awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Limits.VanKampen
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 : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst_2 : HasFiniteCoproducts.{v, u} C
j_1 : Fin (n + 1)
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
Hc : IsColimit c
T : C
f₁ : F' 0 ⟶ T
f₂ :
  (sigmaObj.{0, v, u} fun (j : Fin n) ↦
      obj.{0, v, 0, u} (Discrete.functor F') { as := j.succ : Discrete (Fin (n + 1)) }) ⟶
    T
f₃ :
  Cocone.pt (F :=
      pair (F' 0)
        (sigmaObj.{0, v, u} fun (j : Fin n) ↦
          obj.{0, v, 0, u} (Discrete.functor F') { as := j.succ : Discrete (Fin (n + 1)) }))
      (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) })
        (Sigma.desc fun (b : Fin n) ↦ c.ι.app { as := b.succ : Discrete (Fin (n + 1)) })) ⟶
    T
m₁ : c.ι.app { as := 0 } ≫ f₃ = f₁
m₂ : (Sigma.desc fun (b : Fin n) ↦ c.ι.app { as := b.succ }) ≫ f₃ = f₂
j : Fin (n + 1)
refine_2 :
  Eq (α := F' j ⟶ T) (c.ι.app { as := j } ≫ f₃)
    (Fin.cases (motive := fun (x : Fin (n + 1)) ↦ F' x ⟶ T) f₁
      (fun (i : Fin n) ↦ Sigma.ι.{0, v, u} (C := C) (fun (j : Fin n) ↦ F' j.succ) i ≫ f₂) j)
⊢ Eq (α :=
    obj.{0, v, 0, u} (Discrete.functor F') { as := j : Discrete (Fin (n + 1)) } ⟶
      Cocone.pt.{0, v, 0, u} (J := Discrete (Fin (n + 1))) (F := Discrete.functor F')
        (Cofan.mk T fun (i : Fin (n + 1)) ↦
          Fin.cases (motive := fun (x : Fin (n + 1)) ↦ F' x ⟶ T) f₁
            (fun (i : Fin n) ↦
              Sigma.ι.{0, v, u} (C := C)
                  (fun (j : Fin n) ↦ obj.{0, v, 0, u} (Discrete.functor F') { as := j.succ : Discrete (Fin (n + 1)) })
                  i ≫
                f₂)
            i))
    (c.ι.app { as := j } ≫ f₃)
    (NatTrans.app
      (Cocone.ι
        (Cofan.mk T fun (i : Fin (n + 1)) ↦
          Fin.cases (motive := fun (x : Fin (n + 1)) ↦ F' x ⟶ T) f₁
            (fun (i : Fin n) ↦
              Sigma.ι.{0, v, u} (C := C)
                  (fun (j : Fin n) ↦ obj.{0, v, 0, u} (Discrete.functor F') { as := j.succ : Discrete (Fin (n + 1)) })
                  i ≫
                f₂)
            i))
      { as := j })

Tactic:
simp only [Discrete.functor_obj, Cofan.mk_pt, Functor.const_obj_obj, Cofan.mk_ι_app]

Premises used: CategoryTheory.Discrete.functor_obj, CategoryTheory.Limits.Cofan.mk_ι_app, CategoryTheory.Limits.Cofan.mk_pt, CategoryTheory.Functor.const_obj_obj
-- Module: Mathlib.CategoryTheory.Limits.VanKampen
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 : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst_2 : HasFiniteCoproducts.{v, u} C
j_1 : Fin (n + 1)
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
Hc : IsColimit c
T : C
f₁ : F' 0 ⟶ T
f₂ :
  (sigmaObj.{0, v, u} fun (j : Fin n) ↦
      obj.{0, v, 0, u} (Discrete.functor F') { as := j.succ : Discrete (Fin (n + 1)) }) ⟶
    T
f₃ :
  Cocone.pt (F :=
      pair (F' 0)
        (sigmaObj.{0, v, u} fun (j : Fin n) ↦
          obj.{0, v, 0, u} (Discrete.functor F') { as := j.succ : Discrete (Fin (n + 1)) }))
      (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) })
        (Sigma.desc fun (b : Fin n) ↦ c.ι.app { as := b.succ : Discrete (Fin (n + 1)) })) ⟶
    T
m₁ : c.ι.app { as := 0 } ≫ f₃ = f₁
m₂ : (Sigma.desc fun (b : Fin n) ↦ c.ι.app { as := b.succ }) ≫ f₃ = f₂
j : Fin (n + 1)
refine_2 :
  Eq (α := F' 0 ⟶ T) (c.ι.app { as := 0 } ≫ f₃)
    (Fin.cases (motive := fun (x : Fin (n + 1)) ↦ F' x ⟶ T) f₁
      (fun (i : Fin n) ↦ Sigma.ι.{0, v, u} (C := C) (fun (j : Fin n) ↦ F' j.succ) i ≫ f₂) 0)
refine_2₁ :
  ∀ (i : Fin n),
    Eq (α := F' i.castSucc ⟶ T) (c.ι.app { as := i.castSucc } ≫ f₃)
        (Fin.cases (motive := fun (x : Fin (n + 1)) ↦ F' x ⟶ T) f₁
          (fun (i : Fin n) ↦ Sigma.ι.{0, v, u} (C := C) (fun (j : Fin n) ↦ F' j.succ) i ≫ f₂) i.castSucc) →
      Eq (α := F' i.succ ⟶ T) (c.ι.app { as := i.succ } ≫ f₃)
        (Fin.cases (motive := fun (x : Fin (n + 1)) ↦ F' x ⟶ T) f₁
          (fun (i : Fin n) ↦ Sigma.ι.{0, v, u} (C := C) (fun (j : Fin n) ↦ F' j.succ) i ≫ f₂) i.succ)
⊢ Eq (α := F' j ⟶ T) (c.ι.app { as := j } ≫ f₃)
    (Fin.cases (motive := fun (x : Fin (n + 1)) ↦ F' x ⟶ T) f₁
      (fun (i : Fin n) ↦ Sigma.ι.{0, v, u} (C := C) (fun (j : Fin n) ↦ F' j.succ) i ≫ f₂) j)

Tactic:
induction j using Fin.inductionOn
-- Module: Mathlib.CategoryTheory.Limits.VanKampen
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 : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst_2 : HasFiniteCoproducts.{v, u} C
j : Fin (n + 1)
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
Hc : IsColimit c
T : C
f₁ : F' 0 ⟶ T
f₂ :
  (sigmaObj.{0, v, u} fun (j : Fin n) ↦
      obj.{0, v, 0, u} (Discrete.functor F') { as := j.succ : Discrete (Fin (n + 1)) }) ⟶
    T
f₃ :
  Cocone.pt (F :=
      pair (F' 0)
        (sigmaObj.{0, v, u} fun (j : Fin n) ↦
          obj.{0, v, 0, u} (Discrete.functor F') { as := j.succ : Discrete (Fin (n + 1)) }))
      (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) })
        (Sigma.desc fun (b : Fin n) ↦ c.ι.app { as := b.succ : Discrete (Fin (n + 1)) })) ⟶
    T
m₁ : c.ι.app { as := 0 } ≫ f₃ = f₁
m₂ : (Sigma.desc fun (b : Fin n) ↦ c.ι.app { as := b.succ }) ≫ f₃ = f₂
⊢ Eq (α := F' 0 ⟶ T) (c.ι.app { as := 0 } ≫ f₃)
    (Fin.cases (motive := fun (x : Fin (n + 1)) ↦ F' x ⟶ T) f₁
      (fun (i : Fin n) ↦ Sigma.ι.{0, v, u} (C := C) (fun (j : Fin n) ↦ F' j.succ) i ≫ f₂) 0)

Tactic:
simp only [Fin.cases_zero, m₁]

Premises used: Fin.cases_zero
-- Module: Mathlib.CategoryTheory.Limits.VanKampen
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 : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst_2 : HasFiniteCoproducts.{v, u} C
j : Fin (n + 1)
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
Hc : IsColimit c
T : C
f₁ : F' 0 ⟶ T
f₂ :
  (sigmaObj.{0, v, u} fun (j : Fin n) ↦
      obj.{0, v, 0, u} (Discrete.functor F') { as := j.succ : Discrete (Fin (n + 1)) }) ⟶
    T
f₃ :
  Cocone.pt (F :=
      pair (F' 0)
        (sigmaObj.{0, v, u} fun (j : Fin n) ↦
          obj.{0, v, 0, u} (Discrete.functor F') { as := j.succ : Discrete (Fin (n + 1)) }))
      (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) })
        (Sigma.desc fun (b : Fin n) ↦ c.ι.app { as := b.succ : Discrete (Fin (n + 1)) })) ⟶
    T
m₁ : c.ι.app { as := 0 } ≫ f₃ = f₁
m₂ : (Sigma.desc fun (b : Fin n) ↦ c.ι.app { as := b.succ }) ≫ f₃ = f₂
i_1 : Fin n
h :
  Eq (α := F' i_1.castSucc ⟶ T) (c.ι.app { as := i_1.castSucc } ≫ f₃)
    (Fin.cases (motive := fun (x : Fin (n + 1)) ↦ F' x ⟶ T) f₁
      (fun (i : Fin n) ↦ Sigma.ι.{0, v, u} (C := C) (fun (j : Fin n) ↦ F' j.succ) i ≫ f₂) i_1.castSucc)
⊢ Eq (α := F' i_1.succ ⟶ T) (c.ι.app { as := i_1.succ } ≫ f₃)
    (Fin.cases (motive := fun (x : Fin (n + 1)) ↦ F' x ⟶ T) f₁
      (fun (i : Fin n) ↦ Sigma.ι.{0, v, u} (C := C) (fun (j : Fin n) ↦ F' j.succ) i ≫ f₂) i_1.succ)

Tactic:
simp only [← m₂, colimit.ι_desc_assoc, Discrete.functor_obj,
          Cofan.mk_pt, Cofan.mk_ι_app, Fin.cases_succ]

Premises used: CategoryTheory.Discrete.functor_obj, CategoryTheory.Limits.Cofan.mk_ι_app, CategoryTheory.Limits.Cofan.mk_pt, CategoryTheory.Limits.colimit.ι_desc_assoc, Fin.cases_succ
-- Module: Mathlib.CategoryTheory.Limits.VanKampen
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 : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst_2 : HasFiniteCoproducts.{v, u} C
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
Hc : IsColimit c
t₂' :
  ∀ (j : Discrete WalkingPair),
    IsPullback (P :=
      obj.{0, v, 0, u} (pair (F.obj { as := 0 }) (sigmaObj.{0, v, u} fun (j : Fin n) ↦ F.obj { as := j.succ })) j)
      (NatTrans.app
        (Cocone.ι
          (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) })
            (Sigma.desc fun (b : Fin n) ↦ c.ι.app { as := b.succ })))
        j)
      (NatTrans.app (mapPair (α.app { as := 0 }) (Sigma.desc fun (b : Fin n) ↦ α.app { as := b.succ } ≫ c₁.inj b)) j) i
      (c₂.ι.app j)
⊢ IsPullback (P := F.obj { as := 0 }) (c.ι.app { as := 0 }) (α.app { as := 0 }) i
    ((extendCofan c₁ c₂).ι.app { as := 0 })

Tactic:
t₂' ⟨WalkingPair.left⟩

Premises used: Eq.mpr, CategoryTheory.Discrete.functor, Fin.induction.go, Fin.ofNat, CategoryTheory.Discrete.mk, Fin.induction, CategoryTheory.Limits.pair, Nat.mod, Fin.cases, CategoryTheory.Limits.WalkingPair.left
-- Module: Mathlib.CategoryTheory.Limits.VanKampen
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 : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst_2 : HasFiniteCoproducts.{v, u} C
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
Hc : IsColimit c
t₂' :
  ∀ (j : Discrete WalkingPair),
    IsPullback (P :=
      obj.{0, v, 0, u} (pair (F.obj { as := 0 }) (sigmaObj.{0, v, u} fun (j : Fin n) ↦ F.obj { as := j.succ })) j)
      (NatTrans.app
        (Cocone.ι
          (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) })
            (Sigma.desc fun (b : Fin n) ↦ c.ι.app { as := b.succ })))
        j)
      (NatTrans.app (mapPair (α.app { as := 0 }) (Sigma.desc fun (b : Fin n) ↦ α.app { as := b.succ } ≫ c₁.inj b)) j) i
      (c₂.ι.app j)
⊢ IsPullback (P := F.obj { as := 0 }) (c.ι.app { as := 0 }) (α.app { as := 0 }) i
    ((extendCofan c₁ c₂).ι.app { as := 0 })

Tactic:
exact t₂' ⟨WalkingPair.left⟩

Premises used: CategoryTheory.Discrete.mk, CategoryTheory.Limits.WalkingPair.left
-- Module: Mathlib.CategoryTheory.Limits.VanKampen
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 : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst_2 : HasFiniteCoproducts.{v, u} C
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
Hc : IsColimit c
t₂' :
  ∀ (j : Discrete WalkingPair),
    IsPullback (P :=
      obj.{0, v, 0, u} (pair (F.obj { as := 0 }) (sigmaObj.{0, v, u} fun (j : Fin n) ↦ F.obj { as := j.succ })) j)
      (NatTrans.app
        (Cocone.ι
          (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) })
            (Sigma.desc fun (b : Fin n) ↦ c.ι.app { as := b.succ })))
        j)
      (NatTrans.app (mapPair (α.app { as := 0 }) (Sigma.desc fun (b : Fin n) ↦ α.app { as := b.succ } ≫ c₁.inj b)) j) i
      (c₂.ι.app j)
j_1 : Fin n
h :
  IsPullback (P := F.obj { as := j_1.castSucc }) (c.ι.app { as := j_1.castSucc }) (α.app { as := j_1.castSucc }) i
    ((extendCofan c₁ c₂).ι.app { as := j_1.castSucc })
j : Fin n
refine_3 :
  Eq (α := F.obj { as := j.succ : Discrete (Fin (n + 1)) } ⟶ c₁.pt)
    (α.app { as := j.succ : Discrete (Fin (n + 1)) } ≫ c₁.ι.app { as := j })
    (CategoryStruct.comp (Y :=
      colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ }))
      (Sigma.ι.{0, v, u} (C := C) (fun (j : Fin n) ↦ F.obj { as := j.succ }) j)
      (Sigma.desc fun (j : Fin n) ↦ α.app { as := j.succ } ≫ c₁.inj j))
⊢ Eq (α :=
    obj.{0, v, 0, u} (Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ }) { as := j : Discrete (Fin n) } ⟶
      obj.{0, v, 0, u} (obj (const (Discrete (Fin n))) c₁.pt) { as := j : Discrete (Fin n) })
    (NatTrans.app ((Discrete.natTrans fun (i : Discrete (Fin n)) ↦ α.app { as := i.as.succ }) ≫ c₁.ι) { as := 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 })
            (Cofan.mk (colimit.cocone (Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ })).1
              (Sigma.ι fun (j : Fin n) ↦ F.obj { as := j.succ }))))
        (Cocone.ι
          (Cofan.mk (colimit.cocone (Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ })).1
            (Sigma.ι fun (j : Fin n) ↦ F.obj { as := j.succ })))
        (map (const (Discrete (Fin n))) (Sigma.desc fun (j : Fin n) ↦ α.app { as := j.succ } ≫ c₁.inj j)))
      { as := j })

Tactic:
dsimp
-- Module: Mathlib.CategoryTheory.Limits.VanKampen
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 : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst_2 : HasFiniteCoproducts.{v, u} C
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
Hc : IsColimit c
t₂' :
  ∀ (j : Discrete WalkingPair),
    IsPullback (P :=
      obj.{0, v, 0, u} (pair (F.obj { as := 0 }) (sigmaObj.{0, v, u} fun (j : Fin n) ↦ F.obj { as := j.succ })) j)
      (NatTrans.app
        (Cocone.ι
          (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) })
            (Sigma.desc fun (b : Fin n) ↦ c.ι.app { as := b.succ })))
        j)
      (NatTrans.app (mapPair (α.app { as := 0 }) (Sigma.desc fun (b : Fin n) ↦ α.app { as := b.succ } ≫ c₁.inj b)) j) i
      (c₂.ι.app j)
j_1 : Fin n
h :
  IsPullback (P := F.obj { as := j_1.castSucc }) (c.ι.app { as := j_1.castSucc }) (α.app { as := j_1.castSucc }) i
    ((extendCofan c₁ c₂).ι.app { as := j_1.castSucc })
j : Fin n
refine_3 :
  Eq (α := F.obj { as := j.succ : Discrete (Fin (n + 1)) } ⟶ c₁.pt)
    (α.app { as := j.succ : Discrete (Fin (n + 1)) } ≫ c₁.ι.app { as := j })
    (NatTrans.app (Cocone.ι (Cofan.mk c₁.pt fun (j : Fin n) ↦ α.app { as := j.succ } ≫ c₁.inj j)) { as := j })
⊢ Eq (α := F.obj { as := j.succ : Discrete (Fin (n + 1)) } ⟶ c₁.pt)
    (α.app { as := j.succ : Discrete (Fin (n + 1)) } ≫ c₁.ι.app { as := j })
    (CategoryStruct.comp (Y :=
      colimit.{0, 0, v, u} (J := Discrete (Fin n)) (Discrete.functor fun (j : Fin n) ↦ F.obj { as := j.succ }))
      (Sigma.ι.{0, v, u} (C := C) (fun (j : Fin n) ↦ F.obj { as := j.succ }) j)
      (Sigma.desc fun (j : Fin n) ↦ α.app { as := j.succ } ≫ c₁.inj j))

Tactic:
rw [colimit.ι_desc]

Premises used: CategoryTheory.Limits.colimit.ι_desc
-- Module: Mathlib.CategoryTheory.Limits.VanKampen
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 : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst_2 : HasFiniteCoproducts.{v, u} C
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
Hc : IsColimit c
t₂' :
  ∀ (j : Discrete WalkingPair),
    IsPullback (P :=
      obj.{0, v, 0, u} (pair (F.obj { as := 0 }) (sigmaObj.{0, v, u} fun (j : Fin n) ↦ F.obj { as := j.succ })) j)
      (NatTrans.app
        (Cocone.ι
          (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) })
            (Sigma.desc fun (b : Fin n) ↦ c.ι.app { as := b.succ })))
        j)
      (NatTrans.app (mapPair (α.app { as := 0 }) (Sigma.desc fun (b : Fin n) ↦ α.app { as := b.succ } ≫ c₁.inj b)) j) i
      (c₂.ι.app j)
j_1 : Fin n
h :
  IsPullback (P := F.obj { as := j_1.castSucc }) (c.ι.app { as := j_1.castSucc }) (α.app { as := j_1.castSucc }) i
    ((extendCofan c₁ c₂).ι.app { as := j_1.castSucc })
j : Fin n
⊢ Eq (α := F.obj { as := j.succ : Discrete (Fin (n + 1)) } ⟶ c₁.pt)
    (α.app { as := j.succ : Discrete (Fin (n + 1)) } ≫ c₁.ι.app { as := j })
    (NatTrans.app (Cocone.ι (Cofan.mk c₁.pt fun (j : Fin n) ↦ α.app { as := j.succ } ≫ c₁.inj j)) { as := j })

Tactic:
rfl
-- Module: Mathlib.CategoryTheory.Limits.VanKampen
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 : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst_1 : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst_2 : HasFiniteCoproducts.{v, u} C
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
Hc : IsColimit c
j : Fin (n + 1)
t₂' :
  ∀ (j : Discrete WalkingPair),
    IsPullback (P :=
      obj.{0, v, 0, u} (pair (F.obj { as := 0 }) (sigmaObj.{0, v, u} fun (j : Fin n) ↦ F.obj { as := j.succ })) j)
      (NatTrans.app
        (Cocone.ι
          (BinaryCofan.mk (c.ι.app { as := 0 : Discrete (Fin (n + 1)) })
            (Sigma.desc fun (b : Fin n) ↦ c.ι.app { as := b.succ })))
        j)
      (NatTrans.app (mapPair (α.app { as := 0 }) (Sigma.desc fun (b : Fin n) ↦ α.app { as := b.succ } ≫ c₁.inj b)) j) i
      (c₂.ι.app j)
refine_3 :
  IsPullback (P := F.obj { as := 0 }) (c.ι.app { as := 0 }) (α.app { as := 0 }) i
    ((extendCofan c₁ c₂).ι.app { as := 0 })
refine_3₁ :
  ∀ (j : Fin n),
    IsPullback (P := F.obj { as := j.castSucc }) (c.ι.app { as := j.castSucc }) (α.app { as := j.castSucc }) i
        ((extendCofan c₁ c₂).ι.app { as := j.castSucc }) →
      IsPullback (P := F.obj { as := j.succ }) (c.ι.app { as := j.succ }) (α.app { as := j.succ }) i
        ((extendCofan c₁ c₂).ι.app { as := j.succ })
⊢ IsPullback (P := F.obj { as := j }) (c.ι.app { as := j }) (α.app { as := j }) i
    ((extendCofan c₁ c₂).ι.app { as := j })

Tactic:
induction j using Fin.inductionOn with
  | zero => exact t₂' ⟨WalkingPair.left⟩
  | succ j _ =>
    have t₁' := (@t₁ (Discrete.functor (fun j ↦ F.obj ⟨j.succ⟩)) (Cofan.mk _ _) (Discrete.natTrans
      fun i ↦ α.app _) (Sigma.desc (fun j ↦ α.app _ ≫ c₁.inj _)) ?_
      (.of_discrete _)).mp ⟨coproductIsCoproduct _⟩ ⟨j⟩
    rotate_left
    · ext ⟨j⟩
      dsimp
      rw [colimit.ι_desc]
      rfl
    simpa [Functor.const_obj_obj, Discrete.functor_obj, extendCofan_pt, extendCofan_ι_app,
      Fin.cases_succ, BinaryCofan.mk_pt, colimit.cocone_x, Cofan.mk_pt, Cofan.mk_ι_app,
      BinaryCofan.ι_app_right, BinaryCofan.mk_inr, colimit.ι_desc,
      Discrete.natTrans_app] using t₁'.paste_horiz (t₂' ⟨WalkingPair.right⟩)

Premises used: CategoryTheory.Limits.BinaryCofan.mk_pt, CategoryTheory.Discrete.functor_obj, CategoryTheory.Limits.BinaryCofan.mk_inr, CategoryTheory.Limits.Cofan.mk_ι_app, CategoryTheory.Limits.Cofan.mk_pt, CategoryTheory.Limits.BinaryCofan.ι_app_right, CategoryTheory.Discrete.mk, CategoryTheory.extendCofan_pt, CategoryTheory.extendCofan_ι_app, CategoryTheory.Limits.colimit.ι_desc, CategoryTheory.Limits.colimit.cocone_x, Fin.cases_succ, CategoryTheory.Functor.const_obj_obj, CategoryTheory.Limits.WalkingPair.left, CategoryTheory.Discrete.natTrans_app
-- Module: Mathlib.CategoryTheory.Limits.VanKampen
Goal:
C : Type u
inst : Category.{v, u} C
inst_1 : HasInitial.{v, u} C
ι : Type u_3
X : ι → C
c : Cofan.{u_3, v, u} X
hc : IsVanKampenColimit c
i j : ι
inst_2 : DecidableEq ι
⊢ IsPullback (P := if j = i then X i else initial.{v, u} C)
    (if h : j = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i))
    (if h : j = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X j)) (c.inj i) (c.inj j)

Tactic:
(hc (Cofan.mk (X i) (f := fun k ↦ if k = i then X i else ⊥_ C)
    (fun k ↦ if h : k = i then (eqToHom <| if_pos h) else (eqToHom <| if_neg h) ≫ initial.to _))
    (Discrete.natTrans (fun k ↦ if h : k.1 = i then (eqToHom <| (if_pos h).trans
      (congr_arg X h.symm)) else (eqToHom <| if_neg h) ≫ initial.to _))
    (c.inj i) ?_ (.of_discrete _)).mp ⟨?_⟩ ⟨j⟩

Premises used: CategoryTheory.eqToHom, CategoryTheory.Discrete.functor, CategoryTheory.Limits.initial.to, CategoryTheory.NatTrans.Equifibered.of_discrete, if_pos, Nonempty.intro, CategoryTheory.Discrete.natTrans, dite, CategoryTheory.Discrete.mk, congr_arg, CategoryTheory.Discrete.as, CategoryTheory.Limits.Cofan.inj, CategoryTheory.CategoryStruct.comp, Iff.mp, CategoryTheory.Limits.Cofan.mk, Eq.symm, Eq, if_neg, CategoryTheory.Functor.obj, Eq.trans, ite