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

Goal state before:
pp: 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
x : α
hx : ρ.fst {x} ≠ 0
s : Set Ω
hs : MeasurableSet s
this✝ : IsSFiniteKernel ρCond
this : ∀ (a : α), (ρCond a) (Prod.mk a ⁻¹' {x} ×ˢ s) = {x}.indicator (fun x ↦ (ρCond x) s) a
⊢ (ρCond x) s = (ρ.fst {x})⁻¹ * ∫⁻ (a : α), (ρCond a) (Prod.mk a ⁻¹' {x} ×ˢ s) ∂ρ.fst

Tactic:
this

Goal state after:
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
x : α
hx : ρ.fst {x} ≠ 0
s : Set Ω
hs : MeasurableSet s
this✝ : IsSFiniteKernel ρCond
this : ∀ (a : α), (ρCond a) (Prod.mk a ⁻¹' {x} ×ˢ s) = {x}.indicator (fun x ↦ (ρCond x) s) a
⊢ (ρCond x) s = (ρ.fst {x})⁻¹ * ∫⁻ (a : α), {x}.indicator (fun x ↦ (ρCond x) s) a ∂ρ.fst
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
⊢ MeasurableSet {x | memPartitionSet t n x = s}

Tactic:
[

Goal state after:
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
⊢ MeasurableSet {x | memPartitionSet t n x = s}
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✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
n : ℕ
a : α
x✝ : γ
s : Set β
x : γ × β
⊢ x ∈ countablePartitionSet n x✝ ×ˢ s → x ∈ {p | p.1 ∈ countablePartitionSet n x✝}

Tactic:
simp only [mem_prod, mem_ofPred_eq, and_imp]

Goal state after:
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
x✝ : γ
s : Set β
x : γ × β
⊢ x.1 ∈ countablePartitionSet n x✝ → x.2 ∈ s → x.1 ∈ countablePartitionSet n x✝
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✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this : IsFiniteKernel κ
hu_meas : MeasurableSet u
⊢ ∫ (x : γ) in u, κ.densityProcess ν n a x s ∂ν a = (κ a).real (u ×ˢ s)

Tactic:
[

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this : IsFiniteKernel κ
hu_meas : MeasurableSet u
⊢ ∫ (x : γ) in u, κ.densityProcess ν n a x s ∂ν a = (κ a).real (u ×ˢ s)
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: case iUnion
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set (β × ℝ)
f' : ℕ → Set (β × ℝ)
hf_disj : Pairwise (Disjoint on f')
hf_meas : ∀ (i : ℕ), MeasurableSet (f' i)
hf_eq : ∀ (i : ℕ), ∫⁻ (b : β), ((IsCondKernelCDF.toKernel f hf) (a, b)) (Prod.mk b ⁻¹' f' i) ∂ν a = (κ a) (f' i)
h_eq : ∀ (a : β), Prod.mk a ⁻¹' ⋃ i, f' i = ⋃ i, Prod.mk a ⁻¹' f' i
⊢ ∫⁻ (b : β), ((IsCondKernelCDF.toKernel f hf) (a, b)) (Prod.mk b ⁻¹' ⋃ i, f' i) ∂ν a = (κ a) (⋃ i, f' i)

Tactic:
[

Goal state after:
case iUnion
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set (β × ℝ)
f' : ℕ → Set (β × ℝ)
hf_disj : Pairwise (Disjoint on f')
hf_meas : ∀ (i : ℕ), MeasurableSet (f' i)
hf_eq : ∀ (i : ℕ), ∫⁻ (b : β), ((IsCondKernelCDF.toKernel f hf) (a, b)) (Prod.mk b ⁻¹' f' i) ∂ν a = (κ a) (f' i)
h_eq : ∀ (a : β), Prod.mk a ⁻¹' ⋃ i, f' i = ⋃ i, Prod.mk a ⁻¹' f' i
⊢ ∫⁻ (b : β), ((IsCondKernelCDF.toKernel f hf) (a, b)) (Prod.mk b ⁻¹' ⋃ i, f' i) ∂ν a = (κ a) (⋃ i, f' i)
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
⊢ X ⟂ᵢ[P] Y

Tactic:
have := hXY.isProbabilityMeasure
  rw [indepFun_iff_charFunDual_prod hXY.fst.aemeasurable hXY.snd.aemeasurable]
  intro L
  have : L ∘ (fun ω ↦ (X ω, Y ω)) = (L ∘L (.inl ℝ E F)) ∘ X + (L ∘L (.inr ℝ E F)) ∘ Y := by
    ext; simp [-comp_apply, ← comp_inl_add_comp_inr]
  rw [hXY.charFunDual_map_eq, hXY.fst.charFunDual_map_eq, hXY.snd.charFunDual_map_eq, ← exp_add,
    sub_add_sub_comm, ← add_mul, ← ofReal_add, ← integral_add, ← add_div, ← ofReal_add, this,
    variance_add, h, mul_zero, add_zero]
  · simp
  · exact (hXY.fst.map _).memLp_two
  · exact (hXY.snd.map _).memLp_two
  · exact (hXY.fst.map _).integrable
  · exact (hXY.snd.map _).integrable

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
hκν : κ.fst ≤ ν
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this : IsFiniteKernel κ
hu_meas : MeasurableSet u
x✝ : γ
h0 : (ν a) (countablePartitionSet n x✝) = 0
h0' : (κ a) {p | p.1 ∈ countablePartitionSet n x✝} = 0
x : γ × β
⊢ x ∈ countablePartitionSet n x✝ ×ˢ s → x ∈ {p | p.1 ∈ countablePartitionSet n x✝}

Tactic:
simp only [mem_prod, mem_ofPred_eq, and_imp]

Goal state after:
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
hκν : κ.fst ≤ ν
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this : IsFiniteKernel κ
hu_meas : MeasurableSet u
x✝ : γ
h0 : (ν a) (countablePartitionSet n x✝) = 0
h0' : (κ a) {p | p.1 ∈ countablePartitionSet n x✝} = 0
x : γ × β
⊢ x.1 ∈ countablePartitionSet n x✝ → x.2 ∈ s → x.1 ∈ countablePartitionSet n x✝
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✝ : CountablyGenerated γ
κ : Kernel α (γ × β)
ν : Kernel α γ
hκν : κ.fst ≤ ν
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this✝ : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u, (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ∂ν a =
    ∫⁻ (x : γ) in u, (κ a) (u ×ˢ s) / (ν a) u ∂ν a
⊢ ∫⁻ (a_1 : γ) in u, (κ a) (countablePartitionSet n a_1 ×ˢ s) / (ν a) (countablePartitionSet n a_1) ∂ν a =
    (κ a) (u ×ˢ s)

Tactic:
this

Goal state after:
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
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this✝ : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u, (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ∂ν a =
    ∫⁻ (x : γ) in u, (κ a) (u ×ˢ s) / (ν a) u ∂ν a
⊢ ∫⁻ (x : γ) in u, (κ a) (u ×ˢ s) / (ν a) u ∂ν a = (κ a) (u ×ˢ s)
Module: Mathlib.Probability.Kernel.Disintegration.CDFToKernel

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set (β × ℝ)
f' : ℕ → Set (β × ℝ)
hf_disj : Pairwise (Disjoint on f')
hf_meas : ∀ (i : ℕ), MeasurableSet (f' i)
hf_eq : ∀ (i : ℕ), ∫⁻ (b : β), ((IsCondKernelCDF.toKernel f hf) (a, b)) (Prod.mk b ⁻¹' f' i) ∂ν a = (κ a) (f' i)
h_eq : ∀ (a : β), Prod.mk a ⁻¹' ⋃ i, f' i = ⋃ i, Prod.mk a ⁻¹' f' i
h_disj : ∀ (a : β), Pairwise (Disjoint on fun i ↦ Prod.mk a ⁻¹' f' i)
⊢ ∑' (i : ℕ), ∫⁻ (b : β), ((IsCondKernelCDF.toKernel f hf) (a, b)) (Prod.mk b ⁻¹' f' i) ∂ν a = ∑' (i : ℕ), (κ a) (f' i)

Tactic:
[

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α (β × ℝ)
ν : Kernel α β
f : α × β → StieltjesFunction ℝ
inst✝ : IsFiniteKernel κ
hf : IsCondKernelCDF f κ ν
a : α
s : Set (β × ℝ)
f' : ℕ → Set (β × ℝ)
hf_disj : Pairwise (Disjoint on f')
hf_meas : ∀ (i : ℕ), MeasurableSet (f' i)
hf_eq : ∀ (i : ℕ), ∫⁻ (b : β), ((IsCondKernelCDF.toKernel f hf) (a, b)) (Prod.mk b ⁻¹' f' i) ∂ν a = (κ a) (f' i)
h_eq : ∀ (a : β), Prod.mk a ⁻¹' ⋃ i, f' i = ⋃ i, Prod.mk a ⁻¹' f' i
h_disj : ∀ (a : β), Pairwise (Disjoint on fun i ↦ Prod.mk a ⁻¹' f' i)
⊢ ∑' (i : ℕ), ∫⁻ (b : β), ((IsCondKernelCDF.toKernel f hf) (a, b)) (Prod.mk b ⁻¹' f' i) ∂ν a = ∑' (i : ℕ), (κ a) (f' i)
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case refine_2
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✝¹ : IsProbabilityMeasure P
hX : (fun ω i ↦ X i ω) = ofLp ∘ toLp 2 ∘ fun ω i ↦ X i ω
hY : (fun ω j ↦ Y j ω) = ofLp ∘ toLp 2 ∘ fun ω j ↦ Y j ω
this✝ : Fintype ι := Fintype.ofFinite ι
this : Fintype κ := Fintype.ofFinite κ
x : WithLp 2 (ι → ℝ)
y : WithLp 2 (κ → ℝ)
⊢ cov[fun ω ↦
      ⟪∑ x_1, ((EuclideanSpace.basisFun ι ℝ).repr x).ofLp x_1 • (EuclideanSpace.basisFun ι ℝ) x_1,
        (toLp 2 ∘ fun ω i ↦ X i ω) ω⟫,
      fun ω ↦
      ⟪∑ x, ((EuclideanSpace.basisFun κ ℝ).repr y).ofLp x • (EuclideanSpace.basisFun κ ℝ) x,
        (toLp 2 ∘ fun ω j ↦ Y j ω) ω⟫;
      P] =
    0

Tactic:
[

Goal state after:
case refine_2
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✝¹ : IsProbabilityMeasure P
hX : (fun ω i ↦ X i ω) = ofLp ∘ toLp 2 ∘ fun ω i ↦ X i ω
hY : (fun ω j ↦ Y j ω) = ofLp ∘ toLp 2 ∘ fun ω j ↦ Y j ω
this✝ : Fintype ι := Fintype.ofFinite ι
this : Fintype κ := Fintype.ofFinite κ
x : WithLp 2 (ι → ℝ)
y : WithLp 2 (κ → ℝ)
⊢ cov[fun ω ↦
      ⟪∑ x_1, ((EuclideanSpace.basisFun ι ℝ).repr x).ofLp x_1 • (EuclideanSpace.basisFun ι ℝ) x_1,
        (toLp 2 ∘ fun ω i ↦ X i ω) ω⟫,
      fun ω ↦
      ⟪∑ x, ((EuclideanSpace.basisFun κ ℝ).repr y).ofLp x • (EuclideanSpace.basisFun κ ℝ) x,
        (toLp 2 ∘ fun ω j ↦ Y j ω) ω⟫;
      P] =
    0
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
hκν : κ.fst ≤ ν
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this✝ : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u, (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ∂ν a =
    ∫⁻ (x : γ) in u, (κ a) (u ×ˢ s) / (ν a) u ∂ν a
h0 : (ν a) u = 0
h0' : (κ a) {p | p.1 ∈ u} = 0
p : γ × β
⊢ p ∈ u ×ˢ s → p ∈ {p | p.1 ∈ u}

Tactic:
simp only [mem_prod, mem_ofPred_eq, and_imp]

Goal state after:
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
hκν : κ.fst ≤ ν
hν : IsFiniteKernel ν
n : ℕ
a : α
s : Set β
hs : MeasurableSet s
u : Set γ
hu : u ∈ countablePartition γ n
this✝ : IsFiniteKernel κ
hu_meas : MeasurableSet u
this :
  ∫⁻ (x : γ) in u, (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) ∂ν a =
    ∫⁻ (x : γ) in u, (κ a) (u ×ˢ s) / (ν a) u ∂ν a
h0 : (ν a) u = 0
h0' : (κ a) {p | p.1 ∈ u} = 0
p : γ × β
⊢ p.1 ∈ u → p.2 ∈ s → p.1 ∈ u
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)
⊢ ∫ (x : γ) in ⋃₀ ↑S, κ.densityProcess ν n a x s ∂ν a = (κ a).real (⋃₀ ↑S ×ˢ s)

Tactic:
[

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
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)
⊢ ∫ (x : γ) in ⋃₀ ↑S, κ.densityProcess ν n a x s ∂ν a = (κ a).real (⋃₀ ↑S ×ˢ s)
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✝ : Kernel (α × β) Ω
inst✝³ : Countable α
κCond : α → Kernel β Ω
inst✝² : ∀ (a : α), IsMarkovKernel (κCond a)
h_atom : ∀ (x y : α), x ∈ measurableAtom y → κCond x = κCond y
κ : Kernel α (β × Ω)
inst✝¹ : IsSFiniteKernel κ
inst✝ : ∀ (a : α), (κ a).IsCondKernel (κCond a)
a : α
s : Set (β × Ω)
hs : MeasurableSet s
⊢ ((κ.fst ⊗ₖ condKernelCountable κCond h_atom) a) s = ((κ a).fst ⊗ₘ κCond a) s

Tactic:
[

Goal state after:
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✝ : Kernel (α × β) Ω
inst✝³ : Countable α
κCond : α → Kernel β Ω
inst✝² : ∀ (a : α), IsMarkovKernel (κCond a)
h_atom : ∀ (x y : α), x ∈ measurableAtom y → κCond x = κCond y
κ : Kernel α (β × Ω)
inst✝¹ : IsSFiniteKernel κ
inst✝ : ∀ (a : α), (κ a).IsCondKernel (κCond a)
a : α
s : Set (β × Ω)
hs : MeasurableSet s
⊢ ((κ.fst ⊗ₖ condKernelCountable κCond h_atom) a) s = ((κ a).fst ⊗ₘ κCond a) s
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
a : α
s : Set (β × Ω)
hs : MeasurableSet s
⊢ ∫⁻ (x : β), (κ.condKernel (a, x)) (Prod.mk x ⁻¹' s) ∂κ.fst a = ((κ.fst ⊗ₖ κ.condKernel) a) s

Tactic:
[

Goal state after:
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
a : α
s : Set (β × Ω)
hs : MeasurableSet s
⊢ ∫⁻ (x : β), (κ.condKernel (a, x)) (Prod.mk x ⁻¹' s) ∂κ.fst a = ((κ.fst ⊗ₖ κ.condKernel) a) s
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
⊢ ((κ.fst ⊗ₖ borelMarkovFromReal Ω η) a) t = ((κ.fst ⊗ₖ η.comapRight he) a) t

Tactic:
[

Goal state after:
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
⊢ ((κ.fst ⊗ₖ borelMarkovFromReal Ω η) a) t = ((κ.fst ⊗ₖ η.comapRight he) a) t
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 =
    ∫⁻ (b : β) in s, ∫⁻ (ω : Ω) in Set.univ, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a

Tactic:
[

Goal state after:
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 =
    ∫⁻ (b : β) in s, ∫⁻ (ω : Ω) in Set.univ, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a
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
t : Set Ω
ht : MeasurableSet t
⊢ ∫⁻ (b : β), ∫⁻ (ω : Ω) in t, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a =
    ∫⁻ (b : β) in Set.univ, ∫⁻ (ω : Ω) in t, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a

Tactic:
[

Goal state after:
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
t : Set Ω
ht : MeasurableSet t
⊢ ∫⁻ (b : β), ∫⁻ (ω : Ω) in t, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a =
    ∫⁻ (b : β) in Set.univ, ∫⁻ (ω : Ω) in t, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case h
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
⊢ cexp (↑(∫ (x : Ω), (⇑L ∘ Y) x ∂P) * I - ↑Var[⇑L ∘ Y; P] / 2) =
    cexp (↑(∫ (a : Ω), (⇑L ∘ X) a + (⇑L ∘ (Y - X)) a ∂P) * I - ↑Var[⇑L ∘ X + ⇑L ∘ (Y - X); P] / 2)

Tactic:
congr with ω <;> 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) (Prod.map id e ⁻¹' (univ ×ˢ range e)ᶜ) = 0

Tactic:
this

Goal state after:
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
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✝⁶ : MeasurableSpace Ω
inst✝⁵ : StandardBorelSpace Ω
inst✝⁴ : Nonempty Ω
inst✝³ : CountableOrCountablyGenerated α β
κ : Kernel α (β × Ω)
inst✝² : IsFiniteKernel κ
E : Type u_4
f : β × Ω → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a : α
s : Set β
hs : MeasurableSet s
hf : IntegrableOn f (s ×ˢ Set.univ) (κ a)
⊢ ∫ (b : β) in s, ∫ (ω : Ω), f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a =
    ∫ (b : β) in s, ∫ (ω : Ω) in Set.univ, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a

Tactic:
[

Goal state after:
α : Type u_1
β : Type u_2
Ω : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝⁶ : MeasurableSpace Ω
inst✝⁵ : StandardBorelSpace Ω
inst✝⁴ : Nonempty Ω
inst✝³ : CountableOrCountablyGenerated α β
κ : Kernel α (β × Ω)
inst✝² : IsFiniteKernel κ
E : Type u_4
f : β × Ω → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
a : α
s : Set β
hs : MeasurableSet s
hf : IntegrableOn f (s ×ˢ Set.univ) (κ a)
⊢ ∫ (b : β) in s, ∫ (ω : Ω), f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a =
    ∫ (b : β) in s, ∫ (ω : Ω) in Set.univ, f (b, ω) ∂κ.condKernel (a, b) ∂κ.fst a