awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Category.Profinite.CofilteredLimit (goal)
Goal:
J : Type v
inst : SmallCategory J
inst_1 : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f :
  LocallyConstant
    (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α
S : DiscreteQuotient
  (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) :=
  f.discreteQuotient
ff : Quotient S.toSetoid → α := (⇑f.lift : (a : Quotient f.discreteQuotient.toSetoid) → α)
h : IsEmpty (Quotient S.toSetoid)
this :
  ∃ (j : J),
    IsEmpty
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
j : J
hj :
  IsEmpty (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
⊢ ∃ (g :
    LocallyConstant
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α),
    f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j).hom) g
have refine_1 : ∀ (A : Set α),
  IsOpen (X :=
    (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)))
    ((fun
          (a :
            (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) :
              Type (max u v))) ↦
        hj.elim (p :=
          fun
            (x :
              (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) :
                Type (max u v))) ↦
          α) a) ⁻¹'
      A)
Tactic: refine ⟨⟨hj.elim, fun A => ?_⟩, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Category.Profinite.CofilteredLimit (goal)
Goal:
J : Type v
inst : SmallCategory J
inst_1 : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f :
  LocallyConstant
    (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α
S : DiscreteQuotient
  (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) :=
  f.discreteQuotient
ff : Quotient S.toSetoid → α := (⇑f.lift : (a : Quotient f.discreteQuotient.toSetoid) → α)
h : IsEmpty (Quotient S.toSetoid)
this :
  ∃ (j : J),
    IsEmpty
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
j : J
hj :
  IsEmpty (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
refine_1 :
  ∀ (A : Set α),
    IsOpen (X :=
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)))
      ((fun
            (a :
              (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) :
                Type (max u v))) ↦
          hj.elim (p :=
            fun
              (x :
                (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) :
                  Type (max u v))) ↦
            α) a) ⁻¹'
        A)
⊢ ∃ (g :
    LocallyConstant
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α),
    f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j).hom) g
have refine_2 : f =
  LocallyConstant.comap (TopCat.Hom.hom (C.π.app j).hom)
    {
      toFun :=
        fun
          (a :
            (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) :
              Type (max u v))) ↦
        hj.elim (p :=
          fun
            (x :
              (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) :
                Type (max u v))) ↦
          α) a,
      isLocallyConstant := sorry }
Tactic: refine ⟨⟨hj.elim, fun A => ?_⟩, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Category.Profinite.CofilteredLimit (goal)
Goal:
J : Type v
inst : SmallCategory J
inst_1 : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f :
  LocallyConstant
    (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α
S : DiscreteQuotient
  (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) :=
  f.discreteQuotient
ff : Quotient S.toSetoid → α := (⇑f.lift : (a : Quotient f.discreteQuotient.toSetoid) → α)
h : IsEmpty (Quotient S.toSetoid)
this :
  ∃ (j : J),
    IsEmpty
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
j : J
hj :
  IsEmpty (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
A : Set α
⊢ IsOpen (X :=
    (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)))
    ((fun
          (a :
            (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) :
              Type (max u v))) ↦
        hj.elim (p :=
          fun
            (x :
              (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) :
                Type (max u v))) ↦
          α) a) ⁻¹'
      A)
have h : Eq (α :=
  Set (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)))
  ((fun
        (a :
          (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) :
            Type (max u v))) ↦
      hj.elim (p :=
        fun
          (x :
            (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) :
              Type (max u v))) ↦
        α) a) ⁻¹'
    A)
  ∅
Tactic: convert! isOpen_empty
-- Lean 4 proof step from Mathlib.Topology.Category.Profinite.CofilteredLimit (goal)
Goal:
J : Type v
inst : SmallCategory J
inst_1 : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f :
  LocallyConstant
    (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α
S : DiscreteQuotient
  (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) :=
  f.discreteQuotient
ff : Quotient S.toSetoid → α := (⇑f.lift : (a : Quotient f.discreteQuotient.toSetoid) → α)
h : IsEmpty (Quotient S.toSetoid)
this :
  ∃ (j : J),
    IsEmpty
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
j : J
hj :
  IsEmpty (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
A : Set α
⊢ Eq (α :=
    Set (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)))
    ((fun
          (a :
            (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) :
              Type (max u v))) ↦
        hj.elim (p :=
          fun
            (x :
              (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) :
                Type (max u v))) ↦
          α) a) ⁻¹'
      A)
    ∅
have h : ∀ (x : (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))),
  Membership.mem (γ :=
      Set
        (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)))
      ((fun
            (a :
              (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) :
                Type (max u v))) ↦
          hj.elim (p :=
            fun
              (x :
                (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) :
                  Type (max u v))) ↦
            α) a) ⁻¹'
        A)
      x ↔
    Membership.mem (γ :=
      Set
        (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)))
      ∅ x
Tactic: ext x
-- Lean 4 proof step from Mathlib.Topology.Category.Profinite.CofilteredLimit (goal)
Goal:
J : Type v
inst : SmallCategory J
inst_1 : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f :
  LocallyConstant
    (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α
S : DiscreteQuotient
  (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) :=
  f.discreteQuotient
ff : Quotient S.toSetoid → α := (⇑f.lift : (a : Quotient f.discreteQuotient.toSetoid) → α)
h : IsEmpty (Quotient S.toSetoid)
this :
  ∃ (j : J),
    IsEmpty
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
j : J
hj :
  IsEmpty (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
⊢ f =
    LocallyConstant.comap (TopCat.Hom.hom (C.π.app j).hom)
      {
        toFun :=
          fun
            (a :
              (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) :
                Type (max u v))) ↦
          hj.elim (p :=
            fun
              (x :
                (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) :
                  Type (max u v))) ↦
            α) a,
        isLocallyConstant := ⋯ }
have refine_2 : ∀ (x : (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))),
  Eq.{u_1 + 1} (α := α) ((f : (a : (↑C.pt.toTop : Type (max u v))) → α) x : α)
    ((LocallyConstant.comap (TopCat.Hom.hom (C.π.app j).hom)
            {
              toFun :=
                fun
                  (a :
                    (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) :
                      Type (max u v))) ↦
                hj.elim (p :=
                  fun
                    (x :
                      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) :
                        Type (max u v))) ↦
                  α) a,
              isLocallyConstant := sorry } :
          (a : (↑C.pt.toTop : Type (max u v))) → α)
        x :
      α)
Tactic: ext x
-- Lean 4 proof step from Mathlib.Topology.Category.Profinite.CofilteredLimit (goal)
Goal:
J : Type v
inst : SmallCategory J
inst_1 : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f :
  LocallyConstant
    (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α
S : DiscreteQuotient
  (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) :=
  f.discreteQuotient
ff : Quotient S.toSetoid → α := (⇑f.lift : (a : Quotient f.discreteQuotient.toSetoid) → α)
h : IsEmpty (Quotient S.toSetoid)
⊢ ∃ (j : J) (g :
    LocallyConstant
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α),
    f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j).hom) g
have inl : ∃ (j : J),
  IsEmpty (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
Tactic: suffices ∃ j, IsEmpty (F.obj j) by
      refine this.imp fun j hj => ?_
      refine ⟨⟨hj.elim, fun A => ?_⟩, ?_⟩
      · convert! isOpen_empty
        ext x
        exact hj.elim x
      · ext x
        exact hj.elim' (C.π.app j x)
-- Lean 4 proof step from Mathlib.Topology.Category.Profinite.CofilteredLimit (goal)
Goal:
J : Type v
inst : SmallCategory J
inst_1 : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f :
  LocallyConstant
    (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α
S : DiscreteQuotient
  (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) :=
  f.discreteQuotient
ff : Quotient S.toSetoid → α := (⇑f.lift : (a : Quotient f.discreteQuotient.toSetoid) → α)
h : IsEmpty (Quotient S.toSetoid)
⊢ ∃ (j : J),
    IsEmpty
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
have inl : ∀
  (h :
    ∀ (j : J),
      Nonempty
        (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))),
  False
Tactic: by_contra! h
-- Lean 4 proof step from Mathlib.Topology.Category.Profinite.CofilteredLimit (hypothesis)
Goal:
J : Type v
inst : SmallCategory J
inst_1 : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f :
  LocallyConstant
    (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α
S : DiscreteQuotient
  (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) :=
  f.discreteQuotient
ff : Quotient S.toSetoid → α := (⇑f.lift : (a : Quotient f.discreteQuotient.toSetoid) → α)
h_1 : IsEmpty (Quotient S.toSetoid)
h :
  ∀ (j : J),
    Nonempty
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
⊢ False
have this : ∀ (j : J), Nonempty (↑((F ⋙ toTopCat).obj j) : Type (max u v))
Tactic: haveI : ∀ j : J, Nonempty ((F ⋙ Profinite.toTopCat).obj j) := h
-- Lean 4 proof step from Mathlib.Topology.Category.Profinite.CofilteredLimit (hypothesis)
Goal:
J : Type v
inst : SmallCategory J
inst_1 : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f :
  LocallyConstant
    (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α
S : DiscreteQuotient
  (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) :=
  f.discreteQuotient
ff : Quotient S.toSetoid → α := (⇑f.lift : (a : Quotient f.discreteQuotient.toSetoid) → α)
h_1 : IsEmpty (Quotient S.toSetoid)
h :
  ∀ (j : J),
    Nonempty
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
this : ∀ (j : J), Nonempty (↑((F ⋙ toTopCat).obj j) : Type (max u v))
⊢ False
have this₁ : ∀ (j : J), T2Space (↑((F ⋙ toTopCat).obj j) : Type (max u v))
Tactic: haveI : ∀ j : J, T2Space ((F ⋙ Profinite.toTopCat).obj j) := fun j =>
      (inferInstance : T2Space (F.obj j))
-- Lean 4 proof step from Mathlib.Topology.Category.Profinite.CofilteredLimit (hypothesis)
Goal:
J : Type v
inst : SmallCategory J
inst_1 : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f :
  LocallyConstant
    (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α
S : DiscreteQuotient
  (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) :=
  f.discreteQuotient
ff : Quotient S.toSetoid → α := (⇑f.lift : (a : Quotient f.discreteQuotient.toSetoid) → α)
h_1 : IsEmpty (Quotient S.toSetoid)
h :
  ∀ (j : J),
    Nonempty
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
this_1 : ∀ (j : J), Nonempty (↑((F ⋙ toTopCat).obj j) : Type (max u v))
this : ∀ (j : J), T2Space (↑((F ⋙ toTopCat).obj j) : Type (max u v))
⊢ False
have this₁ : ∀ (j : J), CompactSpace (↑((F ⋙ toTopCat).obj j) : Type (max u v))
Tactic: haveI : ∀ j : J, CompactSpace ((F ⋙ Profinite.toTopCat).obj j) := fun j =>
      (inferInstance : CompactSpace (F.obj j))
-- Lean 4 proof step from Mathlib.Topology.Category.Profinite.CofilteredLimit (hypothesis)
Goal:
J : Type v
inst : SmallCategory J
inst_1 : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f :
  LocallyConstant
    (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α
S : DiscreteQuotient
  (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) :=
  f.discreteQuotient
ff : Quotient S.toSetoid → α := (⇑f.lift : (a : Quotient f.discreteQuotient.toSetoid) → α)
h_1 : IsEmpty (Quotient S.toSetoid)
h :
  ∀ (j : J),
    Nonempty
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
this_1 : ∀ (j : J), Nonempty (↑((F ⋙ toTopCat).obj j) : Type (max u v))
this_2 : ∀ (j : J), T2Space (↑((F ⋙ toTopCat).obj j) : Type (max u v))
this : ∀ (j : J), CompactSpace (↑((F ⋙ toTopCat).obj j) : Type (max u v))
⊢ False
have cond : Nonempty (↑(TopCat.limitCone.{v, u, v} (F ⋙ toTopCat)).pt : Type (max v u))
Tactic: have cond := TopCat.nonempty_limitCone_of_compact_t2_cofiltered_system.{u}
      (F ⋙ Profinite.toTopCat)
-- Lean 4 proof step from Mathlib.Topology.Category.Profinite.CofilteredLimit (goal)
Goal:
J : Type v
inst : SmallCategory J
inst_1 : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f :
  LocallyConstant
    (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α
S : DiscreteQuotient
  (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) :=
  f.discreteQuotient
ff : Quotient S.toSetoid → α := (⇑f.lift : (a : Quotient f.discreteQuotient.toSetoid) → α)
h_1 : IsEmpty (Quotient S.toSetoid)
h :
  ∀ (j : J),
    Nonempty
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
this_1 : ∀ (j : J), Nonempty (↑((F ⋙ toTopCat).obj j) : Type (max u v))
this_2 : ∀ (j : J), T2Space (↑((F ⋙ toTopCat).obj j) : Type (max u v))
this : ∀ (j : J), CompactSpace (↑((F ⋙ toTopCat).obj j) : Type (max u v))
cond : Nonempty (↑(TopCat.limitCone.{v, u, v} (F ⋙ toTopCat)).pt : Type (max v u))
⊢ False
have inl : Nonempty (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
Tactic: suffices Nonempty C.pt from IsEmpty.false (S.proj this.some)
-- Lean 4 proof step from Mathlib.Topology.Category.Profinite.CofilteredLimit (goal)
Goal:
J : Type v
inst : SmallCategory J
inst_1 : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f :
  LocallyConstant
    (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α
S : DiscreteQuotient
  (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) :=
  f.discreteQuotient
ff : Quotient S.toSetoid → α := (⇑f.lift : (a : Quotient f.discreteQuotient.toSetoid) → α)
h_1 : IsEmpty (Quotient S.toSetoid)
h :
  ∀ (j : J),
    Nonempty
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
this_1 : ∀ (j : J), Nonempty (↑((F ⋙ toTopCat).obj j) : Type (max u v))
this_2 : ∀ (j : J), T2Space (↑((F ⋙ toTopCat).obj j) : Type (max u v))
this : ∀ (j : J), CompactSpace (↑((F ⋙ toTopCat).obj j) : Type (max u v))
cond : Nonempty (↑(TopCat.limitCone.{v, u, v} (F ⋙ toTopCat)).pt : Type (max v u))
D : Cone (F ⋙ toTopCat) := toTopCat.mapCone C
⊢ Nonempty (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
have inl : ∀ (hD : IsLimit D),
  Nonempty (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
Tactic: have hD : IsLimit D := isLimitOfPreserves Profinite.toTopCat hC
-- Lean 4 proof step from Mathlib.Topology.Category.Profinite.CofilteredLimit (goal)
Goal:
J : Type v
inst : SmallCategory J
inst_1 : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f :
  LocallyConstant
    (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α
S : DiscreteQuotient
  (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) :=
  f.discreteQuotient
ff : Quotient S.toSetoid → α := (⇑f.lift : (a : Quotient f.discreteQuotient.toSetoid) → α)
h_1 : IsEmpty (Quotient S.toSetoid)
h :
  ∀ (j : J),
    Nonempty
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
this_1 : ∀ (j : J), Nonempty (↑((F ⋙ toTopCat).obj j) : Type (max u v))
this_2 : ∀ (j : J), T2Space (↑((F ⋙ toTopCat).obj j) : Type (max u v))
this : ∀ (j : J), CompactSpace (↑((F ⋙ toTopCat).obj j) : Type (max u v))
cond : Nonempty (↑(TopCat.limitCone.{v, u, v} (F ⋙ toTopCat)).pt : Type (max v u))
D : Cone (F ⋙ toTopCat) := toTopCat.mapCone C
hD : IsLimit D
⊢ Nonempty (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
have inl : ∀ (CD : (TopCat.limitCone (F ⋙ toTopCat)).pt ⟶ D.pt),
  Nonempty (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
Tactic: have CD := (hD.conePointUniqueUpToIso (TopCat.limitConeIsLimit.{v, max u v} _)).inv
-- Lean 4 proof step from Mathlib.Topology.Category.Profinite.CofilteredLimit (goal)
Goal:
J : Type v
inst : SmallCategory J
inst_1 : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f :
  LocallyConstant
    (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α
S : DiscreteQuotient
  (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) :=
  f.discreteQuotient
ff : Quotient S.toSetoid → α := (⇑f.lift : (a : Quotient f.discreteQuotient.toSetoid) → α)
h : Nonempty (Quotient S.toSetoid)
f' : LocallyConstant
  (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
  (Quotient S.toSetoid) :=
  { toFun := S.proj, isLocallyConstant := ⋯ }
⊢ ∃ (j : J) (g :
    LocallyConstant
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α),
    f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j).hom) g
have inr : ∀ (j : J)
  (g' :
    LocallyConstant
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
      (Quotient S.toSetoid))
  (hj : f' = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j).hom) g'),
  ∃ (j : J) (g :
    LocallyConstant
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α),
    f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j).hom) g
Tactic: obtain ⟨j, g', hj⟩ := exists_locallyConstant_finite_nonempty _ hC f'
-- Lean 4 proof step from Mathlib.Topology.Category.Profinite.CofilteredLimit (goal)
Goal:
J : Type v
inst : SmallCategory J
inst_1 : IsCofiltered J
F : J ⥤ Profinite
C : Cone F
α : Type u_1
hC : IsLimit C
f :
  LocallyConstant
    (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α
S : DiscreteQuotient
  (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) :=
  f.discreteQuotient
ff : Quotient S.toSetoid → α := (⇑f.lift : (a : Quotient f.discreteQuotient.toSetoid) → α)
h : Nonempty (Quotient S.toSetoid)
f' : LocallyConstant
  (↑(C.pt.toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
  (Quotient S.toSetoid) :=
  { toFun := S.proj, isLocallyConstant := ⋯ }
j : J
g' :
  LocallyConstant
    (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v))
    (Quotient S.toSetoid)
hj : f' = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j).hom) g'
⊢ ∃ (j : J) (g :
    LocallyConstant
      (↑((F.obj j).toTop (P := fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type (max u v)))) : Type (max u v)) α),
    f = LocallyConstant.comap (TopCat.Hom.hom (C.π.app j).hom) g
have inr : f =
  LocallyConstant.comap (TopCat.Hom.hom (C.π.app j).hom)
    { toFun := ff ∘ (⇑g' : (a : (↑(F.obj j).toTop : Type (max u v))) → Quotient S.toSetoid), isLocallyConstant := sorry }
Tactic: refine ⟨j, ⟨ff ∘ g', g'.isLocallyConstant.comp _⟩, ?_⟩