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

Goal state before:
pp: P P' : MorphismProperty Scheme
Q Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : P.RespectsIso
hQQ' : RingHom.CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ Q) fun {R S} [CommRing R] [CommRing S] ↦ Q'
H₁ : ∀ {R S : CommRingCat} {f : R ⟶ S}, P' (Spec.map f) → Q' (CommRingCat.Hom.hom f)
H₂ : ∀ {R S : CommRingCat} {f : R ⟶ S}, P (Spec.map f) ↔ Q (CommRingCat.Hom.hom f)
R S T : CommRingCat
φ : R ⟶ S
ψ : R ⟶ T
h : P' (Spec.map ψ)
algInst✝¹ : Algebra ↑R ↑S := (CommRingCat.Hom.hom φ).toAlgebra
algInst✝ : Algebra ↑R ↑T := (CommRingCat.Hom.hom ψ).toAlgebra
hf : Q (CommRingCat.Hom.hom (CommRingCat.ofHom Algebra.TensorProduct.includeLeftRingHom))
⊢ Q (algebraMap (↑T) (↑T ⊗[↑R] ↑S))

Tactic:
]

Goal state after:
P P' : MorphismProperty Scheme
Q Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : P.RespectsIso
hQQ' : RingHom.CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ Q) fun {R S} [CommRing R] [CommRing S] ↦ Q'
H₁ : ∀ {R S : CommRingCat} {f : R ⟶ S}, P' (Spec.map f) → Q' (CommRingCat.Hom.hom f)
H₂ : ∀ {R S : CommRingCat} {f : R ⟶ S}, P (Spec.map f) ↔ Q (CommRingCat.Hom.hom f)
R S T : CommRingCat
φ : R ⟶ S
ψ : R ⟶ T
h : P' (Spec.map ψ)
algInst✝¹ : Algebra ↑R ↑S := (CommRingCat.Hom.hom φ).toAlgebra
algInst✝ : Algebra ↑R ↑T := (CommRingCat.Hom.hom ψ).toAlgebra
hf : Q (CommRingCat.Hom.hom (CommRingCat.ofHom Algebra.TensorProduct.includeLeftRingHom))
⊢ Q (algebraMap (↑T) (↑T ⊗[↑R] ↑S))
Module: Mathlib.AlgebraicGeometry.Morphisms.FlatDescent

Goal state before:
pp: case H.inr
R S : CommRingCat
Y : Scheme
φ : R ⟶ S
g : Y ⟶ Spec R
h : (@Surjective ⊓ @Flat) (Spec.map φ)
hfst : IsIso (pullback.fst (Spec.map φ) g)
this✝ : IsAffine Y
this :
  ∀ {Y : Scheme} (g : Y ⟶ Spec R),
    IsIso (pullback.fst (Spec.map φ) g) → IsAffine Y → (∃ T, Y = Spec T) → isomorphisms Scheme g
hY : ¬∃ T, Y = Spec T
⊢ isomorphisms Scheme (Y.isoSpec.inv ≫ g)

Tactic:
]

Goal state after:
case H.inr
R S : CommRingCat
Y : Scheme
φ : R ⟶ S
g : Y ⟶ Spec R
h : (@Surjective ⊓ @Flat) (Spec.map φ)
hfst : IsIso (pullback.fst (Spec.map φ) g)
this✝ : IsAffine Y
this :
  ∀ {Y : Scheme} (g : Y ⟶ Spec R),
    IsIso (pullback.fst (Spec.map φ) g) → IsAffine Y → (∃ T, Y = Spec T) → isomorphisms Scheme g
hY : ¬∃ T, Y = Spec T
⊢ isomorphisms Scheme (Y.isoSpec.inv ≫ g)
Module: Mathlib.AlgebraicGeometry.Morphisms.Descent

Goal state before:
pp: case a
P P' : MorphismProperty Scheme
inst✝³ : P'.RespectsIso
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H : ∀ {R : CommRingCat} {S X : Scheme} (f : Spec R ⟶ S) (g : X ⟶ S), P' f → P (pullback.fst f g) → P g
X Y Z T : Scheme
inst✝ : IsAffine T
p : T ⟶ X
hp : P' p
f : X ⟶ Z
g : Y ⟶ Z
h : P' f
hf : P (pullback.fst f g)
⊢ P' (p ≫ f)

Tactic:
]

Goal state after:
case a
P P' : MorphismProperty Scheme
inst✝³ : P'.RespectsIso
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H : ∀ {R : CommRingCat} {S X : Scheme} (f : Spec R ⟶ S) (g : X ⟶ S), P' f → P (pullback.fst f g) → P g
X Y Z T : Scheme
inst✝ : IsAffine T
p : T ⟶ X
hp : P' p
f : X ⟶ Z
g : Y ⟶ Z
h : P' f
hf : P (pullback.fst f g)
⊢ P' (p ≫ f)
Module: Mathlib.AlgebraicGeometry.Morphisms.Descent

Goal state before:
pp: case a
P P' : MorphismProperty Scheme
inst✝³ : P'.RespectsIso
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H : ∀ {R : CommRingCat} {S X : Scheme} (f : Spec R ⟶ S) (g : X ⟶ S), P' f → P (pullback.fst f g) → P g
X Y Z T : Scheme
inst✝ : IsAffine T
p : T ⟶ X
hp : P' p
f : X ⟶ Z
g : Y ⟶ Z
h : P' f
hf : P (pullback.fst f g)
⊢ P (pullback.fst (T.isoSpec.inv ≫ p) (pullback.fst f g))

Tactic:
]

Goal state after:
case a
P P' : MorphismProperty Scheme
inst✝³ : P'.RespectsIso
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H : ∀ {R : CommRingCat} {S X : Scheme} (f : Spec R ⟶ S) (g : X ⟶ S), P' f → P (pullback.fst f g) → P g
X Y Z T : Scheme
inst✝ : IsAffine T
p : T ⟶ X
hp : P' p
f : X ⟶ Z
g : Y ⟶ Z
h : P' f
hf : P (pullback.fst f g)
⊢ P (pullback.fst (T.isoSpec.inv ≫ p) (pullback.fst f g))
Module: Mathlib.AlgebraicGeometry.Morphisms.FlatDescent

Goal state before:
pp: case H.inr
R S : CommRingCat
Y : Scheme
φ : R ⟶ S
g : Y ⟶ Spec R
h : (@Surjective ⊓ @Flat) (Spec.map φ)
hfst : IsIso (pullback.fst (Spec.map φ) g)
this✝ : IsAffine Y
this :
  ∀ {Y : Scheme} (g : Y ⟶ Spec R),
    IsIso (pullback.fst (Spec.map φ) g) → IsAffine Y → (∃ T, Y = Spec T) → isomorphisms Scheme g
hY : ¬∃ T, Y = Spec T
heq :
  pullback.fst (Spec.map φ) (Y.isoSpec.inv ≫ g) =
    pullback.map (Spec.map φ) (Y.isoSpec.inv ≫ g) (Spec.map φ) g (𝟙 (Spec S)) Y.isoSpec.inv (𝟙 (Spec R)) ⋯ ⋯ ≫
      pullback.fst (Spec.map φ) g
⊢ isomorphisms Scheme (pullback.fst (Spec.map φ) g)

Tactic:
]

Goal state after:
case H.inr
R S : CommRingCat
Y : Scheme
φ : R ⟶ S
g : Y ⟶ Spec R
h : (@Surjective ⊓ @Flat) (Spec.map φ)
hfst : IsIso (pullback.fst (Spec.map φ) g)
this✝ : IsAffine Y
this :
  ∀ {Y : Scheme} (g : Y ⟶ Spec R),
    IsIso (pullback.fst (Spec.map φ) g) → IsAffine Y → (∃ T, Y = Spec T) → isomorphisms Scheme g
hY : ¬∃ T, Y = Spec T
heq :
  pullback.fst (Spec.map φ) (Y.isoSpec.inv ≫ g) =
    pullback.map (Spec.map φ) (Y.isoSpec.inv ≫ g) (Spec.map φ) g (𝟙 (Spec S)) Y.isoSpec.inv (𝟙 (Spec R)) ⋯ ⋯ ≫
      pullback.fst (Spec.map φ) g
⊢ isomorphisms Scheme (pullback.fst (Spec.map φ) g)
Module: Mathlib.AlgebraicGeometry.Morphisms.FlatDescent

Goal state before:
pp: case refine_1
R S : CommRingCat
φ : R ⟶ S
h : (@Surjective ⊓ @Flat) (Spec.map φ)
T : CommRingCat
this : IsAffine (Spec T)
ψ : R ⟶ T
hfst : IsIso (pullback.fst (Spec.map φ) (Spec.map ψ))
R✝ S✝ : CommRingCat
f : R✝ ⟶ S✝
hf : (@Surjective ⊓ @Flat) (Spec.map f)
⊢ Surjective (Spec.map f) ∧ Flat (Spec.map f)

Tactic:
]

Goal state after:
case refine_1
R S : CommRingCat
φ : R ⟶ S
h : (@Surjective ⊓ @Flat) (Spec.map φ)
T : CommRingCat
this : IsAffine (Spec T)
ψ : R ⟶ T
hfst : IsIso (pullback.fst (Spec.map φ) (Spec.map ψ))
R✝ S✝ : CommRingCat
f : R✝ ⟶ S✝
hf : (@Surjective ⊓ @Flat) (Spec.map f)
⊢ Surjective (Spec.map f) ∧ Flat (Spec.map f)
Module: Mathlib.RingTheory.RingHom.OpenImmersion

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
h : Function.Bijective ⇑(algebraMap R S)
⊢ ∃ r, Away r S

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
h : Function.Bijective ⇑(algebraMap R S)
⊢ ∃ r, Away r S
Module: Mathlib.RingTheory.RingHom.OpenImmersion

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : CommSemiring S
inst✝² : Algebra R S
T : Type u_4
inst✝¹ : CommSemiring T
inst✝ : Algebra R T
e : S ≃ₐ[R] T
h : ∃ r, Away r S
⊢ ∃ r, Away r T

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁴ : CommSemiring R
inst✝³ : CommSemiring S
inst✝² : Algebra R S
T : Type u_4
inst✝¹ : CommSemiring T
inst✝ : Algebra R T
e : S ≃ₐ[R] T
h : ∃ r, Away r S
⊢ ∃ r, Away r T
Module: Mathlib.RingTheory.RingHom.OpenImmersion

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝¹² : CommSemiring R
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Algebra R S
R' : Type u_4
S' : Type u_5
inst✝⁹ : CommSemiring R'
inst✝⁸ : CommSemiring S'
inst✝⁷ : Algebra R R'
inst✝⁶ : Algebra S S'
inst✝⁵ : Algebra R' S'
inst✝⁴ : Algebra R S'
inst✝³ : IsScalarTower R R' S'
inst✝² : IsScalarTower R S S'
inst✝¹ : IsPushout R S R' S'
inst✝ : IsStandardOpenImmersion R S
⊢ IsPushout R S R' S'

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝¹² : CommSemiring R
inst✝¹¹ : CommSemiring S
inst✝¹⁰ : Algebra R S
R' : Type u_4
S' : Type u_5
inst✝⁹ : CommSemiring R'
inst✝⁸ : CommSemiring S'
inst✝⁷ : Algebra R R'
inst✝⁶ : Algebra S S'
inst✝⁵ : Algebra R' S'
inst✝⁴ : Algebra R S'
inst✝³ : IsScalarTower R R' S'
inst✝² : IsScalarTower R S S'
inst✝¹ : IsPushout R S R' S'
inst✝ : IsStandardOpenImmersion R S
⊢ IsPushout R S R' S'
Module: Mathlib.RingTheory.RingHom.OpenImmersion

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
⊢ Algebra.IsStandardOpenImmersion R S ↔ Algebra.IsStandardOpenImmersion R S

Tactic:
]

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

Goal state before:
pp: R S T : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.IsStandardOpenImmersion R T
⊢ Algebra.IsStandardOpenImmersion S (TensorProduct R S T)

Tactic:
]

Goal state after:
R S T : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
h : Algebra.IsStandardOpenImmersion R T
⊢ Algebra.IsStandardOpenImmersion S (TensorProduct R S T)
Module: Mathlib.AlgebraicGeometry.Morphisms.Descent

Goal state before:
pp: case inr.refine_1
P P' : MorphismProperty Scheme
inst✝³ : P'.IsStableUnderBaseChange
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H₁ : @IsLocalIso ⊓ @Surjective ≤ P'
inst✝ : IsZariskiLocalAtTarget P
H :
  ∀ {R S : CommRingCat} {Y : Scheme} (φ : R ⟶ S) (g : Y ⟶ Spec R),
    P' (Spec.map φ) → P (pullback.fst (Spec.map φ) g) → P g
R : CommRingCat
X Y✝ : Scheme
f : X ⟶ Spec R
g : Y✝ ⟶ Spec R
hf : (P' ⊓ @QuasiCompact) f
h : P (pullback.fst f g)
this : ∀ {X : Scheme} (f : X ⟶ Spec R), (P' ⊓ @QuasiCompact) f → P (pullback.fst f g) → (∃ T, X = Spec T) → P g
hX : ¬∃ T, X = Spec T
x✝ : CompactSpace ↥X
Y : Scheme
p : Y ⟶ X
hsurj : Surjective p
hP' : IsLocalIso p
hY : IsAffine Y
⊢ (P' ⊓ @QuasiCompact) (p ≫ f)

Tactic:
]

Goal state after:
case inr.refine_1
P P' : MorphismProperty Scheme
inst✝³ : P'.IsStableUnderBaseChange
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H₁ : @IsLocalIso ⊓ @Surjective ≤ P'
inst✝ : IsZariskiLocalAtTarget P
H :
  ∀ {R S : CommRingCat} {Y : Scheme} (φ : R ⟶ S) (g : Y ⟶ Spec R),
    P' (Spec.map φ) → P (pullback.fst (Spec.map φ) g) → P g
R : CommRingCat
X Y✝ : Scheme
f : X ⟶ Spec R
g : Y✝ ⟶ Spec R
hf : (P' ⊓ @QuasiCompact) f
h : P (pullback.fst f g)
this : ∀ {X : Scheme} (f : X ⟶ Spec R), (P' ⊓ @QuasiCompact) f → P (pullback.fst f g) → (∃ T, X = Spec T) → P g
hX : ¬∃ T, X = Spec T
x✝ : CompactSpace ↥X
Y : Scheme
p : Y ⟶ X
hsurj : Surjective p
hP' : IsLocalIso p
hY : IsAffine Y
⊢ (P' ⊓ @QuasiCompact) (p ≫ f)
Module: Mathlib.AlgebraicGeometry.Morphisms.Descent

Goal state before:
pp: case inr.refine_2
P P' : MorphismProperty Scheme
inst✝³ : P'.IsStableUnderBaseChange
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H₁ : @IsLocalIso ⊓ @Surjective ≤ P'
inst✝ : IsZariskiLocalAtTarget P
H :
  ∀ {R S : CommRingCat} {Y : Scheme} (φ : R ⟶ S) (g : Y ⟶ Spec R),
    P' (Spec.map φ) → P (pullback.fst (Spec.map φ) g) → P g
R : CommRingCat
X Y✝ : Scheme
f : X ⟶ Spec R
g : Y✝ ⟶ Spec R
hf : (P' ⊓ @QuasiCompact) f
h : P (pullback.fst f g)
this : ∀ {X : Scheme} (f : X ⟶ Spec R), (P' ⊓ @QuasiCompact) f → P (pullback.fst f g) → (∃ T, X = Spec T) → P g
hX : ¬∃ T, X = Spec T
x✝ : CompactSpace ↥X
Y : Scheme
p : Y ⟶ X
hsurj : Surjective p
hP' : IsLocalIso p
hY : IsAffine Y
⊢ P (pullback.fst (Y.isoSpec.inv ≫ p) (pullback.fst f g))

Tactic:
]

Goal state after:
case inr.refine_2
P P' : MorphismProperty Scheme
inst✝³ : P'.IsStableUnderBaseChange
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H₁ : @IsLocalIso ⊓ @Surjective ≤ P'
inst✝ : IsZariskiLocalAtTarget P
H :
  ∀ {R S : CommRingCat} {Y : Scheme} (φ : R ⟶ S) (g : Y ⟶ Spec R),
    P' (Spec.map φ) → P (pullback.fst (Spec.map φ) g) → P g
R : CommRingCat
X Y✝ : Scheme
f : X ⟶ Spec R
g : Y✝ ⟶ Spec R
hf : (P' ⊓ @QuasiCompact) f
h : P (pullback.fst f g)
this : ∀ {X : Scheme} (f : X ⟶ Spec R), (P' ⊓ @QuasiCompact) f → P (pullback.fst f g) → (∃ T, X = Spec T) → P g
hX : ¬∃ T, X = Spec T
x✝ : CompactSpace ↥X
Y : Scheme
p : Y ⟶ X
hsurj : Surjective p
hP' : IsLocalIso p
hY : IsAffine Y
⊢ P (pullback.fst (Y.isoSpec.inv ≫ p) (pullback.fst f g))
Module: Mathlib.AlgebraicGeometry.Morphisms.FlatDescent

Goal state before:
pp: X Y Z : Scheme
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : HasPullback f g
hf : (@Surjective ⊓ @Flat ⊓ @QuasiCompact) f
hg : IsOpenImmersion (pullback.fst f g)
this✝ : UniversallyOpen g
U : Z.Opens := { carrier := Set.range ⇑g, is_open' := ⋯ }
f' : pullback f U.ι ⟶ ↑U := pullback.snd f U.ι
g' : Y ⟶ ↑U := IsOpenImmersion.lift U.ι g ⋯
this : Surjective g'
⊢ IsOpenImmersion (pullback.fst f' g') ∧ Surjective (pullback.fst f' g')

Tactic:
]

Goal state after:
X Y Z : Scheme
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : HasPullback f g
hf : (@Surjective ⊓ @Flat ⊓ @QuasiCompact) f
hg : IsOpenImmersion (pullback.fst f g)
this✝ : UniversallyOpen g
U : Z.Opens := { carrier := Set.range ⇑g, is_open' := ⋯ }
f' : pullback f U.ι ⟶ ↑U := pullback.snd f U.ι
g' : Y ⟶ ↑U := IsOpenImmersion.lift U.ι g ⋯
this : Surjective g'
⊢ IsOpenImmersion (pullback.fst f' g') ∧ Surjective (pullback.fst f' g')
Module: Mathlib.AlgebraicGeometry.Morphisms.FlatDescent

Goal state before:
pp: X Y Z : Scheme
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : HasPullback f g
hf : (@Surjective ⊓ @Flat ⊓ @QuasiCompact) f
hg : IsOpenImmersion (pullback.fst f g)
this✝ : UniversallyOpen g
U : Z.Opens := { carrier := Set.range ⇑g, is_open' := ⋯ }
f' : pullback f U.ι ⟶ ↑U := pullback.snd f U.ι
g' : Y ⟶ ↑U := IsOpenImmersion.lift U.ι g ⋯
this : Surjective g'
⊢ IsOpenImmersion (pullback.fst f g)

Tactic:
]

Goal state after:
X Y Z : Scheme
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : HasPullback f g
hf : (@Surjective ⊓ @Flat ⊓ @QuasiCompact) f
hg : IsOpenImmersion (pullback.fst f g)
this✝ : UniversallyOpen g
U : Z.Opens := { carrier := Set.range ⇑g, is_open' := ⋯ }
f' : pullback f U.ι ⟶ ↑U := pullback.snd f U.ι
g' : Y ⟶ ↑U := IsOpenImmersion.lift U.ι g ⋯
this : Surjective g'
⊢ IsOpenImmersion (pullback.fst f g)
Module: Mathlib.AlgebraicGeometry.Morphisms.FlatDescent

Goal state before:
pp: X Y Z : Scheme
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : HasPullback f g
hf : (@Surjective ⊓ @Flat ⊓ @QuasiCompact) f
hg : IsOpenImmersion (pullback.fst f g)
this✝¹ : UniversallyOpen g
U : Z.Opens := { carrier := Set.range ⇑g, is_open' := ⋯ }
f' : pullback f U.ι ⟶ ↑U := pullback.snd f U.ι
g' : Y ⟶ ↑U := IsOpenImmersion.lift U.ι g ⋯
this✝ : Surjective g'
this : IsOpenImmersion (pullback.fst f (g' ≫ U.ι))
⊢ IsOpenImmersion ((pullbackLeftPullbackSndIso f U.ι g').hom ≫ pullback.fst f (g' ≫ U.ι))

Tactic:
]

Goal state after:
X Y Z : Scheme
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : HasPullback f g
hf : (@Surjective ⊓ @Flat ⊓ @QuasiCompact) f
hg : IsOpenImmersion (pullback.fst f g)
this✝¹ : UniversallyOpen g
U : Z.Opens := { carrier := Set.range ⇑g, is_open' := ⋯ }
f' : pullback f U.ι ⟶ ↑U := pullback.snd f U.ι
g' : Y ⟶ ↑U := IsOpenImmersion.lift U.ι g ⋯
this✝ : Surjective g'
this : IsOpenImmersion (pullback.fst f (g' ≫ U.ι))
⊢ IsOpenImmersion ((pullbackLeftPullbackSndIso f U.ι g').hom ≫ pullback.fst f (g' ≫ U.ι))
Module: Mathlib.AlgebraicGeometry.Morphisms.FlatDescent

Goal state before:
pp: X Y Z : Scheme
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : HasPullback f g
hf : (@Surjective ⊓ @Flat ⊓ @QuasiCompact) f
hg : IsOpenImmersion (pullback.fst f g)
this✝² : UniversallyOpen g
U : Z.Opens := { carrier := Set.range ⇑g, is_open' := ⋯ }
f' : pullback f U.ι ⟶ ↑U := pullback.snd f U.ι
g' : Y ⟶ ↑U := IsOpenImmersion.lift U.ι g ⋯
this✝¹ : Surjective g'
this✝ : IsOpenImmersion (pullback.fst f (g' ≫ U.ι))
this : IsOpenImmersion (pullback.fst f' g' ≫ pullback.fst f U.ι)
⊢ IsOpenImmersion (pullback.fst f' g')

Tactic:
exact .of_comp _ (pullback.fst _ _)

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

Goal state before:
pp: case inr
P P' : MorphismProperty Scheme
Q Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝³ : P'.IsStableUnderBaseChange
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H₁ : @IsLocalIso ⊓ @Surjective ≤ P'
H₂ : ∀ {R S : CommRingCat} {f : R ⟶ S}, P' (Spec.map f) → Q' (CommRingCat.Hom.hom f)
inst✝ : HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] ↦ Q
hQQ' : RingHom.CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ Q) fun {R S} [CommRing R] [CommRing S] ↦ Q'
R S : CommRingCat
Y : Scheme
φ : R ⟶ S
g : Y ⟶ Spec R
h : P' (Spec.map φ)
hf : P (pullback.fst (Spec.map φ) g)
this : ∀ {Y : Scheme} (g : Y ⟶ Spec R), P (pullback.fst (Spec.map φ) g) → (∃ S, Y = Spec S) → P g
hY : ¬∃ S, Y = Spec S
⊢ ∀ (i : Y.affineCover.I₀), P (Y.affineCover.f i ≫ g)

Tactic:
]

Goal state after:
case inr
P P' : MorphismProperty Scheme
Q Q' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝³ : P'.IsStableUnderBaseChange
inst✝² : P'.IsStableUnderComposition
inst✝¹ : P.IsStableUnderBaseChange
H₁ : @IsLocalIso ⊓ @Surjective ≤ P'
H₂ : ∀ {R S : CommRingCat} {f : R ⟶ S}, P' (Spec.map f) → Q' (CommRingCat.Hom.hom f)
inst✝ : HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] ↦ Q
hQQ' : RingHom.CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ Q) fun {R S} [CommRing R] [CommRing S] ↦ Q'
R S : CommRingCat
Y : Scheme
φ : R ⟶ S
g : Y ⟶ Spec R
h : P' (Spec.map φ)
hf : P (pullback.fst (Spec.map φ) g)
this : ∀ {Y : Scheme} (g : Y ⟶ Spec R), P (pullback.fst (Spec.map φ) g) → (∃ S, Y = Spec S) → P g
hY : ¬∃ S, Y = Spec S
⊢ ∀ (i : Y.affineCover.I₀), P (Y.affineCover.f i ≫ g)
Module: Mathlib.AlgebraicGeometry.Morphisms.FlatDescent

Goal state before:
pp: X Y Z : Scheme
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : HasPullback f g
hf : (@Surjective ⊓ @Flat ⊓ @QuasiCompact) f
hg : IsOpenImmersion (pullback.fst f g)
this✝² : UniversallyOpen g
U : Z.Opens := { carrier := Set.range ⇑g, is_open' := ⋯ }
f' : pullback f U.ι ⟶ ↑U := pullback.snd f U.ι
g' : Y ⟶ ↑U := IsOpenImmersion.lift U.ι g ⋯
this✝¹ : Surjective g'
this✝ : IsIso (pullback.fst f' g')
this : IsIso g'
⊢ IsOpenImmersion (IsOpenImmersion.lift U.ι g ⋯ ≫ U.ι)

Tactic:
]

Goal state after:
X Y Z : Scheme
f : X ⟶ Z
g : Y ⟶ Z
inst✝ : HasPullback f g
hf : (@Surjective ⊓ @Flat ⊓ @QuasiCompact) f
hg : IsOpenImmersion (pullback.fst f g)
this✝² : UniversallyOpen g
U : Z.Opens := { carrier := Set.range ⇑g, is_open' := ⋯ }
f' : pullback f U.ι ⟶ ↑U := pullback.snd f U.ι
g' : Y ⟶ ↑U := IsOpenImmersion.lift U.ι g ⋯
this✝¹ : Surjective g'
this✝ : IsIso (pullback.fst f' g')
this : IsIso g'
⊢ IsOpenImmersion (IsOpenImmersion.lift U.ι g ⋯ ≫ U.ι)
Module: Mathlib.AlgebraicGeometry.Morphisms.FlatDescent

Goal state before:
pp: case refine_1
P : MorphismProperty Scheme
inst✝¹ : P.IsStableUnderBaseChange
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] ↦ Q
h :
  RingHom.CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ Q) fun {R S} [CommRing R] [CommRing S] ↦
    RingHom.FaithfullyFlat
⊢ @Surjective ⊓ @IsLocalIso ≤ @Surjective ⊓ @Flat

Tactic:
]

Goal state after:
case refine_1
P : MorphismProperty Scheme
inst✝¹ : P.IsStableUnderBaseChange
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] ↦ Q
h :
  RingHom.CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ Q) fun {R S} [CommRing R] [CommRing S] ↦
    RingHom.FaithfullyFlat
⊢ @Surjective ⊓ @IsLocalIso ≤ @Surjective ⊓ @Flat
Module: Mathlib.RingTheory.Algebraic.StronglyTranscendental

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
x : S
f : S →ₐ[R] T
hf : Function.Injective ⇑f
h : IsStronglyTranscendental R (f x)
u : S
p : R[X]
hp : (aeval x) p * u = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case refine_2
P : MorphismProperty Scheme
inst✝¹ : P.IsStableUnderBaseChange
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] ↦ Q
h :
  RingHom.CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ Q) fun {R S} [CommRing R] [CommRing S] ↦
    RingHom.FaithfullyFlat
R S : CommRingCat
f : R ⟶ S
hf₁ : Surjective (Spec.map f)
hf₂ : Flat (Spec.map f)
⊢ (CommRingCat.Hom.hom f).Flat ∧ Function.Surjective (PrimeSpectrum.comap (CommRingCat.Hom.hom f))

Tactic:
]

Goal state after:
case refine_2
P : MorphismProperty Scheme
inst✝¹ : P.IsStableUnderBaseChange
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] ↦ Q
h :
  RingHom.CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ Q) fun {R S} [CommRing R] [CommRing S] ↦
    RingHom.FaithfullyFlat
R S : CommRingCat
f : R ⟶ S
hf₁ : Surjective (Spec.map f)
hf₂ : Flat (Spec.map f)
⊢ (CommRingCat.Hom.hom f).Flat ∧ Function.Surjective (PrimeSpectrum.comap (CommRingCat.Hom.hom f))
Module: Mathlib.RingTheory.RingHom.QuasiFinite

Goal state before:
pp: R : Type u_4
S : Type u_5
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
⊢ Algebra.QuasiFinite R S ↔ Algebra.QuasiFinite R S

Tactic:
]

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

Goal state before:
pp: case refine_2
P : MorphismProperty Scheme
inst✝¹ : P.IsStableUnderBaseChange
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] ↦ Q
h :
  RingHom.CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ Q) fun {R S} [CommRing R] [CommRing S] ↦
    RingHom.FaithfullyFlat
R S : CommRingCat
f : R ⟶ S
hf₁ : Surjective (Spec.map f)
hf₂ : (CommRingCat.Hom.hom f).Flat
⊢ (CommRingCat.Hom.hom f).Flat

Tactic:
]

Goal state after:
case refine_2
P : MorphismProperty Scheme
inst✝¹ : P.IsStableUnderBaseChange
Q : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
inst✝ : HasRingHomProperty P fun {R S} [CommRing R] [CommRing S] ↦ Q
h :
  RingHom.CodescendsAlong (fun {R S} [CommRing R] [CommRing S] ↦ Q) fun {R S} [CommRing R] [CommRing S] ↦
    RingHom.FaithfullyFlat
R S : CommRingCat
f : R ⟶ S
hf₁ : Surjective (Spec.map f)
hf₂ : (CommRingCat.Hom.hom f).Flat
⊢ (CommRingCat.Hom.hom f).Flat
Module: Mathlib.RingTheory.RingHom.QuasiFinite

Goal state before:
pp: R S T : Type u_6
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
H : Algebra.QuasiFinite R T
⊢ Algebra.QuasiFinite S (TensorProduct R S T)

Tactic:
]

Goal state after:
R S T : Type u_6
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Algebra R T
H : Algebra.QuasiFinite R T
⊢ Algebra.QuasiFinite S (TensorProduct R S T)
Module: Mathlib.RingTheory.Algebraic.StronglyTranscendental

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
x : S
f : S →ₐ[R] T
hf : Function.Injective ⇑f
h : IsStronglyTranscendental R (f x)
u : S
p : R[X]
hp : (aeval x) p * u = 0
this : map (algebraMap R S) p * C u = 0
⊢ map (algebraMap R S) p * C u = 0

pp: case hf
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
x : S
f : S →ₐ[R] T
hf : Function.Injective ⇑f
h : IsStronglyTranscendental R (f x)
u : S
p : R[X]
hp : (aeval x) p * u = 0
this : (mapRingHom ↑f) (map (algebraMap R S) p * C u) = 0
⊢ Function.Injective ⇑(mapRingHom ↑f)

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
x : S
f : S →ₐ[R] T
hf : Function.Injective ⇑f
h : IsStronglyTranscendental R (f x)
u : S
p : R[X]
hp : (aeval x) p * u = 0
this : map (algebraMap R S) p * C u = 0
⊢ map (algebraMap R S) p * C u = 0

case hf
R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : CommRing T
inst✝ : Algebra R T
x : S
f : S →ₐ[R] T
hf : Function.Injective ⇑f
h : IsStronglyTranscendental R (f x)
u : S
p : R[X]
hp : (aeval x) p * u = 0
this : (mapRingHom ↑f) (map (algebraMap R S) p * C u) = 0
⊢ Function.Injective ⇑(mapRingHom ↑f)
Module: Mathlib.RingTheory.RingHom.QuasiFinite

Goal state before:
pp: ⊢ OfLocalizationFiniteSpanTarget fun {R S} [CommRing R] [CommRing S] ↦ QuasiFinite

Tactic:
]

Goal state after:
⊢ OfLocalizationFiniteSpanTarget fun {R S} [CommRing R] [CommRing S] ↦ QuasiFinite