awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- 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
U : (Opens (↑(↑X : TopCat) : Type u_2))ᵒᵖ
⊢ (Hom.c (𝟙 X)).app U = X.presheaf.map (𝟙 U)
have rw : NatTrans.app (𝟙 X.presheaf) U = 𝟙 (X.presheaf.obj U)
Tactic: rw [id_c, map_id]
-- 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
h : α = β
U : (Opens (↑(↑Y : TopCat) : Type u_2))ᵒᵖ
⊢ (Opens.map β.base).op.obj U = (Opens.map α.base).op.obj U
have subst : (Opens.map α.base).op.obj U = (Opens.map α.base).op.obj U
Tactic: subst h
Cleared hypotheses: β, 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
h : α = β
U : (Opens (↑(↑Y : TopCat) : Type u_2))ᵒᵖ
⊢ α.c.app U = β.c.app U ≫ X.presheaf.map (eqToHom ⋯)
have subst : α.c.app U = α.c.app U ≫ X.presheaf.map (eqToHom sorry)
Tactic: subst h
Cleared hypotheses: β, 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
H : (↑X : TopCat) ≅ (↑Y : TopCat)
α : obj (Presheaf.pushforward C H.hom) X.presheaf ≅ Y.presheaf
⊢ { base := H.hom, c := α.inv : X.Hom Y } ≫ { base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom } = 𝟙 X
have w : ∀ (x : (↑(↑X : TopCat) : Type u_2)),
  Eq (α := (↑(↑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)
            (X := (↑X : TopCat)) (Y := (↑X : TopCat))
            (Hom.base
              ({ base := H.hom, c := α.inv : X.Hom Y } ≫ { base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom })) :
          (a : (↑(↑X : TopCat) : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
        x :
      (↑(↑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)
            (Hom.base (𝟙 X)) :
          (a : (↑(↑X : TopCat) : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
        x :
      (↑(↑X : TopCat) : Type u_2))
Tactic: ext
-- 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 : TopCat) ≅ (↑Y : TopCat)
α : obj (Presheaf.pushforward C H.hom) X.presheaf ≅ Y.presheaf
w :
  ∀ (x : (↑(↑X : TopCat) : Type u_2)),
    Eq (α := (↑(↑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) (X := (↑X : TopCat)) (Y := (↑X : TopCat))
              (Hom.base
                ({ base := H.hom, c := α.inv : X.Hom Y } ≫
                  { base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom })) :
            (a : (↑(↑X : TopCat) : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
          x :
        (↑(↑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) (Hom.base (𝟙 X)) :
            (a : (↑(↑X : TopCat) : Type u_2)) → (↑(↑X : TopCat) : Type u_2))
          x :
        (↑(↑X : TopCat) : Type u_2))
⊢ { base := H.hom, c := α.inv : X.Hom Y } ≫ { base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom } = 𝟙 X
have h : ∀ (U : Opens (↑(↑X : TopCat) : Type u_2)),
  Eq (α := X.presheaf.obj (op U) ⟶ ((Opens.map (Hom.base (𝟙 X))).op ⋙ X.presheaf).obj (op U))
    ((CategoryStruct.comp (Y :=
          obj
            (Presheaf.pushforward C
              (Hom.base
                ({ base := H.hom, c := α.inv : X.Hom Y } ≫
                  { base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom : Y.Hom X })))
            X.presheaf)
          (Hom.c
            ({ base := H.hom, c := α.inv : X.Hom Y } ≫ { base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom }))
          (whiskerRight (eqToHom sorry) X.presheaf)).app
      (op U))
    ((Hom.c (𝟙 X)).app (op U))
Tactic: ext
-- 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 : TopCat) ≅ (↑Y : TopCat)
α : obj (Presheaf.pushforward C H.hom) X.presheaf ≅ Y.presheaf
⊢ { base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom : Y.Hom X } ≫ { base := H.hom, c := α.inv } = 𝟙 Y
have w : ∀ (x : (↑(↑Y : TopCat) : Type u_2)),
  Eq (α := (↑(↑Y : 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)
            (X := (↑Y : TopCat)) (Y := (↑Y : TopCat))
            (Hom.base
              ({ base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom : Y.Hom X } ≫ { base := H.hom, c := α.inv })) :
          (a : (↑(↑Y : TopCat) : Type u_2)) → (↑(↑Y : TopCat) : Type u_2))
        x :
      (↑(↑Y : 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)
            (Hom.base (𝟙 Y)) :
          (a : (↑(↑Y : TopCat) : Type u_2)) → (↑(↑Y : TopCat) : Type u_2))
        x :
      (↑(↑Y : TopCat) : Type u_2))
Tactic: ext
-- 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 : TopCat) ≅ (↑Y : TopCat)
α : obj (Presheaf.pushforward C H.hom) X.presheaf ≅ Y.presheaf
w :
  ∀ (x : (↑(↑Y : TopCat) : Type u_2)),
    Eq (α := (↑(↑Y : 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) (X := (↑Y : TopCat)) (Y := (↑Y : TopCat))
              (Hom.base
                ({ base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom : Y.Hom X } ≫
                  { base := H.hom, c := α.inv })) :
            (a : (↑(↑Y : TopCat) : Type u_2)) → (↑(↑Y : TopCat) : Type u_2))
          x :
        (↑(↑Y : 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) (Hom.base (𝟙 Y)) :
            (a : (↑(↑Y : TopCat) : Type u_2)) → (↑(↑Y : TopCat) : Type u_2))
          x :
        (↑(↑Y : TopCat) : Type u_2))
⊢ { base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom : Y.Hom X } ≫ { base := H.hom, c := α.inv } = 𝟙 Y
have h : ∀ (U : Opens (↑(↑Y : TopCat) : Type u_2)),
  Eq (α := Y.presheaf.obj (op U) ⟶ ((Opens.map (Hom.base (𝟙 Y))).op ⋙ Y.presheaf).obj (op U))
    ((CategoryStruct.comp (Y :=
          obj
            (Presheaf.pushforward C
              (Hom.base
                ({ base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom : Y.Hom X } ≫
                  { base := H.hom, c := α.inv : X.Hom Y })))
            Y.presheaf)
          (Hom.c
            ({ base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom : Y.Hom X } ≫ { base := H.hom, c := α.inv }))
          (whiskerRight (eqToHom sorry) Y.presheaf)).app
      (op U))
    ((Hom.c (𝟙 Y)).app (op U))
Tactic: ext
-- 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 : TopCat) ≅ (↑Y : TopCat)
α : obj (Presheaf.pushforward C H.hom) X.presheaf ≅ Y.presheaf
x : (↑(↑Y : TopCat) : Type u_2)
⊢ Eq (α := (↑(↑Y : 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)
            (X := (↑Y : TopCat)) (Y := (↑Y : TopCat))
            (Hom.base
              ({ base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom : Y.Hom X } ≫ { base := H.hom, c := α.inv })) :
          (a : (↑(↑Y : TopCat) : Type u_2)) → (↑(↑Y : TopCat) : Type u_2))
        x :
      (↑(↑Y : 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)
            (Hom.base (𝟙 Y)) :
          (a : (↑(↑Y : TopCat) : Type u_2)) → (↑(↑Y : TopCat) : Type u_2))
        x :
      (↑(↑Y : TopCat) : Type u_2))
have w : (Hom.hom H.hom : (a : (↑(↑X : TopCat) : Type u_2)) → (↑(↑Y : TopCat) : Type u_2))
    ((Hom.hom H.inv : (a : (↑(↑Y : TopCat) : Type u_2)) → (↑(↑X : TopCat) : Type u_2)) x :
      (↑(↑X : TopCat) : Type u_2)) =
  x
Tactic: dsimp
-- 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 : TopCat) ≅ (↑Y : TopCat)
α : obj (Presheaf.pushforward C H.hom) X.presheaf ≅ Y.presheaf
U : Opens (↑(↑Y : TopCat) : Type u_2)
⊢ Eq (α := Y.presheaf.obj (op U) ⟶ ((Opens.map (Hom.base (𝟙 Y))).op ⋙ Y.presheaf).obj (op U))
    ((CategoryStruct.comp (Y :=
          obj
            (Presheaf.pushforward C
              (Hom.base
                ({ base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom : Y.Hom X } ≫
                  { base := H.hom, c := α.inv : X.Hom Y })))
            Y.presheaf)
          (Hom.c
            ({ base := H.inv, c := Presheaf.toPushforwardOfIso H α.hom : Y.Hom X } ≫ { base := H.hom, c := α.inv }))
          (whiskerRight (eqToHom ⋯) Y.presheaf)).app
      (op U))
    ((Hom.c (𝟙 Y)).app (op U))
have h : Eq (α := Y.presheaf.obj (op U) ⟶ Y.presheaf.obj (op ((Opens.map (𝟙 (↑Y : TopCat))).obj U)))
  ((α.inv.app (op U) ≫ (Presheaf.toPushforwardOfIso H α.hom).app (op ((Opens.map H.hom).obj U))) ≫
    Y.presheaf.map (NatTrans.app (eqToHom sorry) (op U)))
  ((Hom.c (𝟙 Y)).app (op U))
Tactic: dsimp
-- 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 : TopCat) ≅ (↑Y : TopCat)
α : obj (Presheaf.pushforward C H.hom) X.presheaf ≅ Y.presheaf
U : Opens (↑(↑Y : TopCat) : Type u_2)
⊢ Eq (α := Y.presheaf.obj (op U) ⟶ Y.presheaf.obj (op ((Opens.map (𝟙 (↑Y : TopCat))).obj U)))
    ((α.inv.app (op U) ≫ (Presheaf.toPushforwardOfIso H α.hom).app (op ((Opens.map H.hom).obj U))) ≫
      Y.presheaf.map (NatTrans.app (eqToHom ⋯) (op U)))
    ((Hom.c (𝟙 Y)).app (op U))
have h : Eq (α := Y.presheaf.obj (op U) ⟶ Y.presheaf.obj (op ((Opens.map (𝟙 (↑Y : TopCat))).obj U)))
  (α.inv.app (op U) ≫
    X.presheaf.map (eqToHom sorry) ≫
      (obj (Presheaf.pushforward C H.hom) X.presheaf).map (NatTrans.app (eqToHom sorry) (op U)) ≫
        α.hom.app (op ((Opens.map (𝟙 (↑Y : TopCat))).obj U)))
  ((Hom.c (𝟙 Y)).app (op U))
Tactic: simp only [Presheaf.toPushforwardOfIso_app, assoc, ← α.hom.naturality]
-- 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 : TopCat) ≅ (↑Y : TopCat)
α : obj (Presheaf.pushforward C H.hom) X.presheaf ≅ Y.presheaf
U : Opens (↑(↑Y : TopCat) : Type u_2)
⊢ Eq (α := Y.presheaf.obj (op U) ⟶ Y.presheaf.obj (op ((Opens.map (𝟙 (↑Y : TopCat))).obj U)))
    (α.inv.app (op U) ≫
      X.presheaf.map (eqToHom ⋯) ≫
        (obj (Presheaf.pushforward C H.hom) X.presheaf).map (NatTrans.app (eqToHom ⋯) (op U)) ≫
          α.hom.app (op ((Opens.map (𝟙 (↑Y : TopCat))).obj U)))
    ((Hom.c (𝟙 Y)).app (op U))
have h : Eq (α := Y.presheaf.obj (op U) ⟶ Y.presheaf.obj (op ((Opens.map (𝟙 (↑Y : TopCat))).obj U)))
  (α.inv.app (op U) ≫ α.hom.app (op ((Opens.map (𝟙 (↑Y : TopCat))).obj U))) ((Hom.c (𝟙 Y)).app (op U))
Tactic: simp only [eqToHom_map, eqToHom_app, eqToHom_trans_assoc, eqToHom_refl, id_comp]
-- 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
⊢ H.hom.c ≫ Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c = 𝟙 Y.presheaf
have w : ∀ (U : Opens (↑(↑Y : TopCat) : Type u_2)),
  Eq (α := Y.presheaf.obj (op U) ⟶ Y.presheaf.obj (op U))
    (NatTrans.app (H.hom.c ≫ Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c) (op U))
    (NatTrans.app (𝟙 Y.presheaf) (op U))
Tactic: ext U
-- 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)
⊢ Eq (α := Y.presheaf.obj (op U) ⟶ Y.presheaf.obj (op U))
    (NatTrans.app (H.hom.c ≫ Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c) (op U))
    (NatTrans.app (𝟙 Y.presheaf) (op U))
have w : Eq (α := Y.presheaf.obj (op U) ⟶ Y.presheaf.obj (op U))
  (H.hom.c.app (op U) ≫ NatTrans.app (Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c) (op U))
  (NatTrans.app (𝟙 Y.presheaf) (op U))
Tactic: rw [NatTrans.comp_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
H : X ≅ Y
⊢ Eq (α := obj (Presheaf.pushforward C H.hom.base) X.presheaf ⟶ obj (Presheaf.pushforward C H.hom.base) X.presheaf)
    (Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c ≫ H.hom.c :
      obj (Presheaf.pushforward C H.hom.base) X.presheaf ⟶ obj (Presheaf.pushforward C H.hom.base) X.presheaf)
    (𝟙 (obj (Presheaf.pushforward C H.hom.base) X.presheaf) :
      obj (Presheaf.pushforward C H.hom.base) X.presheaf ⟶ obj (Presheaf.pushforward C H.hom.base) X.presheaf)
have w : ∀ (U : Opens (↑(↑Y : TopCat) : Type u_2)),
  Eq.{v_1 + 1} (α :=
    (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U) ⟶
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U))
    (NatTrans.app (Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c ≫ H.hom.c) (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))
    (NatTrans.app (𝟙 (obj (Presheaf.pushforward C H.hom.base) X.presheaf)) (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: ext U
-- 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)
⊢ Eq.{v_1 + 1} (α :=
    (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U) ⟶
      (obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U))
    (NatTrans.app (Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c ≫ H.hom.c) (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))
    (NatTrans.app (𝟙 (obj (Presheaf.pushforward C H.hom.base) X.presheaf)) (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))
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)))
  (NatTrans.app (Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c) (op U) ≫ H.hom.c.app (op U) :
    X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶ X.presheaf.obj (op ((Opens.map H.hom.base).obj U)))
  (NatTrans.app (𝟙 (obj (Presheaf.pushforward C H.hom.base) X.presheaf)) (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: dsimp
-- 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)
⊢ 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)))
    (NatTrans.app (Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c) (op U) ≫ H.hom.c.app (op U) :
      X.presheaf.obj (op ((Opens.map H.hom.base).obj U)) ⟶ X.presheaf.obj (op ((Opens.map H.hom.base).obj U)))
    (NatTrans.app (𝟙 (obj (Presheaf.pushforward C H.hom.base) X.presheaf)) (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))
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)))
  (NatTrans.app (Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c) (op U) ≫ H.hom.c.app (op U) :
    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: rw [NatTrans.id_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
H : X ≅ Y
U : Opens (↑(↑Y : TopCat) : Type u_2)
⊢ 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)))
    (NatTrans.app (Presheaf.pushforwardToOfIso (Iso.symm (mapIso (forget C) H)) H.inv.c) (op U) ≫ H.hom.c.app (op U) :
      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))
have w : H.inv.c.app (op ((Opens.map H.hom.base).obj U)) ≫ eqToHom sorry ≫ H.hom.c.app (op U) =
  𝟙 ((obj (Presheaf.pushforward C H.hom.base) X.presheaf).obj (op U))
Tactic: simp only [Presheaf.pushforwardToOfIso_app, Iso.symm_inv, mapIso_hom, forget_map,
      Iso.symm_hom, mapIso_inv, eqToHom_map, assoc]
-- 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)
⊢ 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 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.hom ≫ H.inv).c.app (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 ≫ H.inv).base) X.presheaf).obj (op ((Opens.map H.hom.base).obj U)))
  ((Hom.c (𝟙 X)).app (op ((Opens.map H.hom.base).obj U)) ≫ X.presheaf.map (eqToHom sorry) :
    X.presheaf.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))))
Tactic: have eq₁ := congr_app H.hom_inv_id (op ((Opens.map H.hom.base).obj U))
-- 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.hom ≫ H.inv).c.app (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 ≫ H.inv).base) X.presheaf).obj (op ((Opens.map H.hom.base).obj U)))
    ((Hom.c (𝟙 X)).app (op ((Opens.map H.hom.base).obj U)) ≫ X.presheaf.map (eqToHom ⋯) :
      X.presheaf.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))))
⊢ 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 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 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 (Quiver.Hom.op (eqToHom sorry)))
Tactic: have eq₂ := H.hom.c.naturality (eqToHom (congr_obj (congr_arg Opens.map
      ((forget C).congr_map H.inv_hom_id.symm)) U)).op
-- 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.hom ≫ H.inv).c.app (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 ≫ H.inv).base) X.presheaf).obj (op ((Opens.map H.hom.base).obj U)))
    ((Hom.c (𝟙 X)).app (op ((Opens.map H.hom.base).obj U)) ≫ X.presheaf.map (eqToHom ⋯) :
      X.presheaf.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 :=
  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 sorry :
    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))))
Tactic: rw [id_c, NatTrans.id_app, id_comp, eqToHom_map, comp_c_app] at eq₁
Cleared hypotheses: eq₁, eq₂