awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
R : CommRingCat
M : Submonoid (↑R : Type u)
x : PrimeSpectrum (Localization M)
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
have dsimp : Function.Bijective (α :=
  (↑(TopCat.Presheaf.stalk (X := Spec.topObj (CommRingCat.of (↑R : Type u)))
        (ObjectProperty.FullSubcategory.obj (P :=
          Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u)))
          (structureSheaf (↑R : Type u)))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := TopCat.carrier)
              (X := Spec.topObj (CommRingCat.of (Localization M))) (Y := Spec.topObj (CommRingCat.of (↑R : Type u)))
              (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (Localization M)))) :
            (a : PrimeSpectrum (Localization M)) → PrimeSpectrum (↑R : Type u))
          x)) :
    Type u))
  (β :=
  (↑(TopCat.Presheaf.stalk
        (ObjectProperty.FullSubcategory.obj (P :=
          Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (Localization M)) : Type u)))
          (structureSheaf (Localization M)))
        x) :
    Type u))
  (Function.comp (β :=
    Localization.AtPrime
      (Ideal.comap (F := (↑R : Type u) →+* Localization M) (algebraMap (↑R : Type u) (Localization M)) x.asIdeal))
    (Function.comp (β := Localization.AtPrime x.asIdeal)
      (⇑(↑(stalkIso (Localization M) x) :
            Localization.AtPrime x.asIdeal →+*
              (↑((structurePresheafInCommRingCat (Localization M)).stalk x) : Type u)) :
        (a : Localization.AtPrime x.asIdeal) →
          (↑(TopCat.Presheaf.stalk (structureSheaf (Localization M)).obj x) : Type u))
      (⇑(Localization.localRingHom
            (Ideal.comap (F := (↑R : Type u) →+* Localization M) (algebraMap (↑R : Type u) (Localization M)) x.asIdeal)
            x.asIdeal (algebraMap (↑R : Type u) (Localization M)) sorry) :
        (a : Localization.AtPrime (Ideal.comap (algebraMap (↑R : Type u) (Localization M)) x.asIdeal)) →
          Localization.AtPrime x.asIdeal))
    (⇑(AlgEquiv.symm (stalkIso (↑R : Type u) (PrimeSpectrum.comap (algebraMap (↑R : Type u) (Localization M)) x))) :
      (a :
          (↑(TopCat.Presheaf.stalk (structureSheaf (↑R : Type u)).obj
                ((ConcreteCategory.hom (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (Localization M)))) :
                    (a : PrimeSpectrum (Localization M)) → PrimeSpectrum (↑R : Type u))
                  x)) :
            Type u)) →
        Localization.AtPrime (Ideal.comap (algebraMap (↑R : Type u) (Localization M)) x.asIdeal)))
Tactic: dsimp
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
R : CommRingCat
M : Submonoid (↑R : Type u)
x : PrimeSpectrum (Localization M)
⊢ Function.Bijective (α :=
    Localization.AtPrime
      (Ideal.comap (F := (↑R : Type u) →+* Localization M) (algebraMap (↑R : Type u) (Localization M)) x.asIdeal))
    (β :=
    (↑(TopCat.Presheaf.stalk
          (ObjectProperty.FullSubcategory.obj (P :=
            Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (Localization M)) : Type u)))
            (structureSheaf (Localization M)))
          x) :
      Type u))
    (Function.comp (β := Localization.AtPrime x.asIdeal)
      (⇑(↑(stalkIso (Localization M) x) :
            Localization.AtPrime x.asIdeal →+*
              (↑((structurePresheafInCommRingCat (Localization M)).stalk x) : Type u)) :
        (a : Localization.AtPrime x.asIdeal) →
          (↑(TopCat.Presheaf.stalk (structureSheaf (Localization M)).obj x) : Type u))
      (⇑(Localization.localRingHom
            (Ideal.comap (F := (↑R : Type u) →+* Localization M) (algebraMap (↑R : Type u) (Localization M)) x.asIdeal)
            x.asIdeal (algebraMap (↑R : Type u) (Localization M)) ⋯) :
        (a : Localization.AtPrime (Ideal.comap (algebraMap (↑R : Type u) (Localization M)) x.asIdeal)) →
          Localization.AtPrime x.asIdeal))
have refine : Function.Bijective (α :=
  Localization.AtPrime
    (Ideal.comap (F := (↑R : Type u) →+* Localization M) (algebraMap (↑R : Type u) (Localization M)) x.asIdeal))
  (β := Localization.AtPrime x.asIdeal)
  (⇑(Localization.localRingHom
        (Ideal.comap (F := (↑R : Type u) →+* Localization M) (algebraMap (↑R : Type u) (Localization M)) x.asIdeal)
        x.asIdeal (algebraMap (↑R : Type u) (Localization M)) sorry) :
    (a : Localization.AtPrime (Ideal.comap (algebraMap (↑R : Type u) (Localization M)) x.asIdeal)) →
      Localization.AtPrime x.asIdeal)
Tactic: refine (stalkIso (Localization M) x).bijective.comp ?_
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
R : CommRingCat
M : Submonoid (↑R : Type u)
x : PrimeSpectrum (Localization M)
this :
  IsIso (X :=
    CommRingCat.of
      (Localization.AtPrime
        (Ideal.comap (F := (↑R : Type u) →+* Localization M) (algebraMap (↑R : Type u) (Localization M)) x.asIdeal)))
    (Y := CommRingCat.of (Localization.AtPrime x.asIdeal))
    (Iso.hom
      (RingEquiv.toCommRingCatIso
        (AlgEquiv.toRingEquiv (IsLocalization.localizationLocalizationAtPrimeIsoLocalization M x.asIdeal))))
⊢ Function.Bijective (α :=
    Localization.AtPrime
      (Ideal.comap (F := (↑R : Type u) →+* Localization M) (algebraMap (↑R : Type u) (Localization M)) x.asIdeal))
    (β := Localization.AtPrime x.asIdeal)
    (⇑(Localization.localRingHom
          (Ideal.comap (F := (↑R : Type u) →+* Localization M) (algebraMap (↑R : Type u) (Localization M)) x.asIdeal)
          x.asIdeal (algebraMap (↑R : Type u) (Localization M)) ⋯) :
      (a : Localization.AtPrime (Ideal.comap (algebraMap (↑R : Type u) (Localization M)) x.asIdeal)) →
        Localization.AtPrime x.asIdeal)
have rwa : ∀
  (this :
    Function.Bijective (α :=
      (↑(CommRingCat.of
            (Localization.AtPrime
              (Ideal.comap (F := (↑R : Type u) →+* Localization M) (algebraMap (↑R : Type u) (Localization M))
                x.asIdeal))) :
        Type u))
      (β := (↑(CommRingCat.of (Localization.AtPrime x.asIdeal)) : Type u))
      (⇑(ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (X :=
            CommRingCat.of
              (Localization.AtPrime
                (Ideal.comap (F := (↑R : Type u) →+* Localization M) (algebraMap (↑R : Type u) (Localization M))
                  x.asIdeal)))
            (Y := CommRingCat.of (Localization.AtPrime x.asIdeal))
            (Iso.hom
              (RingEquiv.toCommRingCatIso
                (AlgEquiv.toRingEquiv (IsLocalization.localizationLocalizationAtPrimeIsoLocalization M x.asIdeal))))) :
        (a :
            (↑(CommRingCat.of
                  (Localization.AtPrime (Ideal.comap (algebraMap (↑R : Type u) (Localization M)) x.asIdeal))) :
              Type u)) →
          (↑(CommRingCat.of (Localization.AtPrime x.asIdeal)) : Type u))),
  Function.Bijective (α :=
    Localization.AtPrime
      (Ideal.comap (F := (↑R : Type u) →+* Localization M) (algebraMap (↑R : Type u) (Localization M)) x.asIdeal))
    (β := Localization.AtPrime x.asIdeal)
    (⇑(Localization.localRingHom
          (Ideal.comap (F := (↑R : Type u) →+* Localization M) (algebraMap (↑R : Type u) (Localization M)) x.asIdeal)
          x.asIdeal (algebraMap (↑R : Type u) (Localization M)) sorry) :
      (a : Localization.AtPrime (Ideal.comap (algebraMap (↑R : Type u) (Localization M)) x.asIdeal)) →
        Localization.AtPrime x.asIdeal)
Tactic: rwa [ConcreteCategory.isIso_iff_bijective] at this
Cleared hypotheses: this
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
R : CommRingCat
M : Submonoid (↑R : Type u)
x : PrimeSpectrum (Localization M)
⊢ Function.Bijective (α :=
    Localization.AtPrime
      (Ideal.comap (F := (↑R : Type u) →+* Localization M) (algebraMap (↑R : Type u) (Localization M)) x.asIdeal))
    (β := Localization.AtPrime x.asIdeal)
    (⇑(Localization.localRingHom
          (Ideal.comap (F := (↑R : Type u) →+* Localization M) (algebraMap (↑R : Type u) (Localization M)) x.asIdeal)
          x.asIdeal (algebraMap (↑R : Type u) (Localization M)) ⋯) :
      (a : Localization.AtPrime (Ideal.comap (algebraMap (↑R : Type u) (Localization M)) x.asIdeal)) →
        Localization.AtPrime x.asIdeal)
have IsIso : IsIso (X :=
  CommRingCat.of
    (Localization.AtPrime
      (Ideal.comap (F := (↑R : Type u) →+* Localization M) (algebraMap (↑R : Type u) (Localization M)) x.asIdeal)))
  (Y := CommRingCat.of (Localization.AtPrime x.asIdeal))
  (Iso.hom
    (RingEquiv.toCommRingCatIso
      (AlgEquiv.toRingEquiv (IsLocalization.localizationLocalizationAtPrimeIsoLocalization M x.asIdeal))))
Tactic: suffices
    IsIso (IsLocalization.localizationLocalizationAtPrimeIsoLocalization M
        x.asIdeal).toRingEquiv.toCommRingCatIso.hom by
    rwa [ConcreteCategory.isIso_iff_bijective] at this
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
R S : CommRingCat
f : R ⟶ S
p : PrimeSpectrum (↑R : Type u)
⊢ f ≫ toPushforwardStalk f p =
    toStalk (↑R : Type u) p ≫
      CategoryTheory.Functor.map (TopCat.Presheaf.stalkFunctor CommRingCat p) (Spec.sheafedSpaceMap f).hom.c
have rw : f ≫ toPushforwardStalk f p =
  CategoryStruct.comp (Y :=
    CommRingCat.of
      (Functor.obj (C := (TopologicalSpace.Opens (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u))ᵒᵖ)
        (ObjectProperty.FullSubcategory.obj (P :=
          Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u)))
          (structureSheafInType (↑R : Type u) (↑R : Type u)))
        (op ⊤)))
    (CommRingCat.ofHom
      (algebraMap (↑R : Type u)
        (Functor.obj (C := (TopologicalSpace.Opens (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u))ᵒᵖ)
          (ObjectProperty.FullSubcategory.obj (P :=
            Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑R : Type u)) : Type u)))
            (structureSheafInType (↑R : Type u) (↑R : Type u)))
          (op ⊤))))
    ((Spec.sheafedSpaceMap f).hom.c.app (op ⊤) ≫
      (Functor.obj (TopCat.Presheaf.pushforward CommRingCat (Spec.sheafedSpaceMap f).hom.base)
            (Spec.sheafedSpaceObj S).presheaf).germ
        ⊤ p trivial)
Tactic: rw [StructureSheaf.toStalk, Category.assoc, TopCat.Presheaf.stalkFunctor_map_germ]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
R S : CommRingCat
f : R ⟶ S
p : PrimeSpectrum (↑R : Type u)
inst : Algebra (↑R : Type u) (↑S : Type u)
⊢ IsLocalizedModule (M' :=
    (↑(TopCat.Presheaf.stalk
          (Functor.obj (C := TopCat.Presheaf CommRingCat (Spec.topObj (CommRingCat.of (↑S : Type u))))
            (TopCat.Presheaf.pushforward CommRingCat
              (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u)))))
            (ObjectProperty.FullSubcategory.obj (P :=
              Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
              (structureSheaf (↑S : Type u))))
          p) :
      Type u))
    p.asIdeal.primeCompl (AlgHom.toLinearMap (toPushforwardStalkAlgHom R S p))
have h₁ : ∀ x ∈ p.asIdeal.primeCompl,
  IsUnit (M :=
    (↑(TopCat.Presheaf.stalk
          (Functor.obj (C := TopCat.Presheaf CommRingCat (Spec.topObj (CommRingCat.of (↑S : Type u))))
            (TopCat.Presheaf.pushforward CommRingCat
              (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u)))))
            (ObjectProperty.FullSubcategory.obj (P :=
              Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
              (structureSheaf (↑S : Type u))))
          p) :
      Type u))
    ((algebraMap (↑R : Type u)
            (↑(TopCat.Presheaf.stalk
                  (Functor.obj (C := TopCat.Presheaf CommRingCat (Spec.topObj (CommRingCat.of (↑S : Type u))))
                    (TopCat.Presheaf.pushforward CommRingCat
                      (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u)))))
                    (ObjectProperty.FullSubcategory.obj (P :=
                      Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
                      (structureSheaf (↑S : Type u))))
                  p) :
              Type u) :
          (a : (↑R : Type u)) →
            (↑(((TopCat.Presheaf.pushforward CommRingCat
                          (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u))))).obj
                      (structureSheaf (↑S : Type u)).obj).stalk
                  p) :
              Type u))
        x :
      (↑(TopCat.Presheaf.stalk
            (Functor.obj (C := TopCat.Presheaf CommRingCat (Spec.topObj (CommRingCat.of (↑S : Type u))))
              (TopCat.Presheaf.pushforward CommRingCat
                (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u)))))
              (ObjectProperty.FullSubcategory.obj (P :=
                Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
                (structureSheaf (↑S : Type u))))
            p) :
        Type u))
Tactic: apply IsLocalizedModule.mkOfAlgebra
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.Spec (goal)
Goal:
R S : CommRingCat
f : R ⟶ S
p : PrimeSpectrum (↑R : Type u)
inst : Algebra (↑R : Type u) (↑S : Type u)
x : (↑R : Type u)
hx : x ∈ p.asIdeal.primeCompl
⊢ IsUnit (M :=
    (↑(TopCat.Presheaf.stalk
          (Functor.obj (C := TopCat.Presheaf CommRingCat (Spec.topObj (CommRingCat.of (↑S : Type u))))
            (TopCat.Presheaf.pushforward CommRingCat
              (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u)))))
            (ObjectProperty.FullSubcategory.obj (P :=
              Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
              (structureSheaf (↑S : Type u))))
          p) :
      Type u))
    ((algebraMap (↑R : Type u)
            (↑(TopCat.Presheaf.stalk
                  (Functor.obj (C := TopCat.Presheaf CommRingCat (Spec.topObj (CommRingCat.of (↑S : Type u))))
                    (TopCat.Presheaf.pushforward CommRingCat
                      (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u)))))
                    (ObjectProperty.FullSubcategory.obj (P :=
                      Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
                      (structureSheaf (↑S : Type u))))
                  p) :
              Type u) :
          (a : (↑R : Type u)) →
            (↑(((TopCat.Presheaf.pushforward CommRingCat
                          (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u))))).obj
                      (structureSheaf (↑S : Type u)).obj).stalk
                  p) :
              Type u))
        x :
      (↑(TopCat.Presheaf.stalk
            (Functor.obj (C := TopCat.Presheaf CommRingCat (Spec.topObj (CommRingCat.of (↑S : Type u))))
              (TopCat.Presheaf.pushforward CommRingCat
                (Spec.topMap (CommRingCat.ofHom (algebraMap (↑R : Type u) (↑S : Type u)))))
              (ObjectProperty.FullSubcategory.obj (P :=
                Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑S : Type u)) : Type u)))
                (structureSheaf (↑S : Type u))))
            p) :
        Type u))
have h₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
Tactic: rw [algebraMap_pushforward_stalk, toPushforwardStalk_comp]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.SpreadingOut (goal)
Goal:
X : Scheme
U : X.Opens
hU : IsAffineOpen U
x : ↥X
hx : x ∈ U
f : (↑Γ(X, U) : Type u)
hf : x ∈ X.basicOpen f
H :
  Function.Injective (α := (↑(X.presheaf.obj (Opposite.op U)) : Type u)) (β := (↑(X.presheaf.stalk x) : Type u))
    (⇑(ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
          CommRingCat.carrier) (Y := X.presheaf.stalk x) (X.presheaf.germ U x hx)) :
      (a : (↑(X.presheaf.obj (Opposite.op U)) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
⊢ Function.Injective (α := (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u)) (β :=
    (↑(X.presheaf.stalk x) : Type u))
    (⇑(ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
          CommRingCat.carrier) (Y := X.presheaf.stalk x) (X.presheaf.germ (X.basicOpen f) x hf)) :
      (a : (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
have rw : ∀
  (H :
    ∀ (x_1 : (↑(X.presheaf.obj (Opposite.op U)) : Type u)),
      Eq (α := (↑(X.presheaf.stalk x) : Type u))
          ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                  CommRingCat.carrier) (Y := X.presheaf.stalk x) (X.presheaf.germ U x hx) :
                (a : (↑(X.presheaf.obj (Opposite.op U)) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
              x_1 :
            (↑(X.presheaf.stalk x) : Type u))
          (0 : (↑(X.presheaf.stalk x) : Type u)) →
        x_1 = 0)
  (x_1 : (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u)),
  Eq (α := (↑(X.presheaf.stalk x) : Type u))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) (Y := X.presheaf.stalk x) (X.presheaf.germ (X.basicOpen f) x hf) :
            (a : (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
          x_1 :
        (↑(X.presheaf.stalk x) : Type u))
      (0 : (↑(X.presheaf.stalk x) : Type u)) →
    x_1 = 0
Tactic: rw [RingHom.injective_iff_ker_eq_bot, RingHom.ker_eq_bot_iff_eq_zero] at H ⊢
Cleared hypotheses: H
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.SpreadingOut (hypothesis)
Goal:
X : Scheme
U : X.Opens
hU : IsAffineOpen U
x : ↥X
hx : x ∈ U
f : (↑Γ(X, U) : Type u)
hf : x ∈ X.basicOpen f
H :
  ∀ (x_1 : (↑(X.presheaf.obj (Opposite.op U)) : Type u)),
    Eq (α := (↑(X.presheaf.stalk x) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (Y := X.presheaf.stalk x) (X.presheaf.germ U x hx) :
              (a : (↑(X.presheaf.obj (Opposite.op U)) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
            x_1 :
          (↑(X.presheaf.stalk x) : Type u))
        (0 : (↑(X.presheaf.stalk x) : Type u)) →
      x_1 = 0
t : (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u)
ht :
  Eq (α := (↑(X.presheaf.stalk x) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Y := X.presheaf.stalk x) (X.presheaf.germ (X.basicOpen f) x hf) :
          (a : (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
        t :
      (↑(X.presheaf.stalk x) : Type u))
    (0 : (↑(X.presheaf.stalk x) : Type u))
⊢ t = 0
have this : IsLocalization.Away f (↑Γ(X, X.basicOpen f) : Type u)
Tactic: have := hU.isLocalization_basicOpen f
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.SpreadingOut (goal)
Goal:
X : Scheme
U : X.Opens
hU : IsAffineOpen U
x : ↥X
hx : x ∈ U
f : (↑Γ(X, U) : Type u)
hf : x ∈ X.basicOpen f
H :
  ∀ (x_1 : (↑(X.presheaf.obj (Opposite.op U)) : Type u)),
    Eq (α := (↑(X.presheaf.stalk x) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (Y := X.presheaf.stalk x) (X.presheaf.germ U x hx) :
              (a : (↑(X.presheaf.obj (Opposite.op U)) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
            x_1 :
          (↑(X.presheaf.stalk x) : Type u))
        (0 : (↑(X.presheaf.stalk x) : Type u)) →
      x_1 = 0
t : (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u)
ht :
  Eq (α := (↑(X.presheaf.stalk x) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Y := X.presheaf.stalk x) (X.presheaf.germ (X.basicOpen f) x hf) :
          (a : (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
        t :
      (↑(X.presheaf.stalk x) : Type u))
    (0 : (↑(X.presheaf.stalk x) : Type u))
this : IsLocalization.Away f (↑Γ(X, X.basicOpen f) : Type u)
⊢ t = 0
have obtain : ∀ (t : (↑Γ(X, U) : Type u)) (s : ↥(Submonoid.powers f))
  (ht :
    Eq (α := (↑(X.presheaf.stalk x) : Type u))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) (Y := X.presheaf.stalk x) (X.presheaf.germ (X.basicOpen f) x hf) :
            (a : (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
          (IsLocalization.mk' (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u) t s) :
        (↑(X.presheaf.stalk x) : Type u))
      (0 : (↑(X.presheaf.stalk x) : Type u))),
  Eq (α := (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u))
    (IsLocalization.mk' (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u) t s) 0
Tactic: obtain ⟨t, s, rfl⟩ := IsLocalization.exists_mk'_eq (.powers f) t
Cleared hypotheses: t, ht
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.SpreadingOut (goal)
Goal:
X : Scheme
U : X.Opens
hU : IsAffineOpen U
x : ↥X
hx : x ∈ U
f : (↑Γ(X, U) : Type u)
hf : x ∈ X.basicOpen f
H :
  ∀ (x_1 : (↑(X.presheaf.obj (Opposite.op U)) : Type u)),
    Eq (α := (↑(X.presheaf.stalk x) : Type u))
        ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                CommRingCat.carrier) (Y := X.presheaf.stalk x) (X.presheaf.germ U x hx) :
              (a : (↑(X.presheaf.obj (Opposite.op U)) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
            x_1 :
          (↑(X.presheaf.stalk x) : Type u))
        (0 : (↑(X.presheaf.stalk x) : Type u)) →
      x_1 = 0
this : IsLocalization.Away f (↑Γ(X, X.basicOpen f) : Type u)
t : (↑Γ(X, U) : Type u)
s : ↥(Submonoid.powers f)
ht :
  Eq (α := (↑(X.presheaf.stalk x) : Type u))
    ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
            CommRingCat.carrier) (Y := X.presheaf.stalk x) (X.presheaf.germ (X.basicOpen f) x hf) :
          (a : (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
        (IsLocalization.mk' (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u) t s) :
      (↑(X.presheaf.stalk x) : Type u))
    (0 : (↑(X.presheaf.stalk x) : Type u))
⊢ Eq (α := (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u))
    (IsLocalization.mk' (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u) t s) 0
have rw : ∀
  (ht :
    Eq (α := (↑(X.presheaf.stalk x) : Type u))
      ((ConcreteCategory.hom (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
              CommRingCat.carrier) (Y := X.presheaf.stalk x) (X.presheaf.germ U x sorry) :
            (a : (↑(X.presheaf.obj (Opposite.op U)) : Type u)) → (↑(X.presheaf.stalk x) : Type u))
          t :
        (↑(X.presheaf.stalk x) : Type u))
      (0 : (↑(X.presheaf.stalk x) : Type u))),
  Eq (α := (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u))
    (IsLocalization.mk' (↑(X.presheaf.obj (Opposite.op (X.basicOpen f))) : Type u) t s) 0
Tactic: rw [← RingHom.mem_ker, IsLocalization.mk'_eq_mul_mk'_one, Ideal.mul_unit_mem_iff_mem,
    RingHom.mem_ker, RingHom.algebraMap_toAlgebra, TopCat.Presheaf.germ_res_apply] at ht
Cleared hypotheses: ht