mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic x) = (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (⋂ i, Iic ↑↑i)

pp: case e_f.hs
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
⊢ Monotone fun r ↦ Iic ↑↑r

pp: case e_f.hsm
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
⊢ ∀ (i : { r' // x < ↑r' }), NullMeasurableSet (Iic ↑↑i) (stieltjesOfMeasurableRat f ⋯ (a, b)).measure

pp: case e_f.hfin
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
⊢ ∃ i, (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic ↑↑i) ≠ ∞

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
⊢ (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic x) = (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (⋂ i, Iic ↑↑i)
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: case e_f.hs
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
⊢ Monotone fun r ↦ Iic ↑↑r

pp: case e_f.hsm
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
⊢ ∀ (i : { r' // x < ↑r' }), NullMeasurableSet (Iic ↑↑i) (stieltjesOfMeasurableRat f ⋯ (a, b)).measure

pp: case e_f.hfin
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
⊢ ∃ i, (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic ↑↑i) ≠ ∞

Tactic:
·

Goal state after:
case e_f.hs
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
⊢ Monotone fun r ↦ Iic ↑↑r
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: case e_f.hsm
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
⊢ ∀ (i : { r' // x < ↑r' }), NullMeasurableSet (Iic ↑↑i) (stieltjesOfMeasurableRat f ⋯ (a, b)).measure

pp: case e_f.hfin
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
⊢ ∃ i, (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic ↑↑i) ≠ ∞

Tactic:
·

Goal state after:
case e_f.hsm
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
⊢ ∀ (i : { r' // x < ↑r' }), NullMeasurableSet (Iic ↑↑i) (stieltjesOfMeasurableRat f ⋯ (a, b)).measure
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: case e_f.hfin
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
⊢ ∃ i, (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic ↑↑i) ≠ ∞

Tactic:
·

Goal state after:
case e_f.hfin
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
⊢ ∃ i, (stieltjesOfMeasurableRat f ⋯ (a, b)).measure (Iic ↑↑i) ≠ ∞
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
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.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)
Module: Mathlib.Probability.BrownianMotion.Basic

Goal state before:
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.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.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case h₂
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)) := ⋯
x y : StrongDual ℝ (E × F)
⊢ ‖(L₂ (x ∘SL inr ℝ E F)) (y ∘SL inr ℝ E F)‖ ≤ ‖L₂‖ * ‖x‖ * ‖y‖

Tactic:
·

Goal state after:
case h₂
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)) := ⋯
x y : StrongDual ℝ (E × F)
⊢ ‖(L₂ (x ∘SL inr ℝ E F)) (y ∘SL inr ℝ E F)‖ ≤ ‖L₂‖ * ‖x‖ * ‖y‖
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

pp: case hX
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
⊢ MemLp (B (t i.succ)) 2 P

pp: case hY
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
⊢ MemLp (B (t i.castSucc)) 2 P

pp: case hZ
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
⊢ MemLp (B (t j.succ)) 2 P

pp: case hT
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
⊢ MemLp (B (t j.castSucc)) 2 P

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.CDFToKernel

Goal state before:
pp: case neg.hf_int
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 : { r' // x < ↑r' }), ∫⁻ (a_1 : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, a_1)) ↑↑b) ∂ν a ≠ ∞

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' }
⊢ Directed (fun x1 x2 ↦ x1 ≥ x2) fun r b ↦ ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r)

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
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' }
q : { r' // x < ↑r' }
⊢ Measurable fun b ↦ ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑q)

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.hf_int
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 : { r' // x < ↑r' }), ∫⁻ (a_1 : β) in s, ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, a_1)) ↑↑b) ∂ν a ≠ ∞
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

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
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' }
⊢ Directed (fun x1 x2 ↦ x1 ≥ x2) fun r b ↦ ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r)

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
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' }
q : { r' // x < ↑r' }
⊢ Measurable fun b ↦ ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑q)

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.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' }
⊢ Directed (fun x1 x2 ↦ x1 ≥ x2) fun r b ↦ ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑r)
Module: Mathlib.Probability.Kernel.Disintegration.CondCDF

Goal state before:
pp: case refine_1
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
h_empty : ρ (s ×ˢ ∅) = 0
h_neg : Tendsto (fun r ↦ ρ (s ×ˢ Iic ↑(-r))) atTop (𝓝 (ρ (⋂ r, s ×ˢ Iic ↑(-r))))
x : α × ℝ
h : ∀ (i : ℚ), x.1 ∈ s ∧ x.2 ≤ ↑(-i)
i : ℚ
h' : x.1 ∈ s ∧ x.2 ≤ ↑(- -i)
⊢ x.2 ≤ ↑i

pp: case refine_2
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
h_empty : ρ (s ×ˢ ∅) = 0
h_neg : Tendsto (fun r ↦ ρ (s ×ˢ Iic ↑(-r))) atTop (𝓝 (ρ (⋂ r, s ×ˢ Iic ↑(-r))))
x : α × ℝ
h : ∀ (i : ℚ), x.1 ∈ s ∧ x.2 ≤ ↑i
i : ℚ
h' : x.1 ∈ s ∧ x.2 ≤ ↑(-i)
⊢ x.2 ≤ ↑(-i)

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
mα : MeasurableSpace α
ρ : Measure (α × ℝ)
inst✝ : IsFiniteMeasure ρ
s : Set α
hs : MeasurableSet s
h_empty : ρ (s ×ˢ ∅) = 0
h_neg : Tendsto (fun r ↦ ρ (s ×ˢ Iic ↑(-r))) atTop (𝓝 (ρ (⋂ r, s ×ˢ Iic ↑(-r))))
x : α × ℝ
h : ∀ (i : ℚ), x.1 ∈ s ∧ x.2 ≤ ↑(-i)
i : ℚ
h' : x.1 ∈ s ∧ x.2 ≤ ↑(- -i)
⊢ x.2 ≤ ↑i
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
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' }
q : { r' // x < ↑r' }
⊢ Measurable fun b ↦ ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑q)

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:
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' }
q : { r' // x < ↑r' }
⊢ Measurable fun b ↦ ENNReal.ofReal (↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑↑q)
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case e_f
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
hX1 : ∀ (i : ι), HasGaussianLaw (X i) P
hX2 : iIndepFun X P
this✝ : IsProbabilityMeasure P
this : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
i : ι
⊢ cexp
      (↑((L ∘SL single ℝ E i) (∫ (x : E i), x ∂Measure.map (X i) P)) * I -
        ↑Var[⇑(L ∘SL single ℝ E i); Measure.map (X i) P] / 2) =
    cexp
      (↑(L (Pi.single i (∫ (x : E i), x ∂Measure.map (X i) P))) * I -
        ↑Var[⇑(L ∘SL single ℝ E i); Measure.map (X i) P] / 2)

pp: case e_f.h
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
hX1 : ∀ (i : ι), HasGaussianLaw (X i) P
hX2 : iIndepFun X P
this✝ : IsProbabilityMeasure P
this : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
i : ι
⊢ MemLp id 2 (Measure.map (X i) P)

pp: case e_f.h
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
hX1 : ∀ (i : ι), HasGaussianLaw (X i) P
hX2 : iIndepFun X P
this✝ : IsProbabilityMeasure P
this : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
i : ι
⊢ Integrable id (Measure.map (X i) P)

Tactic:
·

Goal state after:
case e_f
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
hX1 : ∀ (i : ι), HasGaussianLaw (X i) P
hX2 : iIndepFun X P
this✝ : IsProbabilityMeasure P
this : Fintype ι := Fintype.ofFinite ι
L : StrongDual ℝ ((x : ι) → E x)
i : ι
⊢ cexp
      (↑((L ∘SL single ℝ E i) (∫ (x : E i), x ∂Measure.map (X i) P)) * I -
        ↑Var[⇑(L ∘SL single ℝ E i); Measure.map (X i) P] / 2) =
    cexp
      (↑(L (Pi.single i (∫ (x : E i), x ∂Measure.map (X i) P))) * I -
        ↑Var[⇑(L ∘SL single ℝ E i); Measure.map (X i) P] / 2)