-- Mathlib/Probability/StrongLaw.lean
theorem strong_law_aux2 {c : ℝ} (c_one : 1 < c) :
    ∀ᵐ ω, (fun n : ℕ => ∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i ω -
      𝔼[∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i]) =o[atTop] fun n : ℕ => (⌊c ^ n⌋₊ : ℝ)

Proof state:
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g => IndepFun f g ℙ) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
⊢ ∀ᵐ (ω : Ω),
    (fun n =>
        ∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω -
          ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a) =o[atTop]
      fun n => ↑⌊c ^ n⌋₊

Tactic:
obtain ⟨v, -, v_pos, v_lim⟩ :
    ∃ v : ℕ → ℝ, StrictAnti v ∧ (∀ n : ℕ, 0 < v n) ∧ Tendsto v atTop (𝓝 0) :=
  exists_seq_strictAnti_tendsto (0 : ℝ)

Result:
case intro.intro.intro
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g => IndepFun f g ℙ) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
v : ℕ → ℝ
v_pos : ∀ (n : ℕ), 0 < v n
v_lim : Tendsto v atTop (𝓝 0)
⊢ ∀ᵐ (ω : Ω),
    (fun n =>
        ∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω -
          ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a) =o[atTop]
      fun n => ↑⌊c ^ n⌋₊

Proof state:
case h
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g => IndepFun f g ℙ) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
v : ℕ → ℝ
v_pos : ∀ (n : ℕ), 0 < v n
v_lim : Tendsto v atTop (𝓝 0)
this :
  ∀ (i : ℕ),
    ∀ᵐ (ω : Ω),
      ∀ᶠ (n : ℕ) in atTop,
        |∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a| <
          v i * ↑⌊c ^ n⌋₊
ω : Ω
hω :
  ∀ (i : ℕ),
    ∀ᶠ (n : ℕ) in atTop,
      |∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a| <
        v i * ↑⌊c ^ n⌋₊
ε : ℝ
εpos : 0 < ε
i : ℕ
hi : v i < ε
n : ℕ
hn :
  |∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a| <
    v i * ↑⌊c ^ n⌋₊
⊢ |∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a| ≤
    ε * ↑⌊c ^ n⌋₊

Tactic:
exact hn.le.trans (mul_le_mul_of_nonneg_right hi.le (Nat.cast_nonneg _))

Result:
no goals

Proof state:
case intro.intro.intro
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g => IndepFun f g ℙ) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
v : ℕ → ℝ
v_pos : ∀ (n : ℕ), 0 < v n
v_lim : Tendsto v atTop (𝓝 0)
this :
  ∀ (i : ℕ),
    ∀ᵐ (ω : Ω),
      ∀ᶠ (n : ℕ) in atTop,
        |∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a| <
          v i * ↑⌊c ^ n⌋₊
⊢ ∀ᵐ (ω : Ω),
    (fun n =>
        ∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω -
          ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a) =o[atTop]
      fun n => ↑⌊c ^ n⌋₊

Tactic:
filter_upwards [ae_all_iff.2 this] with ω hω

Result:
case h
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g => IndepFun f g ℙ) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
v : ℕ → ℝ
v_pos : ∀ (n : ℕ), 0 < v n
v_lim : Tendsto v atTop (𝓝 0)
this :
  ∀ (i : ℕ),
    ∀ᵐ (ω : Ω),
      ∀ᶠ (n : ℕ) in atTop,
        |∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a| <
          v i * ↑⌊c ^ n⌋₊
ω : Ω
hω :
  ∀ (i : ℕ),
    ∀ᶠ (n : ℕ) in atTop,
      |∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a| <
        v i * ↑⌊c ^ n⌋₊
⊢ (fun n =>
      ∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω -
        ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a) =o[atTop]
    fun n => ↑⌊c ^ n⌋₊

Proof state:
case intro
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g => IndepFun f g ℙ) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
v : ℕ → ℝ
v_pos : ∀ (n : ℕ), 0 < v n
v_lim : Tendsto v atTop (𝓝 0)
this :
  ∀ (i : ℕ),
    ∀ᵐ (ω : Ω),
      ∀ᶠ (n : ℕ) in atTop,
        |∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a| <
          v i * ↑⌊c ^ n⌋₊
ω : Ω
hω :
  ∀ (i : ℕ),
    ∀ᶠ (n : ℕ) in atTop,
      |∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a| <
        v i * ↑⌊c ^ n⌋₊
ε : ℝ
εpos : 0 < ε
i : ℕ
hi : v i < ε
⊢ ∀ᶠ (x : ℕ) in atTop,
    ‖∑ i ∈ range ⌊c ^ x⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ x⌋₊, truncation (X i) ↑i) a‖ ≤
      ε * ‖↑⌊c ^ x⌋₊‖

Tactic:
filter_upwards [hω i] with n hn

Result:
case h
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise ((fun f g => IndepFun f g ℙ) on X)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
c : ℝ
c_one : 1 < c
v : ℕ → ℝ
v_pos : ∀ (n : ℕ), 0 < v n
v_lim : Tendsto v atTop (𝓝 0)
this :
  ∀ (i : ℕ),
    ∀ᵐ (ω : Ω),
      ∀ᶠ (n : ℕ) in atTop,
        |∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a| <
          v i * ↑⌊c ^ n⌋₊
ω : Ω
hω :
  ∀ (i : ℕ),
    ∀ᶠ (n : ℕ) in atTop,
      |∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a| <
        v i * ↑⌊c ^ n⌋₊
ε : ℝ
εpos : 0 < ε
i : ℕ
hi : v i < ε
n : ℕ
hn :
  |∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a| <
    v i * ↑⌊c ^ n⌋₊
⊢ ‖∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) ↑i) a‖ ≤
    ε * ‖↑⌊c ^ n⌋₊‖