awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
x :
  (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f)) ⋯).toTopCat :
    Type u_1)
z : NumDenSameDeg 𝒜 (Submonoid.powers f)
⊢ ¬IsUnit (M := AtPrime 𝒜 (↑x : (↑Proj.T : Type u_1)).asHomogeneousIdeal.toIdeal)
        ((mapId 𝒜 ⋯ :
              (a : HomogeneousLocalization 𝒜 (Submonoid.powers f)) →
                HomogeneousLocalization 𝒜 (↑x : (↑Proj.T : Type u_1)).asHomogeneousIdeal.toIdeal.primeCompl)
            (HomogeneousLocalization.mk z) :
          HomogeneousLocalization 𝒜 (↑x : (↑Proj.T : Type u_1)).asHomogeneousIdeal.toIdeal.primeCompl) ↔
    HomogeneousLocalization.mk z ∈ ProjIsoSpecTopComponent.ToSpec.carrier x
have rw : (↑(NumDenSameDeg.num
          { deg := z.deg,
            num := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 z.deg) ((GradedRingHom.id 𝒜 : (a : A) → A) (↑z.num : A)) sorry,
            den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 z.deg) ((GradedRingHom.id 𝒜 : (a : A) → A) (↑z.den : A)) sorry,
            den_mem := sorry : NumDenSameDeg 𝒜 (↑x : (↑Proj.T : Type u_1)).asHomogeneousIdeal.toIdeal.primeCompl }) :
      A) ∉
    (↑x : (↑Proj.T : Type u_1)).asHomogeneousIdeal.toIdeal.primeCompl ↔
  (↑z.num : A) ∈ (↑x : (↑Proj.T : Type u_1)).asHomogeneousIdeal
Tactic: rw [← HomogeneousLocalization.isUnit_iff_isUnit_val,
    ProjIsoSpecTopComponent.ToSpec.mk_mem_carrier, HomogeneousLocalization.map_mk,
    HomogeneousLocalization.val_mk, Localization.mk_eq_mk',
    IsLocalization.AtPrime.isUnit_mk'_iff]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
⊢ IsIso (X :=
    (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
            ⋯).toPresheafedSpace :
      TopCat))
    (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f)))
have h : Eq (α :=
  (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
            sorry).toPresheafedSpace :
      TopCat) ⟶
    (↑(Spec A⁰_ f).toPresheafedSpace : TopCat))
  (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) (Iso.hom (projIsoSpecTopComponent f_deg hm))
Tactic: convert! (projIsoSpecTopComponent f_deg hm).isIso_hom
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
t : NumDenSameDeg 𝒜 (Submonoid.powers f)
⊢ (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier) (X :=
            (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                    ⋯).toPresheafedSpace :
              TopCat))
            (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f)))) :
        (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
          (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)) ⁻¹'
      (↑(sbo HomogeneousLocalization.mk t) : Set (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1)) =
    (↑((Opens.comap
                {
                  toFun :=
                    Subtype.val (p := fun (x : (↑Proj.T : Type u_1)) ↦
                      Membership.mem (γ := Opens (↑Proj.T : Type u_1)) (pbo f) x),
                  continuous_toFun := ⋯ : C(↥(pbo f), ProjectiveSpectrum 𝒜) } :
              (a : Opens (ProjectiveSpectrum 𝒜)) → Opens ↥(pbo f))
            (pbo (↑t.num : A)) :
          Opens ↥(pbo f)) :
      Set (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1))
have h : Eq (α :=
  (↑(↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                sorry).toPresheafedSpace :
          TopCat) :
      Type u_1) →
    (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
  (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier) (X :=
        (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                sorry).toPresheafedSpace :
          TopCat))
        (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f)))) :
    (a : (↑(↑(Proj.restrict sorry).toPresheafedSpace : TopCat) : Type u_1)) →
      (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
  (ProjIsoSpecTopComponent.ToSpec.toFun f)
Tactic: convert! (ProjIsoSpecTopComponent.ToSpec.preimage_basicOpen f t)
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
x :
  (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f)) ⋯).toTopCat :
    Type u_1)
⊢ Iso.inv (Scheme.ΓSpecIso (CommRingCat.of (A⁰_ f))) ≫
      (Spec A⁰_ f).presheaf.germ ⊤
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
                (X :=
                (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                        ⋯).toPresheafedSpace :
                  TopCat))
                (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
              (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
                (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
            x)
          ⋯ ≫
        LocallyRingedSpace.Hom.stalkMap (toSpec 𝒜 f) x =
    awayToΓ 𝒜 f ≫
      (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              ⋯).presheaf.Γgerm
        x
have dsimp : Iso.inv (Scheme.ΓSpecIso (CommRingCat.of (A⁰_ f))) ≫
    Presheaf.germ
        (ObjectProperty.FullSubcategory.obj (P :=
          CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (A⁰_ f)) : Type u_1)))
          («Spec».structureSheaf (A⁰_ f)))
        ⊤
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
              (X := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
            (a : (↑((Opens.toTopCat Proj.T).obj (pbo f)) : Type u_1)) → PrimeSpectrum (A⁰_ f))
          x)
        sorry ≫
      LocallyRingedSpace.Hom.stalkMap (toSpec 𝒜 f) x =
  awayToΓ 𝒜 f ≫
    (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
            sorry).presheaf.Γgerm
      x
Tactic: dsimp
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
x :
  (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f)) ⋯).toTopCat :
    Type u_1)
⊢ Iso.inv (Scheme.ΓSpecIso (CommRingCat.of (A⁰_ f))) ≫
      Presheaf.germ
          (ObjectProperty.FullSubcategory.obj (P :=
            CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (A⁰_ f)) : Type u_1)))
            («Spec».structureSheaf (A⁰_ f)))
          ⊤
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
                (X := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
              (a : (↑((Opens.toTopCat Proj.T).obj (pbo f)) : Type u_1)) → PrimeSpectrum (A⁰_ f))
            x)
          ⋯ ≫
        LocallyRingedSpace.Hom.stalkMap (toSpec 𝒜 f) x =
    awayToΓ 𝒜 f ≫
      (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              ⋯).presheaf.Γgerm
        x
have erw : Eq (α :=
  CommRingCat.of (A⁰_ f) ⟶
    (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
            sorry).presheaf.stalk
      x)
  (Iso.inv (Scheme.ΓSpecIso (CommRingCat.of (A⁰_ f))) ≫
    CategoryStruct.comp (Y :=
      (Functor.obj
            (Presheaf.pushforward CommRingCat (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))))
            (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                sorry).presheaf).obj
        (op ⊤))
      (NatTrans.app (PresheafedSpace.Hom.c (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) (op ⊤))
      ((Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              sorry).presheaf.germ
        (Functor.obj (Opens.map (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f)))) ⊤) x sorry))
  (awayToΓ 𝒜 f ≫
    (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
            sorry).presheaf.Γgerm
      x)
Tactic: erw [LocallyRingedSpace.stalkMap_germ (toSpec 𝒜 f) ⊤ x (by simp)]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
x :
  (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f)) ⋯).toTopCat :
    Type u_1)
⊢ Eq (α :=
    CommRingCat.of (A⁰_ f) ⟶
      (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              ⋯).presheaf.stalk
        x)
    (Iso.inv (Scheme.ΓSpecIso (CommRingCat.of (A⁰_ f))) ≫
      CategoryStruct.comp (Y :=
        (Functor.obj
              (Presheaf.pushforward CommRingCat (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))))
              (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                  ⋯).presheaf).obj
          (op ⊤))
        (NatTrans.app (PresheafedSpace.Hom.c (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) (op ⊤))
        ((Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                ⋯).presheaf.germ
          (Functor.obj (Opens.map (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f)))) ⊤) x ⋯))
    (awayToΓ 𝒜 f ≫
      (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              ⋯).presheaf.Γgerm
        x)
have erw : Eq (α :=
  CommRingCat.of (A⁰_ f) ⟶
    (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
            sorry).presheaf.stalk
      x)
  (CategoryStruct.comp (Y :=
    (Functor.obj
          (Presheaf.pushforward CommRingCat (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))))
          (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              sorry).presheaf).obj
      (op ⊤))
    (awayToΓ 𝒜 f ≫
      (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
              sorry).presheaf.map
        (Quiver.Hom.op (homOfLE sorry)))
    ((Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
            sorry).presheaf.germ
      (Functor.obj (Opens.map (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f)))) ⊤) x sorry))
  (awayToΓ 𝒜 f ≫
    (Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
            sorry).presheaf.Γgerm
      x)
Tactic: erw [toOpen_toSpec_val_c_app_assoc]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :=
  RingHom.toAlgebra (mapId 𝒜 ⋯)
⊢ IsLocalization.{u_1, u_1} (R := A⁰_ f)
    (Ideal.primeCompl
      (PrimeSpectrum.asIdeal
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
              (X :=
              (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                      ⋯).toPresheafedSpace :
                TopCat))
              (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
            (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
              (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
          x)))
    (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
have toIsLocalizationMap : Submonoid.IsLocalizationMap.{u_1, u_1} (M := A⁰_ f) (N :=
  AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
  (Ideal.primeCompl
    (PrimeSpectrum.asIdeal
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
            (X :=
            (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                    sorry).toPresheafedSpace :
              TopCat))
            (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
          (a : (↑(↑(Proj.restrict sorry).toPresheafedSpace : TopCat) : Type u_1)) →
            (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
        x)))
  (⇑(algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)) :
    (a : A⁰_ f) → AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
Tactic: constructor
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :=
  RingHom.toAlgebra (mapId 𝒜 ⋯)
⊢ Submonoid.IsLocalizationMap.{u_1, u_1} (M := A⁰_ f) (N :=
    AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
    (Ideal.primeCompl
      (PrimeSpectrum.asIdeal
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
              (X :=
              (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                      ⋯).toPresheafedSpace :
                TopCat))
              (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
            (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
              (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
          x)))
    (⇑(algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)) :
      (a : A⁰_ f) → AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
have toIsLocalizationMap : ∀
  (y :
    ↥(Ideal.primeCompl
        (PrimeSpectrum.asIdeal
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC := carrier)
                (X :=
                (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                        sorry).toPresheafedSpace :
                  TopCat))
                (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
              (a : (↑(↑(Proj.restrict sorry).toPresheafedSpace : TopCat) : Type u_1)) →
                (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
            x)))),
  IsUnit (M := AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
    ((algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :
          (a : A⁰_ f) → AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
        (↑y : A⁰_ f) :
      AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
Tactic: constructor
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :=
  RingHom.toAlgebra (mapId 𝒜 ⋯)
i : ℕ
a : ↥(𝒜 i)
b : A
hb : b ∈ 𝒜 i
hb' : b ∉ (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal
⊢ ∃ (x_1 :
    (A⁰_ f) ×
      ↥(Ideal.primeCompl
          (PrimeSpectrum.asIdeal
            ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u_1}) ↦ C((↑X : Type u_1), (↑Y : Type u_1))) (CC :=
                  carrier) (X :=
                  (↑(Proj.restrict (U := Functor.obj (C := Opens (↑Proj.T : Type u_1)) (Opens.toTopCat Proj.T) (pbo f))
                          ⋯).toPresheafedSpace :
                    TopCat))
                  (PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.toHom (toSpec 𝒜 f))) :
                (a : (↑(↑(Proj.restrict ⋯).toPresheafedSpace : TopCat) : Type u_1)) →
                  (↑(↑(Spec A⁰_ f).toPresheafedSpace : TopCat) : Type u_1))
              x)))),
    HMul.hMul (α := AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) (β :=
        AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
        (HomogeneousLocalization.mk
          { deg := i, num := a, den := Subtype.mk (p := fun (x : A) ↦ x ∈ 𝒜 i) b hb, den_mem := hb' :
            NumDenSameDeg 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal.primeCompl })
        ((algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :
              (a : A⁰_ f) → AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
            (↑x_1.2 : A⁰_ f) :
          AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) =
      (algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :
          (a : A⁰_ f) → AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal)
        x_1.1
have toIsLocalizationMap : (↑a : A) * b ^ (m - 1) ∈ 𝒜 (i * m)
Tactic: refine ⟨⟨HomogeneousLocalization.mk ⟨i * m, ⟨a * b ^ (m - 1), ?_⟩,
        ⟨f ^ i, SetLike.pow_mem_graded _ f_deg⟩, ⟨_, rfl⟩⟩,
      ⟨HomogeneousLocalization.mk ⟨i * m, ⟨b ^ m, mul_comm m i ▸ SetLike.pow_mem_graded _ hb⟩,
        ⟨f ^ i, SetLike.pow_mem_graded _ f_deg⟩, ⟨_, rfl⟩⟩,
        (mk_mem_toSpec_base_apply _ _ _).not.mpr <| x.1.1.toIdeal.primeCompl.pow_mem hb' m⟩⟩,
        val_injective _ ?_⟩
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :=
  RingHom.toAlgebra (mapId 𝒜 ⋯)
i : ℕ
a : ↥(𝒜 i)
b : A
hb : b ∈ 𝒜 i
hb' : b ∉ (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal
⊢ (↑a : A) * b ^ (m - 1) ∈ 𝒜 (i * m)
have h : i * m = i + (m - 1) • i
Tactic: convert! SetLike.mul_mem_graded a.2 (SetLike.pow_mem_graded (m - 1) hb) using 2
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :=
  RingHom.toAlgebra (mapId 𝒜 ⋯)
i : ℕ
a : ↥(𝒜 i)
b : A
hb : b ∈ 𝒜 i
hb' : b ∉ (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal
⊢ i * m = i + (m - 1) • i
have h : i * m = (m - 1 + 1) • i
Tactic: rw [← succ_nsmul', tsub_add_cancel_of_le (by lia), mul_comm, smul_eq_mul]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :=
  RingHom.toAlgebra (mapId 𝒜 ⋯)
i : ℕ
a : ↥(𝒜 i)
b : A
hb : b ∈ 𝒜 i
hb' : b ∉ (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal
h : i * m = (m - 1 + 1) • i
⊢ i * m = i + (m - 1) • i
have h₁ : i * m = m • i
Tactic: rw [← succ_nsmul', tsub_add_cancel_of_le (by lia), mul_comm, smul_eq_mul]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :=
  RingHom.toAlgebra (mapId 𝒜 ⋯)
i : ℕ
a : ↥(𝒜 i)
b : A
hb : b ∈ 𝒜 i
hb' : b ∉ (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal
h : i * m = (m - 1 + 1) • i
h₁ : i * m = m • i
⊢ i * m = i + (m - 1) • i
have h₂ : m * i = m • i
Tactic: rw [← succ_nsmul', tsub_add_cancel_of_le (by lia), mul_comm, smul_eq_mul]
-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme (goal)
Goal:
A : Type u_1
σ : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal.toIdeal) :=
  RingHom.toAlgebra (mapId 𝒜 ⋯)
i : ℕ
a : ↥(𝒜 i)
b : A
hb : b ∈ 𝒜 i
hb' : b ∉ (↑x : ProjectiveSpectrum 𝒜).asHomogeneousIdeal
h : i * m = (m - 1 + 1) • i
h₁ : i * m = m • i
h₂ : m * i = m • i
⊢ i * m = i + (m - 1) • i
have h₃ : m * i = m * i
Tactic: rw [← succ_nsmul', tsub_add_cancel_of_le (by lia), mul_comm, smul_eq_mul]