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

Goal state before:
pp: case hm
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
⊢ (PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm (∫ (x : Ω), I.restrict fun x_1 ↦ X x_1 x ∂P) =
    (PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm (∫ (x : ↥I → ℝ), x ∂projectiveFamily I)

Tactic:
]

Goal state after:
case hm
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
⊢ (PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm (∫ (x : Ω), I.restrict fun x_1 ↦ X x_1 x ∂P) =
    (PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm (∫ (x : ↥I → ℝ), x ∂projectiveFamily I)
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case h
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 : ι
a✝ : i ∈ univ
⊢ ‖L i‖ * ‖x ∘SL single ℝ E i‖ * ‖y ∘SL single ℝ E i‖ ≤ ‖L i‖ * ‖x‖ * ‖y‖

Tactic:
]

Goal state after:
case h
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 : ι
a✝ : i ∈ univ
⊢ ‖L i‖ * ‖x ∘SL single ℝ E i‖ * ‖y ∘SL single ℝ E i‖ ≤ ‖L i‖ * ‖x‖ * ‖y‖
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: case hm
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
i : ↥I
⊢ ∫ (x : Ω), I.restrict (fun x_1 ↦ X x_1 x) i ∂P = 0 i

pp: case hm.hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
i : ↥I
⊢ ∀ (i : ↥I), Integrable (fun x ↦ I.restrict (fun x_1 ↦ X x_1 x) i) P

Tactic:
]

Goal state after:
case hm
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
i : ↥I
⊢ ∫ (x : Ω), I.restrict (fun x_1 ↦ X x_1 x) i ∂P = 0 i

case hm.hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
i : ↥I
⊢ ∀ (i : ↥I), Integrable (fun x ↦ I.restrict (fun x_1 ↦ X x_1 x) i) P
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case hbc
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 : ι
a✝ : i ∈ univ
⊢ ‖x‖ ≤ ‖x‖

Tactic:
]

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

Goal state before:
pp: ⊢ ∃ a, ∀ (b : ℚ), a ≤ b → 1 = defaultRatCDF b

Tactic:
]

Goal state after:
⊢ ∃ a, ∀ (b : ℚ), a ≤ b → 1 = defaultRatCDF b
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: ⊢ ∃ a, ∀ b ≤ a, 0 = defaultRatCDF b

Tactic:
]

Goal state after:
⊢ ∃ a, ∀ b ≤ a, 0 = defaultRatCDF b
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case h₂
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 : ι
a✝ : i ∈ univ
⊢ ‖y‖ ≤ ‖y‖

Tactic:
]

Goal state after:
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 : StrongDual ℝ ((i : ι) → E i)
⊢ 0 ≤ ∑ i, ((L i).toBilinForm (x ∘SL single ℝ E i)) (x ∘SL single ℝ E i)

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 : StrongDual ℝ ((i : ι) → E i)
⊢ 0 ≤ ∑ i, ((L i).toBilinForm (x ∘SL single ℝ E i)) (x ∘SL single ℝ E i)
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: case pos.refine_1
t : ℚ
h_bdd : BddBelow (range fun r ↦ if ↑r < 0 then 0 else 1)
h : t < 0
q : ℚ
htq : t < q
hq_neg : q < 0
⊢ 0 ≤ 0

pp: case pos.refine_1.hc
t : ℚ
h_bdd : BddBelow (range fun r ↦ if ↑r < 0 then 0 else 1)
h : t < 0
q : ℚ
htq : t < q
hq_neg : q < 0
⊢ ↑⟨q, htq⟩ < 0

Tactic:
]

Goal state after:
case pos.refine_1.hc
t : ℚ
h_bdd : BddBelow (range fun r ↦ if ↑r < 0 then 0 else 1)
h : t < 0
q : ℚ
htq : t < q
hq_neg : q < 0
⊢ ↑⟨q, htq⟩ < 0
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: case pos.refine_1.hc
t : ℚ
h_bdd : BddBelow (range fun r ↦ if ↑r < 0 then 0 else 1)
h : t < 0
q : ℚ
htq : t < q
hq_neg : q < 0
⊢ q < 0

Tactic:
]

Goal state after:
case pos.refine_1.hc
t : ℚ
h_bdd : BddBelow (range fun r ↦ if ↑r < 0 then 0 else 1)
h : t < 0
q : ℚ
htq : t < q
hq_neg : q < 0
⊢ q < 0
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: case neg.refine_2
t : ℚ
h_bdd : BddBelow (range fun r ↦ if ↑r < 0 then 0 else 1)
h : ¬t < 0
x : ↑(Ioi t)
⊢ 1 ≤ 1

pp: case neg.refine_2.hnc
t : ℚ
h_bdd : BddBelow (range fun r ↦ if ↑r < 0 then 0 else 1)
h : ¬t < 0
x : ↑(Ioi t)
⊢ ¬↑x < 0

Tactic:
]

Goal state after:
case neg.refine_2.hnc
t : ℚ
h_bdd : BddBelow (range fun r ↦ if ↑r < 0 then 0 else 1)
h : ¬t < 0
x : ↑(Ioi t)
⊢ ¬↑x < 0
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 : IsRatCondKernelCDF f κ ν
a✝ : α
q : ℚ
a : β
ha : IsRatStieltjesPoint f (a✝, a)
⊢ f (a✝, a) q = f (a✝, a) q

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 → ℚ → ℝ
hf : IsRatCondKernelCDF f κ ν
a : α
q : ℚ
s : Set β
hs : MeasurableSet s
⊢ ∫ (b : β) in s, ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑q ∂ν a = (κ a).real (s ×ˢ Iic ↑q)

Tactic:
setIntegral_congr_ae hs (g := fun b ↦ f (a, b) q) ?_,

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 : IsRatCondKernelCDF f κ ν
a : α
q : ℚ
s : Set β
hs : MeasurableSet s
⊢ ∫ (x : β) in s, f (a, x) q ∂ν a = (κ a).real (s ×ˢ Iic ↑q)

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 : IsRatCondKernelCDF f κ ν
a : α
q : ℚ
s : Set β
hs : MeasurableSet s
⊢ ∀ᵐ (x : β) ∂ν a, x ∈ s → ↑(stieltjesOfMeasurableRat f ⋯ (a, x)) ↑q = f (a, x) q
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: case neg.refine_2.hnc
t : ℚ
h_bdd : BddBelow (range fun r ↦ if ↑r < 0 then 0 else 1)
h : 0 ≤ t
x : ↑(Ioi t)
⊢ 0 ≤ ↑x

Tactic:
]

Goal state after:
case neg.refine_2.hnc
t : ℚ
h_bdd : BddBelow (range fun r ↦ if ↑r < 0 then 0 else 1)
h : 0 ≤ t
x : ↑(Ioi t)
⊢ 0 ≤ ↑x
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
pp: case hv
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
⊢ (covarianceBilin
        (Measure.map (⇑(PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm)
          (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P))).toBilinForm =
    (covarianceBilin (Measure.map (⇑(PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm) (projectiveFamily I))).toBilinForm

Tactic:
]

Goal state after:
case hv
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
⊢ (covarianceBilin
        (Measure.map (⇑(PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm)
          (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P))).toBilinForm =
    (covarianceBilin (Measure.map (⇑(PiLp.continuousLinearEquiv 2 ℝ fun a ↦ ℝ).symm) (projectiveFamily I))).toBilinForm
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 : IsRatCondKernelCDF f κ ν
a : α
q : ℚ
s : Set β
hs : MeasurableSet s
⊢ (κ a).real (s ×ˢ Iic ↑q) = (κ a).real (s ×ˢ Iic ↑q)

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 : IsRatCondKernelCDF f κ ν
a : α
q : ℚ
s : Set β
hs : MeasurableSet s
⊢ ∀ᵐ (x : β) ∂ν a, x ∈ s → ↑(stieltjesOfMeasurableRat f ⋯ (a, x)) ↑q = f (a, x) q

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 : IsRatCondKernelCDF f κ ν
a : α
q : ℚ
s : Set β
hs : MeasurableSet s
⊢ ∀ᵐ (x : β) ∂ν a, x ∈ s → ↑(stieltjesOfMeasurableRat f ⋯ (a, x)) ↑q = f (a, x) q
Module: Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes

Goal state before:
pp: α : Type u_1
f : α → ℚ → ℝ
a : α
h : IsRatStieltjesPoint f a
q : ℚ
⊢ f a q = f a q

Tactic:
]

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

Goal state before:
pp: α : Type u_1
f : α → ℚ → ℝ
a : α
⊢ (if IsRatStieltjesPoint f a then (fun p ↦ f p.2) ((), a) else defaultRatCDF) =
    if IsRatStieltjesPoint f a then f a else defaultRatCDF

Tactic:
]

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

Goal state before:
pp: case hv
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
x : WithLp 2 (↥I → ℝ)
⊢ ∑ i, ∑ j, x.ofLp i * x.ofLp j * cov[fun ω ↦ ω i, fun ω ↦ ω j; Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P] =
    ∑ i, ∑ j, x.ofLp i * x.ofLp j * cov[fun ω ↦ ω i, fun ω ↦ ω j; projectiveFamily I]

pp: case hv.hX
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
x : WithLp 2 (↥I → ℝ)
⊢ ∀ (i : ↥I), MemLp (fun ω ↦ ω i) 2 (projectiveFamily I)

pp: case hv.hX
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
x : WithLp 2 (↥I → ℝ)
⊢ ∀ (i : ↥I), MemLp (fun ω ↦ ω i) 2 (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)

Tactic:
]

Goal state after:
case hv
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
x : WithLp 2 (↥I → ℝ)
⊢ ∑ i, ∑ j, x.ofLp i * x.ofLp j * cov[fun ω ↦ ω i, fun ω ↦ ω j; Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P] =
    ∑ i, ∑ j, x.ofLp i * x.ofLp j * cov[fun ω ↦ ω i, fun ω ↦ ω j; projectiveFamily I]

case hv.hX
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
x : WithLp 2 (↥I → ℝ)
⊢ ∀ (i : ↥I), MemLp (fun ω ↦ ω i) 2 (projectiveFamily I)

case hv.hX
Ω : Type u_1
mΩ : MeasurableSpace Ω
X : ℝ≥0 → Ω → ℝ
P : Measure Ω
h1 : IsGaussianProcess X P
h2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0
h3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s
I : Finset ℝ≥0
this : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)
x : WithLp 2 (↥I → ℝ)
⊢ ∀ (i : ↥I), MemLp (fun ω ↦ ω i) 2 (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) 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 → ℚ → ℝ
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
q : ℚ
s : Set β
hs : MeasurableSet s
⊢ ENNReal.ofReal (∫ (x : β) in s, ↑(stieltjesOfMeasurableRat f ⋯ (a, x)) ↑q ∂ν a) = (κ a) (s ×ˢ Iic ↑q)

pp: case hfi
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
q : ℚ
s : Set β
hs : MeasurableSet s
⊢ Integrable (fun b ↦ ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑q) ((ν a).restrict s)

pp: case f_nn
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
q : ℚ
s : Set β
hs : MeasurableSet s
⊢ 0 ≤ᵐ[(ν a).restrict s] fun b ↦ ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑q

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
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
q : ℚ
s : Set β
hs : MeasurableSet s
⊢ ENNReal.ofReal (∫ (x : β) in s, ↑(stieltjesOfMeasurableRat f ⋯ (a, x)) ↑q ∂ν a) = (κ a) (s ×ˢ Iic ↑q)

case hfi
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
q : ℚ
s : Set β
hs : MeasurableSet s
⊢ Integrable (fun b ↦ ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑q) ((ν a).restrict s)

case f_nn
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
q : ℚ
s : Set β
hs : MeasurableSet s
⊢ 0 ≤ᵐ[(ν a).restrict s] fun b ↦ ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑q
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
r : ℚ
⊢ ⨅ q, f a ↑q = ⨅ r_1, f a ↑r_1

Tactic:
]

Goal state after:
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
r : ℚ
⊢ ⨅ q, f a ↑q = ⨅ r_1, f a ↑r_1
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
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
q : ℚ
s : Set β
hs : MeasurableSet s
⊢ (κ a) (s ×ˢ Iic ↑q) = (κ a) (s ×ˢ Iic ↑q)

Tactic:
]

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

Goal state before:
pp: case hfi
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
q : ℚ
s : Set β
hs : MeasurableSet s
⊢ Integrable (fun b ↦ f (a, b) q) (ν a)

Tactic:
]

Goal state after:
case hfi
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
q : ℚ
s : Set β
hs : MeasurableSet s
⊢ Integrable (fun b ↦ f (a, b) q) (ν 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
r : ℝ
this : Nonempty { r' // r < ↑r' }
⊢ 0 ≤ ⨅ q, f a ↑q

Tactic:
]

Goal state after:
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
r : ℝ
this : Nonempty { r' // r < ↑r' }
⊢ 0 ≤ ⨅ q, f a ↑q
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
inst✝ : IsFiniteKernel κ
hf : IsRatCondKernelCDF f κ ν
a : α
q : ℚ
s : Set β
hs : MeasurableSet s
⊢ ∫⁻ (b : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑q) ∂ν a = (κ a) (s ×ˢ Iic ↑q)

Tactic:
by
  rw [← ofReal_integral_eq_lintegral_ofReal]
  · rw [setIntegral_stieltjesOfMeasurableRat_rat hf a q hs, ofReal_measureReal]
  · refine Integrable.restrict ?_
    rw [integrable_congr (stieltjesOfMeasurableRat_ae_eq hf a q)]
    exact hf.integrable a q
  · exact ae_of_all _ (fun x ↦ stieltjesOfMeasurableRat_nonneg _ _ _)

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

Goal state before:
pp: case pos
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
⊢ 0 = (κ a) (s ×ˢ Iic x)

Tactic:
]

Goal state after:
case pos
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
⊢ 0 = (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
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
L₁ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
g : LinearMap.BilinForm ℝ (StrongDual ℝ (E × F)) :=
  LinearMap.mk₂ ℝ (fun x y ↦ (L₁ (x ∘SL inl ℝ E F)) (y ∘SL inl ℝ E F) + (L₂ (x ∘SL inr ℝ E F)) (y ∘SL inr ℝ E F)) ⋯ ⋯ ⋯
    ⋯
x y : StrongDual ℝ (E × F)
⊢ ‖(L₁ (x ∘SL inl ℝ E F)) (y ∘SL inl ℝ E F)‖ + ‖(L₂ (x ∘SL inr ℝ E F)) (y ∘SL inr ℝ E F)‖ ≤
    ‖L₁‖ * ‖x‖ * ‖y‖ + ‖L₂‖ * ‖x‖ * ‖y‖

Tactic:
]

Goal state after:
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
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
L₁ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ
L₂ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ
g : LinearMap.BilinForm ℝ (StrongDual ℝ (E × F)) :=
  LinearMap.mk₂ ℝ (fun x y ↦ (L₁ (x ∘SL inl ℝ E F)) (y ∘SL inl ℝ E F) + (L₂ (x ∘SL inr ℝ E F)) (y ∘SL inr ℝ E F)) ⋯ ⋯ ⋯
    ⋯
x y : StrongDual ℝ (E × F)
⊢ ‖(L₁ (x ∘SL inl ℝ E F)) (y ∘SL inl ℝ E F)‖ + ‖(L₂ (x ∘SL inr ℝ E F)) (y ∘SL inr ℝ E F)‖ ≤
    ‖L₁‖ * ‖x‖ * ‖y‖ + ‖L₂‖ * ‖x‖ * ‖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
⊢ ContinuousWithinAt (stieltjesFunctionAux f a) (Ioi x) x

Tactic:
]

Goal state after:
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x : ℝ
⊢ ContinuousWithinAt (stieltjesFunctionAux f a) (Ioi x) x
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
⊢ ContinuousWithinAt (stieltjesFunctionAux f a) (Ioi x) x ↔
    Tendsto (stieltjesFunctionAux f a) (𝓝[>] x) (𝓝 (⨅ a_1, stieltjesFunctionAux f a ↑a_1))

Tactic:
]

Goal state after:
α : Type u_1
f : α → ℚ → ℝ
inst✝ : MeasurableSpace α
hf : IsMeasurableRatCDF f
a : α
x : ℝ
⊢ ContinuousWithinAt (stieltjesFunctionAux f a) (Ioi x) x ↔
    Tendsto (stieltjesFunctionAux f a) (𝓝[>] x) (𝓝 (⨅ a_1, stieltjesFunctionAux f a ↑a_1))
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
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
q : ℚ
hq : x < ↑q
this : (κ a) (s ×ˢ Iic ↑q) = 0
p : β × ℝ
⊢ p ∈ s ×ˢ Iic x → p ∈ s ×ˢ Iic ↑q

Tactic:
simp only [mem_prod, mem_Iic, and_imp]

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
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
q : ℚ
hq : x < ↑q
this : (κ a) (s ×ˢ Iic ↑q) = 0
p : β × ℝ
⊢ p.1 ∈ s → p.2 ≤ x → p.1 ∈ s ∧ p.2 ≤ ↑q
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
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
q : ℚ
hq : x < ↑q
this : (κ a) (s ×ˢ Iic ↑q) = 0
⊢ (κ a) (s ×ˢ Iic ↑q) = 0

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
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
q : ℚ
hq : x < ↑q
this : (κ a) (s ×ˢ Iic ↑q) = 0
⊢ (κ a) (s ×ˢ Iic ↑q) = 0