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
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)
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))
⊢ Nonempty (IsColimit c)
have refine_4₂ : ∀ (j : Discrete WalkingPair),
  IsPullback (P := obj.{0, v, 0, u} (pair (F.obj { as := 0 }) (pullback c₂.inr i)) j)
    (NatTrans.app (Cocone.ι (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i))) j)
    (NatTrans.app (mapPair (α.app { as := 0 }) (pullback.fst c₂.inr i)) j) i (c₂.ι.app j)
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 }) ⋯)
⊢ 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))
have refine_4 : Eq (α :=
  obj.{0, v, 0, u} (pair (F.obj { as := 0 : Discrete (Fin (n + 1)) }) (pullback c₂.inr i))
      { as := WalkingPair.left : Discrete WalkingPair } ⟶
    obj.{0, v, 0, u} (obj (const (Discrete WalkingPair)) c₂.pt) { as := WalkingPair.left : Discrete WalkingPair })
  (NatTrans.app (mapPair (α.app { as := 0 }) (pullback.fst c₂.inr i) ≫ c₂.ι) { as := WalkingPair.left })
  (NatTrans.app
    (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))
    { as := WalkingPair.left })
Tactic: ext ⟨⟨⟩⟩
-- 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 }) ⋯)
⊢ IsPullback (P :=
    obj.{0, v, 0, u} (pair (F.obj { as := 0 }) (pullback c₂.inr i)) { as := WalkingPair.left : Discrete WalkingPair })
    (NatTrans.app (Cocone.ι (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i)))
      { as := WalkingPair.left })
    (NatTrans.app (mapPair (α.app { as := 0 }) (pullback.fst c₂.inr i)) { as := WalkingPair.left }) i
    (c₂.ι.app { as := WalkingPair.left })
have refine_4 : IsPullback (c.ι.app { as := 0 }) (α.app { as := 0 : Discrete (Fin (n + 1)) }) i c₂.inl
Tactic: simp only [pair_obj_right, Functor.const_obj_obj, pair_obj_left, BinaryCofan.mk_pt,
        BinaryCofan.ι_app_left, BinaryCofan.mk_inl, mapPair_left]
-- 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 }) ⋯)
⊢ IsPullback (P :=
    obj.{0, v, 0, u} (pair (F.obj { as := 0 }) (pullback c₂.inr i)) { as := WalkingPair.right : Discrete WalkingPair })
    (NatTrans.app (Cocone.ι (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i)))
      { as := WalkingPair.right })
    (NatTrans.app (mapPair (α.app { as := 0 }) (pullback.fst c₂.inr i)) { as := WalkingPair.right }) i
    (c₂.ι.app { as := WalkingPair.right })
have refine_4 : IsPullback (P := pullback c₂.inr i) (pullback.snd c₂.inr i) (pullback.fst c₂.inr i) i c₂.inr
Tactic: simp only [pair_obj_right, Functor.const_obj_obj, BinaryCofan.mk_pt, BinaryCofan.ι_app_right,
        BinaryCofan.mk_inr, mapPair_right]
-- 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 }) ⋯)
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)
have refine_4 : ∀
  (H₂ :
    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: obtain ⟨H₂⟩ := t₂'
Cleared hypotheses: t₂'
-- 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 }) ⋯)
H₂ :
  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)
have refine_4 : ∀ (F' : Fin (n + 1) → C) (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 }) sorry))
  (H₂ :
    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: clear_value F'
Cleared hypotheses: F', this, H₁, H₂
-- 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
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 }) ⋯)
H₂ :
  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)
have refine_4 : ∀ (c : Cocone.{0, v, 0, u} (Discrete.functor F'))
  (α : Quiver.Hom (V := Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C) (Discrete.functor 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))
  (H₁ :
    IsColimit.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
      Discrete.functor fun (j : Fin n) ↦
        obj.{0, v, 0, u} (Discrete.functor F') { as := j.succ : Discrete (Fin (n + 1)) })
      (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))
  (H₂ :
    IsColimit (F := pair (obj.{0, v, 0, u} (Discrete.functor F') { 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: subst this
Cleared hypotheses: F, c, α, i, e, hα, H, this, H₁, H₂
-- 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' : Fin (n + 1) → C
c : Cocone.{0, v, 0, u} (Discrete.functor F')
α : Quiver.Hom (V := Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C) (Discrete.functor 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)
H₁ :
  IsColimit.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
    Discrete.functor fun (j : Fin n) ↦ obj.{0, v, 0, u} (Discrete.functor F') { as := j.succ : Discrete (Fin (n + 1)) })
    (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 }) ⋯)
H₂ :
  IsColimit (F := pair (obj.{0, v, 0, u} (Discrete.functor F') { 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)
have refine_4 : ∀ (j : Discrete (Fin (n + 1))),
  NatTrans.app
        (Cocone.ι
          (extendCofan
            (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)
            (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i))))
        j ≫
      Iso.hom
        (Iso.refl
          (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin (n + 1))) (F :=
            Discrete.functor fun (i : Fin (n + 1)) ↦
              obj.{0, v, 0, u} (Discrete.functor F') { as := i : Discrete (Fin (n + 1)) })
            (extendCofan
              (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)
              (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i))))) =
    c.ι.app j
Tactic: refine ⟨IsColimit.ofIsoColimit (extendCofanIsColimit
    (fun i ↦ (Discrete.functor F').obj ⟨i⟩) H₁ H₂) <| Cocone.ext (Iso.refl _) ?_⟩
-- 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' : Fin (n + 1) → C
c : Cocone.{0, v, 0, u} (Discrete.functor F')
α : Quiver.Hom (V := Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C) (Discrete.functor 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)
H₁ :
  IsColimit.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
    Discrete.functor fun (j : Fin n) ↦ obj.{0, v, 0, u} (Discrete.functor F') { as := j.succ : Discrete (Fin (n + 1)) })
    (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 }) ⋯)
H₂ :
  IsColimit (F := pair (obj.{0, v, 0, u} (Discrete.functor F') { as := 0 }) (pullback c₂.inr i))
    (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i))
⊢ ∀ (j : Discrete (Fin (n + 1))),
    NatTrans.app
          (Cocone.ι
            (extendCofan
              (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)) }) ⋯)
              (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i))))
          j ≫
        Iso.hom
          (Iso.refl
            (Cocone.pt.{0, v, 0, u} (J := Discrete (Fin (n + 1))) (F :=
              Discrete.functor fun (i : Fin (n + 1)) ↦
                obj.{0, v, 0, u} (Discrete.functor F') { as := i : Discrete (Fin (n + 1)) })
              (extendCofan
                (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)) }) ⋯)
                (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i))))) =
      c.ι.app j
have refine_4 : ∀ (j : Discrete (Fin (n + 1))),
  Fin.cases (motive := fun (x : Fin (n + 1)) ↦ F' x ⟶ c.pt) (c.ι.app { as := 0 })
        (fun (i_1 : Fin n) ↦
          CategoryStruct.comp (Y := pullback c₂.inr i)
            (pullback.lift (α.app { as := i_1.succ } ≫ c₁.inj i_1) (c.ι.app { as := i_1.succ }) sorry)
            (pullback.snd c₂.inr i))
        j.as ≫
      𝟙 c.pt =
    c.ι.app j
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' : Fin (n + 1) → C
c : Cocone.{0, v, 0, u} (Discrete.functor F')
α : Quiver.Hom (V := Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C) (Discrete.functor 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)
H₁ :
  IsColimit.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
    Discrete.functor fun (j : Fin n) ↦ obj.{0, v, 0, u} (Discrete.functor F') { as := j.succ : Discrete (Fin (n + 1)) })
    (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 }) ⋯)
H₂ :
  IsColimit (F := pair (obj.{0, v, 0, u} (Discrete.functor F') { as := 0 }) (pullback c₂.inr i))
    (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i))
j : Fin (n + 1)
⊢ Eq.{v + 1} (α := F' { as := j : Discrete (Fin (n + 1)) }.as ⟶ c.pt)
    (Fin.cases (motive := fun (x : Fin (n + 1)) ↦ F' x ⟶ c.pt) (c.ι.app { as := 0 })
          (fun (i_1 : Fin n) ↦
            CategoryStruct.comp (Y := pullback c₂.inr i)
              (pullback.lift (α.app { as := i_1.succ } ≫ c₁.inj i_1) (c.ι.app { as := i_1.succ }) ⋯)
              (pullback.snd c₂.inr i))
          { as := j : Discrete (Fin (n + 1)) }.as ≫
        𝟙 c.pt :
      F' { as := j : Discrete (Fin (n + 1)) }.as ⟶ c.pt)
    (c.ι.app { as := j } :
      obj.{0, v, 0, u} (Discrete.functor fun (i : Fin (n + 1)) ↦ F' i) { as := j : Discrete (Fin (n + 1)) } ⟶
        obj.{0, v, 0, u} (obj (const (Discrete (Fin (n + 1)))) c.pt) { as := j : Discrete (Fin (n + 1)) })
have refine_4 : Eq.{v + 1} (α := F' j ⟶ c.pt)
  (Fin.cases (motive := fun (x : Fin (n + 1)) ↦ F' x ⟶ c.pt) (c.ι.app { as := 0 })
      (fun (i : Fin n) ↦ c.ι.app { as := i.succ }) j :
    F' j ⟶ c.pt)
  (c.ι.app { as := j } :
    obj.{0, v, 0, u} (Discrete.functor fun (i : Fin (n + 1)) ↦ F' i) { as := j : Discrete (Fin (n + 1)) } ⟶
      obj.{0, v, 0, u} (obj (const (Discrete (Fin (n + 1)))) c.pt) { as := j : Discrete (Fin (n + 1)) })
Tactic: simp only [limit.lift_π, PullbackCone.mk_pt,
    PullbackCone.mk_π_app, Category.comp_id]
-- 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' : Fin (n + 1) → C
c : Cocone.{0, v, 0, u} (Discrete.functor F')
α : Quiver.Hom (V := Functor.{0, v, 0, u} (Discrete (Fin (n + 1))) C) (Discrete.functor 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)
H₁ :
  IsColimit.{0, v, 0, u} (J := Discrete (Fin n)) (F :=
    Discrete.functor fun (j : Fin n) ↦ obj.{0, v, 0, u} (Discrete.functor F') { as := j.succ : Discrete (Fin (n + 1)) })
    (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 }) ⋯)
H₂ :
  IsColimit (F := pair (obj.{0, v, 0, u} (Discrete.functor F') { as := 0 }) (pullback c₂.inr i))
    (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) }) (pullback.snd c₂.inr i))
j : Fin (n + 1)
⊢ Eq.{v + 1} (α := F' j ⟶ c.pt)
    (Fin.cases (motive := fun (x : Fin (n + 1)) ↦ F' x ⟶ c.pt) (c.ι.app { as := 0 })
        (fun (i : Fin n) ↦ c.ι.app { as := i.succ }) j :
      F' j ⟶ c.pt)
    (c.ι.app { as := j } :
      obj.{0, v, 0, u} (Discrete.functor fun (i : Fin (n + 1)) ↦ F' i) { as := j : Discrete (Fin (n + 1)) } ⟶
        obj.{0, v, 0, u} (obj (const (Discrete (Fin (n + 1)))) c.pt) { as := j : Discrete (Fin (n + 1)) })
have refine_4 : Eq.{v + 1} (α := F' 0 ⟶ c.pt)
  (Fin.cases (motive := fun (x : Fin (n + 1)) ↦ F' x ⟶ c.pt) (c.ι.app { as := 0 })
      (fun (i : Fin n) ↦ c.ι.app { as := i.succ }) 0 :
    F' 0 ⟶ c.pt)
  (c.ι.app { as := 0 } :
    obj.{0, v, 0, u} (C := Discrete (Fin (n + 1))) (Discrete.functor fun (i : Fin (n + 1)) ↦ F' i) { as := 0 } ⟶
      obj.{0, v, 0, u} (C := Discrete (Fin (n + 1))) (obj (const (Discrete (Fin (n + 1)))) c.pt)
        { as := 0 : Discrete (Fin (n + 1)) })
Tactic: induction j using Fin.inductionOn
Cleared hypotheses: j