-- 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:
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.map (α.hom ≫ (mkOfSucc 1).op) (lift s x) = X.map (mkOfSucc 1).op (s.π.app α x)

Tactic:
have : α.hom ≫ (mkOfSucc 1).op = α₀.hom :=
  Quiver.Hom.unop_inj (by ext x; fin_cases x <;> rfl)

Result:
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)
this : α.hom ≫ (mkOfSucc 1).op = α₀.hom
⊢ X.map (α.hom ≫ (mkOfSucc 1).op) (lift s x) = X.map (mkOfSucc 1).op (s.π.app α x)

Proof state:
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, ⋯⟩ ⋯) ⋯
x : Fin ((unop ((Truncated.inclusion 2).op.obj α₀.right)).len + 1)
⊢ ↑((Hom.toOrderHom (α.hom ≫ (Truncated.inclusion 2).op.map (mkOfSucc 1).op).unop) x) =
    ↑((Hom.toOrderHom α₀.hom.unop) x)

Tactic:
fin_cases x <;> rfl

Result:
no goals

Proof state:
case intro.succ
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, ⋯⟩ ⋯ ⋯) ⋯
⊢ X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op (lift s x) = s.π.app (strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) ⋯) x

Tactic:
let α₀ := strArrowMk₂ (mkOfLe (n := n) ⟨i + k, by omega⟩ ⟨j, by omega⟩
  (by simp only [Fin.mk_le_mk]; omega))

Result:
case intro.succ
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, ⋯⟩ ⋯) ⋯
⊢ X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op (lift s x) = s.π.app (strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) ⋯) 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
⊢ X.map α.hom (lift s x) = s.π.app α x

Tactic:
apply StrictSegal.spineInjective

Result:
case a
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
⊢ (spineEquiv 2) (X.map α.hom (lift s x)) = (spineEquiv 2) (s.π.app α x)