mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.QuasiFinite.Weakly

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
p : Ideal R
q : Ideal S
inst✝⁴ : q.IsPrime
inst✝³ : p.IsPrime
inst✝² : q.LiesOver p
Q : Ideal (p.Fiber S)
inst✝¹ : Q.IsPrime
hQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q
inst✝ : QuasiFiniteAt p.ResidueField Q
Sq : Type u_2 := Localization.AtPrime q
this✝ : Algebra (Localization.AtPrime p) (Localization.AtPrime q) := Localization.AtPrime.algebraOfLiesOver p q
φ₁ : p.ResidueField →ₐ[R] Sq ⧸ Ideal.map (algebraMap R Sq) p :=
  Ideal.quotientMapₐ (Ideal.map (algebraMap R Sq) p) (IsScalarTower.toAlgHom R (Localization.AtPrime p) Sq) ⋯
φ₂ : p.Fiber S →ₐ[R] Sq ⧸ Ideal.map (algebraMap R Sq) p :=
  TensorProduct.lift φ₁ (IsScalarTower.toAlgHom R S (Sq ⧸ Ideal.map (algebraMap R Sq) p)) ⋯
y : p.Fiber S
hy : y ∈ Q.primeCompl
r : R
hrp : r ∉ p
s : S
H : r • y = 1 ⊗ₜ[R] s
this : IsUnit ((algebraMap R (Sq ⧸ Ideal.map (algebraMap R Sq) p)) r) ∧ IsUnit (φ₂ y)
⊢ IsUnit (φ₂ ↑⟨y, hy⟩)

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
p : Ideal R
q : Ideal S
inst✝⁴ : q.IsPrime
inst✝³ : p.IsPrime
inst✝² : q.LiesOver p
Q : Ideal (p.Fiber S)
inst✝¹ : Q.IsPrime
hQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q
inst✝ : QuasiFiniteAt p.ResidueField Q
Sq : Type u_2 := Localization.AtPrime q
this✝ : Algebra (Localization.AtPrime p) (Localization.AtPrime q) := Localization.AtPrime.algebraOfLiesOver p q
φ₁ : p.ResidueField →ₐ[R] Sq ⧸ Ideal.map (algebraMap R Sq) p :=
  Ideal.quotientMapₐ (Ideal.map (algebraMap R Sq) p) (IsScalarTower.toAlgHom R (Localization.AtPrime p) Sq) ⋯
φ₂ : p.Fiber S →ₐ[R] Sq ⧸ Ideal.map (algebraMap R Sq) p :=
  TensorProduct.lift φ₁ (IsScalarTower.toAlgHom R S (Sq ⧸ Ideal.map (algebraMap R Sq) p)) ⋯
y : p.Fiber S
hy : y ∈ Q.primeCompl
r : R
hrp : r ∉ p
s : S
H : r • y = 1 ⊗ₜ[R] s
this : IsUnit ((algebraMap R (Sq ⧸ Ideal.map (algebraMap R Sq) p)) r) ∧ IsUnit (φ₂ y)
⊢ IsUnit (φ₂ ↑⟨y, hy⟩)
Module: Mathlib.RingTheory.LocalRing.ResidueField.Polynomial

Goal state before:
pp: case refine_2
R : Type u_1
inst✝¹ : CommRing R
P : Ideal R
inst✝ : P.IsPrime
I : Ideal R[X]
p : P.ResidueField[X]
hp : Ideal.map (mapRingHom (algebraMap R P.ResidueField)) I = Ideal.span {p}
this✝¹ : Algebra (R ⧸ P)[X] P.ResidueField[X] := (mapRingHom (algebraMap (R ⧸ P) P.ResidueField)).toAlgebra
this✝ : IsLocalization (Submonoid.map C (R ⧸ P)⁰) P.ResidueField[X]
this : p ∈ Ideal.map (algebraMap (R ⧸ P)[X] P.ResidueField[X]) (Ideal.map (mapRingHom (algebraMap R (R ⧸ P))) I)
s : ↥(Submonoid.map C (R ⧸ P)⁰)
r : R[X]
hr' : r ∈ I
hr : (mapRingHom (algebraMap R (R ⧸ P))) r ∈ Ideal.map (mapRingHom (algebraMap R (R ⧸ P))) I
e : p * map (algebraMap (R ⧸ P) P.ResidueField) ↑s = map (algebraMap R P.ResidueField) r
⊢ map (algebraMap R P.ResidueField) r * ↑⋯.unit⁻¹ ∈ Ideal.span {map (algebraMap R P.ResidueField) r}

Tactic:
]

Goal state after:
case refine_2
R : Type u_1
inst✝¹ : CommRing R
P : Ideal R
inst✝ : P.IsPrime
I : Ideal R[X]
p : P.ResidueField[X]
hp : Ideal.map (mapRingHom (algebraMap R P.ResidueField)) I = Ideal.span {p}
this✝¹ : Algebra (R ⧸ P)[X] P.ResidueField[X] := (mapRingHom (algebraMap (R ⧸ P) P.ResidueField)).toAlgebra
this✝ : IsLocalization (Submonoid.map C (R ⧸ P)⁰) P.ResidueField[X]
this : p ∈ Ideal.map (algebraMap (R ⧸ P)[X] P.ResidueField[X]) (Ideal.map (mapRingHom (algebraMap R (R ⧸ P))) I)
s : ↥(Submonoid.map C (R ⧸ P)⁰)
r : R[X]
hr' : r ∈ I
hr : (mapRingHom (algebraMap R (R ⧸ P))) r ∈ Ideal.map (mapRingHom (algebraMap R (R ⧸ P))) I
e : p * map (algebraMap (R ⧸ P) P.ResidueField) ↑s = map (algebraMap R P.ResidueField) r
⊢ map (algebraMap R P.ResidueField) r * ↑⋯.unit⁻¹ ∈ Ideal.span {map (algebraMap R P.ResidueField) r}
Module: Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
inst✝ : targetAffineLocally (fun X x f x_1 ↦ IsAffine X ∧ (CommRingCat.Hom.hom (Hom.appTop f)).Finite) f
⊢ LocallyQuasiFinite f

Tactic:
]

Goal state after:
X Y : Scheme
f : X ⟶ Y
inst✝ : targetAffineLocally (fun X x f x_1 ↦ IsAffine X ∧ (CommRingCat.Hom.hom (Hom.appTop f)).Finite) f
⊢ LocallyQuasiFinite f
Module: Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
inst✝ : targetAffineLocally (fun X x f x_1 ↦ IsAffine X ∧ (CommRingCat.Hom.hom (Hom.appTop f)).Finite) f
⊢ affineLocally (fun {R S} [CommRing R] [CommRing S] ↦ RingHom.QuasiFinite) f

Tactic:
]

Goal state after:
X Y : Scheme
f : X ⟶ Y
inst✝ : targetAffineLocally (fun X x f x_1 ↦ IsAffine X ∧ (CommRingCat.Hom.hom (Hom.appTop f)).Finite) f
⊢ affineLocally (fun {R S} [CommRing R] [CommRing S] ↦ RingHom.QuasiFinite) f
Module: Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
inst✝ : IsImmersion f
⊢ LocallyQuasiFinite (Hom.liftCoborder f ≫ (Hom.coborderRange f).ι)

Tactic:
]

Goal state after:
X Y : Scheme
f : X ⟶ Y
inst✝ : IsImmersion f
⊢ LocallyQuasiFinite (Hom.liftCoborder f ≫ (Hom.coborderRange f).ι)
Module: Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski

Goal state before:
pp: X : Scheme
⊢ Monotone toOpens

Tactic:
rintro ⟨U, hU⟩ ⟨V, hV⟩ ⟨f, rfl⟩
  exact X.basicOpen_le _

Goal state after:
Module: Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski

Goal state before:
pp: X : Scheme
U : X.AffineZariskiSite
S : Sieve U
⊢ Sieve.functorPushforward (toOpensFunctor X) S ∈ (Opens.grothendieckTopology ↥X) ((toOpensFunctor X).obj U) ↔
    ∀ x ∈ U.toOpens, ∃ V f, S.arrows f ∧ x ∈ V.toOpens

Tactic:
]

Goal state after:
X : Scheme
U : X.AffineZariskiSite
S : Sieve U
⊢ Sieve.functorPushforward (toOpensFunctor X) S ∈ (Opens.grothendieckTopology ↥X) ((toOpensFunctor X).obj U) ↔
    ∀ x ∈ U.toOpens, ∃ V f, S.arrows f ∧ x ∈ V.toOpens
Module: Mathlib.FieldTheory.RatFunc.Basic

Goal state before:
pp: K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
p : FractionRing K[X]
h : { toFractionRing := p } ≠ 0
hp : p = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.RatFunc.Basic

Goal state before:
pp: case ofFractionRing
K : Type u
inst✝¹ : CommRing K
R : Type u_1
inst✝ : SMul R (FractionRing K[X])
c : R
toFractionRing✝ : FractionRing K[X]
⊢ { toFractionRing := c • toFractionRing✝ }.toFractionRing = c • { toFractionRing := toFractionRing✝ }.toFractionRing

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.RatFunc.Basic

Goal state before:
pp: case H
K : Type u
inst✝ : CommRing K
r : K
y✝ : K[X] × ↥K[X]⁰
⊢ { toFractionRing := Localization.mk (Polynomial.C r * y✝.1) y✝.2 } =
    { toFractionRing := Localization.mk (Polynomial.C r * y✝.1) y✝.2 }

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.RatFunc.Basic

Goal state before:
pp: case pos
K : Type u
inst✝⁴ : CommRing K
R : Type u_1
inst✝³ : IsDomain K
inst✝² : Monoid R
inst✝¹ : DistribMulAction R K[X]
inst✝ : IsScalarTower R K[X] K[X]
c : R
p q : K[X]
this : SMulZeroClass R (FractionRing K[X]) := inferInstance
hq : q = 0
⊢ { toFractionRing := 0 } = { toFractionRing := 0 }

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.RatFunc.Basic

Goal state before:
pp: case neg
K : Type u
inst✝⁴ : CommRing K
R : Type u_1
inst✝³ : IsDomain K
inst✝² : Monoid R
inst✝¹ : DistribMulAction R K[X]
inst✝ : IsScalarTower R K[X] K[X]
c : R
p q : K[X]
this : SMulZeroClass R (FractionRing K[X]) := inferInstance
hq : ¬q = 0
⊢ { toFractionRing := c • Localization.mk p ⟨q, ⋯⟩ } = { toFractionRing := c • Localization.mk p ⟨q, ⋯⟩ }

Tactic:
]

Goal state after:
Module: Mathlib.FieldTheory.RatFunc.Basic

Goal state before:
pp: K : Type u
inst✝⁴ : CommRing K
R : Type u_1
inst✝³ : IsDomain K
inst✝² : Monoid R
inst✝¹ : DistribMulAction R K[X]
inst✝ : IsScalarTower R K[X] K[X]
c : R
p : K[X]
q✝ : K⟮X⟯
q r : K[X]
x✝ : r ≠ 0
⊢ c • p • RatFunc.mk q r = c • p • RatFunc.mk q r

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski

Goal state before:
pp: case mp
X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
f₂ : ↑Γ(X, U.toOpens)
hf₂s : f₂ ∈ s
hW : IsAffineOpen (X.basicOpen f₂)
f₁ : ↑Γ(X, toOpens ⟨X.basicOpen f₂, hW⟩)
hV : IsAffineOpen (X.basicOpen f₁)
f₃ : ↑Γ(X, ↑U)
hf₃ : X.basicOpen f₃ = X.basicOpen f₁
⊢ X.basicOpen f₁ ≤ X.basicOpen f₂

Tactic:
]

Goal state after:
case mp
X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
f₂ : ↑Γ(X, U.toOpens)
hf₂s : f₂ ∈ s
hW : IsAffineOpen (X.basicOpen f₂)
f₁ : ↑Γ(X, toOpens ⟨X.basicOpen f₂, hW⟩)
hV : IsAffineOpen (X.basicOpen f₁)
f₃ : ↑Γ(X, ↑U)
hf₃ : X.basicOpen f₃ = X.basicOpen f₁
⊢ X.basicOpen f₁ ≤ X.basicOpen f₂
Module: Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski

Goal state before:
pp: X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
⊢ (∀ x ∈ U.toOpens, ∃ V f, (Sieve.generate (U.presieveOfSections s)).arrows f ∧ x ∈ V.toOpens) ↔
    ∀ ⦃x : ↥X⦄, x ∈ ↑U → x ∈ ⨆ f, X.basicOpen ↑f

Tactic:
]

Goal state after:
X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
⊢ (∀ x ∈ U.toOpens, ∃ V f, (Sieve.generate (U.presieveOfSections s)).arrows f ∧ x ∈ V.toOpens) ↔
    ∀ ⦃x : ↥X⦄, x ∈ ↑U → x ∈ ⨆ f, X.basicOpen ↑f
Module: Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski

Goal state before:
pp: X : Scheme
U : X.AffineZariskiSite
S : Sieve U
⊢ S ∈ (grothendieckTopology X) U ↔ S ∈ (grothendieckTopology X) U

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski

Goal state before:
pp: X : Scheme
⊢ (∀ (x : ↥X), ∃ i, x ∈ Set.range ⇑({ I₀ := X.AffineZariskiSite, X := fun U ↦ ↑↑U, f := fun U ↦ (↑U).ι }.f i)) ∧
    ∀ (i : { I₀ := X.AffineZariskiSite, X := fun U ↦ ↑↑U, f := fun U ↦ (↑U).ι }.I₀),
      IsOpenImmersion ({ I₀ := X.AffineZariskiSite, X := fun U ↦ ↑↑U, f := fun U ↦ (↑U).ι }.f i)

Tactic:
]

Goal state after:
X : Scheme
⊢ (∀ (x : ↥X), ∃ i, x ∈ Set.range ⇑({ I₀ := X.AffineZariskiSite, X := fun U ↦ ↑↑U, f := fun U ↦ (↑U).ι }.f i)) ∧
    ∀ (i : { I₀ := X.AffineZariskiSite, X := fun U ↦ ↑↑U, f := fun U ↦ (↑U).ι }.I₀),
      IsOpenImmersion ({ I₀ := X.AffineZariskiSite, X := fun U ↦ ↑↑U, f := fun U ↦ (↑U).ι }.f i)
Module: Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski

Goal state before:
pp: X : Scheme
U V : (directedCover X).I₀
x : ↥(pullback ((directedCover X).f U) ((directedCover X).f V))
a : ↥X := (pullback.fst ((directedCover X).f U) ((directedCover X).f V) ≫ (↑U).ι) x
haU : a ∈ ↑U
⊢ (pullback.snd ((directedCover X).f U) ((directedCover X).f V) ≫ (directedCover X).f V) x ∈ ↑V

Tactic:
]

Goal state after:
X : Scheme
U V : (directedCover X).I₀
x : ↥(pullback ((directedCover X).f U) ((directedCover X).f V))
a : ↥X := (pullback.fst ((directedCover X).f U) ((directedCover X).f V) ≫ (↑U).ι) x
haU : a ∈ ↑U
⊢ (pullback.snd ((directedCover X).f U) ((directedCover X).f V) ≫ (directedCover X).f V) x ∈ ↑V
Module: Mathlib.FieldTheory.RatFunc.Basic

Goal state before:
pp: case inl
K : Type u
inst✝⁵ : CommRing K
G₀ : Type u_1
L : Type u_2
R : Type u_3
S : Type u_4
F : Type u_5
inst✝⁴ : CommGroupWithZero G₀
inst✝³ : Field L
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : FunLike F R[X] S[X]
φ : R[X] →*₀ G₀
hφ : R[X]⁰ ≤ Submonoid.comap φ G₀⁰
f : R⟮X⟯
p q p' q' : R[X]
hq : q ∈ R[X]⁰
hq' : q' ∈ R[X]⁰
h : q' * p = q * p'
h✝ : Subsingleton R
⊢ φ q / φ q = φ q / φ q

Tactic:
]

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

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
x : ↥X
hx : QuasiFiniteAt f x
V : X.Opens
hV : IsAffineOpen V
U : Y.Opens
hU : IsAffineOpen U
hVU : V ≤ f ⁻¹ᵁ U
hxV : x ∈ V.carrier
algInst✝¹ : Algebra ↑Γ(Y, U) ↑Γ(X, V) := (CommRingCat.Hom.hom (appLE f U V hVU)).toAlgebra
algInst✝ : Algebra ↑(Y.presheaf.obj (Opposite.op U)) ↑(Y.presheaf.stalk (f x)) :=
  (CommRingCat.Hom.hom (Y.presheaf.germ U (f x) ⋯)).toAlgebra
this : IsLocalization.AtPrime (↑(Y.presheaf.stalk ↑⟨f x, ⋯⟩)) (hU.primeIdealOf ⟨f x, ⋯⟩).asIdeal
⊢ Algebra.QuasiFinite ↑(Y.presheaf.obj (Opposite.op U)) ↑(Y.presheaf.stalk (f x))

Tactic:
]

Goal state after:
X Y : Scheme
f : X ⟶ Y
x : ↥X
hx : QuasiFiniteAt f x
V : X.Opens
hV : IsAffineOpen V
U : Y.Opens
hU : IsAffineOpen U
hVU : V ≤ f ⁻¹ᵁ U
hxV : x ∈ V.carrier
algInst✝¹ : Algebra ↑Γ(Y, U) ↑Γ(X, V) := (CommRingCat.Hom.hom (appLE f U V hVU)).toAlgebra
algInst✝ : Algebra ↑(Y.presheaf.obj (Opposite.op U)) ↑(Y.presheaf.stalk (f x)) :=
  (CommRingCat.Hom.hom (Y.presheaf.germ U (f x) ⋯)).toAlgebra
this : IsLocalization.AtPrime (↑(Y.presheaf.stalk ↑⟨f x, ⋯⟩)) (hU.primeIdealOf ⟨f x, ⋯⟩).asIdeal
⊢ Algebra.QuasiFinite ↑(Y.presheaf.obj (Opposite.op U)) ↑(Y.presheaf.stalk (f x))
Module: Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski

Goal state before:
pp: X : Scheme
U V : X.AffineZariskiSite
f : U ⟶ V
⊢ ⋯.fromSpec = ⋯.fromSpec ≫ 𝟙 X

Tactic:
]

Goal state after:
X : Scheme
U V : X.AffineZariskiSite
f : U ⟶ V
⊢ ⋯.fromSpec = ⋯.fromSpec ≫ 𝟙 X
Module: Mathlib.FieldTheory.RatFunc.Basic

Goal state before:
pp: case inr
K : Type u
inst✝⁵ : CommRing K
G₀ : Type u_1
L : Type u_2
R : Type u_3
S : Type u_4
F : Type u_5
inst✝⁴ : CommGroupWithZero G₀
inst✝³ : Field L
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : FunLike F R[X] S[X]
φ : R[X] →*₀ G₀
hφ : R[X]⁰ ≤ Submonoid.comap φ G₀⁰
f : R⟮X⟯
p q p' q' : R[X]
hq : q ∈ R[X]⁰
hq' : q' ∈ R[X]⁰
h : q' * p = q * p'
h✝ : Nontrivial R
⊢ φ p' * φ q = φ p' * φ q

pp: case inr.hb
K : Type u
inst✝⁵ : CommRing K
G₀ : Type u_1
L : Type u_2
R : Type u_3
S : Type u_4
F : Type u_5
inst✝⁴ : CommGroupWithZero G₀
inst✝³ : Field L
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : FunLike F R[X] S[X]
φ : R[X] →*₀ G₀
hφ : R[X]⁰ ≤ Submonoid.comap φ G₀⁰
f : R⟮X⟯
p q p' q' : R[X]
hq : q ∈ R[X]⁰
hq' : q' ∈ R[X]⁰
h : q' * p = q * p'
h✝ : Nontrivial R
⊢ φ q ≠ 0

pp: case inr.hd
K : Type u
inst✝⁵ : CommRing K
G₀ : Type u_1
L : Type u_2
R : Type u_3
S : Type u_4
F : Type u_5
inst✝⁴ : CommGroupWithZero G₀
inst✝³ : Field L
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : FunLike F R[X] S[X]
φ : R[X] →*₀ G₀
hφ : R[X]⁰ ≤ Submonoid.comap φ G₀⁰
f : R⟮X⟯
p q p' q' : R[X]
hq : q ∈ R[X]⁰
hq' : q' ∈ R[X]⁰
h : q' * p = q * p'
h✝ : Nontrivial R
⊢ φ q' ≠ 0

Tactic:
]

Goal state after:
case inr.hb
K : Type u
inst✝⁵ : CommRing K
G₀ : Type u_1
L : Type u_2
R : Type u_3
S : Type u_4
F : Type u_5
inst✝⁴ : CommGroupWithZero G₀
inst✝³ : Field L
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : FunLike F R[X] S[X]
φ : R[X] →*₀ G₀
hφ : R[X]⁰ ≤ Submonoid.comap φ G₀⁰
f : R⟮X⟯
p q p' q' : R[X]
hq : q ∈ R[X]⁰
hq' : q' ∈ R[X]⁰
h : q' * p = q * p'
h✝ : Nontrivial R
⊢ φ q ≠ 0

case inr.hd
K : Type u
inst✝⁵ : CommRing K
G₀ : Type u_1
L : Type u_2
R : Type u_3
S : Type u_4
F : Type u_5
inst✝⁴ : CommGroupWithZero G₀
inst✝³ : Field L
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : FunLike F R[X] S[X]
φ : R[X] →*₀ G₀
hφ : R[X]⁰ ≤ Submonoid.comap φ G₀⁰
f : R⟮X⟯
p q p' q' : R[X]
hq : q ∈ R[X]⁰
hq' : q' ∈ R[X]⁰
h : q' * p = q * p'
h✝ : Nontrivial R
⊢ φ q' ≠ 0
Module: Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
x : ↥X
hx : QuasiFiniteAt f x
V : X.Opens
hV : IsAffineOpen V
U : Y.Opens
hU : IsAffineOpen U
hVU : V ≤ f ⁻¹ᵁ U
hxV : x ∈ V.carrier
algInst✝¹ : Algebra ↑Γ(Y, U) ↑Γ(X, V) := (CommRingCat.Hom.hom (appLE f U V hVU)).toAlgebra
H : (CommRingCat.Hom.hom (Y.presheaf.germ U (f x) ⋯)).QuasiFinite
algInst✝ : Algebra ↑(X.presheaf.obj (Opposite.op V)) ↑(X.presheaf.stalk x) :=
  (CommRingCat.Hom.hom (X.presheaf.germ V x hxV)).toAlgebra
this : IsLocalization.AtPrime (↑(X.presheaf.stalk ↑⟨x, hxV⟩)) (hV.primeIdealOf ⟨x, hxV⟩).asIdeal
e : ↑(X.presheaf.stalk ↑⟨x, hxV⟩) ≃ₐ[↑Γ(X, V)] Localization.AtPrime (hV.primeIdealOf ⟨x, hxV⟩).asIdeal :=
  IsLocalization.algEquiv (hV.primeIdealOf ⟨x, hxV⟩).asIdeal.primeCompl (↑(X.presheaf.stalk ↑⟨x, hxV⟩))
    (Localization.AtPrime (hV.primeIdealOf ⟨x, hxV⟩).asIdeal)
⊢ (algebraMap (↑Γ(Y, U)) (Localization.AtPrime (hV.primeIdealOf ⟨x, hxV⟩).asIdeal)).QuasiFinite

Tactic:
]

Goal state after:
X Y : Scheme
f : X ⟶ Y
x : ↥X
hx : QuasiFiniteAt f x
V : X.Opens
hV : IsAffineOpen V
U : Y.Opens
hU : IsAffineOpen U
hVU : V ≤ f ⁻¹ᵁ U
hxV : x ∈ V.carrier
algInst✝¹ : Algebra ↑Γ(Y, U) ↑Γ(X, V) := (CommRingCat.Hom.hom (appLE f U V hVU)).toAlgebra
H : (CommRingCat.Hom.hom (Y.presheaf.germ U (f x) ⋯)).QuasiFinite
algInst✝ : Algebra ↑(X.presheaf.obj (Opposite.op V)) ↑(X.presheaf.stalk x) :=
  (CommRingCat.Hom.hom (X.presheaf.germ V x hxV)).toAlgebra
this : IsLocalization.AtPrime (↑(X.presheaf.stalk ↑⟨x, hxV⟩)) (hV.primeIdealOf ⟨x, hxV⟩).asIdeal
e : ↑(X.presheaf.stalk ↑⟨x, hxV⟩) ≃ₐ[↑Γ(X, V)] Localization.AtPrime (hV.primeIdealOf ⟨x, hxV⟩).asIdeal :=
  IsLocalization.algEquiv (hV.primeIdealOf ⟨x, hxV⟩).asIdeal.primeCompl (↑(X.presheaf.stalk ↑⟨x, hxV⟩))
    (Localization.AtPrime (hV.primeIdealOf ⟨x, hxV⟩).asIdeal)
⊢ (algebraMap (↑Γ(Y, U)) (Localization.AtPrime (hV.primeIdealOf ⟨x, hxV⟩).asIdeal)).QuasiFinite
Module: Mathlib.FieldTheory.RatFunc.Basic

Goal state before:
pp: case H.H
K : Type u
inst✝⁵ : CommRing K
G₀ : Type u_1
L : Type u_2
R : Type u_3
S : Type u_4
F : Type u_5
inst✝⁴ : CommGroupWithZero G₀
inst✝³ : Field L
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : FunLike F R[X] S[X]
φ : R[X] →*₀ G₀
hφ : R[X]⁰ ≤ Submonoid.comap φ G₀⁰
y✝¹ y✝ : R[X] × ↥R[X]⁰
⊢ { toFractionRing := Localization.mk (y✝¹.1 * y✝.1) (y✝¹.2 * y✝.2) }.liftOn (fun p q ↦ φ p / φ q) ⋯ =
    { toFractionRing := Localization.mk y✝¹.1 y✝¹.2 }.liftOn (fun p q ↦ φ p / φ q) ⋯ *
      { toFractionRing := Localization.mk y✝.1 y✝.2 }.liftOn (fun p q ↦ φ p / φ q) ⋯

Tactic:
]

Goal state after:
case H.H
K : Type u
inst✝⁵ : CommRing K
G₀ : Type u_1
L : Type u_2
R : Type u_3
S : Type u_4
F : Type u_5
inst✝⁴ : CommGroupWithZero G₀
inst✝³ : Field L
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : FunLike F R[X] S[X]
φ : R[X] →*₀ G₀
hφ : R[X]⁰ ≤ Submonoid.comap φ G₀⁰
y✝¹ y✝ : R[X] × ↥R[X]⁰
⊢ { toFractionRing := Localization.mk (y✝¹.1 * y✝.1) (y✝¹.2 * y✝.2) }.liftOn (fun p q ↦ φ p / φ q) ⋯ =
    { toFractionRing := Localization.mk y✝¹.1 y✝¹.2 }.liftOn (fun p q ↦ φ p / φ q) ⋯ *
      { toFractionRing := Localization.mk y✝.1 y✝.2 }.liftOn (fun p q ↦ φ p / φ q) ⋯
Module: Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski

Goal state before:
pp: X : Scheme
F : X.AffineZariskiSiteᵒᵖ ⥤ CommRingCat
α : (toOpensFunctor X).op ⋙ X.presheaf ⟶ F
U : X.AffineZariskiSite
f : ↑Γ(X, ↑U)
αU : Γ(X, U.toOpens) ⟶ F.obj (Opposite.op U) := α.app (Opposite.op U)
αUf : Γ(X, X.basicOpen f) ⟶ F.obj (Opposite.op (U.basicOpen f)) := α.app (Opposite.op (U.basicOpen f))
algInst✝³ : Algebra ↑Γ(X, U.toOpens) ↑(F.obj (Opposite.op U)) := (CommRingCat.Hom.hom αU).toAlgebra
algInst✝² : Algebra ↑Γ(X, X.basicOpen f) ↑(F.obj (Opposite.op (U.basicOpen f))) := (CommRingCat.Hom.hom αUf).toAlgebra
algInst✝¹ : Algebra ↑(F.obj (Opposite.op U)) ↑(F.obj (Opposite.op (U.basicOpen f))) :=
  (CommRingCat.Hom.hom (F.map ⋯.hom.op)).toAlgebra
algInst✝ : Algebra ↑Γ(X, U.toOpens) ↑(F.obj (Opposite.op (U.basicOpen f))) :=
  ((CommRingCat.Hom.hom (F.map ⋯.hom.op)).comp (CommRingCat.Hom.hom αU)).toAlgebra
scalarTowerInst✝ : IsScalarTower ↑Γ(X, U.toOpens) ↑(F.obj (Opposite.op U)) ↑(F.obj (Opposite.op (U.basicOpen f))) :=
  IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap ↑Γ(X, U.toOpens) ↑(F.obj (Opposite.op (U.basicOpen f)))))
this✝ : IsScalarTower ↑Γ(X, U.toOpens) ↑Γ(X, X.basicOpen f) ↑(F.obj (Opposite.op (U.basicOpen f)))
this : IsLocalization.Away f ↑Γ(X, X.basicOpen f)
⊢ Algebra.IsPushout ↑(X.presheaf.1 (Opposite.op ↑U)) ↑(F.1 (Opposite.op U))
      ↑(X.presheaf.1 (Opposite.op (X.basicOpen f))) ↑(F.1 (Opposite.op (U.basicOpen f))) ↔
    IsLocalization.Away ((ConcreteCategory.hom αU) f) ↑(F.obj (Opposite.op (U.basicOpen f)))

Tactic:
]

Goal state after:
X : Scheme
F : X.AffineZariskiSiteᵒᵖ ⥤ CommRingCat
α : (toOpensFunctor X).op ⋙ X.presheaf ⟶ F
U : X.AffineZariskiSite
f : ↑Γ(X, ↑U)
αU : Γ(X, U.toOpens) ⟶ F.obj (Opposite.op U) := α.app (Opposite.op U)
αUf : Γ(X, X.basicOpen f) ⟶ F.obj (Opposite.op (U.basicOpen f)) := α.app (Opposite.op (U.basicOpen f))
algInst✝³ : Algebra ↑Γ(X, U.toOpens) ↑(F.obj (Opposite.op U)) := (CommRingCat.Hom.hom αU).toAlgebra
algInst✝² : Algebra ↑Γ(X, X.basicOpen f) ↑(F.obj (Opposite.op (U.basicOpen f))) := (CommRingCat.Hom.hom αUf).toAlgebra
algInst✝¹ : Algebra ↑(F.obj (Opposite.op U)) ↑(F.obj (Opposite.op (U.basicOpen f))) :=
  (CommRingCat.Hom.hom (F.map ⋯.hom.op)).toAlgebra
algInst✝ : Algebra ↑Γ(X, U.toOpens) ↑(F.obj (Opposite.op (U.basicOpen f))) :=
  ((CommRingCat.Hom.hom (F.map ⋯.hom.op)).comp (CommRingCat.Hom.hom αU)).toAlgebra
scalarTowerInst✝ : IsScalarTower ↑Γ(X, U.toOpens) ↑(F.obj (Opposite.op U)) ↑(F.obj (Opposite.op (U.basicOpen f))) :=
  IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap ↑Γ(X, U.toOpens) ↑(F.obj (Opposite.op (U.basicOpen f)))))
this✝ : IsScalarTower ↑Γ(X, U.toOpens) ↑Γ(X, X.basicOpen f) ↑(F.obj (Opposite.op (U.basicOpen f)))
this : IsLocalization.Away f ↑Γ(X, X.basicOpen f)
⊢ Algebra.IsPushout ↑(X.presheaf.1 (Opposite.op ↑U)) ↑(F.1 (Opposite.op U))
      ↑(X.presheaf.1 (Opposite.op (X.basicOpen f))) ↑(F.1 (Opposite.op (U.basicOpen f))) ↔
    IsLocalization.Away ((ConcreteCategory.hom αU) f) ↑(F.obj (Opposite.op (U.basicOpen f)))
Module: Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
x : ↥X
hx : QuasiFiniteAt f x
V : X.Opens
hV : IsAffineOpen V
U : Y.Opens
hU : IsAffineOpen U
hVU : V ≤ f ⁻¹ᵁ U
hxV : x ∈ V.carrier
algInst✝¹ : Algebra ↑Γ(Y, U) ↑Γ(X, V) := (CommRingCat.Hom.hom (appLE f U V hVU)).toAlgebra
H : (CommRingCat.Hom.hom (Y.presheaf.germ U (f x) ⋯)).QuasiFinite
algInst✝ : Algebra ↑(X.presheaf.obj (Opposite.op V)) ↑(X.presheaf.stalk x) :=
  (CommRingCat.Hom.hom (X.presheaf.germ V x hxV)).toAlgebra
this : IsLocalization.AtPrime (↑(X.presheaf.stalk ↑⟨x, hxV⟩)) (hV.primeIdealOf ⟨x, hxV⟩).asIdeal
e : ↑(X.presheaf.stalk ↑⟨x, hxV⟩) ≃ₐ[↑Γ(X, V)] Localization.AtPrime (hV.primeIdealOf ⟨x, hxV⟩).asIdeal :=
  IsLocalization.algEquiv (hV.primeIdealOf ⟨x, hxV⟩).asIdeal.primeCompl (↑(X.presheaf.stalk ↑⟨x, hxV⟩))
    (Localization.AtPrime (hV.primeIdealOf ⟨x, hxV⟩).asIdeal)
⊢ ((↑↑e).comp (algebraMap ↑Γ(X, V) ↑(X.presheaf.stalk ↑⟨x, hxV⟩))).comp (algebraMap ↑Γ(Y, U) ↑Γ(X, V)) =
    (e.toRingEquiv.toRingHom.comp (CommRingCat.Hom.hom (X.presheaf.germ V x hxV))).comp
      (CommRingCat.Hom.hom (appLE f U V ⋯))

Tactic:
]

Goal state after:
X Y : Scheme
f : X ⟶ Y
x : ↥X
hx : QuasiFiniteAt f x
V : X.Opens
hV : IsAffineOpen V
U : Y.Opens
hU : IsAffineOpen U
hVU : V ≤ f ⁻¹ᵁ U
hxV : x ∈ V.carrier
algInst✝¹ : Algebra ↑Γ(Y, U) ↑Γ(X, V) := (CommRingCat.Hom.hom (appLE f U V hVU)).toAlgebra
H : (CommRingCat.Hom.hom (Y.presheaf.germ U (f x) ⋯)).QuasiFinite
algInst✝ : Algebra ↑(X.presheaf.obj (Opposite.op V)) ↑(X.presheaf.stalk x) :=
  (CommRingCat.Hom.hom (X.presheaf.germ V x hxV)).toAlgebra
this : IsLocalization.AtPrime (↑(X.presheaf.stalk ↑⟨x, hxV⟩)) (hV.primeIdealOf ⟨x, hxV⟩).asIdeal
e : ↑(X.presheaf.stalk ↑⟨x, hxV⟩) ≃ₐ[↑Γ(X, V)] Localization.AtPrime (hV.primeIdealOf ⟨x, hxV⟩).asIdeal :=
  IsLocalization.algEquiv (hV.primeIdealOf ⟨x, hxV⟩).asIdeal.primeCompl (↑(X.presheaf.stalk ↑⟨x, hxV⟩))
    (Localization.AtPrime (hV.primeIdealOf ⟨x, hxV⟩).asIdeal)
⊢ ((↑↑e).comp (algebraMap ↑Γ(X, V) ↑(X.presheaf.stalk ↑⟨x, hxV⟩))).comp (algebraMap ↑Γ(Y, U) ↑Γ(X, V)) =
    (e.toRingEquiv.toRingHom.comp (CommRingCat.Hom.hom (X.presheaf.germ V x hxV))).comp
      (CommRingCat.Hom.hom (appLE f U V ⋯))
Module: Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski

Goal state before:
pp: X : Scheme
D : Cover.RelativeGluingData (directedCover X) := relativeGluingData ⋯
F : (directedCover X).I₀ ⥤ Scheme := D.functor
this : (D.functor ⋙ forget).IsLocallyDirected
U : (X.openCoverOfIsOpenCover (fun i ↦ ↑i) ⋯).toPreZeroHypercover.1
⊢ ↑(colimit.desc F (cocone X) ⁻¹ᵁ ↑U) = ↑(Hom.opensRange (colimit.ι F U))

Tactic:
]

Goal state after:
X : Scheme
D : Cover.RelativeGluingData (directedCover X) := relativeGluingData ⋯
F : (directedCover X).I₀ ⥤ Scheme := D.functor
this : (D.functor ⋙ forget).IsLocallyDirected
U : (X.openCoverOfIsOpenCover (fun i ↦ ↑i) ⋯).toPreZeroHypercover.1
⊢ ↑(colimit.desc F (cocone X) ⁻¹ᵁ ↑U) = ↑(Hom.opensRange (colimit.ι F U))