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

Tactic:
have J : (ρ + μ) (w n) ≠ ∞ := by
    apply (lt_of_le_of_lt (measure_mono _) (measure_spanningSets_lt_top (ρ + μ) n)).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∞)
J : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (w n) : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) (u m ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)
    ((p • μ : (a : Set α) → ℝ≥0∞) (u m ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)

Tactic:
v.measure_le_of_frequently_le (p • μ) hρ _ fun x hx => ?_

Premises used: VitaliFamily.measure_le_of_frequently_le, HSMul.hSMul
-- 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∞)
J : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (w n) : ℝ≥0∞) (∞ : ℝ≥0∞)
refine :
  ∀ (x : α) (hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n)),
    ∃ᶠ (a : Set α) in v.filterAt x,
      LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((p • μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) (u m ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)
    ((p • μ : (a : Set α) → ℝ≥0∞) (u m ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)

Tactic:
refine v.measure_le_of_frequently_le (p • μ) hρ _ fun x hx => ?_

Premises used: VitaliFamily.measure_le_of_frequently_le, HSMul.hSMul
-- 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∞)
J : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (w n) : ℝ≥0∞) (∞ : ℝ≥0∞)
x : α
hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n)
⊢ 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 (v.limRatio ρ x))

Tactic:
have L : Tendsto (fun a : Set α => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x)) :=
          tendsto_nhds_limUnder hx.1.1.1
-- 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∞)
J : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (w n) : ℝ≥0∞) (∞ : ℝ≥0∞)
x : α
hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n)
L :
  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 (v.limRatio ρ x))
⊢ ∀ᶠ (b : Set α) in v.filterAt x,
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) b : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) b : ℝ≥0∞) <
      (↑p : ℝ≥0∞)

Tactic:
have I : ∀ᶠ b : Set α in v.filterAt x, ρ b / μ b < p := (tendsto_order.1 L).2 _ hx.1.1.2
-- 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_1 : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (u m) : ℝ≥0∞) (∞ : ℝ≥0∞)
J : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (w n) : ℝ≥0∞) (∞ : ℝ≥0∞)
x : α
hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n)
L :
  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 (v.limRatio ρ x))
I :
  ∀ᶠ (b : Set α) in v.filterAt x,
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) b : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) b : ℝ≥0∞) <
      (↑p : ℝ≥0∞)
apply :
  ∀ (a : Set α)
    (ha :
      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∞) <
        (↑p : ℝ≥0∞)),
    LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((p • μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
⊢ ∃ᶠ (a : Set α) in v.filterAt x,
    LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((p • μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)

Tactic:
apply I.frequently.mono fun a ha => ?_

Premises used: Filter.Frequently.mono, Filter.Eventually.frequently
-- 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_1 : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (u m) : ℝ≥0∞) (∞ : ℝ≥0∞)
J : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (w n) : ℝ≥0∞) (∞ : ℝ≥0∞)
x : α
hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n)
L :
  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 (v.limRatio ρ x))
I :
  ∀ᶠ (b : Set α) in v.filterAt x,
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) b : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) b : ℝ≥0∞) <
      (↑p : ℝ≥0∞)
a : Set α
ha :
  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∞) <
    (↑p : ℝ≥0∞)
rw : (ρ : (a : Set α) → ℝ≥0∞) a ≤ HMul.hMul (β := ℝ≥0∞) (↑p : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)
⊢ LE.le (α := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) ((p • μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)

Tactic:
rw [coe_nnreal_smul_apply]

Premises used: MeasureTheory.Measure.coe_nnreal_smul_apply
-- 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_1 : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (u m) : ℝ≥0∞) (∞ : ℝ≥0∞)
J : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (w n) : ℝ≥0∞) (∞ : ℝ≥0∞)
x : α
hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n)
L :
  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 (v.limRatio ρ x))
I :
  ∀ᶠ (b : Set α) in v.filterAt x,
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) b : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) b : ℝ≥0∞) <
      (↑p : ℝ≥0∞)
a : Set α
ha :
  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∞) <
    (↑p : ℝ≥0∞)
⊢ (ρ : (a : Set α) → ℝ≥0∞) a ≤ HMul.hMul (β := ℝ≥0∞) (↑p : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)

Tactic:
(ENNReal.div_le_iff_le_mul ?_ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le

Premises used: ENNReal.div_le_iff_le_mul, LT.lt.le, LT.lt.ne', Iff.mp, LE.le.trans_lt, bot_le, 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
m n : ℕ
I_1 : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (u m) : ℝ≥0∞) (∞ : ℝ≥0∞)
J : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (w n) : ℝ≥0∞) (∞ : ℝ≥0∞)
x : α
hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n)
L :
  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 (v.limRatio ρ x))
I :
  ∀ᶠ (b : Set α) in v.filterAt x,
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) b : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) b : ℝ≥0∞) <
      (↑p : ℝ≥0∞)
a : Set α
ha :
  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∞) <
    (↑p : ℝ≥0∞)
refine : Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) (0 : ℝ≥0∞) ∨ (↑p : ℝ≥0∞) ≠ ∞
⊢ (ρ : (a : Set α) → ℝ≥0∞) a ≤ HMul.hMul (β := ℝ≥0∞) (↑p : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞)

Tactic:
refine (ENNReal.div_le_iff_le_mul ?_ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le

Premises used: ENNReal.div_le_iff_le_mul, LT.lt.le, LT.lt.ne', Iff.mp, LE.le.trans_lt, bot_le, 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
m n : ℕ
I_1 : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (u m) : ℝ≥0∞) (∞ : ℝ≥0∞)
J : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (w n) : ℝ≥0∞) (∞ : ℝ≥0∞)
x : α
hx : x ∈ u m ∩ toMeasurable (ρ + μ) (w n)
L :
  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 (v.limRatio ρ x))
I :
  ∀ᶠ (b : Set α) in v.filterAt x,
    HDiv.hDiv (α := ℝ≥0∞) (β := ℝ≥0∞) ((ρ : (a : Set α) → ℝ≥0∞) b : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) b : ℝ≥0∞) <
      (↑p : ℝ≥0∞)
a : Set α
ha :
  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∞) <
    (↑p : ℝ≥0∞)
⊢ Ne (α := ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) a : ℝ≥0∞) (0 : ℝ≥0∞) ∨ (↑p : ℝ≥0∞) ≠ ∞

Tactic:
simp only [ENNReal.coe_ne_top, Ne, or_true, not_false_iff]

Premises used: Ne, not_false_iff, or_true, ENNReal.coe_ne_top
-- 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∞)
J : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (w n) : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ (p • μ : (a : Set α) → ℝ≥0∞) (u m ∩ toMeasurable (ρ + μ) (w n)) =
    HMul.hMul (β := ℝ≥0∞) (↑p : ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)

Tactic:
simp only [coe_nnreal_smul_apply,
          measure_toMeasurable_add_inter_right (measurableSet_toMeasurable _ _) I]

Premises used: MeasureTheory.Measure.measure_toMeasurable_add_inter_right, MeasureTheory.Measure.coe_nnreal_smul_apply, MeasureTheory.measurableSet_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
m n : ℕ
I : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (u m) : ℝ≥0∞) (∞ : ℝ≥0∞)
J : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (w n) : ℝ≥0∞) (∞ : ℝ≥0∞)
⊢ (ρ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
    HMul.hMul (β := ℝ≥0∞) (↑p : ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)

Tactic:
have A :
    ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
      p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) :=
    calc
      ρ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) =
          ρ (u m ∩ toMeasurable (ρ + μ) (w n)) :=
        measure_toMeasurable_add_inter_left (measurableSet_toMeasurable _ _) I
      _ ≤ (p • μ) (u m ∩ toMeasurable (ρ + μ) (w n)) := by
        refine v.measure_le_of_frequently_le (p • μ) hρ _ fun x hx => ?_
        have L : Tendsto (fun a : Set α => ρ a / μ a) (v.filterAt x) (𝓝 (v.limRatio ρ x)) :=
          tendsto_nhds_limUnder hx.1.1.1
        have I : ∀ᶠ b : Set α in v.filterAt x, ρ b / μ b < p := (tendsto_order.1 L).2 _ hx.1.1.2
        apply I.frequently.mono fun a ha => ?_
        rw [coe_nnreal_smul_apply]
        refine (ENNReal.div_le_iff_le_mul ?_ (Or.inr (bot_le.trans_lt ha).ne')).1 ha.le
        simp only [ENNReal.coe_ne_top, Ne, or_true, not_false_iff]
      _ = p * μ (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) := by
        simp only [coe_nnreal_smul_apply,
          measure_toMeasurable_add_inter_right (measurableSet_toMeasurable _ _) I]
-- 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∞)
J : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (w n) : ℝ≥0∞) (∞ : ℝ≥0∞)
A :
  (ρ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
    HMul.hMul (β := ℝ≥0∞) (↑p : ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)
rw : HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (w n ∩ toMeasurable (ρ + μ) (u m)) : ℝ≥0∞)
| HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ w n) : ℝ≥0∞)

Tactic:
rw [inter_comm]

Premises used: Set.inter_comm
-- 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∞)
J : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (w n) : ℝ≥0∞) (∞ : ℝ≥0∞)
A :
  (ρ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
    HMul.hMul (β := ℝ≥0∞) (↑p : ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)
conv_rhs :
  HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞) =
    HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (w n ∩ toMeasurable (ρ + μ) (u m)) : ℝ≥0∞)
⊢ HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞) =
    HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ w n) : ℝ≥0∞)

Tactic:
conv_rhs => rw [inter_comm]

Premises used: Set.inter_comm
-- 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∞)
J : Ne (α := ℝ≥0∞) ((ρ + μ : (a : Set α) → ℝ≥0∞) (w n) : ℝ≥0∞) (∞ : ℝ≥0∞)
A :
  (ρ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) ≤
    HMul.hMul (β := ℝ≥0∞) (↑p : ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞)
rw :
  HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (w n) ∩ toMeasurable (ρ + μ) (u m)) : ℝ≥0∞) =
    HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (w n ∩ toMeasurable (ρ + μ) (u m)) : ℝ≥0∞)
rw₁ rw₂ :
  HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (w n ∩ toMeasurable (ρ + μ) (u m)) : ℝ≥0∞) =
    HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (w n ∩ toMeasurable (ρ + μ) (u m)) : ℝ≥0∞)
⊢ HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞)
      ((μ : (a : Set α) → ℝ≥0∞) (toMeasurable (ρ + μ) (u m) ∩ toMeasurable (ρ + μ) (w n)) : ℝ≥0∞) =
    HMul.hMul (β := ℝ≥0∞) (↑q : ℝ≥0∞) ((μ : (a : Set α) → ℝ≥0∞) (w n ∩ toMeasurable (ρ + μ) (u m)) : ℝ≥0∞)

Tactic:
rw [inter_comm, measure_toMeasurable_add_inter_right (measurableSet_toMeasurable _ _) J]

Premises used: MeasureTheory.Measure.measure_toMeasurable_add_inter_right, Set.inter_comm, MeasureTheory.measurableSet_toMeasurable