mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski

Goal state before:
pp: case refine_2
X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
x✝¹ : X.AffineZariskiSite
V : X.Opens
hV : IsAffineOpen V
x✝ : ⟨V, hV⟩ ⟶ U
f : ↑Γ(X, U.toOpens)
hf : X.basicOpen f = toOpens ⟨V, hV⟩
⊢ Presieve.ofArrows (fun i ↦ U.basicOpen ↑i) (fun i ↦ homOfLE ⋯) { down := { down := ⋯ } } →
    U.presieveOfSections s { down := { down := ⋯ } }

Tactic:
·

Goal state after:
case refine_2
X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
x✝¹ : X.AffineZariskiSite
V : X.Opens
hV : IsAffineOpen V
x✝ : ⟨V, hV⟩ ⟶ U
f : ↑Γ(X, U.toOpens)
hf : X.basicOpen f = toOpens ⟨V, hV⟩
⊢ Presieve.ofArrows (fun i ↦ U.basicOpen ↑i) (fun i ↦ homOfLE ⋯) { down := { down := ⋯ } } →
    U.presieveOfSections s { down := { down := ⋯ } }
Module: Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite

Goal state before:
pp: case refine_1.refine_2
R S : CommRingCat
φ : R ⟶ S
hf : ∀ (x : ↥(Spec R)), LocallyQuasiFinite (Hom.fiberToSpecResidueField (Spec.map φ) x)
algInst✝ : Algebra ↑R ↑S := (CommRingCat.Hom.hom φ).toAlgebra
x : ↥(Spec R)
hP : x.asIdeal.IsPrime
⊢ (Spec R).fromSpecResidueField x ≫ 𝟙 (Spec R) =
    Spec.map (Spec.residueFieldIso R x).inv ≫ Spec.map (CommRingCat.ofHom (algebraMap (↑R) x.asIdeal.ResidueField))

Tactic:
·

Goal state after:
case refine_1.refine_2
R S : CommRingCat
φ : R ⟶ S
hf : ∀ (x : ↥(Spec R)), LocallyQuasiFinite (Hom.fiberToSpecResidueField (Spec.map φ) x)
algInst✝ : Algebra ↑R ↑S := (CommRingCat.Hom.hom φ).toAlgebra
x : ↥(Spec R)
hP : x.asIdeal.IsPrime
⊢ (Spec R).fromSpecResidueField x ≫ 𝟙 (Spec R) =
    Spec.map (Spec.residueFieldIso R x).inv ≫ Spec.map (CommRingCat.ofHom (algebraMap (↑R) x.asIdeal.ResidueField))
Module: Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite

Goal state before:
pp: case refine_2
R S : CommRingCat
φ : R ⟶ S
hf : ∀ (x : ↥(Spec R)), LocallyQuasiFinite (Hom.fiberToSpecResidueField (Spec.map φ) x)
algInst✝ : Algebra ↑R ↑S := (CommRingCat.Hom.hom φ).toAlgebra
x : ↥(Spec R)
hP : x.asIdeal.IsPrime
⊢ (Arrow.mk (Hom.fiberToSpecResidueField (Spec.map φ) x)).right ≅
    (Arrow.mk (Spec.map (CommRingCat.ofHom (algebraMap x.asIdeal.ResidueField (x.asIdeal.Fiber ↑S))))).right

pp: case refine_3
R S : CommRingCat
φ : R ⟶ S
hf : ∀ (x : ↥(Spec R)), LocallyQuasiFinite (Hom.fiberToSpecResidueField (Spec.map φ) x)
algInst✝ : Algebra ↑R ↑S := (CommRingCat.Hom.hom φ).toAlgebra
x : ↥(Spec R)
hP : x.asIdeal.IsPrime
⊢ (asIso
            (pullback.map (Spec.map φ) ((Spec R).fromSpecResidueField x)
              (Spec.map (CommRingCat.ofHom (algebraMap ↑R ↑S)))
              (Spec.map (CommRingCat.ofHom (algebraMap (↑R) x.asIdeal.ResidueField))) (𝟙 (Spec S))
              (Spec.map (Spec.residueFieldIso R x).inv) (𝟙 (Spec R)) ⋯ ⋯) ≪≫
          pullbackSymmetry (Spec.map (CommRingCat.ofHom (algebraMap ↑R ↑S)))
              (Spec.map (CommRingCat.ofHom (algebraMap (↑R) x.asIdeal.ResidueField))) ≪≫
            pullbackSpecIso (↑R) x.asIdeal.ResidueField ↑S).hom ≫
      (Arrow.mk (Spec.map (CommRingCat.ofHom (algebraMap x.asIdeal.ResidueField (x.asIdeal.Fiber ↑S))))).hom =
    (Arrow.mk (Hom.fiberToSpecResidueField (Spec.map φ) x)).hom ≫ Iso.hom ?refine_2

Tactic:
·

Goal state after:
case refine_2
R S : CommRingCat
φ : R ⟶ S
hf : ∀ (x : ↥(Spec R)), LocallyQuasiFinite (Hom.fiberToSpecResidueField (Spec.map φ) x)
algInst✝ : Algebra ↑R ↑S := (CommRingCat.Hom.hom φ).toAlgebra
x : ↥(Spec R)
hP : x.asIdeal.IsPrime
⊢ (Arrow.mk (Hom.fiberToSpecResidueField (Spec.map φ) x)).right ≅
    (Arrow.mk (Spec.map (CommRingCat.ofHom (algebraMap x.asIdeal.ResidueField (x.asIdeal.Fiber ↑S))))).right
Module: Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski

Goal state before:
pp: case mp
X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
V : X.Opens
hV : IsAffineOpen V
f : ⟨V, hV⟩ ⟶ U
⊢ (Sieve.generate (U.presieveOfSections s)).arrows f → ∃ f ∈ s, ∃ g, X.basicOpen (f * g) = toOpens ⟨V, hV⟩

pp: case mpr
X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
V : X.Opens
hV : IsAffineOpen V
f : ⟨V, hV⟩ ⟶ U
⊢ (∃ f ∈ s, ∃ g, X.basicOpen (f * g) = toOpens ⟨V, hV⟩) → (Sieve.generate (U.presieveOfSections s)).arrows f

Tactic:
·

Goal state after:
case mp
X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
V : X.Opens
hV : IsAffineOpen V
f : ⟨V, hV⟩ ⟶ U
⊢ (Sieve.generate (U.presieveOfSections s)).arrows f → ∃ f ∈ s, ∃ g, X.basicOpen (f * g) = toOpens ⟨V, hV⟩
Module: Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite

Goal state before:
pp: case refine_3
R S : CommRingCat
φ : R ⟶ S
hf : ∀ (x : ↥(Spec R)), LocallyQuasiFinite (Hom.fiberToSpecResidueField (Spec.map φ) x)
algInst✝ : Algebra ↑R ↑S := (CommRingCat.Hom.hom φ).toAlgebra
x : ↥(Spec R)
hP : x.asIdeal.IsPrime
⊢ (asIso
            (pullback.map (Spec.map φ) ((Spec R).fromSpecResidueField x)
              (Spec.map (CommRingCat.ofHom (algebraMap ↑R ↑S)))
              (Spec.map (CommRingCat.ofHom (algebraMap (↑R) x.asIdeal.ResidueField))) (𝟙 (Spec S))
              (Spec.map (Spec.residueFieldIso R x).inv) (𝟙 (Spec R)) ⋯ ⋯) ≪≫
          pullbackSymmetry (Spec.map (CommRingCat.ofHom (algebraMap ↑R ↑S)))
              (Spec.map (CommRingCat.ofHom (algebraMap (↑R) x.asIdeal.ResidueField))) ≪≫
            pullbackSpecIso (↑R) x.asIdeal.ResidueField ↑S).hom ≫
      (Arrow.mk (Spec.map (CommRingCat.ofHom (algebraMap x.asIdeal.ResidueField (x.asIdeal.Fiber ↑S))))).hom =
    (Arrow.mk (Hom.fiberToSpecResidueField (Spec.map φ) x)).hom ≫ (asIso (Spec.map (Spec.residueFieldIso R x).inv)).hom

Tactic:
·

Goal state after:
case refine_3
R S : CommRingCat
φ : R ⟶ S
hf : ∀ (x : ↥(Spec R)), LocallyQuasiFinite (Hom.fiberToSpecResidueField (Spec.map φ) x)
algInst✝ : Algebra ↑R ↑S := (CommRingCat.Hom.hom φ).toAlgebra
x : ↥(Spec R)
hP : x.asIdeal.IsPrime
⊢ (asIso
            (pullback.map (Spec.map φ) ((Spec R).fromSpecResidueField x)
              (Spec.map (CommRingCat.ofHom (algebraMap ↑R ↑S)))
              (Spec.map (CommRingCat.ofHom (algebraMap (↑R) x.asIdeal.ResidueField))) (𝟙 (Spec S))
              (Spec.map (Spec.residueFieldIso R x).inv) (𝟙 (Spec R)) ⋯ ⋯) ≪≫
          pullbackSymmetry (Spec.map (CommRingCat.ofHom (algebraMap ↑R ↑S)))
              (Spec.map (CommRingCat.ofHom (algebraMap (↑R) x.asIdeal.ResidueField))) ≪≫
            pullbackSpecIso (↑R) x.asIdeal.ResidueField ↑S).hom ≫
      (Arrow.mk (Spec.map (CommRingCat.ofHom (algebraMap x.asIdeal.ResidueField (x.asIdeal.Fiber ↑S))))).hom =
    (Arrow.mk (Hom.fiberToSpecResidueField (Spec.map φ) x)).hom ≫ (asIso (Spec.map (Spec.residueFieldIso R x).inv)).hom
Module: Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski

Goal state before:
pp: case mpr
X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
V : X.Opens
hV : IsAffineOpen V
f : ⟨V, hV⟩ ⟶ U
⊢ (∃ f ∈ s, ∃ g, X.basicOpen (f * g) = toOpens ⟨V, hV⟩) → (Sieve.generate (U.presieveOfSections s)).arrows f

Tactic:
·

Goal state after:
case mpr
X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
V : X.Opens
hV : IsAffineOpen V
f : ⟨V, hV⟩ ⟶ U
⊢ (∃ f ∈ s, ∃ g, X.basicOpen (f * g) = toOpens ⟨V, hV⟩) → (Sieve.generate (U.presieveOfSections s)).arrows f
Module: Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite

Goal state before:
pp: case inr
X Y : Scheme
f : X ⟶ Y
inst✝ : LocallyOfFiniteType f
H : ∀ (x : ↥Y), IsDiscrete (⇑f ⁻¹' {x})
this :
  ∀ {X Y : Scheme} {f : X ⟶ Y} [LocallyOfFiniteType f],
    (∀ (x : ↥Y), IsDiscrete (⇑f ⁻¹' {x})) → (∃ R, Y = Spec R) → id (LocallyQuasiFinite f)
hY : ¬∃ R, Y = Spec R
⊢ id (LocallyQuasiFinite f)

pp: X Y : Scheme
f : X ⟶ Y
inst✝ : LocallyOfFiniteType f
H : ∀ (x : ↥Y), IsDiscrete (⇑f ⁻¹' {x})
hY : ∃ R, Y = Spec R
⊢ id (LocallyQuasiFinite f)

Tactic:
·

Goal state after:
case inr
X Y : Scheme
f : X ⟶ Y
inst✝ : LocallyOfFiniteType f
H : ∀ (x : ↥Y), IsDiscrete (⇑f ⁻¹' {x})
this :
  ∀ {X Y : Scheme} {f : X ⟶ Y} [LocallyOfFiniteType f],
    (∀ (x : ↥Y), IsDiscrete (⇑f ⁻¹' {x})) → (∃ R, Y = Spec R) → id (LocallyQuasiFinite f)
hY : ¬∃ R, Y = Spec R
⊢ id (LocallyQuasiFinite f)
Module: Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski

Goal state before:
pp: case mp
X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
x : ↥X
hx : x ∈ U.toOpens
⊢ (∃ V, (∃ f ∈ s, ∃ g, X.basicOpen (f * g) = V.toOpens) ∧ ∃ x_1, x ∈ V.toOpens) → ∃ i ∈ s, x ∈ X.basicOpen i

pp: case mpr
X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
x : ↥X
hx : x ∈ U.toOpens
⊢ (∃ i ∈ s, x ∈ X.basicOpen i) → ∃ V, (∃ f ∈ s, ∃ g, X.basicOpen (f * g) = V.toOpens) ∧ ∃ x_1, x ∈ V.toOpens

Tactic:
·

Goal state after:
case mp
X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
x : ↥X
hx : x ∈ U.toOpens
⊢ (∃ V, (∃ f ∈ s, ∃ g, X.basicOpen (f * g) = V.toOpens) ∧ ∃ x_1, x ∈ V.toOpens) → ∃ i ∈ s, x ∈ X.basicOpen i
Module: Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite

Goal state before:
pp: case inr
X : Scheme
R : CommRingCat
f : X ⟶ Spec R
inst✝ : LocallyOfFiniteType f
H : ∀ (x : ↥(Spec R)), IsDiscrete (⇑f ⁻¹' {x})
this :
  ∀ {X : Scheme} (R : CommRingCat) {f : X ⟶ Spec R} [LocallyOfFiniteType f],
    (∀ (x : ↥(Spec R)), IsDiscrete (⇑f ⁻¹' {x})) → (∃ S, X = Spec S) → id (LocallyQuasiFinite f)
hX : ¬∃ S, X = Spec S
⊢ id (LocallyQuasiFinite f)

pp: X : Scheme
R : CommRingCat
f : X ⟶ Spec R
inst✝ : LocallyOfFiniteType f
H : ∀ (x : ↥(Spec R)), IsDiscrete (⇑f ⁻¹' {x})
hX : ∃ S, X = Spec S
⊢ id (LocallyQuasiFinite f)

Tactic:
·

Goal state after:
case inr
X : Scheme
R : CommRingCat
f : X ⟶ Spec R
inst✝ : LocallyOfFiniteType f
H : ∀ (x : ↥(Spec R)), IsDiscrete (⇑f ⁻¹' {x})
this :
  ∀ {X : Scheme} (R : CommRingCat) {f : X ⟶ Spec R} [LocallyOfFiniteType f],
    (∀ (x : ↥(Spec R)), IsDiscrete (⇑f ⁻¹' {x})) → (∃ S, X = Spec S) → id (LocallyQuasiFinite f)
hX : ¬∃ S, X = Spec S
⊢ id (LocallyQuasiFinite f)
Module: Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski

Goal state before:
pp: case mpr
X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
x : ↥X
hx : x ∈ U.toOpens
⊢ (∃ i ∈ s, x ∈ X.basicOpen i) → ∃ V, (∃ f ∈ s, ∃ g, X.basicOpen (f * g) = V.toOpens) ∧ ∃ x_1, x ∈ V.toOpens

Tactic:
·

Goal state after:
case mpr
X : Scheme
U : X.AffineZariskiSite
s : Set ↑Γ(X, U.toOpens)
x : ↥X
hx : x ∈ U.toOpens
⊢ (∃ i ∈ s, x ∈ X.basicOpen i) → ∃ V, (∃ f ∈ s, ∃ g, X.basicOpen (f * g) = V.toOpens) ∧ ∃ x_1, x ∈ V.toOpens
Module: Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite

Goal state before:
pp: case inr
X Y : Scheme
f : X ⟶ Y
inst✝ : LocallyOfFiniteType f
hf : ∀ (x : ↥Y), (⇑f ⁻¹' {x}).Finite
this :
  ∀ {X Y : Scheme} (f : X ⟶ Y) [LocallyOfFiniteType f],
    (∀ (x : ↥Y), (⇑f ⁻¹' {x}).Finite) → (∃ R, Y = Spec R) → id (LocallyQuasiFinite f)
hY : ¬∃ R, Y = Spec R
⊢ id (LocallyQuasiFinite f)

pp: X Y : Scheme
f : X ⟶ Y
inst✝ : LocallyOfFiniteType f
hf : ∀ (x : ↥Y), (⇑f ⁻¹' {x}).Finite
hY : ∃ R, Y = Spec R
⊢ id (LocallyQuasiFinite f)

Tactic:
·

Goal state after:
case inr
X Y : Scheme
f : X ⟶ Y
inst✝ : LocallyOfFiniteType f
hf : ∀ (x : ↥Y), (⇑f ⁻¹' {x}).Finite
this :
  ∀ {X Y : Scheme} (f : X ⟶ Y) [LocallyOfFiniteType f],
    (∀ (x : ↥Y), (⇑f ⁻¹' {x}).Finite) → (∃ R, Y = Spec R) → id (LocallyQuasiFinite f)
hY : ¬∃ R, Y = Spec R
⊢ id (LocallyQuasiFinite f)
Module: Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite

Goal state before:
pp: case inr
X : Scheme
R : CommRingCat
f : X ⟶ Spec R
inst✝ : LocallyOfFiniteType f
hf : ∀ (x : ↥(Spec R)), (⇑f ⁻¹' {x}).Finite
this :
  ∀ {X : Scheme} (R : CommRingCat) (f : X ⟶ Spec R) [LocallyOfFiniteType f],
    (∀ (x : ↥(Spec R)), (⇑f ⁻¹' {x}).Finite) → (∃ S, X = Spec S) → id (LocallyQuasiFinite f)
hX : ¬∃ S, X = Spec S
⊢ id (LocallyQuasiFinite f)

pp: X : Scheme
R : CommRingCat
f : X ⟶ Spec R
inst✝ : LocallyOfFiniteType f
hf : ∀ (x : ↥(Spec R)), (⇑f ⁻¹' {x}).Finite
hX : ∃ S, X = Spec S
⊢ id (LocallyQuasiFinite f)

Tactic:
·

Goal state after:
case inr
X : Scheme
R : CommRingCat
f : X ⟶ Spec R
inst✝ : LocallyOfFiniteType f
hf : ∀ (x : ↥(Spec R)), (⇑f ⁻¹' {x}).Finite
this :
  ∀ {X : Scheme} (R : CommRingCat) (f : X ⟶ Spec R) [LocallyOfFiniteType f],
    (∀ (x : ↥(Spec R)), (⇑f ⁻¹' {x}).Finite) → (∃ S, X = Spec S) → id (LocallyQuasiFinite f)
hX : ¬∃ S, X = Spec S
⊢ id (LocallyQuasiFinite f)
Module: Mathlib.FieldTheory.RatFunc.Basic

Goal state before:
pp: case H.H
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
y✝¹ y✝ : R[X] × ↥R[X]⁰
⊢ (↑(map φ hφ)).toFun
      ({ toFractionRing := Localization.mk y✝¹.1 y✝¹.2 } + { toFractionRing := Localization.mk y✝.1 y✝.2 }) =
    (↑(map φ hφ)).toFun { toFractionRing := Localization.mk y✝¹.1 y✝¹.2 } +
      (↑(map φ hφ)).toFun { toFractionRing := Localization.mk y✝.1 y✝.2 }

Tactic:
·

Goal state after:
case H.H
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
y✝¹ y✝ : R[X] × ↥R[X]⁰
⊢ (↑(map φ hφ)).toFun
      ({ toFractionRing := Localization.mk y✝¹.1 y✝¹.2 } + { toFractionRing := Localization.mk y✝.1 y✝.2 }) =
    (↑(map φ hφ)).toFun { toFractionRing := Localization.mk y✝¹.1 y✝¹.2 } +
      (↑(map φ hφ)).toFun { toFractionRing := Localization.mk y✝.1 y✝.2 }
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 p / φ q = φ p' / φ q'

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'

Tactic:
·

Goal state after:
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 p / φ 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
⊢ NatTrans.Coequifibered α ↔
    ∀ (U : X.AffineZariskiSite) (f : ↑Γ(X, ↑U)),
      IsPushout (X.presheaf.map (homOfLE ⋯).op) (α.app (Opposite.op U)) (α.app (Opposite.op (U.basicOpen f)))
        (F.map (homOfLE ⋯).op)

pp: X : Scheme
F : X.AffineZariskiSiteᵒᵖ ⥤ CommRingCat
α : (toOpensFunctor X).op ⋙ X.presheaf ⟶ F
⊢ (∀ (U : X.AffineZariskiSite) (f : ↑Γ(X, ↑U)),
      IsPushout (X.presheaf.map (homOfLE ⋯).op) (α.app (Opposite.op U)) (α.app (Opposite.op (U.basicOpen f)))
        (F.map (homOfLE ⋯).op)) ↔
    ∀ (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:
·

Goal state after:
X : Scheme
F : X.AffineZariskiSiteᵒᵖ ⥤ CommRingCat
α : (toOpensFunctor X).op ⋙ X.presheaf ⟶ F
⊢ NatTrans.Coequifibered α ↔
    ∀ (U : X.AffineZariskiSite) (f : ↑Γ(X, ↑U)),
      IsPushout (X.presheaf.map (homOfLE ⋯).op) (α.app (Opposite.op U)) (α.app (Opposite.op (U.basicOpen f)))
        (F.map (homOfLE ⋯).op)
Module: Mathlib.FieldTheory.RatFunc.Basic

Goal state before:
pp: case H.H.refine_3
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
y✝ a' : R[X] × ↥R[X]⁰
h : φ y✝.1 / φ ↑y✝.2 = φ a'.1 / φ ↑a'.2
this : φ y✝.1 * φ ↑a'.2 = φ a'.1 * φ ↑y✝.2
⊢ ↑(a'.1, a'.2).2 * (y✝.1, y✝.2).1 = ↑(y✝.1, y✝.2).2 * (a'.1, a'.2).1

pp: case H.H.refine_1
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
y✝ a' : R[X] × ↥R[X]⁰
h : φ y✝.1 / φ ↑y✝.2 = φ a'.1 / φ ↑a'.2
⊢ φ ↑y✝.2 ≠ 0

pp: case H.H.refine_2
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
y✝ a' : R[X] × ↥R[X]⁰
h : φ y✝.1 / φ ↑y✝.2 = φ a'.1 / φ ↑a'.2
⊢ φ ↑a'.2 ≠ 0

Tactic:
·

Goal state after:
case H.H.refine_3
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
y✝ a' : R[X] × ↥R[X]⁰
h : φ y✝.1 / φ ↑y✝.2 = φ a'.1 / φ ↑a'.2
this : φ y✝.1 * φ ↑a'.2 = φ a'.1 * φ ↑y✝.2
⊢ ↑(a'.1, a'.2).2 * (y✝.1, y✝.2).1 = ↑(y✝.1, y✝.2).2 * (a'.1, a'.2).1
Module: Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski

Goal state before:
pp: case e'_2.e'_1.e'_5.e'_3.e'_5.e'_3.e'_3
X : Scheme
D : Cover.RelativeGluingData (directedCover X) := ⋯
F : (directedCover X).I₀ ⥤ Scheme := ⋯
this : (D.functor ⋙ forget).IsLocallyDirected
U : (X.openCoverOfIsOpenCover (fun i ↦ ↑i) ⋯).toPreZeroHypercover.1
e_1✝ : (cocone X).pt.Opens = X.Opens
e_2✝³ :
  Preorder.smallCategory (TopologicalSpace.Opens ↥(cocone X).pt) ≍ Preorder.smallCategory (TopologicalSpace.Opens ↥X)
e_2✝² : ↑(cocone X).pt.toPresheafedSpace = ↑X.toPresheafedSpace
e_4✝ : (cocone X).pt.toPresheafedSpace = X.toPresheafedSpace
e_2✝¹ : (cocone X).pt.toLocallyRingedSpace = X.toLocallyRingedSpace
e_2✝ : (cocone X).pt = X
⊢ colimit.desc F (cocone X) = D.toBase

pp: case e'_2.e'_1.e'_6
X : Scheme
D : Cover.RelativeGluingData (directedCover X) := ⋯
F : (directedCover X).I₀ ⥤ Scheme := ⋯
this : (D.functor ⋙ forget).IsLocallyDirected
U : (X.openCoverOfIsOpenCover (fun i ↦ ↑i) ⋯).toPreZeroHypercover.1
e_1✝ : (cocone X).pt.Opens = X.Opens
⊢ ↑U = Hom.opensRange ((directedCover X).f U)

Tactic:
·

Goal state after:
case e'_2.e'_1.e'_5.e'_3.e'_5.e'_3.e'_3
X : Scheme
D : Cover.RelativeGluingData (directedCover X) := ⋯
F : (directedCover X).I₀ ⥤ Scheme := ⋯
this : (D.functor ⋙ forget).IsLocallyDirected
U : (X.openCoverOfIsOpenCover (fun i ↦ ↑i) ⋯).toPreZeroHypercover.1
e_1✝ : (cocone X).pt.Opens = X.Opens
e_2✝³ :
  Preorder.smallCategory (TopologicalSpace.Opens ↥(cocone X).pt) ≍ Preorder.smallCategory (TopologicalSpace.Opens ↥X)
e_2✝² : ↑(cocone X).pt.toPresheafedSpace = ↑X.toPresheafedSpace
e_4✝ : (cocone X).pt.toPresheafedSpace = X.toPresheafedSpace
e_2✝¹ : (cocone X).pt.toLocallyRingedSpace = X.toLocallyRingedSpace
e_2✝ : (cocone X).pt = X
⊢ colimit.desc F (cocone X) = D.toBase
Module: Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski

Goal state before:
pp: case e'_2.e'_1.e'_6
X : Scheme
D : Cover.RelativeGluingData (directedCover X) := ⋯
F : (directedCover X).I₀ ⥤ Scheme := ⋯
this : (D.functor ⋙ forget).IsLocallyDirected
U : (X.openCoverOfIsOpenCover (fun i ↦ ↑i) ⋯).toPreZeroHypercover.1
e_1✝ : (cocone X).pt.Opens = X.Opens
⊢ ↑U = Hom.opensRange ((directedCover X).f U)

Tactic:
·

Goal state after:
case e'_2.e'_1.e'_6
X : Scheme
D : Cover.RelativeGluingData (directedCover X) := ⋯
F : (directedCover X).I₀ ⥤ Scheme := ⋯
this : (D.functor ⋙ forget).IsLocallyDirected
U : (X.openCoverOfIsOpenCover (fun i ↦ ↑i) ⋯).toPreZeroHypercover.1
e_1✝ : (cocone X).pt.Opens = X.Opens
⊢ ↑U = Hom.opensRange ((directedCover X).f U)
Module: Mathlib.FieldTheory.RatFunc.Basic

Goal state before:
pp: case H.H.refine_1
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
y✝ a' : R[X] × ↥R[X]⁰
h : φ y✝.1 / φ ↑y✝.2 = φ a'.1 / φ ↑a'.2
⊢ φ ↑y✝.2 ≠ 0

pp: case H.H.refine_2
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
y✝ a' : R[X] × ↥R[X]⁰
h : φ y✝.1 / φ ↑y✝.2 = φ a'.1 / φ ↑a'.2
⊢ φ ↑a'.2 ≠ 0

Tactic:
all_goals exact map_ne_zero_of_mem_nonZeroDivisors _ hφ (SetLike.coe_mem _)

Goal state after:
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] →+* L
hφ : R[X]⁰ ≤ Submonoid.comap φ L⁰
x y : R⟮X⟯
h✝ : Subsingleton R
⊢ (liftMonoidWithZeroHom φ.toMonoidWithZeroHom hφ) (x + y) =
    (liftMonoidWithZeroHom φ.toMonoidWithZeroHom hφ) x + (liftMonoidWithZeroHom φ.toMonoidWithZeroHom hφ) y

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] →+* L
hφ : R[X]⁰ ≤ Submonoid.comap φ L⁰
x y : R⟮X⟯
h✝ : Nontrivial R
⊢ (liftMonoidWithZeroHom φ.toMonoidWithZeroHom hφ) (x + y) =
    (liftMonoidWithZeroHom φ.toMonoidWithZeroHom hφ) x + (liftMonoidWithZeroHom φ.toMonoidWithZeroHom hφ) y

Tactic:
·

Goal state after:
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] →+* L
hφ : R[X]⁰ ≤ Submonoid.comap φ L⁰
x y : R⟮X⟯
h✝ : Subsingleton R
⊢ (liftMonoidWithZeroHom φ.toMonoidWithZeroHom hφ) (x + y) =
    (liftMonoidWithZeroHom φ.toMonoidWithZeroHom hφ) x + (liftMonoidWithZeroHom φ.toMonoidWithZeroHom hφ) y
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
  simp only [QuasiFiniteAt, stalkMap_comp, CommRingCat.hom_comp,
    RingHom.QuasiFinite.respectsIso.cancel_right_isIso]

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

Goal state before:
pp: case inr
X Y : Scheme
f : X ⟶ Y
inst✝ : LocallyOfFiniteType f
x : ↥X
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
⊢ QuasiFiniteAt f x ↔ IsOpen {⟨x, ⋯⟩}

pp: X Y : Scheme
f : X ⟶ Y
inst✝ : LocallyOfFiniteType f
x : ↥X
hY : ∃ R, Y = Spec R
⊢ QuasiFiniteAt f x ↔ IsOpen {⟨x, ⋯⟩}

Tactic:
·

Goal state after:
case inr
X Y : Scheme
f : X ⟶ Y
inst✝ : LocallyOfFiniteType f
x : ↥X
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
⊢ QuasiFiniteAt f x ↔ IsOpen {⟨x, ⋯⟩}