ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AlgebraicGeometry.Scheme.GlueData.oneHypercover_p₁
@[simps]
noncomputable def oneHypercover : Scheme.zariskiTopology.OneHypercover D.glued 
Arguments: (D : AlgebraicGeometry.Scheme.GlueData) (i₁ i₂ : D.J)
Goal: ∀ (x : PUnit.{u + 1}), D.oneHypercover.p₁ x = D.f i₁ i₂
-- AlgebraicGeometry.Scheme.GlueData.oneHypercover_X
@[simps]
noncomputable def oneHypercover : Scheme.zariskiTopology.OneHypercover D.glued 
Arguments: (D : AlgebraicGeometry.Scheme.GlueData)
Goal: ∀ (a : D.J), D.oneHypercover.X a = D.U a
-- AlgebraicGeometry.Scheme.GlueData.oneHypercover_Y
@[simps]
noncomputable def oneHypercover : Scheme.zariskiTopology.OneHypercover D.glued 
Arguments: (D : AlgebraicGeometry.Scheme.GlueData) (i₁ i₂ : D.J)
Goal: ∀ (x : PUnit.{u + 1}), D.oneHypercover.Y x = D.V (i₁, i₂)
-- AlgebraicGeometry.Scheme.GlueData.oneHypercover_p₂
@[simps]
noncomputable def oneHypercover : Scheme.zariskiTopology.OneHypercover D.glued 
Arguments: (D : AlgebraicGeometry.Scheme.GlueData) (i₁ i₂ : D.J) (j : PUnit.{u + 1})
Goal: D.oneHypercover.p₂ j = D.t i₁ i₂ ≫ D.f i₂ i₁
-- AlgebraicGeometry.Scheme.GlueData.oneHypercover_I₀
@[simps]
noncomputable def oneHypercover : Scheme.zariskiTopology.OneHypercover D.glued 
Arguments: (D : AlgebraicGeometry.Scheme.GlueData)
Goal: D.oneHypercover.I₀ = D.J
-- AlgebraicGeometry.Scheme.GlueData.oneHypercover_I₁
@[simps]
noncomputable def oneHypercover : Scheme.zariskiTopology.OneHypercover D.glued 
Arguments: (D : AlgebraicGeometry.Scheme.GlueData)
Goal: ∀ (x x_1 : D.J), D.oneHypercover.I₁ x x_1 = PUnit.{u + 1}
-- AlgebraicGeometry.Scheme.GlueData.sheafValGluedMk_val
@[simp]
lemma sheafValGluedMk_val (j : D.J) : F.val.map (D.ι j).op (D.sheafValGluedMk s h) = s j 
Arguments: (D : AlgebraicGeometry.Scheme.GlueData) {F : CategoryTheory.Sheaf AlgebraicGeometry.Scheme.zariskiTopology (Type v)} (s : (j : D.J) → F.val.obj (Opposite.op (D.U j))) (h : ∀ (i j : D.J), F.val.map (D.f i j).op (s i) = F.val.map ((D.f j i).op ≫ (D.t i j).op) (s j)) (j : D.J)
Goal: F.val.map (D.ι j).op (D.sheafValGluedMk s h) = s j
-- AlgebraicGeometry.IsLocalAtTarget.iff_of_openCover'
class IsLocalAtTarget (P : MorphismProperty Scheme) : Prop where
    respectsIso : P.RespectsIso := by infer_instance
    iff_of_openCover' :
    ∀ {X Y : Scheme.{u}} (f : X ⟶ Y) (𝒰 : Scheme.OpenCover.{u} Y),
      P f ↔ ∀ i, P (𝒰.pullbackHom f i)
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [self : AlgebraicGeometry.IsLocalAtTarget P] {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (𝒰 : Y.OpenCover)
Goal: P f ↔ ∀ (i : (𝒰.pullbackCover f).J), P (𝒰.pullbackHom f i)
-- AlgebraicGeometry.IsLocalAtTarget.respectsIso
class IsLocalAtTarget (P : MorphismProperty Scheme) : Prop where
    respectsIso : P.RespectsIso := by infer_instance
    iff_of_openCover' :
    ∀ {X Y : Scheme.{u}} (f : X ⟶ Y) (𝒰 : Scheme.OpenCover.{u} Y),
      P f ↔ ∀ i, P (𝒰.pullbackHom f i)
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [self : AlgebraicGeometry.IsLocalAtTarget P]
Goal: P.RespectsIso
-- AlgebraicGeometry.IsLocalAtTarget.mk'
protected lemma mk' {P : MorphismProperty Scheme} [P.RespectsIso]
    (restrict : ∀ {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens), P f → P (f ∣_ U))
    (of_sSup_eq_top :
      ∀ {X Y : Scheme.{u}} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), iSup U = ⊤ →
        (∀ i, P (f ∣_ U i)) → P f) :
    IsLocalAtTarget P 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [P.RespectsIso] (restrict : ∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (U : Y.Opens), P f → P (f ∣_ U)) (of_sSup_eq_top :
  ∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens),
    iSup U = ⊤ → (∀ (i : ι), P (f ∣_ U i)) → P f)
Goal: AlgebraicGeometry.IsLocalAtTarget P
-- AlgebraicGeometry.IsLocalAtTarget.of_isPullback
lemma of_isPullback {UX UY : Scheme.{u}} {iY : UY ⟶ Y} [IsOpenImmersion iY]
    {iX : UX ⟶ X} {f' : UX ⟶ UY} (h : IsPullback iX f' f iY) (H : P f) : P f' 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [hP : AlgebraicGeometry.IsLocalAtTarget P] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} {UX UY : AlgebraicGeometry.Scheme} {iY : UY ⟶ Y} [AlgebraicGeometry.IsOpenImmersion iY] {iX : UX ⟶ X} {f' : UX ⟶ UY} (h : CategoryTheory.IsPullback iX f' f iY) (H : P f)
Goal: P f'
-- AlgebraicGeometry.IsLocalAtTarget.restrict
theorem restrict (hf : P f) (U : Y.Opens) : P (f ∣_ U) 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [hP : AlgebraicGeometry.IsLocalAtTarget P] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} (hf : P f) (U : Y.Opens)
Goal: P (f ∣_ U)
-- AlgebraicGeometry.IsLocalAtTarget.of_iSup_eq_top
lemma of_iSup_eq_top {ι} (U : ι → Y.Opens) (hU : iSup U = ⊤)
    (H : ∀ i, P (f ∣_ U i)) : P f 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [hP : AlgebraicGeometry.IsLocalAtTarget P] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} {ι : Sort u_1} (U : ι → Y.Opens) (hU : iSup U = ⊤) (H : ∀ (i : ι), P (f ∣_ U i))
Goal: P f
-- AlgebraicGeometry.IsLocalAtTarget.iff_of_iSup_eq_top
theorem iff_of_iSup_eq_top {ι} (U : ι → Y.Opens) (hU : iSup U = ⊤) :
    P f ↔ ∀ i, P (f ∣_ U i) 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [hP : AlgebraicGeometry.IsLocalAtTarget P] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} {ι : Sort u_1} (U : ι → Y.Opens) (hU : iSup U = ⊤)
Goal: P f ↔ ∀ (i : ι), P (f ∣_ U i)
-- AlgebraicGeometry.IsLocalAtTarget.of_openCover
lemma of_openCover (H : ∀ i, P (𝒰.pullbackHom f i)) : P f 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [hP : AlgebraicGeometry.IsLocalAtTarget P] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} (𝒰 : Y.OpenCover) (H : ∀ (i : (𝒰.pullbackCover f).J), P (𝒰.pullbackHom f i))
Goal: P f
-- AlgebraicGeometry.IsLocalAtTarget.iff_of_openCover
theorem iff_of_openCover (𝒰 : Y.OpenCover) :
    P f ↔ ∀ i, P (𝒰.pullbackHom f i) 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [hP : AlgebraicGeometry.IsLocalAtTarget P] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} (𝒰 : Y.OpenCover)
Goal: P f ↔ ∀ (i : (𝒰.pullbackCover f).J), P (𝒰.pullbackHom f i)
-- AlgebraicGeometry.IsLocalAtSource.iff_of_openCover'
class IsLocalAtSource (P : MorphismProperty Scheme) : Prop where
    respectsIso : P.RespectsIso := by infer_instance
    iff_of_openCover' :
    ∀ {X Y : Scheme.{u}} (f : X ⟶ Y) (𝒰 : Scheme.OpenCover.{u} X),
      P f ↔ ∀ i, P (𝒰.map i ≫ f)
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [self : AlgebraicGeometry.IsLocalAtSource P] {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (𝒰 : X.OpenCover)
Goal: P f ↔ ∀ (i : 𝒰.J), P (𝒰.map i ≫ f)
-- AlgebraicGeometry.IsLocalAtSource.respectsIso
class IsLocalAtSource (P : MorphismProperty Scheme) : Prop where
    respectsIso : P.RespectsIso := by infer_instance
    iff_of_openCover' :
    ∀ {X Y : Scheme.{u}} (f : X ⟶ Y) (𝒰 : Scheme.OpenCover.{u} X),
      P f ↔ ∀ i, P (𝒰.map i ≫ f)
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [self : AlgebraicGeometry.IsLocalAtSource P]
Goal: P.RespectsIso
-- AlgebraicGeometry.IsLocalAtSource.mk'
protected lemma mk' {P : MorphismProperty Scheme} [P.RespectsIso]
    (restrict : ∀ {X Y : Scheme} (f : X ⟶ Y) (U : X.Opens), P f → P (U.ι ≫ f))
    (of_sSup_eq_top :
      ∀ {X Y : Scheme.{u}} (f : X ⟶ Y) {ι : Type u} (U : ι → X.Opens), iSup U = ⊤ →
        (∀ i, P ((U i).ι ≫ f)) → P f) :
    IsLocalAtSource P 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [P.RespectsIso] (restrict : ∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (U : X.Opens), P f → P (U.ι ≫ f)) (of_sSup_eq_top :
  ∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → X.Opens),
    iSup U = ⊤ → (∀ (i : ι), P ((U i).ι ≫ f)) → P f)
Goal: AlgebraicGeometry.IsLocalAtSource P
-- AlgebraicGeometry.IsLocalAtSource.comp
lemma comp {UX : Scheme.{u}} (H : P f) (i : UX ⟶ X) [IsOpenImmersion i] :
    P (i ≫ f) 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsLocalAtSource P] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} {UX : AlgebraicGeometry.Scheme} (H : P f) (i : UX ⟶ X) [AlgebraicGeometry.IsOpenImmersion i]
Goal: P (i ≫ f)
-- AlgebraicGeometry.IsLocalAtSource.of_iSup_eq_top
lemma of_iSup_eq_top {ι} (U : ι → X.Opens) (hU : iSup U = ⊤)
    (H : ∀ i, P ((U i).ι ≫ f)) : P f 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsLocalAtSource P] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} {ι : Sort u_1} (U : ι → X.Opens) (hU : iSup U = ⊤) (H : ∀ (i : ι), P ((U i).ι ≫ f))
Goal: P f
-- AlgebraicGeometry.IsLocalAtSource.iff_of_iSup_eq_top
theorem iff_of_iSup_eq_top {ι} (U : ι → X.Opens) (hU : iSup U = ⊤) :
    P f ↔ ∀ i, P ((U i).ι ≫ f) 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsLocalAtSource P] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} {ι : Sort u_1} (U : ι → X.Opens) (hU : iSup U = ⊤)
Goal: P f ↔ ∀ (i : ι), P ((U i).ι ≫ f)
-- AlgebraicGeometry.IsLocalAtSource.of_openCover
lemma of_openCover (H : ∀ i, P (𝒰.map i ≫ f)) : P f 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsLocalAtSource P] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} (𝒰 : X.OpenCover) (H : ∀ (i : 𝒰.J), P (𝒰.map i ≫ f))
Goal: P f
-- AlgebraicGeometry.IsLocalAtSource.iff_of_openCover
theorem iff_of_openCover :
    P f ↔ ∀ i, P (𝒰.map i ≫ f) 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsLocalAtSource P] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} (𝒰 : X.OpenCover)
Goal: P f ↔ ∀ (i : 𝒰.J), P (𝒰.map i ≫ f)
-- AlgebraicGeometry.IsLocalAtSource.of_isOpenImmersion
lemma of_isOpenImmersion [P.ContainsIdentities] [IsOpenImmersion f] : P f 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsLocalAtSource P] {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [P.ContainsIdentities] [AlgebraicGeometry.IsOpenImmersion f]
Goal: P f
-- AlgebraicGeometry.IsLocalAtSource.isLocalAtTarget
lemma isLocalAtTarget [P.IsMultiplicative]
    (hP : ∀ {X Y Z : Scheme.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) [IsOpenImmersion g], P (f ≫ g) → P f) :
    IsLocalAtTarget P 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsLocalAtSource P] [P.IsMultiplicative] (hP :
  ∀ {X Y Z : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) [inst : AlgebraicGeometry.IsOpenImmersion g],
    P (f ≫ g) → P f)
Goal: AlgebraicGeometry.IsLocalAtTarget P
-- AlgebraicGeometry.AffineTargetMorphismProperty.ext
@[ext]
lemma ext {P Q : AffineTargetMorphismProperty}
    (H : ∀ ⦃X Y : Scheme⦄ (f : X ⟶ Y) [IsAffine Y], P f ↔ Q f) : P = Q 
Arguments: {P Q : AlgebraicGeometry.AffineTargetMorphismProperty} (H : ∀ ⦃X Y : AlgebraicGeometry.Scheme⦄ (f : X ⟶ Y) [inst : AlgebraicGeometry.IsAffine Y], P f ↔ Q f)
Goal: P = Q
-- AlgebraicGeometry.AffineTargetMorphismProperty.toProperty_apply
theorem toProperty_apply (P : AffineTargetMorphismProperty)
    {X Y : Scheme} (f : X ⟶ Y) [i : IsAffine Y] : P.toProperty f ↔ P f 
Arguments: (P : AlgebraicGeometry.AffineTargetMorphismProperty) {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [i : AlgebraicGeometry.IsAffine Y]
Goal: P.toProperty f ↔ P f
-- AlgebraicGeometry.AffineTargetMorphismProperty.cancel_left_of_respectsIso
theorem cancel_left_of_respectsIso
    (P : AffineTargetMorphismProperty) [P.toProperty.RespectsIso]
    {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso f] [IsAffine Z] : P (f ≫ g) ↔ P g 
Arguments: (P : AlgebraicGeometry.AffineTargetMorphismProperty) [P.toProperty.RespectsIso] {X Y Z : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) [CategoryTheory.IsIso f] [AlgebraicGeometry.IsAffine Z]
Goal: P (f ≫ g) ↔ P g
-- AlgebraicGeometry.AffineTargetMorphismProperty.cancel_right_of_respectsIso
theorem cancel_right_of_respectsIso
    (P : AffineTargetMorphismProperty) [P.toProperty.RespectsIso]
    {X Y Z : Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) [IsIso g] [IsAffine Z] [IsAffine Y] :
    P (f ≫ g) ↔ P f 
Arguments: (P : AlgebraicGeometry.AffineTargetMorphismProperty) [P.toProperty.RespectsIso] {X Y Z : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) [CategoryTheory.IsIso g] [AlgebraicGeometry.IsAffine Z] [AlgebraicGeometry.IsAffine Y]
Goal: P (f ≫ g) ↔ P f
-- AlgebraicGeometry.AffineTargetMorphismProperty.arrow_mk_iso_iff
theorem arrow_mk_iso_iff
    (P : AffineTargetMorphismProperty) [P.toProperty.RespectsIso]
    {X Y X' Y' : Scheme} {f : X ⟶ Y} {f' : X' ⟶ Y'}
    (e : Arrow.mk f ≅ Arrow.mk f') {h : IsAffine Y} :
    letI : IsAffine Y' := isAffine_of_isIso (Y := Y) e.inv.right
    P f ↔ P f' 
Arguments: (P : AlgebraicGeometry.AffineTargetMorphismProperty) [P.toProperty.RespectsIso] {X Y X' Y' : AlgebraicGeometry.Scheme} {f : X ⟶ Y} {f' : X' ⟶ Y'} (e : CategoryTheory.Arrow.mk f ≅ CategoryTheory.Arrow.mk f') {h : AlgebraicGeometry.IsAffine Y}
Goal: P f ↔ P f'
-- AlgebraicGeometry.AffineTargetMorphismProperty.respectsIso_mk
theorem respectsIso_mk {P : AffineTargetMorphismProperty}
    (h₁ : ∀ {X Y Z} (e : X ≅ Y) (f : Y ⟶ Z) [IsAffine Z], P f → P (e.hom ≫ f))
    (h₂ : ∀ {X Y Z} (e : Y ≅ Z) (f : X ⟶ Y) [h : IsAffine Y],
      P f → @P _ _ (f ≫ e.hom) (isAffine_of_isIso e.inv)) :
    P.toProperty.RespectsIso 
Arguments: {P : AlgebraicGeometry.AffineTargetMorphismProperty} (h₁ :
  ∀ {X Y Z : AlgebraicGeometry.Scheme} (e : X ≅ Y) (f : Y ⟶ Z) [inst : AlgebraicGeometry.IsAffine Z],
    P f → P (e.hom ≫ f)) (h₂ :
  ∀ {X Y Z : AlgebraicGeometry.Scheme} (e : Y ≅ Z) (f : X ⟶ Y) [h : AlgebraicGeometry.IsAffine Y], P f → P (f ≫ e.hom))
Goal: P.toProperty.RespectsIso
-- AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal.to_basicOpen
class IsLocal (P : AffineTargetMorphismProperty) : Prop where
    respectsIso : P.toProperty.RespectsIso
    to_basicOpen :
    ∀ {X Y : Scheme} [IsAffine Y] (f : X ⟶ Y) (r : Γ(Y, ⊤)), P f → P (f ∣_ Y.basicOpen r)
    of_basicOpenCover :
    ∀ {X Y : Scheme} [IsAffine Y] (f : X ⟶ Y) (s : Finset Γ(Y, ⊤))
      (_ : Ideal.span (s : Set Γ(Y, ⊤)) = ⊤), (∀ r : s, P (f ∣_ Y.basicOpen r.1)) → P f
Arguments: {P : AlgebraicGeometry.AffineTargetMorphismProperty} [self : P.IsLocal] {X Y : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsAffine Y] (f : X ⟶ Y) (r : ↑(Y.presheaf.obj (Opposite.op ⊤)))
Goal: P f → P (f ∣_ Y.basicOpen r)
-- AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal.of_basicOpenCover
class IsLocal (P : AffineTargetMorphismProperty) : Prop where
    respectsIso : P.toProperty.RespectsIso
    to_basicOpen :
    ∀ {X Y : Scheme} [IsAffine Y] (f : X ⟶ Y) (r : Γ(Y, ⊤)), P f → P (f ∣_ Y.basicOpen r)
    of_basicOpenCover :
    ∀ {X Y : Scheme} [IsAffine Y] (f : X ⟶ Y) (s : Finset Γ(Y, ⊤))
      (_ : Ideal.span (s : Set Γ(Y, ⊤)) = ⊤), (∀ r : s, P (f ∣_ Y.basicOpen r.1)) → P f
Arguments: {P : AlgebraicGeometry.AffineTargetMorphismProperty} [self : P.IsLocal] {X Y : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsAffine Y] (f : X ⟶ Y) (s : Finset ↑(Y.presheaf.obj (Opposite.op ⊤)))
Goal: Ideal.span ↑s = ⊤ → (∀ (r : { x // x ∈ s }), P (f ∣_ Y.basicOpen ↑r)) → P f
-- AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal.respectsIso
class IsLocal (P : AffineTargetMorphismProperty) : Prop where
    respectsIso : P.toProperty.RespectsIso
    to_basicOpen :
    ∀ {X Y : Scheme} [IsAffine Y] (f : X ⟶ Y) (r : Γ(Y, ⊤)), P f → P (f ∣_ Y.basicOpen r)
    of_basicOpenCover :
    ∀ {X Y : Scheme} [IsAffine Y] (f : X ⟶ Y) (s : Finset Γ(Y, ⊤))
      (_ : Ideal.span (s : Set Γ(Y, ⊤)) = ⊤), (∀ r : s, P (f ∣_ Y.basicOpen r.1)) → P f
Arguments: {P : AlgebraicGeometry.AffineTargetMorphismProperty} [self : P.IsLocal]
Goal: P.toProperty.RespectsIso
-- AlgebraicGeometry.AffineTargetMorphismProperty.StableUnderBaseChange.mk
lemma StableUnderBaseChange.mk (P : AffineTargetMorphismProperty) [P.toProperty.RespectsIso]
    (H : ∀ ⦃X Y S : Scheme⦄ [IsAffine S] [IsAffine X] (f : X ⟶ S) (g : Y ⟶ S),
      P g → P (pullback.fst f g)) : P.StableUnderBaseChange 
Arguments: (P : AlgebraicGeometry.AffineTargetMorphismProperty) [P.toProperty.RespectsIso] (H :
  ∀ ⦃X Y S : AlgebraicGeometry.Scheme⦄ [inst : AlgebraicGeometry.IsAffine S] [inst_1 : AlgebraicGeometry.IsAffine X]
    (f : X ⟶ S) (g : Y ⟶ S), P g → P (CategoryTheory.Limits.pullback.fst f g))
Goal: P.StableUnderBaseChange
-- AlgebraicGeometry.of_targetAffineLocally_of_isPullback
theorem of_targetAffineLocally_of_isPullback
    {P : AffineTargetMorphismProperty} [P.IsLocal]
    {X Y UX UY : Scheme.{u}} [IsAffine UY] {f : X ⟶ Y} {iY : UY ⟶ Y} [IsOpenImmersion iY]
    {iX : UX ⟶ X} {f' : UX ⟶ UY} (h : IsPullback iX f' f iY) (hf : targetAffineLocally P f) :
    P f' 
Arguments: {P : AlgebraicGeometry.AffineTargetMorphismProperty} [P.IsLocal] {X Y UX UY : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsAffine UY] {f : X ⟶ Y} {iY : UY ⟶ Y} [AlgebraicGeometry.IsOpenImmersion iY] {iX : UX ⟶ X} {f' : UX ⟶ UY} (h : CategoryTheory.IsPullback iX f' f iY) (hf : AlgebraicGeometry.targetAffineLocally P f)
Goal: P f'
-- AlgebraicGeometry.HasAffineProperty.eq_targetAffineLocally'
class HasAffineProperty (P : MorphismProperty Scheme)
    (Q : outParam AffineTargetMorphismProperty) : Prop where
  isLocal_affineProperty : Q.IsLocal
  eq_targetAffineLocally' : P = targetAffineLocally Q
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} {Q : outParam AlgebraicGeometry.AffineTargetMorphismProperty} [self : AlgebraicGeometry.HasAffineProperty P Q]
Goal: P = AlgebraicGeometry.targetAffineLocally Q
-- AlgebraicGeometry.HasAffineProperty.isLocal_affineProperty
class HasAffineProperty (P : MorphismProperty Scheme)
    (Q : outParam AffineTargetMorphismProperty) : Prop where
  isLocal_affineProperty : Q.IsLocal
  eq_targetAffineLocally' : P = targetAffineLocally Q
Arguments: (P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme) {Q : outParam AlgebraicGeometry.AffineTargetMorphismProperty} [self : AlgebraicGeometry.HasAffineProperty P Q]
Goal: AlgebraicGeometry.AffineTargetMorphismProperty.IsLocal Q
-- AlgebraicGeometry.HasAffineProperty.eq_targetAffineLocally
lemma eq_targetAffineLocally : P = targetAffineLocally Q 
Arguments: (P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme) {Q : AlgebraicGeometry.AffineTargetMorphismProperty} [AlgebraicGeometry.HasAffineProperty P Q]
Goal: P = AlgebraicGeometry.targetAffineLocally Q
-- AlgebraicGeometry.HasAffineProperty.of_isLocalAtTarget
lemma of_isLocalAtTarget (P) [IsLocalAtTarget P] :
    HasAffineProperty P (AffineTargetMorphismProperty.of P) 
Arguments: (P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsLocalAtTarget P]
Goal: AlgebraicGeometry.HasAffineProperty P (AlgebraicGeometry.AffineTargetMorphismProperty.of P)
-- AlgebraicGeometry.HasAffineProperty.copy
lemma copy {P P'} {Q Q'} [HasAffineProperty P Q]
    (e : P = P') (e' : Q = Q') : HasAffineProperty P' Q' 
Arguments: {P P' : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} {Q Q' : AlgebraicGeometry.AffineTargetMorphismProperty} [AlgebraicGeometry.HasAffineProperty P Q] (e : P = P') (e' : Q = Q')
Goal: AlgebraicGeometry.HasAffineProperty P' Q'
-- AlgebraicGeometry.HasAffineProperty.of_isPullback
theorem of_isPullback {UX UY : Scheme.{u}} [IsAffine UY] {iY : UY ⟶ Y} [IsOpenImmersion iY]
    {iX : UX ⟶ X} {f' : UX ⟶ UY} (h : IsPullback iX f' f iY) (hf : P f) :
    Q f' 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} {Q : AlgebraicGeometry.AffineTargetMorphismProperty} [AlgebraicGeometry.HasAffineProperty P Q] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} {UX UY : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsAffine UY] {iY : UY ⟶ Y} [AlgebraicGeometry.IsOpenImmersion iY] {iX : UX ⟶ X} {f' : UX ⟶ UY} (h : CategoryTheory.IsPullback iX f' f iY) (hf : P f)
Goal: Q f'
-- AlgebraicGeometry.HasAffineProperty.restrict
theorem restrict (hf : P f) (U : Y.affineOpens) :
    Q (f ∣_ U) 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} {Q : AlgebraicGeometry.AffineTargetMorphismProperty} [AlgebraicGeometry.HasAffineProperty P Q] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} (hf : P f) (U : ↑Y.affineOpens)
Goal: Q (f ∣_ ↑U)
-- AlgebraicGeometry.HasAffineProperty.of_iSup_eq_top
theorem of_iSup_eq_top
    {ι} (U : ι → Y.affineOpens) (hU : ⨆ i, (U i : Y.Opens) = ⊤)
    (hU' : ∀ i, Q (f ∣_ U i)) :
    P f 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} {Q : AlgebraicGeometry.AffineTargetMorphismProperty} [AlgebraicGeometry.HasAffineProperty P Q] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} {ι : Sort u_1} (U : ι → ↑Y.affineOpens) (hU : ⨆ i, ↑(U i) = ⊤) (hU' : ∀ (i : ι), Q (f ∣_ ↑(U i)))
Goal: P f
-- AlgebraicGeometry.HasAffineProperty.iff_of_iSup_eq_top
theorem iff_of_iSup_eq_top
    {ι} (U : ι → Y.affineOpens) (hU : ⨆ i, (U i : Y.Opens) = ⊤) :
    P f ↔ ∀ i, Q (f ∣_ U i) 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} {Q : AlgebraicGeometry.AffineTargetMorphismProperty} [AlgebraicGeometry.HasAffineProperty P Q] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} {ι : Sort u_1} (U : ι → ↑Y.affineOpens) (hU : ⨆ i, ↑(U i) = ⊤)
Goal: P f ↔ ∀ (i : ι), Q (f ∣_ ↑(U i))
-- AlgebraicGeometry.HasAffineProperty.of_openCover
theorem of_openCover
    (𝒰 : Y.OpenCover) [∀ i, IsAffine (𝒰.obj i)] (h𝒰 : ∀ i, Q (𝒰.pullbackHom f i)) :
    P f 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} {Q : AlgebraicGeometry.AffineTargetMorphismProperty} [AlgebraicGeometry.HasAffineProperty P Q] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} (𝒰 : Y.OpenCover) [∀ (i : 𝒰.J), AlgebraicGeometry.IsAffine (𝒰.obj i)] (h𝒰 : ∀ (i : (𝒰.pullbackCover f).J), Q (𝒰.pullbackHom f i))
Goal: P f
-- AlgebraicGeometry.HasAffineProperty.iff_of_openCover
theorem iff_of_openCover (𝒰 : Y.OpenCover) [∀ i, IsAffine (𝒰.obj i)] :
    P f ↔ ∀ i, Q (𝒰.pullbackHom f i) 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} {Q : AlgebraicGeometry.AffineTargetMorphismProperty} [AlgebraicGeometry.HasAffineProperty P Q] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} (𝒰 : Y.OpenCover) [∀ (i : 𝒰.J), AlgebraicGeometry.IsAffine (𝒰.obj i)]
Goal: P f ↔ ∀ (i : (𝒰.pullbackCover f).J), Q (𝒰.pullbackHom f i)
-- AlgebraicGeometry.HasAffineProperty.iff_of_isAffine
theorem iff_of_isAffine [IsAffine Y] : P f ↔ Q f 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} {Q : AlgebraicGeometry.AffineTargetMorphismProperty} [AlgebraicGeometry.HasAffineProperty P Q] {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} [AlgebraicGeometry.IsAffine Y]
Goal: P f ↔ Q f
-- AlgebraicGeometry.HasAffineProperty.iff
protected theorem iff :
    HasAffineProperty P Q ↔ IsLocalAtTarget P ∧ Q = of P 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} {Q : AlgebraicGeometry.AffineTargetMorphismProperty}
Goal: AlgebraicGeometry.HasAffineProperty P Q ↔
  AlgebraicGeometry.IsLocalAtTarget P ∧ Q = AlgebraicGeometry.AffineTargetMorphismProperty.of P
-- _private.Mathlib.AlgebraicGeometry.Morphisms.Basic.0.AlgebraicGeometry.HasAffineProperty.pullback_fst_of_right
private theorem pullback_fst_of_right (hP' : Q.StableUnderBaseChange)
    {X Y S : Scheme} (f : X ⟶ S) (g : Y ⟶ S) [IsAffine S] (H : Q g) :
    P (pullback.fst f g) 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} {Q : AlgebraicGeometry.AffineTargetMorphismProperty} [AlgebraicGeometry.HasAffineProperty P Q] (hP' : Q.StableUnderBaseChange) {X Y S : AlgebraicGeometry.Scheme} (f : X ⟶ S) (g : Y ⟶ S) [AlgebraicGeometry.IsAffine S] (H : Q g)
Goal: P (CategoryTheory.Limits.pullback.fst f g)
-- AlgebraicGeometry.HasAffineProperty.stableUnderBaseChange
theorem stableUnderBaseChange (hP' : Q.StableUnderBaseChange) :
    P.StableUnderBaseChange 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} {Q : AlgebraicGeometry.AffineTargetMorphismProperty} [AlgebraicGeometry.HasAffineProperty P Q] (hP' : Q.StableUnderBaseChange)
Goal: P.StableUnderBaseChange