-- 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)
this :
  Eq.{u_1 + 1} (α :=
    (((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))) → X (↑i : ℕ))
    (_root_.IicProdIoc (X := X) k (k + 1) ∘ Prod.map (_root_.IicProdIoc (X := X) a k) id)
    (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))))
inr :
  Eq (α := Kernel ((i : ↥(Iic a)) → X (↑i : ℕ)) (((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k.succ)) → X (↑i : ℕ))))
    (Kernel.id ×ₖ
      (map (mβ := Prod.instMeasurableSpace)
          (Kernel.id ×ₖ (κ k).map (⇑(piSingleton (X := X) k) : (a : X (k + 1)) → (i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)))
          (IocProdIoc (X := X) a k (k + 1) ∘ Prod.map (restrict₂ (π := X) ⋯) id) ∘ₖ
        partialTraj (X := X) κ a k))
    (Kernel.id ×ₖ
      (map (β := ((i : ↥(Iic k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
          (Kernel.id ×ₖ (κ k).map (⇑(piSingleton (X := X) k) : (a : X (k + 1)) → (i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)))
          (restrict₂ (π := X) ⋯ ∘ _root_.IicProdIoc (X := X) k (k + 1)) ∘ₖ
        partialTraj (X := X) κ a k))
inr₁ :
  Measurable.{u_1, u_1} (α := ((i : ↥(Iic k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ))) (β :=
    ((i : ↥(Ioc a k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ))) (Prod.map (restrict₂ (π := X) ⋯) id)
inr₂ :
  Measurable.{u_1, u_1} (α := ((i : ↥(Ioc a k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)))
    (IocProdIoc (X := X) a k (k + 1))
inr₃ :
  Measurable.{u_1, u_1} (α := (i : ↥(Iic k)) → X (↑i : ℕ)) (β := (i : ↥(Ioc a k)) → X (↑i : ℕ)) (restrict₂ (π := X) ⋯)
inr₄ :
  Measurable.{u_1, u_1} (α := ((i : ↥(Iic k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)))
    (_root_.IicProdIoc (X := X) k (k + 1))
inr₅ :
  Measurable.{u_1, u_1} (α := (i : ↥(Iic (k + 1))) → X (↑i : ℕ)) (β := (i : ↥(Ioc a (k + 1))) → X (↑i : ℕ))
    (restrict₂ (π := X) ⋯)
inr₆ :
  Measurable.{u_1, u_1} (α := ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k)) → X (↑i : ℕ)))
    (_root_.IicProdIoc (X := X) a k)
inr₇ :
  Measurable.{u_1, u_1} (α := (i : ↥(Iic k)) → X (↑i : ℕ)) (β := (i : ↥(Ioc a k)) → X (↑i : ℕ)) (restrict₂ (π := X) ⋯)
inr₈ :
  Measurable.{u_1, u_1} (α := ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a (k + 1))) → X (↑i : ℕ)))
    (_root_.IicProdIoc (X := X) a (k + 1))
inr₉ :
  Measurable.{u_1, u_1} (α := (i : ↥(Iic k.succ)) → X (↑i : ℕ)) (β :=
    ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k.succ)) → X (↑i : ℕ)))
    (⇑(MeasurableEquiv.symm (MeasurableEquiv.IicProdIoc (X := X) ⋯)) :
      (a_1 : (i : ↥(Iic k.succ)) → X (↑i : ℕ)) → ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k.succ)) → X (↑i : ℕ)))
inr₁₀ : Measurable.{u_1, u_1} (α := (i : ↥(Iic a)) → X (↑i : ℕ)) id
inr₁₁ :
  Measurable.{u_1, u_1} (α := ((i : ↥(Ioc a k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)))
    (IocProdIoc (X := X) a k (k + 1))
inr₁₂ :
  Measurable.{u_1, u_1} (α :=
    ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)))
    (Prod.map id (IocProdIoc (X := X) a k (k + 1)))
inr₁₃ :
  Measurable.{u_1, u_1} (α := ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k.succ)) → X (↑i : ℕ))) (β :=
    (i : ↥(Iic k.succ)) → 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))
inr₁₄ :
  Measurable.{u_1, u_1} (α :=
    (((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k)) → X (↑i : ℕ))) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ))) (β :=
    ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)))
    (⇑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)))
inr₁₅ :
  Measurable.{u_1, u_1} (α :=
    ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ))) (β :=
    (i : ↥(Iic k.succ)) → 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)))
inr₁₆ :
  Measurable.{u_1, u_1} (α :=
    (((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k)) → X (↑i : ℕ))) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)))
    (Prod.map (_root_.IicProdIoc (X := X) a k) id)
inr₁₇ :
  Measurable.{u_1, u_1} (α := ((i : ↥(Iic k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)))
    (_root_.IicProdIoc (X := X) k (k + 1))
inr₁₈ :
  Measurable.{u_1, u_1} (α := ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k)) → X (↑i : ℕ)))
    (_root_.IicProdIoc (X := X) a k)
inr₁₉ :
  Measurable.{u_1, u_1} (α := ((i : ↥(Iic k)) → X (↑i : ℕ)) × ((i : ↥(Ioc k (k + 1))) → X (↑i : ℕ)))
    (_root_.IicProdIoc (X := X) k (k + 1))
inr₂₀ :
  Measurable.{u_1, u_1} (α := ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a k)) → X (↑i : ℕ)))
    (_root_.IicProdIoc (X := X) a k)
⊢ Eq (α := Kernel ((i : ↥(Iic a)) → X (↑i : ℕ)) ((i : ↥(Iic (k + 1))) → X (↑i : ℕ))) (partialTraj (X := X) κ a (k + 1))
    (map (β := ((i : ↥(Iic a)) → X (↑i : ℕ)) × ((i : ↥(Ioc a (k + 1))) → X (↑i : ℕ))) (mβ := Prod.instMeasurableSpace)
      (Kernel.id ×ₖ map (partialTraj (X := X) κ a (k + 1)) (restrict₂ (π := X) ⋯))
      (_root_.IicProdIoc (X := X) a (k + 1)))

Tactic:
nth_rw 1 [← partialTraj_comp_partialTraj h k.le_succ, hk, partialTraj_succ_self, comp_map,
      comap_map_comm, comap_prod, id_comap, ← id_map, map_prod_eq, ← map_comp_right, this,
      map_comp_right, id_prod_eq, prodAssoc_prod, map_comp_right, ← map_prod_map, map_id,
      ← map_comp, map_apply_eq_iff_map_symm_apply_eq, fst_prod_comp_id_prod, ← map_comp_right,
      ← coe_IicProdIoc (h.trans k.le_succ), symm_comp_self, map_id,
      deterministic_congr IicProdIoc_comp_restrict₂.symm, ← deterministic_comp_deterministic,
      comp_deterministic_eq_comap, ← comap_prod, ← map_comp, ← comp_map, ← hk,
      ← partialTraj_comp_partialTraj h k.le_succ, partialTraj_succ_self, map_comp, map_comp,
      ← map_comp_right, ← id_map, map_prod_eq, ← map_comp_right]

Premises used: ProbabilityTheory.Kernel.map_comp_right, ProbabilityTheory.Kernel.map_prod_eq, ProbabilityTheory.Kernel.deterministic_comp_deterministic, ProbabilityTheory.Kernel.id_map, ProbabilityTheory.Kernel.prodAssoc_prod, Nat.le_succ, ProbabilityTheory.Kernel.comap_prod, ProbabilityTheory.Kernel.partialTraj_succ_self, MeasurableEquiv.coe_IicProdIoc, ProbabilityTheory.Kernel.partialTraj_comp_partialTraj, ProbabilityTheory.Kernel.map_comp, _private.0.ProbabilityTheory.Kernel.fst_prod_comp_id_prod, ProbabilityTheory.Kernel.comp_map, MeasurableEquiv.symm_comp_self, ProbabilityTheory.Kernel.id_comap, LE.le.trans, ProbabilityTheory.Kernel.map_prod_map, IicProdIoc_comp_restrict₂, ProbabilityTheory.Kernel.comp_deterministic_eq_comap, Eq.symm, ProbabilityTheory.Kernel.id_prod_eq, ProbabilityTheory.Kernel.comap_map_comm, ProbabilityTheory.Kernel.map_apply_eq_iff_map_symm_apply_eq, ProbabilityTheory.Kernel.map_id, ProbabilityTheory.Kernel.deterministic_congr