-- Mathlib/Probability/StrongLaw.lean
theorem strong_law_aux1 {c : ℝ} (c_one : 1 < c) {ε : ℝ} (εpos : 0 < ε) : ∀ᵐ ω, ∀ᶠ n : ℕ in atTop,
    |∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i ω - 𝔼[∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) i]| <
    ε * ⌊c ^ n⌋₊

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
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n => truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n => ∑ i ∈ range n, Y i
hS : S = fun n => ∑ i ∈ range n, Y i
u : ℕ → ℕ := fun n => ⌊c ^ n⌋₊
u_mono : Monotone u
I1 : ∀ (K : ℕ), ∑ j ∈ range K, (↑j ^ 2)⁻¹ * Var[Y j; ℙ] ≤ 2 * ∫ (a : Ω), X 0 a
C : ℝ := c ^ 5 * (c - 1)⁻¹ ^ 3 * (2 * ∫ (a : Ω), X 0 a)
I2 : ∀ (N : ℕ), ∑ i ∈ range N, (↑(u i) ^ 2)⁻¹ * Var[S (u i); ℙ] ≤ C
N : ℕ
⊢ ∑ i ∈ range N, ENNReal.ofReal (Var[S (u i); ℙ] / (↑(u i) * ε) ^ 2) =
    ENNReal.ofReal (∑ i ∈ range N, Var[S (u i); ℙ] / (↑(u i) * ε) ^ 2)

Tactic:
rw [ENNReal.ofReal_sum_of_nonneg fun i _ => ?_]

Result:
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
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n => truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n => ∑ i ∈ range n, Y i
hS : S = fun n => ∑ i ∈ range n, Y i
u : ℕ → ℕ := fun n => ⌊c ^ n⌋₊
u_mono : Monotone u
I1 : ∀ (K : ℕ), ∑ j ∈ range K, (↑j ^ 2)⁻¹ * Var[Y j; ℙ] ≤ 2 * ∫ (a : Ω), X 0 a
C : ℝ := c ^ 5 * (c - 1)⁻¹ ^ 3 * (2 * ∫ (a : Ω), X 0 a)
I2 : ∀ (N : ℕ), ∑ i ∈ range N, (↑(u i) ^ 2)⁻¹ * Var[S (u i); ℙ] ≤ C
N i : ℕ
x✝ : i ∈ range N
⊢ 0 ≤ Var[S (u i); ℙ] / (↑(u i) * ε) ^ 2

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
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n => truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n => ∑ i ∈ range n, Y i
hS : S = fun n => ∑ i ∈ range n, Y i
⊢ ∀ᵐ (ω : Ω),
    ∀ᶠ (n : ℕ) in atTop,
      |∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (range ⌊c ^ n⌋₊).sum Y a| < ε * ↑⌊c ^ n⌋₊

Tactic:
let u : ℕ → ℕ := fun n => ⌊c ^ n⌋₊

Result:
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
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n => truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n => ∑ i ∈ range n, Y i
hS : S = fun n => ∑ i ∈ range n, Y i
u : ℕ → ℕ := fun n => ⌊c ^ n⌋₊
⊢ ∀ᵐ (ω : Ω),
    ∀ᶠ (n : ℕ) in atTop,
      |∑ i ∈ range ⌊c ^ n⌋₊, truncation (X i) (↑i) ω - ∫ (a : Ω), (range ⌊c ^ n⌋₊).sum Y a| < ε * ↑⌊c ^ n⌋₊

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
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n => truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n => ∑ i ∈ range n, Y i
hS : S = fun n => ∑ i ∈ range n, Y i
u : ℕ → ℕ := fun n => ⌊c ^ n⌋₊
u_mono : Monotone u
K j : ℕ
x✝ : j ∈ range K
⊢ Var[truncation (X 0) ↑j; ℙ] ≤ ∫ (a : Ω), (truncation (X 0) ↑j ^ 2) a

Tactic:
exact variance_le_expectation_sq (hX 0).truncation

Result:
no goals

Proof state:
case refine_5
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
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
hX : ∀ (i : ℕ), AEStronglyMeasurable (X i) ℙ
A : ∀ (i : ℝ), StronglyMeasurable ((Set.Ioc (-i) i).indicator id)
Y : ℕ → Ω → ℝ := fun n => truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n => ∑ i ∈ range n, Y i
hS : S = fun n => ∑ i ∈ range n, Y i
u : ℕ → ℕ := fun n => ⌊c ^ n⌋₊
u_mono : Monotone u
I1 : ∀ (K : ℕ), ∑ j ∈ range K, (↑j ^ 2)⁻¹ * Var[Y j; ℙ] ≤ 2 * ∫ (a : Ω), X 0 a
C : ℝ := c ^ 5 * (c - 1)⁻¹ ^ 3 * (2 * ∫ (a : Ω), X 0 a)
N : ℕ
⊢ ∀ a ∈ (range N).sigma fun a => range (u a),
    (↑(u a.fst) ^ 2)⁻¹ * Var[Y a.snd; ℙ] =
      (↑(u ((fun p => ⟨p.snd, p.fst⟩) a).snd) ^ 2)⁻¹ * Var[Y ((fun p => ⟨p.snd, p.fst⟩) a).fst; ℙ]

Tactic:
simp

Result:
no goals

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
ε : ℝ
εpos : 0 < ε
⊢ ∀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| <
        ε * ↑⌊c ^ n⌋₊

Tactic:
have c_pos : 0 < c := zero_lt_one.trans c_one

Result:
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
ε : ℝ
εpos : 0 < ε
c_pos : 0 < c
⊢ ∀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| <
        ε * ↑⌊c ^ n⌋₊