mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
F : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : CompleteSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : SecondCountableTopology E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : CompleteSpace F
inst✝³ : BorelSpace F
inst✝² : SecondCountableTopology F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
X : Ω → E
Y : Ω → F
hXY : HasGaussianLaw (fun ω ↦ (X ω, Y ω)) P
h : ∀ (L₁ : StrongDual ℝ E) (L₂ : StrongDual ℝ F), cov[⇑L₁ ∘ X, ⇑L₂ ∘ Y; P] = 0
this✝ : IsProbabilityMeasure P
L : StrongDual ℝ (E × F)
this : (⇑L ∘ fun ω ↦ (X ω, Y ω)) = ⇑(L ∘SL inl ℝ E F) ∘ X + ⇑(L ∘SL inr ℝ E F) ∘ Y
⊢ Integrable (⇑(L ∘SL inl ℝ E F) ∘ X) P

pp: case hg
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
F : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : CompleteSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : SecondCountableTopology E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : CompleteSpace F
inst✝³ : BorelSpace F
inst✝² : SecondCountableTopology F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
X : Ω → E
Y : Ω → F
hXY : HasGaussianLaw (fun ω ↦ (X ω, Y ω)) P
h : ∀ (L₁ : StrongDual ℝ E) (L₂ : StrongDual ℝ F), cov[⇑L₁ ∘ X, ⇑L₂ ∘ Y; P] = 0
this✝ : IsProbabilityMeasure P
L : StrongDual ℝ (E × F)
this : (⇑L ∘ fun ω ↦ (X ω, Y ω)) = ⇑(L ∘SL inl ℝ E F) ∘ X + ⇑(L ∘SL inr ℝ E F) ∘ Y
⊢ Integrable (⇑(L ∘SL inr ℝ E F) ∘ Y) P

Tactic:
·

Goal state after:
case hf
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
F : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : CompleteSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : SecondCountableTopology E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : CompleteSpace F
inst✝³ : BorelSpace F
inst✝² : SecondCountableTopology F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
X : Ω → E
Y : Ω → F
hXY : HasGaussianLaw (fun ω ↦ (X ω, Y ω)) P
h : ∀ (L₁ : StrongDual ℝ E) (L₂ : StrongDual ℝ F), cov[⇑L₁ ∘ X, ⇑L₂ ∘ Y; P] = 0
this✝ : IsProbabilityMeasure P
L : StrongDual ℝ (E × F)
this : (⇑L ∘ fun ω ↦ (X ω, Y ω)) = ⇑(L ∘SL inl ℝ E F) ∘ X + ⇑(L ∘SL inr ℝ E F) ∘ Y
⊢ Integrable (⇑(L ∘SL inl ℝ E F) ∘ X) P
Module: Mathlib.Probability.Kernel.Disintegration.Basic

Goal state before:
pp: case pos.left
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✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : κ.fst ⊗ₖ κCond = κ
h_sfin : IsSFiniteKernel κCond
h_eq : ∀ (s : Set β), MeasurableSet s → ∫⁻ (b : β), s.indicator (fun b ↦ (κCond (a, b)) univ) b ∂κ.fst a = (κ.fst a) s
h_meas : Measurable fun b ↦ (κCond (a, b)) univ
r : ℝ≥0∞
hr : 1 < r
s : Set β := {b | r ≤ (κCond (a, b)) univ}
hs : MeasurableSet s
⊢ (κ.fst a) {x | r ≤ (κCond (a, x)) univ} = 0

Tactic:
have h_2_le : s.indicator (fun _ ↦ r) ≤ s.indicator (fun b ↦ (κCond (a, b)) Set.univ) := by
      intro b
      by_cases hbs : b ∈ s
      · simpa [hbs]
      · simp [hbs]

Goal state after:
case pos.left
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✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : κ.fst ⊗ₖ κCond = κ
h_sfin : IsSFiniteKernel κCond
h_eq : ∀ (s : Set β), MeasurableSet s → ∫⁻ (b : β), s.indicator (fun b ↦ (κCond (a, b)) univ) b ∂κ.fst a = (κ.fst a) s
h_meas : Measurable fun b ↦ (κCond (a, b)) univ
r : ℝ≥0∞
hr : 1 < r
s : Set β := {b | r ≤ (κCond (a, b)) univ}
hs : MeasurableSet s
h_2_le : (s.indicator fun x ↦ r) ≤ s.indicator fun b ↦ (κCond (a, b)) univ
⊢ (κ.fst a) {x | r ≤ (κCond (a, x)) univ} = 0
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: case hf
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
⊢ ∀ᵐ (x : γ) ∂(ν a).restrict u, (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) < ∞

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
⊢ (∫⁻ (a_1 : γ) in u, (κ a) (countablePartitionSet n a_1 ×ˢ s) / (ν a) (countablePartitionSet n a_1) ∂ν a).toReal =
    (κ a).real (u ×ˢ s)

Tactic:
·

Goal state after:
case hf
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
⊢ ∀ᵐ (x : γ) ∂(ν a).restrict u, (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) < ∞
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
⊢ (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) < ∞

pp: case neg
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
⊢ (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) < ∞

Tactic:
·

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
⊢ (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) < ∞
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case hg
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
F : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : CompleteSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : SecondCountableTopology E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : CompleteSpace F
inst✝³ : BorelSpace F
inst✝² : SecondCountableTopology F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
X : Ω → E
Y : Ω → F
hXY : HasGaussianLaw (fun ω ↦ (X ω, Y ω)) P
h : ∀ (L₁ : StrongDual ℝ E) (L₂ : StrongDual ℝ F), cov[⇑L₁ ∘ X, ⇑L₂ ∘ Y; P] = 0
this✝ : IsProbabilityMeasure P
L : StrongDual ℝ (E × F)
this : (⇑L ∘ fun ω ↦ (X ω, Y ω)) = ⇑(L ∘SL inl ℝ E F) ∘ X + ⇑(L ∘SL inr ℝ E F) ∘ Y
⊢ Integrable (⇑(L ∘SL inr ℝ E F) ∘ Y) P

Tactic:
·

Goal state after:
case hg
Ω : Type u_1
mΩ : MeasurableSpace Ω
P : Measure Ω
E : Type u_2
F : Type u_3
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : CompleteSpace E
inst✝⁸ : BorelSpace E
inst✝⁷ : SecondCountableTopology E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : MeasurableSpace F
inst✝⁴ : CompleteSpace F
inst✝³ : BorelSpace F
inst✝² : SecondCountableTopology F
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
X : Ω → E
Y : Ω → F
hXY : HasGaussianLaw (fun ω ↦ (X ω, Y ω)) P
h : ∀ (L₁ : StrongDual ℝ E) (L₂ : StrongDual ℝ F), cov[⇑L₁ ∘ X, ⇑L₂ ∘ Y; P] = 0
this✝ : IsProbabilityMeasure P
L : StrongDual ℝ (E × F)
this : (⇑L ∘ fun ω ↦ (X ω, Y ω)) = ⇑(L ∘SL inl ℝ E F) ∘ X + ⇑(L ∘SL inr ℝ E F) ∘ Y
⊢ Integrable (⇑(L ∘SL inr ℝ E F) ∘ Y) P
Module: Mathlib.Probability.Kernel.Disintegration.Density

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
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
⊢ (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) < ∞

Tactic:
·

Goal state after:
case neg
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
⊢ (κ a) (countablePartitionSet n x ×ˢ s) / (ν a) (countablePartitionSet n x) < ∞
Module: Mathlib.Probability.Kernel.Disintegration.Basic

Goal state before:
pp: case pos.right
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✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : κ.fst ⊗ₖ κCond = κ
h_sfin : IsSFiniteKernel κCond
h_eq : ∀ (s : Set β), MeasurableSet s → ∫⁻ (b : β), s.indicator (fun b ↦ (κCond (a, b)) univ) b ∂κ.fst a = (κ.fst a) s
h_meas : Measurable fun b ↦ (κCond (a, b)) univ
⊢ ∀ᵐ (b : β) ∂κ.fst a, 1 ≤ (κCond (a, b)) univ

Tactic:
·

Goal state after:
case pos.right
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✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : κ.fst ⊗ₖ κCond = κ
h_sfin : IsSFiniteKernel κCond
h_eq : ∀ (s : Set β), MeasurableSet s → ∫⁻ (b : β), s.indicator (fun b ↦ (κCond (a, b)) univ) b ∂κ.fst a = (κ.fst a) s
h_meas : Measurable fun b ↦ (κCond (a, b)) univ
⊢ ∀ᵐ (b : β) ∂κ.fst a, 1 ≤ (κCond (a, b)) univ
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
⊢ (κ a) (u ×ˢ s) / (ν a) u * (ν a) u = (κ a) (u ×ˢ s)

pp: case neg
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
⊢ (κ a) (u ×ˢ s) / (ν a) u * (ν a) u = (κ a) (u ×ˢ s)

Tactic:
·

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
⊢ (κ a) (u ×ˢ s) / (ν a) u * (ν a) u = (κ a) (u ×ˢ s)
Module: Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence

Goal state before:
pp: case refine_1
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
⊢ HasGaussianLaw (fun ω ↦ ((toLp 2 ∘ fun ω i ↦ X i ω) ω, (toLp 2 ∘ fun ω j ↦ Y j ω) ω)) P

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, (toLp 2 ∘ fun ω i ↦ X i ω) ω⟫, fun ω ↦ ⟪y, (toLp 2 ∘ fun ω j ↦ Y j ω) ω⟫; P] = 0

Tactic:
·

Goal state after:
case refine_1
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
⊢ HasGaussianLaw (fun ω ↦ ((toLp 2 ∘ fun ω i ↦ X i ω) ω, (toLp 2 ∘ fun ω j ↦ Y j ω) ω)) P
Module: Mathlib.Probability.Kernel.Disintegration.StandardBorel

Goal state before:
pp: case refine_1
α : Type u_1
γ : Type u_3
mα : MeasurableSpace α
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × ℝ)
inst✝ : IsFiniteKernel κ
a : α
s : ℕ → ℚ
hs_anti : Antitone s
hs_tendsto : Tendsto s atTop atBot
s' : ℕ → Set ℝ := fun n ↦ Iic ↑(s n)
⊢ Antitone s'

pp: case refine_2
α : Type u_1
γ : Type u_3
mα : MeasurableSpace α
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × ℝ)
inst✝ : IsFiniteKernel κ
a : α
s : ℕ → ℚ
hs_anti : Antitone s
hs_tendsto : Tendsto s atTop atBot
s' : ℕ → Set ℝ := fun n ↦ Iic ↑(s n)
⊢ ⋂ i, s' i = ∅

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
γ : Type u_3
mα : MeasurableSpace α
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × ℝ)
inst✝ : IsFiniteKernel κ
a : α
s : ℕ → ℚ
hs_anti : Antitone s
hs_tendsto : Tendsto s atTop atBot
s' : ℕ → Set ℝ := fun n ↦ Iic ↑(s n)
⊢ Antitone s'
Module: Mathlib.Probability.Kernel.Disintegration.StandardBorel

Goal state before:
pp: case refine_2
α : Type u_1
γ : Type u_3
mα : MeasurableSpace α
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × ℝ)
inst✝ : IsFiniteKernel κ
a : α
s : ℕ → ℚ
hs_anti : Antitone s
hs_tendsto : Tendsto s atTop atBot
s' : ℕ → Set ℝ := fun n ↦ Iic ↑(s n)
⊢ ⋂ i, s' i = ∅

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
γ : Type u_3
mα : MeasurableSpace α
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × ℝ)
inst✝ : IsFiniteKernel κ
a : α
s : ℕ → ℚ
hs_anti : Antitone s
hs_tendsto : Tendsto s atTop atBot
s' : ℕ → Set ℝ := fun n ↦ Iic ↑(s n)
⊢ ⋂ i, s' i = ∅
Module: Mathlib.Probability.Kernel.Disintegration.StandardBorel

Goal state before:
pp: case refine_1
α : Type u_1
γ : Type u_3
mα : MeasurableSpace α
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × ℝ)
inst✝ : IsFiniteKernel κ
a : α
s : ℕ → ℚ
hs_mono : Monotone s
hs_tendsto : Tendsto s atTop atTop
s' : ℕ → Set ℝ := fun n ↦ Iic ↑(s n)
⊢ Monotone s'

pp: case refine_2
α : Type u_1
γ : Type u_3
mα : MeasurableSpace α
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × ℝ)
inst✝ : IsFiniteKernel κ
a : α
s : ℕ → ℚ
hs_mono : Monotone s
hs_tendsto : Tendsto s atTop atTop
s' : ℕ → Set ℝ := fun n ↦ Iic ↑(s n)
⊢ ⋃ i, s' i = univ

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
γ : Type u_3
mα : MeasurableSpace α
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × ℝ)
inst✝ : IsFiniteKernel κ
a : α
s : ℕ → ℚ
hs_mono : Monotone s
hs_tendsto : Tendsto s atTop atTop
s' : ℕ → Set ℝ := fun n ↦ Iic ↑(s n)
⊢ Monotone s'
Module: Mathlib.Probability.Kernel.Disintegration.StandardBorel

Goal state before:
pp: case refine_2
α : Type u_1
γ : Type u_3
mα : MeasurableSpace α
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × ℝ)
inst✝ : IsFiniteKernel κ
a : α
s : ℕ → ℚ
hs_mono : Monotone s
hs_tendsto : Tendsto s atTop atTop
s' : ℕ → Set ℝ := fun n ↦ Iic ↑(s n)
⊢ ⋃ i, s' i = univ

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
γ : Type u_3
mα : MeasurableSpace α
mγ : MeasurableSpace γ
inst✝¹ : CountablyGenerated γ
κ : Kernel α (γ × ℝ)
inst✝ : IsFiniteKernel κ
a : α
s : ℕ → ℚ
hs_mono : Monotone s
hs_tendsto : Tendsto s atTop atTop
s' : ℕ → Set ℝ := fun n ↦ Iic ↑(s n)
⊢ ⋃ i, s' i = univ
Module: Mathlib.Probability.Kernel.Disintegration.Basic

Goal state before:
pp: case pos
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✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : κ.fst ⊗ₖ κCond = κ
h_sfin : IsSFiniteKernel κCond
h_eq : ∀ (s : Set β), MeasurableSet s → ∫⁻ (b : β), s.indicator (fun b ↦ (κCond (a, b)) univ) b ∂κ.fst a = (κ.fst a) s
h_meas : Measurable fun b ↦ (κCond (a, b)) univ
r : ℝ≥0∞
hr : r < 1
s : Set β := {b | (κCond (a, b)) univ ≤ r}
hs : MeasurableSet s
b : β
hbs : b ∈ s
⊢ s.indicator (fun b ↦ (κCond (a, b)) univ) b ≤ s.indicator (fun x ↦ r) b

pp: case neg
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✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : κ.fst ⊗ₖ κCond = κ
h_sfin : IsSFiniteKernel κCond
h_eq : ∀ (s : Set β), MeasurableSet s → ∫⁻ (b : β), s.indicator (fun b ↦ (κCond (a, b)) univ) b ∂κ.fst a = (κ.fst a) s
h_meas : Measurable fun b ↦ (κCond (a, b)) univ
r : ℝ≥0∞
hr : r < 1
s : Set β := {b | (κCond (a, b)) univ ≤ r}
hs : MeasurableSet s
b : β
hbs : b ∉ s
⊢ s.indicator (fun b ↦ (κCond (a, b)) univ) b ≤ s.indicator (fun x ↦ r) b

Tactic:
·

Goal state after:
case pos
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✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : κ.fst ⊗ₖ κCond = κ
h_sfin : IsSFiniteKernel κCond
h_eq : ∀ (s : Set β), MeasurableSet s → ∫⁻ (b : β), s.indicator (fun b ↦ (κCond (a, b)) univ) b ∂κ.fst a = (κ.fst a) s
h_meas : Measurable fun b ↦ (κCond (a, b)) univ
r : ℝ≥0∞
hr : r < 1
s : Set β := {b | (κCond (a, b)) univ ≤ r}
hs : MeasurableSet s
b : β
hbs : b ∈ s
⊢ s.indicator (fun b ↦ (κCond (a, b)) univ) b ≤ s.indicator (fun x ↦ r) b
Module: Mathlib.Probability.Kernel.Disintegration.StandardBorel

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

Tactic:
by
  rw [condKernelUnitReal]
  infer_instance

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

Goal state before:
pp: case neg
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✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : κ.fst ⊗ₖ κCond = κ
h_sfin : IsSFiniteKernel κCond
h_eq : ∀ (s : Set β), MeasurableSet s → ∫⁻ (b : β), s.indicator (fun b ↦ (κCond (a, b)) univ) b ∂κ.fst a = (κ.fst a) s
h_meas : Measurable fun b ↦ (κCond (a, b)) univ
r : ℝ≥0∞
hr : r < 1
s : Set β := {b | (κCond (a, b)) univ ≤ r}
hs : MeasurableSet s
b : β
hbs : b ∉ s
⊢ s.indicator (fun b ↦ (κCond (a, b)) univ) b ≤ s.indicator (fun x ↦ r) b

Tactic:
·

Goal state after:
case neg
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✝¹ : IsFiniteKernel κ.fst
inst✝ : κ.IsCondKernel κCond
a : α
h : κ.fst ⊗ₖ κCond = κ
h_sfin : IsSFiniteKernel κCond
h_eq : ∀ (s : Set β), MeasurableSet s → ∫⁻ (b : β), s.indicator (fun b ↦ (κCond (a, b)) univ) b ∂κ.fst a = (κ.fst a) s
h_meas : Measurable fun b ↦ (κCond (a, b)) univ
r : ℝ≥0∞
hr : r < 1
s : Set β := {b | (κCond (a, b)) univ ≤ r}
hs : MeasurableSet s
b : β
hbs : b ∉ s
⊢ s.indicator (fun b ↦ (κCond (a, b)) univ) b ≤ s.indicator (fun x ↦ r) b
Module: Mathlib.Probability.Kernel.Disintegration.StandardBorel

Goal state before:
pp: case pos
α : Type u_1
mα : MeasurableSpace α
Ω : Type u_5
inst✝² : Nonempty Ω
inst✝¹ : MeasurableSpace Ω
inst✝ : StandardBorelSpace Ω
η : Kernel α ℝ
a : α
s : Set Ω
hs : MeasurableSet s
he : MeasurableEmbedding (embeddingReal Ω)
h : (η a) (range (embeddingReal Ω))ᶜ = 0
⊢ (Measure.comap (embeddingReal Ω) (η a)) s = (η a) (embeddingReal Ω '' s)

pp: case neg
α : Type u_1
mα : MeasurableSpace α
Ω : Type u_5
inst✝² : Nonempty Ω
inst✝¹ : MeasurableSpace Ω
inst✝ : StandardBorelSpace Ω
η : Kernel α ℝ
a : α
s : Set Ω
hs : MeasurableSet s
he : MeasurableEmbedding (embeddingReal Ω)
h : ¬(η a) (range (embeddingReal Ω))ᶜ = 0
⊢ (Measure.comap (embeddingReal Ω) (Measure.dirac (Exists.choose ⋯))) s =
    (embeddingReal Ω '' s).indicator 1 (Exists.choose ⋯)

Tactic:
·

Goal state after:
case pos
α : Type u_1
mα : MeasurableSpace α
Ω : Type u_5
inst✝² : Nonempty Ω
inst✝¹ : MeasurableSpace Ω
inst✝ : StandardBorelSpace Ω
η : Kernel α ℝ
a : α
s : Set Ω
hs : MeasurableSet s
he : MeasurableEmbedding (embeddingReal Ω)
h : (η a) (range (embeddingReal Ω))ᶜ = 0
⊢ (Measure.comap (embeddingReal Ω) (η a)) s = (η a) (embeddingReal Ω '' s)