-- Mathlib/CategoryTheory/Limits/VanKampen.lean
theorem isVanKampenColimit_extendCofan {n : ℕ} (f : Fin (n + 1) → C)
    {c₁ : Cofan fun i : Fin n ↦ f i.succ} {c₂ : BinaryCofan (f 0) c₁.pt}
    (t₁ : IsVanKampenColimit c₁) (t₂ : IsVanKampenColimit c₂)
    [∀ {Z} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i]
    [HasFiniteCoproducts C] :
    IsVanKampenColimit (extendCofan c₁ c₂)

Proof state:
case refine_2.uniq
C : Type u
inst✝² : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst✝ : HasFiniteCoproducts C
j✝ : Fin (n + 1)
F' : Fin (n + 1) → C
c : Cocone (Discrete.functor F')
α : Discrete.functor F' ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
Hc : IsColimit c
T : C
f₁ : F' 0 ⟶ T
f₂ : (∐ fun j => (Discrete.functor F').obj { as := j.succ }) ⟶ T
f₃ : (BinaryCofan.mk (c.ι.app { as := 0 }) (Sigma.desc fun b => c.ι.app { as := b.succ })).pt ⟶ T
m₁ : c.ι.app { as := 0 } ≫ f₃ = f₁
m₂ : (Sigma.desc fun b => c.ι.app { as := b.succ }) ≫ f₃ = f₂
j : Fin (n + 1)
⊢ c.ι.app { as := j } ≫ f₃ =
    (Cofan.mk T fun i =>
            Fin.cases f₁ (fun i => Sigma.ι (fun j => (Discrete.functor F').obj { as := j.succ }) i ≫ f₂) i).ι.app
      { as := j }

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

Result:
case refine_2.uniq
C : Type u
inst✝² : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst✝ : HasFiniteCoproducts C
j✝ : Fin (n + 1)
F' : Fin (n + 1) → C
c : Cocone (Discrete.functor F')
α : Discrete.functor F' ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
Hc : IsColimit c
T : C
f₁ : F' 0 ⟶ T
f₂ : (∐ fun j => (Discrete.functor F').obj { as := j.succ }) ⟶ T
f₃ : (BinaryCofan.mk (c.ι.app { as := 0 }) (Sigma.desc fun b => c.ι.app { as := b.succ })).pt ⟶ T
m₁ : c.ι.app { as := 0 } ≫ f₃ = f₁
m₂ : (Sigma.desc fun b => c.ι.app { as := b.succ }) ≫ f₃ = f₂
j : Fin (n + 1)
⊢ c.ι.app { as := j } ≫ f₃ = Fin.cases f₁ (fun i => Sigma.ι (fun j => F' j.succ) i ≫ f₂) j

Proof state:
C : Type u
inst✝² : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst✝ : HasFiniteCoproducts C
⊢ IsVanKampenColimit (extendCofan c₁ c₂)

Tactic:
intro F c α i e hα

Result:
C : Type u
inst✝² : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst✝ : HasFiniteCoproducts C
F : 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 ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
⊢ Nonempty (IsColimit c) ↔
    ∀ (j : Discrete (Fin (n + 1))), IsPullback (c.ι.app j) (α.app j) i ((extendCofan c₁ c₂).ι.app j)

Proof state:
case refine_2.uniq.succ
C : Type u
inst✝² : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst✝ : HasFiniteCoproducts C
j✝ : Fin (n + 1)
F' : Fin (n + 1) → C
c : Cocone (Discrete.functor F')
α : Discrete.functor F' ⟶ Discrete.functor f
i : c.pt ⟶ (extendCofan c₁ c₂).pt
e : α ≫ (extendCofan c₁ c₂).ι = c.ι ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
Hc : IsColimit c
T : C
f₁ : F' 0 ⟶ T
f₂ : (∐ fun j => (Discrete.functor F').obj { as := j.succ }) ⟶ T
f₃ : (BinaryCofan.mk (c.ι.app { as := 0 }) (Sigma.desc fun b => c.ι.app { as := b.succ })).pt ⟶ T
m₁ : c.ι.app { as := 0 } ≫ f₃ = f₁
m₂ : (Sigma.desc fun b => c.ι.app { as := b.succ }) ≫ f₃ = f₂
j : Fin n
a✝ : c.ι.app { as := j.castSucc } ≫ f₃ = Fin.cases f₁ (fun i => Sigma.ι (fun j => F' j.succ) i ≫ f₂) j.castSucc
⊢ c.ι.app { as := j.succ } ≫ f₃ = Fin.cases f₁ (fun i => Sigma.ι (fun j => F' j.succ) i ≫ f₂) j.succ

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

Result:
no goals

Proof state:
case refine_3.succ.refine_1.w.h.mk
C : Type u
inst✝² : Category.{v, u} C
n : ℕ
f : Fin (n + 1) → C
c₁ : Cofan fun i => f i.succ
c₂ : BinaryCofan (f 0) c₁.pt
t₁ : IsVanKampenColimit c₁
t₂ : IsVanKampenColimit c₂
inst✝¹ : ∀ {Z : C} (i : Z ⟶ c₂.pt), HasPullback c₂.inr i
inst✝ : HasFiniteCoproducts C
F : 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 ≫ (Functor.const (Discrete (Fin (n + 1)))).map i
hα : NatTrans.Equifibered α
Hc : IsColimit c
t₂' :
  ∀ (j : Discrete WalkingPair),
    IsPullback ((BinaryCofan.mk (c.ι.app { as := 0 }) (Sigma.desc fun b => c.ι.app { as := b.succ })).ι.app j)
      ((mapPair (α.app { as := 0 }) (Sigma.desc fun b => α.app { as := b.succ } ≫ c₁.inj b)).app j) i (c₂.ι.app j)
j✝ : Fin n
a✝ :
  IsPullback (c.ι.app { as := j✝.castSucc }) (α.app { as := j✝.castSucc }) i
    ((extendCofan c₁ c₂).ι.app { as := j✝.castSucc })
j : Fin n
⊢ α.app { as := j.succ } ≫ c₁.ι.app { as := j } =
    (Cofan.mk c₁.pt fun j => α.app { as := j.succ } ≫ c₁.inj j).ι.app { as := j }

Tactic:
rfl

Result:
no goals