-- Module: Mathlib.AlgebraicTopology.SimplicialSet.Coskeletal
Goal:
X : SSet
sx : X.StrictSegal
n : ℕ
s :
  Cone (J := StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)))
    (proj (op ⦋n⦌) (Functor.op (inclusion 2)) ⋙ Functor.op (inclusion 2) ⋙ X)
x_1 : s.pt
k : ℕ
hk :
  ∀ (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n) (hk : i + k = j),
    Eq (α := (fun (X : Type u) ↦ X) (X _⦋1⦌))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (X.map (mkOfLe (⟨i, ⋯⟩ : Fin (n + 1)) (⟨j, ⋯⟩ : Fin (n + 1)) hij).op) :
            (a : (fun (X : Type u) ↦ X) (X _⦋n⦌)) → (fun (X : Type u) ↦ X) (X _⦋1⦌))
          (lift sx s x_1) :
        (fun (X : Type u) ↦ X) (X _⦋1⦌))
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
              (s.π.app (strArrowMk₂ (mkOfLe (⟨i, ⋯⟩ : Fin (n + 1)) (⟨j, ⋯⟩ : Fin (n + 1)) hij) fac_aux₁._proof_1)) :
            (a :
                (fun (X : Type u) ↦ X)
                  (((Functor.const (StructuredArrow (op ⦋n⦌) (inclusion 2).op)).obj s.pt).obj
                    (strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) fac_aux₁._proof_1))) →
              (fun (X : Type u) ↦ X)
                ((proj (op ⦋n⦌) (inclusion 2).op ⋙ (inclusion 2).op ⋙ X).obj
                  (strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) fac_aux₁._proof_1)))
          x_1 :
        (fun (X : Type u) ↦ X)
          (obj (proj (op ⦋n⦌) (Functor.op (inclusion 2)) ⋙ Functor.op (inclusion 2) ⋙ X)
            (strArrowMk₂ (mkOfLe (⟨i, ⋯⟩ : Fin (n + 1)) (⟨j, ⋯⟩ : Fin (n + 1)) hij) fac_aux₁._proof_1)))
i j : ℕ
hij : i ≤ j
hj : j ≤ n
hik : i + (k + 1) = j
α : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLeComp ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⟨j, ⋯⟩ ⋯ ⋯) fac_aux₂._proof_11
α₀ : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLe ⟨i + k, ⋯⟩ ⟨j, ⋯⟩ ⋯) lift._proof_3
α₁ : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) lift._proof_3
α₂ : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⋯) lift._proof_3
β₀ : α ⟶ α₀ := homMk (Hom.tr (mkOfSucc 1) ⋯ ⋯).op ⋯
x : Fin ((unop (obj (Functor.op (inclusion 2)) α₁.right)).len + 1)
a :
  (↑((Hom.toOrderHom
              (α.hom ≫
                  CategoryTheory.Functor.map (Functor.op (inclusion 2))
                    (Quiver.Hom.op (Hom.tr (SimplexCategory.δ 1) ⋯ ⋯))).unop :
            (a : Fin ((unop ((inclusion 2).op.obj α₁.right)).len + 1)) → Fin ((unop (op ⦋n⦌)).len + 1))
          ((fun (i : Fin ((unop (obj (Functor.op (inclusion 2)) α₁.right)).len + 1)) ↦ i)
            (⟨0, ⋯⟩ : Fin ((unop (obj (Functor.op (inclusion 2)) α₁.right)).len + 1)))) :
      ℕ) =
    (↑((Hom.toOrderHom α₁.hom.unop :
            (a : Fin ((unop ((inclusion 2).op.obj α₁.right)).len + 1)) → Fin ((unop (op ⦋n⦌)).len + 1))
          ((fun (i : Fin ((unop (obj (Functor.op (inclusion 2)) α₁.right)).len + 1)) ↦ i) ⟨0, ⋯⟩)) :
      ℕ)
a₁ :
  (↑((Hom.toOrderHom
              (α.hom ≫
                  CategoryTheory.Functor.map (Functor.op (inclusion 2))
                    (Quiver.Hom.op (Hom.tr (SimplexCategory.δ 1) ⋯ ⋯))).unop :
            (a : Fin ((unop ((inclusion 2).op.obj α₁.right)).len + 1)) → Fin ((unop (op ⦋n⦌)).len + 1))
          ((fun (i : Fin ((unop (obj (Functor.op (inclusion 2)) α₁.right)).len + 1)) ↦ i)
            (⟨(unop (obj (Functor.op (inclusion 2)) α₁.right)).len, ⋯⟩ :
              Fin ((unop (obj (Functor.op (inclusion 2)) α₁.right)).len + 1)))) :
      ℕ) =
    (↑((Hom.toOrderHom α₁.hom.unop :
            (a : Fin ((unop ((inclusion 2).op.obj α₁.right)).len + 1)) → Fin ((unop (op ⦋n⦌)).len + 1))
          ((fun (i : Fin ((unop (obj (Functor.op (inclusion 2)) α₁.right)).len + 1)) ↦ i)
            ⟨(unop (obj (Functor.op (inclusion 2)) α₁.right)).len, ⋯⟩)) :
      ℕ)
⊢ (↑((Hom.toOrderHom
              (α.hom ≫
                  CategoryTheory.Functor.map (Functor.op (inclusion 2))
                    (Quiver.Hom.op (Hom.tr (SimplexCategory.δ 1) ⋯ ⋯))).unop :
            (a : Fin ((unop ((inclusion 2).op.obj α₁.right)).len + 1)) → Fin ((unop (op ⦋n⦌)).len + 1))
          x) :
      ℕ) =
    (↑((Hom.toOrderHom α₁.hom.unop :
            (a : Fin ((unop ((inclusion 2).op.obj α₁.right)).len + 1)) → Fin ((unop (op ⦋n⦌)).len + 1))
          x) :
      ℕ)

Tactic:
fin_cases x