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
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∞)

Tactic:
le_antisymm ((measure_mono A).trans ?_) bot_le

Premises used: le_antisymm, MeasureTheory.measure_mono, LE.le.trans, bot_le
-- 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∞)
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)
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∞)
⊢ 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∞)

Tactic:
refine le_antisymm ((measure_mono A).trans ?_) bot_le

Premises used: le_antisymm, MeasureTheory.measure_mono, LE.le.trans, bot_le
-- 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∞)
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)
⊢ (μ : (a : Set α) → ℝ≥0∞)
      (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∞)

Tactic:
measure_union_le _ _

Premises used: MeasureTheory.measure_union_le
-- 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∞)
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∞) sᶜ : ℝ≥0∞) (0 : ℝ≥0∞)

Tactic:
have : μ sᶜ = 0 := v.ae_tendsto_div hρ
-- 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∞)
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))
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))
rw₂ rw₃ :
  Eq (α := 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)) : 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))

Tactic:
rw [measure_toMeasurable, this, zero_add]

Premises used: zero_add, MeasureTheory.measure_toMeasurable
-- 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∞)
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)
⊢ LE.le (α := ℝ≥0∞)
    ((μ : (a : Set α) → ℝ≥0∞) (⋃ (m : ℕ), ⋃ (n : ℕ), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)
    (∑' (m : ℕ) (n : ℕ), (μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)

Tactic:
((measure_iUnion_le _).trans (ENNReal.tsum_le_tsum fun m => measure_iUnion_le _))

Premises used: ENNReal.tsum_le_tsum, Inter.inter, MeasureTheory.measure_iUnion_le, HAdd.hAdd, LE.le.trans, MeasureTheory.toMeasurable
-- 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∞)
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∞)
    (∑' (m : ℕ) (n : ℕ), (μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))) 0

Tactic:
simp only [H, tsum_zero]

Premises used: tsum_zero
-- 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∞)
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)
μ₁ :
  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))
μ₂ :
  Eq (α := ℝ≥0∞)
    (∑' (m : ℕ) (n : ℕ), (μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n))) 0
⊢ 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:
calc
      μ (toMeasurable μ sᶜ ∪
        ⋃ (m) (n), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
          μ (toMeasurable μ sᶜ) +
            μ (⋃ (m) (n), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) :=
        measure_union_le _ _
      _ = μ (⋃ (m) (n), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := by
        have : μ sᶜ = 0 := v.ae_tendsto_div hρ; rw [measure_toMeasurable, this, zero_add]
      _ ≤ ∑' (m) (n), μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) :=
        ((measure_iUnion_le _).trans (ENNReal.tsum_le_tsum fun m => measure_iUnion_le _))
      _ = 0 := by simp only [H, tsum_zero]

Premises used: ENNReal.tsum_le_tsum, Compl.compl, Inter.inter, MeasureTheory.measure_iUnion_le, HAdd.hAdd, LE.le.trans, MeasureTheory.toMeasurable, Union.union, DFunLike.coe, Set.iUnion, MeasureTheory.measure_union_le
-- 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
refine_5 :
  ∀ (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∞)

Tactic:
suffices H : ∀ m n : ℕ, μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) = 0 by
    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
    refine le_antisymm ((measure_mono A).trans ?_) bot_le
    calc
      μ (toMeasurable μ sᶜ ∪
        ⋃ (m) (n), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
          μ (toMeasurable μ sᶜ) +
            μ (⋃ (m) (n), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) :=
        measure_union_le _ _
      _ = μ (⋃ (m) (n), toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := by
        have : μ sᶜ = 0 := v.ae_tendsto_div hρ; rw [measure_toMeasurable, this, zero_add]
      _ ≤ ∑' (m) (n), μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) :=
        ((measure_iUnion_le _).trans (ENNReal.tsum_le_tsum fun m => measure_iUnion_le _))
      _ = 0 := by simp only [H, tsum_zero]
-- 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
m n : ℕ
apply : u m ⊆ spanningSets (ρ + μ) m
⊢ Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (u m) : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
apply (lt_of_le_of_lt (measure_mono _) (measure_spanningSets_lt_top (ρ + μ) m)).ne

Premises used: lt_of_le_of_lt, MeasureTheory.measure_spanningSets_lt_top, HAdd.hAdd, MeasureTheory.measure_mono, LT.lt.ne
-- 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
m n : ℕ
⊢ u m ⊆ spanningSets (ρ + μ) m

Tactic:
inter_subset_right

Premises used: 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
m n : ℕ
⊢ u m ⊆ spanningSets (ρ + μ) m

Tactic:
exact inter_subset_right

Premises used: 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
m n : ℕ
⊢ Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (u m) : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
have I : (ρ + μ) (u m) ≠ ∞ := by
    apply (lt_of_le_of_lt (measure_mono _) (measure_spanningSets_lt_top (ρ + μ) m)).ne
    exact 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
m n : ℕ
I : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (u m) : ℝ≥0∞) (∞ : ℝ≥0∞)
apply : w n ⊆ spanningSets (ρ + μ) n
⊢ Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (w n) : ℝ≥0∞) (∞ : ℝ≥0∞)

Tactic:
apply (lt_of_le_of_lt (measure_mono _) (measure_spanningSets_lt_top (ρ + μ) n)).ne

Premises used: lt_of_le_of_lt, MeasureTheory.measure_spanningSets_lt_top, HAdd.hAdd, MeasureTheory.measure_mono, LT.lt.ne
-- 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
m n : ℕ
I : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (u m) : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ w n ⊆ spanningSets (ρ + μ) n

Tactic:
inter_subset_right

Premises used: 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
m n : ℕ
I : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (u m) : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ w n ⊆ spanningSets (ρ + μ) n

Tactic:
exact inter_subset_right

Premises used: Set.inter_subset_right