awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : PresheafedSpace.{v_1, u_1, w} C
Y : TopCat
f : Y ⟶ TopCat.of (↑(↑X : TopCat) : Type w)
h :
  IsOpenEmbedding (X := (↑Y : Type w)) (Y := (↑(TopCat.of (↑(↑X : TopCat) : Type w)) : Type w))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{w}) ↦ C((↑X : Type w), (↑Y : Type w))) (CC := TopCat.carrier) f) :
      (a : (↑Y : Type w)) → (↑(TopCat.of (↑(↑X : TopCat) : Type w)) : Type w))
U : Opens (↑(↑(X.restrict h) : TopCat) : Type w)
⊢ Eq (α := (X.restrict h).presheaf.obj (op U) ⟶ X.presheaf.obj (op ((opensFunctor (X.ofRestrict h)).obj U)))
    (invApp (X.ofRestrict h) U) (𝟙 ((X.restrict h).presheaf.obj (op U)))
have delta : Eq (α := (X.restrict h).presheaf.obj (op U) ⟶ X.presheaf.obj (op ((opensFunctor (X.ofRestrict h)).obj U)))
  ((X.restrict h).presheaf.map (eqToHom sorry) ≫ inv ((X.ofRestrict h).c.app (op ((opensFunctor (X.ofRestrict h)).obj U))))
  (𝟙 ((X.restrict h).presheaf.obj (op U)))
Tactic: delta invApp
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : PresheafedSpace.{v_1, u_1, w} C
Y : TopCat
f : Y ⟶ TopCat.of (↑(↑X : TopCat) : Type w)
h :
  IsOpenEmbedding (X := (↑Y : Type w)) (Y := (↑(TopCat.of (↑(↑X : TopCat) : Type w)) : Type w))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{w}) ↦ C((↑X : Type w), (↑Y : Type w))) (CC := TopCat.carrier) f) :
      (a : (↑Y : Type w)) → (↑(TopCat.of (↑(↑X : TopCat) : Type w)) : Type w))
U : Opens (↑(↑(X.restrict h) : TopCat) : Type w)
⊢ Eq (α := (X.restrict h).presheaf.obj (op U) ⟶ X.presheaf.obj (op ((opensFunctor (X.ofRestrict h)).obj U)))
    ((X.restrict h).presheaf.map (eqToHom ⋯) ≫
      inv ((X.ofRestrict h).c.app (op ((opensFunctor (X.ofRestrict h)).obj U))))
    (𝟙 ((X.restrict h).presheaf.obj (op U)))
have rw : (X.restrict h).presheaf.map (eqToHom sorry) = (X.ofRestrict h).c.app (op ((opensFunctor (X.ofRestrict h)).obj U))
Tactic: rw [IsIso.comp_inv_eq, Category.id_comp]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : PresheafedSpace.{v_1, u_1, w} C
Y : TopCat
f : Y ⟶ TopCat.of (↑(↑X : TopCat) : Type w)
h :
  IsOpenEmbedding (X := (↑Y : Type w)) (Y := (↑(TopCat.of (↑(↑X : TopCat) : Type w)) : Type w))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{w}) ↦ C((↑X : Type w), (↑Y : Type w))) (CC := TopCat.carrier) f) :
      (a : (↑Y : Type w)) → (↑(TopCat.of (↑(↑X : TopCat) : Type w)) : Type w))
U : Opens (↑(↑(X.restrict h) : TopCat) : Type w)
⊢ (X.restrict h).presheaf.map (eqToHom ⋯) = (X.ofRestrict h).c.app (op ((opensFunctor (X.ofRestrict h)).obj U))
have change : X.presheaf.map (h.functor.op.map (eqToHom sorry)) =
  X.presheaf.map (Quiver.Hom.op (sorry.adjunction.counit.app (unop (op ((opensFunctor (X.ofRestrict h)).obj U)))))
Tactic: change X.presheaf.map _ = X.presheaf.map _
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y : PresheafedSpace.{v, u, v} C
f : X ⟶ Y
H : IsOpenImmersion f
inst_1 : HasColimits.{v, u} C
x : (↑(↑X : TopCat) : Type v)
⊢ IsIso (X :=
    Y.presheaf.stalk
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            f.base :
          (a : (↑(↑X : TopCat) : Type v)) → (↑(↑Y : TopCat) : Type v))
        x))
    (Hom.stalkMap f x)
have rw : IsIso (X :=
  Y.presheaf.stalk
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
          (CategoryStruct.comp (Y := Y.restrict (U := (↑X : TopCat)) (f := f.base) sorry)
              (Iso.hom (isoRestrict f) : X ⟶ Y.restrict (U := (↑X : TopCat)) (f := f.base) sorry)
              (Y.ofRestrict sorry : Y.restrict (U := (↑X : TopCat)) (f := f.base) sorry ⟶ Y)).base :
        (a : (↑(↑X : TopCat) : Type v)) → (↑(↑Y : TopCat) : Type v))
      x))
  (Hom.stalkMap (Y.ofRestrict sorry)
        (((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (Y := (↑(Y.restrict (U := (↑X : TopCat)) (f := f.base) sorry) : TopCat))
                (Hom.base (Iso.hom (isoRestrict f)) :
                  (↑X : TopCat) ⟶ (↑(Y.restrict (U := (↑X : TopCat)) (f := f.base) sorry) : TopCat)) :
              C((↑(↑X : TopCat) : Type v), (↑(↑(Y.restrict (U := (↑X : TopCat)) (f := f.base) sorry) : TopCat) : Type v))) :
            (a : (↑(↑X : TopCat) : Type v)) → (↑(↑(Y.restrict sorry) : TopCat) : Type v))
          x) ≫
      Hom.stalkMap (Iso.hom (isoRestrict f)) x :
    Y.presheaf.stalk
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (X := (↑(Y.restrict (U := (↑X : TopCat)) (f := f.base) sorry) : TopCat)) (Hom.base (Y.ofRestrict sorry)) :
            (a : (↑(↑(Y.restrict sorry) : TopCat) : Type v)) → (↑(↑Y : TopCat) : Type v))
          (((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC :=
                    TopCat.carrier) (Y := (↑(Y.restrict (U := (↑X : TopCat)) (f := f.base) sorry) : TopCat))
                    (Hom.base (Iso.hom (isoRestrict f)) :
                      (↑X : TopCat) ⟶ (↑(Y.restrict (U := (↑X : TopCat)) (f := f.base) sorry) : TopCat)) :
                  C((↑(↑X : TopCat) : Type v),
                    (↑(↑(Y.restrict (U := (↑X : TopCat)) (f := f.base) sorry) : TopCat) : Type v))) :
                (a : (↑(↑X : TopCat) : Type v)) → (↑(↑(Y.restrict sorry) : TopCat) : Type v))
              x :
            (↑(↑(Y.restrict (U := (↑X : TopCat)) (f := f.base) sorry) : TopCat) : Type v))) ⟶
      X.presheaf.stalk x)
Tactic: rw [← H.isoRestrict_hom_ofRestrict, PresheafedSpace.stalkMap.comp]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
U : (Opens (↑(↑X : TopCat) : Type u_1))ᵒᵖ
⊢ (Opens.map g.base).op.obj (op (Functor.obj (IsOpenEmbedding.functor (f := f.base) ⋯) (unop U))) =
    Functor.obj (Functor.op (IsOpenEmbedding.functor ⋯))
      (Functor.obj (Functor.op (Opens.map (pullback.fst f.base g.base))) U)
have simp : {
    carrier :=
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
              TopCat.carrier) g.base) :
          (a : (↑(↑Y : TopCat) : Type u_1)) → (↑(↑Z : TopCat) : Type u_1)) ⁻¹'
        (↑{
              carrier :=
                (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                        TopCat.carrier) f.base) :
                    (a : (↑(↑X : TopCat) : Type u_1)) → (↑(↑Z : TopCat) : Type u_1)) ''
                  (↑(unop U) : Set (↑(↑X : TopCat) : Type u_1)),
              is_open' := sorry : Opens (↑(↑Z : TopCat) : Type u_1) } :
          Set (↑(↑Z : TopCat) : Type u_1)),
    is_open' := sorry : Opens (↑(↑Y : TopCat) : Type u_1) } =
  {
    carrier :=
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
              TopCat.carrier) (X := pullback f.base g.base) (pullback.snd f.base g.base)) :
          (a : (↑(pullback f.base g.base) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1)) ''
        (↑{
              carrier :=
                (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                        TopCat.carrier) (X := (↑(Y.restrict (U := pullback f.base g.base) sorry) : TopCat))
                        (pullback.fst f.base g.base)) :
                    (a : (↑(↑(Y.restrict sorry) : TopCat) : Type u_1)) → (↑(↑X : TopCat) : Type u_1)) ⁻¹'
                  (↑(unop U) : Set (↑(↑X : TopCat) : Type u_1)),
              is_open' := sorry : Opens (↑(↑(Y.restrict (U := pullback f.base g.base) sorry) : TopCat) : Type u_1) } :
          Set (↑(pullback f.base g.base) : Type u_1)),
    is_open' := sorry }
Tactic: simp only [IsOpenMap.functor, op_inj_iff, Opens.map,
                    Functor.op_obj]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
U V : (Opens (↑(↑X : TopCat) : Type u_1))ᵒᵖ
i : U ⟶ V
⊢ X.presheaf.map i ≫
      CategoryStruct.comp (Y := Z.presheaf.obj (op ((opensFunctor f).obj (unop V)))) (invApp f (unop V))
        (g.c.app (op (Functor.obj (IsOpenEmbedding.functor (f := f.base) ⋯) (unop V))) ≫ Y.presheaf.map (eqToHom ⋯)) =
    CategoryStruct.comp (Y := Z.presheaf.obj (op ((opensFunctor f).obj (unop U)))) (invApp f (unop U))
        (g.c.app (op (Functor.obj (IsOpenEmbedding.functor (f := f.base) ⋯) (unop U))) ≫ Y.presheaf.map (eqToHom ⋯)) ≫
      (Functor.obj (TopCat.Presheaf.pushforward C (pullback.fst f.base g.base))
            (Y.restrict (U := pullback f.base g.base) ⋯).presheaf).map
        i
have op : ∀ (unop : Opens (↑(↑X : TopCat) : Type u_1)) (i : op unop ⟶ V),
  X.presheaf.map i ≫
      CategoryStruct.comp (Y := Z.presheaf.obj (op ((opensFunctor f).obj (Opposite.unop V))))
        (invApp f (Opposite.unop V))
        (g.c.app (op (Functor.obj (IsOpenEmbedding.functor (f := f.base) sorry) (Opposite.unop V))) ≫
          Y.presheaf.map (eqToHom sorry)) =
    CategoryStruct.comp (Y := Z.presheaf.obj (op ((opensFunctor f).obj (Opposite.unop (op unop)))))
        (invApp f (Opposite.unop (op unop)))
        (g.c.app (op (Functor.obj (IsOpenEmbedding.functor (f := f.base) sorry) (Opposite.unop (op unop)))) ≫
          Y.presheaf.map (eqToHom sorry)) ≫
      (Functor.obj (TopCat.Presheaf.pushforward C (pullback.fst f.base g.base))
            (Y.restrict (U := pullback f.base g.base) sorry).presheaf).map
        i
Tactic: induction U
Cleared hypotheses: U, i
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
V : (Opens (↑(↑X : TopCat) : Type u_1))ᵒᵖ
unop : Opens (↑(↑X : TopCat) : Type u_1)
i : op unop ⟶ V
⊢ X.presheaf.map i ≫
      CategoryStruct.comp (Y := Z.presheaf.obj (op ((opensFunctor f).obj (Opposite.unop V))))
        (invApp f (Opposite.unop V))
        (g.c.app (op (Functor.obj (IsOpenEmbedding.functor (f := f.base) ⋯) (Opposite.unop V))) ≫
          Y.presheaf.map (eqToHom ⋯)) =
    CategoryStruct.comp (Y := Z.presheaf.obj (op ((opensFunctor f).obj (Opposite.unop (op unop)))))
        (invApp f (Opposite.unop (op unop)))
        (g.c.app (op (Functor.obj (IsOpenEmbedding.functor (f := f.base) ⋯) (Opposite.unop (op unop)))) ≫
          Y.presheaf.map (eqToHom ⋯)) ≫
      (Functor.obj (TopCat.Presheaf.pushforward C (pullback.fst f.base g.base))
            (Y.restrict (U := pullback f.base g.base) ⋯).presheaf).map
        i
have op : ∀ (unop_1 : Opens (↑(↑X : TopCat) : Type u_1)) (i : op unop ⟶ op unop_1),
  X.presheaf.map i ≫
      CategoryStruct.comp (Y := Z.presheaf.obj (op ((opensFunctor f).obj (Opposite.unop (op unop_1)))))
        (invApp f (Opposite.unop (op unop_1)))
        (g.c.app (op (Functor.obj (IsOpenEmbedding.functor (f := f.base) sorry) (Opposite.unop (op unop_1)))) ≫
          Y.presheaf.map (eqToHom sorry)) =
    CategoryStruct.comp (Y := Z.presheaf.obj (op ((opensFunctor f).obj (Opposite.unop (op unop)))))
        (invApp f (Opposite.unop (op unop)))
        (g.c.app (op (Functor.obj (IsOpenEmbedding.functor (f := f.base) sorry) (Opposite.unop (op unop)))) ≫
          Y.presheaf.map (eqToHom sorry)) ≫
      (Functor.obj (TopCat.Presheaf.pushforward C (pullback.fst f.base g.base))
            (Y.restrict (U := pullback f.base g.base) sorry).presheaf).map
        i
Tactic: induction V
Cleared hypotheses: V, i
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
⊢ Eq (α := Y.restrict (U := pullback f.base g.base) ⋯ ⟶ Z)
    (pullbackConeOfLeftFst f g ≫ f : Y.restrict (U := pullback f.base g.base) ⋯ ⟶ Z)
    (Y.ofRestrict ⋯ ≫ g : Y.restrict (U := pullback f.base g.base) ⋯ ⟶ Z)
have refine_1 : Eq (α := (↑(Y.restrict (U := pullback f.base g.base) sorry) : TopCat) ⟶ (↑Z : TopCat))
  (Hom.base (pullbackConeOfLeftFst f g ≫ f) : (↑(Y.restrict (U := pullback f.base g.base) sorry) : TopCat) ⟶ (↑Z : TopCat))
  (Hom.base (Y.ofRestrict sorry ≫ g) : (↑(Y.restrict (U := pullback f.base g.base) sorry) : TopCat) ⟶ (↑Z : TopCat))
Tactic: refine PresheafedSpace.Hom.ext _ _ ?_ <| NatTrans.ext <| funext fun U => ?_
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
refine_1 :
  Eq (α := (↑(Y.restrict (U := pullback f.base g.base) ⋯) : TopCat) ⟶ (↑Z : TopCat))
    (Hom.base (pullbackConeOfLeftFst f g ≫ f) :
      (↑(Y.restrict (U := pullback f.base g.base) ⋯) : TopCat) ⟶ (↑Z : TopCat))
    (Hom.base (Y.ofRestrict ⋯ ≫ g) : (↑(Y.restrict (U := pullback f.base g.base) ⋯) : TopCat) ⟶ (↑Z : TopCat))
⊢ Eq (α := Y.restrict (U := pullback f.base g.base) ⋯ ⟶ Z)
    (pullbackConeOfLeftFst f g ≫ f : Y.restrict (U := pullback f.base g.base) ⋯ ⟶ Z)
    (Y.ofRestrict ⋯ ≫ g : Y.restrict (U := pullback f.base g.base) ⋯ ⟶ Z)
have refine_2 : ∀ (U : (Opens (↑(↑Z : TopCat) : Type u_1))ᵒᵖ),
  Eq (α :=
    Z.presheaf.obj U ⟶
      (Functor.op (Opens.map (Hom.base (Y.ofRestrict sorry ≫ g))) ⋙
            (Y.restrict (U := pullback f.base g.base) sorry).presheaf).obj
        U)
    (NatTrans.app
        (CategoryStruct.comp (Y :=
          Functor.obj
            (TopCat.Presheaf.pushforward C (Hom.base (pullbackConeOfLeftFst f g ≫ f)) :
              TopCat.Presheaf.{u_1, v, u} C (↑(Y.restrict (U := pullback f.base g.base) sorry) : TopCat) ⥤
                TopCat.Presheaf.{u_1, v, u} C (↑Z : TopCat))
            (Y.restrict (U := pullback f.base g.base) sorry).presheaf)
          (Hom.c (pullbackConeOfLeftFst f g ≫ f) :
            Z.presheaf ⟶
              Functor.obj
                (TopCat.Presheaf.pushforward C (Hom.base (pullbackConeOfLeftFst f g ≫ f)) :
                  TopCat.Presheaf.{u_1, v, u} C (↑(Y.restrict (U := pullback f.base g.base) sorry) : TopCat) ⥤
                    TopCat.Presheaf.{u_1, v, u} C (↑Z : TopCat))
                (Y.restrict (U := pullback f.base g.base) sorry).presheaf)
          (Functor.whiskerRight (eqToHom sorry) (Y.restrict (U := pullback f.base g.base) sorry).presheaf :
            Functor.op (Opens.map (Hom.base (pullbackConeOfLeftFst f g ≫ f))) ⋙
                (Y.restrict (U := pullback f.base g.base) sorry).presheaf ⟶
              Functor.op (Opens.map (Hom.base (Y.ofRestrict sorry ≫ g))) ⋙
                (Y.restrict (U := pullback f.base g.base) sorry).presheaf))
        U :
      Z.presheaf.obj U ⟶
        (Functor.op (Opens.map (Hom.base (Y.ofRestrict sorry ≫ g))) ⋙
              (Y.restrict (U := pullback f.base g.base) sorry).presheaf).obj
          U)
    (NatTrans.app (Hom.c (Y.ofRestrict sorry ≫ g)) U :
      Z.presheaf.obj U ⟶
        (Functor.op (Opens.map (Hom.base (Y.ofRestrict sorry ≫ g))) ⋙
              (Y.restrict (U := pullback f.base g.base) sorry).presheaf).obj
          U)
Tactic: refine PresheafedSpace.Hom.ext _ _ ?_ <| NatTrans.ext <| funext fun U => ?_
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
U : (Opens (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1))ᵒᵖ
⊢ op
      {
        carrier :=
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                  TopCat.carrier) s.snd.base) :
              (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1)) ⁻¹'
            (↑(⋯.functor.obj (unop U)) : Set (↑(↑Y : TopCat) : Type u_1)),
        is_open' := ⋯ : Opens (↑(↑s.pt : TopCat) : Type u_1) } =
    op
      {
        carrier :=
          (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                  TopCat.carrier) (Y := (↑(pullbackConeOfLeft f g).pt : TopCat))
                  (pullback.lift s.fst.base s.snd.base ⋯)) :
              (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1)) ⁻¹'
            (↑(unop U) : Set (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1)),
        is_open' := ⋯ }
have e_unop : Eq (α := Set (↑(↑s.pt : TopCat) : Type u_1))
  ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := TopCat.carrier)
          s.snd.base) :
      (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1)) ⁻¹'
    (↑(sorry.functor.obj (unop U)) : Set (↑(↑Y : TopCat) : Type u_1)))
  ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := TopCat.carrier)
          (Y := (↑(pullbackConeOfLeft f g).pt : TopCat)) (pullback.lift s.fst.base s.snd.base sorry)) :
      (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1)) ⁻¹'
    (↑(unop U) : Set (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1)))
Tactic: congr 2
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (hypothesis)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
U : (Opens (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1))ᵒᵖ
s' : PullbackCone f.base g.base := PullbackCone.mk s.fst.base s.snd.base ⋯
⊢ Eq (α := Set (↑(↑s.pt : TopCat) : Type u_1))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
            TopCat.carrier) s.snd.base) :
        (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1)) ⁻¹'
      (↑(⋯.functor.obj (unop U)) : Set (↑(↑Y : TopCat) : Type u_1)))
    ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
            TopCat.carrier) (Y := (↑(pullbackConeOfLeft f g).pt : TopCat)) (pullback.lift s.fst.base s.snd.base ⋯)) :
        (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1)) ⁻¹'
      (↑(unop U) : Set (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1)))
have this : CategoryStruct.comp (Y := limit (cospan f.base g.base)) (limit.lift (cospan f.base g.base) s')
    (limit.π (cospan f.base g.base) WalkingCospan.right) =
  s.snd.base
Tactic: have : _ = s.snd.base := limit.lift_π s' WalkingCospan.right
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
U : (Opens (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1))ᵒᵖ
s' : PullbackCone f.base g.base := PullbackCone.mk s.fst.base s.snd.base ⋯
this :
  CategoryStruct.comp (Y := limit (cospan f.base g.base)) (limit.lift (cospan f.base g.base) s')
      (limit.π (cospan f.base g.base) WalkingCospan.right) =
    s.snd.base
| (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := TopCat.carrier)
          s.snd.base) :
      (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1)) ⁻¹'
    (↑(⋯.functor.obj (unop U)) : Set (↑(↑Y : TopCat) : Type u_1))
have rw : (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := TopCat.carrier)
        (CategoryStruct.comp (Y := limit (cospan f.base g.base)) (limit.lift (cospan f.base g.base) s')
          (limit.π (cospan f.base g.base) WalkingCospan.right))) :
    (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1)) ⁻¹'
  (↑(sorry.functor.obj (unop U)) : Set (↑(↑Y : TopCat) : Type u_1))
Tactic: rw [← this]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.OpenImmersion (goal)
Goal:
C : Type u
inst : Category.{v, u} C
X Y Z : PresheafedSpace.{v, u, u_1} C
f : X ⟶ Z
hf : IsOpenImmersion f
g : Y ⟶ Z
s : PullbackCone f g
U : (Opens (↑(↑(pullbackConeOfLeft f g).pt : TopCat) : Type u_1))ᵒᵖ
s' : PullbackCone f.base g.base := PullbackCone.mk s.fst.base s.snd.base ⋯
this :
  CategoryStruct.comp (Y := limit (cospan f.base g.base)) (limit.lift (cospan f.base g.base) s')
      (limit.π (cospan f.base g.base) WalkingCospan.right) =
    s.snd.base
| (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := TopCat.carrier)
          (CategoryStruct.comp (Y := limit (cospan f.base g.base)) (limit.lift (cospan f.base g.base) s')
            (limit.π (cospan f.base g.base) WalkingCospan.right))) :
      (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1)) ⁻¹'
    (↑(⋯.functor.obj (unop U)) : Set (↑(↑Y : TopCat) : Type u_1))
have dsimp : Set.preimage (β := (↑(↑Y : TopCat) : Type u_1))
  (Function.comp (β := (↑(limit (cospan f.base g.base)) : Type u_1))
    (⇑(TopCat.Hom.hom (limit.π (cospan f.base g.base) WalkingCospan.right)) :
      (a : (↑(limit (cospan f.base g.base)) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1))
    (⇑(TopCat.Hom.hom (limit.lift (cospan f.base g.base) (PullbackCone.mk s.fst.base s.snd.base sorry)) :
          C((↑(↑s.pt : TopCat) : Type u_1), (↑(limit (cospan f.base g.base)) : Type u_1))) :
      (a : (↑(↑s.pt : TopCat) : Type u_1)) → (↑(limit (cospan f.base g.base)) : Type u_1)))
  ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := TopCat.carrier)
          (X := pullback f.base g.base) (pullback.snd f.base g.base)) :
      (a : (↑(pullback f.base g.base) : Type u_1)) → (↑(↑Y : TopCat) : Type u_1)) ''
    (↑(unop U) : Set (↑(pullback f.base g.base) : Type u_1)))
Tactic: dsimp [s']