-- Mathlib/MeasureTheory/Covering/Differentiation.lean
theorem ae_tendsto_div : ∀ᵐ x ∂μ, ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)

Proof state:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left✝ : 0 ∉ w
w_top : ⊤ ∉ w
I : ∀ x ∈ w, x ≠ ⊤
c : ℝ≥0∞
hc : c ∈ w
d : ℝ≥0∞
hd : d ∈ w
hcd : c < d
⊢ ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ (a : Set α) in v.filterAt x, d < ρ a / μ a)

Tactic:
lift c to ℝ≥0 using I c hc

Result:
case intro
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left✝ : 0 ∉ w
w_top : ⊤ ∉ w
I : ∀ x ∈ w, x ≠ ⊤
d : ℝ≥0∞
hd : d ∈ w
c : ℝ≥0
hc : ↑c ∈ w
hcd : ↑c < d
⊢ ∀ᵐ (x : α) ∂μ, ¬((∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < ↑c) ∧ ∃ᶠ (a : Set α) in v.filterAt x, d < ρ a / μ a)

Proof state:
case h
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left✝ : 0 ∉ w
w_top : ⊤ ∉ w
I : ∀ x ∈ w, x ≠ ⊤
A :
  ∀ c ∈ w,
    ∀ d ∈ w,
      c < d →
        ∀ᵐ (x : α) ∂μ,
          ¬((∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ (a : Set α) in v.filterAt x, d < ρ a / μ a)
B :
  ∀ᵐ (x : α) ∂μ,
    ∀ c ∈ w,
      ∀ d ∈ w,
        c < d → ¬((∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ (a : Set α) in v.filterAt x, d < ρ a / μ a)
x : α
hx :
  ∀ c ∈ w,
    ∀ d ∈ w, c < d → ¬((∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ (a : Set α) in v.filterAt x, d < ρ a / μ a)
⊢ ∃ c, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 c)

Tactic:
exact tendsto_of_no_upcrossings w_dense hx

Result:
no goals

Proof state:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left✝ : 0 ∉ w
w_top : ⊤ ∉ w
I : ∀ x ∈ w, x ≠ ⊤
A :
  ∀ c ∈ w,
    ∀ d ∈ w,
      c < d →
        ∀ᵐ (x : α) ∂μ,
          ¬((∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ (a : Set α) in v.filterAt x, d < ρ a / μ a)
⊢ ∀ᵐ (x : α) ∂μ,
    ∀ c ∈ w,
      ∀ d ∈ w,
        c < d → ¬((∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ (a : Set α) in v.filterAt x, d < ρ a / μ a)

Tactic:
rw [ae_ball_iff w_count]

Result:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left✝ : 0 ∉ w
w_top : ⊤ ∉ w
I : ∀ x ∈ w, x ≠ ⊤
A :
  ∀ c ∈ w,
    ∀ d ∈ w,
      c < d →
        ∀ᵐ (x : α) ∂μ,
          ¬((∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ (a : Set α) in v.filterAt x, d < ρ a / μ a)
⊢ ∀ i ∈ w,
    ∀ᵐ (x : α) ∂μ,
      ∀ d ∈ w,
        i < d → ¬((∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < i) ∧ ∃ᶠ (a : Set α) in v.filterAt x, d < ρ a / μ a)

Proof state:
case intro.intro.hd
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left✝ : 0 ∉ w
w_top : ⊤ ∉ w
I : ∀ x ∈ w, x ≠ ⊤
c : ℝ≥0
hc : ↑c ∈ w
d : ℝ≥0
hd : ↑d ∈ w
hcd : ↑c < ↑d
⊢ ∀ (x : α),
    (∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < ↑c) →
      (∃ᶠ (x : Set α) in v.filterAt x, ↑d < ρ x / μ x) → ∃ᶠ (a : Set α) in v.filterAt x, ↑d * μ a ≤ ρ a

Tactic:
intro x _ h2x

Result:
case intro.intro.hd
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left✝ : 0 ∉ w
w_top : ⊤ ∉ w
I : ∀ x ∈ w, x ≠ ⊤
c : ℝ≥0
hc : ↑c ∈ w
d : ℝ≥0
hd : ↑d ∈ w
hcd : ↑c < ↑d
x : α
a✝ : ∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < ↑c
h2x : ∃ᶠ (x : Set α) in v.filterAt x, ↑d < ρ x / μ x
⊢ ∃ᶠ (a : Set α) in v.filterAt x, ↑d * μ a ≤ ρ a

Proof state:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left✝ : 0 ∉ w
w_top : ⊤ ∉ w
I : ∀ x ∈ w, x ≠ ⊤
c : ℝ≥0
hc : ↑c ∈ w
d : ℝ≥0
hd : ↑d ∈ w
hcd : ↑c < ↑d
x : α
h1x : ∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < ↑c
a✝ : ∃ᶠ (x : Set α) in v.filterAt x, ↑d < ρ x / μ x
a : Set α
ha : ρ a / μ a < ↑c
⊢ μ a ≠ 0 ∨ ↑c ≠ ⊤

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

Result:
no goals

Proof state:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left✝ : 0 ∉ w
w_top : ⊤ ∉ w
I : ∀ x ∈ w, x ≠ ⊤
A :
  ∀ c ∈ w,
    ∀ d ∈ w,
      c < d →
        ∀ᵐ (x : α) ∂μ,
          ¬((∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ (a : Set α) in v.filterAt x, d < ρ a / μ a)
⊢ ∀ i ∈ w,
    ∀ᵐ (x : α) ∂μ,
      ∀ d ∈ w,
        i < d → ¬((∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < i) ∧ ∃ᶠ (a : Set α) in v.filterAt x, d < ρ a / μ a)

Tactic:
intro x hx

Result:
α : Type u_1
inst✝⁴ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
inst✝³ : SecondCountableTopology α
inst✝² : BorelSpace α
inst✝¹ : IsLocallyFiniteMeasure μ
ρ : Measure α
inst✝ : IsLocallyFiniteMeasure ρ
hρ : ρ ≪ μ
w : Set ℝ≥0∞
w_count : w.Countable
w_dense : Dense w
left✝ : 0 ∉ w
w_top : ⊤ ∉ w
I : ∀ x ∈ w, x ≠ ⊤
A :
  ∀ c ∈ w,
    ∀ d ∈ w,
      c < d →
        ∀ᵐ (x : α) ∂μ,
          ¬((∃ᶠ (a : Set α) in v.filterAt x, ρ a / μ a < c) ∧ ∃ᶠ (a : Set α) in v.filterAt x, d < ρ a / μ a)
x : ℝ≥0∞
hx : x ∈ w
⊢ ∀ᵐ (x_1 : α) ∂μ,
    ∀ d ∈ w,
      x < d → ¬((∃ᶠ (a : Set α) in v.filterAt x_1, ρ a / μ a < x) ∧ ∃ᶠ (a : Set α) in v.filterAt x_1, d < ρ a / μ a)