1 : ℕ), toMeasurable (ρ + μ) (u i) ∩ toMeasurable (ρ + μ) (w i_1) ⊆
  toMeasurable μ sᶜ ∪ ⋃ (i : ℕ), ⋃ (i_1 : ℕ), toMeasurable (ρ + μ) (u i) ∩ toMeasurable (ρ + μ) (w i_1)
Tactic: simp_rw [iUnion_inter, inter_iUnion]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (hypothesis)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
H :
  ∀ (m n : ℕ),
    Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)
      (0 : ℝ≥0∞)
⊢ Eq (α := ℝ≥0∞)
    ((μ : (a : Set α) → ℝ≥0∞)
        ((toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩
          (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n))) :
      ℝ≥0∞)
    (0 : ℝ≥0∞)
have A : (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩
    (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n)) ⊆
  toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
Tactic: have A :
      (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n)) ∩
          (toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n)) ⊆
        toMeasurable μ sᶜ ∪
          ⋃ (m) (n), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n) := by
      simp only [inter_union_distrib_left, union_inter_distrib_right, true_and,
        subset_union_left, union_subset_iff, inter_self]
      refine ⟨?_, ?_, ?_⟩
      · exact inter_subset_right.trans subset_union_left
      · exact inter_subset_left.trans subset_union_left
      · simp_rw [iUnion_inter, inter_iUnion]; exact subset_union_right
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
H :
  ∀ (m n : ℕ),
    Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)
      (0 : ℝ≥0∞)
A :
  (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩
      (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n)) ⊆
    toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
⊢ Eq (α := ℝ≥0∞)
    ((μ : (a : Set α) → ℝ≥0∞)
        ((toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩
          (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n))) :
      ℝ≥0∞)
    (0 : ℝ≥0∞)
have refine : LE.le (α := ℝ≥0∞)
  ((μ : (a : Set α) → ℝ≥0∞)
      (toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) :
    ℝ≥0∞)
  (0 : ℝ≥0∞)
Tactic: refine le_antisymm ((measure_mono A).trans ?_) bot_le
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (hypothesis)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
H :
  ∀ (m n : ℕ),
    Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)
      (0 : ℝ≥0∞)
A :
  (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩
      (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n)) ⊆
    toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable μ sᶜ) : ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞) (⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) :
        ℝ≥0∞) =
    (μ : (a : Set α) → ℝ≥0∞) (⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
have this : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) sᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
Tactic: have : μ sᶜ = 0 := v.ae_tendsto_div hρ
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
H :
  ∀ (m n : ℕ),
    Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)
      (0 : ℝ≥0∞)
A :
  (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩
      (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n)) ⊆
    toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
this : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) sᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable μ sᶜ) : ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞) (⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) :
        ℝ≥0∞) =
    (μ : (a : Set α) → ℝ≥0∞) (⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
have rw : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) sᶜ : ℝ≥0∞)
    ((μ : (a : Set α) → ℝ≥0∞) (⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞) =
  (μ : (a : Set α) → ℝ≥0∞) (⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
Tactic: rw [measure_toMeasurable, this, zero_add]
-- Lean 4 proof step from Mathlib.MeasureTheory.Covering.Differentiation (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst_1 : SecondCountableTopology α
inst_2 : BorelSpace α
inst_3 : IsLocallyFiniteMeasure μ
ρ : Measure α
inst_4 : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
p q : ℝ≥0
hpq : p < q
s : Set α :=
  {x : α |
    ∃ (c : ℝ≥0∞),
      Tendsto
        (fun (a : Set α) ↦
          HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞))
        (v.filterAt x) (𝓝 c)}
o : ℕ → Set α := spanningSets (ρ + μ)
u : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | v.limRatio ρ x < (↑p : ℝ≥0∞)} ∩ o n
w : (n : ℕ) → Set α := fun (n : ℕ) ↦ s ∩ {x : α | (↑q : ℝ≥0∞) < v.limRatio ρ x} ∩ o n
H :
  ∀ (m n : ℕ),
    Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)
      (0 : ℝ≥0∞)
A :
  (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (u n)) ∩
      (toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n)) ⊆
    toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)
this : Eq (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) sᶜ : ℝ≥0∞) (0 : ℝ≥0∞)
rw :
  HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) sᶜ : ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞) (⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) :
        ℝ≥0∞) =
    (μ : (a : Set α) → ℝ≥0∞) (⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
⊢ HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable μ sᶜ) : ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞) (⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) :
        ℝ≥0∞) =
    (μ : (a : Set α) → ℝ≥0∞) (⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
have rw₁ : HAdd.hAdd (α := ℝ≥0∞) (β := ℝ≥0∞) (0 : ℝ≥0∞)
    ((μ : (a : Set α) → ℝ≥0∞) (⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞) =
  (μ : (a : Set α) → ℝ≥0∞) (⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))
Tactic: rw [measure_toMeasurable, this, zero_add]