-- 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.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

Tactic:
let α₁ := strArrowMk₂ (mkOfLe (n := n) ⟨i, 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, ⋯⟩ ⋯) ⋯
α₁ : StructuredArrow (op ⦋n⦌) (Truncated.inclusion 2).op := strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ ⋯) ⋯
⊢ X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op (lift s x) = s.π.app (strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) ⋯) x

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, ⋯⟩ ⋯ ⋯) ⋯
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 α₁.hom (lift s x) = s.π.app α₁ x

Tactic:
have : X.map α.hom (lift s x) = s.π.app α x := by
  apply StrictSegal.spineInjective
  apply Path.ext'
  intro t
  dsimp only [spineEquiv]
  rw [Equiv.coe_fn_mk, spine_arrow, spine_arrow,
      ← FunctorToTypes.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 [this, h₂, ← congr_fun (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 [this, h₀, ← congr_fun (s.w β₀) x]
      rfl

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, ⋯⟩ ⋯) ⋯
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
this : X.map α.hom (lift s x) = s.π.app α x
⊢ X.map α₁.hom (lift s x) = s.π.app α₁ 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 j : ℕ) (hij : i ≤ j) (hj : j ≤ n),
    i + 0 = j → X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op (lift s x) = s.π.app (strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) ⋯) x

Tactic:
rintro i j hij hj hik

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 j : ℕ
hij : i ≤ j
hj : j ≤ n
hik : i + 0 = 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 ⋯
⊢ X.map α₀.hom (lift s x) = s.π.app α₀ x

Tactic:
subst hik

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 : ℕ
hij : i ≤ i + (k + 1)
hj : i + (k + 1) ≤ n
α : StructuredArrow (op ⦋n⦌) (Truncated.inclusion 2).op :=
  strArrowMk₂ (mkOfLeComp ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⟨i + (k + 1), ⋯⟩ ⋯ ⋯) ⋯
α₀ : StructuredArrow (op ⦋n⦌) (Truncated.inclusion 2).op := strArrowMk₂ (mkOfLe ⟨i + k, ⋯⟩ ⟨i + (k + 1), ⋯⟩ ⋯) ⋯
α₁ : StructuredArrow (op ⦋n⦌) (Truncated.inclusion 2).op := strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨i + (k + 1), ⋯⟩ ⋯) ⋯
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 ⋯
⊢ X.map α₀.hom (lift s x) = s.π.app α₀ 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
⊢ i + k < n + 1

Tactic:
omega

Result:
no goals