mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Kernel.Composition.KernelLemmas

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
T : Type u_4
mX : MeasurableSpace X
mY : MeasurableSpace Y
mZ : MeasurableSpace Z
mT : MeasurableSpace T
κ : Kernel X Y
X' : Type u_5
mX' : MeasurableSpace X'
η : Kernel X' Z
inst✝¹ : IsSFiniteKernel η
ξ : Kernel Z T
inst✝ : IsSFiniteKernel ξ
⊢ Kernel.id ∥ₖ ξ ∘ₖ (κ ∥ₖ η) = κ ∥ₖ (ξ ∘ₖ η)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.KernelLemmas

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
T : Type u_4
mX : MeasurableSpace X
mY : MeasurableSpace Y
mZ : MeasurableSpace Z
mT : MeasurableSpace T
κ : Kernel X Y
X' : Type u_5
mX' : MeasurableSpace X'
η : Kernel X' Z
inst✝¹ : IsSFiniteKernel η
ξ : Kernel Z T
inst✝ : IsSFiniteKernel ξ
this : swap T Y ∘ₖ (ξ ∥ₖ Kernel.id) ∘ₖ (η ∥ₖ κ) = swap T Y ∘ₖ (ξ ∘ₖ η ∥ₖ κ)
⊢ ξ ∥ₖ Kernel.id ∘ₖ (η ∥ₖ κ) = swap Y T ∘ₖ (swap T Y ∘ₖ (ξ ∥ₖ Kernel.id) ∘ₖ (η ∥ₖ κ))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.KernelLemmas

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
T : Type u_4
mX : MeasurableSpace X
mY : MeasurableSpace Y
mZ : MeasurableSpace Z
mT : MeasurableSpace T
κ : Kernel X Y
X' : Type u_5
mX' : MeasurableSpace X'
η : Kernel X' Z
inst✝¹ : IsSFiniteKernel η
ξ : Kernel Z T
inst✝ : IsSFiniteKernel ξ
this : swap T Y ∘ₖ (ξ ∥ₖ Kernel.id) ∘ₖ (η ∥ₖ κ) = swap T Y ∘ₖ (ξ ∘ₖ η ∥ₖ κ)
⊢ swap Y T ∘ₖ (swap T Y ∘ₖ (ξ ∥ₖ Kernel.id) ∘ₖ (η ∥ₖ κ)) = swap Y T ∘ₖ (swap T Y ∘ₖ (ξ ∘ₖ η ∥ₖ κ))

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
h2s : ((κ ⊗ₖ η) a) s ≠ ∞
t : Set (β × γ) := toMeasurable ((κ ⊗ₖ η) a) s
this : ∀ (b : β), (η (a, b)) (Prod.mk b ⁻¹' s) ≤ (η (a, b)) (Prod.mk b ⁻¹' t)
ht : MeasurableSet t
⊢ ((κ ⊗ₖ η) a) t ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.KernelLemmas

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
T : Type u_4
mX : MeasurableSpace X
mY : MeasurableSpace Y
mZ : MeasurableSpace Z
mT : MeasurableSpace T
κ : Kernel X Y
X' : Type u_5
mX' : MeasurableSpace X'
η : Kernel X' Z
inst✝¹ : IsSFiniteKernel η
ξ : Kernel Z T
inst✝ : IsSFiniteKernel ξ
this : swap T Y ∘ₖ (ξ ∥ₖ Kernel.id) ∘ₖ (η ∥ₖ κ) = swap T Y ∘ₖ (ξ ∘ₖ η ∥ₖ κ)
⊢ swap Y T ∘ₖ (swap T Y ∘ₖ (ξ ∘ₖ η ∥ₖ κ)) = ξ ∘ₖ η ∥ₖ κ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.KernelLemmas

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
T : Type u_4
mX : MeasurableSpace X
mY : MeasurableSpace Y
mZ : MeasurableSpace Z
mT : MeasurableSpace T
κ : Kernel X Y
X' : Type u_5
mX' : MeasurableSpace X'
η : Kernel X' Z
inst✝¹ : IsSFiniteKernel η
ξ : Kernel Z T
inst✝ : IsSFiniteKernel ξ
this : swap T Y ∘ₖ (ξ ∥ₖ Kernel.id) ∘ₖ (η ∥ₖ κ) = swap T Y ∘ₖ (ξ ∘ₖ η ∥ₖ κ)
⊢ ξ ∥ₖ Kernel.id ∘ₖ (η ∥ₖ κ) = ξ ∘ₖ η ∥ₖ κ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
h2s : ((κ ⊗ₖ η) a) s ≠ ∞
t : Set (β × γ) := toMeasurable ((κ ⊗ₖ η) a) s
this : ∀ (b : β), (η (a, b)) (Prod.mk b ⁻¹' s) ≤ (η (a, b)) (Prod.mk b ⁻¹' t)
ht : MeasurableSet t
h2t : ((κ ⊗ₖ η) a) t ≠ ∞
⊢ ∀ᵐ (b : β) ∂κ a, (η (a, b)) (Prod.mk b ⁻¹' t) < ∞

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.KernelLemmas

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
T : Type u_4
mX : MeasurableSpace X
mY : MeasurableSpace Y
mZ : MeasurableSpace Z
mT : MeasurableSpace T
κ : Kernel X Y
X' : Type u_5
mX' : MeasurableSpace X'
η : Kernel X' Z
inst✝¹ : IsSFiniteKernel η
ξ : Kernel Z T
inst✝ : IsSFiniteKernel ξ
⊢ ξ ∥ₖ Kernel.id ∘ₖ (η ∥ₖ κ) = ξ ∘ₖ η ∥ₖ κ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureCompProd

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
⊢ μ ⊗ₘ κ ⟂ₘ ν ⊗ₘ η ↔ ∀ (ξ : Measure α), SFinite ξ → ξ ≪ μ → ξ ≪ ν → ξ ⊗ₘ κ ⟂ₘ ξ ⊗ₘ η

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
h2s : ((κ ⊗ₖ η) a) s ≠ ∞
⊢ ∀ᵐ (b : β) ∂κ a, (η (a, b)) (Prod.mk b ⁻¹' s) < ∞

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
κ : Kernel α β
η : Kernel γ δ
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsSFiniteKernel η
⊢ (prodMkRight α η ×ₖ prodMkLeft γ κ).comap Prod.swap ⋯ = (prodMkRight γ κ ×ₖ prodMkLeft α η).map Prod.swap

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.KernelLemmas

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
mX : MeasurableSpace X
mY : MeasurableSpace Y
mZ : MeasurableSpace Z
κ : Kernel X Y
X' : Type u_5
Y' : Type u_6
Z' : Type u_7
mX' : MeasurableSpace X'
mY' : MeasurableSpace Y'
mZ' : MeasurableSpace Z'
inst✝³ : IsSFiniteKernel κ
η : Kernel Y Z
inst✝² : IsSFiniteKernel η
κ' : Kernel X' Y'
inst✝¹ : IsSFiniteKernel κ'
η' : Kernel Y' Z'
inst✝ : IsSFiniteKernel η'
⊢ η ∥ₖ η' ∘ₖ (κ ∥ₖ κ') = η ∘ₖ κ ∥ₖ (η' ∘ₖ κ')

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.KernelLemmas

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
mX : MeasurableSpace X
mY : MeasurableSpace Y
mZ : MeasurableSpace Z
κ : Kernel X Y
Y' : Type u_6
Z' : Type u_7
mY' : MeasurableSpace Y'
mZ' : MeasurableSpace Z'
inst✝³ : IsSFiniteKernel κ
η : Kernel Y Z
inst✝² : IsSFiniteKernel η
κ' : Kernel X Y'
inst✝¹ : IsSFiniteKernel κ'
η' : Kernel Y' Z'
inst✝ : IsSFiniteKernel η'
⊢ η ∥ₖ η' ∘ₖ (κ ×ₖ κ') = η ∘ₖ κ ×ₖ (η' ∘ₖ κ')

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureCompProd

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SigmaFinite μ
inst✝ : SigmaFinite ν
hκη : μ ⊗ₘ κ ≪ ν ⊗ₘ η
⊢ μ ⊗ₘ κ ≪ μ ⊗ₘ η

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
hs : MeasurableSet s
⊢ ((κ ⊗ₖ η) a) s = 0 ↔ (fun b ↦ (η (a, b)) (Prod.mk b ⁻¹' s)) =ᵐ[κ a] 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
⊢ ⇑κ ∘ₘ μ = (κ ∘ₖ Kernel.const Unit μ) ()

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
inst✝¹ : Countable α
inst✝ : MeasurableSingletonClass α
s : Set β
hs : MeasurableSet s
⊢ AEMeasurable (⇑κ) μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
inst✝¹ : Countable α
inst✝ : MeasurableSingletonClass α
⊢ ⇑κ ∘ₘ μ = sum fun ω ↦ μ {ω} • κ ω

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
inst✝¹ : SFinite μ
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
⊢ (μ ⊗ₘ κ).snd = ⇑κ ∘ₘ μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
h : ((κ ⊗ₖ η) a) s = 0
⊢ (fun b ↦ (η (a, b)) (Prod.mk b ⁻¹' s)) =ᵐ[κ a] 0

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
p : β → Prop
h : ∀ᵐ (ω : β) ∂⇑κ ∘ₘ μ, p ω
⊢ ∀ᵐ (ω' : α) ∂μ, ∀ᵐ (ω : β) ∂κ ω', p ω

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
p : β → Prop
hp : MeasurableSet {z | p z}
h : ∀ᵐ (y : α) ∂μ, ∀ᵐ (z : β) ∂κ y, p z
⊢ ∀ᵐ (z : β) ∂⇑κ ∘ₘ μ, p z

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

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
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : SFinite μ
inst✝ : IsSFiniteKernel κ
⊢ SFinite (⇑κ ∘ₘ μ)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.KernelLemmas

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
T : Type u_4
mX : MeasurableSpace X
mY : MeasurableSpace Y
mZ : MeasurableSpace Z
mT : MeasurableSpace T
κ : Kernel X Y
η : Kernel Z T
⊢ Kernel.id ∥ₖ κ ∘ₖ (η ∥ₖ Kernel.id) = η ∥ₖ Kernel.id ∘ₖ (Kernel.id ∥ₖ κ)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

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
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteKernel κ
⊢ IsFiniteMeasure (⇑κ ∘ₘ μ)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

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
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : IsProbabilityMeasure μ
inst✝ : IsMarkovKernel κ
⊢ IsProbabilityMeasure (⇑κ ∘ₘ μ)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

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
μ ν : Measure α
κ η : Kernel α β
inst✝¹ : IsZeroOrProbabilityMeasure μ
inst✝ : IsZeroOrMarkovKernel κ
⊢ IsZeroOrProbabilityMeasure (⇑κ ∘ₘ μ)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.KernelLemmas

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
T : Type u_4
mX : MeasurableSpace X
mY : MeasurableSpace Y
mZ : MeasurableSpace Z
mT : MeasurableSpace T
κ : Kernel X Y
η : Kernel Z T
inst✝ : IsSFiniteKernel κ
⊢ Kernel.id ∥ₖ κ ∘ₖ (η ∥ₖ Kernel.id) = η ∥ₖ κ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

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
μ : Measure α
κ : Kernel α β
f : β → γ
hf : Measurable f
⊢ map f (⇑κ ∘ₘ μ) = ⇑(κ.map f) ∘ₘ μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
⊢ ⇑(Kernel.discard α) ∘ₘ μ = μ Set.univ • dirac ()

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
κ : Kernel α β
η : Kernel α γ
inst✝¹ : IsSFiniteKernel κ
inst✝ : IsSFiniteKernel η
⊢ (κ ×ₖ η).map Prod.swap = η ×ₖ κ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
f : α → β
hf : AEMeasurable f μ
⊢ ⇑(Kernel.copy β) ∘ₘ map f μ = map (Function.prod f f) μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
inst✝¹ : SFinite μ
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
⊢ μ ⊗ₘ κ = ⇑(Kernel.id ×ₖ κ) ∘ₘ μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel α γ
inst✝ : IsSFiniteKernel η
⊢ swap β γ ∘ₖ (κ ×ₖ η) = η ×ₖ κ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
⊢ μ ⊗ₘ Kernel.id = ⇑(Kernel.copy α) ∘ₘ μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
f : α → β
g : α → γ
hf : Measurable f
hg : Measurable g
⊢ deterministic f hf ×ₖ deterministic g hg = deterministic (fun a ↦ (f a, g a)) ⋯

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
⊢ Kernel.id = deterministic Prod.fst ⋯ ×ₖ deterministic Prod.snd ⋯

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
κ : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel α γ
inst✝¹ : IsSFiniteKernel η
ξ : Kernel α δ
inst✝ : IsSFiniteKernel ξ
a : α
⊢ Measurable ⇑MeasurableEquiv.prodAssoc

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
a : α
κ : Kernel α β
η : Kernel (α × β) γ
p : β × γ → Prop
hp : MeasurableSet {x | p x}
h : ∀ᵐ (b : β) ∂κ a, ∀ᵐ (c : γ) ∂η (a, b), p (b, c)
⊢ ∀ᵐ (bc : β × γ) ∂(κ ⊗ₖ η) a, p bc

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
κ : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel α γ
inst✝¹ : IsSFiniteKernel η
ξ : Kernel α δ
inst✝ : IsSFiniteKernel ξ
⊢ (κ ×ₖ ξ ×ₖ η).map ⇑MeasurableEquiv.prodAssoc = κ ×ₖ (ξ ×ₖ η)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
κ : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel α γ
inst✝¹ : IsSFiniteKernel η
ξ : Kernel α δ
inst✝ : IsSFiniteKernel ξ
⊢ Measurable ⇑MeasurableEquiv.prodAssoc

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
κ : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel α γ
inst✝¹ : IsSFiniteKernel η
ξ : Kernel α δ
inst✝ : IsSFiniteKernel ξ
⊢ Measurable ⇑MeasurableEquiv.prodAssoc.symm

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
δ : Type u_5
mγ : MeasurableSpace γ
mδ : MeasurableSpace δ
κ : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel α γ
inst✝¹ : IsSFiniteKernel η
ξ : Kernel α δ
inst✝ : IsSFiniteKernel ξ
⊢ (κ ×ₖ (ξ ×ₖ η)).map ⇑MeasurableEquiv.prodAssoc.symm = κ ×ₖ ξ ×ₖ η

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: case pos.e_f
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
κ : Kernel α β
η : Kernel (α × β) γ
inst✝¹ : SFinite μ
inst✝ : IsSFiniteKernel η
hκ : IsSFiniteKernel κ
s : Set γ
hs : MeasurableSet s
a : α
⊢ ∫⁻ (b : β), (η (a, b)) s ∂κ a = ((κ ⊗ₖ η) a) (Prod.snd ⁻¹' s)

Tactic:
rw [Kernel.compProd_apply]

Goal state after:
case pos.e_f
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
κ : Kernel α β
η : Kernel (α × β) γ
inst✝¹ : SFinite μ
inst✝ : IsSFiniteKernel η
hκ : IsSFiniteKernel κ
s : Set γ
hs : MeasurableSet s
a : α
⊢ ∫⁻ (b : β), (η (a, b)) s ∂κ a = ∫⁻ (b : β), (η (a, b)) (Prod.mk b ⁻¹' Prod.snd ⁻¹' s) ∂κ a

case pos.e_f.hs
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
μ : Measure α
κ : Kernel α β
η : Kernel (α × β) γ
inst✝¹ : SFinite μ
inst✝ : IsSFiniteKernel η
hκ : IsSFiniteKernel κ
s : Set γ
hs : MeasurableSet s
a : α
⊢ MeasurableSet (Prod.snd ⁻¹' s)
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

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
μ : Measure α
κ : Kernel α β
η : Kernel (α × β) γ
inst✝¹ : SFinite μ
inst✝ : IsSFiniteKernel η
⊢ ⇑η ∘ₘ μ ⊗ₘ κ = (⇑(κ ⊗ₖ η) ∘ₘ μ).snd

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
δ : Type u_6
mδ : MeasurableSpace δ
κ : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel β γ
inst✝¹ : IsSFiniteKernel η
μ : Measure δ
inst✝ : SFinite μ
⊢ η ×ₖ const β μ ∘ₖ κ = η ∘ₖ κ ×ₖ const α μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

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
μ : Measure α
κ : Kernel α β
η : Kernel β γ
inst✝¹ : SFinite μ
inst✝ : IsSFiniteKernel κ
⊢ ⇑(Kernel.prodMkLeft α η) ∘ₘ μ ⊗ₘ κ = ⇑η ∘ₘ ⇑κ ∘ₘ μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
s : Set β
t : Set γ
hs : MeasurableSet s
ht : MeasurableSet t
a : α
u : Set (β × γ)
hu : MeasurableSet u
b : β
⊢ (η (a, b)) {c | (b, c) ∈ u ∧ b ∈ s ∧ c ∈ t} = s.indicator (fun b ↦ (η (a, b)) ({c | (b, c) ∈ u} ∩ t)) b

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
γ : Type u_4
mγ : MeasurableSpace γ
δ : Type u_6
mδ : MeasurableSpace δ
κ : Kernel α β
inst✝² : IsSFiniteKernel κ
μ : Measure γ
inst✝¹ : SFinite μ
η : Kernel β δ
inst✝ : IsSFiniteKernel η
⊢ const β μ ×ₖ η ∘ₖ κ = const α μ ×ₖ (η ∘ₖ κ)

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
inst✝ : SFinite μ
f : α → β
hf : Measurable f
⊢ μ ⊗ₘ Kernel.deterministic f hf = map (fun a ↦ (a, f a)) μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
s : Set β
t : Set γ
hs : MeasurableSet s
ht : MeasurableSet t
⊢ κ.restrict hs ⊗ₖ η.restrict ht = (κ ⊗ₖ η).restrict ⋯

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
s : Set β
hs : MeasurableSet s
⊢ κ.restrict hs ⊗ₖ η = (κ ⊗ₖ η).restrict ⋯

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
t : Set γ
ht : MeasurableSet t
⊢ κ ⊗ₖ η.restrict ht = (κ ⊗ₖ η).restrict ⋯

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ : Kernel α β
⊢ ⇑κ ∘ₘ (μ + ν) = ⇑κ ∘ₘ μ + ⇑κ ∘ₘ ν

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ η : Kernel α β
⊢ ⇑(κ + η) ∘ₘ μ = ⇑κ ∘ₘ μ + ⇑η ∘ₘ μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ η : Kernel α β
⊢ (⇑κ + ⇑η) ∘ₘ μ = ⇑κ ∘ₘ μ + ⇑η ∘ₘ μ

Tactic:
rw [← FunLike.coe_add, add_comp]

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ η : Kernel α β
⊢ (⇑κ + ⇑η) ∘ₘ μ = ⇑κ ∘ₘ μ + ⇑η ∘ₘ μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
a : ℝ≥0∞
⊢ ⇑κ ∘ₘ (a • μ) = a • ⇑κ ∘ₘ μ

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.MeasurableIntegral

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝ : NormedAddCommGroup E
f : β → E
hf : StronglyMeasurable f
⊢ MeasurableSet {a | Integrable f (κ a)}

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.MeasurableIntegral

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : IsSFiniteKernel κ
f : α → β → E
hf : StronglyMeasurable (uncurry f)
⊢ MeasurableSet {x | Integrable (f x) (κ x)}

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
γ : Type u_3
mα : MeasurableSpace α
mγ : MeasurableSpace γ
μ ν : Measure α
hμν : μ ≪ ν
κ : Kernel α γ
⊢ ⇑κ ∘ₘ μ ≪ ⇑κ ∘ₘ ν

Tactic:
by

Goal state after:
Module: Mathlib.Probability.Kernel.MeasurableIntegral

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : β → E
hf : StronglyMeasurable f
hE : CompleteSpace E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : TopologicalSpace.SeparableSpace ↑(range f ∪ {0})
⊢ 0 ∈ range f ∪ {0}

Tactic:
by

Goal state after: