-- 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 : 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

Tactic:
exact fac_aux₁ _ _ _ _ hj

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, ⋯⟩ ⋯ ⋯) ⋯
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 ((Truncated.inclusion 2).op.map β₁.right) (s.π.app α x) =
    (s.π.app α ≫ (proj (op ⦋n⦌) (Truncated.inclusion 2).op ⋙ (Truncated.inclusion 2).op ⋙ X).map β₁) x

Tactic:
dsimp

Result:
no goals

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
i j : ℕ
hij : i ≤ j
hj : j ≤ n
⊢ i < n + 1

Tactic:
omega

Result:
no goals

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, ⋯⟩ ⋯ ⋯) ⋯
⊢ j < n + 1

Tactic:
omega

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),
    i + (k + 1) = j →
      X.map (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij).op (lift s x) = s.π.app (strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) ⋯) x

Tactic:
intro i j hij hj hik

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
⊢ 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, ⋯⟩ ⋯) ⋯
⊢ i + k < n + 1

Tactic:
omega

Result:
no goals

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, ⋯⟩ ⋯ ⋯) ⋯
⊢ i + k ≤ j

Tactic:
omega

Result:
no goals

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 (⦋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:
rw [nat, op_comp, Functor.map_comp]

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)
⊢ (X.map (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩).op ≫ X.map (⦋1⦌.const ⦋0⦌ 0).op) (lift s x) =
    X.map ((Truncated.inclusion 2).map α.right.unop).op (s.π.app (strArrowMk₂ (⦋0⦌.const ⦋n⦌ ⟨i, ⋯⟩) ⋯) x)