mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel ν
a : α
q : ℚ
A : Set β
hA : MeasurableSet A
h : ∀ (r : ↑(Ioi q)), ∫ (t : β) in A, ⨅ r', f (a, t) ↑r' ∂ν a ≤ (κ a).real (A ×ˢ Iic ↑↑r)
⊢ ⨅ r, (κ a).real (A ×ˢ Iic ↑↑r) = (⨅ r, (κ a) (A ×ˢ Iic ↑↑r)).toReal

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel ν
a : α
q : ℚ
A : Set β
hA : MeasurableSet A
h : ∀ (r : ↑(Ioi q)), ∫ (t : β) in A, ⨅ r', f (a, t) ↑r' ∂ν a ≤ (κ a).real (A ×ˢ Iic ↑↑r)
⊢ ⨅ r, (κ a).real (A ×ˢ Iic ↑↑r) = (⨅ r, (κ a) (A ×ˢ Iic ↑↑r)).toReal
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel ν
a : α
q : ℚ
A : Set β
hA : MeasurableSet A
⊢ ∫ (c : β) in A, f (a, c) q ∂ν a ≤ ∫ (t : β) in A, ⨅ r, f (a, t) ↑r ∂ν a

Tactic:
]

Goal state after:
case refine_2
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel ν
a : α
q : ℚ
A : Set β
hA : MeasurableSet A
⊢ ∫ (c : β) in A, f (a, c) q ∂ν a ≤ ∫ (t : β) in A, ⨅ r, f (a, t) ↑r ∂ν a
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
F : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : CompleteSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : SecondCountableTopology E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : CompleteSpace F
inst✝³ : BorelSpace F
inst✝² : SecondCountableTopology F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
X : Ω → E
Y : Ω → F
hX : HasGaussianLaw X P
hY : HasGaussianLaw Y P
hXY : X ⟂ᵢ[P] Y
this : IsProbabilityMeasure P
L : StrongDual ℝ (E × F)
⊢ charFunDual (Measure.map X P) (L ∘SL inl ℝ E F) * charFunDual (Measure.map Y P) (L ∘SL inr ℝ E F) =
    cexp
      (↑(L (∫ (x : E), x ∂Measure.map X P, ∫ (y : F), y ∂Measure.map Y P)) * I -
        ↑((((covarianceBilinDual (Measure.map X P)).diagonalStrongDualProd (covarianceBilinDual (Measure.map Y P))) L)
              L) /
          2)

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
F : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : CompleteSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : SecondCountableTopology E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : CompleteSpace F
inst✝³ : BorelSpace F
inst✝² : SecondCountableTopology F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
X : Ω → E
Y : Ω → F
hX : HasGaussianLaw X P
hY : HasGaussianLaw Y P
hXY : X ⟂ᵢ[P] Y
this : IsProbabilityMeasure P
L : StrongDual ℝ (E × F)
⊢ charFunDual (Measure.map X P) (L ∘SL inl ℝ E F) * charFunDual (Measure.map Y P) (L ∘SL inr ℝ E F) =
    cexp
      (↑(L (∫ (x : E), x ∂Measure.map X P, ∫ (y : F), y ∂Measure.map Y P)) * I -
        ↑((((covarianceBilinDual (Measure.map X P)).diagonalStrongDualProd (covarianceBilinDual (Measure.map Y P))) L)
              L) /
          2)
Module: Mathlib.Probability.CentralLimitTheorem

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
t : ℝ
⊢ Tendsto (fun n ↦ (√↑n)⁻¹ * t) atTop (𝓝 (0 * t))

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
inst✝ : IsProbabilityMeasure P
X : Ω → ℝ
hX : AEMeasurable X P
h0 : ∫ (x : Ω), X x ∂P = 0
h1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1
t : ℝ
⊢ Tendsto (fun n ↦ (√↑n)⁻¹ * t) atTop (𝓝 (0 * t))
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel ν
a : α
⊢ ∀ (i : ℚ), ∀ᵐ (a_1 : β) ∂ν a, ⨅ r, f (a, a_1) ↑r = f (a, a_1) i

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel ν
a : α
⊢ ∀ (i : ℚ), ∀ᵐ (a_1 : β) ∂ν a, ⨅ r, f (a, a_1) ↑r = f (a, a_1) i
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: case refine_3
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → ℚ → ℝ
hf : IsRatCondKernelCDFAux f κ ν
inst✝¹ : IsFiniteKernel κ
inst✝ : IsFiniteKernel ν
a : α
q : ℚ
s : Set β
hs : MeasurableSet s
a✝ : (ν a) s < ∞
⊢ (κ a).real (s ×ˢ Iic ↑q) = (κ a).real (s ×ˢ Iic ↑q)

Tactic:
]

Goal state after:
Module: Mathlib.Probability.CentralLimitTheorem

Goal state before:
pp: 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
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 * ((∑ x ∈ Finset.range n, X x ω - ∑ x ∈ Finset.range n, ∫ (x : Ω), X 0 x ∂P) / √Var[X 0; P])

Tactic:
]

Goal state after:
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
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 * ((∑ x ∈ Finset.range n, X x ω - ∑ x ∈ Finset.range n, ∫ (x : Ω), X 0 x ∂P) / √Var[X 0; P])
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: case h
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ : ℝ≥0
mX : ∀ (t : ℝ≥0), AEMeasurable (B t) P
s t : ℝ≥0
ht : t ≤ t₀
this : IsProbabilityMeasure P
⊢ 0 = 0

pp: case h
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ : ℝ≥0
mX : ∀ (t : ℝ≥0), AEMeasurable (B t) P
s t : ℝ≥0
ht : t ≤ t₀
this : IsProbabilityMeasure P
⊢ ↑⟨t, ht⟩ ≤ t₀

pp: case h
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ : ℝ≥0
mX : ∀ (t : ℝ≥0), AEMeasurable (B t) P
s t : ℝ≥0
ht : t ≤ t₀
this : IsProbabilityMeasure P
⊢ ↑⟨t, ht⟩ ≤ t₀ + s

pp: case h.hX
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ : ℝ≥0
mX : ∀ (t : ℝ≥0), AEMeasurable (B t) P
s t : ℝ≥0
ht : t ≤ t₀
this : IsProbabilityMeasure P
⊢ MemLp (B (t₀ + s)) 2 P

pp: case h.hY
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ : ℝ≥0
mX : ∀ (t : ℝ≥0), AEMeasurable (B t) P
s t : ℝ≥0
ht : t ≤ t₀
this : IsProbabilityMeasure P
⊢ MemLp (B t₀) 2 P

pp: case h.hZ
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ : ℝ≥0
mX : ∀ (t : ℝ≥0), AEMeasurable (B t) P
s t : ℝ≥0
ht : t ≤ t₀
this : IsProbabilityMeasure P
⊢ MemLp (B ↑⟨t, ht⟩) 2 P

Tactic:
]

Goal state after:
case h
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ : ℝ≥0
mX : ∀ (t : ℝ≥0), AEMeasurable (B t) P
s t : ℝ≥0
ht : t ≤ t₀
this : IsProbabilityMeasure P
⊢ ↑⟨t, ht⟩ ≤ t₀

case h
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ : ℝ≥0
mX : ∀ (t : ℝ≥0), AEMeasurable (B t) P
s t : ℝ≥0
ht : t ≤ t₀
this : IsProbabilityMeasure P
⊢ ↑⟨t, ht⟩ ≤ t₀ + s

case h.hX
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ : ℝ≥0
mX : ∀ (t : ℝ≥0), AEMeasurable (B t) P
s t : ℝ≥0
ht : t ≤ t₀
this : IsProbabilityMeasure P
⊢ MemLp (B (t₀ + s)) 2 P

case h.hY
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ : ℝ≥0
mX : ∀ (t : ℝ≥0), AEMeasurable (B t) P
s t : ℝ≥0
ht : t ≤ t₀
this : IsProbabilityMeasure P
⊢ MemLp (B t₀) 2 P

case h.hZ
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ : ℝ≥0
mX : ∀ (t : ℝ≥0), AEMeasurable (B t) P
s t : ℝ≥0
ht : t ≤ t₀
this : IsProbabilityMeasure P
⊢ MemLp (B ↑⟨t, ht⟩) 2 P
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
hf : IsCondKernelCDF f κ ν
a : α
x : ℝ
⊢ ∫ (b : β), ↑(f (a, b)) x ∂ν a = ∫ (b : β), ↑(f (a, b)) x ∂ν a

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
inst✝ : IsFiniteKernel κ
f : α × β → StieltjesFunction ℝ
hf : IsCondKernelCDF f κ ν
a : α
s : Set β
hs : MeasurableSet s
x : ℝ
⊢ (κ a) (s ×ˢ Iic x) = (κ a) (s ×ˢ Iic x)

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
inst✝ : IsFiniteKernel κ
f : α × β → StieltjesFunction ℝ
hf : IsCondKernelCDF f κ ν
a : α
x : ℝ
⊢ ∫⁻ (b : β), ENNReal.ofReal (↑(f (a, b)) x) ∂ν a = ∫⁻ (b : β), ENNReal.ofReal (↑(f (a, b)) x) ∂ν a

Tactic:
]

Goal state after:
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: case refine_2
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t : ℝ≥0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Probability.CentralLimitTheorem

Goal state before:
pp: case convert_6
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]
⊢ (∫ (a : Ω), X 0 a ∂P - ∫ (a : Ω), ∫ (x : Ω), X 0 x ∂P ∂P) / √Var[X 0; P] = 0

Tactic:
]

Goal state after:
case convert_6
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]
⊢ (∫ (a : Ω), X 0 a ∂P - ∫ (a : Ω), ∫ (x : Ω), X 0 x ∂P ∂P) / √Var[X 0; P] = 0
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
x✝ : MeasurableSpace β
f : α × β → StieltjesFunction ℝ
κ : Kernel α (β × ℝ)
ν : Kernel α β
hf : IsCondKernelCDF f κ ν
p : α × β
x : ℝ
⊢ ENNReal.ofReal (↑(f p) x - 0) = ENNReal.ofReal (↑(f p) x)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
x✝ : MeasurableSpace β
f : α × β → StieltjesFunction ℝ
κ : Kernel α (β × ℝ)
ν : Kernel α β
hf : IsCondKernelCDF f κ ν
p : α × β
x : ℝ
⊢ ENNReal.ofReal (↑(f p) x - 0) = ENNReal.ofReal (↑(f p) x)
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: case refine_3
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
s t : ℝ≥0
hst : s ≤ t
this : IsProbabilityMeasure P
⊢ ↑s * (↑t * ↑(min (1 / s) (1 / t))) = ↑s

Tactic:
]

Goal state after:
case refine_3
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
s t : ℝ≥0
hst : s ≤ t
this : IsProbabilityMeasure P
⊢ ↑s * (↑t * ↑(min (1 / s) (1 / t))) = ↑s
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: case refine_3.inr
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
s t : ℝ≥0
hst : s ≤ t
this✝ : IsProbabilityMeasure P
hs : s ≠ 0
this : 0 < t
⊢ ↑s * (↑t * ↑(1 / t)) = ↑s

pp: case refine_3.inr
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
s t : ℝ≥0
hst : s ≤ t
this✝ : IsProbabilityMeasure P
hs : s ≠ 0
this : 0 < t
⊢ 1 / t ≤ 1 / s

Tactic:
]

Goal state after:
case refine_3.inr
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
s t : ℝ≥0
hst : s ≤ t
this✝ : IsProbabilityMeasure P
hs : s ≠ 0
this : 0 < t
⊢ ↑s * (↑t * ↑(1 / t)) = ↑s

case refine_3.inr
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
s t : ℝ≥0
hst : s ≤ t
this✝ : IsProbabilityMeasure P
hs : s ≠ 0
this : 0 < t
⊢ 1 / t ≤ 1 / s
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, ((IsCondKernelCDF.toKernel f hf) (a, b)) (⋃ r, Iic ↑r) ∂ν a = (κ a) (⋃ i, s ×ˢ Iic ↑i)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, ((IsCondKernelCDF.toKernel f hf) (a, b)) (⋃ r, Iic ↑r) ∂ν a = (κ a) (⋃ i, s ×ˢ Iic ↑i)
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]
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.Lemmas

Goal state before:
pp: case e_f
β : Type u_2
γ : Type u_3
δ : Type u_4
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
β' : Type u_5
ε : Type u_6
mβ' : MeasurableSpace β'
mε : MeasurableSpace ε
κ : Kernel γ β
inst✝² : IsSFiniteKernel κ
η : Kernel ε β'
inst✝¹ : IsSFiniteKernel η
ξ : Kernel (β × ε) δ
inst✝ : IsSFiniteKernel ξ
f : γ → ε
hf : Measurable f
ω : γ
s : Set (δ × β')
hs : MeasurableSet s
b : β
⊢ ∫⁻ (b_1 : δ), (η (b, f ω).2) (Prod.mk b_1 ⁻¹' s) ∂ξ (b, f ω) = ∫⁻ (c : δ), (η (f ω)) (Prod.mk c ⁻¹' s) ∂ξ (b, f ω)

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set β
hs : MeasurableSet s
h_dir : Directed (fun x y ↦ x ⊆ y) fun q ↦ Iic ↑q
h_dir_prod : Directed (fun x y ↦ x ⊆ y) fun q ↦ s ×ˢ Iic ↑q
⊢ ⨆ b, ∫⁻ (a_1 : β) in s, ((IsCondKernelCDF.toKernel f hf) (a, a_1)) (Iic ↑b) ∂ν a = ⨆ i, (κ a) (s ×ˢ Iic ↑i)

pp: case hf
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set β
hs : MeasurableSet s
h_dir : Directed (fun x y ↦ x ⊆ y) fun q ↦ Iic ↑q
h_dir_prod : Directed (fun x y ↦ x ⊆ y) fun q ↦ s ×ˢ Iic ↑q
⊢ ∀ (b : ℚ), AEMeasurable (fun b_1 ↦ ((IsCondKernelCDF.toKernel f hf) (a, b_1)) (Iic ↑b)) ((ν a).restrict s)

pp: case h_directed
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set β
hs : MeasurableSet s
h_dir : Directed (fun x y ↦ x ⊆ y) fun q ↦ Iic ↑q
h_dir_prod : Directed (fun x y ↦ x ⊆ y) fun q ↦ s ×ˢ Iic ↑q
⊢ Directed (fun x1 x2 ↦ x1 ≤ x2) fun i b ↦ ((IsCondKernelCDF.toKernel f hf) (a, b)) (Iic ↑i)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set β
hs : MeasurableSet s
h_dir : Directed (fun x y ↦ x ⊆ y) fun q ↦ Iic ↑q
h_dir_prod : Directed (fun x y ↦ x ⊆ y) fun q ↦ s ×ˢ Iic ↑q
⊢ ⨆ b, ∫⁻ (a_1 : β) in s, ((IsCondKernelCDF.toKernel f hf) (a, a_1)) (Iic ↑b) ∂ν a = ⨆ i, (κ a) (s ×ˢ Iic ↑i)

case hf
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set β
hs : MeasurableSet s
h_dir : Directed (fun x y ↦ x ⊆ y) fun q ↦ Iic ↑q
h_dir_prod : Directed (fun x y ↦ x ⊆ y) fun q ↦ s ×ˢ Iic ↑q
⊢ ∀ (b : ℚ), AEMeasurable (fun b_1 ↦ ((IsCondKernelCDF.toKernel f hf) (a, b_1)) (Iic ↑b)) ((ν a).restrict s)

case h_directed
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set β
hs : MeasurableSet s
h_dir : Directed (fun x y ↦ x ⊆ y) fun q ↦ Iic ↑q
h_dir_prod : Directed (fun x y ↦ x ⊆ y) fun q ↦ s ×ˢ Iic ↑q
⊢ Directed (fun x1 x2 ↦ x1 ≤ x2) fun i b ↦ ((IsCondKernelCDF.toKernel f hf) (a, b)) (Iic ↑i)
Module: Mathlib.Probability.Kernel.Composition.Lemmas

Goal state before:
pp: case e_f
β : Type u_2
γ : Type u_3
δ : Type u_4
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
β' : Type u_5
ε : Type u_6
mβ' : MeasurableSpace β'
mε : MeasurableSpace ε
κ : Kernel γ β
inst✝² : IsSFiniteKernel κ
η : Kernel ε β'
inst✝¹ : IsSFiniteKernel η
ξ : Kernel (ε × β) δ
inst✝ : IsSFiniteKernel ξ
f : γ → ε
hf : Measurable f
ω : γ
s : Set (δ × β')
hs : MeasurableSet s
b : β
⊢ ∫⁻ (b_1 : δ), (η (f ω, b).1) (Prod.mk b_1 ⁻¹' s) ∂ξ (f ω, b) = ∫⁻ (c : δ), (η (f ω)) (Prod.mk c ⁻¹' s) ∂ξ (f ω, b)

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
⊢ (κ a) univ = (κ a) univ

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: case e_f
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set β
hs : MeasurableSet s
t✝ t : Set ℝ
ht : MeasurableSet t
iht : ∫⁻ (b : β) in s, ((IsCondKernelCDF.toKernel f hf) (a, b)) t ∂ν a = (κ a) (s ×ˢ t)
x : β
⊢ ((IsCondKernelCDF.toKernel f hf) (a, x)) univ - ((IsCondKernelCDF.toKernel f hf) (a, x)) t =
    ((IsCondKernelCDF.toKernel f hf) (a, x)) univ - ((IsCondKernelCDF.toKernel f hf) (a, x)) t

Tactic:
]

Goal state after:
Module: Mathlib.Probability.CentralLimitTheorem

Goal state before:
pp: 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 Var[X 0; P].toNNReal) P'
hX : MemLp (X 0) 2 P
hindep : iIndepFun X P
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) P P
h : Var[X 0; P] = 0
n : ℕ
x✝ : Ω
⊢ ∫ (x : Ω), X 0 x ∂P = ∫ (x : Ω), X 0 x ∂P

Tactic:
]

Goal state after:
Module: Mathlib.Probability.CentralLimitTheorem

Goal state before:
pp: 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 Var[X 0; P].toNNReal) P'
hX : MemLp (X 0) 2 P
hindep : iIndepFun X P
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) P P
h : Var[X 0; P] = 0
n : ℕ
⊢ Var[X 0; P] = 0

Tactic:
]

Goal state after:
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 Var[X 0; P].toNNReal) P'
hX : MemLp (X 0) 2 P
hindep : iIndepFun X P
hident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) P P
h : Var[X 0; P] = 0
n : ℕ
⊢ Var[X 0; P] = 0
Module: Mathlib.Probability.Kernel.Composition.Lemmas

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
inst✝ : SFinite μ
hκ : IsSFiniteKernel κ
⊢ ⇑(Kernel.id ∥ₖ κ ∘ₖ Kernel.copy α) ∘ₘ μ = ⇑(Kernel.id ∥ₖ κ ∘ₖ Kernel.copy α) ∘ₘ μ

Tactic:
]

Goal state after:
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: case hg_fin
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set β
hs : MeasurableSet s
t✝ t : Set ℝ
ht : MeasurableSet t
iht : ∫⁻ (b : β) in s, ((IsCondKernelCDF.toKernel f hf) (a, b)) t ∂ν a = (κ a) (s ×ˢ t)
⊢ (κ a) (s ×ˢ t) ≠ ∞

Tactic:
]

Goal state after:
case hg_fin
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set β
hs : MeasurableSet s
t✝ t : Set ℝ
ht : MeasurableSet t
iht : ∫⁻ (b : β) in s, ((IsCondKernelCDF.toKernel f hf) (a, b)) t ∂ν a = (κ a) (s ×ˢ t)
⊢ (κ a) (s ×ˢ t) ≠ ∞
Module: Mathlib.Probability.Kernel.Composition.Lemmas

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
κ : Kernel β γ
inst✝ : IsSFiniteKernel κ
s : Set (α × γ)
hs : MeasurableSet s
⊢ ∫⁻ (x : α), (⇑κ ∘ₘ ν) (Prod.mk x ⁻¹' s) ∂μ = ∫⁻ (a : α × β), ((Kernel.id ∥ₖ κ) a) s ∂μ.prod ν

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
ν : Measure β
inst✝¹ : SFinite ν
κ : Kernel β γ
inst✝ : IsSFiniteKernel κ
s : Set (α × γ)
hs : MeasurableSet s
⊢ ∫⁻ (x : α), (⇑κ ∘ₘ ν) (Prod.mk x ⁻¹' s) ∂μ = ∫⁻ (a : α × β), ((Kernel.id ∥ₖ κ) a) s ∂μ.prod ν