-- Mathlib/Topology/Category/Profinite/CofilteredLimit.lean
theorem exists_locallyConstant_finite_aux {α : Type*} [Finite α] (hC : IsLimit C)
    (f : LocallyConstant C.pt α) : ∃ (j : J) (g : LocallyConstant (F.obj j) (α → Fin 2)),
      (f.map fun a b => if a = b then (0 : Fin 2) else 1) = g.comap (C.π.app _).hom

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

Tactic:
let G : Finset J := Finset.univ.image j

Result:
case intro
J : Type v
inst✝² : SmallCategory J
inst✝¹ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝ : Finite α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
val✝ : Fintype α
ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1
ff : α → LocallyConstant (↑C.pt.toTop) (Fin 2) := (LocallyConstant.map ι f).flip
j : α → J
g : (a : α) → LocallyConstant (↑(F.obj (j a)).toTop) (Fin 2)
h : ∀ (a : α), ff a = LocallyConstant.comap (TopCat.Hom.hom (C.π.app (j a))) (g a)
G : Finset J := Finset.image j Finset.univ
⊢ ∃ j g,
    LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j)) g

Proof state:
J : Type v
inst✝² : SmallCategory J
inst✝¹ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝ : Finite α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
val✝ : Fintype α
ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1
ff : α → LocallyConstant (↑C.pt.toTop) (Fin 2) := (LocallyConstant.map ι f).flip
j : α → J
g : (a : α) → LocallyConstant (↑(F.obj (j a)).toTop) (Fin 2)
h : ∀ (a : α), ff a = LocallyConstant.comap (TopCat.Hom.hom (C.π.app (j a))) (g a)
G : Finset J := Finset.image j Finset.univ
j0 : J
hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)
hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ
fs : (a : α) → j0 ⟶ j a := fun a => ⋯.some
gg : α → LocallyConstant (↑(F.obj j0).toTop) (Fin 2) :=
  fun a => LocallyConstant.comap (TopCat.Hom.hom (F.map (fs a))) (g a)
ggg : LocallyConstant (↑(F.obj j0).toTop) (α → Fin 2) := LocallyConstant.unflip gg
⊢ LocallyConstant.comap (TopCat.Hom.hom (C.π.app j0)) ggg =
    LocallyConstant.unflip (LocallyConstant.comap (TopCat.Hom.hom (C.π.app j0)) ggg).flip

Tactic:
simp

Result:
no goals

Proof state:
case intro.intro
J : Type v
inst✝² : SmallCategory J
inst✝¹ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝ : Finite α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
val✝ : Fintype α
ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1
ff : α → LocallyConstant (↑C.pt.toTop) (Fin 2) := (LocallyConstant.map ι f).flip
j : α → J
g : (a : α) → LocallyConstant (↑(F.obj (j a)).toTop) (Fin 2)
h : ∀ (a : α), ff a = LocallyConstant.comap (TopCat.Hom.hom (C.π.app (j a))) (g a)
G : Finset J := Finset.image j Finset.univ
j0 : J
hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)
hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ
fs : (a : α) → j0 ⟶ j a := fun a => ⋯.some
gg : α → LocallyConstant (↑(F.obj j0).toTop) (Fin 2) :=
  fun a => LocallyConstant.comap (TopCat.Hom.hom (F.map (fs a))) (g a)
ggg : LocallyConstant (↑(F.obj j0).toTop) (α → Fin 2) := LocallyConstant.unflip gg
⊢ ∃ j g,
    LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j)) g

Tactic:
refine ⟨j0, ggg, ?_⟩

Result:
case intro.intro
J : Type v
inst✝² : SmallCategory J
inst✝¹ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝ : Finite α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
val✝ : Fintype α
ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1
ff : α → LocallyConstant (↑C.pt.toTop) (Fin 2) := (LocallyConstant.map ι f).flip
j : α → J
g : (a : α) → LocallyConstant (↑(F.obj (j a)).toTop) (Fin 2)
h : ∀ (a : α), ff a = LocallyConstant.comap (TopCat.Hom.hom (C.π.app (j a))) (g a)
G : Finset J := Finset.image j Finset.univ
j0 : J
hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)
hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ
fs : (a : α) → j0 ⟶ j a := fun a => ⋯.some
gg : α → LocallyConstant (↑(F.obj j0).toTop) (Fin 2) :=
  fun a => LocallyConstant.comap (TopCat.Hom.hom (F.map (fs a))) (g a)
ggg : LocallyConstant (↑(F.obj j0).toTop) (α → Fin 2) := LocallyConstant.unflip gg
⊢ LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j0)) ggg

Proof state:
case intro.intro.e_f.h.h
J : Type v
inst✝² : SmallCategory J
inst✝¹ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝ : Finite α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
val✝ : Fintype α
ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1
ff : α → LocallyConstant (↑C.pt.toTop) (Fin 2) := (LocallyConstant.map ι f).flip
j : α → J
g : (a : α) → LocallyConstant (↑(F.obj (j a)).toTop) (Fin 2)
h : ∀ (a : α), ff a = LocallyConstant.comap (TopCat.Hom.hom (C.π.app (j a))) (g a)
G : Finset J := Finset.image j Finset.univ
j0 : J
hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)
hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ
fs : (a : α) → j0 ⟶ j a := fun a => ⋯.some
gg : α → LocallyConstant (↑(F.obj j0).toTop) (Fin 2) :=
  fun a => LocallyConstant.comap (TopCat.Hom.hom (F.map (fs a))) (g a)
ggg : LocallyConstant (↑(F.obj j0).toTop) (α → Fin 2) := LocallyConstant.unflip gg
a : α
x : ↑C.pt.toTop
⊢ (LocallyConstant.comap (TopCat.Hom.hom (C.π.app (j a))) (g a)) x =
    (g a) ((ConcreteCategory.hom (C.π.app j0 ≫ F.map (fs a))) x)

Tactic:
rw [C.w]

Result:
case intro.intro.e_f.h.h
J : Type v
inst✝² : SmallCategory J
inst✝¹ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝ : Finite α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
val✝ : Fintype α
ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1
ff : α → LocallyConstant (↑C.pt.toTop) (Fin 2) := (LocallyConstant.map ι f).flip
j : α → J
g : (a : α) → LocallyConstant (↑(F.obj (j a)).toTop) (Fin 2)
h : ∀ (a : α), ff a = LocallyConstant.comap (TopCat.Hom.hom (C.π.app (j a))) (g a)
G : Finset J := Finset.image j Finset.univ
j0 : J
hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)
hj : ∀ (a : α), j a ∈ Finset.image j Finset.univ
fs : (a : α) → j0 ⟶ j a := fun a => ⋯.some
gg : α → LocallyConstant (↑(F.obj j0).toTop) (Fin 2) :=
  fun a => LocallyConstant.comap (TopCat.Hom.hom (F.map (fs a))) (g a)
ggg : LocallyConstant (↑(F.obj j0).toTop) (α → Fin 2) := LocallyConstant.unflip gg
a : α
x : ↑C.pt.toTop
⊢ (LocallyConstant.comap (TopCat.Hom.hom (C.π.app (j a))) (g a)) x = (g a) ((ConcreteCategory.hom (C.π.app (j a))) x)

Proof state:
J : Type v
inst✝² : SmallCategory J
inst✝¹ : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
inst✝ : Finite α
hC : IsLimit C
f : LocallyConstant (↑C.pt.toTop) α
val✝ : Fintype α
ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1
ff : α → LocallyConstant (↑C.pt.toTop) (Fin 2) := (LocallyConstant.map ι f).flip
j : α → J
g : (a : α) → LocallyConstant (↑(F.obj (j a)).toTop) (Fin 2)
h : ∀ (a : α), ff a = LocallyConstant.comap (TopCat.Hom.hom (C.π.app (j a))) (g a)
G : Finset J := Finset.image j Finset.univ
j0 : J
hj0 : ∀ {X : J}, X ∈ G → Nonempty (j0 ⟶ X)
a : α
⊢ j a ∈ Finset.image j Finset.univ

Tactic:
simp only [Finset.mem_image, Finset.mem_univ, true_and, exists_apply_eq_apply]

Result:
no goals