-- Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
theorem exists_locallyConstant_finite_nonempty {α : Type*} [Finite α] [Nonempty α]
    (hC : IsLimit C) (f : LocallyConstant C.pt α) :
    ∃ (j : J) (g : LocallyConstant (F.obj j) α), f = g.comap (C.π.app _).hom

Proof state:
case intro.intro.h.inj
J : Type v
inst✝³ : SmallCategory J
inst✝² : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝¹ : Finite α
inst✝ : Nonempty α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)
h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j)) gg
ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1
σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default
x : ↑C.pt.toTop
h1 : ι (f x) = gg ((ConcreteCategory.hom (C.π.app j)) x)
h2 : ∃ a, ι a = gg ((ConcreteCategory.hom (C.π.app j)) x)
a b : α
hh : ι a = ι b
hhh : 0 = if b = a then 0 else 1
⊢ a = b

Tactic:
split_ifs at hhh with hh1

Result:
case pos
J : Type v
inst✝³ : SmallCategory J
inst✝² : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝¹ : Finite α
inst✝ : Nonempty α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)
h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j)) gg
ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1
σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default
x : ↑C.pt.toTop
h1 : ι (f x) = gg ((ConcreteCategory.hom (C.π.app j)) x)
h2 : ∃ a, ι a = gg ((ConcreteCategory.hom (C.π.app j)) x)
a b : α
hh : ι a = ι b
hh1 : b = a
hhh : 0 = 0
⊢ a = b

case neg
J : Type v
inst✝³ : SmallCategory J
inst✝² : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝¹ : Finite α
inst✝ : Nonempty α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)
h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j)) gg
ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1
σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default
x : ↑C.pt.toTop
h1 : ι (f x) = gg ((ConcreteCategory.hom (C.π.app j)) x)
h2 : ∃ a, ι a = gg ((ConcreteCategory.hom (C.π.app j)) x)
a b : α
hh : ι a = ι b
hh1 : ¬b = a
hhh : 0 = 1
⊢ a = b

Proof state:
case intro.intro.h
J : Type v
inst✝³ : SmallCategory J
inst✝² : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝¹ : Finite α
inst✝ : Nonempty α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)
h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j)) gg
ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1
σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default
x : ↑C.pt.toTop
⊢ f x = if h : ∃ a, ι a = gg ((TopCat.Hom.hom (C.π.app j)) x) then h.choose else default

Tactic:
have h1 : ι (f x) = gg (C.π.app j x) := by
  change f.map (fun a b => if a = b then (0 : Fin 2) else 1) x = _
  rw [h]
  rfl

Result:
case intro.intro.h
J : Type v
inst✝³ : SmallCategory J
inst✝² : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝¹ : Finite α
inst✝ : Nonempty α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)
h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j)) gg
ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1
σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default
x : ↑C.pt.toTop
h1 : ι (f x) = gg ((ConcreteCategory.hom (C.π.app j)) x)
⊢ f x = if h : ∃ a, ι a = gg ((TopCat.Hom.hom (C.π.app j)) x) then h.choose else default

Proof state:
case intro.intro
J : Type v
inst✝³ : SmallCategory J
inst✝² : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝¹ : Finite α
inst✝ : Nonempty α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)
h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j)) gg
ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1
⊢ ∃ j g, f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j)) g

Tactic:
let σ : (α → Fin 2) → α := fun f => if h : ∃ a : α, ι a = f then h.choose else default

Result:
case intro.intro
J : Type v
inst✝³ : SmallCategory J
inst✝² : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝¹ : Finite α
inst✝ : Nonempty α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)
h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j)) gg
ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1
σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default
⊢ ∃ j g, f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j)) g

Proof state:
J : Type v
inst✝³ : SmallCategory J
inst✝² : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝¹ : Finite α
inst✝ : Nonempty α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)
h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j)) gg
ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1
σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default
x : ↑C.pt.toTop
⊢ ι (f x) = gg ((ConcreteCategory.hom (C.π.app j)) x)

Tactic:
change f.map (fun a b => if a = b then (0 : Fin 2) else 1) x = _

Result:
J : Type v
inst✝³ : SmallCategory J
inst✝² : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝¹ : Finite α
inst✝ : Nonempty α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toTop) (α → Fin 2)
h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j)) gg
ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1
σ : (α → Fin 2) → α := fun f => if h : ∃ a, ι a = f then h.choose else default
x : ↑C.pt.toTop
⊢ (LocallyConstant.map (fun a b => if a = b then 0 else 1) f) x = gg ((ConcreteCategory.hom (C.π.app j)) x)