-- Mathlib/MeasureTheory/Covering/LiminfLimsup.lean
theorem blimsup_cthickening_ae_le_of_eventually_mul_le_aux (p : ℕ → Prop) {s : ℕ → Set α}
    (hs : ∀ i, IsClosed (s i)) {r₁ r₂ : ℕ → ℝ} (hr : Tendsto r₁ atTop (𝓝[>] 0)) (hrp : 0 ≤ r₁)
    {M : ℝ} (hM : 0 < M) (hM' : M < 1) (hMr : ∀ᶠ i in atTop, M * r₁ i ≤ r₂ i) :
    (blimsup (fun i => cthickening (r₁ i) (s i)) atTop p : Set α) ≤ᵐ[μ]
      (blimsup (fun i => cthickening (r₂ i) (s i)) atTop p : Set α)

Proof state:
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)
j : ℕ
hB : ↑C⁻¹ * μ (B j) ≠ ⊤
hj₁ : Disjoint (b j) (W ∩ B j)
hj₂ : ↑C⁻¹ * μ (B j) ≤ μ (b j)
⊢ ↑C⁻¹ * μ (B j) + μ (W ∩ B j) ≤ μ (B j)

Tactic:
refine le_trans (add_le_add_right hj₂ _) ?_

Result:
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
h₄ : ∀ᶠ (j : ℕ) in atTop, μ (B j) ≤ ↑C * μ (b j)
j : ℕ
hB : ↑C⁻¹ * μ (B j) ≠ ⊤
hj₁ : Disjoint (b j) (W ∩ B j)
hj₂ : ↑C⁻¹ * μ (B j) ≤ μ (b j)
⊢ μ (b j) + μ (W ∩ B j) ≤ μ (B j)

Proof state:
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
⊢ ∀ᶠ (j : ℕ) in atTop, μ (W ∩ closedBall (w j) (r₁ (f j))) / μ (closedBall (w j) (r₁ (f j))) ≤ ↑(1 - C⁻¹)

Tactic:
have h₃ : ∀ᶠ j in atTop, Disjoint (b j) (W ∩ B j) := by
  apply hMr.mp
  rw [eventually_atTop]
  refine
    ⟨i, fun j hj hj' => Disjoint.inf_right (B j) <| Disjoint.inf_right' (blimsup Y₁ atTop p) ?_⟩
  change Disjoint (b j) (Z i)ᶜ
  rw [disjoint_compl_right_iff_subset]
  refine (closedBall_subset_cthickening (hw j) (M * r₁ (f j))).trans
    ((cthickening_mono hj' _).trans fun a ha => ?_)
  simp only [Z, mem_iUnion, exists_prop]
  exact ⟨f j, ⟨hf₁ j, hj.le.trans (hf₂ j)⟩, ha⟩

Result:
case intro.intro.intro.intro.intro
α : Type u_1
inst✝⁵ : PseudoMetricSpace α
inst✝⁴ : SecondCountableTopology α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : IsLocallyFiniteMeasure μ
inst✝ : IsUnifLocDoublingMeasure μ
p : ℕ → Prop
s : ℕ → Set α
hs : ∀ (i : ℕ), IsClosed (s i)
r₁ r₂ : ℕ → ℝ
hrp : 0 ≤ r₁
M : ℝ
hM : 0 < M
hM' : M < 1
Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i)
Y₂ : ℕ → Set α := fun i => cthickening (r₂ i) (s i)
Z : ℕ → Set α := fun i => ⋃ j, ⋃ (_ : p j ∧ i ≤ j), Y₂ j
i : ℕ
W : Set α := blimsup Y₁ atTop p \ Z i
contra : ¬μ W = 0
d : α
hd' :
  ∀ {ι : Type} {l : Filter ι} (w : ι → α) (δ : ι → ℝ),
    Tendsto δ l (𝓝[>] 0) →
      (∀ᶠ (j : ι) in l, d ∈ closedBall (w j) (2 * δ j)) →
        Tendsto (fun j => μ (W ∩ closedBall (w j) (δ j)) / μ (closedBall (w j) (δ j))) l (𝓝 1)
hd : d ∈ blimsup Y₁ atTop p
f : ℕ → ℕ
hf₁ : ∀ (x : ℕ), p (f x)
hf₂ : ∀ (j : ℕ), j ≤ f j
hf₃ : Tendsto f atTop atTop
hr : Tendsto (r₁ ∘ f) atTop (𝓝[>] 0)
hMr : ∀ᶠ (j : ℕ) in atTop, M * r₁ (f j) ≤ r₂ (f j)
w : ℕ → α
hw : ∀ (j : ℕ), w j ∈ s (f j)
hw' : ∀ (j : ℕ), d ∈ closedBall (w j) (2 * r₁ (f j))
C : ℝ≥0 := IsUnifLocDoublingMeasure.scalingConstantOf μ M⁻¹
hC : C ≠ 0
b : ℕ → Set α := fun j => closedBall (w j) (M * r₁ (f j))
B : ℕ → Set α := fun j => closedBall (w j) (r₁ (f j))
h₁ : ∀ (j : ℕ), b j ⊆ B j
h₂ : ∀ (j : ℕ), W ∩ B j ⊆ B j
h₃ : ∀ᶠ (j : ℕ) in atTop, Disjoint (b j) (W ∩ B j)
⊢ ∀ᶠ (j : ℕ) in atTop, μ (W ∩ closedBall (w j) (r₁ (f j))) / μ (closedBall (w j) (r₁ (f j))) ≤ ↑(1 - C⁻¹)