ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AlgebraicGeometry.HasAffineProperty.isLocalAtSource
lemma isLocalAtSource
    (H : ∀ {X Y : Scheme.{u}} (f : X ⟶ Y) [IsAffine Y] (𝒰 : Scheme.OpenCover.{u} X),
        Q f ↔ ∀ i, Q (𝒰.map i ≫ f)) : IsLocalAtSource P 
Arguments: {P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme} {Q : AlgebraicGeometry.AffineTargetMorphismProperty} [AlgebraicGeometry.HasAffineProperty P Q] (H :
  ∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [inst : AlgebraicGeometry.IsAffine Y] (𝒰 : X.OpenCover),
    Q f ↔ ∀ (i : 𝒰.J), Q (𝒰.map i ≫ f))
Goal: AlgebraicGeometry.IsLocalAtSource P
-- AlgebraicGeometry.IsAffineHom.isAffine_preimage
@[mk_iff]
class IsAffineHom {X Y : Scheme} (f : X ⟶ Y) : Prop where
  isAffine_preimage : ∀ U : Y.Opens, IsAffineOpen U → IsAffineOpen (f ⁻¹ᵁ U)
Arguments: {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} [self : AlgebraicGeometry.IsAffineHom f] (U : Y.Opens)
Goal: AlgebraicGeometry.IsAffineOpen U → AlgebraicGeometry.IsAffineOpen ((TopologicalSpace.Opens.map f.val.base).obj U)
-- AlgebraicGeometry.isAffineHom_iff
@[mk_iff]
class IsAffineHom {X Y : Scheme} (f : X ⟶ Y) : Prop where
  isAffine_preimage : ∀ U : Y.Opens, IsAffineOpen U → IsAffineOpen (f ⁻¹ᵁ U)
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y)
Goal: AlgebraicGeometry.IsAffineHom f ↔
  ∀ (U : Y.Opens),
    AlgebraicGeometry.IsAffineOpen U → AlgebraicGeometry.IsAffineOpen ((TopologicalSpace.Opens.map f.val.base).obj U)
-- AlgebraicGeometry.isAffineOpen.preimage
lemma isAffineOpen.preimage {X Y : Scheme} {U : Y.Opens} (hU : IsAffineOpen U)
    (f : X ⟶ Y) [IsAffineHom f] :
    IsAffineOpen (f ⁻¹ᵁ U) 
Arguments: {X Y : AlgebraicGeometry.Scheme} {U : Y.Opens} (hU : AlgebraicGeometry.IsAffineOpen U) (f : X ⟶ Y) [AlgebraicGeometry.IsAffineHom f]
Goal: AlgebraicGeometry.IsAffineOpen ((TopologicalSpace.Opens.map f.val.base).obj U)
-- AlgebraicGeometry.affinePreimage_coe
@[simps]
def affinePreimage {X Y : Scheme} (f : X ⟶ Y) [IsAffineHom f] (U : Y.affineOpens) :
    X.affineOpens 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsAffineHom f] (U : ↑Y.affineOpens)
Goal: ↑(AlgebraicGeometry.affinePreimage f U) = (TopologicalSpace.Opens.map f.val.base).obj ↑U
-- AlgebraicGeometry.isAffineOpen_of_isAffineOpen_basicOpen_aux
lemma isAffineOpen_of_isAffineOpen_basicOpen_aux (s : Set Γ(X, ⊤))
    (hs : Ideal.span s = ⊤) (hs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)) :
    QuasiSeparatedSpace X 
Arguments: {X : AlgebraicGeometry.Scheme} (s : Set ↑(X.presheaf.obj (Opposite.op ⊤))) (hs : Ideal.span s = ⊤) (hs₂ : ∀ i ∈ s, AlgebraicGeometry.IsAffineOpen (X.basicOpen i))
Goal: QuasiSeparatedSpace ↑↑X.toPresheafedSpace
-- AlgebraicGeometry.isAffine_of_isAffineOpen_basicOpen
lemma isAffine_of_isAffineOpen_basicOpen (s : Set Γ(X, ⊤))
    (hs : Ideal.span s = ⊤) (hs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)) :
    IsAffine X 
Arguments: {X : AlgebraicGeometry.Scheme} (s : Set ↑(X.presheaf.obj (Opposite.op ⊤))) (hs : Ideal.span s = ⊤) (hs₂ : ∀ i ∈ s, AlgebraicGeometry.IsAffineOpen (X.basicOpen i))
Goal: AlgebraicGeometry.IsAffine X
-- AlgebraicGeometry.isAffineOpen_of_isAffineOpen_basicOpen
lemma isAffineOpen_of_isAffineOpen_basicOpen (U) (s : Set Γ(X, U))
    (hs : Ideal.span s = ⊤) (hs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)) :
    IsAffineOpen U 
Arguments: {X : AlgebraicGeometry.Scheme} (U : X.Opens) (s : Set ↑(X.presheaf.obj (Opposite.op U))) (hs : Ideal.span s = ⊤) (hs₂ : ∀ i ∈ s, AlgebraicGeometry.IsAffineOpen (X.basicOpen i))
Goal: AlgebraicGeometry.IsAffineOpen U
-- AlgebraicGeometry.isAffineHom_stableUnderBaseChange
lemma isAffineHom_stableUnderBaseChange :
    MorphismProperty.StableUnderBaseChange @IsAffineHom 
Goal: CategoryTheory.MorphismProperty.StableUnderBaseChange @AlgebraicGeometry.IsAffineHom
-- AlgebraicGeometry.isAffine_of_isAffineHom
lemma isAffine_of_isAffineHom [IsAffineHom f] [IsAffine Y] : IsAffine X 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsAffineHom f] [AlgebraicGeometry.IsAffine Y]
Goal: AlgebraicGeometry.IsAffine X
-- AlgebraicGeometry.HasAffineProperty.diagonal_of_openCover
theorem HasAffineProperty.diagonal_of_openCover (P) {Q} [HasAffineProperty P Q]
    {X Y : Scheme.{u}} (f : X ⟶ Y) (𝒰 : Scheme.OpenCover.{u} Y) [∀ i, IsAffine (𝒰.obj i)]
    (𝒰' : ∀ i, Scheme.OpenCover.{u} (pullback f (𝒰.map i))) [∀ i j, IsAffine ((𝒰' i).obj j)]
    (h𝒰' : ∀ i j k,
      Q (pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i))) :
    P.diagonal 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)] (𝒰' : (i : 𝒰.J) → (CategoryTheory.Limits.pullback f (𝒰.map i)).OpenCover) [∀ (i : 𝒰.J) (j : (𝒰' i).J), AlgebraicGeometry.IsAffine ((𝒰' i).obj j)] (h𝒰' :
  ∀ (i : 𝒰.J) (j k : (𝒰' i).J),
    Q (CategoryTheory.Limits.pullback.mapDesc ((𝒰' i).map j) ((𝒰' i).map k) (𝒰.pullbackHom f i)))
Goal: P.diagonal f
-- AlgebraicGeometry.HasAffineProperty.diagonal_of_openCover_diagonal
theorem HasAffineProperty.diagonal_of_openCover_diagonal
    (P) {Q} [HasAffineProperty P Q]
    {X Y : Scheme.{u}} (f : X ⟶ Y) (𝒰 : Scheme.OpenCover.{u} Y) [∀ i, IsAffine (𝒰.obj i)]
    (h𝒰 : ∀ i, Q.diagonal (𝒰.pullbackHom f i)) :
    P.diagonal 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.diagonal (𝒰.pullbackHom f i))
Goal: P.diagonal f
-- AlgebraicGeometry.HasAffineProperty.diagonal_of_diagonal_of_isPullback
theorem HasAffineProperty.diagonal_of_diagonal_of_isPullback
    (P) {Q} [HasAffineProperty P Q]
    {X Y U V : Scheme.{u}} {f : X ⟶ Y} {g : U ⟶ Y}
    [IsAffine U] [IsOpenImmersion g]
    {iV : V ⟶ X} {f' : V ⟶ U} (h : IsPullback iV f' f g) (H : P.diagonal f) :
    Q.diagonal f' 
Arguments: (P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme) {Q : AlgebraicGeometry.AffineTargetMorphismProperty} [AlgebraicGeometry.HasAffineProperty P Q] {X Y U V : AlgebraicGeometry.Scheme} {f : X ⟶ Y} {g : U ⟶ Y} [AlgebraicGeometry.IsAffine U] [AlgebraicGeometry.IsOpenImmersion g] {iV : V ⟶ X} {f' : V ⟶ U} (h : CategoryTheory.IsPullback iV f' f g) (H : P.diagonal f)
Goal: Q.diagonal f'
-- AlgebraicGeometry.HasAffineProperty.diagonal_iff
theorem HasAffineProperty.diagonal_iff
    (P) {Q} [HasAffineProperty P Q] {X Y} {f : X ⟶ Y} [IsAffine Y] :
    Q.diagonal f ↔ P.diagonal 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: Q.diagonal f ↔ P.diagonal f
-- AlgebraicGeometry.universally_isLocalAtTarget
theorem universally_isLocalAtTarget (P : MorphismProperty Scheme)
    (hP₂ : ∀ {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.universally 
Arguments: (P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme) (hP₂ :
  ∀ {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.universally
-- AlgebraicGeometry.topologically_isStableUnderComposition
lemma topologically_isStableUnderComposition
    (hP : ∀ {α β γ : Type u} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
      (f : α → β) (g : β → γ) (_ : P f) (_ : P g), P (g ∘ f)) :
    (topologically P).IsStableUnderComposition 
Arguments: (P : {α β : Type u} → [inst : TopologicalSpace α] → [inst : TopologicalSpace β] → (α → β) → Prop) (hP :
  ∀ {α β γ : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] [inst_2 : TopologicalSpace γ] (f : α → β)
    (g : β → γ), P f → P g → P (g ∘ f))
Goal: (AlgebraicGeometry.topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P).IsStableUnderComposition
-- AlgebraicGeometry.topologically_iso_le
lemma topologically_iso_le
    (hP : ∀ {α β : Type u} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β), P f) :
    MorphismProperty.isomorphisms Scheme ≤ (topologically P) 
Arguments: (P : {α β : Type u} → [inst : TopologicalSpace α] → [inst : TopologicalSpace β] → (α → β) → Prop) (hP : ∀ {α β : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α ≃ₜ β), P ⇑f)
Goal: CategoryTheory.MorphismProperty.isomorphisms AlgebraicGeometry.Scheme ≤
  AlgebraicGeometry.topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P
-- AlgebraicGeometry.topologically_respectsIso
lemma topologically_respectsIso
    (hP₁ : ∀ {α β : Type u} [TopologicalSpace α] [TopologicalSpace β] (f : α ≃ₜ β), P f)
    (hP₂ : ∀ {α β γ : Type u} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
      (f : α → β) (g : β → γ) (_ : P f) (_ : P g), P (g ∘ f)) :
      (topologically P).RespectsIso 
Arguments: (P : {α β : Type u} → [inst : TopologicalSpace α] → [inst : TopologicalSpace β] → (α → β) → Prop) (hP₁ : ∀ {α β : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α ≃ₜ β), P ⇑f) (hP₂ :
  ∀ {α β γ : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] [inst_2 : TopologicalSpace γ] (f : α → β)
    (g : β → γ), P f → P g → P (g ∘ f))
Goal: (AlgebraicGeometry.topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P).RespectsIso
-- AlgebraicGeometry.topologically_isLocalAtTarget
lemma topologically_isLocalAtTarget
    [(topologically P).RespectsIso]
    (hP₂ : ∀ {α β : Type u} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) (s : Set β),
      P f → P (s.restrictPreimage f))
    (hP₃ : ∀ {α β : Type u} [TopologicalSpace α] [TopologicalSpace β] (f : α → β) {ι : Type u}
      (U : ι → TopologicalSpace.Opens β) (_ : iSup U = ⊤) (_ : Continuous f),
      (∀ i, P ((U i).carrier.restrictPreimage f)) → P f) :
    IsLocalAtTarget (topologically P) 
Arguments: (P : {α β : Type u} → [inst : TopologicalSpace α] → [inst : TopologicalSpace β] → (α → β) → Prop) [(AlgebraicGeometry.topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P).RespectsIso] (hP₂ :
  ∀ {α β : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α → β) (s : Set β),
    P f → P (s.restrictPreimage f)) (hP₃ :
  ∀ {α β : Type u} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α → β) {ι : Type u}
    (U : ι → TopologicalSpace.Opens β),
    iSup U = ⊤ → Continuous f → (∀ (i : ι), P ((U i).carrier.restrictPreimage f)) → P f)
Goal: AlgebraicGeometry.IsLocalAtTarget
  (AlgebraicGeometry.topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] => P)
-- AlgebraicGeometry.stalkwise_respectsIso
lemma stalkwise_respectsIso (hP : RingHom.RespectsIso P) :
    (stalkwise P).RespectsIso 
Arguments: {P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop} (hP : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => P)
Goal: (AlgebraicGeometry.stalkwise fun {R S} [CommRing R] [CommRing S] => P).RespectsIso
-- AlgebraicGeometry.stalkwiseIsLocalAtTarget_of_respectsIso
lemma stalkwiseIsLocalAtTarget_of_respectsIso (hP : RingHom.RespectsIso P) :
    IsLocalAtTarget (stalkwise P) 
Arguments: {P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop} (hP : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] => P)
Goal: AlgebraicGeometry.IsLocalAtTarget (AlgebraicGeometry.stalkwise fun {R S} [CommRing R] [CommRing S] => P)
-- AlgebraicGeometry.AffineTargetMorphismProperty.stableUnderBaseChange_of_stableUnderBaseChangeOnAffine_of_isLocalAtTarget
lemma stableUnderBaseChange_of_stableUnderBaseChangeOnAffine_of_isLocalAtTarget
    (P : MorphismProperty Scheme) [IsLocalAtTarget P]
    (hP₂ : (of P).StableUnderBaseChange) :
    P.StableUnderBaseChange 
Arguments: (P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsLocalAtTarget P] (hP₂ : (AlgebraicGeometry.AffineTargetMorphismProperty.of P).StableUnderBaseChange)
Goal: P.StableUnderBaseChange
-- AlgebraicGeometry.IsReduced.component_reduced
class IsReduced : Prop where
  component_reduced : ∀ U, IsReduced Γ(X, U) := by infer_instance
Arguments: {X : AlgebraicGeometry.Scheme} [self : AlgebraicGeometry.IsReduced X] (U : X.Opens)
Goal: IsReduced ↑(X.presheaf.obj (Opposite.op U))
-- AlgebraicGeometry.isReduced_of_isReduced_stalk
theorem isReduced_of_isReduced_stalk [∀ x : X, _root_.IsReduced (X.presheaf.stalk x)] :
    IsReduced X 
Arguments: (X : AlgebraicGeometry.Scheme) [∀ (x : ↑↑X.toPresheafedSpace), IsReduced ↑(X.presheaf.stalk x)]
Goal: AlgebraicGeometry.IsReduced X
-- AlgebraicGeometry.isReduced_of_isOpenImmersion
theorem isReduced_of_isOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [H : IsOpenImmersion f]
    [IsReduced Y] : IsReduced X 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [H : AlgebraicGeometry.IsOpenImmersion f] [AlgebraicGeometry.IsReduced Y]
Goal: AlgebraicGeometry.IsReduced X
-- AlgebraicGeometry.affine_isReduced_iff
theorem affine_isReduced_iff (R : CommRingCat) :
    IsReduced (Spec R) ↔ _root_.IsReduced R 
Arguments: (R : CommRingCat)
Goal: AlgebraicGeometry.IsReduced (AlgebraicGeometry.Spec R) ↔ IsReduced ↑R
-- AlgebraicGeometry.isReduced_of_isAffine_isReduced
theorem isReduced_of_isAffine_isReduced [IsAffine X] [_root_.IsReduced Γ(X, ⊤)] :
    IsReduced X 
Arguments: (X : AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsAffine X] [IsReduced ↑(X.presheaf.obj (Opposite.op ⊤))]
Goal: AlgebraicGeometry.IsReduced X
-- AlgebraicGeometry.reduce_to_affine_global
@[elab_as_elim]
theorem reduce_to_affine_global (P : ∀ {X : Scheme} (_ : X.Opens), Prop)
    {X : Scheme} (U : X.Opens)
    (h₁ : ∀ (X : Scheme) (U : X.Opens),
      (∀ x : U, ∃ (V : _) (_ : x.1 ∈ V) (_ : V ⟶ U), P V) → P U)
    (h₂ : ∀ (X Y) (f : X ⟶ Y) [hf : IsOpenImmersion f],
      ∃ (U : Set X) (V : Set Y) (hU : U = ⊤) (hV : V = Set.range f.1.base),
        P ⟨U, hU.symm ▸ isOpen_univ⟩ → P ⟨V, hV.symm ▸ hf.base_open.isOpen_range⟩)
    (h₃ : ∀ R : CommRingCat, P (X := Spec R) ⊤) : P U 
Arguments: (P : {X : AlgebraicGeometry.Scheme} → X.Opens → Prop) {X : AlgebraicGeometry.Scheme} (U : X.Opens) (h₁ : ∀ (X : AlgebraicGeometry.Scheme) (U : X.Opens), (∀ (x : ↥U), ∃ V, ∃ (_ : ↑x ∈ V), ∃ x, P V) → P U) (h₂ :
  ∀ (X Y : AlgebraicGeometry.Scheme) (f : X ⟶ Y) [hf : AlgebraicGeometry.IsOpenImmersion f],
    ∃ U V,
      ∃ (hU : U = ⊤) (hV : V = Set.range ⇑f.val.base),
        P { carrier := U, is_open' := ⋯ } → P { carrier := V, is_open' := ⋯ }) (h₃ : ∀ (R : CommRingCat), P ⊤)
Goal: P U
-- AlgebraicGeometry.reduce_to_affine_nbhd
theorem reduce_to_affine_nbhd (P : ∀ (X : Scheme) (_ : X), Prop)
    (h₁ : ∀ R x, P (Spec R) x)
    (h₂ : ∀ {X Y} (f : X ⟶ Y) [IsOpenImmersion f] (x : X), P X x → P Y (f.1.base x)) :
    ∀ (X : Scheme) (x : X), P X x 
Arguments: (P : (X : AlgebraicGeometry.Scheme) → ↑↑X.toPresheafedSpace → Prop) (h₁ : ∀ (R : CommRingCat) (x : ↑↑(AlgebraicGeometry.Spec R).toPresheafedSpace), P (AlgebraicGeometry.Spec R) x) (h₂ :
  ∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [inst : AlgebraicGeometry.IsOpenImmersion f]
    (x : ↑↑X.toPresheafedSpace), P X x → P Y (f.val.base x)) (X : AlgebraicGeometry.Scheme) (x : ↑↑X.toPresheafedSpace)
Goal: P X x
-- AlgebraicGeometry.eq_zero_of_basicOpen_eq_bot
theorem eq_zero_of_basicOpen_eq_bot {X : Scheme} [hX : IsReduced X] {U : X.Opens}
    (s : Γ(X, U)) (hs : X.basicOpen s = ⊥) : s = 0 
Arguments: {X : AlgebraicGeometry.Scheme} [hX : AlgebraicGeometry.IsReduced X] {U : X.Opens} (s : ↑(X.presheaf.obj (Opposite.op U))) (hs : X.basicOpen s = ⊥)
Goal: s = 0
-- AlgebraicGeometry.basicOpen_eq_bot_iff
@[simp]
theorem basicOpen_eq_bot_iff {X : Scheme} [IsReduced X] {U : X.Opens}
    (s : Γ(X, U)) : X.basicOpen s = ⊥ ↔ s = 0 
Arguments: {X : AlgebraicGeometry.Scheme} [AlgebraicGeometry.IsReduced X] {U : X.Opens} (s : ↑(X.presheaf.obj (Opposite.op U)))
Goal: X.basicOpen s = ⊥ ↔ s = 0
-- AlgebraicGeometry.IsIntegral.nonempty
class IsIntegral : Prop where
  nonempty : Nonempty X := by infer_instance
  component_integral : ∀ (U : X.Opens) [Nonempty U], IsDomain Γ(X, U) := by infer_instance
Arguments: {X : AlgebraicGeometry.Scheme} [self : AlgebraicGeometry.IsIntegral X]
Goal: Nonempty ↑↑X.toPresheafedSpace
-- AlgebraicGeometry.IsIntegral.component_integral
class IsIntegral : Prop where
  nonempty : Nonempty X := by infer_instance
  component_integral : ∀ (U : X.Opens) [Nonempty U], IsDomain Γ(X, U) := by infer_instance
Arguments: {X : AlgebraicGeometry.Scheme} [self : AlgebraicGeometry.IsIntegral X] (U : X.Opens) [Nonempty ↑↑(↑U).toPresheafedSpace]
Goal: IsDomain ↑(X.presheaf.obj (Opposite.op U))
-- AlgebraicGeometry.isIntegral_of_irreducibleSpace_of_isReduced
theorem isIntegral_of_irreducibleSpace_of_isReduced [IsReduced X] [H : IrreducibleSpace X] :
    IsIntegral X 
Arguments: (X : AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsReduced X] [H : IrreducibleSpace ↑↑X.toPresheafedSpace]
Goal: AlgebraicGeometry.IsIntegral X
-- AlgebraicGeometry.isIntegral_iff_irreducibleSpace_and_isReduced
theorem isIntegral_iff_irreducibleSpace_and_isReduced :
    IsIntegral X ↔ IrreducibleSpace X ∧ IsReduced X 
Arguments: (X : AlgebraicGeometry.Scheme)
Goal: AlgebraicGeometry.IsIntegral X ↔ IrreducibleSpace ↑↑X.toPresheafedSpace ∧ AlgebraicGeometry.IsReduced X
-- AlgebraicGeometry.isIntegral_of_isOpenImmersion
theorem isIntegral_of_isOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [H : IsOpenImmersion f]
    [IsIntegral Y] [Nonempty X] : IsIntegral X 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [H : AlgebraicGeometry.IsOpenImmersion f] [AlgebraicGeometry.IsIntegral Y] [Nonempty ↑↑X.toPresheafedSpace]
Goal: AlgebraicGeometry.IsIntegral X
-- AlgebraicGeometry.affine_isIntegral_iff
theorem affine_isIntegral_iff (R : CommRingCat) :
    IsIntegral (Spec R) ↔ IsDomain R 
Arguments: (R : CommRingCat)
Goal: AlgebraicGeometry.IsIntegral (AlgebraicGeometry.Spec R) ↔ IsDomain ↑R
-- AlgebraicGeometry.isIntegral_of_isAffine_of_isDomain
theorem isIntegral_of_isAffine_of_isDomain [IsAffine X] [Nonempty X] [IsDomain Γ(X, ⊤)] :
    IsIntegral X 
Arguments: (X : AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsAffine X] [Nonempty ↑↑X.toPresheafedSpace] [IsDomain ↑(X.presheaf.obj (Opposite.op ⊤))]
Goal: AlgebraicGeometry.IsIntegral X
-- AlgebraicGeometry.map_injective_of_isIntegral
theorem map_injective_of_isIntegral [IsIntegral X] {U V : X.Opens} (i : U ⟶ V)
    [H : Nonempty U] : Function.Injective (X.presheaf.map i.op) 
Arguments: (X : AlgebraicGeometry.Scheme) [AlgebraicGeometry.IsIntegral X] {U V : X.Opens} (i : U ⟶ V) [H : Nonempty ↑↑(↑U).toPresheafedSpace]
Goal: Function.Injective ⇑(X.presheaf.map i.op)
-- AlgebraicGeometry.Scheme.local_affine
structure Scheme extends LocallyRingedSpace where
  local_affine :
    ∀ x : toLocallyRingedSpace,
      ∃ (U : OpenNhds x) (R : CommRingCat),
        Nonempty
          (toLocallyRingedSpace.restrict U.openEmbedding ≅ Spec.toLocallyRingedSpace.obj (op R))
Arguments: (self : AlgebraicGeometry.Scheme) (x : ↑self.toTopCat)
Goal: ∃ U R, Nonempty (self.restrict ⋯ ≅ AlgebraicGeometry.Spec.toLocallyRingedSpace.obj (Opposite.op R))
-- AlgebraicGeometry.Scheme.Hom.continuous
@[continuity, fun_prop]
lemma Hom.continuous {X Y : Scheme} (f : X ⟶ Y) : Continuous f.1.base 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y)
Goal: Continuous ⇑f.val.base
-- AlgebraicGeometry.Scheme.Hom.naturality
@[reassoc]
lemma naturality (i : op U' ⟶ op U) :
    Y.presheaf.map i ≫ f.app U = f.app U' ≫ X.presheaf.map ((Opens.map f.1.base).map i.unop).op 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) {U U' : Y.Opens} (i : Opposite.op U' ⟶ Opposite.op U)
Goal: Y.presheaf.map i ≫ f.app U = f.app U' ≫ X.presheaf.map ((TopologicalSpace.Opens.map f.val.base).map i.unop).op
-- AlgebraicGeometry.Scheme.Hom.appLE_map
@[reassoc (attr := simp)]
lemma appLE_map (e : V ≤ f ⁻¹ᵁ U) (i : op V ⟶ op V') :
    f.appLE U V e ≫ X.presheaf.map i = f.appLE U V' (i.unop.le.trans e) 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) {U : Y.Opens} {V V' : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.val.base).obj U) (i : Opposite.op V ⟶ Opposite.op V')
Goal: f.appLE U V e ≫ X.presheaf.map i = f.appLE U V' ⋯
-- AlgebraicGeometry.Scheme.Hom.appLE_map'
@[reassoc]
lemma appLE_map' (e : V ≤ f ⁻¹ᵁ U) (i : V = V') :
    f.appLE U V' (i ▸ e) ≫ X.presheaf.map (eqToHom i).op = f.appLE U V e 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) {U : Y.Opens} {V V' : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.val.base).obj U) (i : V = V')
Goal: f.appLE U V' ⋯ ≫ X.presheaf.map (CategoryTheory.eqToHom i).op = f.appLE U V e
-- AlgebraicGeometry.Scheme.Hom.map_appLE
@[reassoc (attr := simp)]
lemma map_appLE (e : V ≤ f ⁻¹ᵁ U) (i : op U' ⟶ op U) :
    Y.presheaf.map i ≫ f.appLE U V e =
      f.appLE U' V (e.trans ((Opens.map f.1.base).map i.unop).le) 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) {U U' : Y.Opens} {V : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.val.base).obj U) (i : Opposite.op U' ⟶ Opposite.op U)
Goal: Y.presheaf.map i ≫ f.appLE U V e = f.appLE U' V ⋯
-- AlgebraicGeometry.Scheme.Hom.map_appLE'
@[reassoc]
lemma map_appLE' (e : V ≤ f ⁻¹ᵁ U) (i : U' = U) :
    Y.presheaf.map (eqToHom i).op ≫ f.appLE U' V (i ▸ e) = f.appLE U V e 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) {U U' : Y.Opens} {V : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.val.base).obj U) (i : U' = U)
Goal: Y.presheaf.map (CategoryTheory.eqToHom i).op ≫ f.appLE U' V ⋯ = f.appLE U V e
-- AlgebraicGeometry.Scheme.Hom.app_eq_appLE
lemma app_eq_appLE {U : Y.Opens} :
    f.app U = f.appLE U _ le_rfl 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) {U : Y.Opens}
Goal: f.app U = f.appLE U ((TopologicalSpace.Opens.map f.val.base).obj U) ⋯
-- AlgebraicGeometry.Scheme.Hom.appLE_eq_app
lemma appLE_eq_app {U : Y.Opens} :
    f.appLE U (f ⁻¹ᵁ U) le_rfl = f.app U 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) {U : Y.Opens}
Goal: f.appLE U ((TopologicalSpace.Opens.map f.val.base).obj U) ⋯ = f.app U
-- AlgebraicGeometry.Scheme.Hom.appLE_congr
lemma appLE_congr (e : V ≤ f ⁻¹ᵁ U) (e₁ : U = U') (e₂ : V = V')
    (P : ∀ {R S : Type u} [CommRing R] [CommRing S] (_ : R →+* S), Prop) :
    P (f.appLE U V e) ↔ P (f.appLE U' V' (e₁ ▸ e₂ ▸ e)) 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) {U U' : Y.Opens} {V V' : X.Opens} (e : V ≤ (TopologicalSpace.Opens.map f.val.base).obj U) (e₁ : U = U') (e₂ : V = V') (P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop)
Goal: P (f.appLE U V e) ↔ P (f.appLE U' V' ⋯)
-- AlgebraicGeometry.Scheme.Hom.ext
@[ext]
protected lemma ext {f g : X ⟶ Y} (h_base : f.1.base = g.1.base)
    (h_app : ∀ U, f.app U ≫ X.presheaf.map
      (eqToHom congr((Opens.map $h_base.symm).obj U)).op = g.app U) : f = g 
Arguments: {X Y : AlgebraicGeometry.Scheme} {f g : X ⟶ Y} (h_base : f.val.base = g.val.base) (h_app :
  ∀ (U : Y.Opens),
    AlgebraicGeometry.Scheme.Hom.app f U ≫ X.presheaf.map (CategoryTheory.eqToHom ⋯).op =
      AlgebraicGeometry.Scheme.Hom.app g U)
Goal: f = g
-- AlgebraicGeometry.Scheme.Hom.preimage_iSup
lemma preimage_iSup {ι} (U : ι → Opens Y) : f ⁻¹ᵁ iSup U = ⨆ i, f ⁻¹ᵁ U i 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) {ι : Sort u_1} (U : ι → Y.Opens)
Goal: (TopologicalSpace.Opens.map f.val.base).obj (iSup U) = ⨆ i, (TopologicalSpace.Opens.map f.val.base).obj (U i)
-- AlgebraicGeometry.Scheme.Hom.preimage_iSup_eq_top
lemma preimage_iSup_eq_top {ι} {U : ι → Opens Y} (hU : iSup U = ⊤) :
    ⨆ i, f ⁻¹ᵁ U i = ⊤ 
Arguments: {X Y : AlgebraicGeometry.Scheme} (f : X.Hom Y) {ι : Sort u_1} {U : ι → Y.Opens} (hU : iSup U = ⊤)
Goal: ⨆ i, (TopologicalSpace.Opens.map f.val.base).obj (U i) = ⊤
-- AlgebraicGeometry.Scheme.preimage_comp
@[simp]
lemma preimage_comp {X Y Z : Scheme.{u}} (f : X ⟶ Y) (g : Y ⟶ Z) (U) :
    (f ≫ g) ⁻¹ᵁ U = f ⁻¹ᵁ g ⁻¹ᵁ U 
Arguments: {X Y Z : AlgebraicGeometry.Scheme} (f : X ⟶ Y) (g : Y ⟶ Z) (U : Z.Opens)
Goal: (TopologicalSpace.Opens.map (f ≫ g).val.base).obj U =
  (TopologicalSpace.Opens.map f.val.base).obj ((TopologicalSpace.Opens.map g.val.base).obj U)