awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
s : Cocone F
m : (colimitCocone F).pt ⟶ s.pt
w : ∀ (j : J), (colimitCocone F).ι.app j ≫ m = s.ι.app j
j : J
x : (↑((F ⋙ forget C).obj j) : Type v)
⊢ Eq (α := (↑(↑s.pt : TopCat) : Type v))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (colimit.ι (F ⋙ forget C) j ≫ m.base) :
          (a : (↑((F ⋙ forget C).obj j) : Type v)) → (↑(↑s.pt : TopCat) : Type v))
        x :
      (↑(↑s.pt : TopCat) : Type v))
    (((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
              (Y := (↑s.pt : TopCat))
              (CategoryStruct.comp (Y := Limits.colimit (F ⋙ forget C)) (colimit.ι (F ⋙ forget C) j)
                  (colimit.desc (F ⋙ forget C) (mapCocone (forget C) s)) :
                (F ⋙ forget C).obj j ⟶ (↑s.pt : TopCat)) :
            C((↑((F ⋙ forget C).obj j) : Type v), (↑(↑s.pt : TopCat) : Type v))) :
          (a : (↑((F ⋙ forget C).obj j) : Type v)) → (↑(↑s.pt : TopCat) : Type v))
        x :
      (↑(↑s.pt : TopCat) : Type v))
have w₁ : Eq (α := (↑(↑s.pt : TopCat) : Type v))
  ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
          (colimit.ι (F ⋙ forget C) j ≫ m.base) :
        (a : (↑((F ⋙ forget C).obj j) : Type v)) → (↑(↑s.pt : TopCat) : Type v))
      x :
    (↑(↑s.pt : TopCat) : Type v))
  (((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier) (X :=
            (F ⋙ forget C).obj j) (Y := (↑s.pt : TopCat))
            (map (forget C) ((colimitCocone F).ι.app j ≫ m) : obj (forget C) (F.obj j) ⟶ obj (forget C) s.pt) :
          C((↑((F ⋙ forget C).obj j) : Type v), (↑(↑s.pt : TopCat) : Type v))) :
        (a : (↑((F ⋙ forget C).obj j) : Type v)) → (↑(↑s.pt : TopCat) : Type v))
      x :
    (↑(↑s.pt : TopCat) : Type v))
Tactic: rw [colimit.ι_desc, mapCocone_ι_app, ← w j]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (hypothesis)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
s : Cocone F
m : (colimitCocone F).pt ⟶ s.pt
w : ∀ (j : J), (colimitCocone F).ι.app j ≫ m = s.ι.app j
⊢ m = desc F s
have t : m.base = colimit.desc (F ⋙ forget C) (mapCocone (forget C) s)
Tactic: have t :
      m.base =
        colimit.desc (F ⋙ PresheafedSpace.forget C) ((PresheafedSpace.forget C).mapCocone s) := by
      dsimp
      -- `colimit.hom_ext` used to be automatically applied by `ext` before https://github.com/leanprover-community/mathlib4/pull/21302
      apply colimit.hom_ext fun j => ?_
      ext
      rw [colimit.ι_desc, mapCocone_ι_app, ← w j]
      simp
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
s : Cocone F
m : (colimitCocone F).pt ⟶ s.pt
w : ∀ (j : J), (colimitCocone F).ι.app j ≫ m = s.ι.app j
t : m.base = colimit.desc (F ⋙ forget C) (mapCocone (forget C) s)
⊢ m = desc F s
have w₁ : m.base = Hom.base (desc F s)
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
s : Cocone F
m : (colimitCocone F).pt ⟶ s.pt
w : ∀ (j : J), (colimitCocone F).ι.app j ≫ m = s.ι.app j
t : m.base = colimit.desc (F ⋙ forget C) (mapCocone (forget C) s)
w₁ : m.base = Hom.base (desc F s)
⊢ m = desc F s
have h : Eq (α := s.pt.presheaf ⟶ Functor.op (Opens.map (Hom.base (desc F s))) ⋙ (colimitCocone F).pt.presheaf)
  (m.c ≫ whiskerRight (eqToHom sorry) (colimitCocone F).pt.presheaf) (Hom.c (desc F s))
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
⊢ IsColimit (F := F ⋙ forget C) (mapCocone (forget C) (colimitCocone F))
have apply : colimit.cocone (F ⋙ forget C) ≅ mapCocone (forget C) (colimitCocone F)
Tactic: apply IsColimit.ofIsoColimit (colimit.isColimit _)
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
⊢ colimit.cocone (F ⋙ forget C) ≅ mapCocone (forget C) (colimitCocone F)
have φ : (colimit.cocone (F ⋙ forget C)).pt ≅ (colimit.cocone (F ⋙ forget C)).pt
Tactic: fapply Cocone.ext
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
φ : (colimit.cocone (F ⋙ forget C)).pt ≅ (colimit.cocone (F ⋙ forget C)).pt
⊢ colimit.cocone (F ⋙ forget C) ≅ mapCocone (forget C) (colimitCocone F)
have w : autoParam
  (∀ (j : J),
    (colimit.cocone (F ⋙ forget C)).ι.app j ≫ Iso.hom (Iso.refl (colimit.cocone (F ⋙ forget C)).pt) =
      NatTrans.app (Cocone.ι (mapCocone (forget C) (colimitCocone F))) j)
  Cocone.ext._auto_1
Tactic: fapply Cocone.ext
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
inst_5 : HasColimit F
U : Opens (↑(↑(Limits.colimit F) : TopCat) : Type v)
⊢ (Limits.colimit F).presheaf.obj (op U) ≅ limit (componentwiseDiagram F U)
have colimitCoconeIsColimit : obj
    (obj
      (pushforward C
        (Hom.base
          (Iso.hom
            (Iso.symm
              (colimit.isoColimitCocone
                { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F })))))
      { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F }.cocone.pt.presheaf)
    (op U) ≅
  limit (componentwiseDiagram F U)
Tactic: refine
    ((sheafIsoOfIso (colimit.isoColimitCocone ⟨_, colimitCoconeIsColimit F⟩).symm).app
          (op U)).trans
      ?_
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
inst_5 : HasColimit F
U : Opens (↑(↑(Limits.colimit F) : TopCat) : Type v)
⊢ obj
      (obj
        (pushforward C
          (Hom.base
            (Iso.hom
              (Iso.symm
                (colimit.isoColimitCocone
                  { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F })))))
        { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F }.cocone.pt.presheaf)
      (op U) ≅
    limit (componentwiseDiagram F U)
have refine : Functor.comp (D := CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C)
    (Functor.leftOp (pushforwardDiagramToColimit F) : Jᵒᵖ ⥤ Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C)))
    (obj (evaluation (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C)
        (obj
          (Functor.op
            (Opens.map
              (Hom.base
                (Iso.hom
                  (Iso.symm
                    (colimit.isoColimitCocone
                      { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F }))))))
          (op U)) :
      CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C) ≅
  componentwiseDiagram F U
Tactic: refine (limitObjIsoLimitCompEvaluation _ _).trans (Limits.lim.mapIso ?_)
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
inst_5 : HasColimit F
U : Opens (↑(↑(Limits.colimit F) : TopCat) : Type v)
⊢ Functor.comp (D := CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C)
      (Functor.leftOp (pushforwardDiagramToColimit F) : Jᵒᵖ ⥤ Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C)))
      (obj (evaluation (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C)
          (obj
            (Functor.op
              (Opens.map
                (Hom.base
                  (Iso.hom
                    (Iso.symm
                      (colimit.isoColimitCocone
                        { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F }))))))
            (op U)) :
        CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C) ≅
    componentwiseDiagram F U
have app : (X : Jᵒᵖ) →
  obj
      (Functor.comp (D := CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C)
        (Functor.leftOp (pushforwardDiagramToColimit F) : Jᵒᵖ ⥤ Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C)))
        (obj (evaluation (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C)
            (obj
              (Functor.op
                (Opens.map
                  (Hom.base
                    (Iso.hom
                      (Iso.symm
                        (colimit.isoColimitCocone
                          { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F }))))))
              (op U)) :
          CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C))
      X ≅
    (componentwiseDiagram F U).obj X
Tactic: fapply NatIso.ofComponents
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
inst_5 : HasColimit F
U : Opens (↑(↑(Limits.colimit F) : TopCat) : Type v)
X : Jᵒᵖ
⊢ obj
      (Functor.comp (D := CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C)
        (Functor.leftOp (pushforwardDiagramToColimit F) : Jᵒᵖ ⥤ Presheaf.{v, v, u} C (Limits.colimit (F ⋙ forget C)))
        (obj (evaluation (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C)
            (obj
              (Functor.op
                (Opens.map
                  (Hom.base
                    (Iso.hom
                      (Iso.symm
                        (colimit.isoColimitCocone
                          { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F }))))))
              (op U)) :
          CategoryTheory.Functor.{v, v, v, u} (Opens (↑(Limits.colimit (F ⋙ forget C)) : Type v))ᵒᵖ C ⥤ C))
      X ≅
    (componentwiseDiagram F U).obj X
have app : Eq (α := (Opens (↑(↑(F.obj (unop X)) : TopCat) : Type v))ᵒᵖ)
  (obj (Functor.op (Opens.map (colimit.ι (F ⋙ forget C) (unop X))))
    (obj
      (Functor.op
        (Opens.map
          (Hom.base
            (Iso.hom
              (Iso.symm
                (colimit.isoColimitCocone
                  { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F }))))))
      (op U)))
  (op (obj (Opens.map (Hom.base (colimit.ι F (unop X)))) U))
Tactic: refine (F.obj (unop X)).presheaf.mapIso (eqToIso ?_)
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
inst_5 : HasColimit F
U : Opens (↑(↑(Limits.colimit F) : TopCat) : Type v)
X : Jᵒᵖ
⊢ Eq (α := (Opens (↑(↑(F.obj (unop X)) : TopCat) : Type v))ᵒᵖ)
    (obj (Functor.op (Opens.map (colimit.ι (F ⋙ forget C) (unop X))))
      (obj
        (Functor.op
          (Opens.map
            (Hom.base
              (Iso.hom
                (Iso.symm
                  (colimit.isoColimitCocone
                    { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F : ColimitCocone F }))))))
        (op U)))
    (op (obj (Opens.map (Hom.base (colimit.ι F (unop X)))) U))
have app : Eq (α := Set (↑((F ⋙ forget C).obj (unop X)) : Type v))
  ((fun (x : (↑((F ⋙ forget C).obj (unop X)) : Type v)) ↦
      (ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
            (Y := (↑(Limits.colimit F) : TopCat))
            (Hom.base
              (Iso.hom
                (Iso.symm
                  (colimit.isoColimitCocone { cocone := colimitCocone F, isColimit := colimitCoconeIsColimit F })))) :
          (a : (↑(↑(colimitCocone F).pt : TopCat) : Type v)) → (↑(↑(Limits.colimit F) : TopCat) : Type v))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier)
                (X := (F ⋙ forget C).obj (unop X)) (Y := Limits.colimit (F ⋙ forget C))
                (colimit.ι (F ⋙ forget C) (unop X)) :
              (a : (↑((F ⋙ forget C).obj (unop X)) : Type v)) → (↑(Limits.colimit (F ⋙ forget C)) : Type v))
            x :
          (↑(Limits.colimit (F ⋙ forget C)) : Type v))) ⁻¹'
    (↑U : Set (↑(↑(Limits.colimit F) : TopCat) : Type v)))
  ((⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{v}) ↦ C((↑X : Type v), (↑Y : Type v))) (CC := TopCat.carrier) (X :=
          (↑(F.obj (unop X)) : TopCat)) (Y := (↑(Limits.colimit F) : TopCat)) (Hom.base (colimit.ι F (unop X)))) :
      (a : (↑(↑(F.obj (unop X)) : TopCat) : Type v)) → (↑(↑(Limits.colimit F) : TopCat) : Type v)) ⁻¹'
    (↑U : Set (↑(↑(Limits.colimit F) : TopCat) : Type v)))
Tactic: simp only [Functor.op_obj, op_inj_iff, Opens.map_coe, SetLike.ext'_iff,
      Set.preimage_preimage]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
U : Opens (↑(↑(Limits.colimit F) : TopCat) : Type v)
j : J
⊢ CategoryStruct.comp (Y := limit (componentwiseDiagram F U))
      (Iso.hom (colimitPresheafObjIsoComponentwiseLimit F U) :
        (Limits.colimit F).presheaf.obj (op U) ⟶ limit (componentwiseDiagram F U))
      (limit.π (componentwiseDiagram F U) (op j) :
        limit (componentwiseDiagram F U) ⟶ (componentwiseDiagram F U).obj (op j)) =
    NatTrans.app (Hom.c (colimit.ι F j)) (op U)
have rw : Eq (α := limit (componentwiseDiagram F U) ⟶ (componentwiseDiagram F U).obj (op j))
  (limit.π (componentwiseDiagram F U) (op j) : limit (componentwiseDiagram F U) ⟶ (componentwiseDiagram F U).obj (op j))
  (Iso.inv (colimitPresheafObjIsoComponentwiseLimit F U) ≫ NatTrans.app (Hom.c (colimit.ι F j)) (op U) :
    limit (componentwiseDiagram F U) ⟶ (componentwiseDiagram F U).obj (op j))
Tactic: rw [← Iso.eq_inv_comp, colimitPresheafObjIsoComponentwiseLimit_inv_ι_app]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
inst_3 : ∀ (X : TopCat.{v}), HasLimitsOfShape Jᵒᵖ (Presheaf C X)
inst_4 : HasLimitsOfShape.{v', u', v, u} Jᵒᵖ C
F : J ⥤ PresheafedSpace.{v, u, v} C
U : Opens (↑(↑(Limits.colimit F) : TopCat) : Type v)
j : J
rw :
  Eq (α := limit (componentwiseDiagram F U) ⟶ (componentwiseDiagram F U).obj (op j))
    (limit.π (componentwiseDiagram F U) (op j) :
      limit (componentwiseDiagram F U) ⟶ (componentwiseDiagram F U).obj (op j))
    (Iso.inv (colimitPresheafObjIsoComponentwiseLimit F U) ≫ NatTrans.app (Hom.c (colimit.ι F j)) (op U) :
      limit (componentwiseDiagram F U) ⟶ (componentwiseDiagram F U).obj (op j))
⊢ CategoryStruct.comp (Y := limit (componentwiseDiagram F U))
      (Iso.hom (colimitPresheafObjIsoComponentwiseLimit F U) :
        (Limits.colimit F).presheaf.obj (op U) ⟶ limit (componentwiseDiagram F U))
      (limit.π (componentwiseDiagram F U) (op j) :
        limit (componentwiseDiagram F U) ⟶ (componentwiseDiagram F U).obj (op j)) =
    NatTrans.app (Hom.c (colimit.ι F j)) (op U)
have rw₁ : Eq (α := limit (componentwiseDiagram F U) ⟶ (componentwiseDiagram F U).obj (op j))
  (limit.π (componentwiseDiagram F U) (op j)) (limit.π (componentwiseDiagram F U) (op j))
Tactic: rw [← Iso.eq_inv_comp, colimitPresheafObjIsoComponentwiseLimit_inv_ι_app]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
α β : X.Hom Y
w : α.base = β.base
⊢ (Opens.map α.base).op = (Opens.map β.base).op
have rw : (Opens.map β.base).op = (Opens.map β.base).op
Tactic: rw [w]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
α β : X.Hom Y
w : α.base = β.base
h : α.c ≫ whiskerRight (eqToHom ⋯) X.presheaf = β.c
⊢ α = β
have rcases : ∀ (base : (↑X : TopCat) ⟶ (↑Y : TopCat)) (c : Y.presheaf ⟶ obj (Presheaf.pushforward C base) X.presheaf)
  (w : { base := base, c := c : X.Hom Y }.base = β.base)
  (h : { base := base, c := c : X.Hom Y }.c ≫ whiskerRight (eqToHom sorry) X.presheaf = β.c),
  { base := base, c := c : X.Hom Y } = β
Tactic: rcases α with ⟨base, c⟩
Cleared hypotheses: α, w, h
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
β : X.Hom Y
base : (↑X : TopCat) ⟶ (↑Y : TopCat)
c : Y.presheaf ⟶ obj (Presheaf.pushforward C base) X.presheaf
w : { base := base, c := c : X.Hom Y }.base = β.base
h : { base := base, c := c : X.Hom Y }.c ≫ whiskerRight (eqToHom ⋯) X.presheaf = β.c
⊢ { base := base, c := c : X.Hom Y } = β
have rcases : ∀ (base' : (↑X : TopCat) ⟶ (↑Y : TopCat)) (c' : Y.presheaf ⟶ obj (Presheaf.pushforward C base') X.presheaf)
  (w : { base := base, c := c : X.Hom Y }.base = { base := base', c := c' }.base)
  (h : { base := base, c := c : X.Hom Y }.c ≫ whiskerRight (eqToHom sorry) X.presheaf = { base := base', c := c' }.c),
  { base := base, c := c : X.Hom Y } = { base := base', c := c' }
Tactic: rcases β with ⟨base', c'⟩
Cleared hypotheses: β, w, h
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
base : (↑X : TopCat) ⟶ (↑Y : TopCat)
c : Y.presheaf ⟶ obj (Presheaf.pushforward C base) X.presheaf
base' : (↑X : TopCat) ⟶ (↑Y : TopCat)
c' : Y.presheaf ⟶ obj (Presheaf.pushforward C base') X.presheaf
w : base = base'
h : { base := base, c := c : X.Hom Y }.c ≫ whiskerRight (eqToHom ⋯) X.presheaf = { base := base', c := c' : X.Hom Y }.c
⊢ { base := base, c := c : X.Hom Y } = { base := base', c := c' }
have subst : ∀ (c' : Y.presheaf ⟶ obj (Presheaf.pushforward C base) X.presheaf)
  (h :
    { base := base, c := c : X.Hom Y }.c ≫ whiskerRight (eqToHom sorry) X.presheaf = { base := base, c := c' : X.Hom Y }.c),
  { base := base, c := c : X.Hom Y } = { base := base, c := c' }
Tactic: subst w
Cleared hypotheses: base', c', w, h
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
base : (↑X : TopCat) ⟶ (↑Y : TopCat)
c c' : Y.presheaf ⟶ obj (Presheaf.pushforward C base) X.presheaf
h : c ≫ whiskerRight (𝟙 (Opens.map base).op) X.presheaf = c'
⊢ { base := base, c := c : X.Hom Y } = { base := base, c := c' }
have h₁ : c = c'
Tactic: erw [whiskerRight_id', comp_id] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
base : (↑X : TopCat) ⟶ (↑Y : TopCat)
c c' : Y.presheaf ⟶ obj (Presheaf.pushforward C base) X.presheaf
h : c = c'
⊢ { base := base, c := c : X.Hom Y } = { base := base, c := c' }
have subst : { base := base, c := c : X.Hom Y } = { base := base, c := c }
Tactic: subst h
Cleared hypotheses: c', h
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
α β : X.Hom Y
w : α.base = β.base
h : α.c ≍ β.c
⊢ α = β
have mk : ∀ (base : (↑X : TopCat) ⟶ (↑Y : TopCat)) (c : Y.presheaf ⟶ obj (Presheaf.pushforward C base) X.presheaf)
  (w : { base := base, c := c : X.Hom Y }.base = β.base) (h : { base := base, c := c : X.Hom Y }.c ≍ β.c),
  { base := base, c := c : X.Hom Y } = β
Tactic: cases α
Cleared hypotheses: α, w, h
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
β : X.Hom Y
base : (↑X : TopCat) ⟶ (↑Y : TopCat)
c : Y.presheaf ⟶ obj (Presheaf.pushforward C base) X.presheaf
w : { base := base, c := c : X.Hom Y }.base = β.base
h : { base := base, c := c : X.Hom Y }.c ≍ β.c
⊢ { base := base, c := c : X.Hom Y } = β
have mk : ∀ (base_1 : (↑X : TopCat) ⟶ (↑Y : TopCat)) (c_1 : Y.presheaf ⟶ obj (Presheaf.pushforward C base_1) X.presheaf)
  (w : { base := base, c := c : X.Hom Y }.base = { base := base_1, c := c_1 }.base)
  (h : { base := base, c := c : X.Hom Y }.c ≍ { base := base_1, c := c_1 : X.Hom Y }.c),
  { base := base, c := c : X.Hom Y } = { base := base_1, c := c_1 }
Tactic: cases β
Cleared hypotheses: β, w, h
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X Y : PresheafedSpace.{v_1, u_1, u_2} C
α β : X ⟶ Y
w : α.base = β.base
⊢ (Opens.map α.base).op = (Opens.map β.base).op
have rw : (Opens.map β.base).op = (Opens.map β.base).op
Tactic: rw [w]