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

Goal state before:
pp: S : Type u_1
Ω : Type u_3
mΩ : MeasurableSpace Ω
α : Type u_4
mα : MeasurableSpace α
κ : Kernel α Ω
P : Measure α
T : S → Type u_5
𝓧 : (i : S) → T i → Type u_6
inst✝ : (i : S) → (j : T i) → MeasurableSpace (𝓧 i j)
X X' : (i : S) → (j : T i) → Ω → 𝓧 i j
h1 : iIndepFun (fun i ω j ↦ X i j ω) κ P
h2 : ∀ (i : S) (j : T i), ∀ᵐ (a : α) ∂P, X i j =ᵐ[κ a] X' i j
s : Finset S
f : S → Set Ω
g : (i : S) → Set ((fun i ↦ (j : T i) → 𝓧 i j) i)
mg : ∀ i ∈ s, MeasurableSet (g i)
hg : ∀ i ∈ s, (fun i ω j ↦ X' i j ω) i ⁻¹' g i = f i
h3 : ⋂ i ∈ s, f i = ⋂ i ∈ s, (fun i ω j ↦ X' i j ω) i ⁻¹' g i
h3' : ∀ (a : α), ∏ i ∈ s, (κ a) (f i) = ∏ i ∈ s, (κ a) ((fun i ω j ↦ X' i j ω) i ⁻¹' g i)
⊢ ∀ᵐ (a : α) ∂P, (κ a) (⋂ i ∈ s, f i) = ∏ i ∈ s, (κ a) (f i)

Tactic:
[

Goal state after:
S : Type u_1
Ω : Type u_3
mΩ : MeasurableSpace Ω
α : Type u_4
mα : MeasurableSpace α
κ : Kernel α Ω
P : Measure α
T : S → Type u_5
𝓧 : (i : S) → T i → Type u_6
inst✝ : (i : S) → (j : T i) → MeasurableSpace (𝓧 i j)
X X' : (i : S) → (j : T i) → Ω → 𝓧 i j
h1 : iIndepFun (fun i ω j ↦ X i j ω) κ P
h2 : ∀ (i : S) (j : T i), ∀ᵐ (a : α) ∂P, X i j =ᵐ[κ a] X' i j
s : Finset S
f : S → Set Ω
g : (i : S) → Set ((fun i ↦ (j : T i) → 𝓧 i j) i)
mg : ∀ i ∈ s, MeasurableSet (g i)
hg : ∀ i ∈ s, (fun i ω j ↦ X' i j ω) i ⁻¹' g i = f i
h3 : ⋂ i ∈ s, f i = ⋂ i ∈ s, (fun i ω j ↦ X' i j ω) i ⁻¹' g i
h3' : ∀ (a : α), ∏ i ∈ s, (κ a) (f i) = ∏ i ∈ s, (κ a) ((fun i ω j ↦ X' i j ω) i ⁻¹' g i)
⊢ ∀ᵐ (a : α) ∂P, (κ a) (⋂ i ∈ s, f i) = ∏ i ∈ s, (κ a) (f i)
Module: Mathlib.Probability.Independence.Process.Basic

Goal state before:
pp: S : Type u_1
Ω : Type u_3
mΩ : MeasurableSpace Ω
α : Type u_4
mα : MeasurableSpace α
κ : Kernel α Ω
P : Measure α
T : S → Type u_5
𝓧 : (i : S) → T i → Type u_6
inst✝ : (i : S) → (j : T i) → MeasurableSpace (𝓧 i j)
X X' : (i : S) → (j : T i) → Ω → 𝓧 i j
h1 : iIndepFun (fun i ω j ↦ X i j ω) κ P
h2 : ∀ (i : S) (j : T i), ∀ᵐ (a : α) ∂P, X i j =ᵐ[κ a] X' i j
s : Finset S
f : S → Set Ω
g : (i : S) → Set ((fun i ↦ (j : T i) → 𝓧 i j) i)
mg : ∀ i ∈ s, MeasurableSet (g i)
hg : ∀ i ∈ s, (fun i ω j ↦ X' i j ω) i ⁻¹' g i = f i
h3 : ⋂ i ∈ s, f i = ⋂ i ∈ s, (fun i ω j ↦ X' i j ω) i ⁻¹' g i
h3' : ∀ (a : α), ∏ i ∈ s, (κ a) (f i) = ∏ i ∈ s, (κ a) ((fun i ω j ↦ X' i j ω) i ⁻¹' g i)
I : (i : S) → Set (T i)
u : (i : S) → Set ((a : ↑(I i)) → 𝓧 i ↑a)
hI : ∀ i ∈ s, (I i).Countable
hu : ∀ i ∈ s, g i = (I i).domRestrict ⁻¹' u i
h4 :
  ∀ (f : (i : S) → (j : T i) → Ω → 𝓧 i j),
    ⋂ i ∈ s, (fun i ω j ↦ f i j ω) i ⁻¹' g i = ⋂ i ∈ s, (fun i ω j ↦ f i j ω) i ⁻¹' (I i).domRestrict ⁻¹' u i
h4' :
  ∀ (a : α) (f : (i : S) → (j : T i) → Ω → 𝓧 i j),
    ∏ i ∈ s, (κ a) ((fun i ω j ↦ f i j ω) i ⁻¹' g i) =
      ∏ i ∈ s, (κ a) ((fun i ω j ↦ f i j ω) i ⁻¹' (I i).domRestrict ⁻¹' u i)
h5 : ∀ᵐ (a : α) ∂P, (κ a) (⋂ i ∈ s, (fun ω j ↦ X i j ω) ⁻¹' g i) = ∏ x ∈ s, (κ a) ((fun ω j ↦ X x j ω) ⁻¹' g x)
⊢ ∀ᵐ (a : α) ∂P, (κ a) (⋂ i ∈ s, (fun ω j ↦ X' i j ω) ⁻¹' g i) = ∏ x ∈ s, (κ a) ((fun ω j ↦ X' x j ω) ⁻¹' g x)

Tactic:
[

Goal state after:
S : Type u_1
Ω : Type u_3
mΩ : MeasurableSpace Ω
α : Type u_4
mα : MeasurableSpace α
κ : Kernel α Ω
P : Measure α
T : S → Type u_5
𝓧 : (i : S) → T i → Type u_6
inst✝ : (i : S) → (j : T i) → MeasurableSpace (𝓧 i j)
X X' : (i : S) → (j : T i) → Ω → 𝓧 i j
h1 : iIndepFun (fun i ω j ↦ X i j ω) κ P
h2 : ∀ (i : S) (j : T i), ∀ᵐ (a : α) ∂P, X i j =ᵐ[κ a] X' i j
s : Finset S
f : S → Set Ω
g : (i : S) → Set ((fun i ↦ (j : T i) → 𝓧 i j) i)
mg : ∀ i ∈ s, MeasurableSet (g i)
hg : ∀ i ∈ s, (fun i ω j ↦ X' i j ω) i ⁻¹' g i = f i
h3 : ⋂ i ∈ s, f i = ⋂ i ∈ s, (fun i ω j ↦ X' i j ω) i ⁻¹' g i
h3' : ∀ (a : α), ∏ i ∈ s, (κ a) (f i) = ∏ i ∈ s, (κ a) ((fun i ω j ↦ X' i j ω) i ⁻¹' g i)
I : (i : S) → Set (T i)
u : (i : S) → Set ((a : ↑(I i)) → 𝓧 i ↑a)
hI : ∀ i ∈ s, (I i).Countable
hu : ∀ i ∈ s, g i = (I i).domRestrict ⁻¹' u i
h4 :
  ∀ (f : (i : S) → (j : T i) → Ω → 𝓧 i j),
    ⋂ i ∈ s, (fun i ω j ↦ f i j ω) i ⁻¹' g i = ⋂ i ∈ s, (fun i ω j ↦ f i j ω) i ⁻¹' (I i).domRestrict ⁻¹' u i
h4' :
  ∀ (a : α) (f : (i : S) → (j : T i) → Ω → 𝓧 i j),
    ∏ i ∈ s, (κ a) ((fun i ω j ↦ f i j ω) i ⁻¹' g i) =
      ∏ i ∈ s, (κ a) ((fun i ω j ↦ f i j ω) i ⁻¹' (I i).domRestrict ⁻¹' u i)
h5 : ∀ᵐ (a : α) ∂P, (κ a) (⋂ i ∈ s, (fun ω j ↦ X i j ω) ⁻¹' g i) = ∏ x ∈ s, (κ a) ((fun ω j ↦ X x j ω) ⁻¹' g x)
⊢ ∀ᵐ (a : α) ∂P, (κ a) (⋂ i ∈ s, (fun ω j ↦ X' i j ω) ⁻¹' g i) = ∏ x ∈ s, (κ a) ((fun ω j ↦ X' x j ω) ⁻¹' g x)
Module: Mathlib.Probability.Independence.Process.Basic

Goal state before:
pp: case inr
S : Type u_1
Ω : Type u_3
mΩ : MeasurableSpace Ω
α : Type u_4
mα : MeasurableSpace α
κ : Kernel α Ω
P : Measure α
T : S → Type u_5
𝓧 : (i : S) → T i → Type u_6
inst✝ : (i : S) → (j : T i) → MeasurableSpace (𝓧 i j)
X : (i : S) → (j : T i) → Ω → 𝓧 i j
hX : ∀ (i : S) (j : T i), Measurable (X i j)
h : ∀ (I : Finset S) (J : (i : ↥I) → Finset (T ↑i)), iIndepFun (fun i ω j ↦ X (↑i) (↑j) ω) κ P
hμ : P ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᵐ[P] ⇑η
hη : IsMarkovKernel η
π : S → Set (Set Ω) := fun i ↦ {s | ∃ t ∈ squareCylinders fun j ↦ {s | MeasurableSet s}, (fun ω j ↦ X i j ω) ⁻¹' t = s}
π_pi : ∀ (i : S), IsPiSystem (π i)
π_gen : ∀ (i : S), MeasurableSpace.comap (fun ω j ↦ X i j ω) MeasurableSpace.pi = generateFrom (π i)
I : Finset S
s : S → Set Ω
J : (i : S) → Finset (T i)
t : (i : S) → (i_1 : T i) → Set (𝓧 i i_1)
ht : ∀ i ∈ I, ∀ (i_1 : T i), MeasurableSet (t i i_1)
hs : ∀ i ∈ I, (fun ω j ↦ X i j ω) ⁻¹' (↑(J i)).pi (t i) = s i
⊢ ∀ᵐ (a : α) ∂P, (η a) (⋂ i ∈ I, s i) = ∏ i ∈ I, (η a) (s i)

Tactic:
[

Goal state after:
case inr
S : Type u_1
Ω : Type u_3
mΩ : MeasurableSpace Ω
α : Type u_4
mα : MeasurableSpace α
κ : Kernel α Ω
P : Measure α
T : S → Type u_5
𝓧 : (i : S) → T i → Type u_6
inst✝ : (i : S) → (j : T i) → MeasurableSpace (𝓧 i j)
X : (i : S) → (j : T i) → Ω → 𝓧 i j
hX : ∀ (i : S) (j : T i), Measurable (X i j)
h : ∀ (I : Finset S) (J : (i : ↥I) → Finset (T ↑i)), iIndepFun (fun i ω j ↦ X (↑i) (↑j) ω) κ P
hμ : P ≠ 0
η : Kernel α Ω
η_eq : ⇑κ =ᵐ[P] ⇑η
hη : IsMarkovKernel η
π : S → Set (Set Ω) := fun i ↦ {s | ∃ t ∈ squareCylinders fun j ↦ {s | MeasurableSet s}, (fun ω j ↦ X i j ω) ⁻¹' t = s}
π_pi : ∀ (i : S), IsPiSystem (π i)
π_gen : ∀ (i : S), MeasurableSpace.comap (fun ω j ↦ X i j ω) MeasurableSpace.pi = generateFrom (π i)
I : Finset S
s : S → Set Ω
J : (i : S) → Finset (T i)
t : (i : S) → (i_1 : T i) → Set (𝓧 i i_1)
ht : ∀ i ∈ I, ∀ (i_1 : T i), MeasurableSet (t i i_1)
hs : ∀ i ∈ I, (fun ω j ↦ X i j ω) ⁻¹' (↑(J i)).pi (t i) = s i
⊢ ∀ᵐ (a : α) ∂P, (η a) (⋂ i ∈ I, s i) = ∏ i ∈ I, (η a) (s i)
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
f : α → StieltjesFunction ℝ
hf : ∀ (q : ℝ), Measurable fun a ↦ ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
⊢ ∀ s ∈ range Iic, Measurable fun a ↦ (f a).measure s

Tactic:
[

Goal state after:
α : Type u_1
x✝ : MeasurableSpace α
f : α → StieltjesFunction ℝ
hf : ∀ (q : ℝ), Measurable fun a ↦ ↑(f a) q
hf_bot : ∀ (a : α), Tendsto (↑(f a)) atBot (𝓝 0)
hf_top : ∀ (a : α), Tendsto (↑(f a)) atTop (𝓝 1)
this : ∀ (a : α), IsProbabilityMeasure (f a).measure
⊢ ∀ s ∈ range Iic, Measurable fun a ↦ (f a).measure s
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : Measurable f
⊢ MeasurableSet {a | ∀ (q r : ℚ), q ≤ r → f a q ≤ f a r}

Tactic:
[

Goal state after:
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : Measurable f
⊢ MeasurableSet {a | ∀ (q r : ℚ), q ≤ r → f a q ≤ f a r}
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : Measurable f
h1 : MeasurableSet {a | Monotone (f a)}
h2 : MeasurableSet {a | Tendsto (f a) atTop (𝓝 1)}
h3 : MeasurableSet {a | Tendsto (f a) atBot (𝓝 0)}
h4 : MeasurableSet {a | ∀ (t : ℚ), ⨅ r, f a ↑r = f a t}
this :
  {a | IsRatStieltjesPoint f a} =
    {a | Monotone (f a)} ∩ {a | Tendsto (f a) atTop (𝓝 1)} ∩ {a | Tendsto (f a) atBot (𝓝 0)} ∩
      {a | ∀ (t : ℚ), ⨅ r, f a ↑r = f a t}
⊢ MeasurableSet {a | IsRatStieltjesPoint f a}

Tactic:
this

Goal state after:
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : Measurable f
h1 : MeasurableSet {a | Monotone (f a)}
h2 : MeasurableSet {a | Tendsto (f a) atTop (𝓝 1)}
h3 : MeasurableSet {a | Tendsto (f a) atBot (𝓝 0)}
h4 : MeasurableSet {a | ∀ (t : ℚ), ⨅ r, f a ↑r = f a t}
this :
  {a | IsRatStieltjesPoint f a} =
    {a | Monotone (f a)} ∩ {a | Tendsto (f a) atTop (𝓝 1)} ∩ {a | Tendsto (f a) atBot (𝓝 0)} ∩
      {a | ∀ (t : ℚ), ⨅ r, f a ↑r = f a t}
⊢ MeasurableSet
    ({a | Monotone (f a)} ∩ {a | Tendsto (f a) atTop (𝓝 1)} ∩ {a | Tendsto (f a) atBot (𝓝 0)} ∩
      {a | ∀ (t : ℚ), ⨅ r, f a ↑r = f a t})
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
E : ι → Type u_3
inst✝¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
L✝ L : (i : ι) → StrongDual ℝ (E i) →L[ℝ] StrongDual ℝ (E i) →L[ℝ] ℝ
g : LinearMap.BilinForm ℝ (StrongDual ℝ ((i : ι) → E i)) :=
  LinearMap.mk₂ ℝ (fun x y ↦ ∑ i, ((L i) (x ∘SL single ℝ E i)) (y ∘SL single ℝ E i)) ⋯ ⋯ ⋯ ⋯
x y : StrongDual ℝ ((i : ι) → E i)
⊢ ∑ i, ‖((L i) (x ∘SL single ℝ E i)) (y ∘SL single ℝ E i)‖ ≤ (∑ i, ‖L i‖) * ‖x‖ * ‖y‖

Tactic:
sum_mul,

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
E : ι → Type u_3
inst✝¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
L✝ L : (i : ι) → StrongDual ℝ (E i) →L[ℝ] StrongDual ℝ (E i) →L[ℝ] ℝ
g : LinearMap.BilinForm ℝ (StrongDual ℝ ((i : ι) → E i)) :=
  LinearMap.mk₂ ℝ (fun x y ↦ ∑ i, ((L i) (x ∘SL single ℝ E i)) (y ∘SL single ℝ E i)) ⋯ ⋯ ⋯ ⋯
x y : StrongDual ℝ ((i : ι) → E i)
⊢ ∑ i, ‖((L i) (x ∘SL single ℝ E i)) (y ∘SL single ℝ E i)‖ ≤ (∑ i, ‖L i‖ * ‖x‖) * ‖y‖
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: ι : Type u_2
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
E : ι → Type u_3
inst✝¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
L : (i : ι) → StrongDual ℝ (E i) →L[ℝ] StrongDual ℝ (E i) →L[ℝ] ℝ
hL : ∀ (i : ι), (L i).toBilinForm.IsPosSemidef
x y : StrongDual ℝ ((i : ι) → E i)
⊢ ((diagonalStrongDualPi L).toBilinForm x) y = ((diagonalStrongDualPi L).toBilinForm y) x

Tactic:
[

Goal state after:
ι : Type u_2
inst✝³ : Fintype ι
inst✝² : DecidableEq ι
E : ι → Type u_3
inst✝¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
L : (i : ι) → StrongDual ℝ (E i) →L[ℝ] StrongDual ℝ (E i) →L[ℝ] ℝ
hL : ∀ (i : ι), (L i).toBilinForm.IsPosSemidef
x y : StrongDual ℝ ((i : ι) → E i)
⊢ ((diagonalStrongDualPi L).toBilinForm x) y = ((diagonalStrongDualPi L).toBilinForm y) x
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: α : Type u_1
f : α → ℚ → ℝ
a : α
⊢ stieltjesFunctionAux (fun p ↦ f p.2) ((), a) = stieltjesFunctionAux f a

Tactic:
[

Goal state after:
α : Type u_1
f : α → ℚ → ℝ
a : α
⊢ stieltjesFunctionAux (fun p ↦ f p.2) ((), a) = stieltjesFunctionAux f a
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x y : ℝ
hxy : x ≤ y
this : Nonempty { r' // y < ↑r' }
⊢ stieltjesFunctionAux f a x ≤ stieltjesFunctionAux f a y

Tactic:
[

Goal state after:
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x y : ℝ
hxy : x ≤ y
this : Nonempty { r' // y < ↑r' }
⊢ stieltjesFunctionAux f a x ≤ stieltjesFunctionAux f a y
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x : ℝ
r : ℚ
hrx : x < ↑r
⊢ ⨅ r, ↑(hf.stieltjesFunction a) ↑↑r ≤ 1

Tactic:
[

Goal state after:
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x : ℝ
r : ℚ
hrx : x < ↑r
⊢ ⨅ r, ↑(hf.stieltjesFunction a) ↑↑r ≤ 1
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 → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
⊢ ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) =
    ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r)

Tactic:
[

Goal state after:
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 → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
b : β
⊢ ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) =
    ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r)
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
x : ℝ
this : (fun a ↦ ↑(hf.stieltjesFunction a) x) = fun a ↦ ⨅ r, f a ↑r
⊢ Measurable fun a ↦ ↑(hf.stieltjesFunction a) x

Tactic:
this

Goal state after:
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
x : ℝ
this : (fun a ↦ ↑(hf.stieltjesFunction a) x) = fun a ↦ ⨅ r, f a ↑r
⊢ Measurable fun a ↦ ⨅ r, f a ↑r
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
this : IsProbabilityMeasure P
n : ℕ
t : Fin (n + 1) → ℝ≥0
ht : Monotone t
i j : Fin n
hij : i ≠ j
⊢ cov[B (t i.succ), B (t j.succ); P] - cov[B (t i.succ), B (t j.castSucc); P] - cov[B (t i.castSucc), B (t j.succ); P] +
      cov[B (t i.castSucc), B (t j.castSucc); P] =
    0

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
this : IsProbabilityMeasure P
n : ℕ
t : Fin (n + 1) → ℝ≥0
ht : Monotone t
i j : Fin n
hij : i ≠ j
⊢ cov[B (t i.succ), B (t j.succ); P] - cov[B (t i.succ), B (t j.castSucc); P] - cov[B (t i.castSucc), B (t j.succ); P] +
      cov[B (t i.castSucc), B (t j.castSucc); P] =
    0
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : Measurable f
a : α
⊢ stieltjesOfMeasurableRat (fun p ↦ f p.2) ⋯ ((), a) = stieltjesOfMeasurableRat f hf a

Tactic:
[

Goal state after:
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : Measurable f
a : α
⊢ stieltjesOfMeasurableRat (fun p ↦ f p.2) ⋯ ((), a) = stieltjesOfMeasurableRat f hf a
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
t : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
⊢ ⨅ r, (ρ.IicSnd ↑↑r) s = (ρ.IicSnd ↑t) s

Tactic:
[

Goal state after:
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
t : ℚ
s : Set α
hs : MeasurableSet s
inst✝ : IsFiniteMeasure ρ
⊢ ⨅ r, (ρ.IicSnd ↑↑r) s = (ρ.IicSnd ↑t) s
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
s : Set α
hs : MeasurableSet s
⊢ Tendsto (fun r ↦ (ρ.IicSnd ↑r) s) atTop (𝓝 (ρ.fst s))

Tactic:
[

Goal state after:
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
s : Set α
hs : MeasurableSet s
⊢ Tendsto (fun r ↦ (ρ.IicSnd ↑r) s) atTop (𝓝 (ρ.fst s))
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
⊢ Tendsto (fun r ↦ (ρ.IicSnd ↑r) s) atBot (𝓝 0)

Tactic:
[

Goal state after:
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
⊢ Tendsto (fun r ↦ (ρ.IicSnd ↑r) s) atBot (𝓝 0)
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
L₁ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
h₁ : L₁.toBilinForm.IsPosSemidef
h₂ : L₂.toBilinForm.IsPosSemidef
x y : StrongDual ℝ (E × F)
⊢ ((L₁.diagonalStrongDualProd L₂).toBilinForm x) y = ((L₁.diagonalStrongDualProd L₂).toBilinForm y) x

Tactic:
[

Goal state after:
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
L₁ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
h₁ : L₁.toBilinForm.IsPosSemidef
h₂ : L₂.toBilinForm.IsPosSemidef
x y : StrongDual ℝ (E × F)
⊢ ((L₁.diagonalStrongDualProd L₂).toBilinForm x) y = ((L₁.diagonalStrongDualProd L₂).toBilinForm y) x
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: case inr
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
this✝ : IsProbabilityMeasure P
n : ℕ
t : Fin (n + 1) → ℝ≥0
ht : Monotone t
i j : Fin n
hij : i ≠ j
this :
  ∀ (i j : Fin n),
    i ≠ j →
      i < j →
        ↑(min (t i.succ) (t j.succ)) - ↑(min (t i.succ) (t j.castSucc)) - ↑(min (t i.castSucc) (t j.succ)) +
            ↑(min (t i.castSucc) (t j.castSucc)) =
          0
h : ¬i < j
⊢ ↑(min (t i.succ) (t j.succ)) - ↑(min (t i.succ) (t j.castSucc)) - ↑(min (t i.castSucc) (t j.succ)) +
      ↑(min (t i.castSucc) (t j.castSucc)) =
    0

Tactic:
[

Goal state after:
case inr
Ω : Type u_1
mΩ : MeasurableSpace Ω
B : ℝ≥0 → Ω → ℝ
P : Measure Ω
hB : IsPreBrownianReal B P
this✝ : IsProbabilityMeasure P
n : ℕ
t : Fin (n + 1) → ℝ≥0
ht : Monotone t
i j : Fin n
hij : i ≠ j
this :
  ∀ (i j : Fin n),
    i ≠ j →
      i < j →
        ↑(min (t i.succ) (t j.succ)) - ↑(min (t i.succ) (t j.castSucc)) - ↑(min (t i.castSucc) (t j.succ)) +
            ↑(min (t i.castSucc) (t j.castSucc)) =
          0
h : ¬i < j
⊢ ↑(min (t i.succ) (t j.succ)) - ↑(min (t i.succ) (t j.castSucc)) - ↑(min (t i.castSucc) (t j.succ)) +
      ↑(min (t i.castSucc) (t j.castSucc)) =
    0
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: case neg.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 → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
  ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
    ∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
i j : { r' // x < ↑r' }
hij : i ≤ j
b : β
⊢ ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑i) ≤
    ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑j)

Tactic:
[

Goal state after:
case neg.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 → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
  ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
    ∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
i j : { r' // x < ↑r' }
hij : i ≤ j
b : β
⊢ ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑i) ≤
    ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑j)
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: case neg
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
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
  ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
    ∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ⨅ b, ∫⁻ (a_1 : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, a_1)) ↑↑b) ∂ν a = (κ a) (s ×ˢ Iic x)

Tactic:
[

Goal state after:
case neg
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
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
x : ℝ
s : Set β
hs : MeasurableSet s
hρ_zero : ¬(ν a).restrict s = 0
h :
  ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) x) ∂ν a =
    ∫⁻ (b : β) in s, ⨅ r, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r) ∂ν a
h_nonempty : Nonempty { r' // x < ↑r' }
⊢ ⨅ b, ∫⁻ (a_1 : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, a_1)) ↑↑b) ∂ν a = (κ a) (s ×ˢ Iic x)
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝⁶ : Finite ι
E : ι → Type u_3
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → MeasurableSpace (E i)
inst✝³ : ∀ (i : ι), CompleteSpace (E i)
inst✝² : ∀ (i : ι), BorelSpace (E i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX : HasGaussianLaw (fun ω i ↦ X i ω) P
h : ∀ (i j : ι), i ≠ j → ∀ (L₁ : StrongDual ℝ (E i)) (L₂ : StrongDual ℝ (E j)), cov[⇑L₁ ∘ X i, ⇑L₂ ∘ X j; P] = 0
⊢ iIndepFun X P

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝⁶ : Finite ι
E : ι → Type u_3
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → MeasurableSpace (E i)
inst✝³ : ∀ (i : ι), CompleteSpace (E i)
inst✝² : ∀ (i : ι), BorelSpace (E i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX : HasGaussianLaw (fun ω i ↦ X i ω) P
h : ∀ (i j : ι), i ≠ j → ∀ (L₁ : StrongDual ℝ (E i)) (L₂ : StrongDual ℝ (E j)), cov[⇑L₁ ∘ X i, ⇑L₂ ∘ X j; P] = 0
⊢ iIndepFun X P
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝⁶ : Finite ι
E : ι → Type u_3
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → MeasurableSpace (E i)
inst✝³ : ∀ (i : ι), CompleteSpace (E i)
inst✝² : ∀ (i : ι), BorelSpace (E i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX : HasGaussianLaw (fun ω i ↦ X i ω) P
h :
  ∀ (i j : ι),
    i ≠ j → ∀ (L₁ : StrongDual ℝ (E i)) (L₂ : StrongDual ℝ (E j)), cov[fun x ↦ L₁ (X i x), fun x ↦ L₂ (X j x); P] = 0
this✝¹ : IsProbabilityMeasure P
this✝ : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
this : ∀ (ω : Ω), (L fun x ↦ X x ω) = ∑ i, (L ∘SL single ℝ E i) (X i ω)
⊢ charFunDual (Measure.map (fun ω x ↦ X x ω) P) L =
    ∏ i, charFunDual (Measure.map (fun x ↦ X i x) P) (L ∘SL single ℝ E i)

Tactic:
[

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
ι : Type u_2
inst✝⁶ : Finite ι
E : ι → Type u_3
inst✝⁵ : (i : ι) → NormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → MeasurableSpace (E i)
inst✝³ : ∀ (i : ι), CompleteSpace (E i)
inst✝² : ∀ (i : ι), BorelSpace (E i)
inst✝¹ : ∀ (i : ι), SecondCountableTopology (E i)
inst✝ : (i : ι) → NormedSpace ℝ (E i)
X : (i : ι) → Ω → E i
hX : HasGaussianLaw (fun ω i ↦ X i ω) P
h :
  ∀ (i j : ι),
    i ≠ j → ∀ (L₁ : StrongDual ℝ (E i)) (L₂ : StrongDual ℝ (E j)), cov[fun x ↦ L₁ (X i x), fun x ↦ L₂ (X j x); P] = 0
this✝¹ : IsProbabilityMeasure P
this✝ : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
this : ∀ (ω : Ω), (L fun x ↦ X x ω) = ∑ i, (L ∘SL single ℝ E i) (X i ω)
⊢ charFunDual (Measure.map (fun ω x ↦ X x ω) P) L =
    ∏ i, charFunDual (Measure.map (fun x ↦ X i x) P) (L ∘SL single ℝ E i)