mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.Morphisms.Constructors

Goal state before:
pp: case restrict
P : {α β : Type u} → [TopologicalSpace α] → [TopologicalSpace β] → (α → β) → Prop
inst✝ : (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),
    Continuous f → IsOpen s → 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 → Opens β),
    IsOpenCover U → Continuous f → (∀ (i : ι), P ((U i).carrier.restrictPreimage f)) → P f
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens),
    topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) f →
      topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) (f ∣_ U)

pp: case of_sSup_eq_top
P : {α β : Type u} → [TopologicalSpace α] → [TopologicalSpace β] → (α → β) → Prop
inst✝ : (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),
    Continuous f → IsOpen s → 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 → Opens β),
    IsOpenCover U → Continuous f → (∀ (i : ι), P ((U i).carrier.restrictPreimage f)) → P f
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens),
    iSup U = ⊤ →
      (∀ (i : ι), topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) (f ∣_ U i)) →
        topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) f

Tactic:
·

Goal state after:
case restrict
P : {α β : Type u} → [TopologicalSpace α] → [TopologicalSpace β] → (α → β) → Prop
inst✝ : (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),
    Continuous f → IsOpen s → 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 → Opens β),
    IsOpenCover U → Continuous f → (∀ (i : ι), P ((U i).carrier.restrictPreimage f)) → P f
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens),
    topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) f →
      topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) (f ∣_ U)
Module: Mathlib.AlgebraicGeometry.Morphisms.Constructors

Goal state before:
pp: case of_sSup_eq_top
P : {α β : Type u} → [TopologicalSpace α] → [TopologicalSpace β] → (α → β) → Prop
inst✝ : (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),
    Continuous f → IsOpen s → 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 → Opens β),
    IsOpenCover U → Continuous f → (∀ (i : ι), P ((U i).carrier.restrictPreimage f)) → P f
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens),
    iSup U = ⊤ →
      (∀ (i : ι), topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) (f ∣_ U i)) →
        topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) f

Tactic:
·

Goal state after:
case of_sSup_eq_top
P : {α β : Type u} → [TopologicalSpace α] → [TopologicalSpace β] → (α → β) → Prop
inst✝ : (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),
    Continuous f → IsOpen s → 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 → Opens β),
    IsOpenCover U → Continuous f → (∀ (i : ι), P ((U i).carrier.restrictPreimage f)) → P f
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens),
    iSup U = ⊤ →
      (∀ (i : ι), topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) (f ∣_ U i)) →
        topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) f
Module: Mathlib.RingTheory.RingHom.Locally

Goal state before:
pp: case refine_3
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPl : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P
R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
s : Set S
hsone : Ideal.span s = ⊤
hs : ∀ t ∈ s, (fun {R S} [CommRing R] [CommRing S] ↦ P) ((algebraMap S (Localization.Away t)).comp f)
rₐ : (a : ↑s) → Localization.Away ↑a := fun a ↦ (algebraMap S (Localization.Away ↑a)) (f r)
Sₐ : ↑s → Type u := fun a ↦ Localization.Away (rₐ a)
this✝² : ∀ (a : ↑s), IsLocalization.Away (((algebraMap S (Localization.Away ↑a)).comp f) r) (Sₐ a)
this✝¹ :
  ∀ (a : ↑s),
    IsLocalization (Algebra.algebraMapSubmonoid (Localization.Away ↑a) (Submonoid.map f (Submonoid.powers r))) (Sₐ a)
H : ∀ (a : ↑s), Submonoid.powers (f r) ≤ Submonoid.comap (algebraMap S (Localization.Away ↑a)) (Submonoid.powers (rₐ a))
this✝ : (a : ↑s) → Algebra S' (Sₐ a) :=
  fun a ↦ (IsLocalization.map (Sₐ a) (algebraMap S (Localization.Away ↑a)) ⋯).toAlgebra
this : ∀ (a : ↑s), IsScalarTower S S' (Sₐ a)
a : ↑s
⊢ P ((algebraMap S' (Sₐ a)).comp (IsLocalization.Away.map R' S' f r))

Tactic:
·

Goal state after:
case refine_3
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPl : LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P
R S : Type u
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
f : R →+* S
r : R
R' S' : Type u
inst✝⁵ : CommRing R'
inst✝⁴ : CommRing S'
inst✝³ : Algebra R R'
inst✝² : Algebra S S'
inst✝¹ : IsLocalization.Away r R'
inst✝ : IsLocalization.Away (f r) S'
s : Set S
hsone : Ideal.span s = ⊤
hs : ∀ t ∈ s, (fun {R S} [CommRing R] [CommRing S] ↦ P) ((algebraMap S (Localization.Away t)).comp f)
rₐ : (a : ↑s) → Localization.Away ↑a := fun a ↦ (algebraMap S (Localization.Away ↑a)) (f r)
Sₐ : ↑s → Type u := fun a ↦ Localization.Away (rₐ a)
this✝² : ∀ (a : ↑s), IsLocalization.Away (((algebraMap S (Localization.Away ↑a)).comp f) r) (Sₐ a)
this✝¹ :
  ∀ (a : ↑s),
    IsLocalization (Algebra.algebraMapSubmonoid (Localization.Away ↑a) (Submonoid.map f (Submonoid.powers r))) (Sₐ a)
H : ∀ (a : ↑s), Submonoid.powers (f r) ≤ Submonoid.comap (algebraMap S (Localization.Away ↑a)) (Submonoid.powers (rₐ a))
this✝ : (a : ↑s) → Algebra S' (Sₐ a) :=
  fun a ↦ (IsLocalization.map (Sₐ a) (algebraMap S (Localization.Away ↑a)) ⋯).toAlgebra
this : ∀ (a : ↑s), IsScalarTower S S' (Sₐ a)
a : ↑s
⊢ P ((algebraMap S' (Sₐ a)).comp (IsLocalization.Away.map R' S' f r))
Module: Mathlib.AlgebraicGeometry.Morphisms.Constructors

Goal state before:
pp: case restrict
P : {α β : Type u} → [TopologicalSpace α] → [TopologicalSpace β] → (α → β) → Prop
inst✝ : (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P).RespectsIso
hP₁ :
  ∀ {X Y : Type u} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (f : X → Y),
    Continuous f → ∀ (U : Opens X), P f → P (f ∘ Subtype.val)
hP₂ :
  ∀ {X Y : Type u} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (f : X → Y),
    Continuous f → ∀ {ι : Type u} (U : ι → Opens X), IsOpenCover U → (∀ (i : ι), P (f ∘ Subtype.val)) → P f
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) (U : X.Opens),
    topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) f →
      topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) (U.ι ≫ f)

pp: case of_sSup_eq_top
P : {α β : Type u} → [TopologicalSpace α] → [TopologicalSpace β] → (α → β) → Prop
inst✝ : (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P).RespectsIso
hP₁ :
  ∀ {X Y : Type u} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (f : X → Y),
    Continuous f → ∀ (U : Opens X), P f → P (f ∘ Subtype.val)
hP₂ :
  ∀ {X Y : Type u} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (f : X → Y),
    Continuous f → ∀ {ι : Type u} (U : ι → Opens X), IsOpenCover U → (∀ (i : ι), P (f ∘ Subtype.val)) → P f
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → X.Opens),
    iSup U = ⊤ →
      (∀ (i : ι), topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) ((U i).ι ≫ f)) →
        topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) f

Tactic:
·

Goal state after:
case restrict
P : {α β : Type u} → [TopologicalSpace α] → [TopologicalSpace β] → (α → β) → Prop
inst✝ : (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P).RespectsIso
hP₁ :
  ∀ {X Y : Type u} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (f : X → Y),
    Continuous f → ∀ (U : Opens X), P f → P (f ∘ Subtype.val)
hP₂ :
  ∀ {X Y : Type u} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (f : X → Y),
    Continuous f → ∀ {ι : Type u} (U : ι → Opens X), IsOpenCover U → (∀ (i : ι), P (f ∘ Subtype.val)) → P f
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) (U : X.Opens),
    topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) f →
      topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) (U.ι ≫ f)
Module: Mathlib.AlgebraicGeometry.Morphisms.Constructors

Goal state before:
pp: case of_sSup_eq_top
P : {α β : Type u} → [TopologicalSpace α] → [TopologicalSpace β] → (α → β) → Prop
inst✝ : (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P).RespectsIso
hP₁ :
  ∀ {X Y : Type u} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (f : X → Y),
    Continuous f → ∀ (U : Opens X), P f → P (f ∘ Subtype.val)
hP₂ :
  ∀ {X Y : Type u} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (f : X → Y),
    Continuous f → ∀ {ι : Type u} (U : ι → Opens X), IsOpenCover U → (∀ (i : ι), P (f ∘ Subtype.val)) → P f
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → X.Opens),
    iSup U = ⊤ →
      (∀ (i : ι), topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) ((U i).ι ≫ f)) →
        topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) f

Tactic:
·

Goal state after:
case of_sSup_eq_top
P : {α β : Type u} → [TopologicalSpace α] → [TopologicalSpace β] → (α → β) → Prop
inst✝ : (topologically fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P).RespectsIso
hP₁ :
  ∀ {X Y : Type u} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (f : X → Y),
    Continuous f → ∀ (U : Opens X), P f → P (f ∘ Subtype.val)
hP₂ :
  ∀ {X Y : Type u} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y] (f : X → Y),
    Continuous f → ∀ {ι : Type u} (U : ι → Opens X), IsOpenCover U → (∀ (i : ι), P (f ∘ Subtype.val)) → P f
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → X.Opens),
    iSup U = ⊤ →
      (∀ (i : ι), topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) ((U i).ι ≫ f)) →
        topologically (fun {α β} [TopologicalSpace α] [TopologicalSpace β] ↦ P) f
Module: Mathlib.AlgebraicGeometry.Morphisms.SurjectiveOnStalks

Goal state before:
pp: case h₁
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
⊢ RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ RingHom.SurjectiveOnStalks

pp: case h₂
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
⊢ ∀ ⦃R S T : Type u_1⦄ [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S]
    [inst_4 : Algebra R T],
    (algebraMap R T).SurjectiveOnStalks → (algebraMap S (TensorProduct R S T)).SurjectiveOnStalks

Tactic:
·

Goal state after:
case h₁
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
⊢ RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ RingHom.SurjectiveOnStalks
Module: Mathlib.AlgebraicGeometry.Morphisms.Constructors

Goal state before:
pp: case restrict
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
hP' : (RingHom.toMorphismProperty fun {R S} [CommRing R] [CommRing S] ↦ P).RespectsIso
this : (stalkwise fun {R S} [CommRing R] [CommRing S] ↦ P).RespectsIso := ⋯
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens),
    stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) f → stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) (f ∣_ U)

pp: case of_sSup_eq_top
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
hP' : (RingHom.toMorphismProperty fun {R S} [CommRing R] [CommRing S] ↦ P).RespectsIso
this : (stalkwise fun {R S} [CommRing R] [CommRing S] ↦ P).RespectsIso := ⋯
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens),
    iSup U = ⊤ →
      (∀ (i : ι), stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) (f ∣_ U i)) →
        stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) f

Tactic:
·

Goal state after:
case restrict
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
hP' : (RingHom.toMorphismProperty fun {R S} [CommRing R] [CommRing S] ↦ P).RespectsIso
this : (stalkwise fun {R S} [CommRing R] [CommRing S] ↦ P).RespectsIso := ⋯
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) (U : Y.Opens),
    stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) f → stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) (f ∣_ U)
Module: Mathlib.AlgebraicGeometry.Morphisms.Constructors

Goal state before:
pp: case of_sSup_eq_top
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
hP' : (RingHom.toMorphismProperty fun {R S} [CommRing R] [CommRing S] ↦ P).RespectsIso
this : (stalkwise fun {R S} [CommRing R] [CommRing S] ↦ P).RespectsIso := ⋯
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens),
    iSup U = ⊤ →
      (∀ (i : ι), stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) (f ∣_ U i)) →
        stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) f

Tactic:
·

Goal state after:
case of_sSup_eq_top
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
hP' : (RingHom.toMorphismProperty fun {R S} [CommRing R] [CommRing S] ↦ P).RespectsIso
this : (stalkwise fun {R S} [CommRing R] [CommRing S] ↦ P).RespectsIso := ⋯
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens),
    iSup U = ⊤ →
      (∀ (i : ι), stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) (f ∣_ U i)) →
        stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) f
Module: Mathlib.AlgebraicGeometry.Morphisms.SurjectiveOnStalks

Goal state before:
pp: case h₁
X Y : Scheme
f : X ⟶ Y
inst✝ : SurjectiveOnStalks f
hf : Function.Injective ⇑f
⊢ Function.Injective
    ⇑(ConcreteCategory.hom
        ((Scheme.forgetToLocallyRingedSpace ⋙ LocallyRingedSpace.forgetToSheafedSpace).map f).hom.base)

pp: case h₂
X Y : Scheme
f : X ⟶ Y
inst✝ : SurjectiveOnStalks f
hf : Function.Injective ⇑f
⊢ ∀ (x : ↑↑((Scheme.forgetToLocallyRingedSpace ⋙ LocallyRingedSpace.forgetToSheafedSpace).obj X).toPresheafedSpace),
    Epi
      (PresheafedSpace.Hom.stalkMap
        ((Scheme.forgetToLocallyRingedSpace ⋙ LocallyRingedSpace.forgetToSheafedSpace).map f).hom x)

Tactic:
·

Goal state after:
case h₁
X Y : Scheme
f : X ⟶ Y
inst✝ : SurjectiveOnStalks f
hf : Function.Injective ⇑f
⊢ Function.Injective
    ⇑(ConcreteCategory.hom
        ((Scheme.forgetToLocallyRingedSpace ⋙ LocallyRingedSpace.forgetToSheafedSpace).map f).hom.base)
Module: Mathlib.AlgebraicGeometry.Morphisms.SurjectiveOnStalks

Goal state before:
pp: case h₂
X Y : Scheme
f : X ⟶ Y
inst✝ : SurjectiveOnStalks f
hf : Function.Injective ⇑f
⊢ ∀ (x : ↑↑((Scheme.forgetToLocallyRingedSpace ⋙ LocallyRingedSpace.forgetToSheafedSpace).obj X).toPresheafedSpace),
    Epi
      (PresheafedSpace.Hom.stalkMap
        ((Scheme.forgetToLocallyRingedSpace ⋙ LocallyRingedSpace.forgetToSheafedSpace).map f).hom x)

Tactic:
·

Goal state after:
case h₂
X Y : Scheme
f : X ⟶ Y
inst✝ : SurjectiveOnStalks f
hf : Function.Injective ⇑f
⊢ ∀ (x : ↑↑((Scheme.forgetToLocallyRingedSpace ⋙ LocallyRingedSpace.forgetToSheafedSpace).obj X).toPresheafedSpace),
    Epi
      (PresheafedSpace.Hom.stalkMap
        ((Scheme.forgetToLocallyRingedSpace ⋙ LocallyRingedSpace.forgetToSheafedSpace).map f).hom x)
Module: Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties

Goal state before:
pp: case h₁
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
⊢ ∀ {X Y Z : Scheme} (e : X ≅ Y) (f : Y ⟶ Z) [inst : IsAffine Z],
    sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ P) f →
      sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ P) (e.hom ≫ f)

pp: case h₂
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
⊢ ∀ {X Y Z : Scheme} (e : Y ≅ Z) (f : X ⟶ Y) [inst : IsAffine Y],
    sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ P) f →
      sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ P) (f ≫ e.hom)

Tactic:
·

Goal state after:
case h₁
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
⊢ ∀ {X Y Z : Scheme} (e : X ≅ Y) (f : Y ⟶ Z) [inst : IsAffine Z],
    sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ P) f →
      sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ P) (e.hom ≫ f)
Module: Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties

Goal state before:
pp: case h₂
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
⊢ ∀ {X Y Z : Scheme} (e : Y ≅ Z) (f : X ⟶ Y) [inst : IsAffine Y],
    sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ P) f →
      sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ P) (f ≫ e.hom)

Tactic:
·

Goal state after:
case h₂
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
⊢ ∀ {X Y Z : Scheme} (e : Y ≅ Z) (f : X ⟶ Y) [inst : IsAffine Y],
    sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ P) f →
      sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ P) (f ≫ e.hom)
Module: Mathlib.AlgebraicGeometry.Morphisms.Constructors

Goal state before:
pp: case restrict
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
this : (stalkwise fun {R S} [CommRing R] [CommRing S] ↦ P).RespectsIso := ⋯
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) (U : X.Opens),
    stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) f →
      stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) (U.ι ≫ f)

pp: case of_sSup_eq_top
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
this : (stalkwise fun {R S} [CommRing R] [CommRing S] ↦ P).RespectsIso := ⋯
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → X.Opens),
    iSup U = ⊤ →
      (∀ (i : ι), stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) ((U i).ι ≫ f)) →
        stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) f

Tactic:
·

Goal state after:
case restrict
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
this : (stalkwise fun {R S} [CommRing R] [CommRing S] ↦ P).RespectsIso := ⋯
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) (U : X.Opens),
    stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) f →
      stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) (U.ι ≫ f)
Module: Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties

Goal state before:
pp: P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
X Y : Scheme
f : X ⟶ Y
U : Y.Opens
hU : IsAffineOpen U
⊢ (∀ (U_1 : ↑(↑(f ⁻¹ᵁ U)).affineOpens), P (CommRingCat.Hom.hom (Hom.appLE (f ∣_ U) ⊤ ↑U_1 ⋯))) ↔
    ∀ (V : ↑X.affineOpens) (e : ↑V ≤ f ⁻¹ᵁ U), P (CommRingCat.Hom.hom (Hom.appLE f U (↑V) e))

Tactic:
simp only [morphismRestrict_appLE]

Goal state after:
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
X Y : Scheme
f : X ⟶ Y
U : Y.Opens
hU : IsAffineOpen U
⊢ (∀ (U_1 : ↑(↑(f ⁻¹ᵁ U)).affineOpens), P (CommRingCat.Hom.hom (Hom.appLE f (U.ι ''ᵁ ⊤) ((f ⁻¹ᵁ U).ι ''ᵁ ↑U_1) ⋯))) ↔
    ∀ (V : ↑X.affineOpens) (e : ↑V ≤ f ⁻¹ᵁ U), P (CommRingCat.Hom.hom (Hom.appLE f U (↑V) e))
Module: Mathlib.AlgebraicGeometry.Morphisms.Constructors

Goal state before:
pp: case of_sSup_eq_top
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
this : (stalkwise fun {R S} [CommRing R] [CommRing S] ↦ P).RespectsIso := ⋯
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → X.Opens),
    iSup U = ⊤ →
      (∀ (i : ι), stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) ((U i).ι ≫ f)) →
        stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) f

Tactic:
·

Goal state after:
case of_sSup_eq_top
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
this : (stalkwise fun {R S} [CommRing R] [CommRing S] ↦ P).RespectsIso := ⋯
⊢ ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → X.Opens),
    iSup U = ⊤ →
      (∀ (i : ι), stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) ((U i).ι ≫ f)) →
        stalkwise (fun {R S} [CommRing R] [CommRing S] ↦ P) f
Module: Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties

Goal state before:
pp: case respectsIso
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
h₂ : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P
h₃ : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ P
⊢ (sourceAffineLocally fun {R S} [CommRing R] [CommRing S] ↦ P).toProperty.RespectsIso

pp: case to_basicOpen
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
h₂ : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P
h₃ : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ P
⊢ ∀ {X Y : Scheme} [inst : IsAffine Y] (f : X ⟶ Y) (r : ↑Γ(Y, ⊤)),
    sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ P) f →
      sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ P) (f ∣_ Y.basicOpen r)

pp: case of_basicOpenCover
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
h₂ : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P
h₃ : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ P
⊢ ∀ {X Y : Scheme} [inst : IsAffine Y] (f : X ⟶ Y) (s : Finset ↑Γ(Y, ⊤)),
    Ideal.span ↑s = ⊤ →
      (∀ (r : ↥s), sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ P) (f ∣_ Y.basicOpen ↑r)) →
        sourceAffineLocally (fun {R S} [CommRing R] [CommRing S] ↦ P) f

Tactic:
·

Goal state after:
case respectsIso
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
h₁ : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ P
h₂ : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ P
h₃ : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ P
⊢ (sourceAffineLocally fun {R S} [CommRing R] [CommRing S] ↦ P).toProperty.RespectsIso