Theorem: ProbabilityTheory.strong_law_aux3
File: Mathlib/Probability/StrongLaw.lean

Proof steps:
state_before: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise fun i j => IndepFun (X i) (X j) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
⊢ (fun n => (∫ (a : Ω), (∑ i ∈ range n, truncation (X i) ↑i) a) - ↑n * ∫ (a : Ω), X 0 a) =o[atTop] Nat.cast
state_after: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise fun i j => IndepFun (X i) (X j) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : Tendsto (fun i => ∫ (a : Ω), truncation (X i) (↑i) a) atTop (𝓝 (∫ (a : Ω), X 0 a))
⊢ (fun n => (∫ (a : Ω), (∑ i ∈ range n, truncation (X i) ↑i) a) - ↑n * ∫ (a : Ω), X 0 a) =o[atTop] Nat.cast
tactic: have A : Tendsto (fun i => 𝔼[truncation (X i) i]) atTop (𝓝 𝔼[X 0]) := by
    convert (tendsto_integral_truncation hint).comp tendsto_natCast_atTop_atTop using 1
    ext i
    exact (hident i).truncation.integral_eq
state_before: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise fun i j => IndepFun (X i) (X j) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : Tendsto (fun i => ∫ (a : Ω), truncation (X i) (↑i) a) atTop (𝓝 (∫ (a : Ω), X 0 a))
⊢ (fun n => (∫ (a : Ω), (∑ i ∈ range n, truncation (X i) ↑i) a) - ↑n * ∫ (a : Ω), X 0 a) =o[atTop] Nat.cast
state_after: case h.e'_7
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise fun i j => IndepFun (X i) (X j) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : Tendsto (fun i => ∫ (a : Ω), truncation (X i) (↑i) a) atTop (𝓝 (∫ (a : Ω), X 0 a))
⊢ (fun n => (∫ (a : Ω), (∑ i ∈ range n, truncation (X i) ↑i) a) - ↑n * ∫ (a : Ω), X 0 a) = fun n =>
    ∑ i ∈ range n, ((∫ (a : Ω), truncation (X i) (↑i) a) - ∫ (a : Ω), X 0 a)
tactic: convert Asymptotics.isLittleO_sum_range_of_tendsto_zero (tendsto_sub_nhds_zero_iff.2 A) using 1
state_before: case h.e'_7
Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise fun i j => IndepFun (X i) (X j) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : Tendsto (fun i => ∫ (a : Ω), truncation (X i) (↑i) a) atTop (𝓝 (∫ (a : Ω), X 0 a))
⊢ (fun n => (∫ (a : Ω), (∑ i ∈ range n, truncation (X i) ↑i) a) - ↑n * ∫ (a : Ω), X 0 a) = fun n =>
    ∑ i ∈ range n, ((∫ (a : Ω), truncation (X i) (↑i) a) - ∫ (a : Ω), X 0 a)
state_after: case 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 i j => IndepFun (X i) (X j) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : Tendsto (fun i => ∫ (a : Ω), truncation (X i) (↑i) a) atTop (𝓝 (∫ (a : Ω), X 0 a))
n : ℕ
⊢ (∫ (a : Ω), (∑ i ∈ range n, truncation (X i) ↑i) a) - ↑n * ∫ (a : Ω), X 0 a =
    ∑ i ∈ range n, ((∫ (a : Ω), truncation (X i) (↑i) a) - ∫ (a : Ω), X 0 a)
tactic: ext1 n
state_before: case 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 i j => IndepFun (X i) (X j) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : Tendsto (fun i => ∫ (a : Ω), truncation (X i) (↑i) a) atTop (𝓝 (∫ (a : Ω), X 0 a))
n : ℕ
⊢ (∫ (a : Ω), (∑ i ∈ range n, truncation (X i) ↑i) a) - ↑n * ∫ (a : Ω), X 0 a =
    ∑ i ∈ range n, ((∫ (a : Ω), truncation (X i) (↑i) a) - ∫ (a : Ω), X 0 a)
state_after: case 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 i j => IndepFun (X i) (X j) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : Tendsto (fun i => ∫ (a : Ω), truncation (X i) (↑i) a) atTop (𝓝 (∫ (a : Ω), X 0 a))
n : ℕ
⊢ ∫ (a : Ω), ∑ c ∈ range n, truncation (X c) (↑c) a = ∑ x ∈ range n, ∫ (a : Ω), truncation (X x) (↑x) a
tactic: simp only [sum_sub_distrib, sum_const, card_range, nsmul_eq_mul, sum_apply, sub_left_inj]
state_before: case 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 i j => IndepFun (X i) (X j) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : Tendsto (fun i => ∫ (a : Ω), truncation (X i) (↑i) a) atTop (𝓝 (∫ (a : Ω), X 0 a))
n : ℕ
⊢ ∫ (a : Ω), ∑ c ∈ range n, truncation (X c) (↑c) a = ∑ x ∈ range n, ∫ (a : Ω), truncation (X x) (↑x) a
state_after: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise fun i j => IndepFun (X i) (X j) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : Tendsto (fun i => ∫ (a : Ω), truncation (X i) (↑i) a) atTop (𝓝 (∫ (a : Ω), X 0 a))
n i : ℕ
x✝ : i ∈ range n
⊢ Integrable (truncation (X i) ↑i) ℙ
tactic: rw [integral_finset_sum _ fun i _ => ?_]
state_before: Ω : Type u_1
inst✝¹ : MeasureSpace Ω
inst✝ : IsProbabilityMeasure ℙ
X : ℕ → Ω → ℝ
hint : Integrable (X 0) ℙ
hindep : Pairwise fun i j => IndepFun (X i) (X j) ℙ
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ
hnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω
A : Tendsto (fun i => ∫ (a : Ω), truncation (X i) (↑i) a) atTop (𝓝 (∫ (a : Ω), X 0 a))
n i : ℕ
x✝ : i ∈ range n
⊢ Integrable (truncation (X i) ↑i) ℙ
state_after: no goals
tactic: exact ((hident i).symm.integrable_snd hint).1.integrable_truncation