awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.VanKampen (goal)
Goal:
C : Type u
inst : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryCofan.{v, u} (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ map (const (Discrete (Fin (n + 1)))) i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
h_obj : ∀ (X : Discrete (Fin (n + 1))), F.obj X = obj.{0, v, 0, u} (Discrete.functor F') X
⊢ F = Discrete.functor F'
have h_map : ∀ (X Y : Discrete (Fin (n + 1))) (f : X ⟶ Y),
  HEq (F.map f) (β := obj.{0, v, 0, u} (Discrete.functor F') X ⟶ obj.{0, v, 0, u} (Discrete.functor F') Y)
    (map (Discrete.functor F') f)
Tactic: apply Functor.hext
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.VanKampen (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryCofan.{v, u} (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ map (const (Discrete (Fin (n + 1)))) i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
⊢ Nonempty (IsColimit c)
have this : F = Discrete.functor F'
Tactic: have : F = Discrete.functor F' := by
    apply Functor.hext
    · exact fun i ↦ rfl
    · rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩
      simp [F']
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.VanKampen (goal)
Goal:
C : Type u
inst : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryCofan.{v, u} (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ map (const (Discrete (Fin (n + 1)))) i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
⊢ Nonempty (IsColimit c)
have refine_4 : ∀
  (t₁' :
    Nonempty
      (IsColimit.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ })
        (Cofan.mk (pullback c₂.inr i) fun (j : Fin n) ↦
          pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) sorry))),
  Nonempty (IsColimit c)
Tactic: have t₁' := @t₁ (Discrete.functor (fun j ↦ F.obj ⟨j.succ⟩))
    (Cofan.mk (pullback c₂.inr i) fun j ↦ pullback.lift (α.app _ ≫ c₁.inj _) (c.ι.app _) ?_)
    (Discrete.natTrans fun i ↦ α.app _) (pullback.fst _ _) ?_ (.of_discrete _) ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.VanKampen (goal)
Goal:
C : Type u
inst : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryCofan.{v, u} (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ map (const (Discrete (Fin (n + 1)))) i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
refine_4 :
  ∀
    (t₁' :
      Nonempty
        (IsColimit.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ })
          (Cofan.mk (pullback c₂.inr i) fun (j : Fin n) ↦
            pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯))),
    Nonempty (IsColimit c)
⊢ Nonempty (IsColimit c)
have refine_1 : ∀ (j : Fin n), (α.app { as := j.succ : Discrete (Fin (n + 1)) } ≫ c₁.inj j) ≫ c₂.inr = c.ι.app { as := j.succ } ≫ i
Tactic: have t₁' := @t₁ (Discrete.functor (fun j ↦ F.obj ⟨j.succ⟩))
    (Cofan.mk (pullback c₂.inr i) fun j ↦ pullback.lift (α.app _ ≫ c₁.inj _) (c.ι.app _) ?_)
    (Discrete.natTrans fun i ↦ α.app _) (pullback.fst _ _) ?_ (.of_discrete _) ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.VanKampen (goal)
Goal:
C : Type u
inst : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryCofan.{v, u} (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ map (const (Discrete (Fin (n + 1)))) i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
refine_4 :
  ∀
    (t₁' :
      Nonempty
        (IsColimit.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ })
          (Cofan.mk (pullback c₂.inr i) fun (j : Fin n) ↦
            pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯))),
    Nonempty (IsColimit c)
refine_1 :
  ∀ (j : Fin n), (α.app { as := j.succ : Discrete (Fin (n + 1)) } ≫ c₁.inj j) ≫ c₂.inr = c.ι.app { as := j.succ } ≫ i
⊢ Nonempty (IsColimit c)
have refine_2 : Eq (α :=
  Quiver.Hom (V := Functor.{0, v, 0, u} (Discrete (Fin n)) C)
    (Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ }) (obj (const (Discrete (Fin n))) c₁.pt))
  ((Discrete.natTrans fun (i : Discrete (Fin n)) ↦ α.app { as := i.as.succ }) ≫ c₁.ι)
  (CategoryStruct.comp (Y :=
    obj (const (Discrete (Fin n)))
      (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
        Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ : Discrete (Fin (n + 1)) })
        (Cofan.mk (pullback c₂.inr i) fun (j : Fin n) ↦
          pullback.lift (α.app { as := j.succ : Discrete (Fin (n + 1)) } ≫ c₁.inj j)
            (c.ι.app { as := j.succ : Discrete (Fin (n + 1)) }) sorry)))
    (Cocone.ι
      (Cofan.mk (pullback c₂.inr i) fun (j : Fin n) ↦
        pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) sorry))
    (map (const (Discrete (Fin n))) (pullback.fst c₂.inr i)))
Tactic: have t₁' := @t₁ (Discrete.functor (fun j ↦ F.obj ⟨j.succ⟩))
    (Cofan.mk (pullback c₂.inr i) fun j ↦ pullback.lift (α.app _ ≫ c₁.inj _) (c.ι.app _) ?_)
    (Discrete.natTrans fun i ↦ α.app _) (pullback.fst _ _) ?_ (.of_discrete _) ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.VanKampen (goal)
Goal:
C : Type u
inst : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryCofan.{v, u} (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ map (const (Discrete (Fin (n + 1)))) i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
⊢ Eq (α :=
    Quiver.Hom (V := Functor.{0, v, 0, u} (Discrete (Fin n)) C)
      (Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ }) (obj (const (Discrete (Fin n))) c₁.pt))
    ((Discrete.natTrans fun (i : Discrete (Fin n)) ↦ α.app { as := i.as.succ }) ≫ c₁.ι)
    (CategoryStruct.comp (Y :=
      obj (const (Discrete (Fin n)))
        (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
          Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ : Discrete (Fin (n + 1)) })
          (Cofan.mk (pullback c₂.inr i) fun (j : Fin n) ↦
            pullback.lift (α.app { as := j.succ : Discrete (Fin (n + 1)) } ≫ c₁.inj j)
              (c.ι.app { as := j.succ : Discrete (Fin (n + 1)) }) ⋯)))
      (Cocone.ι
        (Cofan.mk (pullback c₂.inr i) fun (j : Fin n) ↦
          pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯))
      (map (const (Discrete (Fin n))) (pullback.fst c₂.inr i)))
have refine_2 : ∀ (j : Discrete (Fin n)),
  Eq (α :=
    obj.{0, v, 0, u} (Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ }) j ⟶
      obj.{0, v, 0, u} (obj (const (Discrete (Fin n))) c₁.pt) j)
    (NatTrans.app ((Discrete.natTrans fun (i : Discrete (Fin n)) ↦ α.app { as := i.as.succ }) ≫ c₁.ι) j)
    (NatTrans.app
      (CategoryStruct.comp (Y :=
        obj (const (Discrete (Fin n)))
          (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
            Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ : Discrete (Fin (n + 1)) })
            (Cofan.mk (pullback c₂.inr i) fun (j : Fin n) ↦
              pullback.lift (α.app { as := j.succ : Discrete (Fin (n + 1)) } ≫ c₁.inj j)
                (c.ι.app { as := j.succ : Discrete (Fin (n + 1)) }) sorry)))
        (Cocone.ι
          (Cofan.mk (pullback c₂.inr i) fun (j : Fin n) ↦
            pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) sorry))
        (map (const (Discrete (Fin n))) (pullback.fst c₂.inr i)))
      j)
Tactic: ext j
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.VanKampen (goal)
Goal:
C : Type u
inst : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryCofan.{v, u} (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ map (const (Discrete (Fin (n + 1)))) i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
j : Discrete (Fin n)
⊢ Eq (α :=
    obj.{0, v, 0, u} (Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ }) j ⟶
      obj.{0, v, 0, u} (obj (const (Discrete (Fin n))) c₁.pt) j)
    (NatTrans.app ((Discrete.natTrans fun (i : Discrete (Fin n)) ↦ α.app { as := i.as.succ }) ≫ c₁.ι) j)
    (NatTrans.app
      (CategoryStruct.comp (Y :=
        obj (const (Discrete (Fin n)))
          (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
            Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ : Discrete (Fin (n + 1)) })
            (Cofan.mk (pullback c₂.inr i) fun (j : Fin n) ↦
              pullback.lift (α.app { as := j.succ : Discrete (Fin (n + 1)) } ≫ c₁.inj j)
                (c.ι.app { as := j.succ : Discrete (Fin (n + 1)) }) ⋯)))
        (Cocone.ι
          (Cofan.mk (pullback c₂.inr i) fun (j : Fin n) ↦
            pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯))
        (map (const (Discrete (Fin n))) (pullback.fst c₂.inr i)))
      j)
have refine_2 : α.app { as := j.as.succ : Discrete (Fin (n + 1)) } ≫ c₁.ι.app j =
  CategoryStruct.comp (Y := pullback c₂.inr i)
    (pullback.lift (α.app { as := j.as.succ : Discrete (Fin (n + 1)) } ≫ c₁.inj j.as) (c.ι.app { as := j.as.succ }) sorry)
    (pullback.fst c₂.inr i)
Tactic: dsimp
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.VanKampen (goal)
Goal:
C : Type u
inst : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryCofan.{v, u} (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ map (const (Discrete (Fin (n + 1)))) i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
j : Discrete (Fin n)
⊢ IsPullback (P := obj.{0, v, 0, u} (Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ }) j) (X :=
    obj.{0, v, 0, u}
      (obj (const (Discrete (Fin n)))
        (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
          Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ : Discrete (Fin (n + 1)) })
          (Cofan.mk (pullback c₂.inr i) fun (j : Fin n) ↦
            pullback.lift (α.app { as := j.succ : Discrete (Fin (n + 1)) } ≫ c₁.inj j)
              (c.ι.app { as := j.succ : Discrete (Fin (n + 1)) }) ⋯)))
      j)
    (NatTrans.app
      (Cocone.ι
        (Cofan.mk (pullback c₂.inr i) fun (j : Fin n) ↦
          pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯))
      j)
    (NatTrans.app (Discrete.natTrans fun (i : Discrete (Fin n)) ↦ α.app { as := i.as.succ }) j) (pullback.fst c₂.inr i)
    (c₁.ι.app j)
have refine_3 : IsPullback (X := pullback c₂.inr i)
  (pullback.lift (α.app { as := j.as.succ : Discrete (Fin (n + 1)) } ≫ c₁.inj j.as) (c.ι.app { as := j.as.succ }) sorry)
  (α.app { as := j.as.succ : Discrete (Fin (n + 1)) }) (pullback.fst c₂.inr i) (c₁.ι.app j)
Tactic: simp only [pair_obj_right, Functor.const_obj_obj, Discrete.functor_obj,
      Cofan.mk_pt, Cofan.mk_ι_app, Discrete.natTrans_app]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.VanKampen (goal)
Goal:
C : Type u
inst : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryCofan.{v, u} (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ map (const (Discrete (Fin (n + 1)))) i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
j : Discrete (Fin n)
⊢ IsPullback (X := pullback c₂.inr i)
    (pullback.lift (α.app { as := j.as.succ : Discrete (Fin (n + 1)) } ≫ c₁.inj j.as) (c.ι.app { as := j.as.succ }) ⋯)
    (α.app { as := j.as.succ : Discrete (Fin (n + 1)) }) (pullback.fst c₂.inr i) (c₁.ι.app j)
have refine_3 : IsPullback
  (CategoryStruct.comp (Y := pullback c₂.inr i)
    (pullback.lift (α.app { as := j.as.succ : Discrete (Fin (n + 1)) } ≫ c₁.inj j.as) (c.ι.app { as := j.as.succ }) sorry)
    (pullback.snd c₂.inr i))
  (α.app { as := j.as.succ : Discrete (Fin (n + 1)) }) i (c₁.ι.app j ≫ c₂.inr)
Tactic: refine IsPullback.of_right ?_ ?_ (IsPullback.of_hasPullback (BinaryCofan.inr c₂) i).flip
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.VanKampen (goal)
Goal:
C : Type u
inst : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryCofan.{v, u} (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ map (const (Discrete (Fin (n + 1)))) i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
j : Discrete (Fin n)
refine_3 :
  IsPullback
    (CategoryStruct.comp (Y := pullback c₂.inr i)
      (pullback.lift (α.app { as := j.as.succ : Discrete (Fin (n + 1)) } ≫ c₁.inj j.as) (c.ι.app { as := j.as.succ }) ⋯)
      (pullback.snd c₂.inr i))
    (α.app { as := j.as.succ : Discrete (Fin (n + 1)) }) i (c₁.ι.app j ≫ c₂.inr)
⊢ IsPullback (X := pullback c₂.inr i)
    (pullback.lift (α.app { as := j.as.succ : Discrete (Fin (n + 1)) } ≫ c₁.inj j.as) (c.ι.app { as := j.as.succ }) ⋯)
    (α.app { as := j.as.succ : Discrete (Fin (n + 1)) }) (pullback.fst c₂.inr i) (c₁.ι.app j)
have refine_3₁ : CategoryStruct.comp (Y := pullback c₂.inr i)
    (pullback.lift (α.app { as := j.as.succ : Discrete (Fin (n + 1)) } ≫ c₁.inj j.as) (c.ι.app { as := j.as.succ }) sorry)
    (pullback.fst c₂.inr i) =
  α.app { as := j.as.succ : Discrete (Fin (n + 1)) } ≫ c₁.ι.app j
Tactic: refine IsPullback.of_right ?_ ?_ (IsPullback.of_hasPullback (BinaryCofan.inr c₂) i).flip
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.VanKampen (goal)
Goal:
C : Type u
inst : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryCofan.{v, u} (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ map (const (Discrete (Fin (n + 1)))) i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
j : Discrete (Fin n)
⊢ IsPullback
    (CategoryStruct.comp (Y := pullback c₂.inr i)
      (pullback.lift (α.app { as := j.as.succ : Discrete (Fin (n + 1)) } ≫ c₁.inj j.as) (c.ι.app { as := j.as.succ }) ⋯)
      (pullback.snd c₂.inr i))
    (α.app { as := j.as.succ : Discrete (Fin (n + 1)) }) i (c₁.ι.app j ≫ c₂.inr)
have refine_3 : IsPullback (c.ι.app { as := j.as.succ }) (α.app { as := j.as.succ : Discrete (Fin (n + 1)) }) i (c₁.ι.app j ≫ c₂.inr)
Tactic: simp only [Functor.const_obj_obj, pair_obj_right, limit.lift_π,
        PullbackCone.mk_pt, PullbackCone.mk_π_app]
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.VanKampen (goal)
Goal:
C : Type u
inst : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryCofan.{v, u} (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ map (const (Discrete (Fin (n + 1)))) i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
t₁' :
  Nonempty
    (IsColimit.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ })
      (Cofan.mk (pullback c₂.inr i) fun (j : Fin n) ↦
        pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯))
⊢ Nonempty (IsColimit c)
have refine_4 : ∀
  (H₁ :
    IsColimit.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ })
      (Cofan.mk (pullback c₂.inr i) fun (j : Fin n) ↦
        pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) sorry)),
  Nonempty (IsColimit c)
Tactic: obtain ⟨H₁⟩ := t₁'
Cleared hypotheses: t₁'
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.VanKampen (goal)
Goal:
C : Type u
inst : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryCofan.{v, u} (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ map (const (Discrete (Fin (n + 1)))) i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
H₁ :
  IsColimit.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ })
    (Cofan.mk (pullback c₂.inr i) fun (j : Fin n) ↦
      pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯)
⊢ Nonempty (IsColimit c)
have refine_4 : ∀
  (t₂' :
    Nonempty
      (IsColimit (F := pair (F.obj { as := 0 }) (pullback c₂.inr i))
        (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i)))),
  Nonempty (IsColimit c)
Tactic: have t₂' := @t₂ (pair (F.obj ⟨0⟩) (pullback c₂.inr i))
    (BinaryCofan.mk (c.ι.app ⟨0⟩) (pullback.snd _ _)) (mapPair (α.app _) (pullback.fst _ _)) i ?_
    (.of_discrete _) ?_
-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.VanKampen (goal)
Goal:
C : Type u
inst : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan.{0, v, u} (C := C) fun (i : Fin n) ↦ f i.succ
c₂ : BinaryCofan.{v, u} (f 0) c₁.pt
t₁ : IsUniversalColimit c₁
t₂ : IsUniversalColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
F : Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C
c : Cocone F
α : F ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ map (const (Discrete (Fin (n + 1)))) i
hα : NatTrans.Equifibered α
H : ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)
F' : Fin (n + 1) → C := F.obj ∘ Discrete.mk
this : F = Discrete.functor F'
H₁ :
  IsColimit.{0, v, 0, u} (J := Discrete (Fin n)) (F := Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ })
    (Cofan.mk (pullback c₂.inr i) fun (j : Fin n) ↦
      pullback.lift (α.app { as := j.succ } ≫ c₁.inj j) (c.ι.app { as := j.succ }) ⋯)
refine_4 :
  ∀
    (t₂' :
      Nonempty
        (IsColimit (F := pair (F.obj { as := 0 }) (pullback c₂.inr i))
          (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i)))),
    Nonempty (IsColimit c)
⊢ Nonempty (IsColimit c)
have refine_4₁ : Eq (α :=
  Quiver.Hom (V := Functor.{0, v, 0, u} (Discrete WalkingPair) C) (pair (F.obj { as := 0 }) (pullback c₂.inr i))
    (obj (const (Discrete WalkingPair)) c₂.pt))
  (mapPair (α.app { as := 0 }) (pullback.fst c₂.inr i) ≫ c₂.ι)
  (CategoryStruct.comp (Y :=
    obj (const (Discrete WalkingPair))
      (Cocone.pt (F := pair (F.obj { as := 0 : Discrete (Fin (n + 1)) }) (pullback c₂.inr i))
        (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i))))
    (Cocone.ι (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i)))
    (map (const (Discrete WalkingPair)) i))
Tactic: have t₂' := @t₂ (pair (F.obj ⟨0⟩) (pullback c₂.inr i))
    (BinaryCofan.mk (c.ι.app ⟨0⟩) (pullback.snd _ _)) (mapPair (α.app _) (pullback.fst _ _)) i ?_
    (.of_discrete _) ?_