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

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
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
inst✝ : IsSFiniteKernel κ
⊢ IsSFiniteKernel (Kernel.prodMkRight γ κ)

Tactic:
by rw [Kernel.prodMkRight]; infer_instance

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

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
δ : Type u_4
mδ : MeasurableSpace δ
γ : Type u_5
inst✝ : MeasurableSpace γ
κ : Kernel α β
f : β → δ
hf : Measurable f
⊢ (prodMkLeft γ κ).map f = prodMkLeft γ (κ.map f)

pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
δ : Type u_4
mδ : MeasurableSpace δ
γ : Type u_5
inst✝ : MeasurableSpace γ
κ : Kernel α β
f : β → δ
hf : ¬Measurable f
⊢ (prodMkLeft γ κ).map f = prodMkLeft γ (κ.map f)

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
δ : Type u_4
mδ : MeasurableSpace δ
γ : Type u_5
inst✝ : MeasurableSpace γ
κ : Kernel α β
f : β → δ
hf : Measurable f
⊢ (prodMkLeft γ κ).map f = prodMkLeft γ (κ.map f)
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 : Measure α
κ : Kernel α β
inst✝¹ : SFinite μ
inst✝ : IsSFiniteKernel κ
f : α × β → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : Set β
ht : MeasurableSet t
⊢ ∫⁻ (x : α × β) in s ×ˢ t, f x ∂μ ⊗ₘ κ = ∫⁻ (a : α) in s, ∫⁻ (b : β) in t, f (a, b) ∂κ a ∂μ

Tactic:
compProd,

Goal state after:
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
inst✝¹ : SFinite μ
inst✝ : IsSFiniteKernel κ
f : α × β → ℝ≥0∞
hf : Measurable f
s : Set α
hs : MeasurableSet s
t : Set β
ht : MeasurableSet t
⊢ ∫⁻ (x : α × β) in s ×ˢ t, f x ∂(Kernel.const Unit μ ⊗ₖ Kernel.prodMkLeft Unit κ) () =
    ∫⁻ (a : α) in s, ∫⁻ (b : β) in t, f (a, b) ∂κ a ∂μ
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)
hs : MeasurableSet s
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
⊢ ((κ ⊗ₖ η) a) s = ∫⁻ (b : β), (η (a, b)) (Prod.mk b ⁻¹' s) ∂κ a

Tactic:
compProd,

Goal state after:
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)
hs : MeasurableSet s
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
⊢ ((swap γ β ∘ₖ (η ∥ₖ Kernel.id) ∘ₖ deterministic ⇑MeasurableEquiv.prodAssoc.symm ⋯ ∘ₖ (Kernel.id ∥ₖ copy β) ∘ₖ
            (Kernel.id ∥ₖ κ) ∘ₖ
          copy α)
        a)
      s =
    ∫⁻ (b : β), (η (a, b)) (Prod.mk b ⁻¹' s) ∂κ a
Module: Mathlib.Probability.Kernel.Composition.MapComap

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
δ : Type u_4
mδ : MeasurableSpace δ
γ : Type u_5
inst✝ : MeasurableSpace γ
κ : Kernel α β
f : β → δ
hf : ¬Measurable f
⊢ (prodMkLeft γ κ).map f = prodMkLeft γ (κ.map f)

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
δ : Type u_4
mδ : MeasurableSpace δ
γ : Type u_5
inst✝ : MeasurableSpace γ
κ : Kernel α β
f : β → δ
hf : ¬Measurable f
⊢ (prodMkLeft γ κ).map f = prodMkLeft γ (κ.map f)
Module: Mathlib.Probability.Kernel.Composition.MapComap

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
γ : Type u_5
mγ : MeasurableSpace γ
f : β → δ
hf : Measurable f
⊢ (prodMkRight γ κ).map f = prodMkRight γ (κ.map f)

pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
γ : Type u_5
mγ : MeasurableSpace γ
f : β → δ
hf : ¬Measurable f
⊢ (prodMkRight γ κ).map f = prodMkRight γ (κ.map f)

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
γ : Type u_5
mγ : MeasurableSpace γ
f : β → δ
hf : Measurable f
⊢ (prodMkRight γ κ).map f = prodMkRight γ (κ.map f)
Module: Mathlib.Probability.Kernel.Composition.MapComap

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
γ : Type u_5
mγ : MeasurableSpace γ
f : β → δ
hf : ¬Measurable f
⊢ (prodMkRight γ κ).map f = prodMkRight γ (κ.map f)

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
γ : Type u_5
mγ : MeasurableSpace γ
f : β → δ
hf : ¬Measurable f
⊢ (prodMkRight γ κ).map f = prodMkRight γ (κ.map f)
Module: Mathlib.Probability.Kernel.Composition.CompMap

Goal state before:
pp: case pos
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 β γ
f : γ → δ
hf : Measurable f
⊢ (η ∘ₖ κ).map f = η.map f ∘ₖ κ

pp: case neg
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 β γ
f : γ → δ
hf : ¬Measurable f
⊢ (η ∘ₖ κ).map f = η.map f ∘ₖ κ

Tactic:
·

Goal state after:
case pos
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 β γ
f : γ → δ
hf : Measurable f
⊢ (η ∘ₖ κ).map f = η.map f ∘ₖ κ
Module: Mathlib.Probability.Kernel.Composition.MapComap

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
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α (β × γ)
inst✝ : IsMarkovKernel κ
⊢ IsMarkovKernel κ.swapRight

Tactic:
by
  rw [Kernel.swapRight_eq]; exact IsMarkovKernel.map _ measurable_swap

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

Goal state before:
pp: case pos
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 β γ
f : γ → δ
hf : Measurable f
a : α
s : Set δ
hs : MeasurableSet s
⊢ ∫⁻ (b : β), (η b) (f ⁻¹' s) ∂κ a = ∫⁻ (b : β), ((η.map f) b) s ∂κ a

pp: case pos.hs
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 β γ
f : γ → δ
hf : Measurable f
a : α
s : Set δ
hs : MeasurableSet s
⊢ MeasurableSet (f ⁻¹' s)

Tactic:
·

Goal state after:
case pos
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 β γ
f : γ → δ
hf : Measurable f
a : α
s : Set δ
hs : MeasurableSet s
⊢ ∫⁻ (b : β), (η b) (f ⁻¹' s) ∂κ a = ∫⁻ (b : β), ((η.map f) b) s ∂κ a
Module: Mathlib.Probability.Kernel.Composition.CompMap

Goal state before:
pp: case pos.hs
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 β γ
f : γ → δ
hf : Measurable f
a : α
s : Set δ
hs : MeasurableSet s
⊢ MeasurableSet (f ⁻¹' s)

Tactic:
·

Goal state after:
case pos.hs
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 β γ
f : γ → δ
hf : Measurable f
a : α
s : Set δ
hs : MeasurableSet s
⊢ MeasurableSet (f ⁻¹' s)
Module: Mathlib.Probability.Kernel.Composition.MeasureCompProd

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
γ : Type u_3
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
hμ : ¬SFinite μ
⊢ map (⇑MeasurableEquiv.prodAssoc.symm) (μ ⊗ₘ (κ ⊗ₖ η)) = μ ⊗ₘ κ ⊗ₘ η

pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
γ : Type u_3
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
hμ : SFinite μ
⊢ map (⇑MeasurableEquiv.prodAssoc.symm) (μ ⊗ₘ (κ ⊗ₖ η)) = μ ⊗ₘ κ ⊗ₘ η

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
γ : Type u_3
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
hμ : ¬SFinite μ
⊢ map (⇑MeasurableEquiv.prodAssoc.symm) (μ ⊗ₘ (κ ⊗ₖ η)) = μ ⊗ₘ κ ⊗ₘ η
Module: Mathlib.Probability.Kernel.Composition.CompMap

Goal state before:
pp: case neg
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 β γ
f : γ → δ
hf : ¬Measurable f
⊢ (η ∘ₖ κ).map f = η.map f ∘ₖ κ

Tactic:
·

Goal state after:
case neg
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 β γ
f : γ → δ
hf : ¬Measurable f
⊢ (η ∘ₖ κ).map f = η.map f ∘ₖ κ
Module: Mathlib.Probability.Kernel.Composition.MeasureCompProd

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
γ : Type u_3
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
hμ : SFinite μ
hκ : ¬IsSFiniteKernel κ
⊢ map (⇑MeasurableEquiv.prodAssoc.symm) (μ ⊗ₘ (κ ⊗ₖ η)) = μ ⊗ₘ κ ⊗ₘ η

pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
γ : Type u_3
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
hμ : SFinite μ
hκ : IsSFiniteKernel κ
⊢ map (⇑MeasurableEquiv.prodAssoc.symm) (μ ⊗ₘ (κ ⊗ₖ η)) = μ ⊗ₘ κ ⊗ₘ η

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
γ : Type u_3
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
hμ : SFinite μ
hκ : ¬IsSFiniteKernel κ
⊢ map (⇑MeasurableEquiv.prodAssoc.symm) (μ ⊗ₘ (κ ⊗ₖ η)) = μ ⊗ₘ κ ⊗ₘ η
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
s : Set (β × γ)
hs : MeasurableSet s
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
⊢ Measurable fun x ↦
    ∫⁻ (y : β),
      ((swap γ β ∘ₖ (η ∥ₖ Kernel.id) ∘ₖ deterministic ⇑MeasurableEquiv.prodAssoc.symm ⋯ ∘ₖ (Kernel.id ∥ₖ copy β))
          (x, y))
        s ∂κ (a, a).2

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)
hs : MeasurableSet s
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
⊢ ∫⁻ (y : β),
      ((swap γ β ∘ₖ (η ∥ₖ Kernel.id) ∘ₖ deterministic ⇑MeasurableEquiv.prodAssoc.symm ⋯ ∘ₖ (Kernel.id ∥ₖ copy β))
          ((a, a).1, y))
        s ∂κ (a, a).2 =
    ∫⁻ (b : β), (η (a, b)) (Prod.mk b ⁻¹' s) ∂κ a

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
s : Set (β × γ)
hs : MeasurableSet s
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
⊢ Measurable fun x ↦
    ∫⁻ (y : β),
      ((swap γ β ∘ₖ (η ∥ₖ Kernel.id) ∘ₖ deterministic ⇑MeasurableEquiv.prodAssoc.symm ⋯ ∘ₖ (Kernel.id ∥ₖ copy β))
          (x, y))
        s ∂κ (a, a).2
Module: Mathlib.Probability.Kernel.Composition.MapComap

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
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hg : Measurable g
hf : Measurable f
⊢ (κ.map fun x ↦ (f x, g x)).fst = κ.map f

pp: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hg : Measurable g
hf : ¬Measurable f
⊢ (κ.map fun x ↦ (f x, g x)).fst = κ.map f

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
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hg : Measurable g
hf : Measurable f
⊢ (κ.map fun x ↦ (f x, g x)).fst = κ.map f
Module: Mathlib.Probability.Kernel.Composition.MapComap

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
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hg : Measurable g
hf : Measurable f
x : α
s : Set γ
hs : MeasurableSet s
⊢ (κ x) ((fun x ↦ (f x, g x)) ⁻¹' {p | p.1 ∈ s}) = (κ x) (f ⁻¹' s)

pp: case pos
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hg : Measurable g
hf : Measurable f
x : α
s : Set γ
hs : MeasurableSet s
⊢ MeasurableSet {p | p.1 ∈ 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
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hg : Measurable g
hf : Measurable f
x : α
s : Set γ
hs : MeasurableSet s
⊢ (κ x) ((fun x ↦ (f x, g x)) ⁻¹' {p | p.1 ∈ s}) = (κ x) (f ⁻¹' s)
Module: Mathlib.Probability.Kernel.Composition.MapComap

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
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hg : Measurable g
hf : Measurable f
x : α
s : Set γ
hs : MeasurableSet s
⊢ MeasurableSet {p | p.1 ∈ 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
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hg : Measurable g
hf : Measurable f
x : α
s : Set γ
hs : MeasurableSet s
⊢ MeasurableSet {p | p.1 ∈ s}
Module: Mathlib.Probability.Kernel.Composition.MapComap

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
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hg : Measurable g
hf : ¬Measurable f
⊢ (κ.map fun x ↦ (f x, g x)).fst = κ.map f

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
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hg : Measurable g
hf : ¬Measurable f
⊢ (κ.map fun x ↦ (f x, g x)).fst = κ.map f
Module: Mathlib.Probability.Kernel.Composition.MeasureCompProd

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
γ : Type u_3
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
hμ : SFinite μ
hκ : IsSFiniteKernel κ
hη : ¬IsSFiniteKernel η
⊢ map (⇑MeasurableEquiv.prodAssoc.symm) (μ ⊗ₘ (κ ⊗ₖ η)) = μ ⊗ₘ κ ⊗ₘ η

pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
γ : Type u_3
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
hμ : SFinite μ
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
⊢ map (⇑MeasurableEquiv.prodAssoc.symm) (μ ⊗ₘ (κ ⊗ₖ η)) = μ ⊗ₘ κ ⊗ₘ η

Tactic:
·

Goal state after:
case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
γ : Type u_3
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
hμ : SFinite μ
hκ : IsSFiniteKernel κ
hη : ¬IsSFiniteKernel η
⊢ map (⇑MeasurableEquiv.prodAssoc.symm) (μ ⊗ₘ (κ ⊗ₖ η)) = μ ⊗ₘ κ ⊗ₘ η
Module: Mathlib.Probability.Kernel.Composition.CompMap

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
f : β → γ
hf : Measurable f
x : α
s : Set δ
ms : MeasurableSet s
⊢ ∫⁻ (a : β), (η (f a)) s ∂κ x = ∫⁻ (b : β), ((η.comap f hf) b) s ∂κ x

Tactic:
simp_rw [comap_apply']

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

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
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hf : Measurable f
hg : Measurable g
⊢ (κ.map fun x ↦ (f x, g x)).snd = κ.map g

pp: case neg
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hf : Measurable f
hg : ¬Measurable g
⊢ (κ.map fun x ↦ (f x, g x)).snd = κ.map g

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
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hf : Measurable f
hg : Measurable g
⊢ (κ.map fun x ↦ (f x, g x)).snd = κ.map g
Module: Mathlib.Probability.Kernel.Composition.MeasureCompProd

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
γ : Type u_3
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
hμ : SFinite μ
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
s : Set ((α × β) × γ)
hs : MeasurableSet s
⊢ Measurable fun a ↦ (η a) (Prod.mk a ⁻¹' s)

pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
γ : Type u_3
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
hμ : SFinite μ
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
s : Set ((α × β) × γ)
hs : MeasurableSet s
⊢ ∫⁻ (a : α), ((κ ⊗ₖ η) a) (Prod.mk a ⁻¹' ⇑MeasurableEquiv.prodAssoc.symm ⁻¹' s) ∂μ =
    ∫⁻ (a : α), ∫⁻ (b : β), (η (a, b)) (Prod.mk (a, b) ⁻¹' s) ∂κ a ∂μ

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
γ : Type u_3
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
hμ : SFinite μ
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
s : Set ((α × β) × γ)
hs : MeasurableSet s
⊢ Measurable fun a ↦ (η a) (Prod.mk a ⁻¹' s)
Module: Mathlib.Probability.Kernel.Composition.MapComap

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
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hf : Measurable f
hg : Measurable g
x : α
s : Set δ
hs : MeasurableSet s
⊢ (κ x) ((fun x ↦ (f x, g x)) ⁻¹' Prod.snd ⁻¹' s) = (κ x) (g ⁻¹' s)

pp: case pos.hs
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hf : Measurable f
hg : Measurable g
x : α
s : Set δ
hs : MeasurableSet s
⊢ MeasurableSet (Prod.snd ⁻¹' 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
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hf : Measurable f
hg : Measurable g
x : α
s : Set δ
hs : MeasurableSet s
⊢ (κ x) ((fun x ↦ (f x, g x)) ⁻¹' Prod.snd ⁻¹' s) = (κ x) (g ⁻¹' s)
Module: Mathlib.Probability.Kernel.Composition.MapComap

Goal state before:
pp: case pos.hs
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hf : Measurable f
hg : Measurable g
x : α
s : Set δ
hs : MeasurableSet s
⊢ MeasurableSet (Prod.snd ⁻¹' s)

Tactic:
·

Goal state after:
case pos.hs
α : Type u_1
β : Type u_2
γ : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hf : Measurable f
hg : Measurable g
x : α
s : Set δ
hs : MeasurableSet s
⊢ MeasurableSet (Prod.snd ⁻¹' s)
Module: Mathlib.Probability.Kernel.Composition.MapComap

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
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hf : Measurable f
hg : ¬Measurable g
⊢ (κ.map fun x ↦ (f x, g x)).snd = κ.map g

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
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
f : β → γ
g : β → δ
hf : Measurable f
hg : ¬Measurable g
⊢ (κ.map fun x ↦ (f x, g x)).snd = κ.map g
Module: Mathlib.Probability.Kernel.Composition.MeasureCompProd

Goal state before:
pp: case pos.e_f
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
γ : Type u_3
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
hμ : SFinite μ
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
s : Set ((α × β) × γ)
hs : MeasurableSet s
a : α
⊢ ∫⁻ (b : β), (η (a, b)) (Prod.mk b ⁻¹' Prod.mk a ⁻¹' ⇑MeasurableEquiv.prodAssoc.symm ⁻¹' s) ∂κ a =
    ∫⁻ (b : β), (η (a, b)) (Prod.mk (a, b) ⁻¹' s) ∂κ a

pp: case pos.e_f.hs
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
γ : Type u_3
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
hμ : SFinite μ
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
s : Set ((α × β) × γ)
hs : MeasurableSet s
a : α
⊢ MeasurableSet (Prod.mk a ⁻¹' ⇑MeasurableEquiv.prodAssoc.symm ⁻¹' s)

Tactic:
·

Goal state after:
case pos.e_f
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
γ : Type u_3
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
hμ : SFinite μ
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
s : Set ((α × β) × γ)
hs : MeasurableSet s
a : α
⊢ ∫⁻ (b : β), (η (a, b)) (Prod.mk b ⁻¹' Prod.mk a ⁻¹' ⇑MeasurableEquiv.prodAssoc.symm ⁻¹' s) ∂κ a =
    ∫⁻ (b : β), (η (a, b)) (Prod.mk (a, b) ⁻¹' s) ∂κ a
Module: Mathlib.Probability.Kernel.Composition.MeasureCompProd

Goal state before:
pp: case pos.e_f.hs
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
γ : Type u_3
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
hμ : SFinite μ
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
s : Set ((α × β) × γ)
hs : MeasurableSet s
a : α
⊢ MeasurableSet (Prod.mk a ⁻¹' ⇑MeasurableEquiv.prodAssoc.symm ⁻¹' s)

Tactic:
·

Goal state after:
case pos.e_f.hs
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ : Kernel α β
γ : Type u_3
mγ : MeasurableSpace γ
η : Kernel (α × β) γ
hμ : SFinite μ
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
s : Set ((α × β) × γ)
hs : MeasurableSet s
a : α
⊢ MeasurableSet (Prod.mk a ⁻¹' ⇑MeasurableEquiv.prodAssoc.symm ⁻¹' s)
Module: Mathlib.Probability.Kernel.Composition.MeasureCompProd

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
inst✝ : SFinite ν
hμν : μ ≪ ν
κ : Kernel α β
hκ : IsSFiniteKernel κ
⊢ μ ⊗ₘ κ ≪ ν ⊗ₘ κ

pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
inst✝ : SFinite ν
hμν : μ ≪ ν
κ : Kernel α β
hκ : ¬IsSFiniteKernel κ
⊢ μ ⊗ₘ κ ≪ ν ⊗ₘ κ

Tactic:
·

Goal state after:
case pos
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
inst✝ : SFinite ν
hμν : μ ≪ ν
κ : Kernel α β
hκ : IsSFiniteKernel κ
⊢ μ ⊗ₘ κ ≪ ν ⊗ₘ κ