awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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)
neg : ∀ (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)

Tactic:
by_cases h : x ∈ s
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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)

Tactic:
Or.inr (mem_iUnion.2 ⟨spanningSetsIndex (ρ + μ) x, ?_⟩)

Premises used: Iff.mpr, Set.union, Set.Mem, setOf, Set.mem_iUnion, MeasureTheory.spanningSetsIndex, HAdd.hAdd, Exists.intro, Or.inr
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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
pos : x ∈ toMeasurable (ρ + μ) (w (spanningSetsIndex (ρ + μ) x))
⊢ x ∈ toMeasurable μ sᶜ ∪ ⋃ (n : ℕ), toMeasurable (ρ + μ) (w n)

Tactic:
refine Or.inr (mem_iUnion.2 ⟨spanningSetsIndex (ρ + μ) x, ?_⟩)

Premises used: Iff.mpr, Set.mem_iUnion, MeasureTheory.spanningSetsIndex, HAdd.hAdd, Exists.intro, Or.inr
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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 (ρ + μ) (w (spanningSetsIndex (ρ + μ) x))

Tactic:
subset_toMeasurable _ _ ⟨⟨h, hx⟩, mem_spanningSetsIndex _ _⟩

Premises used: MeasureTheory.subset_toMeasurable, And.intro, MeasureTheory.mem_spanningSetsIndex
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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 (ρ + μ) (w (spanningSetsIndex (ρ + μ) x))

Tactic:
exact subset_toMeasurable _ _ ⟨⟨h, hx⟩, mem_spanningSetsIndex _ _⟩
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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)

Tactic:
Or.inl (subset_toMeasurable μ sᶜ h)

Premises used: Set.union, Set.Mem, Compl.compl, setOf, MeasureTheory.subset_toMeasurable, Or.inl
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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)

Tactic:
exact Or.inl (subset_toMeasurable μ sᶜ h)

Premises used: Compl.compl, MeasureTheory.subset_toMeasurable, Or.inl
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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∞)
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)
⊢ (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:
simp only [inter_union_distrib_left, union_inter_distrib_right, true_and,
        subset_union_left, union_subset_iff, inter_self]

Premises used: Set.union_subset_iff, Set.inter_self, Set.subset_union_left, Set.union_inter_distrib_right, Set.inter_union_distrib_left, true_and
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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)

Tactic:
⟨?_, ?_, ?_⟩

Premises used: And.intro
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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)
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)
⊢ (⋃ (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:
refine ⟨?_, ?_, ?_⟩
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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)

Tactic:
inter_subset_right.trans subset_union_left

Premises used: HasSubset.Subset.trans, Set.subset_union_left, Set.inter_subset_right
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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)

Tactic:
exact inter_subset_right.trans subset_union_left

Premises used: HasSubset.Subset.trans, Set.subset_union_left, Set.inter_subset_right
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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) (w n) ⊆
    toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)

Tactic:
inter_subset_left.trans subset_union_left

Premises used: HasSubset.Subset.trans, Set.inter_subset_left, Set.subset_union_left
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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) (w n) ⊆
    toMeasurable μ sᶜ ∪ ⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)

Tactic:
exact inter_subset_left.trans subset_union_left

Premises used: HasSubset.Subset.trans, Set.inter_subset_left, Set.subset_union_left
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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_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)
⊢ (⋃ (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_rw [iUnion_inter, inter_iUnion]

Premises used: Set.iUnion_inter, Set.inter_iUnion
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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∞)
⊢ ⋃ (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:
subset_union_right

Premises used: Set.subset_union_right
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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∞)
⊢ ⋃ (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:
exact subset_union_right

Premises used: Set.subset_union_right
-- Module: Mathlib.MeasureTheory.Covering.Differentiation
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)

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