mathlib-initiative/mathlib-tactics huggingface dataset:
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✝ : IsSFiniteKernel κ
η : Kernel (α × β) γ
⊢ 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 =
    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

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ
hs : MeasurableSet s
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ ENNReal.ofReal (∫ (x : α) in s, rexp (f x) / ∫ (x : α), rexp (f x) ∂μ ∂μ) =
    ENNReal.ofReal (∫ (a : α) in s, rexp (f a) / ∫ (x : α), rexp (f x) ∂μ ∂μ)

pp: case pos.hfi
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ
hs : MeasurableSet s
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ Integrable (fun a ↦ rexp (f a) / ∫ (x : α), rexp (f x) ∂μ) (μ.restrict s)

pp: case pos.f_nn
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ
hs : MeasurableSet s
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ 0 ≤ᵐ[μ.restrict s] fun a ↦ rexp (f a) / ∫ (x : α), rexp (f x) ∂μ

Tactic:
]

Goal state after:
case pos.hfi
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ
hs : MeasurableSet s
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ Integrable (fun a ↦ rexp (f a) / ∫ (x : α), rexp (f x) ∂μ) (μ.restrict s)

case pos.f_nn
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ
hs : MeasurableSet s
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ 0 ≤ᵐ[μ.restrict s] fun a ↦ rexp (f a) / ∫ (x : α), rexp (f x) ∂μ
Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun x ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hν : ¬ν = 0
this✝ : NeZero ν
μ' : Measure 𝓧 := (ν univ)⁻¹ • μ
ν' : Measure 𝓧 := (ν univ)⁻¹ • ν
this : IsFiniteMeasure μ'
hμν' : μ' ≪ ν'
h_rnDeriv_eq : μ'.rnDeriv ν' =ᵐ[ν] μ.rnDeriv ν
x : 𝓧
hx : μ'.rnDeriv ν' x = μ.rnDeriv ν x
⊢ f (μ.rnDeriv ν x).toReal = f (μ.rnDeriv ν x).toReal

Tactic:
]

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
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
⊢ 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 =
    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

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun x ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hν : ¬ν = 0
this✝ : NeZero ν
μ' : Measure 𝓧 := (ν univ)⁻¹ • μ
ν' : Measure 𝓧 := (ν univ)⁻¹ • ν
this : IsFiniteMeasure μ'
hμν' : μ' ≪ ν'
h_rnDeriv_eq : μ'.rnDeriv ν' =ᵐ[ν] μ.rnDeriv ν
h_eq : ∫ (x : 𝓧), f (μ'.rnDeriv ν' x).toReal ∂ν' = (ν.real univ)⁻¹ * ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
⊢ (ν univ)⁻¹ ≠ ∞

Tactic:
by simp [hν]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ
s : Set α
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ ENNReal.ofReal (∫ (x : α) in s, rexp (f x) / ∫ (x : α), rexp (f x) ∂μ ∂μ) =
    ENNReal.ofReal (∫ (a : α) in s, rexp (f a) / ∫ (x : α), rexp (f x) ∂μ ∂μ)

pp: case pos.hfi
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ
s : Set α
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ Integrable (fun a ↦ rexp (f a) / ∫ (x : α), rexp (f x) ∂μ) (μ.restrict s)

pp: case pos.f_nn
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ
s : Set α
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ 0 ≤ᵐ[μ.restrict s] fun a ↦ rexp (f a) / ∫ (x : α), rexp (f x) ∂μ

Tactic:
]

Goal state after:
case pos.hfi
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ
s : Set α
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ Integrable (fun a ↦ rexp (f a) / ∫ (x : α), rexp (f x) ∂μ) (μ.restrict s)

case pos.f_nn
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
inst✝ : SFinite μ
f : α → ℝ
s : Set α
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ 0 ≤ᵐ[μ.restrict s] fun a ↦ rexp (f a) / ∫ (x : α), rexp (f x) ∂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)
κ : Kernel α β
inst✝¹ : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsSFiniteKernel η
a : α
hs : MeasurableSet s
⊢ ((Kernel.sum fun n ↦ Kernel.sum fun m ↦ κ.seq n ⊗ₖ η.seq m) a) s = ∑' (n : ℕ) (m : ℕ), ((κ.seq n ⊗ₖ η.seq m) a) 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
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
⊢ ((Kernel.sum fun n ↦ Kernel.sum fun m ↦ κ.seq n ⊗ₖ η.seq m) a) s = ∑' (n : ℕ) (m : ℕ), ((κ.seq n ⊗ₖ η.seq m) a) s
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 : IsZeroOrMarkovKernel κ
η : Kernel (α × β) γ
inst✝ : IsZeroOrMarkovKernel η
⊢ IsZeroOrMarkovKernel
    (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)

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
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsZeroOrMarkovKernel κ
η : Kernel (α × β) γ
inst✝ : IsZeroOrMarkovKernel η
⊢ IsZeroOrMarkovKernel
    (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)
Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv

Goal state before:
pp: 𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun x ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hν : ¬ν = 0
this✝ : NeZero ν
μ' : Measure 𝓧 := (ν univ)⁻¹ • μ
ν' : Measure 𝓧 := (ν univ)⁻¹ • ν
this : IsFiniteMeasure μ'
hμν' : μ' ≪ ν'
h_rnDeriv_eq : μ'.rnDeriv ν' =ᵐ[ν] μ.rnDeriv ν
h_eq : ∫ (x : 𝓧), f (μ'.rnDeriv ν' x).toReal ∂ν' = (ν.real univ)⁻¹ * ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
x : 𝓧
hx : μ'.rnDeriv ν' x = μ.rnDeriv ν x
⊢ f (μ.rnDeriv ν x).toReal = f (μ.rnDeriv ν x).toReal

Tactic:
]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
inst✝ : NeZero μ
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ 1 = 1

pp: case h0
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
inst✝ : NeZero μ
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ ENNReal.ofReal (∫ (x : α), rexp (f x) ∂μ) ≠ 0

pp: case ht
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
inst✝ : NeZero μ
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ ENNReal.ofReal (∫ (x : α), rexp (f x) ∂μ) ≠ ∞

Tactic:
]

Goal state after:
case h0
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
inst✝ : NeZero μ
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ ENNReal.ofReal (∫ (x : α), rexp (f x) ∂μ) ≠ 0

case ht
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
inst✝ : NeZero μ
hf : Integrable (fun x ↦ rexp (f x)) μ
⊢ ENNReal.ofReal (∫ (x : α), rexp (f x) ∂μ) ≠ ∞
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
κ : Kernel α β
η : Kernel (α × β) γ
inst✝ : IsFiniteKernel η
a : α
hκ : ¬IsSFiniteKernel κ
⊢ (0 a) Set.univ ≤ (κ a) Set.univ * η.bound

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
κ : Kernel α β
η : Kernel (α × β) γ
inst✝ : IsFiniteKernel η
a : α
hκ : ¬IsSFiniteKernel κ
⊢ (0 a) Set.univ ≤ (κ a) Set.univ * η.bound
Module: Mathlib.Probability.Kernel.Composition.CompProd

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
κ : Kernel α β
η : Kernel (α × β) γ
inst✝ : IsFiniteKernel η
a : α
hκ : IsSFiniteKernel κ
⊢ ∫⁻ (b : β), (η (a, b)) (Prod.mk b ⁻¹' Set.univ) ∂κ a ≤ (κ a) Set.univ * η.bound

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
κ : Kernel α β
η : Kernel (α × β) γ
inst✝ : IsFiniteKernel η
a : α
hκ : IsSFiniteKernel κ
⊢ ∫⁻ (b : β), (η (a, b)) (Prod.mk b ⁻¹' Set.univ) ∂κ a ≤ (κ a) Set.univ * η.bound
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : AEMeasurable f μ
⊢ ∫⁻ (a : α) in s, ((fun x ↦ ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ)) * g) a ∂μ =
    ∫⁻ (x : α) in s, ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * g x ∂μ

pp: case pos.hf
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : AEMeasurable f μ
⊢ AEMeasurable (fun x ↦ ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ)) (μ.restrict s)

pp: case pos.hs
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : AEMeasurable f μ
⊢ MeasurableSet s

pp: case pos.h'f
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : AEMeasurable f μ
⊢ ∀ᵐ (x : α) ∂μ.restrict s, ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) < ∞

Tactic:
]

Goal state after:
case pos
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : AEMeasurable f μ
⊢ ∫⁻ (a : α) in s, ((fun x ↦ ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ)) * g) a ∂μ =
    ∫⁻ (x : α) in s, ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * g x ∂μ

case pos.hf
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : AEMeasurable f μ
⊢ AEMeasurable (fun x ↦ ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ)) (μ.restrict s)

case pos.hs
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : AEMeasurable f μ
⊢ MeasurableSet s

case pos.h'f
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : AEMeasurable f μ
⊢ ∀ᵐ (x : α) ∂μ.restrict s, ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂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)
κ : Kernel α β
inst✝¹ : IsFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsFiniteKernel η
⊢ IsFiniteKernel
    (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)

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
s : Set (β × γ)
κ : Kernel α β
inst✝¹ : IsFiniteKernel κ
η : Kernel (α × β) γ
inst✝ : IsFiniteKernel η
⊢ IsFiniteKernel
    (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)
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
η : Kernel (α × β) γ
⊢ IsSFiniteKernel
    (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)

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
s : Set (β × γ)
κ : Kernel α β
η : Kernel (α × β) γ
⊢ IsSFiniteKernel
    (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)
Module: Mathlib.Probability.Kernel.Composition.IntegralCompProd

Goal state before:
pp: case h_ae
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝³ : NormedAddCommGroup E
a : α
κ : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝¹ : IsSFiniteKernel η
inst✝ : NormedSpace ℝ E
f✝ : β × γ → E
hE : CompleteSpace E
f g : β × γ → E
hfg : f =ᵐ[(κ ⊗ₖ η) a] g
a✝ : Integrable f ((κ ⊗ₖ η) a)
hf : ∫ (z : β × γ), f z ∂(κ ⊗ₖ η) a = ∫ (x : β), ∫ (y : γ), f (x, y) ∂η (a, x) ∂κ a
⊢ ∫ (z : β × γ), g z ∂(κ ⊗ₖ η) a = ∫ (x : β), ∫ (y : γ), g (x, y) ∂η (a, x) ∂κ a

Tactic:
convert! hf using 1

Goal state after:
case e'_2
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝³ : NormedAddCommGroup E
a : α
κ : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝¹ : IsSFiniteKernel η
inst✝ : NormedSpace ℝ E
f✝ : β × γ → E
hE : CompleteSpace E
f g : β × γ → E
hfg : f =ᵐ[(κ ⊗ₖ η) a] g
a✝ : Integrable f ((κ ⊗ₖ η) a)
hf : ∫ (z : β × γ), f z ∂(κ ⊗ₖ η) a = ∫ (x : β), ∫ (y : γ), f (x, y) ∂η (a, x) ∂κ a
⊢ ∫ (z : β × γ), g z ∂(κ ⊗ₖ η) a = ∫ (z : β × γ), f z ∂(κ ⊗ₖ η) a

case e'_3
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝³ : NormedAddCommGroup E
a : α
κ : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝¹ : IsSFiniteKernel η
inst✝ : NormedSpace ℝ E
f✝ : β × γ → E
hE : CompleteSpace E
f g : β × γ → E
hfg : f =ᵐ[(κ ⊗ₖ η) a] g
a✝ : Integrable f ((κ ⊗ₖ η) a)
hf : ∫ (z : β × γ), f z ∂(κ ⊗ₖ η) a = ∫ (x : β), ∫ (y : γ), f (x, y) ∂η (a, x) ∂κ a
⊢ ∫ (x : β), ∫ (y : γ), g (x, y) ∂η (a, x) ∂κ a = ∫ (x : β), ∫ (y : γ), f (x, y) ∂η (a, x) ∂κ a
Module: Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv

Goal state before:
pp: case neg.refine_2
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun x ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hν : ¬ν = 0
this✝ : NeZero ν
μ' : Measure 𝓧 := (ν univ)⁻¹ • μ
ν' : Measure 𝓧 := (ν univ)⁻¹ • ν
this : IsFiniteMeasure μ'
hμν' : μ' ≪ ν'
h_rnDeriv_eq : μ'.rnDeriv ν' =ᵐ[ν] μ.rnDeriv ν
h_eq : ∫ (x : 𝓧), f (μ'.rnDeriv ν' x).toReal ∂ν' = (ν.real univ)⁻¹ * ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
h : ν.real univ * f (μ'.real univ) ≤ ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
⊢ ν.real univ * f (μ.real univ / ν.real univ) ≤ ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν

pp: case neg.refine_2
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun x ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hν : ¬ν = 0
this✝ : NeZero ν
μ' : Measure 𝓧 := (ν univ)⁻¹ • μ
ν' : Measure 𝓧 := (ν univ)⁻¹ • ν
this : IsFiniteMeasure μ'
hμν' : μ' ≪ ν'
h_rnDeriv_eq : μ'.rnDeriv ν' =ᵐ[ν] μ.rnDeriv ν
h_eq : ∫ (x : 𝓧), f (μ'.rnDeriv ν' x).toReal ∂ν' = (ν.real univ)⁻¹ * ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
h : f (μ'.real univ) ≤ (∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν) * (ν.real univ)⁻¹
⊢ 0 < (ν.real univ)⁻¹

Tactic:
]

Goal state after:
case neg.refine_2
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun x ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hν : ¬ν = 0
this✝ : NeZero ν
μ' : Measure 𝓧 := (ν univ)⁻¹ • μ
ν' : Measure 𝓧 := (ν univ)⁻¹ • ν
this : IsFiniteMeasure μ'
hμν' : μ' ≪ ν'
h_rnDeriv_eq : μ'.rnDeriv ν' =ᵐ[ν] μ.rnDeriv ν
h_eq : ∫ (x : 𝓧), f (μ'.rnDeriv ν' x).toReal ∂ν' = (ν.real univ)⁻¹ * ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
h : ν.real univ * f (μ'.real univ) ≤ ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
⊢ ν.real univ * f (μ.real univ / ν.real univ) ≤ ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν

case neg.refine_2
𝓧 : Type u_1
m𝓧 : MeasurableSpace 𝓧
μ ν : Measure 𝓧
f : ℝ → ℝ
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hf_cvx : ConvexOn ℝ (Ici 0) f
hf_cont : ContinuousWithinAt f (Ici 0) 0
hf_int : Integrable (fun x ↦ f (μ.rnDeriv ν x).toReal) ν
hμν : μ ≪ ν
hν : ¬ν = 0
this✝ : NeZero ν
μ' : Measure 𝓧 := (ν univ)⁻¹ • μ
ν' : Measure 𝓧 := (ν univ)⁻¹ • ν
this : IsFiniteMeasure μ'
hμν' : μ' ≪ ν'
h_rnDeriv_eq : μ'.rnDeriv ν' =ᵐ[ν] μ.rnDeriv ν
h_eq : ∫ (x : 𝓧), f (μ'.rnDeriv ν' x).toReal ∂ν' = (ν.real univ)⁻¹ * ∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν
h : f (μ'.real univ) ≤ (∫ (x : 𝓧), f (μ.rnDeriv ν x).toReal ∂ν) * (ν.real univ)⁻¹
⊢ 0 < (ν.real univ)⁻¹
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: case pos.hf
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : AEMeasurable f μ
⊢ AEMeasurable (fun x ↦ ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ)) (μ.restrict s)

Tactic:
refine AEMeasurable.restrict ?_
      exact ((measurable_exp.comp_aemeasurable hf).div_const _).ennreal_ofReal

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: case pos.hs
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : AEMeasurable f μ
⊢ MeasurableSet s

Tactic:
exact hs

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Tilted

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : ¬AEMeasurable f μ
hf' : ¬Integrable (fun x ↦ rexp (f x)) μ
⊢ 0 = ∫⁻ (x : α) in s, ENNReal.ofReal (rexp (f x) / 0) * g x ∂μ

Tactic:
]

Goal state after:
case neg
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
f : α → ℝ
g : α → ℝ≥0∞
s : Set α
hs : MeasurableSet s
hf : ¬AEMeasurable f μ
hf' : ¬Integrable (fun x ↦ rexp (f x)) μ
⊢ 0 = ∫⁻ (x : α) in s, ENNReal.ofReal (rexp (f x) / 0) * g x ∂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
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
η : Kernel (α × β) γ
ξ : Kernel (α × β × γ) δ
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
hξ : ¬IsSFiniteKernel ξ
h_sfin : IsSFiniteKernel (ξ.comap ⇑MeasurableEquiv.prodAssoc ⋯)
this : ξ = (ξ.comap ⇑MeasurableEquiv.prodAssoc ⋯).comap ⇑MeasurableEquiv.prodAssoc.symm ⋯
⊢ IsSFiniteKernel ((ξ.comap ⇑MeasurableEquiv.prodAssoc ⋯).comap ⇑MeasurableEquiv.prodAssoc.symm ⋯)

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
δ : Type u_4
mδ : MeasurableSpace δ
κ : Kernel α β
η : Kernel (α × β) γ
ξ : Kernel (α × β × γ) δ
hκ : IsSFiniteKernel κ
hη : IsSFiniteKernel η
hξ : ¬IsSFiniteKernel ξ
h_sfin : IsSFiniteKernel (ξ.comap ⇑MeasurableEquiv.prodAssoc ⋯)
this : ξ = (ξ.comap ⇑MeasurableEquiv.prodAssoc ⋯).comap ⇑MeasurableEquiv.prodAssoc.symm ⋯
⊢ IsSFiniteKernel ((ξ.comap ⇑MeasurableEquiv.prodAssoc ⋯).comap ⇑MeasurableEquiv.prodAssoc.symm ⋯)
Module: Mathlib.Probability.Kernel.Composition.IntegralCompProd

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝³ : NormedAddCommGroup E
a : α
κ : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝¹ : IsSFiniteKernel η
inst✝ : NormedSpace ℝ E
f : β × γ → E
s : Set β
t : Set γ
hs : MeasurableSet s
ht : MeasurableSet t
hf : IntegrableOn f (s ×ˢ t) ((κ ⊗ₖ η) a)
⊢ ∫ (x : β), ∫ (y : γ), f (x, y) ∂(η.restrict ht) (a, x) ∂(κ.restrict hs) a =
    ∫ (x : β) in s, ∫ (y : γ) in t, f (x, y) ∂η (a, x) ∂κ a

pp: α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝³ : NormedAddCommGroup E
a : α
κ : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝¹ : IsSFiniteKernel η
inst✝ : NormedSpace ℝ E
f : β × γ → E
s : Set β
t : Set γ
hs : MeasurableSet s
ht : MeasurableSet t
hf : IntegrableOn f (s ×ˢ t) ((κ ⊗ₖ η) a)
⊢ Integrable f ((κ.restrict hs ⊗ₖ η.restrict ht) a)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝³ : NormedAddCommGroup E
a : α
κ : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝¹ : IsSFiniteKernel η
inst✝ : NormedSpace ℝ E
f : β × γ → E
s : Set β
t : Set γ
hs : MeasurableSet s
ht : MeasurableSet t
hf : IntegrableOn f (s ×ˢ t) ((κ ⊗ₖ η) a)
⊢ ∫ (x : β), ∫ (y : γ), f (x, y) ∂(η.restrict ht) (a, x) ∂(κ.restrict hs) a =
    ∫ (x : β) in s, ∫ (y : γ) in t, f (x, y) ∂η (a, x) ∂κ a

α : Type u_1
β : Type u_2
γ : Type u_3
E : Type u_4
mα : MeasurableSpace α
mβ : MeasurableSpace β
mγ : MeasurableSpace γ
inst✝³ : NormedAddCommGroup E
a : α
κ : Kernel α β
inst✝² : IsSFiniteKernel κ
η : Kernel (α × β) γ
inst✝¹ : IsSFiniteKernel η
inst✝ : NormedSpace ℝ E
f : β × γ → E
s : Set β
t : Set γ
hs : MeasurableSet s
ht : MeasurableSet t
hf : IntegrableOn f (s ×ˢ t) ((κ ⊗ₖ η) a)
⊢ Integrable f ((κ.restrict hs ⊗ₖ η.restrict ht) a)