Module: Mathlib.Probability.CentralLimitTheorem

Goal state before:
pp: case convert_7
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
P : Measure Ω
P' : Measure Ω'
X : ℕ → Ω → ℝ
Y : Ω' → ℝ
inst✝¹ : IsProbabilityMeasure P
inst✝ : IsProbabilityMeasure P'
hY : HasLaw Y (gaussianReal 0 1) P'
hX : Var[X 0; P] ≠ 0
hindep : iIndepFun X P
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) P P
mX0 : AEMeasurable (X 0) P
intX0 : Integrable (X 0) P
this :
  ∀ (n : ℕ) (ω : Ω),
    (√(↑n * Var[X 0; P]))⁻¹ * (∑ k ∈ Finset.range n, X k ω - ↑n * ∫ (x : Ω), X 0 x ∂P) =
      (√↑n)⁻¹ * ∑ k ∈ Finset.range n, (X k ω - ∫ (x : Ω), X 0 x ∂P) / √Var[X 0; P]
⊢ ∫ (x : Ω), ((fun ω ↦ (X 0 ω - ∫ (x : Ω), X 0 x ∂P) / √Var[X 0; P]) ^ 2) x ∂P = 1

pp: case convert_8
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
P : Measure Ω
P' : Measure Ω'
X : ℕ → Ω → ℝ
Y : Ω' → ℝ
inst✝¹ : IsProbabilityMeasure P
inst✝ : IsProbabilityMeasure P'
hY : HasLaw Y (gaussianReal 0 1) P'
hX : Var[X 0; P] ≠ 0
hindep : iIndepFun X P
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) P P
mX0 : AEMeasurable (X 0) P
intX0 : Integrable (X 0) P
this :
  ∀ (n : ℕ) (ω : Ω),
    (√(↑n * Var[X 0; P]))⁻¹ * (∑ k ∈ Finset.range n, X k ω - ↑n * ∫ (x : Ω), X 0 x ∂P) =
      (√↑n)⁻¹ * ∑ k ∈ Finset.range n, (X k ω - ∫ (x : Ω), X 0 x ∂P) / √Var[X 0; P]
⊢ iIndepFun (fun k ω ↦ (X k ω - ∫ (x : Ω), X 0 x ∂P) / √Var[X 0; P]) P

pp: case convert_9
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
P : Measure Ω
P' : Measure Ω'
X : ℕ → Ω → ℝ
Y : Ω' → ℝ
inst✝¹ : IsProbabilityMeasure P
inst✝ : IsProbabilityMeasure P'
hY : HasLaw Y (gaussianReal 0 1) P'
hX : Var[X 0; P] ≠ 0
hindep : iIndepFun X P
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) P P
mX0 : AEMeasurable (X 0) P
intX0 : Integrable (X 0) P
this :
  ∀ (n : ℕ) (ω : Ω),
    (√(↑n * Var[X 0; P]))⁻¹ * (∑ k ∈ Finset.range n, X k ω - ↑n * ∫ (x : Ω), X 0 x ∂P) =
      (√↑n)⁻¹ * ∑ k ∈ Finset.range n, (X k ω - ∫ (x : Ω), X 0 x ∂P) / √Var[X 0; P]
⊢ ∀ (i : ℕ),
    IdentDistrib (fun ω ↦ (X i ω - ∫ (x : Ω), X 0 x ∂P) / √Var[X 0; P])
      (fun ω ↦ (X 0 ω - ∫ (x : Ω), X 0 x ∂P) / √Var[X 0; P]) P P

Tactic:
· simp only [Pi.pow_apply, div_pow]
    rw [integral_div, ← variance_eq_integral mX0, Real.sq_sqrt (variance_nonneg _ _), div_self hX]

Goal state after:
case convert_8
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
P : Measure Ω
P' : Measure Ω'
X : ℕ → Ω → ℝ
Y : Ω' → ℝ
inst✝¹ : IsProbabilityMeasure P
inst✝ : IsProbabilityMeasure P'
hY : HasLaw Y (gaussianReal 0 1) P'
hX : Var[X 0; P] ≠ 0
hindep : iIndepFun X P
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) P P
mX0 : AEMeasurable (X 0) P
intX0 : Integrable (X 0) P
this :
  ∀ (n : ℕ) (ω : Ω),
    (√(↑n * Var[X 0; P]))⁻¹ * (∑ k ∈ Finset.range n, X k ω - ↑n * ∫ (x : Ω), X 0 x ∂P) =
      (√↑n)⁻¹ * ∑ k ∈ Finset.range n, (X k ω - ∫ (x : Ω), X 0 x ∂P) / √Var[X 0; P]
⊢ iIndepFun (fun k ω ↦ (X k ω - ∫ (x : Ω), X 0 x ∂P) / √Var[X 0; P]) P

case convert_9
Ω : Type u_1
Ω' : Type u_2
mΩ : MeasurableSpace Ω
mΩ' : MeasurableSpace Ω'
P : Measure Ω
P' : Measure Ω'
X : ℕ → Ω → ℝ
Y : Ω' → ℝ
inst✝¹ : IsProbabilityMeasure P
inst✝ : IsProbabilityMeasure P'
hY : HasLaw Y (gaussianReal 0 1) P'
hX : Var[X 0; P] ≠ 0
hindep : iIndepFun X P
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) P P
mX0 : AEMeasurable (X 0) P
intX0 : Integrable (X 0) P
this :
  ∀ (n : ℕ) (ω : Ω),
    (√(↑n * Var[X 0; P]))⁻¹ * (∑ k ∈ Finset.range n, X k ω - ↑n * ∫ (x : Ω), X 0 x ∂P) =
      (√↑n)⁻¹ * ∑ k ∈ Finset.range n, (X k ω - ∫ (x : Ω), X 0 x ∂P) / √Var[X 0; P]
⊢ ∀ (i : ℕ),
    IdentDistrib (fun ω ↦ (X i ω - ∫ (x : Ω), X 0 x ∂P) / √Var[X 0; P])
      (fun ω ↦ (X 0 ω - ∫ (x : Ω), X 0 x ∂P) / √Var[X 0; P]) P P