-- 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
⊢ ∀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:
set S := fun n => ∑ i ∈ range n, Y i with hS

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
⊢ ∀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:
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
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
⊢ ε⁻¹ ^ 2 * ∑ i ∈ range N, (↑(u i))⁻¹ ^ 2 * Var[S (u i); ℙ] ≤ ε⁻¹ ^ 2 * C

Tactic:
refine mul_le_mul_of_nonneg_left ?_ (sq_nonneg _)

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
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, (↑(u i))⁻¹ ^ 2 * Var[S (u i); ℙ] ≤ C

Proof state:
case h.e'_3.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
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 j : ℕ
hj✝ : j ∈ range (u (N - 1))
hj : 0 < j
⊢ filter (fun i => j < u i) (range N) = filter (fun i => ↑j < ↑⌊c ^ i⌋₊) (range N)

Tactic:
simp only [u, Nat.cast_lt]

Result:
no goals

Proof state:
case h.e'_2.h.h.e'_3.h.e'_4.h.e'_6.h.e'_7.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
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
I3 : ∀ (N : ℕ), ∑ i ∈ range N, ℙ {ω | ↑(u i) * ε ≤ |S (u i) ω - ∫ (a : Ω), S (u i) a|} ≤ ENNReal.ofReal (ε⁻¹ ^ 2 * C)
I4 : ∑' (i : ℕ), ℙ {ω | ↑(u i) * ε ≤ |S (u i) ω - ∫ (a : Ω), S (u i) a|} < ⊤
ω : Ω
hω : ∀ᶠ (n : ℕ) in atTop, |∑ c ∈ range (u n), Y c ω - ∫ (a : Ω), ∑ c ∈ range (u n), Y c a| < ε * ↑(u n)
x✝¹ : ℕ
x✝ : Ω
⊢ (range ⌊c ^ x✝¹⌋₊).sum Y x✝ = ∑ c ∈ range (u x✝¹), Y c x✝

Tactic:
simp only [Y, S, u, C, sum_apply]

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
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[Y j; ℙ] ≤ ∫ (a : Ω), (truncation (X 0) ↑j ^ 2) a

Tactic:
rw [(hident j).truncation.variance_eq]

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
K j : ℕ
x✝ : j ∈ range K
⊢ Var[truncation (X 0) ↑j; ℙ] ≤ ∫ (a : Ω), (truncation (X 0) ↑j ^ 2) a