-- Module: Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj
Goal:
X : ℕ → Type u_1
mX : (n : ℕ) → MeasurableSpace (X n)
κ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X (↑i : ℕ)) (X (n + 1))
inst : ∀ (n : ℕ), IsSFiniteKernel (κ n)
a b k : ℕ
h : a ≤ k
hk :
  partialTraj (X := X) κ a k =
    map (β := ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k)) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ (partialTraj (X := X) κ a k).map (restrict₂ (π := X) ⋯)) (_root_.IicProdIoc (X := X) a k)
x : (((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k)) → X (↑i : ℕ))) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ))
i : ↥(Iic (k + 1))
h₁ :
  Eq.{u_1 + 1} (α := X (↑i : ℕ))
    (if h : (↑i : ℕ) ≤ k then
      if h_1 : (↑i : ℕ) ≤ a then x.1.1 (Subtype.mk (p := fun (x : ℕ) ↦ x ∈ Iic a) (↑i : ℕ) ⋯)
      else x.1.2 (Subtype.mk (p := fun (x : ℕ) ↦ x ∈ Ioc a k) (↑i : ℕ) ⋯)
    else x.2 (Subtype.mk (p := fun (x : ℕ) ↦ x ∈ Ioc k (k + 1)) (↑i : ℕ) ⋯))
    (if h : (↑i : ℕ) ≤ a then
      ((prodAssoc :
              (a_1 :
                  (((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k)) → X (↑i : ℕ))) ×
                    ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ))) →
                ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)))
            x).1
        (Subtype.mk (p := fun (x : ℕ) ↦ x ∈ Iic a) (↑i : ℕ) ⋯)
    else
      if h_1 : (↑i : ℕ) ≤ k then
        ((prodAssoc :
                  (a_1 :
                      (((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k)) → X (↑i : ℕ))) ×
                        ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ))) →
                    ((i : ↥(Iic a)) → X (↑i : ℕ)) ×
                      ((i : ↥(Ioc a k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)))
                x).2.1
          (Subtype.mk (p := fun (x : ℕ) ↦ x ∈ Ioc a k) (↑i : ℕ) ⋯)
      else
        ((prodAssoc :
                  (a_1 :
                      (((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k)) → X (↑i : ℕ))) ×
                        ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ))) →
                    ((i : ↥(Iic a)) → X (↑i : ℕ)) ×
                      ((i : ↥(Ioc a k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)))
                x).2.2
          (Subtype.mk (p := fun (x : ℕ) ↦ x ∈ Ioc k (k + 1)) (↑i : ℕ) ⋯))
⊢ (_root_.IicProdIoc (X := X) k (k + 1) ∘ Prod.map (_root_.IicProdIoc (X := X) a k) id) x i =
    Function.comp (β :=
      ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)))
      ((⇑(MeasurableEquiv.IicProdIoc (X := X) ⋯) :
          (a : ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k.succ)) → X (↑i : ℕ))) →
            (i : ↥(Iic k.succ)) → X (↑i : ℕ)) ∘
        Prod.map id (IocProdIoc (X := X) a k (k + 1)))
      (⇑prodAssoc :
        (a_1 :
            (((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k)) → X (↑i : ℕ))) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ))) →
          ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)))
      x i

Tactic:
simp only [IicProdIoc_def, MeasurableEquiv.IicProdIoc, MeasurableEquiv.coe_mk,
        Equiv.coe_fn_mk, Function.comp_apply, Prod.map_fst, Prod.map_snd, id_eq,
        Nat.succ_eq_add_one, IocProdIoc]

Premises used: Prod.map_snd, Nat.succ_eq_add_one, MeasurableEquiv.coe_mk, IicProdIoc_def, Prod.map_fst, Equiv.coe_fn_mk, Function.comp_apply, MeasurableEquiv.IicProdIoc, IocProdIoc, id_eq