-- Mathlib/AlgebraicTopology/SimplicialSet/Coskeletal.lean
lemma fac_aux₂ {n : ℕ}
    (s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X))
    (x : s.pt) (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n) :
    X.map (mkOfLe ⟨i, by omega⟩ ⟨j, by omega⟩ hij).op (lift s x) =
      s.π.app (strArrowMk₂ (mkOfLe ⟨i, by omega⟩ ⟨j, by omega⟩ hij)) x

Proof state:
case intro.zero
X : SSet
inst✝ : X.StrictSegal
n : ℕ
s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X)
x : s.pt
i : ℕ
hij : i ≤ i
hj : i ≤ n
this : mkOfLe ⟨i, ⋯⟩ ⟨i, ⋯⟩ ⋯ = ⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩
α : strArrowMk₂ (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯ ⟶ strArrowMk₂ (⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯ :=
  homMk (⦋1⦌.const ⦋0⦌ 0).op ⋯
⊢ X.map (⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩).op (lift s x) =
    s.π.app (strArrowMk₂ (⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯) x

Tactic:
have nat := congr_fun (s.π.naturality α) x

Result:
case intro.zero
X : SSet
inst✝ : X.StrictSegal
n : ℕ
s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X)
x : s.pt
i : ℕ
hij : i ≤ i
hj : i ≤ n
this : mkOfLe ⟨i, ⋯⟩ ⟨i, ⋯⟩ ⋯ = ⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩
α : strArrowMk₂ (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯ ⟶ strArrowMk₂ (⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯ :=
  homMk (⦋1⦌.const ⦋0⦌ 0).op ⋯
nat :
  (((Functor.const (StructuredArrow (op ⦋n⦌) (Truncated.inclusion 2).op)).obj s.pt).map α ≫
        s.π.app (strArrowMk₂ (⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯))
      x =
    (s.π.app (strArrowMk₂ (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯) ≫
        (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X).map α)
      x
⊢ X.map (⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩).op (lift s x) =
    s.π.app (strArrowMk₂ (⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯) x

Proof state:
case intro.zero
X : SSet
inst✝ : X.StrictSegal
n : ℕ
s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X)
x : s.pt
i : ℕ
hij : i ≤ i
hj : i ≤ n
this : mkOfLe ⟨i, ⋯⟩ ⟨i, ⋯⟩ ⋯ = ⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩
α : strArrowMk₂ (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯ ⟶ strArrowMk₂ (⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯ :=
  homMk (⦋1⦌.const ⦋0⦌ 0).op ⋯
nat :
  s.π.app (strArrowMk₂ (⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯) x =
    X.map ((Truncated.inclusion 2).map α.right.unop).op (s.π.app (strArrowMk₂ (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯) x)
⊢ X.map (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩).op (lift s x) = s.π.app (StructuredArrow.mk (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩).op) x

Tactic:
rw [lift, StrictSegal.spineToSimplex_vertex]

Result:
case intro.zero
X : SSet
inst✝ : X.StrictSegal
n : ℕ
s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X)
x : s.pt
i : ℕ
hij : i ≤ i
hj : i ≤ n
this : mkOfLe ⟨i, ⋯⟩ ⟨i, ⋯⟩ ⋯ = ⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩
α : strArrowMk₂ (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯ ⟶ strArrowMk₂ (⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯ :=
  homMk (⦋1⦌.const ⦋0⦌ 0).op ⋯
nat :
  s.π.app (strArrowMk₂ (⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯) x =
    X.map ((Truncated.inclusion 2).map α.right.unop).op (s.π.app (strArrowMk₂ (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯) x)
⊢ {
          vertex := fun i =>
            s.π.app
              (StructuredArrow.mk
                (((Truncated.inclusion 2).obj (unop (op { obj := ⦋0⦌, property := lift.proof_1 }))).const ⦋n⦌ i).op)
              x,
          arrow := fun i => s.π.app (StructuredArrow.mk (mkOfLe i.castSucc i.succ ⋯).op) x, arrow_src := ⋯,
          arrow_tgt := ⋯ }.vertex
      ⟨i, ⋯⟩ =
    s.π.app (StructuredArrow.mk (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩).op) x

Proof state:
X : SSet
inst✝ : X.StrictSegal
n : ℕ
s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X)
x : s.pt
k : ℕ
hk :
  ∀ (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n),
    i + k = j → X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op (lift s x) = s.π.app (strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) ⋯) x
i j : ℕ
hij : i ≤ j
hj : j ≤ n
hik : i + (k + 1) = j
α : StructuredArrow (op ⦋n⦌) (Truncated.inclusion 2).op := strArrowMk₂ (mkOfLeComp ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⟨j, ⋯⟩ ⋯ ⋯) ⋯
α₀ : StructuredArrow (op ⦋n⦌) (Truncated.inclusion 2).op := strArrowMk₂ (mkOfLe ⟨i + k, ⋯⟩ ⟨j, ⋯⟩ ⋯) ⋯
α₁ : StructuredArrow (op ⦋n⦌) (Truncated.inclusion 2).op := strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ ⋯) ⋯
α₂ : StructuredArrow (op ⦋n⦌) (Truncated.inclusion 2).op := strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⋯) ⋯
β₀ : α ⟶ α₀ := homMk (mkOfSucc 1).op ⋯
β₁ : α ⟶ α₁ := homMk (δ 1).op ⋯
β₂ : α ⟶ α₂ := homMk (mkOfSucc 0).op ⋯
h₀ : X.map α₀.hom (lift s x) = s.π.app α₀ x
h₂ : X.map α₂.hom (lift s x) = s.π.app α₂ x
t : Fin (1 + 1)
⊢ (α.hom ≫ (mkOfSucc 1).op).unop = α₀.hom.unop

Tactic:
ext x

Result:
case a.h.h.h
X : SSet
inst✝ : X.StrictSegal
n : ℕ
s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X)
x✝ : s.pt
k : ℕ
hk :
  ∀ (i j : ℕ) (hij : i ≤ j) (hj : j ≤ n),
    i + k = j → X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op (lift s x✝) = s.π.app (strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) ⋯) x✝
i j : ℕ
hij : i ≤ j
hj : j ≤ n
hik : i + (k + 1) = j
α : StructuredArrow (op ⦋n⦌) (Truncated.inclusion 2).op := strArrowMk₂ (mkOfLeComp ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⟨j, ⋯⟩ ⋯ ⋯) ⋯
α₀ : StructuredArrow (op ⦋n⦌) (Truncated.inclusion 2).op := strArrowMk₂ (mkOfLe ⟨i + k, ⋯⟩ ⟨j, ⋯⟩ ⋯) ⋯
α₁ : StructuredArrow (op ⦋n⦌) (Truncated.inclusion 2).op := strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ ⋯) ⋯
α₂ : StructuredArrow (op ⦋n⦌) (Truncated.inclusion 2).op := strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⋯) ⋯
β₀ : α ⟶ α₀ := homMk (mkOfSucc 1).op ⋯
β₁ : α ⟶ α₁ := homMk (δ 1).op ⋯
β₂ : α ⟶ α₂ := homMk (mkOfSucc 0).op ⋯
h₀ : X.map α₀.hom (lift s x✝) = s.π.app α₀ x✝
h₂ : X.map α₂.hom (lift s x✝) = s.π.app α₂ x✝
t : Fin (1 + 1)
x : Fin ((unop (op ⦋1⦌)).len + 1)
⊢ ↑((Hom.toOrderHom (α.hom ≫ (mkOfSucc 1).op).unop) x) = ↑((Hom.toOrderHom α₀.hom.unop) x)

Proof state:
case intro.zero
X : SSet
inst✝ : X.StrictSegal
n : ℕ
s : Cone (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X)
x : s.pt
i : ℕ
hij : i ≤ i
hj : i ≤ n
this : mkOfLe ⟨i, ⋯⟩ ⟨i, ⋯⟩ ⋯ = ⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩
α : strArrowMk₂ (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯ ⟶ strArrowMk₂ (⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯ :=
  homMk (⦋1⦌.const ⦋0⦌ 0).op ⋯
nat :
  s.π.app (strArrowMk₂ (⦋1⦌.const ⦋0⦌ 0 ≫ ⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯) x =
    X.map ((Truncated.inclusion 2).map α.right.unop).op (s.π.app (strArrowMk₂ (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯) x)
⊢ {
          vertex := fun i =>
            s.π.app
              (StructuredArrow.mk
                (((Truncated.inclusion 2).obj (unop (op { obj := ⦋0⦌, property := lift.proof_1 }))).const ⦋n⦌ i).op)
              x,
          arrow := fun i => s.π.app (StructuredArrow.mk (mkOfLe i.castSucc i.succ ⋯).op) x, arrow_src := ⋯,
          arrow_tgt := ⋯ }.vertex
      ⟨i, ⋯⟩ =
    s.π.app (StructuredArrow.mk (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩).op) x

Tactic:
congr

Result:
no goals