-- Mathlib/Probability/Kernel/Composition/CompProd.lean
theorem lintegral_compProd' (κ : Kernel α β) [IsSFiniteKernel κ] (η : Kernel (α × β) γ)
    [IsSFiniteKernel η] (a : α) {f : β → γ → ℝ≥0∞} (hf : Measurable (Function.uncurry f)) :
    ∫⁻ bc, f bc.1 bc.2 ∂(κ ⊗ₖ η) a = ∫⁻ b, ∫⁻ c, f b c ∂η (a, b) ∂κ a

Proof state:
case e_s.h.refine_2
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
a : α
f✝ : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f✝)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f✝)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f✝ a b
h_mono : Monotone F
this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, (F n) (b, c) ∂η (a, b) = ⨆ n, ∫⁻ (c : γ), (F n) (b, c) ∂η (a, b)
h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), f' (b, c) ∂η (a, b)
n : ℕ
f f' : SimpleFunc (β × γ) ℝ≥0∞
a✝ : Disjoint (Function.support ⇑f) (Function.support ⇑f')
hf_eq : ∫⁻ (a : β × γ), f a ∂(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), f (a_1, c) ∂η (a, a_1) ∂κ a
hf'_eq : ∫⁻ (a : β × γ), f' a ∂(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), f' (a_1, c) ∂η (a, a_1) ∂κ a
⊢ ∫⁻ (a : β × γ), f a + f' a ∂(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), f (a_1, c) + f' (a_1, c) ∂η (a, a_1) ∂κ a

Tactic:
change
  ∫⁻ x, (f : β × γ → ℝ≥0∞) x + f' x ∂(κ ⊗ₖ η) a =
    ∫⁻ b, ∫⁻ c : γ, f (b, c) + f' (b, c) ∂η (a, b) ∂κ a

Result:
case e_s.h.refine_2
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
a : α
f✝ : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f✝)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f✝)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f✝ a b
h_mono : Monotone F
this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, (F n) (b, c) ∂η (a, b) = ⨆ n, ∫⁻ (c : γ), (F n) (b, c) ∂η (a, b)
h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), f' (b, c) ∂η (a, b)
n : ℕ
f f' : SimpleFunc (β × γ) ℝ≥0∞
a✝ : Disjoint (Function.support ⇑f) (Function.support ⇑f')
hf_eq : ∫⁻ (a : β × γ), f a ∂(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), f (a_1, c) ∂η (a, a_1) ∂κ a
hf'_eq : ∫⁻ (a : β × γ), f' a ∂(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), f' (a_1, c) ∂η (a, a_1) ∂κ a
⊢ ∫⁻ (x : β × γ), f x + f' x ∂(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), f (b, c) + f' (b, c) ∂η (a, b) ∂κ a

Proof state:
case e_s.h.refine_2.e_f.h.hf
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
a : α
f✝ : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f✝)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f✝)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f✝ a b
h_mono : Monotone F
this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, (F n) (b, c) ∂η (a, b) = ⨆ n, ∫⁻ (c : γ), (F n) (b, c) ∂η (a, b)
h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), f' (b, c) ∂η (a, b)
n : ℕ
f f' : SimpleFunc (β × γ) ℝ≥0∞
a✝ : Disjoint (Function.support ⇑f) (Function.support ⇑f')
hf_eq : ∫⁻ (a : β × γ), f a ∂(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), f (a_1, c) ∂η (a, a_1) ∂κ a
hf'_eq : ∫⁻ (a : β × γ), f' a ∂(κ ⊗ₖ η) a = ∫⁻ (a_1 : β), ∫⁻ (c : γ), f' (a_1, c) ∂η (a, a_1) ∂κ a
b : β
⊢ Measurable fun c => f (b, c)

Tactic:
exact (SimpleFunc.measurable _).comp measurable_prod_mk_left

Result:
no goals

Proof state:
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
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
h_mono : Monotone F
this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, (F n) (b, c) ∂η (a, b) = ⨆ n, ∫⁻ (c : γ), (F n) (b, c) ∂η (a, b)
f' : SimpleFunc (β × γ) ℝ≥0∞
⊢ (fun b => ∫⁻ (c : γ), f' (b, c) ∂η (a, b)) = (fun ab => ∫⁻ (c : γ), f' (ab.2, c) ∂η ab) ∘ fun b => (a, b)

Tactic:
ext1 ab

Result:
case h
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
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
h_mono : Monotone F
this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, (F n) (b, c) ∂η (a, b) = ⨆ n, ∫⁻ (c : γ), (F n) (b, c) ∂η (a, b)
f' : SimpleFunc (β × γ) ℝ≥0∞
ab : β
⊢ ∫⁻ (c : γ), f' (ab, c) ∂η (a, ab) = ((fun ab => ∫⁻ (c : γ), f' (ab.2, c) ∂η ab) ∘ fun b => (a, b)) ab

Proof state:
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
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
h_mono : Monotone F
this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, (F n) (b, c) ∂η (a, b) = ⨆ n, ∫⁻ (c : γ), (F n) (b, c) ∂η (a, b)
h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), f' (b, c) ∂η (a, b)
⊢ ⨆ n, ∫⁻ (a : β × γ), (F n) a ∂(κ ⊗ₖ η) a = ⨆ n, ∫⁻ (a_1 : β), ∫⁻ (c : γ), (F n) (a_1, c) ∂η (a, a_1) ∂κ a

Tactic:
congr

Result:
case e_s
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
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
h_mono : Monotone F
this : ∀ (b : β), ∫⁻ (c : γ), ⨆ n, (F n) (b, c) ∂η (a, b) = ⨆ n, ∫⁻ (c : γ), (F n) (b, c) ∂η (a, b)
h_some_meas_integral : ∀ (f' : SimpleFunc (β × γ) ℝ≥0∞), Measurable fun b => ∫⁻ (c : γ), f' (b, c) ∂η (a, b)
⊢ (fun n => ∫⁻ (a : β × γ), (F n) a ∂(κ ⊗ₖ η) a) = fun n => ∫⁻ (a_1 : β), ∫⁻ (c : γ), (F n) (a_1, c) ∂η (a, a_1) ∂κ a

Proof state:
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
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
⊢ ∫⁻ (bc : β × γ), ⨆ n, (F n) (bc.1, bc.2) ∂(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), ⨆ n, (F n) (b, c) ∂η (a, b) ∂κ a

Tactic:
have h_mono : Monotone F := fun i j hij b =>
  SimpleFunc.monotone_eapprox (Function.uncurry f) hij _

Result:
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
a : α
f : β → γ → ℝ≥0∞
hf : Measurable (Function.uncurry f)
F : ℕ → SimpleFunc (β × γ) ℝ≥0∞ := SimpleFunc.eapprox (Function.uncurry f)
h : ∀ (a : β) (b : γ), ⨆ n, (F n) (a, b) = f a b
h_mono : Monotone F
⊢ ∫⁻ (bc : β × γ), ⨆ n, (F n) (bc.1, bc.2) ∂(κ ⊗ₖ η) a = ∫⁻ (b : β), ∫⁻ (c : γ), ⨆ n, (F n) (b, c) ∂η (a, b) ∂κ a