awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
H : X ≅ Y
U : Opens (↑(↑Y : TopCat) : Type u_2)
eq₁ :
  Eq (α :=
    X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
      (obj (Presheaf.pushforward C (H.hom ≫ H.inv).base) X.presheaf).obj (op ((Opens.map H.hom.base).obj U)))
    (H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫
        H.hom.c.app (op ((Opens.map H.inv.base).obj (unop (op ((Opens.map H.hom.base).obj U))))) :
      X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
        (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj
          (op ((Opens.map H.inv.base).obj (unop (op ((Opens.map H.hom.base).obj U))))))
    (eqToHom ⋯ :
      X.presheaf.obj ((Opens.map (Hom.base (𝟙 X))).op.obj (op ((Opens.map H.hom.base).obj U))) ⟶
        X.presheaf.obj ((Opens.map (H.hom ≫ H.inv).base).op.obj (op ((Opens.map H.hom.base).obj U))))
eq₂ :
  Eq (α :=
    Y.presheaf.obj (op (obj (Opens.map (map (forget C) (H.inv ≫ H.hom))) U)) ⟶
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op (obj (Opens.map (map (forget C) (𝟙 Y))) U)))
    (Y.presheaf.map (Quiver.Hom.op (eqToHom ⋯)) ≫ H.hom.c.app (op (obj (Opens.map (map (forget C) (𝟙 Y))) U)))
    (H.hom.c.app (op (obj (Opens.map (map (forget C) (H.inv ≫ H.hom))) U)) ≫
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).map (Quiver.Hom.op (eqToHom ⋯)))
⊢ H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫ eqToHom ⋯ ≫ H.hom.c.app (op U) =
    𝟙 ((obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U))
have eq₃ : Eq (α :=
  Y.presheaf.obj (op (obj (Opens.map (map (forget C) (H.inv ≫ H.hom))) U)) ⟶
    (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op (obj (Opens.map (map (forget C) (𝟙 Y))) U)))
  (eqToHom sorry ≫ H.hom.c.app (op (obj (Opens.map (map (forget C) (𝟙 Y))) U)))
  (H.hom.c.app (op (obj (Opens.map (map (forget C) (H.inv ≫ H.hom))) U)) ≫
    (obj (Presheaf.pushforward C H.hom.base) X.presheaf).map (eqToHom sorry))
Tactic: rw [eqToHom_op, eqToHom_map] at eq₂
Cleared hypotheses: eq₂
-- 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
H : X ≅ Y
U : Opens (↑(↑Y : TopCat) : Type u_2)
eq₁ :
  Eq (α :=
    X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
      (obj (Presheaf.pushforward C (H.hom ≫ H.inv).base) X.presheaf).obj (op ((Opens.map H.hom.base).obj U)))
    (H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫
        H.hom.c.app (op ((Opens.map H.inv.base).obj (unop (op ((Opens.map H.hom.base).obj U))))) :
      X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶
        (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj
          (op ((Opens.map H.inv.base).obj (unop (op ((Opens.map H.hom.base).obj U))))))
    (eqToHom ⋯ :
      X.presheaf.obj ((Opens.map (Hom.base (𝟙 X))).op.obj (op ((Opens.map H.hom.base).obj U))) ⟶
        X.presheaf.obj ((Opens.map (H.hom ≫ H.inv).base).op.obj (op ((Opens.map H.hom.base).obj U))))
eq₂ :
  Eq (α :=
    Y.presheaf.obj (op (obj (Opens.map (map (forget C) (H.inv ≫ H.hom))) U)) ⟶
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op (obj (Opens.map (map (forget C) (𝟙 Y))) U)))
    (eqToHom ⋯ ≫ H.hom.c.app (op (obj (Opens.map (map (forget C) (𝟙 Y))) U)))
    (H.hom.c.app (op (obj (Opens.map (map (forget C) (H.inv ≫ H.hom))) U)) ≫
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).map (eqToHom ⋯))
⊢ H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫ eqToHom ⋯ ≫ H.hom.c.app (op U) =
    𝟙 ((obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U))
have w : Eq.{v_1 + 1} (α :=
  X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶ X.presheaf.obj (op ((Opens.map H.hom.base).obj U)))
  (CategoryStruct.comp (Y :=
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj
        (op ((Opens.map H.inv.base).obj ((Opens.map H.hom.base).obj U))))
      (eqToHom sorry) ((obj (Presheaf.pushforward C H.hom.base) X.presheaf).map (eqToHom sorry)) :
    X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶ X.presheaf.obj (op ((Opens.map H.hom.base).obj U)))
  (𝟙 ((obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U)) :
    (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U) ⟶
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U))
Tactic: erw [eq₂, reassoc_of% eq₁]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
U_1 : TopCat
X : PresheafedSpace.{v_1, u_1, u_2} C
f_1 : U_1 ⟶ (↑X : TopCat)
h :
  IsOpenEmbedding (X := (↑U_1 : Type u_2)) (Y := (↑(↑X : TopCat) : Type u_2))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC :=
          TopCat.carrier) f_1) :
      (a : (↑U_1 : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
U V : (Opens (↑(↑X : TopCat) : Type u_2))ᵒᵖ
f : U ⟶ V
⊢ X.presheaf.map f ≫ X.presheaf.map (Quiver.Hom.op (⋯.adjunction.counit.app (unop V))) =
    X.presheaf.map (Quiver.Hom.op (⋯.adjunction.counit.app (unop U))) ≫
      X.presheaf.map (h.functor.op.map ((Opens.map f_1).op.map f))
have rw : X.presheaf.map (f ≫ Quiver.Hom.op (sorry.adjunction.counit.app (unop V))) =
  X.presheaf.map (Quiver.Hom.op (sorry.adjunction.counit.app (unop U)) ≫ h.functor.op.map ((Opens.map f_1).op.map f))
Tactic: rw [← map_comp, ← map_comp]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
U : TopCat
X : PresheafedSpace.{v_1, u_1, u_2} C
f : U ⟶ (↑X : TopCat)
hf :
  IsOpenEmbedding (X := (↑U : Type u_2)) (Y := (↑(↑X : TopCat) : Type u_2))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC :=
          TopCat.carrier) f) :
      (a : (↑U : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
⊢ Mono (X.ofRestrict hf)
have this : Mono f
Tactic: haveI : Mono f := (TopCat.mono_iff_injective _).mpr hf.injective
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
U : TopCat
X : PresheafedSpace.{v_1, u_1, u_2} C
f : U ⟶ (↑X : TopCat)
hf :
  IsOpenEmbedding (X := (↑U : Type u_2)) (Y := (↑(↑X : TopCat) : Type u_2))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC :=
          TopCat.carrier) f) :
      (a : (↑U : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
this : Mono f
⊢ Mono (X.ofRestrict hf)
have right_cancellation : ∀ {Z : PresheafedSpace.{v_1, u_1, u_2} C} (g h : Z ⟶ X.restrict hf), g ≫ X.ofRestrict hf = h ≫ X.ofRestrict hf → g = h
Tactic: constructor
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
U : TopCat
X : PresheafedSpace.{v_1, u_1, u_2} C
f : U ⟶ (↑X : TopCat)
hf :
  IsOpenEmbedding (X := (↑U : Type u_2)) (Y := (↑(↑X : TopCat) : Type u_2))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC :=
          TopCat.carrier) f) :
      (a : (↑U : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
this : Mono f
Z : PresheafedSpace.{v_1, u_1, u_2} C
g₁ g₂ : Z ⟶ X.restrict hf
eq : g₁ ≫ X.ofRestrict hf = g₂ ≫ X.ofRestrict hf
⊢ g₁ = g₂
have right_cancellation : g₁.base = g₂.base
Tactic: ext1
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
U : TopCat
X : PresheafedSpace.{v_1, u_1, u_2} C
f : U ⟶ (↑X : TopCat)
hf :
  IsOpenEmbedding (X := (↑U : Type u_2)) (Y := (↑(↑X : TopCat) : Type u_2))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC :=
          TopCat.carrier) f) :
      (a : (↑U : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
this : Mono f
Z : PresheafedSpace.{v_1, u_1, u_2} C
g₁ g₂ : Z ⟶ X.restrict hf
eq : g₁ ≫ X.ofRestrict hf = g₂ ≫ X.ofRestrict hf
right_cancellation : g₁.base = g₂.base
⊢ g₁ = g₂
have right_cancellation₁ : g₁.c ≫ whiskerRight (eqToHom sorry) Z.presheaf = g₂.c
Tactic: ext1
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
U : TopCat
X : PresheafedSpace.{v_1, u_1, u_2} C
f : U ⟶ (↑X : TopCat)
hf :
  IsOpenEmbedding (X := (↑U : Type u_2)) (Y := (↑(↑X : TopCat) : Type u_2))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC :=
          TopCat.carrier) f) :
      (a : (↑U : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
this : Mono f
Z : PresheafedSpace.{v_1, u_1, u_2} C
g₁ g₂ : Z ⟶ X.restrict hf
eq : g₁ ≫ X.ofRestrict hf = g₂ ≫ X.ofRestrict hf
⊢ g₁.base = g₂.base
have this₁ : (g₁ ≫ X.ofRestrict hf).base = (g₂ ≫ X.ofRestrict hf).base
Tactic: have := congr_arg PresheafedSpace.Hom.base eq
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
U : TopCat
X : PresheafedSpace.{v_1, u_1, u_2} C
f : U ⟶ (↑X : TopCat)
hf :
  IsOpenEmbedding (X := (↑U : Type u_2)) (Y := (↑(↑X : TopCat) : Type u_2))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC :=
          TopCat.carrier) f) :
      (a : (↑U : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
this : Mono f
Z : PresheafedSpace.{v_1, u_1, u_2} C
g₁ g₂ : Z ⟶ X.restrict hf
eq : g₁ ≫ X.ofRestrict hf = g₂ ≫ X.ofRestrict hf
⊢ g₁.c ≫ whiskerRight (eqToHom ⋯) Z.presheaf = g₂.c
have right_cancellation : ∀ (V : Opens (↑(↑(X.restrict hf) : TopCat) : Type u_2)),
  Eq (α := (X.restrict hf).presheaf.obj (op V) ⟶ ((Opens.map g₂.base).op ⋙ Z.presheaf).obj (op V))
    ((g₁.c ≫ whiskerRight (eqToHom sorry) Z.presheaf).app (op V)) (g₂.c.app (op V))
Tactic: ext V
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
U : TopCat
X : PresheafedSpace.{v_1, u_1, u_2} C
f : U ⟶ (↑X : TopCat)
hf :
  IsOpenEmbedding (X := (↑U : Type u_2)) (Y := (↑(↑X : TopCat) : Type u_2))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC :=
          TopCat.carrier) f) :
      (a : (↑U : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
this : Mono f
Z : PresheafedSpace.{v_1, u_1, u_2} C
g₁ g₂ : Z ⟶ X.restrict hf
eq : g₁ ≫ X.ofRestrict hf = g₂ ≫ X.ofRestrict hf
V : Opens (↑(↑(X.restrict hf) : TopCat) : Type u_2)
⊢ (Opens.map (X.ofRestrict hf).base).obj (hf.functor.obj V) = V
have h : (↑((Opens.map (X.ofRestrict hf).base).obj (hf.functor.obj V)) : Set (↑(↑(X.restrict hf) : TopCat) : Type u_2)) =
  (↑V : Set (↑(↑(X.restrict hf) : TopCat) : Type u_2))
Tactic: ext1
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
U : TopCat
X : PresheafedSpace.{v_1, u_1, u_2} C
f : U ⟶ (↑X : TopCat)
hf :
  IsOpenEmbedding (X := (↑U : Type u_2)) (Y := (↑(↑X : TopCat) : Type u_2))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC :=
          TopCat.carrier) f) :
      (a : (↑U : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
this : Mono f
Z : PresheafedSpace.{v_1, u_1, u_2} C
g₁ g₂ : Z ⟶ X.restrict hf
eq : g₁ ≫ X.ofRestrict hf = g₂ ≫ X.ofRestrict hf
V : Opens (↑(↑(X.restrict hf) : TopCat) : Type u_2)
⊢ Eq (α := (X.restrict hf).presheaf.obj (op V) ⟶ ((Opens.map g₂.base).op ⋙ Z.presheaf).obj (op V))
    ((g₁.c ≫ whiskerRight (eqToHom ⋯) Z.presheaf).app (op V)) (g₂.c.app (op V))
have hV : (Opens.map (X.ofRestrict hf).base).obj (hf.functor.obj V) = V
Tactic: have hV : (Opens.map (X.ofRestrict hf).base).obj (hf.functor.obj V) = V := by
      ext1
      exact Set.preimage_image_eq _ hf.injective
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
U : TopCat
X : PresheafedSpace.{v_1, u_1, u_2} C
f : U ⟶ (↑X : TopCat)
hf :
  IsOpenEmbedding (X := (↑U : Type u_2)) (Y := (↑(↑X : TopCat) : Type u_2))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC :=
          TopCat.carrier) f) :
      (a : (↑U : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
this : Mono f
Z : PresheafedSpace.{v_1, u_1, u_2} C
g₁ g₂ : Z ⟶ X.restrict hf
eq : g₁ ≫ X.ofRestrict hf = g₂ ≫ X.ofRestrict hf
V : Opens (↑(↑(X.restrict hf) : TopCat) : Type u_2)
hV : (Opens.map (X.ofRestrict hf).base).obj (hf.functor.obj V) = V
⊢ Eq (α := (X.restrict hf).presheaf.obj (op V) ⟶ ((Opens.map g₂.base).op ⋙ Z.presheaf).obj (op V))
    ((g₁.c ≫ whiskerRight (eqToHom ⋯) Z.presheaf).app (op V)) (g₂.c.app (op V))
have this₁ : IsIso (X := (Opens.map f ⋙ sorry.functor).obj (unop (op (hf.functor.obj V))))
  (sorry.adjunction.counit.app (unop (op (hf.functor.obj V))))
Tactic: haveI :
      IsIso (hf.isOpenMap.adjunction.counit.app (unop (op (hf.functor.obj V)))) :=
        NatIso.isIso_app_of_isIso
          (whiskerLeft hf.functor hf.isOpenMap.adjunction.counit) V
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
U : TopCat
X : PresheafedSpace.{v_1, u_1, u_2} C
f : U ⟶ (↑X : TopCat)
hf :
  IsOpenEmbedding (X := (↑U : Type u_2)) (Y := (↑(↑X : TopCat) : Type u_2))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC :=
          TopCat.carrier) f) :
      (a : (↑U : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
this_1 : Mono f
Z : PresheafedSpace.{v_1, u_1, u_2} C
g₁ g₂ : Z ⟶ X.restrict hf
eq : g₁ ≫ X.ofRestrict hf = g₂ ≫ X.ofRestrict hf
V : Opens (↑(↑(X.restrict hf) : TopCat) : Type u_2)
hV : (Opens.map (X.ofRestrict hf).base).obj (hf.functor.obj V) = V
this :
  IsIso (X := (Opens.map f ⋙ ⋯.functor).obj (unop (op (hf.functor.obj V))))
    (⋯.adjunction.counit.app (unop (op (hf.functor.obj V))))
⊢ Eq (α := (X.restrict hf).presheaf.obj (op V) ⟶ ((Opens.map g₂.base).op ⋙ Z.presheaf).obj (op V))
    ((g₁.c ≫ whiskerRight (eqToHom ⋯) Z.presheaf).app (op V)) (g₂.c.app (op V))
have this₁ : Eq (α :=
  X.presheaf.obj (op (hf.functor.obj V)) ⟶
    (obj (Presheaf.pushforward C (g₁ ≫ X.ofRestrict hf).base) Z.presheaf).obj (op (hf.functor.obj V)))
  ((g₁ ≫ X.ofRestrict hf).c.app (op (hf.functor.obj V)))
  ((g₂ ≫ X.ofRestrict hf).c.app (op (hf.functor.obj V)) ≫ Z.presheaf.map (eqToHom sorry))
Tactic: have := PresheafedSpace.congr_app eq (op (hf.functor.obj V))
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
U : TopCat
X : PresheafedSpace.{v_1, u_1, u_2} C
f : U ⟶ (↑X : TopCat)
hf :
  IsOpenEmbedding (X := (↑U : Type u_2)) (Y := (↑(↑X : TopCat) : Type u_2))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC :=
          TopCat.carrier) f) :
      (a : (↑U : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
this_1 : Mono f
Z : PresheafedSpace.{v_1, u_1, u_2} C
g₁ g₂ : Z ⟶ X.restrict hf
eq : g₁ ≫ X.ofRestrict hf = g₂ ≫ X.ofRestrict hf
V : Opens (↑(↑(X.restrict hf) : TopCat) : Type u_2)
hV : (Opens.map (X.ofRestrict hf).base).obj (hf.functor.obj V) = V
this_2 :
  IsIso (X := (Opens.map f ⋙ ⋯.functor).obj (unop (op (hf.functor.obj V))))
    (⋯.adjunction.counit.app (unop (op (hf.functor.obj V))))
this :
  Eq (α :=
    X.presheaf.obj (op (hf.functor.obj V)) ⟶
      (obj (Presheaf.pushforward C (g₁ ≫ X.ofRestrict hf).base) Z.presheaf).obj (op (hf.functor.obj V)))
    ((g₁ ≫ X.ofRestrict hf).c.app (op (hf.functor.obj V)))
    ((g₂ ≫ X.ofRestrict hf).c.app (op (hf.functor.obj V)) ≫ Z.presheaf.map (eqToHom ⋯))
⊢ Eq (α := (X.restrict hf).presheaf.obj (op V) ⟶ ((Opens.map g₂.base).op ⋙ Z.presheaf).obj (op V))
    ((g₁.c ≫ whiskerRight (eqToHom ⋯) Z.presheaf).app (op V)) (g₂.c.app (op V))
have this₁ : Eq (α :=
  (obj (Presheaf.pushforward C (X.ofRestrict hf).base) (X.restrict hf).presheaf).obj (op (hf.functor.obj V)) ⟶
    (obj (Presheaf.pushforward C g₁.base) Z.presheaf).obj
      (op ((Opens.map (X.ofRestrict hf).base).obj (unop (op (hf.functor.obj V))))))
  (g₁.c.app (op ((Opens.map (X.ofRestrict hf).base).obj (unop (op (hf.functor.obj V))))))
  (g₂.c.app (op ((Opens.map (X.ofRestrict hf).base).obj (unop (op (hf.functor.obj V))))) ≫ Z.presheaf.map (eqToHom sorry))
Tactic: rw [PresheafedSpace.comp_c_app, PresheafedSpace.comp_c_app,
      PresheafedSpace.ofRestrict_c_app, Category.assoc, cancel_epi] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
U : TopCat
X : PresheafedSpace.{v_1, u_1, u_2} C
f : U ⟶ (↑X : TopCat)
hf :
  IsOpenEmbedding (X := (↑U : Type u_2)) (Y := (↑(↑X : TopCat) : Type u_2))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC :=
          TopCat.carrier) f) :
      (a : (↑U : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
this_1 : Mono f
Z : PresheafedSpace.{v_1, u_1, u_2} C
g₁ g₂ : Z ⟶ X.restrict hf
eq : g₁ ≫ X.ofRestrict hf = g₂ ≫ X.ofRestrict hf
V : Opens (↑(↑(X.restrict hf) : TopCat) : Type u_2)
hV : (Opens.map (X.ofRestrict hf).base).obj (hf.functor.obj V) = V
this_2 :
  IsIso (X := (Opens.map f ⋙ ⋯.functor).obj (unop (op (hf.functor.obj V))))
    (⋯.adjunction.counit.app (unop (op (hf.functor.obj V))))
this :
  Eq (α :=
    (obj (Presheaf.pushforward C (X.ofRestrict hf).base) (X.restrict hf).presheaf).obj (op (hf.functor.obj V)) ⟶
      (obj (Presheaf.pushforward C g₁.base) Z.presheaf).obj
        (op ((Opens.map (X.ofRestrict hf).base).obj (unop (op (hf.functor.obj V))))))
    (g₁.c.app (op ((Opens.map (X.ofRestrict hf).base).obj (unop (op (hf.functor.obj V))))))
    (g₂.c.app (op ((Opens.map (X.ofRestrict hf).base).obj (unop (op (hf.functor.obj V))))) ≫ Z.presheaf.map (eqToHom ⋯))
⊢ Eq (α := (X.restrict hf).presheaf.obj (op V) ⟶ ((Opens.map g₂.base).op ⋙ Z.presheaf).obj (op V))
    ((g₁.c ≫ whiskerRight (eqToHom ⋯) Z.presheaf).app (op V)) (g₂.c.app (op V))
have h : (X.restrict hf).presheaf.map (Quiver.Hom.op (eqToHom hV)) ≫
    g₁.c.app (op ((Opens.map (X.ofRestrict hf).base).obj (unop (op (hf.functor.obj V))))) =
  (X.restrict hf).presheaf.map (Quiver.Hom.op (eqToHom hV)) ≫
    g₂.c.app (op ((Opens.map (X.ofRestrict hf).base).obj (unop (op (hf.functor.obj V))))) ≫ Z.presheaf.map (eqToHom sorry)
Tactic: have h : _ ≫ _ = _ ≫ _ ≫ _ :=
      congr_arg (fun f => (X.restrict hf).presheaf.map (eqToHom hV).op ≫ f) this
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
U : TopCat
X : PresheafedSpace.{v_1, u_1, u_2} C
f : U ⟶ (↑X : TopCat)
hf :
  IsOpenEmbedding (X := (↑U : Type u_2)) (Y := (↑(↑X : TopCat) : Type u_2))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC :=
          TopCat.carrier) f) :
      (a : (↑U : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
this_1 : Mono f
Z : PresheafedSpace.{v_1, u_1, u_2} C
g₁ g₂ : Z ⟶ X.restrict hf
eq : g₁ ≫ X.ofRestrict hf = g₂ ≫ X.ofRestrict hf
V : Opens (↑(↑(X.restrict hf) : TopCat) : Type u_2)
hV : (Opens.map (X.ofRestrict hf).base).obj (hf.functor.obj V) = V
this_2 :
  IsIso (X := (Opens.map f ⋙ ⋯.functor).obj (unop (op (hf.functor.obj V))))
    (⋯.adjunction.counit.app (unop (op (hf.functor.obj V))))
this :
  Eq (α :=
    (obj (Presheaf.pushforward C (X.ofRestrict hf).base) (X.restrict hf).presheaf).obj (op (hf.functor.obj V)) ⟶
      (obj (Presheaf.pushforward C g₁.base) Z.presheaf).obj
        (op ((Opens.map (X.ofRestrict hf).base).obj (unop (op (hf.functor.obj V))))))
    (g₁.c.app (op ((Opens.map (X.ofRestrict hf).base).obj (unop (op (hf.functor.obj V))))))
    (g₂.c.app (op ((Opens.map (X.ofRestrict hf).base).obj (unop (op (hf.functor.obj V))))) ≫ Z.presheaf.map (eqToHom ⋯))
h :
  (X.restrict hf).presheaf.map (Quiver.Hom.op (eqToHom hV)) ≫
      g₁.c.app (op ((Opens.map (X.ofRestrict hf).base).obj (unop (op (hf.functor.obj V))))) =
    (X.restrict hf).presheaf.map (Quiver.Hom.op (eqToHom hV)) ≫
      g₂.c.app (op ((Opens.map (X.ofRestrict hf).base).obj (unop (op (hf.functor.obj V))))) ≫ Z.presheaf.map (eqToHom ⋯)
⊢ Eq (α := (X.restrict hf).presheaf.obj (op V) ⟶ ((Opens.map g₂.base).op ⋙ Z.presheaf).obj (op V))
    ((g₁.c ≫ whiskerRight (eqToHom ⋯) Z.presheaf).app (op V)) (g₂.c.app (op V))
have h₁ : g₁.c.app (op V) ≫ (obj (Presheaf.pushforward C g₁.base) Z.presheaf).map (Quiver.Hom.op (eqToHom hV)) =
  g₂.c.app (op V) ≫
    (obj (Presheaf.pushforward C g₂.base) Z.presheaf).map (Quiver.Hom.op (eqToHom hV)) ≫ Z.presheaf.map (eqToHom sorry)
Tactic: simp only [g₁.c.naturality, g₂.c.naturality_assoc] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (hypothesis)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
U : TopCat
X : PresheafedSpace.{v_1, u_1, u_2} C
f : U ⟶ (↑X : TopCat)
hf :
  IsOpenEmbedding (X := (↑U : Type u_2)) (Y := (↑(↑X : TopCat) : Type u_2))
    (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_2}) ↦ C((↑X : Type u_2), (↑Y : Type u_2))) (CC :=
          TopCat.carrier) f) :
      (a : (↑U : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
this_1 : Mono f
Z : PresheafedSpace.{v_1, u_1, u_2} C
g₁ g₂ : Z ⟶ X.restrict hf
eq : g₁ ≫ X.ofRestrict hf = g₂ ≫ X.ofRestrict hf
V : Opens (↑(↑(X.restrict hf) : TopCat) : Type u_2)
hV : (Opens.map (X.ofRestrict hf).base).obj (hf.functor.obj V) = V
this_2 :
  IsIso (X := (Opens.map f ⋙ ⋯.functor).obj (unop (op (hf.functor.obj V))))
    (⋯.adjunction.counit.app (unop (op (hf.functor.obj V))))
this :
  Eq (α :=
    (obj (Presheaf.pushforward C (X.ofRestrict hf).base) (X.restrict hf).presheaf).obj (op (hf.functor.obj V)) ⟶
      (obj (Presheaf.pushforward C g₁.base) Z.presheaf).obj
        (op ((Opens.map (X.ofRestrict hf).base).obj (unop (op (hf.functor.obj V))))))
    (g₁.c.app (op ((Opens.map (X.ofRestrict hf).base).obj (unop (op (hf.functor.obj V))))))
    (g₂.c.app (op ((Opens.map (X.ofRestrict hf).base).obj (unop (op (hf.functor.obj V))))) ≫ Z.presheaf.map (eqToHom ⋯))
h :
  g₁.c.app (op V) ≫ (obj (Presheaf.pushforward C g₁.base) Z.presheaf).map (Quiver.Hom.op (eqToHom hV)) =
    g₂.c.app (op V) ≫
      (obj (Presheaf.pushforward C g₂.base) Z.presheaf).map (Quiver.Hom.op (eqToHom hV)) ≫ Z.presheaf.map (eqToHom ⋯)
⊢ Eq (α := (X.restrict hf).presheaf.obj (op V) ⟶ ((Opens.map g₂.base).op ⋙ Z.presheaf).obj (op V))
    ((g₁.c ≫ whiskerRight (eqToHom ⋯) Z.presheaf).app (op V)) (g₂.c.app (op V))
have h₁ : Eq (α := (X.restrict hf).presheaf.obj (op V) ⟶ (obj (Presheaf.pushforward C g₂.base) Z.presheaf).obj (op V))
  (g₁.c.app (op V) ≫ eqToHom sorry) (g₂.c.app (op V))
Tactic: simp only [eqToHom_op, eqToHom_map, eqToHom_trans,
      ← IsIso.comp_inv_eq, inv_eqToHom, Category.assoc] 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 : PresheafedSpace.{v_1, u_1, u_2} C
⊢ Functor.op (IsOpenEmbedding.functor ⋯) ⋙ X.presheaf =
    obj (Presheaf.pushforward C (Iso.inv (Opens.inclusionTopIso (↑X : TopCat)))) X.presheaf
have rw : Functor.op (Opens.map (Iso.inv (Opens.inclusionTopIso (↑X : TopCat)))) ⋙ X.presheaf =
  obj (Presheaf.pushforward C (Iso.inv (Opens.inclusionTopIso (↑X : TopCat)))) X.presheaf
Tactic: rw [Opens.inclusion'_top_functor X.carrier]
-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace (goal)
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
X : PresheafedSpace.{v_1, u_1, u_2} C
⊢ X.presheaf =
    obj (Presheaf.pushforward C (Hom.base (X.ofRestrict ⋯)))
      (X.restrict (U := obj (C := Opens (↑(↑X : TopCat) : Type u_2)) (Opens.toTopCat (↑X : TopCat)) ⊤) ⋯).presheaf
have rw : X.presheaf =
  obj (Presheaf.pushforward C (Iso.inv (Opens.inclusionTopIso (↑X : TopCat)) ≫ Hom.base (X.ofRestrict sorry))) X.presheaf
Tactic: rw [restrict_top_presheaf, ← Presheaf.Pushforward.comp_eq]