awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
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
hik : i + (k + 1) = j
α : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLeComp ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⟨j, ⋯⟩ ⋯ ⋯) fac_aux₂._proof_11
α₀ : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLe ⟨i + k, ⋯⟩ ⟨j, ⋯⟩ ⋯) lift._proof_3
α₁ : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) lift._proof_3
α₂ : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⋯) lift._proof_3
β₀ : α ⟶ α₀ := homMk (Hom.tr (mkOfSucc 1) ⋯ ⋯).op ⋯
β₁ : α ⟶ α₁ := homMk (Quiver.Hom.op (Hom.tr (SimplexCategory.δ 1) ⋯ ⋯)) ⋯
a :
  ∀ (x : Fin ((unop (obj (Functor.op (inclusion 2)) α₂.right)).len + 1)),
    (↑((Hom.toOrderHom
                (α.hom ≫ CategoryTheory.Functor.map (Functor.op (inclusion 2)) (Hom.tr (mkOfSucc 0) ⋯ ⋯).op).unop :
              (a : Fin ((unop ((inclusion 2).op.obj α₂.right)).len + 1)) → Fin ((unop (op ⦋n⦌)).len + 1))
            x) :
        ℕ) =
      (↑((Hom.toOrderHom α₂.hom.unop :
              (a : Fin ((unop ((inclusion 2).op.obj α₂.right)).len + 1)) → Fin ((unop (op ⦋n⦌)).len + 1))
            x) :
        ℕ)
⊢ (α.hom ≫ CategoryTheory.Functor.map (Functor.op (inclusion 2)) (Hom.tr (mkOfSucc 0) ⋯ ⋯).op).unop = α₂.hom.unop

Tactic:
ext x
-- 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
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
hik : i + (k + 1) = j
α : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLeComp ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⟨j, ⋯⟩ ⋯ ⋯) fac_aux₂._proof_11
α₀ : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLe ⟨i + k, ⋯⟩ ⟨j, ⋯⟩ ⋯) lift._proof_3
α₁ : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) lift._proof_3
α₂ : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⋯) lift._proof_3
β₀ : α ⟶ α₀ := homMk (Hom.tr (mkOfSucc 1) ⋯ ⋯).op ⋯
β₁ : α ⟶ α₁ := homMk (Quiver.Hom.op (Hom.tr (SimplexCategory.δ 1) ⋯ ⋯)) ⋯
⊢ (↑((Hom.toOrderHom (α.hom ≫ CategoryTheory.Functor.map (Functor.op (inclusion 2)) (Hom.tr (mkOfSucc 0) ⋯ ⋯).op).unop :
            (a : Fin ((unop ((inclusion 2).op.obj α₂.right)).len + 1)) → Fin ((unop (op ⦋n⦌)).len + 1))
          ((fun (i : Fin ((unop (obj (Functor.op (inclusion 2)) α₂.right)).len + 1)) ↦ i)
            (⟨0, ⋯⟩ : Fin ((unop (obj (Functor.op (inclusion 2)) α₂.right)).len + 1)))) :
      ℕ) =
    (↑((Hom.toOrderHom α₂.hom.unop :
            (a : Fin ((unop ((inclusion 2).op.obj α₂.right)).len + 1)) → Fin ((unop (op ⦋n⦌)).len + 1))
          ((fun (i : Fin ((unop (obj (Functor.op (inclusion 2)) α₂.right)).len + 1)) ↦ i) ⟨0, ⋯⟩)) :
      ℕ)

Tactic:
rfl
-- 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
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
hik : i + (k + 1) = j
α : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLeComp ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⟨j, ⋯⟩ ⋯ ⋯) fac_aux₂._proof_11
α₀ : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLe ⟨i + k, ⋯⟩ ⟨j, ⋯⟩ ⋯) lift._proof_3
α₁ : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) lift._proof_3
α₂ : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⋯) lift._proof_3
β₀ : α ⟶ α₀ := homMk (Hom.tr (mkOfSucc 1) ⋯ ⋯).op ⋯
β₁ : α ⟶ α₁ := homMk (Quiver.Hom.op (Hom.tr (SimplexCategory.δ 1) ⋯ ⋯)) ⋯
⊢ (↑((Hom.toOrderHom (α.hom ≫ CategoryTheory.Functor.map (Functor.op (inclusion 2)) (Hom.tr (mkOfSucc 0) ⋯ ⋯).op).unop :
            (a : Fin ((unop ((inclusion 2).op.obj α₂.right)).len + 1)) → Fin ((unop (op ⦋n⦌)).len + 1))
          ((fun (i : Fin ((unop (obj (Functor.op (inclusion 2)) α₂.right)).len + 1)) ↦ i)
            (⟨(unop (obj (Functor.op (inclusion 2)) α₂.right)).len, ⋯⟩ :
              Fin ((unop (obj (Functor.op (inclusion 2)) α₂.right)).len + 1)))) :
      ℕ) =
    (↑((Hom.toOrderHom α₂.hom.unop :
            (a : Fin ((unop ((inclusion 2).op.obj α₂.right)).len + 1)) → Fin ((unop (op ⦋n⦌)).len + 1))
          ((fun (i : Fin ((unop (obj (Functor.op (inclusion 2)) α₂.right)).len + 1)) ↦ i)
            ⟨(unop (obj (Functor.op (inclusion 2)) α₂.right)).len, ⋯⟩)) :
      ℕ)

Tactic:
rfl
-- 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
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
hik : i + (k + 1) = j
α : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLeComp ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⟨j, ⋯⟩ ⋯ ⋯) fac_aux₂._proof_11
α₀ : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLe ⟨i + k, ⋯⟩ ⟨j, ⋯⟩ ⋯) lift._proof_3
α₁ : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) lift._proof_3
α₂ : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⋯) lift._proof_3
β₀ : α ⟶ α₀ := homMk (Hom.tr (mkOfSucc 1) ⋯ ⋯).op ⋯
β₁ : α ⟶ α₁ := homMk (Quiver.Hom.op (Hom.tr (SimplexCategory.δ 1) ⋯ ⋯)) ⋯
β₂ : α ⟶ α₂ := homMk (Hom.tr (mkOfSucc 0) ⋯ ⋯).op ⋯
h₀ :
  Eq (α := (fun (X : Type u) ↦ X) (X.obj (obj (Functor.op (inclusion 2)) α₀.right)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (X.map α₀.hom) :
          (a : (fun (X : Type u) ↦ X) (X _⦋n⦌)) → (fun (X : Type u) ↦ X) (X.obj ((inclusion 2).op.obj α₀.right)))
        (lift sx s x) :
      (fun (X : Type u) ↦ X) (X.obj (obj (Functor.op (inclusion 2)) α₀.right)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (s.π.app α₀) :
          (a : (fun (X : Type u) ↦ X) (((Functor.const (StructuredArrow (op ⦋n⦌) (inclusion 2).op)).obj s.pt).obj α₀)) →
            (fun (X : Type u) ↦ X) ((proj (op ⦋n⦌) (inclusion 2).op ⋙ (inclusion 2).op ⋙ X).obj α₀))
        x :
      (fun (X : Type u) ↦ X) (obj (proj (op ⦋n⦌) (Functor.op (inclusion 2)) ⋙ Functor.op (inclusion 2) ⋙ X) α₀))
⊢ i ≤ i + k

Tactic:
simp
-- 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
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
hik : i + (k + 1) = j
α : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLeComp ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⟨j, ⋯⟩ ⋯ ⋯) fac_aux₂._proof_11
α₀ : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLe ⟨i + k, ⋯⟩ ⟨j, ⋯⟩ ⋯) lift._proof_3
α₁ : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨j, ⋯⟩ hij) lift._proof_3
α₂ : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)) :=
  strArrowMk₂ (mkOfLe ⟨i, ⋯⟩ ⟨i + k, ⋯⟩ ⋯) lift._proof_3
β₀ : α ⟶ α₀ := homMk (Hom.tr (mkOfSucc 1) ⋯ ⋯).op ⋯
β₁ : α ⟶ α₁ := homMk (Quiver.Hom.op (Hom.tr (SimplexCategory.δ 1) ⋯ ⋯)) ⋯
β₂ : α ⟶ α₂ := homMk (Hom.tr (mkOfSucc 0) ⋯ ⋯).op ⋯
h₀ :
  Eq (α := (fun (X : Type u) ↦ X) (X.obj (obj (Functor.op (inclusion 2)) α₀.right)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (X.map α₀.hom) :
          (a : (fun (X : Type u) ↦ X) (X _⦋n⦌)) → (fun (X : Type u) ↦ X) (X.obj ((inclusion 2).op.obj α₀.right)))
        (lift sx s x) :
      (fun (X : Type u) ↦ X) (X.obj (obj (Functor.op (inclusion 2)) α₀.right)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (s.π.app α₀) :
          (a : (fun (X : Type u) ↦ X) (((Functor.const (StructuredArrow (op ⦋n⦌) (inclusion 2).op)).obj s.pt).obj α₀)) →
            (fun (X : Type u) ↦ X) ((proj (op ⦋n⦌) (inclusion 2).op ⋙ (inclusion 2).op ⋙ X).obj α₀))
        x :
      (fun (X : Type u) ↦ X) (obj (proj (op ⦋n⦌) (Functor.op (inclusion 2)) ⋙ Functor.op (inclusion 2) ⋙ X) α₀))
⊢ i + k ≤ n

Tactic:
lia
-- 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
φ : ⦋1⦌ ⟶ ⦋n⦌
⊢ LE.le (α := Fin (⦋1⦌.len + 1)) 0 1

Tactic:
decide
-- 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
φ : ⦋1⦌ ⟶ ⦋n⦌
obtain :
  ∀ (i j : Fin (n + 1)) (hij : i ≤ j),
    Eq (α := (fun (X : Type u) ↦ X) (X _⦋1⦌))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (X.map (mkOfLe i j 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 j 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 j hij) fac_aux₁._proof_1)))
⊢ Eq (α := (fun (X : Type u) ↦ X) (X _⦋1⦌))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (X.map φ.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 φ fac_aux₁._proof_1)) :
          (a :
              (fun (X : Type u) ↦ X)
                (((Functor.const (StructuredArrow (op ⦋n⦌) (inclusion 2).op)).obj s.pt).obj
                  (strArrowMk₂ φ fac_aux₁._proof_1))) →
            (fun (X : Type u) ↦ X)
              ((proj (op ⦋n⦌) (inclusion 2).op ⋙ (inclusion 2).op ⋙ X).obj (strArrowMk₂ φ fac_aux₁._proof_1)))
        x :
      (fun (X : Type u) ↦ X)
        (obj (proj (op ⦋n⦌) (Functor.op (inclusion 2)) ⋙ Functor.op (inclusion 2) ⋙ X)
          (strArrowMk₂ φ fac_aux₁._proof_1)))

Tactic:
obtain ⟨i, j, hij, rfl⟩ : ∃ i j hij, φ = mkOfLe i j hij :=
    ⟨φ.toOrderHom 0, φ.toOrderHom 1, φ.toOrderHom.monotone (by decide),
      Hom.ext_one_left _ _ rfl rfl⟩

Premises used: rfl
-- 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
i j : Fin (n + 1)
hij : i ≤ j
⊢ (↑j : ℕ) ≤ n

Tactic:
lia
-- 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
i j : Fin (n + 1)
hij : i ≤ j
⊢ Eq (α := (fun (X : Type u) ↦ X) (X _⦋1⦌))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (X.map (mkOfLe i j 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 j 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 j hij) fac_aux₁._proof_1)))

Tactic:
exact fac_aux₂ _ _ _ _ _ _ (by lia)

Premises used: SSet.StrictSegal.isPointwiseRightKanExtensionAt.fac_aux₂
-- 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)
j : StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2))
h :
  ∀ (x : s.pt),
    TypeCat.Fun.toFun
        (ConcreteCategory.hom (FC := TypeCat.Fun)
          (CategoryStruct.comp (Y :=
            Cone.pt (J := StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)))
              (RightExtension.coneAt (rightExtensionInclusion X 2) (op ⦋n⦌)))
            (↾fun (x : s.pt) ↦ lift sx s x :
              s.pt ⟶
                Cone.pt (J :=
                  StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)))
                  (RightExtension.coneAt (rightExtensionInclusion X 2) (op ⦋n⦌)))
            (NatTrans.app (Cone.π (RightExtension.coneAt (rightExtensionInclusion X 2) (op ⦋n⦌))) j :
              obj
                  (obj
                    (Functor.const
                      (StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2))))
                    (Cone.pt (J :=
                      StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)))
                      (RightExtension.coneAt (rightExtensionInclusion X 2) (op ⦋n⦌))))
                  j ⟶
                obj (proj (op ⦋n⦌) (Functor.op (inclusion 2)) ⋙ Functor.op (inclusion 2) ⋙ X) j)))
        x =
      (ConcreteCategory.hom (FC := TypeCat.Fun) (s.π.app j)).toFun x
⊢ CategoryStruct.comp (Y :=
      Cone.pt (J := StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)))
        (RightExtension.coneAt (rightExtensionInclusion X 2) (op ⦋n⦌)))
      (↾fun (x : s.pt) ↦ lift sx s x :
        s.pt ⟶
          Cone.pt (J := StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)))
            (RightExtension.coneAt (rightExtensionInclusion X 2) (op ⦋n⦌)))
      (NatTrans.app (Cone.π (RightExtension.coneAt (rightExtensionInclusion X 2) (op ⦋n⦌))) j :
        obj
            (obj
              (Functor.const
                (StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2))))
              (Cone.pt (J := StructuredArrow (C := (SimplexCategory.Truncated 2)ᵒᵖ) (op ⦋n⦌) (Functor.op (inclusion 2)))
                (RightExtension.coneAt (rightExtensionInclusion X 2) (op ⦋n⦌))))
            j ⟶
          obj (proj (op ⦋n⦌) (Functor.op (inclusion 2)) ⋙ Functor.op (inclusion 2) ⋙ X) j) =
    s.π.app j

Tactic:
ext x
-- 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
i : SimplexCategory
hi : i.len ≤ 2
f :
  unop
      (obj (C := (SimplexCategory.Truncated 2)ᵒᵖ) (Functor.op (inclusion 2))
        (op (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) i hi))) ⟶
    unop (op ⦋n⦌)
⊢ ∃ (j : ℕ), ⦋j⦌ = i

Tactic:
⟨_, i.mk_len⟩

Premises used: SimplexCategory.mk_len, Exists.intro
-- 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
i : ℕ
hi : i ≤ 2
f :
  unop
      (obj (C := (SimplexCategory.Truncated 2)ᵒᵖ) (Functor.op (inclusion 2))
        (op (ObjectProperty.FullSubcategory.mk (P := fun (a : SimplexCategory) ↦ a.len ≤ 2) ⦋i⦌ hi))) ⟶
    unop (op ⦋n⦌)
k : Fin (i + 1)
⊢ Fin (⦋i⦌.len + 1)

Tactic:
exact k