Theorem: Profinite.exists_locallyConstant_finite_nonempty
File: Mathlib/Topology/Category/Profinite/CofilteredLimit.lean

tactic: inhabit α
state_before: J : Type u
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.toCompHaus.toTop) α
⊢ ∃ j g, f = LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g

tactic: obtain ⟨j, gg, h⟩ := exists_locallyConstant_finite_aux _ hC f
state_before: J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
⊢ ∃ j g, f = LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g

tactic: let ι : α → α → Fin 2 := fun a b => if a = b then 0 else 1
state_before: case intro.intro
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (C.π.app j)) gg
⊢ ∃ j g, f = LocallyConstant.comap ((forget Profinite).map (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
state_before: case intro.intro
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 ((forget Profinite).map (C.π.app j)) g

tactic: refine' ⟨j, gg.map σ, _⟩
state_before: case intro.intro
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
⊢ ∃ j g, f = LocallyConstant.comap ((forget Profinite).map (C.π.app j)) g

tactic: ext x
state_before: case intro.intro
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
⊢ f = LocallyConstant.comap ((forget Profinite).map (C.π.app j)) (LocallyConstant.map σ gg)

tactic: rw [LocallyConstant.coe_comap ((forget Profinite).map _) _ (C.π.app j).continuous]
state_before: case intro.intro.h
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
x : ↑C.pt.toCompHaus.toTop
⊢ ↑f x = ↑(LocallyConstant.comap ((forget Profinite).map (C.π.app j)) (LocallyConstant.map σ gg)) x

tactic: dsimp
state_before: case intro.intro.h
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
x : ↑C.pt.toCompHaus.toTop
⊢ ↑f x = (↑(LocallyConstant.map σ gg) ∘ (forget Profinite).map (C.π.1 j)) x

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, LocallyConstant.coe_comap ((forget Profinite).map _) _ (C.π.app j).continuous]
  rfl
state_before: case intro.intro.h
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
x : ↑C.pt.toCompHaus.toTop
⊢ ↑f x =
    if h : ∃ a, (fun b => if a = b then 0 else 1) = ↑gg ((forget Profinite).map (C.π.1 j) x) then Exists.choose h
    else default

tactic: have h2 : ∃ a : α, ι a = gg (C.π.app j x) := ⟨f x, h1⟩
state_before: case intro.intro.h
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
x : ↑C.pt.toCompHaus.toTop
h1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)
⊢ ↑f x =
    if h : ∃ a, (fun b => if a = b then 0 else 1) = ↑gg ((forget Profinite).map (C.π.1 j) x) then Exists.choose h
    else default

tactic: rw [dif_pos h2]
state_before: case intro.intro.h
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
x : ↑C.pt.toCompHaus.toTop
h1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)
h2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)
⊢ ↑f x =
    if h : ∃ a, (fun b => if a = b then 0 else 1) = ↑gg ((forget Profinite).map (C.π.1 j) x) then Exists.choose h
    else default

tactic: apply_fun ι
state_before: case intro.intro.h
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
x : ↑C.pt.toCompHaus.toTop
h1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)
h2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)
⊢ ↑f x = Exists.choose h2

tactic: change f.map (fun a b => if a = b then (0 : Fin 2) else 1) x = _
state_before: J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
x : ↑C.pt.toCompHaus.toTop
⊢ ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)

tactic: rw [h, LocallyConstant.coe_comap ((forget Profinite).map _) _ (C.π.app j).continuous]
state_before: J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
x : ↑C.pt.toCompHaus.toTop
⊢ ↑(LocallyConstant.map (fun a b => if a = b then 0 else 1) f) x = ↑gg ((forget Profinite).map (C.π.app j) x)

tactic: rfl
state_before: J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
x : ↑C.pt.toCompHaus.toTop
⊢ (↑gg ∘ (forget Profinite).map (C.π.1 j)) x = ↑gg ((forget Profinite).map (C.π.app j) x)

tactic: rw [h2.choose_spec]
state_before: case intro.intro.h
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
x : ↑C.pt.toCompHaus.toTop
h1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)
h2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)
⊢ ι (↑f x) = ι (Exists.choose h2)

tactic: exact h1
state_before: case intro.intro.h
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
x : ↑C.pt.toCompHaus.toTop
h1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)
h2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)
⊢ ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)

tactic: intro a b hh
state_before: case intro.intro.h.inj
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
x : ↑C.pt.toCompHaus.toTop
h1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)
h2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)
⊢ Function.Injective ι

tactic: have hhh := congr_fun hh a
state_before: case intro.intro.h.inj
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
x : ↑C.pt.toCompHaus.toTop
h1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)
h2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)
a b : α
hh : ι a = ι b
⊢ a = b

tactic: dsimp at hhh
state_before: case intro.intro.h.inj
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
x : ↑C.pt.toCompHaus.toTop
h1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)
h2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)
a b : α
hh : ι a = ι b
hhh : ι a a = ι b a
⊢ a = b

tactic: rw [if_pos rfl] at hhh
state_before: case intro.intro.h.inj
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
x : ↑C.pt.toCompHaus.toTop
h1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)
h2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)
a b : α
hh : ι a = ι b
hhh : (if a = a then 0 else 1) = if b = a then 0 else 1
⊢ a = b

tactic: split_ifs at hhh with hh1
state_before: case intro.intro.h.inj
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
x : ↑C.pt.toCompHaus.toTop
h1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)
h2 : ∃ a, ι a = ↑gg ((forget Profinite).map (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: exact hh1.symm
state_before: case intro.intro.h.inj.inl
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
x : ↑C.pt.toCompHaus.toTop
h1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)
h2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)
a b : α
hh : ι a = ι b
hh1 : b = a
hhh : 0 = 0
⊢ a = b

tactic: exact False.elim (bot_ne_top hhh)
state_before: case intro.intro.h.inj.inr
J : Type u
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.toCompHaus.toTop) α
inhabited_h : Inhabited α
j : J
gg : LocallyConstant (↑(F.obj j).toCompHaus.toTop) (α → Fin 2)
h :
  LocallyConstant.map (fun a b => if a = b then 0 else 1) f =
    LocallyConstant.comap ((forget Profinite).map (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 Exists.choose h else default
x : ↑C.pt.toCompHaus.toTop
h1 : ι (↑f x) = ↑gg ((forget Profinite).map (C.π.app j) x)
h2 : ∃ a, ι a = ↑gg ((forget Profinite).map (C.π.app j) x)
a b : α
hh : ι a = ι b
hh1 : ¬b = a
hhh : 0 = 1
⊢ a = b