awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.GrothendieckTopology (goal)
Goal:
X Y : TopCat
𝒰 : precoverage.ZeroHypercover Y
heq :
  ∀ (i : 𝒰.I₀) (y : (↑(𝒰.X i) : Type u_1)),
    Eq (α := (↑Y : Type u_1))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
              (𝒰.f i) :
            (a : (↑(𝒰.X i) : Type u_1)) → (↑Y : Type u_1))
          y :
        (↑Y : Type u_1))
      ((↑(Subtype.mk (p := fun (x : (↑Y : Type u_1)) ↦
              Membership.mem (γ := Set (↑Y : Type u_1))
                (Set.range (ι := (↑(𝒰.X i) : Type u_1))
                  (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                        carrier) (𝒰.f i)) :
                    (a : (↑(𝒰.X i) : Type u_1)) → (↑Y : Type u_1)))
                x)
              ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                    carrier) (𝒰.f i) :
                  (a : (↑(𝒰.X i) : Type u_1)) → (↑Y : Type u_1))
                y)
              ⋯) :
          (↑Y : Type u_1)) :
        (↑Y : Type u_1))
x_1 : (i : 𝒰.I₀) → Functor.obj (Functor.obj yoneda X) (Opposite.op (𝒰.X i))
hx : Presieve.Arrows.Compatible (Functor.obj yoneda X) 𝒰.f x_1
x : (↑Y : Type u_1)
⊢ ∃ (i : 𝒰.I₀),
    (fun (i : 𝒰.I₀) ↦
          (Set.range (ι := (↑(𝒰.X i) : Type u_1))
              (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                    carrier) (𝒰.f i)) :
                (a : (↑(𝒰.X i) : Type u_1)) → (↑Y : Type u_1)) :
            Set (↑Y : Type u_1)))
        i ∈
      nhds x
have refine_1 : ∀ (i : 𝒰.I₀)
  (hi :
    Membership.mem (γ := Set (↑Y : Type u_1))
      (Set.range (ι := (↑(𝒰.X i) : Type u_1))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
              (𝒰.f i)) :
          (a : (↑(𝒰.X i) : Type u_1)) → (↑Y : Type u_1)))
      x),
  ∃ (i : 𝒰.I₀),
    (fun (i : 𝒰.I₀) ↦
          (Set.range (ι := (↑(𝒰.X i) : Type u_1))
              (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                    carrier) (𝒰.f i)) :
                (a : (↑(𝒰.X i) : Type u_1)) → (↑Y : Type u_1)) :
            Set (↑Y : Type u_1)))
        i ∈
      nhds x
Tactic: obtain ⟨i, hi⟩ := exists_mem_zeroHypercover_range 𝒰 x
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.GrothendieckTopology (goal)
Goal:
⊢ precoverage ≤ Precoverage.comap uliftFunctor precoverage
have refine : ∀ (X : TopCat.{u}) (E : precoverage.ZeroHypercover X),
  E.presieve₀ ∈ (Precoverage.comap uliftFunctor precoverage).coverings X
Tactic: refine Precoverage.le_of_zeroHypercover fun X E ↦ ?_
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.GrothendieckTopology (goal)
Goal:
X : TopCat
E : precoverage.ZeroHypercover X
⊢ E.presieve₀ ∈ (Precoverage.comap uliftFunctor precoverage).coverings X
have refine_1 : Membership.mem (γ := Set (Presieve (uliftFunctor.obj X)))
  (Precoverage.coverings
    (Precoverage.comap
      (forget TopCat (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC := carrier))
      Types.jointlySurjectivePrecoverage)
    (uliftFunctor.obj X))
  (Presieve.map uliftFunctor E.presieve₀)
Tactic: refine ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.GrothendieckTopology (goal)
Goal:
X : TopCat
E : precoverage.ZeroHypercover X
refine_1 :
  Membership.mem (γ := Set (Presieve (uliftFunctor.obj X)))
    (Precoverage.coverings
      (Precoverage.comap
        (forget TopCat (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC := carrier))
        Types.jointlySurjectivePrecoverage)
      (uliftFunctor.obj X))
    (Presieve.map uliftFunctor E.presieve₀)
⊢ E.presieve₀ ∈ (Precoverage.comap uliftFunctor precoverage).coverings X
have refine_2 : Membership.mem (γ := Set (Presieve (uliftFunctor.obj X))) (isOpenEmbedding.precoverage.coverings (uliftFunctor.obj X))
  (Presieve.map uliftFunctor E.presieve₀)
Tactic: refine ⟨?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.GrothendieckTopology (goal)
Goal:
X : TopCat
E : precoverage.ZeroHypercover X
⊢ Membership.mem (γ := Set (Presieve (uliftFunctor.obj X)))
    (Precoverage.coverings
      (Precoverage.comap
        (forget TopCat (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC := carrier))
        Types.jointlySurjectivePrecoverage)
      (uliftFunctor.obj X))
    (Presieve.map uliftFunctor E.presieve₀)
have refine_1 : ∀
  (x :
    ToType (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC := carrier)
      (uliftFunctor.obj X)),
  ∃ (i : E.I₀) (y :
    ToType (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC := carrier)
      (uliftFunctor.obj (E.X i))),
    (ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC := carrier)
            (uliftFunctor.map (E.f i)) :
          (a : (↑(uliftFunctor.obj (E.X i)) : Type (max v u))) → (↑(uliftFunctor.obj X) : Type (max v u)))
        y =
      x
Tactic: simp only [Presieve.map_ofArrows, Precoverage.mem_comap_iff,
      Types.ofArrows_mem_jointlySurjectivePrecoverage_iff, ConcreteCategory.hom_ofHom,
      Set.mem_range, TypeCat.Fun.coe_mk]
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.GrothendieckTopology (goal)
Goal:
X : TopCat
E : precoverage.ZeroHypercover X
x : (↑X : Type u)
⊢ ∃ (i : E.I₀) (y :
    ToType (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC := carrier)
      (uliftFunctor.obj (E.X i))),
    (ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC := carrier)
            (uliftFunctor.map (E.f i)) :
          (a : (↑(uliftFunctor.obj (E.X i)) : Type (max v u))) → (↑(uliftFunctor.obj X) : Type (max v u)))
        y =
      { down := x : ULift.{v, u} (↑X : Type u) }
have refine_1 : ∀ (i : E.I₀) (y : (↑(E.X i) : Type u)),
  ∃ (i_1 : E.I₀) (y_1 :
    ToType (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC := carrier)
      (uliftFunctor.obj (E.X i_1))),
    Eq (α :=
      ToType (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC := carrier)
        (uliftFunctor.obj X))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC :=
              carrier) (uliftFunctor.map (E.f i_1)) :
            (a : (↑(uliftFunctor.obj (E.X i_1)) : Type (max v u))) → (↑(uliftFunctor.obj X) : Type (max v u)))
          y_1 :
        (↑(uliftFunctor.obj X) : Type (max v u)))
      ({
          down :=
            (ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                  (E.f i) :
                (a : (↑(E.X i) : Type u)) → (↑X : Type u))
              y } :
        ULift.{v, u} (↑X : Type u))
Tactic: obtain ⟨i, y, rfl⟩ := exists_mem_zeroHypercover_range E x
Cleared hypotheses: x
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.GrothendieckTopology (goal)
Goal:
X : TopCat
E : precoverage.ZeroHypercover X
i : E.I₀
y : (↑(E.X i) : Type u)
⊢ ∃ (i_1 : E.I₀) (y_1 :
    ToType (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC := carrier)
      (uliftFunctor.obj (E.X i_1))),
    Eq (α :=
      ToType (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC := carrier)
        (uliftFunctor.obj X))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC :=
              carrier) (uliftFunctor.map (E.f i_1)) :
            (a : (↑(uliftFunctor.obj (E.X i_1)) : Type (max v u))) → (↑(uliftFunctor.obj X) : Type (max v u)))
          y_1 :
        (↑(uliftFunctor.obj X) : Type (max v u)))
      ({
          down :=
            (ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                  (E.f i) :
                (a : (↑(E.X i) : Type u)) → (↑X : Type u))
              y } :
        ULift.{v, u} (↑X : Type u))
have h : Eq (α :=
  ToType (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC := carrier)
    (uliftFunctor.obj X))
  ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max v u)), (↑Y : Type (max v u)))) (CC := carrier)
          (uliftFunctor.map (E.f i)) :
        (a : (↑(uliftFunctor.obj (E.X i)) : Type (max v u))) → (↑(uliftFunctor.obj X) : Type (max v u)))
      { down := y } :
    (↑(uliftFunctor.obj X) : Type (max v u)))
  ({
      down :=
        (ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (E.f i) :
            (a : (↑(E.X i) : Type u)) → (↑X : Type u))
          y } :
    ULift.{v, u} (↑X : Type u))
Tactic: use i, ⟨y⟩
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.GrothendieckTopology (goal)
Goal:
X : TopCat
E : precoverage.ZeroHypercover X
⊢ Membership.mem (γ := Set (Presieve (uliftFunctor.obj X))) (isOpenEmbedding.precoverage.coverings (uliftFunctor.obj X))
    (Presieve.map uliftFunctor E.presieve₀)
have refine_2 : ∀ (i : E.I₀),
  Topology.IsOpenEmbedding (X := (↑(uliftFunctor.obj (E.X i)) : Type (max u v))) (Y :=
    (↑(uliftFunctor.obj X) : Type (max u v)))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC := carrier)
          (uliftFunctor.map (E.f i))) :
      (a : (↑(uliftFunctor.obj (E.X i)) : Type (max u v))) → (↑(uliftFunctor.obj X) : Type (max u v)))
Tactic: simp only [Presieve.map_ofArrows, MorphismProperty.ofArrows_mem_precoverage,
      isOpenEmbedding_iff]
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.GrothendieckTopology (goal)
Goal:
⊢ uliftFunctor.IsContinuous grothendieckTopology grothendieckTopology
have h : precoverage ≤ Precoverage.comap uliftFunctor precoverage
Tactic: apply Functor.isContinuous_toGrothendieck_of_pullbacksPreservedBy
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Limits.Basic (goal)
Goal:
J : Type v
inst : Category.{w, v} J
F : J ⥤ TopCat
S : Cone F
x : (↑S.1 : Type (max u v))
i j : J
f : i ⟶ j
⊢ Eq (α := (↑(F.obj j) : Type (max u v)))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC := carrier)
          (F.map f) :
        (a : (↑(F.obj i) : Type (max u v))) → (↑(F.obj j) : Type (max u v)))
      ((fun (x_1 : J) ↦
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
                  carrier) (S.π.app x_1) :
                (a : (↑(((Functor.const J).obj S.pt).obj x_1) : Type (max u v))) → (↑(F.obj x_1) : Type (max u v)))
              x :
            (↑(F.obj x_1) : Type (max u v))))
        i))
    ((fun (x_1 : J) ↦
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
                carrier) (S.π.app x_1) :
              (a : (↑(((Functor.const J).obj S.pt).obj x_1) : Type (max u v))) → (↑(F.obj x_1) : Type (max u v)))
            x :
          (↑(F.obj x_1) : Type (max u v))))
      j)
have dsimp : Eq (α := (↑(F.obj j) : Type (max u v)))
  ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC := carrier)
          (F.map f) :
        (a : (↑(F.obj i) : Type (max u v))) → (↑(F.obj j) : Type (max u v)))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
            carrier) (S.π.app i) :
          (a : (↑S.pt : Type (max u v))) → (↑(F.obj i) : Type (max u v)))
        x) :
    (↑(F.obj j) : Type (max u v)))
  ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC := carrier)
          (S.π.app j) :
        (a : (↑S.pt : Type (max u v))) → (↑(F.obj j) : Type (max u v)))
      x :
    (↑(F.obj j) : Type (max u v)))
Tactic: dsimp
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Limits.Basic (goal)
Goal:
J : Type v
inst : Category.{w, v} J
F : J ⥤ TopCat
S : Cone F
x : (↑S.1 : Type (max u v))
i j : J
f : i ⟶ j
⊢ Eq (α := (↑(F.obj j) : Type (max u v)))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC := carrier)
            (F.map f) :
          (a : (↑(F.obj i) : Type (max u v))) → (↑(F.obj j) : Type (max u v)))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
              carrier) (S.π.app i) :
            (a : (↑S.pt : Type (max u v))) → (↑(F.obj i) : Type (max u v)))
          x) :
      (↑(F.obj j) : Type (max u v)))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC := carrier)
            (S.π.app j) :
          (a : (↑S.pt : Type (max u v))) → (↑(F.obj j) : Type (max u v)))
        x :
      (↑(F.obj j) : Type (max u v)))
have rw : Eq (α := (↑(F.obj j) : Type (max u v)))
  ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC := carrier)
          (F.map f) :
        (a : (↑(F.obj i) : Type (max u v))) → (↑(F.obj j) : Type (max u v)))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC :=
            carrier) (S.π.app i) :
          (a : (↑S.pt : Type (max u v))) → (↑(F.obj i) : Type (max u v)))
        x) :
    (↑(F.obj j) : Type (max u v)))
  ((ConcreteCategory.hom (FC := fun (X Y : TopCat) ↦ C((↑X : Type (max u v)), (↑Y : Type (max u v)))) (CC := carrier)
          (S.π.app i ≫ F.map f) :
        (a : (↑S.pt : Type (max u v))) → (↑(F.obj j) : Type (max u v)))
      x :
    (↑(F.obj j) : Type (max u v)))
Tactic: rw [← S.w f]
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Limits.Basic (goal)
Goal:
J : Type v
inst : Category.{w, v} J
F : CategoryTheory.Functor.{w, u, v, u + 1} J TopCat
c : Cone (F ⋙ forget (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier))
j : J
⊢ Continuous (X := conePtOfConeForget c) (Y := (↑(F.1 j) : Type u))
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (c.π.app j)) :
      (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj j)) → (fun (X : Type u) ↦ X) ((F ⋙ forget).obj j))
have rw : topologicalSpaceConePtOfConeForget c ≤
  induced (Y := (↑(F.1 j) : Type u))
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (c.π.app j)) :
      (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj j)) → (fun (X : Type u) ↦ X) ((F ⋙ forget).obj j))
    (str (F.1 j))
Tactic: rw [continuous_iff_le_induced]
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Limits.Basic (goal)
Goal:
J : Type v
inst : Category.{w, v} J
F : CategoryTheory.Functor.{w, u, v, u + 1} J TopCat
c : Cone (F ⋙ forget (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier))
j j' : J
φ : j ⟶ j'
⊢ CategoryTheory.Functor.map (Functor.obj (Functor.const J) (of (conePtOfConeForget c))) φ ≫
      ofHom
        {
          toFun :=
            (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (c.π.app j')) :
              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj j')) →
                (fun (X : Type u) ↦ X) ((F ⋙ forget).obj j')),
          continuous_toFun := ⋯ : C(conePtOfConeForget c, (↑(F.1 j') : Type u)) } =
    ofHom
        {
          toFun :=
            (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (c.π.app j)) :
              (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj j)) →
                (fun (X : Type u) ↦ X) ((F ⋙ forget).obj j)),
          continuous_toFun := ⋯ : C(conePtOfConeForget c, (↑(F.1 j) : Type u)) } ≫
      F.map φ
have w : ∀ (x : (↑(Functor.obj.{w, u, v, u + 1} (Functor.obj (Functor.const J) (of (conePtOfConeForget c))) j) : Type u)),
  Eq (α := (↑(F.obj j') : Type u))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (X :=
            Functor.obj.{w, u, v, u + 1} (Functor.obj (Functor.const J) (of (conePtOfConeForget c))) j) (Y := F.obj j')
            (CategoryTheory.Functor.map (Functor.obj (Functor.const J) (of (conePtOfConeForget c))) φ ≫
              ofHom
                {
                  toFun :=
                    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (c.π.app j')) :
                      (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj j')) →
                        (fun (X : Type u) ↦ X) ((F ⋙ forget).obj j')),
                  continuous_toFun := sorry : C(conePtOfConeForget c, (↑(F.1 j') : Type u)) }) :
          (a : (↑(((Functor.const J).obj (of (conePtOfConeForget c))).obj j) : Type u)) → (↑(F.obj j') : Type u))
        x :
      (↑(F.obj j') : Type u))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
            (ofHom
                {
                  toFun :=
                    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (c.π.app j)) :
                      (a : (fun (X : Type u) ↦ X) (((Functor.const J).obj c.pt).obj j)) →
                        (fun (X : Type u) ↦ X) ((F ⋙ forget).obj j)),
                  continuous_toFun := sorry : C(conePtOfConeForget c, (↑(F.1 j) : Type u)) } ≫
              F.map φ) :
          (a : (↑(((Functor.const J).obj (of (conePtOfConeForget c))).obj j) : Type u)) → (↑(F.obj j') : Type u))
        x :
      (↑(F.obj j') : Type u))
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Limits.Basic (goal)
Goal:
J : Type v
inst : Category.{w, v} J
F : CategoryTheory.Functor.{w, u, v, u + 1} J TopCat
c_1 c : Cone (F ⋙ forget (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier))
hc : IsLimit c
⊢ IsLimit (coneOfConeForget c)
have refine : ∀ (s : Cone F),
  Continuous (X := (↑s.1 : Type u)) (Y := conePtOfConeForget c)
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
          (hc.lift
            (Functor.mapCone (forget (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier))
              s))) :
      (a : (fun (X : Type u) ↦ X) (forget.mapCone s).pt) → (fun (X : Type u) ↦ X) c.pt)
Tactic: refine IsLimit.ofFaithful forget (ht := hc)
    (fun s ↦ ofHom (ContinuousMap.mk (hc.lift ((forget).mapCone s)) ?_)) (fun _ ↦ rfl)
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Limits.Basic (goal)
Goal:
J : Type v
inst : Category.{w, v} J
F : CategoryTheory.Functor.{w, u, v, u + 1} J TopCat
c_1 c : Cone (F ⋙ forget (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier))
hc : IsLimit c
s : Cone F
⊢ Continuous (X := (↑s.1 : Type u)) (Y := conePtOfConeForget c)
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
          (hc.lift
            (Functor.mapCone (forget (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier))
              s))) :
      (a : (fun (X : Type u) ↦ X) (forget.mapCone s).pt) → (fun (X : Type u) ↦ X) c.pt)
have rw : coinduced (X := (↑s.1 : Type u))
    (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
          (hc.lift
            (Functor.mapCone (forget (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier))
              s))) :
      (a : (fun (X : Type u) ↦ X) (forget.mapCone s).pt) → (fun (X : Type u) ↦ X) c.pt)
    (str s.1) ≤
  topologicalSpaceConePtOfConeForget c
Tactic: rw [continuous_iff_coinduced_le]
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Limits.Basic (goal)
Goal:
J : Type v
inst : Category.{w, v} J
F : CategoryTheory.Functor.{w, u, v, u + 1} J TopCat
c_1 c : Cone (F ⋙ forget (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier))
hc : IsLimit c
s : Cone F
⊢ coinduced (X := (↑s.1 : Type u))
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun)
            (hc.lift
              (Functor.mapCone (forget (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier))
                s))) :
        (a : (fun (X : Type u) ↦ X) (forget.mapCone s).pt) → (fun (X : Type u) ↦ X) c.pt)
      (str s.1) ≤
    topologicalSpaceConePtOfConeForget c
have dsimp : LE.le (α := TopologicalSpace (conePtOfConeForget c))
  (coinduced (X := (↑s.1 : Type u))
    (⇑(@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
          (hc.lift
            (Functor.mapCone (forget (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier))
              s))) :
      (a : ToType s.pt) → c.pt)
    (str s.1))
  (⨅ (j : J),
    induced (Y := ToType (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (F.obj j))
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (c.π.app j)) : (a : c.pt) → ToType (F.obj j)) (F.obj j).str)
Tactic: dsimp [topologicalSpaceConePtOfConeForget]
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Limits.Basic (goal)
Goal:
J : Type v
inst : Category.{w, v} J
F : CategoryTheory.Functor.{w, u, v, u + 1} J TopCat
c_1 c : Cone (F ⋙ forget (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier))
hc : IsLimit c
s : Cone F
⊢ LE.le (α := TopologicalSpace (conePtOfConeForget c))
    (coinduced (X := (↑s.1 : Type u))
      (⇑(@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
            (hc.lift
              (Functor.mapCone (forget (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier))
                s))) :
        (a : ToType s.pt) → c.pt)
      (str s.1))
    (⨅ (j : J),
      induced (Y := ToType (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (F.obj j))
        (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (c.π.app j)) : (a : c.pt) → ToType (F.obj j)) (F.obj j).str)
have rw : ∀ (i : J),
  LE.le (α := TopologicalSpace (conePtOfConeForget c))
    (coinduced (X := (↑s.1 : Type u))
      (⇑(@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
            (hc.lift
              (Functor.mapCone (forget (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier))
                s))) :
        (a : ToType s.pt) → c.pt)
      (str s.1))
    (induced (Y := ToType (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (F.obj i))
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (c.π.app i)) : (a : c.pt) → ToType (F.obj i)) (F.obj i).str)
Tactic: rw [le_iInf_iff]
-- Lean 4 proof step from Mathlib.Topology.Category.TopCat.Limits.Basic (goal)
Goal:
J : Type v
inst : Category.{w, v} J
F : CategoryTheory.Functor.{w, u, v, u + 1} J TopCat
c_1 c : Cone (F ⋙ forget (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier))
hc : IsLimit c
s : Cone F
j : J
⊢ LE.le (α := TopologicalSpace (conePtOfConeForget c))
    (coinduced (X := (↑s.1 : Type u))
      (⇑(@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
            (hc.lift
              (Functor.mapCone (forget (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier))
                s))) :
        (a : ToType s.pt) → c.pt)
      (str s.1))
    (induced (Y := ToType (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (F.obj j))
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (c.π.app j)) : (a : c.pt) → ToType (F.obj j)) (F.obj j).str)
have rw : LE.le (α := TopologicalSpace (↑s.1 : Type u)) (str s.1)
  (induced (Y := ToType (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (F.obj j))
    (Function.comp (β := conePtOfConeForget c)
      (⇑(ConcreteCategory.hom (FC := TypeCat.Fun) (c.π.app j)) : (a : c.pt) → ToType (F.obj j))
      (⇑(@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
            (hc.lift
              (Functor.mapCone (forget (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier))
                s))) :
        (a : ToType s.pt) → c.pt))
    (F.obj j).str)
Tactic: rw [coinduced_le_iff_le_induced, induced_compose]