-- 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 : s.pt
k : ℕ
zero :
  ∀ (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n) (hk : i + 0 = 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) :
        (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 :
        (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)))
succ :
  ∀ (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) :
            (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 :
            (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) (hk : i + (k + 1) = 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) :
        (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 :
        (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) (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) :
        (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 :
        (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)))

Tactic:
induction k with
  | zero =>
      rintro i j hij hj hik
      obtain rfl : i = j := hik
      have : mkOfLe ⟨i, Nat.lt_add_one_of_le hj⟩ ⟨i, Nat.lt_add_one_of_le hj⟩ (by rfl) =
        ⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, Nat.lt_add_one_of_le hj⟩ := Hom.ext_one_left _ _
      rw [this]
      let α : (strArrowMk₂ (⦋0⦌.const ⦋n⦌ ⟨i, Nat.lt_add_one_of_le hj⟩)) ⟶
        (strArrowMk₂ (⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, Nat.lt_add_one_of_le hj⟩)) :=
            StructuredArrow.homMk ((Hom.tr (⦋1⦌.const ⦋0⦌ 0)).op) (by simp; rfl)
      conv_rhs => dsimp; rw [dsimp% s.π.naturality_apply α x]
      rw [op_comp, Functor.map_comp]
      simp only [types_comp_apply]
      refine congrArg (X.map (⦋1⦌.const ⦋0⦌ 0).op) ?_
      unfold strArrowMk₂
      rw [lift, StrictSegal.spineToSimplex_vertex]
      congr
  | succ k hk =>
      intro i j hij hj hik
      let α := strArrowMk₂ (mkOfLeComp (n := n) ⟨i, by omega⟩ ⟨i + k, by omega⟩
          ⟨j, by omega⟩ (by simp) (by simp only [Fin.mk_le_mk]; omega))
      let α₀ := strArrowMk₂ (mkOfLe (n := n) ⟨i + k, by omega⟩ ⟨j, by omega⟩
        (by simp only [Fin.mk_le_mk]; omega))
      let α₁ := strArrowMk₂ (mkOfLe (n := n) ⟨i, by omega⟩ ⟨j, by omega⟩ hij)
      let α₂ := strArrowMk₂ (mkOfLe (n := n) ⟨i, by omega⟩ ⟨i + k, by omega⟩ (by simp))
      let β₀ : α ⟶ α₀ := StructuredArrow.homMk ((Hom.tr (mkOfSucc 1)).op) (Quiver.Hom.unop_inj
        (by ext x; fin_cases x <;> rfl))
      let β₁ : α ⟶ α₁ := StructuredArrow.homMk ((Hom.tr (δ 1)).op) (Quiver.Hom.unop_inj
        (by ext x; fin_cases x <;> rfl))
      let β₂ : α ⟶ α₂ := StructuredArrow.homMk ((Hom.tr (mkOfSucc 0)).op) (Quiver.Hom.unop_inj
        (by ext x; fin_cases x <;> rfl))
      have h₀ : X.map α₀.hom (lift sx s x) = s.π.app α₀ x := by
        subst hik
        exact fac_aux₁ _ _ _ _ hj
      have h₂ : X.map α₂.hom (lift sx s x) = s.π.app α₂ x :=
        hk i (i + k) (by simp) (by lia) rfl
      change X.map α₁.hom (lift sx s x) = s.π.app α₁ x
      have : X.map α.hom (lift sx s x) = s.π.app α x := by
        apply sx.spineInjective
        apply Path.ext'
        intro t
        dsimp [spineEquiv, α]
        rw [← Functor.map_comp_apply]
        match t with
        | 0 =>
            have : α.hom ≫ (mkOfSucc 0).op = α₂.hom :=
              Quiver.Hom.unop_inj (by ext x; fin_cases x <;> rfl)
            rw [dsimp% [α] this]
            dsimp [α₂] at h₂ ⊢
            rw [h₂, ← dsimp% [α₂] ConcreteCategory.congr_hom (s.w β₂) x]
            rfl
        | 1 =>
            have : α.hom ≫ (mkOfSucc 1).op = α₀.hom :=
              Quiver.Hom.unop_inj (by ext x; fin_cases x <;> rfl)
            rw [dsimp% [α] this]
            dsimp [α₀] at h₀ ⊢
            rw [h₀, ← dsimp% [α₀] ConcreteCategory.congr_hom (s.w β₀) x]
            rfl
      rw [← StructuredArrow.w β₁, Functor.map_comp_apply]
      dsimp [fromPUnit] at this ⊢
      rw [this, ← s.w β₁]
      dsimp

Premises used: CategoryTheory.op_comp, SSet.StrictSegal.isPointwiseRightKanExtensionAt.lift, congrArg, CategoryTheory.ConcreteCategory.hom, Quiver.Hom.op, CategoryTheory.Functor.map_comp_apply, CategoryTheory.Functor.map, SimplexCategory.const, SimplexCategory.mk, SSet.StrictSegal.isPointwiseRightKanExtensionAt.strArrowMk₂, CategoryTheory.Functor.map_comp, CategoryTheory.Functor.fromPUnit, OfNat.ofNat, DFunLike.coe, rfl, SSet.StrictSegal.spineToSimplex_vertex, CategoryTheory.types_comp_apply