mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.FieldTheory.RatFunc.Basic

Goal state before:
pp: K : Type u
inst✝⁶ : CommRing K
G₀ : Type u_1
L : Type u_2
R : Type u_3
S : Type u_4
F : Type u_5
inst✝⁵ : CommGroupWithZero G₀
inst✝⁴ : Field L
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : FunLike F R[X] S[X]
inst✝ : MonoidHomClass F R[X] S[X]
φ : F
hφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰
p : R[X]
q : ↥R[X]⁰
p' : R[X]
q' : ↥R[X]⁰
hq : φ ↑q ∈ S[X]⁰
hq' : φ ↑q' ∈ S[X]⁰
⊢ φ ↑(q * q') ∈ S[X]⁰

Tactic:
by

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

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
inst✝ : LocallyOfFiniteType f
⊢ LocallyQuasiFinite f ↔ ∀ (x : ↥Y), IsDiscrete (⇑f ⁻¹' {x})

Tactic:
by

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

Goal state before:
pp: X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
x : ↥X
hx : x ∈ U.toOpens
f : ↑Γ(X, ↑U)
hfs : f ∈ s
hxf : x ∈ X.basicOpen f
⊢ x ∈ (U.basicOpen (f * 1)).toOpens

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝⁶ : CommRing K
G₀ : Type u_1
L : Type u_2
R : Type u_3
S : Type u_4
F : Type u_5
inst✝⁵ : CommGroupWithZero G₀
inst✝⁴ : Field L
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : FunLike F R[X] S[X]
inst✝ : MonoidHomClass F R[X] S[X]
φ : F
hφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰
x y : R⟮X⟯
⊢ (x * y).liftOn (fun n d ↦ if h : φ d ∈ S[X]⁰ then { toFractionRing := Localization.mk (φ n) ⟨φ d, h⟩ } else 0) ⋯ =
    x.liftOn (fun n d ↦ if h : φ d ∈ S[X]⁰ then { toFractionRing := Localization.mk (φ n) ⟨φ d, h⟩ } else 0) ⋯ *
      y.liftOn (fun n d ↦ if h : φ d ∈ S[X]⁰ then { toFractionRing := Localization.mk (φ n) ⟨φ d, h⟩ } else 0) ⋯

Tactic:
by

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

Goal state before:
pp: X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
⊢ Sieve.generate (U.presieveOfSections s) ∈ (grothendieckTopology X) U ↔ Ideal.span s = ⊤

Tactic:
by

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

Goal state before:
pp: R : Type u_3
S : Type u_4
F : Type u_5
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : FunLike F R[X] S[X]
inst✝ : MonoidHomClass F R[X] S[X]
φ : F
hφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰
n : R[X]
d : ↥R[X]⁰
⊢ (map φ hφ) { toFractionRing := Localization.mk n d } = { toFractionRing := Localization.mk (φ n) ⟨φ ↑d, ⋯⟩ }

Tactic:
by

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

Goal state before:
pp: X : Scheme
U : X.AffineZariskiSite
S : Sieve U
⊢ S ∈ (grothendieckTopology X) U ↔ Ideal.span (sectionsOfPresieve S.arrows) = ⊤

Tactic:
by

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

Goal state before:
pp: X : Scheme
⊢ { I₀ := X.AffineZariskiSite, X := fun U ↦ ↑↑U, f := fun U ↦ (↑U).ι }.presieve₀ ∈
    (precoverage IsOpenImmersion).coverings X

Tactic:
by

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

Goal state before:
pp: R : Type u_3
S : Type u_4
F : Type u_5
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : FunLike F R[X] S[X]
inst✝ : MonoidHomClass F R[X] S[X]
φ : F
hφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰
hf : Function.Injective ⇑φ
⊢ Function.Injective ⇑(map φ hφ)

Tactic:
by

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

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
inst✝ : LocallyOfFiniteType f
hf : ∀ (x : ↥Y), (⇑f ⁻¹' {x}).Finite
⊢ LocallyQuasiFinite f

Tactic:
by

Goal state after:
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
⊢ a ∈ ↑V

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝⁶ : CommRing K
G₀ : Type u_1
L : Type u_2
R : Type u_3
S : Type u_4
F : Type u_5
inst✝⁵ : CommGroupWithZero G₀
inst✝⁴ : Field L
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : FunLike F R[X] S[X]
inst✝ : RingHomClass F R[X] S[X]
φ : F
hφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰
⊢ (↑(map φ hφ)).toFun 0 = 0

Tactic:
by

Goal state after:
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
haV : a ∈ ↑V
f : ↑Γ(X, ↑U)
g : ↑Γ(X, ↑V)
e : X.basicOpen f = X.basicOpen g
hxf : a ∈ X.basicOpen f
⊢ ↑(basicOpen U f) = ↑(basicOpen V g)

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝⁶ : CommRing K
G₀ : Type u_1
L : Type u_2
R : Type u_3
S : Type u_4
F : Type u_5
inst✝⁵ : CommGroupWithZero G₀
inst✝⁴ : Field L
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : FunLike F R[X] S[X]
inst✝ : RingHomClass F R[X] S[X]
φ : F
hφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰
⊢ ∀ (x y : R⟮X⟯), (↑(map φ hφ)).toFun (x + y) = (↑(map φ hφ)).toFun x + (↑(map φ hφ)).toFun y

Tactic:
by

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✝ : Algebra ↑Γ(Y, U) ↑Γ(X, V) := (CommRingCat.Hom.hom (appLE f U V hVU)).toAlgebra
⊢ (CommRingCat.Hom.hom (Y.presheaf.germ U (f x) ⋯)).QuasiFinite

Tactic:
by

Goal state after:
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))
⊢ ∃ k hki hkj y, (pullback.lift (X.homOfLE ⋯) (X.homOfLE ⋯) ⋯) y = x

Tactic:
by

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

Goal state before:
pp: X : Scheme
U V : X.AffineZariskiSite
f : U ⟶ V
⊢ (toOpensFunctor X ⋙ Functor.rightOp X.presheaf ⋙ Scheme.Spec).map f ≫ ⋯.fromSpec =
    ⋯.fromSpec ≫ ((Functor.const X.AffineZariskiSite).obj X).map f

Tactic:
by

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

Goal state before:
pp: 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'
⊢ φ p / φ q = φ p' / φ q'

Tactic:
by

Goal state after:
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))
⊢ IsPushout (X.presheaf.map (homOfLE ⋯).op) αU αUf (F.map (homOfLE ⋯).op) ↔
    IsLocalization.Away ((ConcreteCategory.hom αU) f) ↑(F.obj (Opposite.op (U.basicOpen f)))

Tactic:
algebraize [(X.presheaf.map (homOfLE (X.basicOpen_le f)).op).hom, αU.hom, αUf.hom,
    (F.map (U.basicOpen_le f).hom.op).hom, (F.map (U.basicOpen_le f).hom.op).hom.comp αU.hom]

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)))))
⊢ IsPushout (X.presheaf.map (homOfLE ⋯).op) αU αUf (F.map (homOfLE ⋯).op) ↔
    IsLocalization.Away ((ConcreteCategory.hom αU) f) ↑(F.obj (Opposite.op (U.basicOpen f)))
Module: Mathlib.FieldTheory.RatFunc.Basic

Goal state before:
pp: 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
⊢ RatFunc.liftOn 0 (fun p q ↦ φ p / φ q) ⋯ = 0

Tactic:
by

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

Goal state before:
pp: 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
⊢ RatFunc.liftOn 1 (fun p q ↦ φ p / φ q) ⋯ = 1

Tactic:
by

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

Goal state before:
pp: 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
x y : R⟮X⟯
⊢ (x * y).liftOn (fun p q ↦ φ p / φ q) ⋯ = x.liftOn (fun p q ↦ φ p / φ q) ⋯ * y.liftOn (fun p q ↦ φ p / φ q) ⋯

Tactic:
by

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

Goal state before:
pp: X : Scheme
F : X.AffineZariskiSiteᵒᵖ ⥤ CommRingCat
α : (toOpensFunctor X).op ⋙ X.presheaf ⟶ F
⊢ NatTrans.Coequifibered α ↔
    ∀ (U : X.AffineZariskiSite) (f : ↑Γ(X, ↑U)),
      IsLocalization.Away ((ConcreteCategory.hom (α.app (Opposite.op U))) f) ↑(F.obj (Opposite.op (U.basicOpen f)))

Tactic:
by

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

Goal state before:
pp: X : Scheme
F : X.AffineZariskiSiteᵒᵖ ⥤ CommRingCat
α : (toOpensFunctor X).op ⋙ X.presheaf ⟶ F
H : NatTrans.Coequifibered α
⊢ ∀ ⦃U V : X.AffineZariskiSite⦄ (f : U ⟶ V), IsOpenImmersion ((F.rightOp ⋙ Scheme.Spec).map f)

Tactic:
by

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

Goal state before:
pp: X : Scheme
F : X.AffineZariskiSiteᵒᵖ ⥤ CommRingCat
α : (toOpensFunctor X).op ⋙ X.presheaf ⟶ F
H : NatTrans.Coequifibered α
U : X.AffineZariskiSite
r : ↑Γ(X, ↑U)
⊢ Hom.opensRange ((relativeGluingData H).functor.map (homOfLE ⋯)) =
    PrimeSpectrum.basicOpen ((ConcreteCategory.hom (α.app (Opposite.op U))) r)

Tactic:
by

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

Goal state before:
pp: X : Scheme
F : X.AffineZariskiSiteᵒᵖ ⥤ CommRingCat
α : (toOpensFunctor X).op ⋙ X.presheaf ⟶ F
H : NatTrans.Coequifibered α
U : X.AffineZariskiSite
⊢ (relativeGluingData H).toBase ⁻¹ᵁ ↑U = Hom.opensRange ((relativeGluingData H).cover.f U)

Tactic:
by

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
⊢ (CommRingCat.Hom.hom (appLE f U V hVU)).QuasiFiniteAt (hV.primeIdealOf ⟨x, hxV⟩).asIdeal

Tactic:
by

Goal state after:
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
⊢ Set.range ⇑(pullback.fst (colimit.desc F (cocone X)) (↑U).ι) =
    Set.range ⇑((IsAffineOpen.isoSpec ⋯).hom ≫ colimit.ι F U)

Tactic:
by

Goal state after:
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
⊢ IsIso ((colimit.isColimit F).desc (cocone X))

Tactic:
by

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

Goal state before:
pp: G₀ : Type u_1
R : Type u_3
inst✝² : CommGroupWithZero G₀
inst✝¹ : CommRing R
inst✝ : Nontrivial R
φ : R[X] →*₀ G₀
hφ : Function.Injective ⇑φ
hφ' : R[X]⁰ ≤ Submonoid.comap φ G₀⁰
⊢ Function.Injective ⇑(liftMonoidWithZeroHom φ hφ')

Tactic:
by

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

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
inst✝ : LocallyQuasiFinite f
x : ↥X
⊢ QuasiFiniteAt f x

Tactic:
by

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

Goal state before:
pp: X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
x : ↥X
inst✝ : IsOpenImmersion f
⊢ QuasiFiniteAt (f ≫ g) x ↔ QuasiFiniteAt g (f x)

Tactic:
by

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

Goal state before:
pp: X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
x : ↥X
inst✝ : LocallyQuasiFinite g
⊢ QuasiFiniteAt (f ≫ g) x ↔ QuasiFiniteAt f x

Tactic:
by

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

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
x : ↥X
⊢ QuasiFiniteAt f x ↔ LocallyQuasiFinite (X.fromSpecStalk x ≫ f)

Tactic:
by

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

Goal state before:
pp: 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] →+* L
hφ : R[X]⁰ ≤ Submonoid.comap φ L⁰
x y : R⟮X⟯
⊢ (↑(liftMonoidWithZeroHom φ.toMonoidWithZeroHom hφ)).toFun (x + y) =
    (↑(liftMonoidWithZeroHom φ.toMonoidWithZeroHom hφ)).toFun x +
      (↑(liftMonoidWithZeroHom φ.toMonoidWithZeroHom hφ)).toFun y

Tactic:
by

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

Goal state before:
pp: L : Type u_2
R : Type u_3
inst✝¹ : Field L
inst✝ : CommRing R
φ : R[X] →+* L
hφ : R[X]⁰ ≤ Submonoid.comap φ L⁰
x : R[X]
⊢ (liftRingHom φ hφ) { toFractionRing := (algebraMap R[X] (FractionRing R[X])) x } = φ x

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
⊢ ∀ (a b : K⟮X⟯), a / b = a * b⁻¹

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
⊢ 0⁻¹ = 0

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝³ : CommRing K
inst✝² : IsDomain K
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Algebra R K[X]
⊢ RatFunc.mk ((algebraMap R K[X]) 1) 1 = 1

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝³ : CommRing K
inst✝² : IsDomain K
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Algebra R K[X]
x y : R
⊢ RatFunc.mk ((algebraMap R K[X]) (x * y)) 1 =
    RatFunc.mk ((algebraMap R K[X]) x) 1 * RatFunc.mk ((algebraMap R K[X]) y) 1

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝³ : CommRing K
inst✝² : IsDomain K
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Algebra R K[X]
⊢ RatFunc.mk ((algebraMap R K[X]) 0) 1 = 0

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝³ : CommRing K
inst✝² : IsDomain K
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Algebra R K[X]
x y : R
⊢ RatFunc.mk ((algebraMap R K[X]) (x + y)) 1 =
    RatFunc.mk ((algebraMap R K[X]) x) 1 + RatFunc.mk ((algebraMap R K[X]) y) 1

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝³ : CommRing K
inst✝² : IsDomain K
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Algebra R K[X]
c : R
x : K⟮X⟯
⊢ c • x =
    { toFun := fun x ↦ RatFunc.mk ((algebraMap R K[X]) x) 1, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯,
          map_add' := ⋯ }
        c *
      x

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
x : K[X]
⊢ { toFractionRing := (algebraMap K[X] (FractionRing K[X])) x } = (algebraMap K[X] K⟮X⟯) x

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝³ : CommRing K
inst✝² : IsDomain K
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Algebra R K[X]
r : R
⊢ (toFractionRingRingEquiv K).toFun ((algebraMap R K⟮X⟯) r) = (algebraMap R (FractionRing K[X])) r

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
p q : K[X]
⊢ RatFunc.mk p q = (algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝⁴ : CommRing K
inst✝³ : IsDomain K
R : Type u_1
inst✝² : Monoid R
inst✝¹ : DistribMulAction R K[X]
inst✝ : IsScalarTower R K[X] K[X]
c : R
p q : K[X]
⊢ (algebraMap K[X] K⟮X⟯) (c • p) / (algebraMap K[X] K⟮X⟯) q = c • ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q)

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝³ : CommRing K
inst✝² : IsDomain K
R : Type u_1
inst✝¹ : CommSemiring R
inst✝ : Algebra R K[X]
x : R
⊢ (algebraMap R K⟮X⟯) x = (algebraMap K[X] K⟮X⟯) ((algebraMap R K[X]) x) / (algebraMap K[X] K⟮X⟯) 1

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝⁵ : CommRing K
inst✝⁴ : IsDomain K
R : Type u_1
F : Type u_2
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : FunLike F K[X] R[X]
inst✝ : MonoidHomClass F K[X] R[X]
φ : F
hφ : K[X]⁰ ≤ Submonoid.comap φ R[X]⁰
p q : K[X]
hq : q ≠ 0
⊢ (map φ hφ) ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q) =
    (algebraMap R[X] R⟮X⟯) (φ p) / (algebraMap R[X] R⟮X⟯) (φ q)

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝⁵ : CommRing K
inst✝⁴ : IsDomain K
R : Type u_1
F : Type u_2
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : FunLike F K[X] R[X]
inst✝ : MonoidWithZeroHomClass F K[X] R[X]
φ : F
hφ : K[X]⁰ ≤ Submonoid.comap φ R[X]⁰
p : K[X]
⊢ 0 = (algebraMap K[X] K⟮X⟯) 0 / (algebraMap K[X] K⟮X⟯) 1

Tactic:
by

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

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
inst✝ : LocallyOfFiniteType f
this :
  ∀ {X Y : Scheme} {f : X ⟶ Y} [LocallyOfFiniteType f] {x : ↥X},
    (∃ R, Y = Spec R) → (QuasiFiniteAt f x ↔ IsOpen {⟨x, ⋯⟩})
hY : ¬∃ R, Y = Spec R
i : Y.affineCover.I₀
x : ↥(pullback f (Y.affineCover.f i))
hy : (Y.affineCover.f i) ((pullback.snd f (Y.affineCover.f i)) x) = f ((pullback.fst f (Y.affineCover.f i)) x)
ι : Y.affineCover.X i ⟶ Y := Y.affineCover.f i
⊢ ⇑(pullback.snd f ι) ⁻¹' {(pullback.snd f ι) x} = ⇑(pullback.fst f ι) ⁻¹' ⇑f ⁻¹' {f ((pullback.fst f ι) x)}

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝⁵ : CommRing K
inst✝⁴ : IsDomain K
R : Type u_1
F : Type u_2
inst✝³ : CommRing R
inst✝² : IsDomain R
inst✝¹ : FunLike F K[X] R[X]
inst✝ : MonoidWithZeroHomClass F K[X] R[X]
φ : F
hφ : K[X]⁰ ≤ Submonoid.comap φ R[X]⁰
p q : K[X]
⊢ (map φ hφ) ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q) =
    (algebraMap R[X] R⟮X⟯) (φ p) / (algebraMap R[X] R⟮X⟯) (φ q)

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝² : CommRing K
inst✝¹ : IsDomain K
L : Type u_1
inst✝ : CommGroupWithZero L
φ : K[X] →*₀ L
hφ : K[X]⁰ ≤ Submonoid.comap φ L⁰
p q : K[X]
⊢ (liftMonoidWithZeroHom φ hφ) ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q) = φ p / φ q

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝² : CommRing K
inst✝¹ : IsDomain K
L : Type u_1
inst✝ : CommGroupWithZero L
φ : K[X] →*₀ L
hφ : K[X]⁰ ≤ Submonoid.comap φ L⁰
p q : K[X]
⊢ (liftMonoidWithZeroHom φ hφ) ((algebraMap K[X] K⟮X⟯) p) / (liftMonoidWithZeroHom φ hφ) ((algebraMap K[X] K⟮X⟯) q) =
    φ p / φ q

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝² : CommRing K
inst✝¹ : IsDomain K
L : Type u_1
inst✝ : Field L
φ : K[X] →+* L
hφ : K[X]⁰ ≤ Submonoid.comap φ L⁰
x : K[X]
⊢ (liftRingHom φ hφ) ((algebraMap K[X] K⟮X⟯) x) = φ x

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
⊢ Function.Injective ⇑(algebraMap K[X] K⟮X⟯)

Tactic:
by

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

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
inst✝ : LocallyOfFiniteType f
this :
  ∀ {X Y : Scheme} {f : X ⟶ Y} [LocallyOfFiniteType f] {x : ↥X},
    (∃ R, Y = Spec R) → (QuasiFiniteAt f x ↔ IsOpen {⟨x, ⋯⟩})
hY : ¬∃ R, Y = Spec R
i : Y.affineCover.I₀
x : ↥(pullback f (Y.affineCover.f i))
hy : (Y.affineCover.f i) ((pullback.snd f (Y.affineCover.f i)) x) = f ((pullback.fst f (Y.affineCover.f i)) x)
ι : Y.affineCover.X i ⟶ Y := Y.affineCover.f i
H : ⇑(pullback.snd f ι) ⁻¹' {(pullback.snd f ι) x} = ⇑(pullback.fst f ι) ⁻¹' ⇑f ⁻¹' {f ((pullback.fst f ι) x)}
f' : ↑(⇑(pullback.snd f ι) ⁻¹' {(pullback.snd f ι) x}) → ↑(⇑f ⁻¹' {f ((pullback.fst f ι) x)}) :=
  Set.MapsTo.restrict (⇑(pullback.fst f ι)) (⇑(pullback.snd f ι) ⁻¹' {(pullback.snd f ι) x})
    (⇑f ⁻¹' {f ((pullback.fst f ι) x)}) ⋯
⊢ Topology.IsOpenEmbedding f'

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝⁸ : CommRing K
inst✝⁷ : IsDomain K
L : Type u_1
R : Type u_2
S : Type u_3
inst✝⁶ : Field L
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : CommSemiring S
inst✝² : Algebra S K[X]
inst✝¹ : Algebra S L
inst✝ : Algebra S R[X]
φ✝ : K[X] →ₐ[S] L
hφ✝ : K[X]⁰ ≤ Submonoid.comap φ✝ L⁰
φ : K[X] →ₐ[S] R[X]
hφ : K[X]⁰ ≤ Submonoid.comap φ R[X]⁰
r : S
⊢ (↑↑(mapRingHom φ hφ)).toFun ((algebraMap S K⟮X⟯) r) = (algebraMap S R⟮X⟯) r

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝⁸ : CommRing K
inst✝⁷ : IsDomain K
L : Type u_1
R : Type u_2
S : Type u_3
inst✝⁶ : Field L
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : CommSemiring S
inst✝² : Algebra S K[X]
inst✝¹ : Algebra S L
inst✝ : Algebra S R[X]
φ : K[X] →ₐ[S] L
hφ : K[X]⁰ ≤ Submonoid.comap φ L⁰
r : S
⊢ (↑↑(liftRingHom φ.toRingHom hφ)).toFun ((algebraMap S K⟮X⟯) r) = (algebraMap S L) r

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
y : ↥K[X]⁰
⊢ IsUnit ((algebraMap K[X] K⟮X⟯) ↑y)

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
⊢ ∀ (z : K⟮X⟯), ∃ x, z * (algebraMap K[X] K⟮X⟯) ↑x.2 = (algebraMap K[X] K⟮X⟯) x.1

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
x y : K[X]
⊢ (algebraMap K[X] K⟮X⟯) x = (algebraMap K[X] K⟮X⟯) y → ∃ c, ↑c * x = ↑c * y

Tactic:
by

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

Goal state before:
pp: K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
x : K[X]
hx : x ≠ 0
⊢ (algebraMap K[X] K⟮X⟯) x ≠ 0

Tactic:
by

Goal state after: