awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
⊢ Function.Injective (α := (↑(D.mapGlueData (forget C)).glued : Type v)) (β := (↑(↑D.glued : TopCat) : Type v))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (X := (D.mapGlueData (forget C)).glued) (Y := (↑D.glued : TopCat))
            (Iso.inv (D.gluedIso (forget C)) : (D.mapGlueData (forget C)).glued ⟶ Functor.obj (forget C) D.glued) :
          C((↑(D.mapGlueData (forget C)).glued : Type v), (↑(↑D.glued : TopCat) : Type v))) :
      (a : (↑(D.mapGlueData (forget C)).glued : Type v)) → (↑(↑D.glued : TopCat) : Type v))
have h : Mono (X := (D.mapGlueData (forget C)).glued) (Y := (↑D.glued : TopCat))
  (Iso.inv (D.gluedIso (forget C)) : (D.mapGlueData (forget C)).glued ⟶ Functor.obj (forget C) D.glued)
Tactic: rw [← TopCat.mono_iff_injective]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j k : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
have h : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Tactic: erw [← comp_c_app, ← comp_c_app]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i j k : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
have h : Eq (α :=
  (D.U i).presheaf.obj (op U) ⟶
    (Functor.obj (TopCat.Presheaf.pushforward C (D.f j k).base) (D.V (j, k)).presheaf).obj
      (op
        (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
          (Opens.map (Hom.base (D.ι j)) : Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U j) : TopCat) : Type v))
          (Functor.obj
              (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) sorry :
                Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
              U :
            Opens (↑(↑D.glued : TopCat) : Type v)))))
  (CategoryStruct.comp (Y :=
    (Functor.obj (TopCat.Presheaf.pushforward C (Hom.base (pullback.fst (D.f j i) (D.f j k))))
          (pullback (D.f j i) (D.f j k)).presheaf).obj
      (op ((Opens.map (D.t j i).base).1 (unop ((Opens.map (D.f i j).base).op.obj (op U))))))
    (NatTrans.app (Hom.c (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j)) (op U) :
      (D.U i).presheaf.obj (op U) ⟶
        (Functor.obj (TopCat.Presheaf.pushforward C (Hom.base (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j)))
              (pullback (D.f j i) (D.f j k)).presheaf).obj
          (op U))
    (invApp (pullback.snd (D.f j i) (D.f j k))
        ((Opens.map (Hom.base (pullback.fst (D.f j i) (D.f j k))) :
              Opens (↑(↑(D.V (j, i)) : TopCat) : Type v) ⥤
                Opens (↑(↑(pullback (D.f j i) (D.f j k)) : TopCat) : Type v)).1
          ((Opens.map (D.t j i).base).1 (unop ((Opens.map (D.f i j).base).op.obj (op U))))) ≫
      (D.V (j, k)).presheaf.map (eqToHom sorry ≫ (Opens.map (D.f j k).base).op.map (Quiver.Hom.op (eqToHom sorry)))))
  (CategoryStruct.comp (Y :=
    (Functor.obj (TopCat.Presheaf.pushforward C (Hom.base (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j)))
          (pullback (D.f j i) (D.f j k)).presheaf).obj
      (op U))
    (NatTrans.app (Hom.c (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j)) (op U))
    (invApp (pullback.snd (D.f j i) (D.f j k))
        ((Opens.map (Hom.base (pullback.fst (D.f j i) (D.f j k) ≫ D.t j i ≫ D.f i j)) :
              Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑(pullback (D.f j i) (D.f j k)) : TopCat) : Type v)).1
          U) ≫
      (D.V (j, k)).presheaf.map (eqToHom sorry)))
Tactic: simp_rw [Category.assoc]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
j : WalkingMultispan (MultispanShape.prod D.J)
⊢ (D.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op j)
have left : (j k : D.J) → (D.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op (WalkingMultispan.left (j, k)))
Tactic: rcases j with (⟨j, k⟩ | j)
Cleared hypotheses: j
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
j : WalkingMultispan (MultispanShape.prod D.J)
left : (j k : D.J) → (D.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op (WalkingMultispan.left (j, k)))
⊢ (D.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op j)
have right : (j : (MultispanShape.prod D.J).R) →
  (D.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op (WalkingMultispan.right j))
Tactic: rcases j with (⟨j, k⟩ | j)
Cleared hypotheses: j
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
j k : D.J
⊢ (D.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op (WalkingMultispan.left (j, k)))
have left : (Opens.map (D.f j k).base).op.obj
    (op
      (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
        (Opens.map (Hom.base (D.ι j)) : Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U j) : TopCat) : Type v))
        (Functor.obj
            (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) sorry :
              Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
            U :
          Opens (↑(↑D.glued : TopCat) : Type v)))) =
  op
    (Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
      (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.left (j, k)))))) :
        Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
          Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.left (j, k))))) : TopCat) : Type v))
      (Functor.obj
          (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) sorry :
            Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
          U :
        Opens (↑(↑D.glued : TopCat) : Type v)))
Tactic: refine
      D.opensImagePreimageMap i j U ≫ (D.f j k).c.app _ ≫ (D.V (j, k)).presheaf.map (eqToHom ?_)
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
j k : D.J
⊢ op
      ((Opens.map (D.f j k).base).obj
        (unop
          (op
            (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
              (Opens.map (Hom.base (D.ι j)) :
                Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U j) : TopCat) : Type v))
              (Functor.obj
                  (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                    Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                  U :
                Opens (↑(↑D.glued : TopCat) : Type v)))))) =
    op
      (Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
        (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.left (j, k)))))) :
          Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
            Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.left (j, k))))) : TopCat) : Type v))
        (Functor.obj
            (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
              Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
            U :
          Opens (↑(↑D.glued : TopCat) : Type v)))
have left : (Opens.map (D.f j k).base).obj
    (unop
      (op
        (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
          (Opens.map (Hom.base (D.ι j)) : Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U j) : TopCat) : Type v))
          (Functor.obj
              (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) sorry :
                Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
              U :
            Opens (↑(↑D.glued : TopCat) : Type v))))) =
  Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
    (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.left (j, k)))))) :
      Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
        Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.left (j, k))))) : TopCat) : Type v))
    (Functor.obj
        (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) sorry :
          Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
        U :
      Opens (↑(↑D.glued : TopCat) : Type v))
Tactic: congr 1
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
j k : D.J
⊢ (Opens.map (D.f j k).base).obj
      (unop
        (op
          (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
            (Opens.map (Hom.base (D.ι j)) :
              Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U j) : TopCat) : Type v))
            (Functor.obj
                (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                  Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                U :
              Opens (↑(↑D.glued : TopCat) : Type v))))) =
    Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
      (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.left (j, k)))))) :
        Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
          Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.left (j, k))))) : TopCat) : Type v))
      (Functor.obj
          (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
            Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
          U :
        Opens (↑(↑D.glued : TopCat) : Type v))
have left : (↑((Opens.map (D.f j k).base).obj
        (unop
          (op
            (Functor.obj (C := Opens (↑(↑D.glued : TopCat) : Type v))
              (Opens.map (Hom.base (D.ι j)) :
                Opens (↑(↑D.glued : TopCat) : Type v) ⥤ Opens (↑(↑(D.U j) : TopCat) : Type v))
              (Functor.obj
                  (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) sorry :
                    Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
                  U :
                Opens (↑(↑D.glued : TopCat) : Type v)))))) :
    Set (↑(↑(D.V (j, k)) : TopCat) : Type v)) =
  (↑(Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
          (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.left (j, k)))))) :
            Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
              Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.left (j, k))))) : TopCat) : Type v))
          (Functor.obj
              (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) sorry :
                Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
              U :
            Opens (↑(↑D.glued : TopCat) : Type v)) :
        Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.left (j, k))))) : TopCat) : Type v)) :
    Set (↑(↑(D.V (j, k)) : TopCat) : Type v))
Tactic: ext1
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
j k : D.J
⊢ Eq (α := Set (↑(↑(D.V (j, k)) : TopCat) : Type v))
    (Set.preimage (β := (↑(↑D.glued : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (Y := (↑D.glued : TopCat)) (Hom.base (D.f j k ≫ D.ι j))) :
        (a : (↑(↑(D.V (j, k)) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (Y := (↑D.glued : TopCat)) (Hom.base (D.ι i))) :
          (a : (↑(↑(D.U i) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v)) ''
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
    (Set.preimage (β := (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (X := (↑(D.diagram.multispan.obj (WalkingMultispan.left (j, k))) : TopCat)) (Y :=
            (↑(Limits.colimit D.diagram.multispan) : TopCat))
            (Hom.base (colimit.ι D.diagram.multispan (WalkingMultispan.left (j, k))))) :
        (a : (↑(↑(D.diagram.multispan.obj (WalkingMultispan.left (j, k))) : TopCat) : Type v)) →
          (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (Y := (↑D.glued : TopCat)) (Hom.base (D.ι i))) :
          (a : (↑(↑(D.U i) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v)) ''
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
have left : Eq (α := (D.V (j, k)).Hom D.glued) (D.f j k ≫ D.ι j) (colimit.ι D.diagram.multispan (WalkingMultispan.left (j, k)))
Tactic: congr 4
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
⊢ ∃ (eq :
    op U =
      op
        (Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
          (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i))))) :
            Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
              Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v))
          (Functor.obj
              (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
              U :
            Opens (↑(↑D.glued : TopCat) : Type v)))),
    D.ιInvApp U ≫ D.diagramOverOpenπ U i = (D.U i).presheaf.map (eqToHom eq)
have w : op U =
  op
    (Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
      (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i))))) :
        Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
          Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v))
      (Functor.obj
          (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) sorry :
            Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
          U :
        Opens (↑(↑D.glued : TopCat) : Type v)))
Tactic: fconstructor
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
w :
  op U =
    op
      (Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
        (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i))))) :
          Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
            Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v))
        (Functor.obj
            (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
              Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
            U :
          Opens (↑(↑D.glued : TopCat) : Type v)))
⊢ ∃ (eq :
    op U =
      op
        (Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
          (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i))))) :
            Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
              Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v))
          (Functor.obj
              (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
                Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
              U :
            Opens (↑(↑D.glued : TopCat) : Type v)))),
    D.ιInvApp U ≫ D.diagramOverOpenπ U i = (D.U i).presheaf.map (eqToHom eq)
have h : Eq (α := (D.U i).presheaf.obj (op U) ⟶ (D.diagramOverOpen U).obj (op (WalkingMultispan.right i)))
  (D.ιInvApp U ≫ D.diagramOverOpenπ U i) ((D.U i).presheaf.map (eqToHom sorry))
Tactic: fconstructor
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
⊢ op U =
    op
      (Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
        (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i))))) :
          Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
            Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v))
        (Functor.obj
            (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
              Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
            U :
          Opens (↑(↑D.glued : TopCat) : Type v)))
have w : U =
  Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
    (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i))))) :
      Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
        Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v))
    (Functor.obj
        (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) sorry :
          Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
        U :
      Opens (↑(↑D.glued : TopCat) : Type v))
Tactic: congr
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
⊢ U =
    Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
      (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i))))) :
        Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
          Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v))
      (Functor.obj
          (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) ⋯ :
            Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
          U :
        Opens (↑(↑D.glued : TopCat) : Type v))
have w : (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)) =
  (↑(Functor.obj (C := Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
          (Opens.map (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i))))) :
            Opens (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v) ⥤
              Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v))
          (Functor.obj
              (IsOpenEmbedding.functor (f := Hom.base (D.ι i)) sorry :
                Opens (↑(↑(D.U i) : TopCat) : Type v) ⥤ Opens (↑(↑D.glued : TopCat) : Type v))
              U :
            Opens (↑(↑D.glued : TopCat) : Type v)) :
        Opens (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v)) :
    Set (↑(↑(D.U i) : TopCat) : Type v))
Tactic: ext1
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing (goal)
Goal:
C : Type u
inst : Category.{v, u} C
D : GlueData.{v, u} C
inst_1 : HasLimits.{v, u} C
i : D.J
U : Opens (↑(↑(D.U i) : TopCat) : Type v)
⊢ (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)) =
    Set.preimage (β := (↑(↑D.glued : TopCat) : Type v))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (X := (↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat)) (Y :=
            (↑(Limits.colimit D.diagram.multispan) : TopCat))
            (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i)))))) :
        (a : (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v)) →
          (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
      ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (Y := (↑D.glued : TopCat)) (Hom.base (D.ι i))) :
          (a : (↑(↑(D.U i) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v)) ''
        (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)))
have w : ∀ (x : (↑(↑(D.U i) : TopCat) : Type v)),
  x ∈ (↑U : Set (↑(↑(D.U i) : TopCat) : Type v)) ↔
    Membership.mem (γ := Set (↑(↑(D.U i) : TopCat) : Type v))
      (Set.preimage (β := (↑(↑D.glued : TopCat) : Type v))
        (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (X := (↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat)) (Y :=
              (↑(Limits.colimit D.diagram.multispan) : TopCat))
              (Hom.base (colimit.ι D.diagram.multispan (unop (op (WalkingMultispan.right i)))))) :
          (a : (↑(↑(D.diagram.multispan.obj (unop (op (WalkingMultispan.right i)))) : TopCat) : Type v)) →
            (↑(↑(Limits.colimit D.diagram.multispan) : TopCat) : Type v))
        ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (Y := (↑D.glued : TopCat)) (Hom.base (D.ι i))) :
            (a : (↑(↑(D.U i) : TopCat) : Type v)) → (↑(↑D.glued : TopCat) : Type v)) ''
          (↑U : Set (↑(↑(D.U i) : TopCat) : Type v))))
      x
Tactic: ext1 x