mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: case refine_2.hY
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ s t : ℝ≥0
hst : s ≤ t
this : IsProbabilityMeasure P
⊢ MemLp (B t₀) 2 P

Tactic:
simp

Goal state after:
case refine_2.hY
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ s t : ℝ≥0
hst : s ≤ t
this : IsProbabilityMeasure P
⊢ MemLp (B t₀) 2 P
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: case refine_2.hZ
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ s t : ℝ≥0
hst : s ≤ t
this : IsProbabilityMeasure P
⊢ MemLp (fun ω ↦ B (t₀ + t) ω - B t₀ ω) 2 P

Tactic:
simp

Goal state after:
case refine_2.hZ
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
t₀ s t : ℝ≥0
hst : s ≤ t
this : IsProbabilityMeasure P
⊢ MemLp (fun ω ↦ B (t₀ + t) ω - B t₀ ω) 2 P
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
this :
  (fun n ↦ charFun (Measure.map X P) ((√↑n)⁻¹ * t) - (1 + -((↑(√↑n)⁻¹ * ↑t) ^ 2 / 2))) =o[atTop] fun n ↦
    ((√↑n)⁻¹ * t) ^ 2
⊢ (fun n ↦ ‖1 / ↑n‖) = fun n ↦ ‖1 / ↑n‖

Tactic:
simp

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

Goal state before:
pp: 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
t : ℝ≥0
x : ↥{t₀, t₀ + t} → ℝ
⊢ t₀ + t ∈ {t₀, t₀ + t}

Tactic:
simp

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

Goal state before:
pp: 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
t : ℝ≥0
x : ↥{t₀, t₀ + t} → ℝ
⊢ t₀ ∈ {t₀, t₀ + t}

Tactic:
simp

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

Goal state before:
pp: 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
t : ℝ≥0
x y : ↥{t₀, t₀ + t} → ℝ
⊢ (x + y) ⟨t₀ + t, ⋯⟩ - (x + y) ⟨t₀, ⋯⟩ = x ⟨t₀ + t, ⋯⟩ - x ⟨t₀, ⋯⟩ + (y ⟨t₀ + t, ⋯⟩ - y ⟨t₀, ⋯⟩)

Tactic:
simp

Goal state after:
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
t : ℝ≥0
x y : ↥{t₀, t₀ + t} → ℝ
⊢ x ⟨t₀ + t, ⋯⟩ + y ⟨t₀ + t, ⋯⟩ - (x ⟨t₀, ⋯⟩ + y ⟨t₀, ⋯⟩) = x ⟨t₀ + t, ⋯⟩ - x ⟨t₀, ⋯⟩ + (y ⟨t₀ + t, ⋯⟩ - y ⟨t₀, ⋯⟩)
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: 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
t : ℝ≥0
c : ℝ
x : ↥{t₀, t₀ + t} → ℝ
⊢ (c • x) ⟨t₀ + t, ⋯⟩ - (c • x) ⟨t₀, ⋯⟩ = (RingHom.id ℝ) c • (x ⟨t₀ + t, ⋯⟩ - x ⟨t₀, ⋯⟩)

Tactic:
simp

Goal state after:
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
t : ℝ≥0
c : ℝ
x : ↥{t₀, t₀ + t} → ℝ
⊢ c * x ⟨t₀ + t, ⋯⟩ - c * x ⟨t₀, ⋯⟩ = c * (x ⟨t₀ + t, ⋯⟩ - x ⟨t₀, ⋯⟩)
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: 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
t : ℝ≥0
⊢ ∀ (ω : Ω),
    Sum.elim (fun s ω ↦ B (t₀ + s) ω - B t₀ ω) (fun t ↦ B ↑t) (Sum.inl t) ω =
      { toFun := fun x ↦ x ⟨t₀ + t, ⋯⟩ - x ⟨t₀, ⋯⟩, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ }
        ({t₀, t₀ + t}.restrict fun x ↦ B x ω)

Tactic:
simp

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

Goal state before:
pp: 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
t : ℝ≥0
ht : t ∈ Set.Iic t₀
x : ↥{t} → ℝ
⊢ t ∈ {t}

Tactic:
simp

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

Goal state before:
pp: 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
t : ℝ≥0
ht : t ∈ Set.Iic t₀
x y : ↥{t} → ℝ
⊢ (x + y) ⟨t, ⋯⟩ = x ⟨t, ⋯⟩ + y ⟨t, ⋯⟩

Tactic:
simp

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

Goal state before:
pp: 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
t : ℝ≥0
ht : t ∈ Set.Iic t₀
c : ℝ
x : ↥{t} → ℝ
⊢ (c • x) ⟨t, ⋯⟩ = (RingHom.id ℝ) c • x ⟨t, ⋯⟩

Tactic:
simp

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

Goal state before:
pp: 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
t : ℝ≥0
ht : t ∈ Set.Iic t₀
⊢ ∀ (ω : Ω),
    Sum.elim (fun s ω ↦ B (t₀ + s) ω - B t₀ ω) (fun t ↦ B ↑t) (Sum.inr ⟨t, ht⟩) ω =
      { toFun := fun x ↦ x ⟨t, ⋯⟩, map_add' := ⋯, map_smul' := ⋯, cont := ⋯ } ({t}.restrict fun x ↦ B x ω)

Tactic:
simp

Goal state after:
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)
⊢ (∫ (x : E), x ∂Measure.map X P, ∫ (y : F), y ∂Measure.map Y P) =
    (inl ℝ E F) (∫ (x : E), x ∂Measure.map X P) + (inr ℝ E F) (∫ (y : F), y ∂Measure.map Y P)

Tactic:
simp

Goal state after:
Module: Mathlib.Probability.CentralLimitTheorem

Goal state before:
pp: case e'_7
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
this : Tendsto (fun n ↦ (√↑n)⁻¹ * t) atTop (𝓝 0)
x✝ : ℕ
⊢ charFun (Measure.map X P) ((√↑x✝)⁻¹ * t) - (1 + -((↑(√↑x✝)⁻¹ * ↑t) ^ 2 / 2)) =
    ((fun t ↦ charFun (Measure.map X P) t - (1 - ↑t ^ 2 / 2)) ∘ fun n ↦ (√↑n)⁻¹ * t) x✝

Tactic:
simp

Goal state after:
case e'_7
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
this : Tendsto (fun n ↦ (√↑n)⁻¹ * t) atTop (𝓝 0)
x✝ : ℕ
⊢ 1 + -(((↑√↑x✝)⁻¹ * ↑t) ^ 2 / 2) = 1 - ((↑√↑x✝)⁻¹ * ↑t) ^ 2 / 2
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
⊢ 1 ≤ 2

Tactic:
simp

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
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]
⊢ Integrable (fun a ↦ ∫ (x : Ω), X 0 x ∂P) P

Tactic:
simp

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

Goal state before:
pp: case refine_1
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
⊢ IsGaussianProcess (fun t ω ↦ ↑t * B (1 / t) ω) P

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
⊢ ∫ (x : Ω), ↑t * B (1 / t) x ∂P = 0

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
⊢ cov[fun ω ↦ ↑s * B (1 / s) ω, fun ω ↦ ↑t * B (1 / t) ω; P] = ↑s

Tactic:
· exact (IsGaussianProcess.comp_right hB.isGaussianProcess _).smul _

Goal state after:
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
⊢ ∫ (x : Ω), ↑t * B (1 / t) x ∂P = 0

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
⊢ cov[fun ω ↦ ↑s * B (1 / s) ω, fun ω ↦ ↑t * B (1 / t) ω; P] = ↑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
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 = (κ a) (univ ×ˢ Iic x)

Tactic:
by
  rw [← hf.setLIntegral _ MeasurableSet.univ, Measure.restrict_univ]

Goal state after:
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case e_a
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)
this :
  (∫ (x : E), x ∂Measure.map X P, ∫ (y : F), y ∂Measure.map Y P) =
    (inl ℝ E F) (∫ (x : E), x ∂Measure.map X P) + (inr ℝ E F) (∫ (y : F), y ∂Measure.map Y P)
⊢ cexp (↑((L ∘SL inl ℝ E F) (∫ (x : E), x ∂Measure.map X P)) * I - ↑Var[⇑(L ∘SL inl ℝ E F); Measure.map X P] / 2) =
    cexp (↑(L ((inl ℝ E F) (∫ (x : E), x ∂Measure.map X P))) * I - ↑Var[⇑(L ∘SL inl ℝ E F); Measure.map X P] / 2)

Tactic:
simp

Goal state after:
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:
simp

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:
simp

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

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

Tactic:
simp

Goal state after:
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case e_a
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)
this :
  (∫ (x : E), x ∂Measure.map X P, ∫ (y : F), y ∂Measure.map Y P) =
    (inl ℝ E F) (∫ (x : E), x ∂Measure.map X P) + (inr ℝ E F) (∫ (y : F), y ∂Measure.map Y P)
⊢ cexp (↑((L ∘SL inr ℝ E F) (∫ (x : F), x ∂Measure.map Y P)) * I - ↑Var[⇑(L ∘SL inr ℝ E F); Measure.map Y P] / 2) =
    cexp (↑(L ((inr ℝ E F) (∫ (y : F), y ∂Measure.map Y P))) * I - ↑Var[⇑(L ∘SL inr ℝ E F); Measure.map Y P] / 2)

Tactic:
simp

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

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel β γ
inst✝ : IsSFiniteKernel η
hμ : ¬SFinite μ
⊢ ⇑(Kernel.id ∥ₖ η) ∘ₘ μ ⊗ₘ κ = μ ⊗ₘ (η ∘ₖ κ)

Tactic:
simp [hμ]

Goal state after:
Module: Mathlib.Probability.CentralLimitTheorem

Goal state before:
pp: case e'_6.e'_1
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
⊢ 0 = 0 / √Var[X 0; P]

Tactic:
simp

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

Goal state before:
pp: case inl.e_6
α : Type u_1
Ω : Type u_3
mα : MeasurableSpace α
mΩ : MeasurableSpace Ω
ρ : Measure (α × Ω)
ρCond : Kernel α Ω
inst✝² : ρ.IsCondKernel ρCond
inst✝¹ : IsFiniteMeasure ρ
inst✝ : MeasurableSingletonClass α
s : Set Ω
hs : MeasurableSet s
this : IsSFiniteKernel ρCond
a : α
hx : ρ.fst {a} ≠ 0
y : Ω
⊢ y ∈ Prod.mk a ⁻¹' Prod.mk a '' s ↔ y ∈ s

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mΩ : MeasurableSpace Ω
κ : Kernel α (β × Ω)
κCond✝ κCond : Kernel (α × β) Ω
⊢ fst 0 ⊗ₖ κCond = 0

Tactic:
simp

Goal state after:
Module: Mathlib.Probability.Process.PartitionFiltration

Goal state before:
pp: α : Type u_1
m✝ : MeasurableSpace α
t : ℕ → Set α
ht : ∀ (n : ℕ), MeasurableSet (t n)
n : ℕ
m : MeasurableSpace ↑(memPartition t n)
s : Set α
hs : s ∈ memPartition t n
this : MeasurableSet {x | memPartitionSet t n x = s}
x : α
⊢ x ∈ (fun a ↦ ⟨memPartitionSet t n a, ⋯⟩) ⁻¹' {⟨s, hs⟩} ↔ x ∈ {x | memPartitionSet t n x = s}

Tactic:
simp

Goal state after:
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
hXY : HasGaussianLaw (fun ω ↦ (X ω, Y ω)) P
h : ∀ (L₁ : StrongDual ℝ E) (L₂ : StrongDual ℝ F), cov[⇑L₁ ∘ X, ⇑L₂ ∘ Y; P] = 0
this✝ : IsProbabilityMeasure P
L : StrongDual ℝ (E × F)
this : (⇑L ∘ fun ω ↦ (X ω, Y ω)) = ⇑(L ∘SL inl ℝ E F) ∘ X + ⇑(L ∘SL inr ℝ E F) ∘ Y
⊢ cexp
      (↑(∫ (x : Ω), (⇑(L ∘SL inl ℝ E F) ∘ X + ⇑(L ∘SL inr ℝ E F) ∘ Y) x ∂P) * I -
        ↑(Var[⇑(L ∘SL inl ℝ E F) ∘ X; P] + Var[⇑(L ∘SL inr ℝ E F) ∘ Y; P]) / 2) =
    cexp
      (↑(∫ (a : Ω), (⇑(L ∘SL inl ℝ E F) ∘ X) a + (⇑(L ∘SL inr ℝ E F) ∘ Y) a ∂P) * I -
        ↑(Var[⇑(L ∘SL inl ℝ E F) ∘ X; P] + Var[⇑(L ∘SL inr ℝ E F) ∘ Y; P]) / 2)

Tactic:
simp

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

Goal state before:
pp: case refine_2
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
n : ℕ
a : α
s : Set β
⊢ (ν a) univ ^ (ENNReal.toReal 1)⁻¹ * ENNReal.ofReal 1 ≤ (ν a) univ

Tactic:
simp

Goal state after:
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_4
κ : Type u_5
inst✝¹ : Finite ι
inst✝ : Finite κ
X : ι → Ω → ℝ
Y : κ → Ω → ℝ
hXY : HasGaussianLaw (fun ω ↦ (fun i ↦ X i ω, fun j ↦ Y j ω)) P
h : ∀ (i : ι) (j : κ), cov[X i, Y j; P] = 0
this : IsProbabilityMeasure P
x✝¹ : Ω
x✝ : ι
⊢ X x✝ x✝¹ = (ofLp ∘ toLp 2 ∘ fun ω i ↦ X i ω) x✝¹ x✝

Tactic:
simp

Goal state after:
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_4
κ : Type u_5
inst✝¹ : Finite ι
inst✝ : Finite κ
X : ι → Ω → ℝ
Y : κ → Ω → ℝ
hXY : HasGaussianLaw (fun ω ↦ (fun i ↦ X i ω, fun j ↦ Y j ω)) P
h : ∀ (i : ι) (j : κ), cov[X i, Y j; P] = 0
this : IsProbabilityMeasure P
hX : (fun ω i ↦ X i ω) = ofLp ∘ toLp 2 ∘ fun ω i ↦ X i ω
x✝¹ : Ω
x✝ : κ
⊢ Y x✝ x✝¹ = (ofLp ∘ toLp 2 ∘ fun ω j ↦ Y j ω) x✝¹ x✝

Tactic:
simp

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

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
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
this : IsFiniteKernel κ
S : Finset (Set γ)
hS_subset : ↑S ⊆ countablePartition γ n
hA : MeasurableSet (⋃₀ ↑S)
u v : ↥S
huv : u ≠ v
⊢ ↑u ∈ ↑S

Tactic:
simp

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

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
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
this : IsFiniteKernel κ
S : Finset (Set γ)
hS_subset : ↑S ⊆ countablePartition γ n
hA : MeasurableSet (⋃₀ ↑S)
u v : ↥S
huv : u ≠ v
⊢ ↑v ∈ ↑S

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
Ω : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝³ : CountablyGenerated γ
mΩ : MeasurableSpace Ω
inst✝² : StandardBorelSpace Ω
inst✝¹ : Nonempty Ω
κ : Kernel Unit (α × ℝ)
inst✝ : IsFiniteKernel κ
a✝¹ : Unit
s✝ : Set (α × ℝ)
a✝ : MeasurableSet s✝
⊢ ((const Unit (κ ())) a✝¹) s✝ = (κ a✝¹) s✝

Tactic:
simp

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

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
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
this : IsFiniteKernel κ
S : Finset (Set γ)
hS_subset : ↑S ⊆ countablePartition γ n
hA : MeasurableSet (⋃₀ ↑S)
h_disj : Pairwise (Disjoint on fun i ↦ ↑i)
u : ↑↑S
⊢ ↑u ∈ ↑S

Tactic:
simp

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

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
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
inst✝ : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
this : IsFiniteKernel κ
S : Finset (Set γ)
hS_subset : ↑S ⊆ countablePartition γ n
hA : MeasurableSet (⋃₀ ↑S)
h_disj : Pairwise (Disjoint on fun i ↦ ↑i)
x✝ : ↑↑S
⊢ ↑x✝ ∈ ↑S

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁴ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
inst✝¹ : CountableOrCountablyGenerated α β
κ : Kernel α (β × Ω)
inst✝ : IsFiniteKernel κ
f : β × Ω → ℝ≥0∞
hf : Measurable f
a : α
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, ∫⁻ (ω : Ω), f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a = ∫⁻ (x : β × Ω) in s ×ˢ Set.univ, f x ∂κ a

Tactic:
by
  rw [← setLIntegral_condKernel hf a hs MeasurableSet.univ]; simp_rw [Measure.restrict_univ]

Goal state after:
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case h.e_a.e_a.e_f
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : SecondCountableTopology E
X Y : Ω → E
hX : HasGaussianLaw X P
hY : HasGaussianLaw Y P
h : X ⟂ᵢ[P] Y - X
this : IsProbabilityMeasure P
L : StrongDual ℝ E
ω : Ω
⊢ (⇑L ∘ Y) ω = (⇑L ∘ X) ω + (⇑L ∘ (Y - X)) ω

Tactic:
simp

Goal state after:
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case h.e_a.e_a.e_X
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
inst✝ : SecondCountableTopology E
X Y : Ω → E
hX : HasGaussianLaw X P
hY : HasGaussianLaw Y P
h : X ⟂ᵢ[P] Y - X
this : IsProbabilityMeasure P
L : StrongDual ℝ E
ω : Ω
⊢ (⇑L ∘ Y) ω = (⇑L ∘ X + ⇑L ∘ (Y - X)) ω

Tactic:
simp

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

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
κ' : Kernel α (γ × β)
hκκ' : κ ≤ κ'
hκ'ν : κ'.fst ≤ ν
n : ℕ
a : α
x : γ
s : Set β
h0 : (ν a) (countablePartitionSet n x) = 0
⊢ ((κ a) (countablePartitionSet n x ×ˢ s)).toReal / ENNReal.toReal 0 ≤
    ((κ' a) (countablePartitionSet n x ×ˢ s)).toReal / ENNReal.toReal 0

Tactic:
simp

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

Goal state before:
pp: α : Type u_1
β : Type u_2
Ω : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mΩ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
κ : Kernel α (β × Ω)
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) ℝ
inst✝ : IsSFiniteKernel η
hη : (κ.map (Prod.map id (embeddingReal Ω))).fst ⊗ₖ η = κ.map (Prod.map id (embeddingReal Ω))
e : Ω → ℝ := embeddingReal Ω
he : MeasurableEmbedding (embeddingReal Ω) := measurableEmbedding_embeddingReal Ω
κ' : Kernel α (β × ℝ) := κ.map (Prod.map id e)
hη' : κ'.fst ⊗ₖ η = κ'
h_prod_embed : MeasurableEmbedding (Prod.map id e)
h_fst : κ'.fst = κ.fst
a : α
t : Set (β × Ω)
ht : MeasurableSet t
this : Prod.map id e ⁻¹' (univ ×ˢ range e)ᶜ = ∅
⊢ (κ a) ∅ = 0

Tactic:
simp

Goal state after: