-- 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:
case intro.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 ≠ ⊤
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:
apply v.null_of_frequently_le_of_frequently_ge hρ (ENNReal.coe_lt_coe.1 hcd)

Result:
case intro.intro.hc
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 ∈
      {x |
          (fun x =>
              ¬((∃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}ᶜ,
    ∃ᶠ (a : Set α) in v.filterAt x, ρ a ≤ ↑c * μ a

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 ∈
      {x |
          (fun x =>
              ¬((∃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}ᶜ,
    ∃ᶠ (a : Set α) in v.filterAt x, ↑d * μ a ≤ ρ 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 ∈
      {x |
          (fun x =>
              ¬((∃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}ᶜ,
    ∃ᶠ (a : Set α) in v.filterAt x, ↑d * μ a ≤ ρ a

Tactic:
simp only [and_imp, exists_prop, not_frequently, not_and, not_lt, not_le, not_eventually,
  mem_setOf_eq, mem_compl_iff, not_forall]

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),
    (∃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

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
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

Tactic:
apply h2x.mono fun a ha => ?_

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 ≠ ⊤
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
a : Set α
ha : ↑d < ρ a / μ a
⊢ ↑d * μ a ≤ ρ a

Proof state:
case intro.intro.hc
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, ρ a ≤ ↑c * μ a

Tactic:
intro x h1x _

Result:
case intro.intro.hc
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
⊢ ∃ᶠ (a : Set α) in v.filterAt x, ρ a ≤ ↑c * μ 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 ≠ ⊤
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)

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)
x : ℝ≥0∞
hx : x ∈ w
⊢ ∀ i ∈ w,
    ∀ᵐ (x_1 : α) ∂μ,
      x < i → ¬((∃ᶠ (a : Set α) in v.filterAt x_1, ρ a / μ a < x) ∧ ∃ᶠ (a : Set α) in v.filterAt x_1, i < ρ a / μ a)