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

Goal state before:
pp: X : Scheme
U✝ : X.Opens
hU✝ : IsCompact U✝.carrier
S : X.Opens
hS : IsCompact S.carrier
U : ↑X.affineOpens
hU :
  IsQuasiSeparated S.carrier →
    ∀ (f : ↑Γ(X, S)) (x : ↑Γ(X, X.basicOpen f)),
      ∃ n y,
        (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) y =
          (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) f ^ n * x
hSU : IsQuasiSeparated (S ⊔ ↑U).carrier
f : ↑Γ(X, S ⊔ ↑U)
x : ↑Γ(X, X.basicOpen f)
n₁ : ℕ
y₁ : ↑Γ(X, S)
hy₁ :
  (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) y₁ =
    (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) ((ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) f) ^
        n₁ *
      (ConcreteCategory.hom (X.presheaf.map (homOfLE ?m.181).op)) x
n₂ : ℕ
y₂ : ↑Γ(X, ↑U)
hy₂ :
  (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) y₂ =
    (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) ((ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) f) ^
        n₂ *
      (ConcreteCategory.hom (X.presheaf.map (homOfLE ?m.240).op)) x
s : Set ↑X.affineOpens
hs' : s.Finite
this : Finite ↑s
val✝ : Fintype ↑s
hs : S ⊓ ↑U = ⨆ i, ↑↑i
hs₁ : ∀ (i : ↑s), ↑↑i ≤ S
i : ↑s
⊢ ↑↑i ≤ S ⊓ ↑U

Tactic:
hs

Goal state after:
X : Scheme
U✝ : X.Opens
hU✝ : IsCompact U✝.carrier
S : X.Opens
hS : IsCompact S.carrier
U : ↑X.affineOpens
hU :
  IsQuasiSeparated S.carrier →
    ∀ (f : ↑Γ(X, S)) (x : ↑Γ(X, X.basicOpen f)),
      ∃ n y,
        (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) y =
          (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) f ^ n * x
hSU : IsQuasiSeparated (S ⊔ ↑U).carrier
f : ↑Γ(X, S ⊔ ↑U)
x : ↑Γ(X, X.basicOpen f)
n₁ : ℕ
y₁ : ↑Γ(X, S)
hy₁ :
  (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) y₁ =
    (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) ((ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) f) ^
        n₁ *
      (ConcreteCategory.hom (X.presheaf.map (homOfLE ?m.181).op)) x
n₂ : ℕ
y₂ : ↑Γ(X, ↑U)
hy₂ :
  (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) y₂ =
    (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) ((ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) f) ^
        n₂ *
      (ConcreteCategory.hom (X.presheaf.map (homOfLE ?m.240).op)) x
s : Set ↑X.affineOpens
hs' : s.Finite
this : Finite ↑s
val✝ : Fintype ↑s
hs : S ⊓ ↑U = ⨆ i, ↑↑i
hs₁ : ∀ (i : ↑s), ↑↑i ≤ S
i : ↑s
⊢ ↑↑i ≤ ⨆ i, ↑↑i
Module: Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties

Goal state before:
pp: case ofLocalizationSpan
P : MorphismProperty Scheme
inst✝³ : IsZariskiLocalAtTarget P
inst✝² : IsZariskiLocalAtSource P
hP : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] f ↦ P (Spec.map (CommRingCat.ofHom f))
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set R
⊢ Ideal.span s = ⊤ →
    (∀ (r : ↑s),
        (fun {R S} [CommRing R] [CommRing S] f ↦ P (Spec.map (CommRingCat.ofHom f))) (Localization.awayMap f ↑r)) →
      (fun {R S} [CommRing R] [CommRing S] f ↦ P (Spec.map (CommRingCat.ofHom f))) f

Tactic:
hs

Goal state after:
case ofLocalizationSpan
P : MorphismProperty Scheme
inst✝³ : IsZariskiLocalAtTarget P
inst✝² : IsZariskiLocalAtSource P
hP : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] f ↦ P (Spec.map (CommRingCat.ofHom f))
R S : Type u
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set R
hs : Ideal.span s = ⊤
⊢ (∀ (r : ↑s),
      (fun {R S} [CommRing R] [CommRing S] f ↦ P (Spec.map (CommRingCat.ofHom f))) (Localization.awayMap f ↑r)) →
    (fun {R S} [CommRing R] [CommRing S] f ↦ P (Spec.map (CommRingCat.ofHom f))) f
Module: Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties

Goal state before:
pp: case ofLocalizationSpan
P : MorphismProperty Scheme
inst✝¹ : IsZariskiLocalAtTarget P
inst✝ : IsZariskiLocalAtSource P
hP : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] f ↦ P (Spec.map (CommRingCat.ofHom f))
⊢ RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] f ↦ P (Spec.map (CommRingCat.ofHom f))

pp: case StableUnderCompositionWithLocalizationAwayTarget
P : MorphismProperty Scheme
inst✝¹ : IsZariskiLocalAtTarget P
inst✝ : IsZariskiLocalAtSource P
hP : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] f ↦ P (Spec.map (CommRingCat.ofHom f))
⊢ RingHom.StableUnderCompositionWithLocalizationAwayTarget fun {R S} [CommRing R] [CommRing S] f ↦
    P (Spec.map (CommRingCat.ofHom f))

Tactic:
· intro R S _ _ f s hs H
    apply IsZariskiLocalAtTarget.of_iSup_eq_top _ (PrimeSpectrum.iSup_basicOpen_eq_top_iff
      (f := fun i : s ↦ (i : R)).mpr (by simpa))
    intro i
    exact (MorphismProperty.arrow_mk_iso_iff P (SpecMapRestrictBasicOpenIso
      (CommRingCat.ofHom f) i.1)).mpr (H i)

Goal state after:
case StableUnderCompositionWithLocalizationAwayTarget
P : MorphismProperty Scheme
inst✝¹ : IsZariskiLocalAtTarget P
inst✝ : IsZariskiLocalAtSource P
hP : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] f ↦ P (Spec.map (CommRingCat.ofHom f))
⊢ RingHom.StableUnderCompositionWithLocalizationAwayTarget fun {R S} [CommRing R] [CommRing S] f ↦
    P (Spec.map (CommRingCat.ofHom f))
Module: Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated

Goal state before:
pp: case iUV
X : Scheme
U✝ : X.Opens
hU✝ : IsCompact U✝.carrier
S : X.Opens
hS : IsCompact S.carrier
U : ↑X.affineOpens
hU :
  IsQuasiSeparated S.carrier →
    ∀ (f : ↑Γ(X, S)) (x : ↑Γ(X, X.basicOpen f)),
      ∃ n y,
        (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) y =
          (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) f ^ n * x
hSU : IsQuasiSeparated (S ⊔ ↑U).carrier
f : ↑Γ(X, S ⊔ ↑U)
x : ↑Γ(X, X.basicOpen f)
n₁ : ℕ
y₁ : ↑Γ(X, S)
hy₁ :
  (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) y₁ =
    (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) ((ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) f) ^
        n₁ *
      (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) x
n₂ : ℕ
y₂ : ↑Γ(X, ↑U)
hy₂ :
  (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) y₂ =
    (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) ((ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) f) ^
        n₂ *
      (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) x
s : Set ↑X.affineOpens
hs' : s.Finite
this : Finite ↑s
val✝ : Fintype ↑s
hs : S ⊓ ↑U = ⨆ i, ↑↑i
hs₁ : ∀ (i : ↑s), ↑↑i ≤ S
hs₂ : ∀ (i : ↑s), ↑↑i ≤ ↑U
n : ↑s → ℕ
hn : ∀ (i : ↑s) (m : ℕ), n i ≤ m → (((f |_ S) ⋯ ^ (m + n₂) * y₁) |_ ↑↑i) ⋯ = (((f |_ ↑U) ⋯ ^ (m + n₁) * y₂) |_ ↑↑i) ⋯
i : ↑s
⊢ ↑↑i ≤ S ⊓ ↑U

Tactic:
hs

Goal state after:
case iUV
X : Scheme
U✝ : X.Opens
hU✝ : IsCompact U✝.carrier
S : X.Opens
hS : IsCompact S.carrier
U : ↑X.affineOpens
hU :
  IsQuasiSeparated S.carrier →
    ∀ (f : ↑Γ(X, S)) (x : ↑Γ(X, X.basicOpen f)),
      ∃ n y,
        (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) y =
          (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) f ^ n * x
hSU : IsQuasiSeparated (S ⊔ ↑U).carrier
f : ↑Γ(X, S ⊔ ↑U)
x : ↑Γ(X, X.basicOpen f)
n₁ : ℕ
y₁ : ↑Γ(X, S)
hy₁ :
  (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) y₁ =
    (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) ((ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) f) ^
        n₁ *
      (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) x
n₂ : ℕ
y₂ : ↑Γ(X, ↑U)
hy₂ :
  (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) y₂ =
    (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) ((ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) f) ^
        n₂ *
      (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) x
s : Set ↑X.affineOpens
hs' : s.Finite
this : Finite ↑s
val✝ : Fintype ↑s
hs : S ⊓ ↑U = ⨆ i, ↑↑i
hs₁ : ∀ (i : ↑s), ↑↑i ≤ S
hs₂ : ∀ (i : ↑s), ↑↑i ≤ ↑U
n : ↑s → ℕ
hn : ∀ (i : ↑s) (m : ℕ), n i ≤ m → (((f |_ S) ⋯ ^ (m + n₂) * y₁) |_ ↑↑i) ⋯ = (((f |_ ↑U) ⋯ ^ (m + n₁) * y₂) |_ ↑↑i) ⋯
i : ↑s
⊢ ↑↑i ≤ ⨆ i, ↑↑i
Module: Mathlib.AlgebraicGeometry.Properties

Goal state before:
pp: X : Scheme
R : CommRingCat
H : _root_.IsReduced ↑R
x : ↥(Spec R)
this : _root_.IsReduced ↑(CommRingCat.of (Localization.AtPrime x.asIdeal))
⊢ _root_.IsReduced ↑((Spec R).presheaf.stalk x)

Tactic:
exact isReduced_of_injective (Spec.stalkIso R x).hom.hom
    (Spec.stalkIso R x).commRingCatIsoToRingEquiv.injective

Goal state after:
Module: Mathlib.AlgebraicGeometry.Properties

Goal state before:
pp: case h₁
X✝ X : Scheme
U : X.Opens
H✝ :
  ∀ (x : ↥U),
    ∃ V,
      ∃ (_ : ↑x ∈ V),
        ∃ x,
          ∀ [hX : IsReduced X] (s : ↑Γ(X, V)),
            X.basicOpen s = ⊥ → ∀ (x : ↥X) (hx : x ∈ V), (ConcreteCategory.hom (X.sheaf.presheaf.germ V x hx)) s = 0
hX : IsReduced X
s : ↑Γ(X, U)
hs : X.basicOpen s = ⊥
x : ↥X
hx✝ : x ∈ U
V : X.Opens
hx : ↑⟨x, hx✝⟩ ∈ V
i : V ⟶ U
H :
  V ⊓ X.basicOpen s = ⊥ →
    ∀ (x : ↥X) (hx : x ∈ V),
      (ConcreteCategory.hom (X.sheaf.presheaf.germ V x hx)) ((ConcreteCategory.hom (X.presheaf.map i.op)) s) = 0
⊢ (ConcreteCategory.hom (X.sheaf.presheaf.germ U x hx✝)) s = 0

Tactic:
hs

Goal state after:
case h₁
X✝ X : Scheme
U : X.Opens
H✝ :
  ∀ (x : ↥U),
    ∃ V,
      ∃ (_ : ↑x ∈ V),
        ∃ x,
          ∀ [hX : IsReduced X] (s : ↑Γ(X, V)),
            X.basicOpen s = ⊥ → ∀ (x : ↥X) (hx : x ∈ V), (ConcreteCategory.hom (X.sheaf.presheaf.germ V x hx)) s = 0
hX : IsReduced X
s : ↑Γ(X, U)
hs : X.basicOpen s = ⊥
x : ↥X
hx✝ : x ∈ U
V : X.Opens
hx : ↑⟨x, hx✝⟩ ∈ V
i : V ⟶ U
H :
  V ⊓ ⊥ = ⊥ →
    ∀ (x : ↥X) (hx : x ∈ V),
      (ConcreteCategory.hom (X.sheaf.presheaf.germ V x hx)) ((ConcreteCategory.hom (X.presheaf.map i.op)) s) = 0
⊢ (ConcreteCategory.hom (X.sheaf.presheaf.germ U x hx✝)) s = 0
Module: Mathlib.AlgebraicGeometry.Properties

Goal state before:
pp: case h₂
X✝ X Y : Scheme
f : X ⟶ Y
inst✝ : IsOpenImmersion f
H :
  ∀ [hX : IsReduced X] (s : ↑Γ(X, f ⁻¹ᵁ Scheme.Hom.opensRange f)),
    X.basicOpen s = ⊥ →
      ∀ (x : ↥X) (hx : x ∈ f ⁻¹ᵁ Scheme.Hom.opensRange f),
        (ConcreteCategory.hom (X.sheaf.presheaf.germ (f ⁻¹ᵁ Scheme.Hom.opensRange f) x hx)) s = 0
hX : IsReduced Y
s : ↑Γ(Y, Scheme.Hom.opensRange f)
hs : Y.basicOpen s = ⊥
x : ↥X
this : IsReduced X
⊢ f ⁻¹ᵁ Y.basicOpen s = ⊥

Tactic:
hs

Goal state after:
case h₂
X✝ X Y : Scheme
f : X ⟶ Y
inst✝ : IsOpenImmersion f
H :
  ∀ [hX : IsReduced X] (s : ↑Γ(X, f ⁻¹ᵁ Scheme.Hom.opensRange f)),
    X.basicOpen s = ⊥ →
      ∀ (x : ↥X) (hx : x ∈ f ⁻¹ᵁ Scheme.Hom.opensRange f),
        (ConcreteCategory.hom (X.sheaf.presheaf.germ (f ⁻¹ᵁ Scheme.Hom.opensRange f) x hx)) s = 0
hX : IsReduced Y
s : ↑Γ(Y, Scheme.Hom.opensRange f)
hs : Y.basicOpen s = ⊥
x : ↥X
this : IsReduced X
⊢ f ⁻¹ᵁ ⊥ = ⊥
Module: Mathlib.AlgebraicGeometry.Properties

Goal state before:
pp: X : Scheme
x : ↥X
hx : closure {x} ∈ irreducibleComponents ↥X
inst✝ : IsReduced X
⊢ IsField ↑(X.presheaf.stalk x)

Tactic:
wlog hX : ∃ R, X = Spec R

Goal state after:
case inr
X : Scheme
x : ↥X
hx : closure {x} ∈ irreducibleComponents ↥X
inst✝ : IsReduced X
this :
  ∀ (X : Scheme) (x : ↥X),
    closure {x} ∈ irreducibleComponents ↥X → ∀ [IsReduced X], (∃ R, X = Spec R) → IsField ↑(X.presheaf.stalk x)
hX : ¬∃ R, X = Spec R
⊢ IsField ↑(X.presheaf.stalk x)

X✝ X : Scheme
x : ↥X
hx : closure {x} ∈ irreducibleComponents ↥X
inst✝ : IsReduced X
hX : ∃ R, X = Spec R
⊢ IsField ↑(X.presheaf.stalk x)
Module: Mathlib.AlgebraicGeometry.Properties

Goal state before:
pp: X : Scheme
inst✝ : IsIntegral X
U : X.Opens
⊢ _root_.IsReduced ↑Γ(X, U)

Tactic:
rcases U.1.eq_empty_or_nonempty with h | h

Goal state after:
case inl
X : Scheme
inst✝ : IsIntegral X
U : X.Opens
h : U.carrier = ∅
⊢ _root_.IsReduced ↑Γ(X, U)

case inr
X : Scheme
inst✝ : IsIntegral X
U : X.Opens
h : U.carrier.Nonempty
⊢ _root_.IsReduced ↑Γ(X, U)
Module: Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties

Goal state before:
pp: case refine_1
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
X Y : Scheme
f : X ⟶ Y
hQ : StableUnderCompositionWithLocalizationAwaySource fun {R S} [CommRing R] [CommRing S] ↦ Q
hQi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q
inst✝ : HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] ↦ Locally fun {R S} [CommRing R] [CommRing S] ↦ Q
this : P.Respects IsOpenImmersion
hf : P f
x : ↥X
U : Opens ↥Y
hU : U ∈ Y.affineOpens
hfx : f x ∈ U
right✝ : U ≤ ⊤
⊢ ∃ U V, ∃ (_ : x ∈ ↑V) (e : ↑V ≤ f ⁻¹ᵁ ↑U), Q (CommRingCat.Hom.hom (Scheme.Hom.appLE f (↑U) (↑V) e))

Tactic:
obtain ⟨V, hV, hx, e⟩ := Opens.isBasis_iff_nbhd.mp X.isBasis_affineOpens
      (show x ∈ f ⁻¹ᵁ U from hfx)

Goal state after:
case refine_1
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
X Y : Scheme
f : X ⟶ Y
hQ : StableUnderCompositionWithLocalizationAwaySource fun {R S} [CommRing R] [CommRing S] ↦ Q
hQi : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q
inst✝ : HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] ↦ Locally fun {R S} [CommRing R] [CommRing S] ↦ Q
this : P.Respects IsOpenImmersion
hf : P f
x : ↥X
U : Opens ↥Y
hU : U ∈ Y.affineOpens
hfx : f x ∈ U
right✝ : U ≤ ⊤
V : Opens ↥X
hV : V ∈ X.affineOpens
hx : x ∈ V
e : V ≤ f ⁻¹ᵁ U
⊢ ∃ U V, ∃ (_ : x ∈ ↑V) (e : ↑V ≤ f ⁻¹ᵁ ↑U), Q (CommRingCat.Hom.hom (Scheme.Hom.appLE f (↑U) (↑V) e))
Module: Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated

Goal state before:
pp: X : Scheme
U : Opens ↥X
hU : IsCompact U.carrier
hU' : IsQuasiSeparated U.carrier
f g s : ↑Γ(X, U)
hfg : (f |_ X.basicOpen s) ⋯ = (g |_ X.basicOpen s) ⋯
⊢ ∃ n, s ^ n * f = s ^ n * g

Tactic:
by
  obtain ⟨n, hc⟩ := (isLocalization_basicOpen_of_qcqs hU hU' s).exists_of_eq s hfg
  use n

Goal state after:
Module: Mathlib.AlgebraicGeometry.Properties

Goal state before:
pp: X : Scheme
z : ↥X
n : ℕ
hz : coheight z ≤ ↑n
⊢ Ring.KrullDimLE n ↑(X.presheaf.stalk z)

Tactic:
Ring.krullDimLE_iff,

Goal state after:
X : Scheme
z : ↥X
n : ℕ
hz : coheight z ≤ ↑n
⊢ ringKrullDim ↑(X.presheaf.stalk z) ≤ ↑n
Module: Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties

Goal state before:
pp: case inr
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P Q
X Y : Scheme
f : X ⟶ Y
Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hQ' : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q'
hQ :
  ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (f : R →+* S),
    Q f → ∀ (J : Ideal S) (x : J.IsPrime), Q' (Localization.localRingHom (Ideal.comap f J) J f ⋯)
hf : P f
x : ↥X
this :
  ∀ {X Y : Scheme} {f : X ⟶ Y},
    P f → ∀ (x : ↥X), IsAffine X ∧ IsAffine Y → Q' (CommRingCat.Hom.hom (Scheme.Hom.stalkMap f x))
h : ¬(IsAffine X ∧ IsAffine Y)
U : Opens ↥Y
hU : U ∈ Y.affineOpens
hfx : f x ∈ U
right✝ : U ≤ ⊤
⊢ Q' (CommRingCat.Hom.hom (Scheme.Hom.stalkMap f x))

Tactic:
obtain ⟨V, hV, hx, e⟩ := Opens.isBasis_iff_nbhd.mp X.isBasis_affineOpens
      (show x ∈ f ⁻¹ᵁ U from hfx)

Goal state after:
case inr
P : MorphismProperty Scheme
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P Q
X Y : Scheme
f : X ⟶ Y
Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hQ' : RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q'
hQ :
  ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] (f : R →+* S),
    Q f → ∀ (J : Ideal S) (x : J.IsPrime), Q' (Localization.localRingHom (Ideal.comap f J) J f ⋯)
hf : P f
x : ↥X
this :
  ∀ {X Y : Scheme} {f : X ⟶ Y},
    P f → ∀ (x : ↥X), IsAffine X ∧ IsAffine Y → Q' (CommRingCat.Hom.hom (Scheme.Hom.stalkMap f x))
h : ¬(IsAffine X ∧ IsAffine Y)
U : Opens ↥Y
hU : U ∈ Y.affineOpens
hfx : f x ∈ U
right✝ : U ≤ ⊤
V : Opens ↥X
hV : V ∈ X.affineOpens
hx : x ∈ V
e : V ≤ f ⁻¹ᵁ U
⊢ Q' (CommRingCat.Hom.hom (Scheme.Hom.stalkMap f x))
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Basic

Goal state before:
pp: case a
X : Scheme
I : X.IdealSheafData
Z : Closeds ↥X
U : ↑X.affineOpens
f : ↑Γ(X, ↑U)
F : Γ(X, ↑U) ⟶ Γ(X, X.basicOpen f) := ⋯
⊢ PrimeSpectrum.vanishingIdeal (⇑(IsAffineOpen.fromSpec ⋯) ⁻¹' ↑Z) ≤
    Ideal.map (CommRingCat.Hom.hom (X.presheaf.map (homOfLE ⋯).op))
      (PrimeSpectrum.vanishingIdeal (⇑(IsAffineOpen.fromSpec ⋯) ⁻¹' ↑Z))

Tactic:
· let : Algebra Γ(X, U) Γ(X, X.affineBasicOpen f) := F.hom.toAlgebra
        have : IsLocalization.Away f Γ(X, X.basicOpen f) :=
          U.2.isLocalization_of_eq_basicOpen _ _ rfl
        intro x hx
        dsimp only at hx ⊢
        have : Topology.IsOpenEmbedding (Spec.map F) :=
          localization_away_isOpenEmbedding Γ(X, X.basicOpen f) f
        rw [← U.2.map_fromSpec (X.affineBasicOpen f).2 (homOfLE (X.basicOpen_le f)).op,
          Scheme.Hom.comp_base, TopCat.coe_comp, Set.preimage_comp] at hx
        generalize U.2.fromSpec ⁻¹' Z = Z' at hx ⊢
        replace hx : x ∈ vanishingIdeal (Spec.map F ⁻¹' Z') := hx
        obtain ⟨I, hI, e⟩ :=
          (isClosed_iff_zeroLocus_radical_ideal _).mp (isClosed_closure (s := Z'))
        rw [← vanishingIdeal_closure,
          ← this.isOpenMap.preimage_closure_eq_closure_preimage this.continuous, e] at hx
        rw [← vanishingIdeal_closure, e]
        erw [preimage_comap_zeroLocus] at hx
        rwa [← PrimeSpectrum.zeroLocus_span, ← Ideal.map, vanishingIdeal_zeroLocus_eq_radical,
          ← RingHom.algebraMap_toAlgebra (X.presheaf.map _).hom,
          ← IsLocalization.map_radical (.powers f), ← vanishingIdeal_zeroLocus_eq_radical] at hx

Goal state after:
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Basic

Goal state before:
pp: X Y Z : Scheme
f : X ⟶ Y
g : Y.Hom Z
U : ↑Z.affineOpens
⊢ RingHom.ker (CommRingCat.Hom.hom (g.app ↑U)) ≤ RingHom.ker (CommRingCat.Hom.hom (app (f ≫ g) ↑U))

Tactic:
rw [Scheme.Hom.comp_app f g U, CommRingCat.hom_comp, ← RingHom.comap_ker]

Goal state after:
X Y Z : Scheme
f : X ⟶ Y
g : Y.Hom Z
U : ↑Z.affineOpens
⊢ RingHom.ker (CommRingCat.Hom.hom (g.app ↑U)) ≤
    Ideal.comap (CommRingCat.Hom.hom (g.app ↑U)) (RingHom.ker (CommRingCat.Hom.hom (app f (g ⁻¹ᵁ ↑U))))
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Basic

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
x : ↥X
U : TopologicalSpace.Opens ↥Y
hU : U ∈ Y.affineOpens
hxU : f x ∈ ↑U
s : ↑Γ(Y, U)
⊢ s ∈ (ker f).ideal ⟨U, hU⟩ → f x ∉ Y.basicOpen s

Tactic:
hs

Goal state after:
X Y : Scheme
f : X ⟶ Y
x : ↥X
U : TopologicalSpace.Opens ↥Y
hU : U ∈ Y.affineOpens
hxU : f x ∈ ↑U
s : ↑Γ(Y, U)
hs : s ∈ (ker f).ideal ⟨U, hU⟩
⊢ f x ∉ Y.basicOpen s
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Basic

Goal state before:
pp: X Y : Scheme
f : X.Hom Y
inst✝ : QuasiCompact f
𝒰 : X.OpenCover
U : ↑Y.affineOpens
s : ↑Γ(Y, ↑U)
⊢ s ∈ ⨅ i, (ker (𝒰.f i ≫ f)).ideal U → s ∈ f.ker.ideal U

Tactic:
hs

Goal state after:
X Y : Scheme
f : X.Hom Y
inst✝ : QuasiCompact f
𝒰 : X.OpenCover
U : ↑Y.affineOpens
s : ↑Γ(Y, ↑U)
hs : s ∈ ⨅ i, (ker (𝒰.f i ≫ f)).ideal U
⊢ s ∈ f.ker.ideal U
Module: Mathlib.AlgebraicGeometry.Morphisms.Affine

Goal state before:
pp: case refine_2
X : Scheme
s : Set ↑Γ(X, ⊤)
hs : Ideal.span s = ⊤
hs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)
this✝ : QuasiSeparatedSpace ↥X
this : CompactSpace ↥X
⊢ Ideal.span s = ⊤

Tactic:
hs

Goal state after:
case refine_2
X : Scheme
s : Set ↑Γ(X, ⊤)
hs : Ideal.span s = ⊤
hs₂ : ∀ i ∈ s, IsAffineOpen (X.basicOpen i)
this✝ : QuasiSeparatedSpace ↥X
this : CompactSpace ↥X
⊢ ⊤ = ⊤
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Basic

Goal state before:
pp: X Y : Scheme
S R : CommRingCat
φ : S ⟶ R
inst✝ : QuasiCompact (Spec.map φ)
⊢ ↑(ker (Spec.map φ)).support ⊆ closure (Set.range ⇑(Spec.map φ))

Tactic:
ker_of_isAffine,

Goal state after:
X Y : Scheme
S R : CommRingCat
φ : S ⟶ R
inst✝ : QuasiCompact (Spec.map φ)
⊢ ↑(ofIdealTop (RingHom.ker (CommRingCat.Hom.hom (appTop (Spec.map φ))))).support ⊆ closure (Set.range ⇑(Spec.map φ))
Module: Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme

Goal state before:
pp: case h₁.h₀
X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ pullback.fst (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) ≫
      pullback.snd (I.glueDataObjι i) (X.homOfLE ⋯) ≫ (↑i ⊓ ↑j).ι =
    pullback.fst (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) ≫
      pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯) ≫ I.glueDataObjι i ≫ (↑i).ι

Tactic:
pullback.condition_assoc (f := I.glueDataObjι i),

Goal state after:
case h₁.h₀
X : Scheme
I : X.IdealSheafData
i j k : ↑X.affineOpens
⊢ pullback.fst (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) ≫
      pullback.snd (I.glueDataObjι i) (X.homOfLE ⋯) ≫ (↑i ⊓ ↑j).ι =
    pullback.fst (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) (pullback.fst (I.glueDataObjι i) (X.homOfLE ⋯)) ≫
      pullback.snd (I.glueDataObjι i) (X.homOfLE ⋯) ≫ X.homOfLE ⋯ ≫ (↑i).ι
Module: Mathlib.AlgebraicGeometry.Morphisms.AffineAnd

Goal state before:
pp: case hs
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hPi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q
hQl : RingHom.LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ Q
hQs : RingHom.OfLocalizationSpan fun {R S} [CommRing R] [CommRing S] ↦ Q
X Y : Scheme
x✝ : IsAffine Y
f : X ⟶ Y
s : Finset ↑Γ(Y, ⊤)
hs : Ideal.span ↑s = ⊤
hf : ∀ (r : ↥s), IsAffine ↑(f ⁻¹ᵁ Y.basicOpen ↑r) ∧ Q (CommRingCat.Hom.hom (Scheme.Hom.appTop (f ∣_ Y.basicOpen ↑r)))
⊢ Ideal.span (⇑(ConcreteCategory.hom (Scheme.Hom.appTop f)) '' ↑s) = ⊤

Tactic:
apply_fun Ideal.map (f.appTop).hom at hs
        rwa [Ideal.map_span, Ideal.map_top] at hs

Goal state after:
Module: Mathlib.AlgebraicGeometry.Morphisms.AffineAnd

Goal state before:
pp: Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hQi : RingHom.RespectsIso fun {R S} [CommRing R] [CommRing S] ↦ Q
X Y : Scheme
f : X ⟶ Y
⊢ targetAffineLocally (affineAnd fun {R S} [CommRing R] [CommRing S] ↦ Q) f ↔
    ∀ (U : Y.Opens), IsAffineOpen U → IsAffineOpen (f ⁻¹ᵁ U) ∧ Q (CommRingCat.Hom.hom (Scheme.Hom.app f U))

Tactic:
simp only [targetAffineLocally, affineAnd_apply, morphismRestrict_app, CommRingCat.hom_comp,
    hQi.cancel_right_isIso]
  refine ⟨fun hf U hU ↦ ?_, fun h U ↦ ?_⟩
  · obtain ⟨hfU, hf⟩ := hf ⟨U, hU⟩
    use hfU
    have hf : Q (Scheme.Hom.app f (((⟨U, hU⟩ : Y.affineOpens) : Y.Opens).ι ''ᵁ ⊤)).hom := hf
    rwa [Scheme.Opens.ι_image_top] at hf
  · refine ⟨(h U U.2).1, ?_⟩
    change Q (Scheme.Hom.app f ((U : Y.Opens).ι ''ᵁ ⊤)).hom
    rw [Scheme.Opens.ι_image_top]
    exact (h U U.2).2

Goal state after:
Module: Mathlib.RingTheory.Finiteness.FiniteTypeLocal

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
s✝ : Set S
hs✝ : Ideal.span s✝ = ⊤
s : Finset S
h₁ : ↑s ⊆ s✝
hs : Ideal.span ↑s = ⊤
t : (r : ↥s) → Finset (Localization.Away ↑r)
ht : ∀ (r : ↥s), adjoin R ↑(t r) = ⊤
⊢ 1 ∈ Ideal.span ↑s

Tactic:
hs

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
s✝ : Set S
hs✝ : Ideal.span s✝ = ⊤
s : Finset S
h₁ : ↑s ⊆ s✝
hs : Ideal.span ↑s = ⊤
t : (r : ↥s) → Finset (Localization.Away ↑r)
ht : ∀ (r : ↥s), adjoin R ↑(t r) = ⊤
⊢ 1 ∈ ⊤
Module: Mathlib.AlgebraicGeometry.Morphisms.Affine

Goal state before:
pp: X Y Z : Scheme
f : X ⟶ Z
g : Y ⟶ Z
U : X.Opens
hU : IsAffineOpen U
V : Y.Opens
hV : IsCompact ↑V
W : Z.Opens
hW : IsAffineOpen W
hUW : U ≤ f ⁻¹ᵁ W
hVW : V ≤ g ⁻¹ᵁ W
this✝¹ : IsAffine ↑U
this✝ : IsAffine ↑W
this : CompactSpace ↥↑V
f' : ↑U ⟶ ↑W := Scheme.Hom.resLE f W U hUW
q : ↑V ⟶ ↑W := IsOpenImmersion.lift W.ι (V.ι ≫ g) ⋯
⊢ f' ≫ W.ι = U.ι ≫ f

Tactic:
by simp [f']

Goal state after:
Module: Mathlib.RingTheory.RingHom.FiniteType

Goal state before:
pp: ⊢ OfLocalizationSpan @FiniteType

Tactic:
refine OfLocalizationSpan.mk _ finiteType_respectsIso (fun s hs h ↦ ?_)
  simp_rw [finiteType_algebraMap] at h ⊢
  exact Algebra.FiniteType.of_span_eq_top_source s hs h

Goal state after:
Module: Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
inst✝ : IsClosedImmersion f
this : IsClosedImmersion (Scheme.Hom.toImage f)
⊢ Function.Surjective ⇑(Scheme.Hom.toImage f)

Tactic:
by
      rw [← Set.range_eq_univ, ← f.toImage.isClosedEmbedding.isClosed_range.closure_eq]
      exact f.toImage.denseRange.closure_eq

Goal state after: