awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.GammaSpecAdjunction (goal)
Goal:
X Y : LocallyRingedSpace
f : X ⟶ Y
w :
  PresheafedSpace.Hom.base (Hom.toHom (Functor.obj (𝟭 LocallyRingedSpace) X).toΓSpec) ≫
      (Spec.locallyRingedSpaceMap (Γ.rightOp.map f).unop).base =
    PresheafedSpace.Hom.base (Hom.toHom (CategoryTheory.Functor.map (𝟭 LocallyRingedSpace) f ≫ Y.toΓSpec))
⊢ X.toΓSpec ≫ (Γ.rightOp ⋙ Spec.toLocallyRingedSpace).map f =
    CategoryTheory.Functor.map (𝟭 LocallyRingedSpace) f ≫ Y.toΓSpec
have h : ∀ (r : (↑(unop (Γ.rightOp.obj Y)) : Type u)),
  Eq (α :=
    unop (Γ.rightOp.obj Y) ⟶
      (Functor.obj (𝟭 LocallyRingedSpace) X).presheaf.obj
        (op
          (Functor.obj (C :=
            Opens (↑(↑(Spec.locallyRingedSpaceObj (unop (Γ.rightOp.obj Y))).toPresheafedSpace : TopCat) : Type u))
            (Opens.map
              (PresheafedSpace.Hom.base (Hom.toHom (CategoryTheory.Functor.map (𝟭 LocallyRingedSpace) f ≫ Y.toΓSpec))))
            (basicOpen r))))
    ((Γ.rightOp.map f).unop ≫ (Functor.obj (𝟭 LocallyRingedSpace) X).presheaf.map (Quiver.Hom.op (homOfLE sorry)))
    (CommRingCat.ofHom
        (algebraMap (↑(unop (Γ.rightOp.obj Y)) : Type u)
          (Functor.obj (C := (Opens (↑(PrimeSpectrum.Top (↑(unop (Γ.rightOp.obj Y)) : Type u)) : Type u))ᵒᵖ)
            (ObjectProperty.FullSubcategory.obj (P :=
              Presheaf.IsSheaf
                (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑(unop (Γ.rightOp.obj Y)) : Type u)) : Type u)))
              (structureSheafInType (↑(unop (Γ.rightOp.obj Y)) : Type u) (↑(unop (Γ.rightOp.obj Y)) : Type u)))
            (op (basicOpen r)))) ≫
      NatTrans.app (PresheafedSpace.Hom.c (Hom.toHom (CategoryTheory.Functor.map (𝟭 LocallyRingedSpace) f ≫ Y.toΓSpec)))
        (op (basicOpen r)))
Tactic: apply LocallyRingedSpace.comp_ring_hom_ext
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.GammaSpecAdjunction (goal)
Goal:
X Y : LocallyRingedSpace
f : X ⟶ Y
⊢ PresheafedSpace.Hom.base (Hom.toHom (Functor.obj (𝟭 LocallyRingedSpace) X).toΓSpec) ≫
      (Spec.locallyRingedSpaceMap (Γ.rightOp.map f).unop).base =
    PresheafedSpace.Hom.base (Hom.toHom (CategoryTheory.Functor.map (𝟭 LocallyRingedSpace) f ≫ Y.toΓSpec))
have w : ∀ (x : (↑(↑(Functor.obj (𝟭 LocallyRingedSpace) X).toPresheafedSpace : TopCat) : Type u)),
  Eq (α := (↑(↑(Spec.locallyRingedSpaceObj (unop (Γ.rightOp.obj Y))).toPresheafedSpace : TopCat) : Type u))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
            (PresheafedSpace.Hom.base (Hom.toHom (Functor.obj (𝟭 LocallyRingedSpace) X).toΓSpec) ≫
              (Spec.locallyRingedSpaceMap (Γ.rightOp.map f).unop).base) :
          (a : (↑(↑((𝟭 LocallyRingedSpace).obj X).toPresheafedSpace : TopCat) : Type u)) →
            (↑(↑(Spec.locallyRingedSpaceObj (unop (Γ.rightOp.obj Y))).toPresheafedSpace : TopCat) : Type u))
        x :
      (↑(↑(Spec.locallyRingedSpaceObj (unop (Γ.rightOp.obj Y))).toPresheafedSpace : TopCat) : Type u))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) (X :=
            (↑(Functor.obj (𝟭 LocallyRingedSpace) X).toPresheafedSpace : TopCat)) (Y :=
            (↑(Spec.locallyRingedSpaceObj (unop (Γ.rightOp.obj Y))).toPresheafedSpace : TopCat))
            (PresheafedSpace.Hom.base (Hom.toHom (CategoryTheory.Functor.map (𝟭 LocallyRingedSpace) f ≫ Y.toΓSpec))) :
          (a : (↑(↑((𝟭 LocallyRingedSpace).obj X).toPresheafedSpace : TopCat) : Type u)) →
            (↑(↑(Spec.locallyRingedSpaceObj (unop (Γ.rightOp.obj Y))).toPresheafedSpace : TopCat) : Type u))
        x :
      (↑(↑(Spec.locallyRingedSpaceObj (unop (Γ.rightOp.obj Y))).toPresheafedSpace : TopCat) : Type u))
Tactic: ext1 x
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.GammaSpecAdjunction (goal)
Goal:
X Y : LocallyRingedSpace
f : X ⟶ Y
x : (↑(↑(Functor.obj (𝟭 LocallyRingedSpace) X).toPresheafedSpace : TopCat) : Type u)
⊢ Eq (α := (↑(↑(Spec.locallyRingedSpaceObj (unop (Γ.rightOp.obj Y))).toPresheafedSpace : TopCat) : Type u))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
            (PresheafedSpace.Hom.base (Hom.toHom (Functor.obj (𝟭 LocallyRingedSpace) X).toΓSpec) ≫
              (Spec.locallyRingedSpaceMap (Γ.rightOp.map f).unop).base) :
          (a : (↑(↑((𝟭 LocallyRingedSpace).obj X).toPresheafedSpace : TopCat) : Type u)) →
            (↑(↑(Spec.locallyRingedSpaceObj (unop (Γ.rightOp.obj Y))).toPresheafedSpace : TopCat) : Type u))
        x :
      (↑(↑(Spec.locallyRingedSpaceObj (unop (Γ.rightOp.obj Y))).toPresheafedSpace : TopCat) : Type u))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier) (X :=
            (↑(Functor.obj (𝟭 LocallyRingedSpace) X).toPresheafedSpace : TopCat)) (Y :=
            (↑(Spec.locallyRingedSpaceObj (unop (Γ.rightOp.obj Y))).toPresheafedSpace : TopCat))
            (PresheafedSpace.Hom.base (Hom.toHom (CategoryTheory.Functor.map (𝟭 LocallyRingedSpace) f ≫ Y.toΓSpec))) :
          (a : (↑(↑((𝟭 LocallyRingedSpace).obj X).toPresheafedSpace : TopCat) : Type u)) →
            (↑(↑(Spec.locallyRingedSpaceObj (unop (Γ.rightOp.obj Y))).toPresheafedSpace : TopCat) : Type u))
        x :
      (↑(↑(Spec.locallyRingedSpaceObj (unop (Γ.rightOp.obj Y))).toPresheafedSpace : TopCat) : Type u))
have w : Eq (α := PrimeSpectrum (↑(Y.presheaf.obj (op ⊤)) : Type u))
  (((TopCat.Hom.hom (Spec.topMap (f.c.app (op ⊤))) :
          C((↑(Spec.topObj (X.presheaf.obj (op ⊤))) : Type u), (↑(Spec.topObj (Y.presheaf.obj (op ⊤))) : Type u))) :
        (a : PrimeSpectrum (↑(X.presheaf.obj (op ⊤)) : Type u)) → PrimeSpectrum (↑(Y.presheaf.obj (op ⊤)) : Type u))
      ((TopCat.Hom.hom X.toΓSpecBase :
          (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → PrimeSpectrum (↑(X.presheaf.obj (op ⊤)) : Type u))
        x) :
    PrimeSpectrum (↑(Y.presheaf.obj (op ⊤)) : Type u))
  ((TopCat.Hom.hom Y.toΓSpecBase :
        (a : (↑(↑Y.toPresheafedSpace : TopCat) : Type u)) → PrimeSpectrum (↑(Y.presheaf.obj (op ⊤)) : Type u))
      ((TopCat.Hom.hom f.base :
          (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
        x) :
    PrimeSpectrum (↑(Y.presheaf.obj (op ⊤)) : Type u))
Tactic: dsimp
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.GammaSpecAdjunction (goal)
Goal:
X Y : LocallyRingedSpace
f : X ⟶ Y
x : (↑(↑(Functor.obj (𝟭 LocallyRingedSpace) X).toPresheafedSpace : TopCat) : Type u)
⊢ Eq (α := PrimeSpectrum (↑(Y.presheaf.obj (op ⊤)) : Type u))
    (((TopCat.Hom.hom (Spec.topMap (f.c.app (op ⊤))) :
            C((↑(Spec.topObj (X.presheaf.obj (op ⊤))) : Type u), (↑(Spec.topObj (Y.presheaf.obj (op ⊤))) : Type u))) :
          (a : PrimeSpectrum (↑(X.presheaf.obj (op ⊤)) : Type u)) → PrimeSpectrum (↑(Y.presheaf.obj (op ⊤)) : Type u))
        ((TopCat.Hom.hom X.toΓSpecBase :
            (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → PrimeSpectrum (↑(X.presheaf.obj (op ⊤)) : Type u))
          x) :
      PrimeSpectrum (↑(Y.presheaf.obj (op ⊤)) : Type u))
    ((TopCat.Hom.hom Y.toΓSpecBase :
          (a : (↑(↑Y.toPresheafedSpace : TopCat) : Type u)) → PrimeSpectrum (↑(Y.presheaf.obj (op ⊤)) : Type u))
        ((TopCat.Hom.hom f.base :
            (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
          x) :
      PrimeSpectrum (↑(Y.presheaf.obj (op ⊤)) : Type u))
have w : PrimeSpectrum.comap (CommRingCat.Hom.hom (f.c.app (op ⊤))) (X.toΓSpecFun x) =
  Y.toΓSpecFun
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
          f.base :
        (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
      x)
Tactic: change PrimeSpectrum.comap (f.c.app (op ⊤)).hom (X.toΓSpecFun x) = Y.toΓSpecFun (f.base x)
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.GammaSpecAdjunction (goal)
Goal:
X Y : LocallyRingedSpace
f : X ⟶ Y
x : (↑(↑(Functor.obj (𝟭 LocallyRingedSpace) X).toPresheafedSpace : TopCat) : Type u)
⊢ PrimeSpectrum.comap (CommRingCat.Hom.hom (f.c.app (op ⊤))) (X.toΓSpecFun x) =
    Y.toΓSpecFun
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
            f.base :
          (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
        x)
have w : PrimeSpectrum.comap (CommRingCat.Hom.hom (f.c.app (op ⊤)))
    (PrimeSpectrum.comap (S := (↑(X.presheaf.stalk x) : Type u)) (CommRingCat.Hom.hom (X.presheaf.Γgerm x))
      (IsLocalRing.closedPoint (↑(X.presheaf.stalk x) : Type u))) =
  PrimeSpectrum.comap (S :=
    (↑(Y.presheaf.stalk
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
                f.base :
              (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
            x)) :
      Type u))
    (CommRingCat.Hom.hom
      (Y.presheaf.Γgerm
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
              f.base :
            (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
          x)))
    (IsLocalRing.closedPoint
      (↑(Y.presheaf.stalk
            ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                  TopCat.carrier) f.base :
                (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
              x)) :
        Type u))
Tactic: dsimp [toΓSpecFun]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.GammaSpecAdjunction (goal)
Goal:
X Y : LocallyRingedSpace
f : X ⟶ Y
x : (↑(↑(Functor.obj (𝟭 LocallyRingedSpace) X).toPresheafedSpace : TopCat) : Type u)
⊢ PrimeSpectrum.comap (CommRingCat.Hom.hom (f.c.app (op ⊤)))
      (PrimeSpectrum.comap (S := (↑(X.presheaf.stalk x) : Type u)) (CommRingCat.Hom.hom (X.presheaf.Γgerm x))
        (IsLocalRing.closedPoint (↑(X.presheaf.stalk x) : Type u))) =
    PrimeSpectrum.comap (S :=
      (↑(Y.presheaf.stalk
            ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                  TopCat.carrier) f.base :
                (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
              x)) :
        Type u))
      (CommRingCat.Hom.hom
        (Y.presheaf.Γgerm
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
                f.base :
              (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
            x)))
      (IsLocalRing.closedPoint
        (↑(Y.presheaf.stalk
              ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                    TopCat.carrier) f.base :
                  (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
                x)) :
          Type u))
have w : PrimeSpectrum.comap (S := (↑(X.presheaf.stalk x) : Type u)) (CommRingCat.Hom.hom (f.c.app (op ⊤) ≫ X.presheaf.Γgerm x))
    (IsLocalRing.closedPoint (↑(X.presheaf.stalk x) : Type u)) =
  PrimeSpectrum.comap
    (CommRingCat.Hom.hom
      (Y.presheaf.Γgerm
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
                f.base :
              (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
            x) ≫
        Hom.stalkMap f x))
    (IsLocalRing.closedPoint (↑(X.presheaf.stalk x) : Type u))
Tactic: rw [← IsLocalRing.comap_closedPoint (f.stalkMap x).hom, ←
        PrimeSpectrum.comap_comp_apply, ← PrimeSpectrum.comap_comp_apply,
        ← CommRingCat.hom_comp, ← CommRingCat.hom_comp]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.GammaSpecAdjunction (goal)
Goal:
X Y : LocallyRingedSpace
f : X ⟶ Y
x : (↑(↑(Functor.obj (𝟭 LocallyRingedSpace) X).toPresheafedSpace : TopCat) : Type u)
⊢ PrimeSpectrum.comap (S := (↑(X.presheaf.stalk x) : Type u))
      (CommRingCat.Hom.hom (f.c.app (op ⊤) ≫ X.presheaf.Γgerm x))
      (IsLocalRing.closedPoint (↑(X.presheaf.stalk x) : Type u)) =
    PrimeSpectrum.comap
      (CommRingCat.Hom.hom
        (Y.presheaf.Γgerm
            ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC :=
                  TopCat.carrier) f.base :
                (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
              x) ≫
          Hom.stalkMap f x))
      (IsLocalRing.closedPoint (↑(X.presheaf.stalk x) : Type u))
have w : f.c.app (op ⊤) ≫ X.presheaf.Γgerm x =
  Y.presheaf.Γgerm
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
            f.base :
          (a : (↑(↑X.toPresheafedSpace : TopCat) : Type u)) → (↑(↑Y.toPresheafedSpace : TopCat) : Type u))
        x) ≫
    Hom.stalkMap f x
Tactic: congr 2
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.GammaSpecAdjunction (goal)
Goal:
X Y : LocallyRingedSpace
f : X ⟶ Y
r : (↑(unop (Γ.rightOp.obj Y)) : Type u)
⊢ Eq (α :=
    unop (Γ.rightOp.obj Y) ⟶
      (Functor.obj (𝟭 LocallyRingedSpace) X).presheaf.obj
        (op
          (Functor.obj (C :=
            Opens (↑(↑(Spec.locallyRingedSpaceObj (unop (Γ.rightOp.obj Y))).toPresheafedSpace : TopCat) : Type u))
            (Opens.map
              (PresheafedSpace.Hom.base (Hom.toHom (CategoryTheory.Functor.map (𝟭 LocallyRingedSpace) f ≫ Y.toΓSpec))))
            (basicOpen r))))
    ((Γ.rightOp.map f).unop ≫ (Functor.obj (𝟭 LocallyRingedSpace) X).presheaf.map (Quiver.Hom.op (homOfLE ⋯)))
    (CommRingCat.ofHom
        (algebraMap (↑(unop (Γ.rightOp.obj Y)) : Type u)
          (Functor.obj (C := (Opens (↑(PrimeSpectrum.Top (↑(unop (Γ.rightOp.obj Y)) : Type u)) : Type u))ᵒᵖ)
            (ObjectProperty.FullSubcategory.obj (P :=
              Presheaf.IsSheaf
                (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑(unop (Γ.rightOp.obj Y)) : Type u)) : Type u)))
              (structureSheafInType (↑(unop (Γ.rightOp.obj Y)) : Type u) (↑(unop (Γ.rightOp.obj Y)) : Type u)))
            (op (basicOpen r)))) ≫
      NatTrans.app (PresheafedSpace.Hom.c (Hom.toHom (CategoryTheory.Functor.map (𝟭 LocallyRingedSpace) f ≫ Y.toΓSpec)))
        (op (basicOpen r)))
have h : Eq (α :=
  unop (Γ.rightOp.obj Y) ⟶
    (Functor.obj (𝟭 LocallyRingedSpace) X).presheaf.obj
      (op
        (Functor.obj (C :=
          Opens (↑(↑(Spec.locallyRingedSpaceObj (unop (Γ.rightOp.obj Y))).toPresheafedSpace : TopCat) : Type u))
          (Opens.map
            (PresheafedSpace.Hom.base (Hom.toHom (CategoryTheory.Functor.map (𝟭 LocallyRingedSpace) f ≫ Y.toΓSpec))))
          (basicOpen r))))
  ((Γ.rightOp.map f).unop ≫ (Functor.obj (𝟭 LocallyRingedSpace) X).presheaf.map (Quiver.Hom.op (homOfLE sorry)))
  ((CommRingCat.ofHom
        (algebraMap (↑(unop (Γ.rightOp.obj Y)) : Type u)
          (Functor.obj (C := (Opens (↑(PrimeSpectrum.Top (↑(unop (Γ.rightOp.obj Y)) : Type u)) : Type u))ᵒᵖ)
            (ObjectProperty.FullSubcategory.obj (P :=
              Presheaf.IsSheaf
                (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑(unop (Γ.rightOp.obj Y)) : Type u)) : Type u)))
              (structureSheafInType (↑(unop (Γ.rightOp.obj Y)) : Type u) (↑(unop (Γ.rightOp.obj Y)) : Type u)))
            (op (basicOpen r)))) ≫
      Y.toΓSpec.c.app (op (basicOpen r))) ≫
    NatTrans.app (PresheafedSpace.Hom.c (Hom.toHom (CategoryTheory.Functor.map (𝟭 LocallyRingedSpace) f)))
      (op ((Opens.map Y.toΓSpec.base).obj (unop (op (basicOpen r))))))
Tactic: rw [LocallyRingedSpace.comp_c_app, ← Category.assoc]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.GammaSpecAdjunction (goal)
Goal:
X Y : LocallyRingedSpace
f : X ⟶ Y
r : (↑(unop (Γ.rightOp.obj Y)) : Type u)
⊢ Eq (α :=
    unop (Γ.rightOp.obj Y) ⟶
      (Functor.obj (𝟭 LocallyRingedSpace) X).presheaf.obj
        (op
          (Functor.obj (C :=
            Opens (↑(↑(Spec.locallyRingedSpaceObj (unop (Γ.rightOp.obj Y))).toPresheafedSpace : TopCat) : Type u))
            (Opens.map
              (PresheafedSpace.Hom.base (Hom.toHom (CategoryTheory.Functor.map (𝟭 LocallyRingedSpace) f ≫ Y.toΓSpec))))
            (basicOpen r))))
    ((Γ.rightOp.map f).unop ≫ (Functor.obj (𝟭 LocallyRingedSpace) X).presheaf.map (Quiver.Hom.op (homOfLE ⋯)))
    ((CommRingCat.ofHom
          (algebraMap (↑(unop (Γ.rightOp.obj Y)) : Type u)
            (Functor.obj (C := (Opens (↑(PrimeSpectrum.Top (↑(unop (Γ.rightOp.obj Y)) : Type u)) : Type u))ᵒᵖ)
              (ObjectProperty.FullSubcategory.obj (P :=
                Presheaf.IsSheaf
                  (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑(unop (Γ.rightOp.obj Y)) : Type u)) : Type u)))
                (structureSheafInType (↑(unop (Γ.rightOp.obj Y)) : Type u) (↑(unop (Γ.rightOp.obj Y)) : Type u)))
              (op (basicOpen r)))) ≫
        Y.toΓSpec.c.app (op (basicOpen r))) ≫
      NatTrans.app (PresheafedSpace.Hom.c (Hom.toHom (CategoryTheory.Functor.map (𝟭 LocallyRingedSpace) f)))
        (op ((Opens.map Y.toΓSpec.base).obj (unop (op (basicOpen r))))))
have h : (Γ.rightOp.map f).unop ≫ (Functor.obj (𝟭 LocallyRingedSpace) X).presheaf.map (Quiver.Hom.op (homOfLE sorry)) =
  f.c.app (op ⊤) ≫ (Functor.obj (pushforward CommRingCat f.base) X.presheaf).map (Y.toΓSpecMapBasicOpen r).leTop.op
Tactic: erw [Y.toΓSpecSheafedSpace_app_spec, f.c.naturality]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.GammaSpecAdjunction (goal)
Goal:
R : CommRingCat
⊢ identityToΓSpec.app (Spec.toLocallyRingedSpace.obj (op R)) ≫
      Spec.toLocallyRingedSpace.map (SpecΓIdentity.inv.app R).op =
    𝟙 (Functor.obj (𝟭 LocallyRingedSpace) (Spec.toLocallyRingedSpace.obj (op R)))
have w : PresheafedSpace.Hom.base
      (Hom.toHom (Functor.obj (𝟭 LocallyRingedSpace) (Spec.toLocallyRingedSpace.obj (op R))).toΓSpec) ≫
    (Spec.locallyRingedSpaceMap (SpecΓIdentity.inv.app R).op.unop).base =
  (Hom.toHom (𝟙 (Functor.obj (𝟭 LocallyRingedSpace) (Spec.toLocallyRingedSpace.obj (op R))))).base
Tactic: apply LocallyRingedSpace.comp_ring_hom_ext
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.GammaSpecAdjunction (goal)
Goal:
R : CommRingCat
w :
  PresheafedSpace.Hom.base
        (Hom.toHom (Functor.obj (𝟭 LocallyRingedSpace) (Spec.toLocallyRingedSpace.obj (op R))).toΓSpec) ≫
      (Spec.locallyRingedSpaceMap (SpecΓIdentity.inv.app R).op.unop).base =
    (Hom.toHom (𝟙 (Functor.obj (𝟭 LocallyRingedSpace) (Spec.toLocallyRingedSpace.obj (op R))))).base
⊢ identityToΓSpec.app (Spec.toLocallyRingedSpace.obj (op R)) ≫
      Spec.toLocallyRingedSpace.map (SpecΓIdentity.inv.app R).op =
    𝟙 (Functor.obj (𝟭 LocallyRingedSpace) (Spec.toLocallyRingedSpace.obj (op R)))
have h : ∀ (r : (↑(unop (op (Functor.obj (𝟭 CommRingCat) R))) : Type u_1)),
  Eq (α :=
    unop (op (Functor.obj (𝟭 CommRingCat) R)) ⟶
      (Functor.obj (𝟭 LocallyRingedSpace) (Spec.toLocallyRingedSpace.obj (op R))).presheaf.obj
        (op
          ((Opens.map
                (Hom.toHom (𝟙 (Functor.obj (𝟭 LocallyRingedSpace) (Spec.toLocallyRingedSpace.obj (op R))))).base).obj
            (basicOpen r))))
    ((SpecΓIdentity.inv.app R).op.unop ≫
      (Functor.obj (𝟭 LocallyRingedSpace) (Spec.toLocallyRingedSpace.obj (op R))).presheaf.map
        (Quiver.Hom.op (homOfLE sorry)))
    (CommRingCat.ofHom
        (algebraMap (↑(unop (op (Functor.obj (𝟭 CommRingCat) R))) : Type u_1)
          (Functor.obj (C :=
            (Opens (↑(PrimeSpectrum.Top (↑(unop (op (Functor.obj (𝟭 CommRingCat) R))) : Type u_1)) : Type u_1))ᵒᵖ)
            (ObjectProperty.FullSubcategory.obj (P :=
              Presheaf.IsSheaf
                (Opens.grothendieckTopology
                  (↑(PrimeSpectrum.Top (↑(unop (op (Functor.obj (𝟭 CommRingCat) R))) : Type u_1)) : Type u_1)))
              (structureSheafInType (↑(unop (op (Functor.obj (𝟭 CommRingCat) R))) : Type u_1)
                (↑(unop (op (Functor.obj (𝟭 CommRingCat) R))) : Type u_1)))
            (op (basicOpen r)))) ≫
      (Hom.toHom (𝟙 (Functor.obj (𝟭 LocallyRingedSpace) (Spec.toLocallyRingedSpace.obj (op R))))).c.app
        (op (basicOpen r)))
Tactic: apply LocallyRingedSpace.comp_ring_hom_ext
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.GammaSpecAdjunction (goal)
Goal:
R : CommRingCat
⊢ PresheafedSpace.Hom.base
        (Hom.toHom (Functor.obj (𝟭 LocallyRingedSpace) (Spec.toLocallyRingedSpace.obj (op R))).toΓSpec) ≫
      (Spec.locallyRingedSpaceMap (SpecΓIdentity.inv.app R).op.unop).base =
    (Hom.toHom (𝟙 (Functor.obj (𝟭 LocallyRingedSpace) (Spec.toLocallyRingedSpace.obj (op R))))).base
have w : ∀ (p : PrimeSpectrum (↑R : Type u_1)),
  Eq (α :=
    (↑(↑(Spec.locallyRingedSpaceObj (unop (op (Functor.obj (𝟭 CommRingCat) R)))).toPresheafedSpace : TopCat) :
      Type u_1))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := TopCat.carrier)
            (PresheafedSpace.Hom.base
                (Hom.toHom (Functor.obj (𝟭 LocallyRingedSpace) (Spec.toLocallyRingedSpace.obj (op R))).toΓSpec) ≫
              (Spec.locallyRingedSpaceMap (SpecΓIdentity.inv.app R).op.unop).base) :
          (a :
              (↑(↑((𝟭 LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R))).toPresheafedSpace : TopCat) :
                Type u_1)) →
            (↑(↑(Spec.locallyRingedSpaceObj (unop (op ((𝟭 CommRingCat).obj R)))).toPresheafedSpace : TopCat) :
              Type u_1))
        p :
      (↑(↑(Spec.locallyRingedSpaceObj (unop (op (Functor.obj (𝟭 CommRingCat) R)))).toPresheafedSpace : TopCat) :
        Type u_1))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := TopCat.carrier)
            (Hom.toHom (𝟙 (Functor.obj (𝟭 LocallyRingedSpace) (Spec.toLocallyRingedSpace.obj (op R))))).base :
          (a :
              (↑(↑((𝟭 LocallyRingedSpace).obj (Spec.toLocallyRingedSpace.obj (op R))).toPresheafedSpace : TopCat) :
                Type u_1)) →
            (↑(↑(Spec.locallyRingedSpaceObj (unop (op ((𝟭 CommRingCat).obj R)))).toPresheafedSpace : TopCat) :
              Type u_1))
        p :
      (↑(↑(Spec.locallyRingedSpaceObj (unop (op (Functor.obj (𝟭 CommRingCat) R)))).toPresheafedSpace : TopCat) :
        Type u_1))
Tactic: ext (p : PrimeSpectrum R)