theorem strong_law_aux1 {c : ℝ} (c_one : 1 < c) {ε : ℝ} (εpos : 0 < ε) : ∀ᵐ ω, ∀ᶠ n : ℕ in atTop,
    |∑ i in range ⌊c ^ n⌋₊, truncation (X i) i ω - 𝔼[∑ i in range ⌊c ^ n⌋₊, truncation (X i) i]| <
    ε * ⌊c ^ n⌋₊ := by
  /- Let `S n = ∑ i in range n, Y i` where `Y i = truncation (X i) i`. We should show that
    `|S k - 𝔼[S k]| / k ≤ ε` along the sequence of powers of `c`. For this, we apply Borel-Cantelli:
    it suffices to show that the converse probabilites are summable. From Chebyshev inequality, this
    will follow from a variance control `∑' Var[S (c^i)] / (c^i)^2 < ∞`. This is checked in `I2`
    using pairwise independence to expand the variance of the sum as the sum of the variances,
    and then a straightforward but tedious computation (essentially boiling down to the fact that
    the sum of `1/(c ^ i)^2` beyong a threshold `j` is comparable to `1/j^2`).
    Note that we have written `c^i` in the above proof sketch, but rigorously one should put integer
    parts everywhere, making things more painful. We write `u i = ⌊c^i⌋₊` for brevity. -/
  have c_pos : 0 < c := zero_lt_one.trans c_one
  have hX : ∀ i, AEStronglyMeasurable (X i) ℙ := fun i =>
    (hident i).symm.aestronglyMeasurable_snd hint.1
  have A : ∀ i, StronglyMeasurable (indicator (Set.Ioc (-i : ℝ) i) id) := fun i =>
    stronglyMeasurable_id.indicator measurableSet_Ioc
  set Y := fun n : ℕ => truncation (X n) n
  set S := fun n => ∑ i in range n, Y i with hS
  let u : ℕ → ℕ := fun n => ⌊c ^ n⌋₊
  have u_mono : Monotone u := fun i j hij => Nat.floor_mono (pow_le_pow_right c_one.le hij)
  have I1 : ∀ K, ∑ j in range K, ((j : ℝ) ^ 2)⁻¹ * Var[Y j] ≤ 2 * 𝔼[X 0] := by
    intro K
    calc
      ∑ j in range K, ((j : ℝ) ^ 2)⁻¹ * Var[Y j] ≤
          ∑ j in range K, ((j : ℝ) ^ 2)⁻¹ * 𝔼[truncation (X 0) j ^ 2] := by
        apply sum_le_sum fun j _ => ?_
        refine' mul_le_mul_of_nonneg_left _ (inv_nonneg.2 (sq_nonneg _))
        rw [(hident j).truncation.variance_eq]
        exact variance_le_expectation_sq (hX 0).truncation
      _ ≤ 2 * 𝔼[X 0] := sum_variance_truncation_le hint (hnonneg 0) K
  let C := c ^ 5 * (c - 1)⁻¹ ^ 3 * (2 * 𝔼[X 0])
  have I2 : ∀ N, ∑ i in range N, ((u i : ℝ) ^ 2)⁻¹ * Var[S (u i)] ≤ C := by
    intro N
    calc
      ∑ i in range N, ((u i : ℝ) ^ 2)⁻¹ * Var[S (u i)] =
          ∑ i in range N, ((u i : ℝ) ^ 2)⁻¹ * ∑ j in range (u i), Var[Y j] := by
        congr 1 with i
        congr 1
        rw [hS, IndepFun.variance_sum]
        · intro j _
          exact (hident j).aestronglyMeasurable_fst.memℒp_truncation
        · intro k _ l _ hkl
          exact (hindep hkl).comp (A k).measurable (A l).measurable
      _ = ∑ j in range (u (N - 1)),
          (∑ i in (range N).filter fun i => j < u i, ((u i : ℝ) ^ 2)⁻¹) * Var[Y j] := by
        simp_rw [mul_sum, sum_mul, sum_sigma']
        refine' sum_bij' (fun (p : Σ _ : ℕ, ℕ) _ => (⟨p.2, p.1⟩ : Σ _ : ℕ, ℕ)) _ (fun a _ => rfl)
          (fun (p : Σ _ : ℕ, ℕ) _ => (⟨p.2, p.1⟩ : Σ _ : ℕ, ℕ)) _ _ _
        · rintro ⟨i, j⟩ hij
          simp only [mem_sigma, mem_range] at hij
          simp only [hij.1, hij.2, mem_sigma, mem_range, mem_filter, and_true_iff]
          exact hij.2.trans_le (u_mono (Nat.le_sub_one_of_lt hij.1))
        · rintro ⟨i, j⟩ hij
          simp only [mem_sigma, mem_range, mem_filter] at hij
          simp only [hij.2.1, hij.2.2, mem_sigma, mem_range, and_self_iff]
        · rintro ⟨i, j⟩ hij; rfl
        · rintro ⟨i, j⟩ hij; rfl
      _ ≤ ∑ j in range (u (N - 1)), c ^ 5 * (c - 1)⁻¹ ^ 3 / ↑j ^ 2 * Var[Y j] := by
        apply sum_le_sum fun j hj => ?_
        rcases @eq_zero_or_pos _ _ j with (rfl | hj)
        · simp only [Nat.cast_zero, zero_pow', Ne.def, bit0_eq_zero, Nat.one_ne_zero,
            not_false_iff, div_zero, zero_mul]
          simp only [Nat.cast_zero, truncation_zero, variance_zero, mul_zero, le_rfl]
        apply mul_le_mul_of_nonneg_right _ (variance_nonneg _ _)
        convert sum_div_nat_floor_pow_sq_le_div_sq N (Nat.cast_pos.2 hj) c_one using 2
        · simp only [Nat.cast_lt]
        · simp only [one_div]
      _ = c ^ 5 * (c - 1)⁻¹ ^ 3 * ∑ j in range (u (N - 1)), ((j : ℝ) ^ 2)⁻¹ * Var[Y j] := by
        simp_rw [mul_sum, div_eq_mul_inv, mul_assoc]
      _ ≤ c ^ 5 * (c - 1)⁻¹ ^ 3 * (2 * 𝔼[X 0]) := by
        apply mul_le_mul_of_nonneg_left (I1 _)
        apply mul_nonneg (pow_nonneg c_pos.le _)
        exact pow_nonneg (inv_nonneg.2 (sub_nonneg.2 c_one.le)) _
  have I3 : ∀ N, ∑ i in range N, ℙ {ω | (u i * ε : ℝ) ≤ |S (u i) ω - 𝔼[S (u i)]|} ≤
      ENNReal.ofReal (ε⁻¹ ^ 2 * C) := by
    intro N
    calc
      ∑ i in range N, ℙ {ω | (u i * ε : ℝ) ≤ |S (u i) ω - 𝔼[S (u i)]|} ≤
          ∑ i in range N, ENNReal.ofReal (Var[S (u i)] / (u i * ε) ^ 2) := by
        refine' sum_le_sum fun i _ => _
        apply meas_ge_le_variance_div_sq
        · exact memℒp_finset_sum' _ fun j _ => (hident j).aestronglyMeasurable_fst.memℒp_truncation
        · apply mul_pos (Nat.cast_pos.2 _) εpos
          refine' zero_lt_one.trans_le _
          apply Nat.le_floor
          

Proof state:
case h
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0)
hindep : Pairwise fun i j => IndepFun (X i) (X j)
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0)
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 (indicator (Set.Ioc (-i) i) id)
Y : ℕ → Ω → ℝ := fun n => truncation (X n) ↑n
S : ℕ → Ω → ℝ := fun n => ∑ i in range n, Y i
hS : S = fun n => ∑ i in range n, Y i
u : ℕ → ℕ := fun n => ⌊c ^ n⌋₊
u_mono : Monotone u
I1 : ∀ (K : ℕ), ∑ j in range K, (↑j ^ 2)⁻¹ * variance (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 in range N, (↑(u i) ^ 2)⁻¹ * variance (S (u i)) ℙ ≤ C
N i : ℕ
x✝ : i ∈ range N
⊢ ↑1 ≤ c ^ i

Next tactic:
rw [Nat.cast_one]