mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.Group.Affine

Goal state before:
pp: case h₀
R✝¹ R✝ A : CommRingCat
X M G : Scheme
R S T : Type u
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : CommRing T
inst✝¹ : Algebra R S
inst✝ : Bialgebra R T
this : includeLeftRingHom = algebraMap S (S ⊗[R] T)
⊢ pullback.snd (pullback.snd (Spec.map (ofHom (algebraMap R T))) (Spec.map (ofHom (algebraMap R S))))
        (pullback.snd (Spec.map (ofHom (algebraMap R T))) (Spec.map (ofHom (algebraMap R S)))) ≫
      pullback.snd (Spec.map (ofHom (algebraMap R T))) (Spec.map (ofHom (algebraMap R S))) =
    pullback.snd (pullback (Spec.map (ofHom (algebraMap R T))) (Spec.map (ofHom (algebraMap R S))) ↘ Spec (of S))
        (pullback (Spec.map (ofHom (algebraMap R T))) (Spec.map (ofHom (algebraMap R S))) ↘ Spec (of S)) ≫
      pullback.snd (Spec.map (ofHom (algebraMap R T))) (Spec.map (ofHom (algebraMap R S)))

Tactic:
rfl

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.Adjunction.Basic

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
a✝ b c : B
f : a✝ ⟶ b
g : b ⟶ a✝
a : B
⊢ leftZigzag (ρ_ (𝟙 a)).inv (ρ_ (𝟙 a)).hom = (λ_ (𝟙 a)).hom ≫ (ρ_ (𝟙 a)).inv

Tactic:
by bicategory_coherence

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.Adjunction.Basic

Goal state before:
pp: B : Type u₁
inst✝¹ : Bicategory B
C : Type u₂
inst✝ : Bicategory C
a✝ b c : B
f : a✝ ⟶ b
g : b ⟶ a✝
a : B
⊢ rightZigzag (ρ_ (𝟙 a)).inv (ρ_ (𝟙 a)).hom = (ρ_ (𝟙 a)).hom ≫ (λ_ (𝟙 a)).inv

Tactic:
by bicategory_coherence

Goal state after:
Module: Mathlib.CategoryTheory.Bicategory.Adjunction.Adj

Goal state before:
pp: B : Type u
inst✝ : Bicategory B
a b : Adj B
α β : a ⟶ b
p : Hom₂ α β
⊢ (conjugateEquiv β.adj α.adj).symm p.τr = p.τl

Tactic:
← Hom₂.conjugateEquiv_τl,

Goal state after:
B : Type u
inst✝ : Bicategory B
a b : Adj B
α β : a ⟶ b
p : Hom₂ α β
⊢ (conjugateEquiv β.adj α.adj).symm ((conjugateEquiv β.adj α.adj) p.τl) = p.τl
Module: Mathlib.RingTheory.Flat.Rank

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Flat R S
inst✝ : Module.Finite R S
h : (∀ (p : PrimeSpectrum R), 0 < rankAtStalk S p) ∧ ∀ (p : PrimeSpectrum R), rankAtStalk S p ≤ 1
p : PrimeSpectrum R
⊢ rankAtStalk S p = 1

Tactic:
grind

Goal state after:
Module: Mathlib.RingTheory.Flat.Rank

Goal state before:
pp: R : Type u_3
S : Type u_4
inst✝¹³ : CommRing R
inst✝¹² : CommRing S
inst✝¹¹ : Algebra R S
R' : Type u_5
S' : Type u_6
inst✝¹⁰ : CommRing R'
inst✝⁹ : CommRing S'
inst✝⁸ : Algebra R R'
inst✝⁷ : Algebra S S'
inst✝⁶ : Algebra R' S'
inst✝⁵ : Algebra R S'
inst✝⁴ : IsScalarTower R R' S'
inst✝³ : IsScalarTower R S S'
inst✝² : IsPushout R S R' S'
inst✝¹ : Module.Flat R S
inst✝ : Module.Finite R S
x : PrimeSpectrum R'
⊢ Module.rankAtStalk S' x = Module.rankAtStalk S (PrimeSpectrum.comap (algebraMap R R') x)

Tactic:
have : IsPushout R R' S S' := Algebra.IsPushout.symm inferInstance
  have := Module.rankAtStalk_eq_of_equiv (Algebra.IsPushout.equiv R R' S S').symm.toLinearEquiv
  rw [Module.rankAtStalk_eq_of_equiv (Algebra.IsPushout.equiv R R' S S').symm.toLinearEquiv,
    Module.rankAtStalk_baseChange]

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

Goal state before:
pp: X S Y T : Scheme
f : X ⟶ S
inst✝³ : IsAffine S
inst✝² : IsAffine T
f' : Y ⟶ T
g' : Y ⟶ X
g : T ⟶ S
h : IsPullback g' f' f g
inst✝¹ : Flat f
inst✝ : IsFinite f
t : ↥T
this✝ : IsAffine X
this : IsPushout (Scheme.Hom.appTop f) (Scheme.Hom.appTop g) (Scheme.Hom.appTop g') (Scheme.Hom.appTop f')
⊢ (CommRingCat.Hom.hom (Scheme.Hom.appTop f)).finrank
      (PrimeSpectrum.comap (CommRingCat.Hom.hom (Scheme.Hom.appTop g)) (T.isoSpec.hom t)) =
    (CommRingCat.Hom.hom (Scheme.Hom.appTop f)).finrank ((T.isoSpec.hom ≫ Spec.map (Scheme.Hom.appTop g)) t)

Tactic:
rfl

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

Goal state before:
pp: X S T : Scheme
f : X ⟶ S
g : T ⟶ S
inst✝² : IsAffine T
t : ↥T
inst✝¹ : Flat f
inst✝ : IsFinite f
i : Spec (S.affineOpenCover.X (S.affineOpenCover.idx (g t))) ⟶ S := S.affineOpenCover.f (S.affineOpenCover.idx (g t))
R : CommRingCat
u : Spec R ⟶ pullback i g
hu : IsOpenImmersion u
z : ↥(Spec R)
hyl : (pullback.fst (S.affineOpenCover.f (S.affineOpenCover.idx (g t))) g) (u z) = Exists.choose ⋯
hyr : (pullback.snd (S.affineOpenCover.f (S.affineOpenCover.idx (g t))) g) (u z) = t
⊢ u ≫ pullback.fst i g ≫ i = u ≫ pullback.fst i g ≫ S.affineOpenCover.f (S.affineOpenCover.idx (g t))

Tactic:
rfl

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

Goal state before:
pp: case h
R S : CommRingCat
x : ↥(Spec S)
φ : R ⟶ S
inst✝¹ : (CommRingCat.Hom.hom φ).Flat
inst✝ : (CommRingCat.Hom.hom φ).Finite
algInst✝ : Algebra ↑R ↑S := ⋯
algebraizeInst✝¹ : Module.Flat ↑R ↑S
algebraizeInst✝ : Module.Finite ↑R ↑S
⊢ PrimeSpectrum.comap (algebraMap ↑R ↑S) x = (Spec.map φ) x

Tactic:
rfl

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

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
inst✝ : IsIso f
R : CommRingCat
g : Spec R ⟶ Y
w✝ : IsOpenImmersion g
y : ↥(Spec R)
this : Nontrivial ↑R
⊢ (RingHom.id ↑R).finrank y = 1

pp: case hf₁
X Y : Scheme
f : X ⟶ Y
inst✝ : IsIso f
R : CommRingCat
g : Spec R ⟶ Y
w✝ : IsOpenImmersion g
y : ↥(Spec R)
this : Nontrivial ↑R
⊢ (CommRingCat.Hom.hom (𝟙 R)).Finite

pp: case hf₂
X Y : Scheme
f : X ⟶ Y
inst✝ : IsIso f
R : CommRingCat
g : Spec R ⟶ Y
w✝ : IsOpenImmersion g
y : ↥(Spec R)
this : Nontrivial ↑R
⊢ (CommRingCat.Hom.hom (𝟙 R)).Flat

Tactic:
← Algebra.algebraMap_self,

Goal state after:
X Y : Scheme
f : X ⟶ Y
inst✝ : IsIso f
R : CommRingCat
g : Spec R ⟶ Y
w✝ : IsOpenImmersion g
y : ↥(Spec R)
this : Nontrivial ↑R
⊢ (algebraMap ↑R ↑R).finrank y = 1

case hf₁
X Y : Scheme
f : X ⟶ Y
inst✝ : IsIso f
R : CommRingCat
g : Spec R ⟶ Y
w✝ : IsOpenImmersion g
y : ↥(Spec R)
this : Nontrivial ↑R
⊢ (CommRingCat.Hom.hom (𝟙 R)).Finite

case hf₂
X Y : Scheme
f : X ⟶ Y
inst✝ : IsIso f
R : CommRingCat
g : Spec R ⟶ Y
w✝ : IsOpenImmersion g
y : ↥(Spec R)
this : Nontrivial ↑R
⊢ (CommRingCat.Hom.hom (𝟙 R)).Flat
Module: Mathlib.AlgebraicGeometry.Morphisms.SmoothFiber

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
R S : CommRingCat
φ : R ⟶ S
h : ∀ (y : ↥(Spec R)), Smooth (Scheme.Hom.fiberToSpecResidueField (Spec.map φ) y)
inst✝¹ : (CommRingCat.Hom.hom φ).FinitePresentation
inst✝ : (CommRingCat.Hom.hom φ).Flat
algInst✝ : Algebra ↑R ↑S := (CommRingCat.Hom.hom φ).toAlgebra
algebraizeInst✝¹ : Algebra.FinitePresentation ↑R ↑S
algebraizeInst✝ : Module.Flat ↑R ↑S
p : Ideal ↑R
hp : p.IsPrime
⊢ (algebraMap p.ResidueField (p.Fiber ↑S)).FormallySmooth

Tactic:
refine RingHom.Smooth.formallySmooth ?_

Goal state after:
X Y : Scheme
f : X ⟶ Y
R S : CommRingCat
φ : R ⟶ S
h : ∀ (y : ↥(Spec R)), Smooth (Scheme.Hom.fiberToSpecResidueField (Spec.map φ) y)
inst✝¹ : (CommRingCat.Hom.hom φ).FinitePresentation
inst✝ : (CommRingCat.Hom.hom φ).Flat
algInst✝ : Algebra ↑R ↑S := (CommRingCat.Hom.hom φ).toAlgebra
algebraizeInst✝¹ : Algebra.FinitePresentation ↑R ↑S
algebraizeInst✝ : Module.Flat ↑R ↑S
p : Ideal ↑R
hp : p.IsPrime
⊢ (algebraMap p.ResidueField (p.Fiber ↑S)).Smooth
Module: Mathlib.AlgebraicGeometry.PointsPi

Goal state before:
pp: ι : Type u
R : ι → CommRingCat
U : (Spec (CommRingCat.of ((i : ι) → ↑(R i)))).Opens
V : Set ↥(Spec (CommRingCat.of ((i : ι) → ↑(R i))))
hV : ↑(Scheme.Hom.opensRange (sigmaSpec R)) ⊆ V
hV' : IsCompact V
hVU : V ⊆ ↑U
s : Set ↑(CommRingCat.of ((i : ι) → ↑(R i)))
hs : U.carrierᶜ = zeroLocus s
t : Set ↑(CommRingCat.of ((i : ι) → ↑(R i)))
hts : t ⊆ s
ht : t.Finite
ht' : V ⊆ (zeroLocus t)ᶜ
i : ι
⊢ Set.range (comap (Pi.evalRingHom (fun x ↦ ↑(R x)) i)) ⊆
    Set.range ⇑(Spec.map (CommRingCat.ofHom (Pi.evalRingHom (fun x ↦ ↑(R x)) i)))

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.PointsPi

Goal state before:
pp: ι : Type u
R : ι → CommRingCat
X : Scheme
inst✝¹ : CompactSpace ↥X
inst✝ : ∀ (i : ι), IsLocalRing ↑(R i)
f : (i : ι) → Spec (R i) ⟶ X
𝒰 : X.OpenCover := X.affineCover.finiteSubcover
j : ι → 𝒰.I₀
hj : ∀ (i : ι), Set.range ⇑(f i) ⊆ ↑(Scheme.Hom.opensRange (𝒰.f (j i)))
g : (j₀ : 𝒰.I₀) → Spec (CommRingCat.of ((i : { i // j i = j₀ }) → ↑(R ↑i))) ⟶ 𝒰.X j₀
hg :
  ∀ (j₀ : 𝒰.I₀) (x : { i // j i = j₀ }),
    Spec.map (CommRingCat.ofHom (Pi.evalRingHom (fun x ↦ ↑(R ↑x)) x)) ≫ g j₀ = IsOpenImmersion.lift (𝒰.f j₀) (f ↑x) ⋯
R' : 𝒰.I₀ → CommRingCat := fun j₀ ↦ CommRingCat.of ((i : { i // j i = j₀ }) → ↑(R ↑i))
x✝ : (j₀ : 𝒰.I₀) → ↑(R' j₀)
i : ι
⊢ (fun f x i ↦ f ↑i) ((fun f i ↦ f (j i) ⟨i, ⋯⟩) x✝) (j i) ⟨i, ⋯⟩ = x✝ (j i) ⟨i, ⋯⟩

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology

Goal state before:
pp: A : Type u_1
σ : Type u_2
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
I : ProjectiveSpectrum 𝒜
⊢ f ∉ ↑I.asHomogeneousIdeal ↔ f ∉ I.asHomogeneousIdeal

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology

Goal state before:
pp: A : Type u_1
σ : Type u_2
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
U : Set (ProjectiveSpectrum 𝒜)
⊢ IsOpen U ↔ ∃ s, zeroLocus 𝒜 s = Uᶜ

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology

Goal state before:
pp: A : Type u_1
σ : Type u_2
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
r : A
x : ProjectiveSpectrum 𝒜
⊢ x ∈ ↑(basicOpen 𝒜 r) ↔ r ∉ ↑x.asHomogeneousIdeal

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.Modules.Sheaf

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
inst✝ : IsOpenImmersion f
M : Y.Modules
U V : X.Opens
hUV : Opposite.op V ⟶ Opposite.op U
⊢ (M.restrict f).presheaf.map hUV ≫ (restrictAppIso f M U).hom =
    (restrictAppIso f M V).hom ≫ M.presheaf.map (homOfLE ⋯).op

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁴ : CommRing A
inst✝³ : SetLike σ A
𝒜 : ι → σ
x : Submonoid A
α : Type u_4
inst✝² : SMul α A
inst✝¹ : IsScalarTower α A A
inst✝ : SMulMemClass σ α A
n : α
x✝ : NumDenSameDeg 𝒜 x
⊢ Localization.mk (n • ↑x✝.num) ⟨↑(n • x✝).den, ⋯⟩ = Localization.mk (n • ↑x✝.num) ⟨↑x✝.den, ⋯⟩

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝² : CommRing A
inst✝¹ : SetLike σ A
inst✝ : NegMemClass σ A
𝒜 : ι → σ
x : Submonoid A
y : NumDenSameDeg 𝒜 x
⊢ Localization.mk ↑(-y).num ⟨↑(-y).den, ⋯⟩ = Localization.mk (-↑y.num) ⟨↑y.den, ⋯⟩

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

Goal state before:
pp: A : Type u_1
σ : Type u_2
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
x : ↑(Proj.restrict ⋯).toTopCat
z : HomogeneousLocalization.NumDenSameDeg 𝒜 (Submonoid.powers f)
⊢ ↑z.num ∈ (↑x).asHomogeneousIdeal.toIdeal ↔ ↑z.num ∈ (↑x).asHomogeneousIdeal

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: case e'_2
ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
x : Submonoid A
z : HomogeneousLocalization 𝒜 x
n : ℕ
c1 c2 : NumDenSameDeg 𝒜 x
h : Localization.mk ↑c1.num ⟨↑c1.den, ⋯⟩ = Localization.mk ↑c2.num ⟨↑c2.den, ⋯⟩
⊢ Localization.mk (↑c1.num ^ n) ⟨↑c1.den ^ n, ⋯⟩ = Localization.mk (↑c1.num ^ n) (⟨↑c1.den, ⋯⟩ ^ n)

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: case e'_3
ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
x : Submonoid A
z : HomogeneousLocalization 𝒜 x
n : ℕ
c1 c2 : NumDenSameDeg 𝒜 x
h : Localization.mk ↑c1.num ⟨↑c1.den, ⋯⟩ = Localization.mk ↑c2.num ⟨↑c2.den, ⋯⟩
⊢ Localization.mk (↑c2.num ^ n) ⟨↑c2.den ^ n, ⋯⟩ = Localization.mk (↑c2.num ^ n) (⟨↑c2.den, ⋯⟩ ^ n)

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.Modules.Sheaf

Goal state before:
pp: X Y Z T : Scheme
M N K : X.Modules
φ : M ⟶ N
U✝ V : X.Opens
f : X ⟶ Y
g : Y ⟶ Z
inst✝¹ : IsOpenImmersion f
inst✝ : IsOpenImmersion g
U : X.Opensᵒᵖ
x : ↑(X.ringCatSheaf.obj.obj U)
⊢ X.presheaf.map ((eqToHom ⋯).op ≫ (eqToHom ⋯).op) = X.presheaf.map (𝟙 (Opposite.op (Opposite.unop U)))

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: case e'_2
ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
x : Submonoid A
c1 c2 : NumDenSameDeg 𝒜 x
h : Localization.mk ↑c1.num ⟨↑c1.den, ⋯⟩ = Localization.mk ↑c2.num ⟨↑c2.den, ⋯⟩
c3 c4 : NumDenSameDeg 𝒜 x
h' : Localization.mk ↑c3.num ⟨↑c3.den, ⋯⟩ = Localization.mk ↑c4.num ⟨↑c4.den, ⋯⟩
⊢ Localization.mk (↑c1.den * ↑c3.num + ↑c3.den * ↑c1.num) ⟨↑c1.den * ↑c3.den, ⋯⟩ =
    Localization.mk (↑⟨↑c1.den, ⋯⟩ * ↑c3.num + ↑⟨↑c3.den, ⋯⟩ * ↑c1.num) (⟨↑c1.den, ⋯⟩ * ⟨↑c3.den, ⋯⟩)

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: case e'_3
ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
x : Submonoid A
c1 c2 : NumDenSameDeg 𝒜 x
h : Localization.mk ↑c1.num ⟨↑c1.den, ⋯⟩ = Localization.mk ↑c2.num ⟨↑c2.den, ⋯⟩
c3 c4 : NumDenSameDeg 𝒜 x
h' : Localization.mk ↑c3.num ⟨↑c3.den, ⋯⟩ = Localization.mk ↑c4.num ⟨↑c4.den, ⋯⟩
⊢ Localization.mk (↑c2.den * ↑c4.num + ↑c4.den * ↑c2.num) ⟨↑c2.den * ↑c4.den, ⋯⟩ =
    Localization.mk (↑⟨↑c2.den, ⋯⟩ * ↑c4.num + ↑⟨↑c4.den, ⋯⟩ * ↑c2.num) (⟨↑c2.den, ⋯⟩ * ⟨↑c4.den, ⋯⟩)

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: case e'_2
ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
x : Submonoid A
c1 c2 : NumDenSameDeg 𝒜 x
h : Localization.mk ↑c1.num ⟨↑c1.den, ⋯⟩ = Localization.mk ↑c2.num ⟨↑c2.den, ⋯⟩
c3 c4 : NumDenSameDeg 𝒜 x
h' : Localization.mk ↑c3.num ⟨↑c3.den, ⋯⟩ = Localization.mk ↑c4.num ⟨↑c4.den, ⋯⟩
⊢ Localization.mk (↑c1.num * ↑c3.num) ⟨↑c1.den * ↑c3.den, ⋯⟩ =
    Localization.mk (↑c1.num * ↑c3.num) (⟨↑c1.den, ⋯⟩ * ⟨↑c3.den, ⋯⟩)

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: case e'_3
ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
x : Submonoid A
c1 c2 : NumDenSameDeg 𝒜 x
h : Localization.mk ↑c1.num ⟨↑c1.den, ⋯⟩ = Localization.mk ↑c2.num ⟨↑c2.den, ⋯⟩
c3 c4 : NumDenSameDeg 𝒜 x
h' : Localization.mk ↑c3.num ⟨↑c3.den, ⋯⟩ = Localization.mk ↑c4.num ⟨↑c4.den, ⋯⟩
⊢ Localization.mk (↑c2.num * ↑c4.num) ⟨↑c2.den * ↑c4.den, ⋯⟩ =
    Localization.mk (↑c2.num * ↑c4.num) (⟨↑c2.den, ⋯⟩ * ⟨↑c4.den, ⋯⟩)

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
x : Submonoid A
y1 y2 : NumDenSameDeg 𝒜 x
⊢ Localization.mk ↑(y1 + y2).num ⟨↑(y1 + y2).den, ⋯⟩ =
    Localization.mk (↑⟨↑y1.den, ⋯⟩ * ↑y2.num + ↑⟨↑y2.den, ⋯⟩ * ↑y1.num) (⟨↑y1.den, ⋯⟩ * ⟨↑y2.den, ⋯⟩)

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
x : Submonoid A
y1 y2 : NumDenSameDeg 𝒜 x
⊢ Localization.mk ↑(y1 * y2).num ⟨↑(y1 * y2).den, ⋯⟩ = Localization.mk (↑y1.num * ↑y2.num) (⟨↑y1.den, ⋯⟩ * ⟨↑y2.den, ⋯⟩)

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
x : Submonoid A
y1 y2 : HomogeneousLocalization 𝒜 x
⊢ (y1 - y2).val = (y1 + -y2).val

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁵ : CommRing A
inst✝⁴ : SetLike σ A
inst✝³ : AddSubgroupClass σ A
inst✝² : AddCommMonoid ι
inst✝¹ : DecidableEq ι
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
x : Submonoid A
y : NumDenSameDeg 𝒜 x
n : ℕ
⊢ Localization.mk ↑(y ^ n).num ⟨↑(y ^ n).den, ⋯⟩ = Localization.mk (↑y.num ^ n) (⟨↑y.den, ⋯⟩ ^ n)

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

Goal state before:
pp: A : Type u_1
σ : Type u_2
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
i : ℕ
x : A⁰_ f
hx : (algebraMap (A⁰_ f) (Localization.Away f)) x = Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩
h : x ∈ q.asIdeal
⊢ (algebraMap (A⁰_ f) (Localization.Away f)) x = HomogeneousLocalization.val x

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

Goal state before:
pp: case e_a
A : Type u_1
σ : Type u_2
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a : A
n : ℕ
hn : a ∈ 𝒜 (n * m)
⊢ Localization.mk (a ^ m) (⟨f ^ n, ⋯⟩ ^ m) = Localization.mk (a ^ m) ⟨(f ^ n) ^ m, ⋯⟩

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝¹³ : CommRing A
inst✝¹² : SetLike σ A
inst✝¹¹ : AddSubgroupClass σ A
inst✝¹⁰ : AddCommMonoid ι
inst✝⁹ : DecidableEq ι
𝒜 : ι → σ
inst✝⁸ : GradedRing 𝒜
B : Type u_4
τ : Type u_5
inst✝⁷ : CommRing B
inst✝⁶ : SetLike τ B
inst✝⁵ : AddSubgroupClass τ B
ℬ : ι → τ
inst✝⁴ : GradedRing ℬ
C : Type u_6
ψ : Type u_7
inst✝³ : CommRing C
inst✝² : SetLike ψ C
inst✝¹ : AddSubgroupClass ψ C
𝒞 : ι → ψ
inst✝ : GradedRing 𝒞
P : Submonoid A
Q : Submonoid B
g : 𝒜 →+*ᵍ ℬ
comap_le : P ≤ Submonoid.comap g Q
⊢ Quotient.mk'' { deg := NumDenSameDeg.deg 1, num := ⟨1, ⋯⟩, den := ⟨1, ⋯⟩, den_mem := ⋯ } = 1

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝¹³ : CommRing A
inst✝¹² : SetLike σ A
inst✝¹¹ : AddSubgroupClass σ A
inst✝¹⁰ : AddCommMonoid ι
inst✝⁹ : DecidableEq ι
𝒜 : ι → σ
inst✝⁸ : GradedRing 𝒜
B : Type u_4
τ : Type u_5
inst✝⁷ : CommRing B
inst✝⁶ : SetLike τ B
inst✝⁵ : AddSubgroupClass τ B
ℬ : ι → τ
inst✝⁴ : GradedRing ℬ
C : Type u_6
ψ : Type u_7
inst✝³ : CommRing C
inst✝² : SetLike ψ C
inst✝¹ : AddSubgroupClass ψ C
𝒞 : ι → ψ
inst✝ : GradedRing 𝒞
P : Submonoid A
Q : Submonoid B
g : 𝒜 →+*ᵍ ℬ
comap_le : P ≤ Submonoid.comap g Q
x y : NumDenSameDeg 𝒜 P
⊢ Quotient.mk'' { deg := (x * y).deg, num := ⟨g ↑x.num * g ↑y.num, ⋯⟩, den := ⟨g ↑x.den * g ↑y.den, ⋯⟩, den_mem := ⋯ } =
    mk
      ({ deg := x.deg, num := ⟨g ↑x.num, ⋯⟩, den := ⟨g ↑x.den, ⋯⟩, den_mem := ⋯ } *
        { deg := y.deg, num := ⟨g ↑y.num, ⋯⟩, den := ⟨g ↑y.den, ⋯⟩, den_mem := ⋯ })

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝¹³ : CommRing A
inst✝¹² : SetLike σ A
inst✝¹¹ : AddSubgroupClass σ A
inst✝¹⁰ : AddCommMonoid ι
inst✝⁹ : DecidableEq ι
𝒜 : ι → σ
inst✝⁸ : GradedRing 𝒜
B : Type u_4
τ : Type u_5
inst✝⁷ : CommRing B
inst✝⁶ : SetLike τ B
inst✝⁵ : AddSubgroupClass τ B
ℬ : ι → τ
inst✝⁴ : GradedRing ℬ
C : Type u_6
ψ : Type u_7
inst✝³ : CommRing C
inst✝² : SetLike ψ C
inst✝¹ : AddSubgroupClass ψ C
𝒞 : ι → ψ
inst✝ : GradedRing 𝒞
P : Submonoid A
Q : Submonoid B
g : 𝒜 →+*ᵍ ℬ
comap_le : P ≤ Submonoid.comap g Q
⊢ Quotient.mk'' { deg := 0, num := ⟨0, ⋯⟩, den := ⟨1, ⋯⟩, den_mem := ⋯ } = 0

Tactic:
rfl

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝¹³ : CommRing A
inst✝¹² : SetLike σ A
inst✝¹¹ : AddSubgroupClass σ A
inst✝¹⁰ : AddCommMonoid ι
inst✝⁹ : DecidableEq ι
𝒜 : ι → σ
inst✝⁸ : GradedRing 𝒜
B : Type u_4
τ : Type u_5
inst✝⁷ : CommRing B
inst✝⁶ : SetLike τ B
inst✝⁵ : AddSubgroupClass τ B
ℬ : ι → τ
inst✝⁴ : GradedRing ℬ
C : Type u_6
ψ : Type u_7
inst✝³ : CommRing C
inst✝² : SetLike ψ C
inst✝¹ : AddSubgroupClass ψ C
𝒞 : ι → ψ
inst✝ : GradedRing 𝒞
P : Submonoid A
Q : Submonoid B
g : 𝒜 →+*ᵍ ℬ
comap_le : P ≤ Submonoid.comap g Q
x y : NumDenSameDeg 𝒜 P
⊢ Quotient.mk''
      { deg := (x + y).deg, num := ⟨g ↑x.den * g ↑y.num + g ↑y.den * g ↑x.num, ⋯⟩, den := ⟨g ↑x.den * g ↑y.den, ⋯⟩,
        den_mem := ⋯ } =
    mk
      ({ deg := x.deg, num := ⟨g ↑x.num, ⋯⟩, den := ⟨g ↑x.den, ⋯⟩, den_mem := ⋯ } +
        { deg := y.deg, num := ⟨g ↑y.num, ⋯⟩, den := ⟨g ↑y.den, ⋯⟩, den_mem := ⋯ })

Tactic:
rfl

Goal state after:
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

Goal state before:
pp: A : Type u_1
σ : Type u_2
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
⊢ HomogeneousLocalization.mk
      { deg := m * i + m * i, num := ⟨(proj 𝒜 i) (a + b) ^ m * (proj 𝒜 i) (a + b) ^ m, ⋯⟩, den := ⟨f ^ i * f ^ i, ⋯⟩,
        den_mem := ⋯ } ∈
    q.asIdeal

Tactic:
simp_rw [← pow_add, map_add, add_pow, mul_comm, ← nsmul_eq_mul]

Goal state after:
A : Type u_1
σ : Type u_2
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
m : ℕ
f_deg : f ∈ 𝒜 m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i : ℕ
⊢ HomogeneousLocalization.mk
      { deg := m * i + m * i,
        num := ⟨∑ x ∈ range (m + m + 1), (m + m).choose x • ((proj 𝒜 i) a ^ x * (proj 𝒜 i) b ^ (m + m - x)), ⋯⟩,
        den := ⟨f ^ (i + i), ⋯⟩, den_mem := ⋯ } ∈
    q.asIdeal
Module: Mathlib.AlgebraicGeometry.Modules.Sheaf

Goal state before:
pp: case refine_2
X✝ Y Z T : Scheme
M N K : X✝.Modules
φ : M ⟶ N
U✝ V✝ : X✝.Opens
X : Scheme
U V : X.Opens
f : V ⟶ U
this✝⁴ :
  (Hom.opensFunctor (X.homOfLE ⋯)).IsContinuous (Opens.grothendieckTopology ↥↑V) (Opens.grothendieckTopology ↑U.carrier)
this✝³ :
  Functor.IsDenseSubsite (Opens.grothendieckTopology ↥U) ((Opens.grothendieckTopology ↥X).over U)
    (Opens.overEquivalence U).inverse
this✝² : (Hom.opensFunctor (X.homOfLE ⋯)).IsContinuous (Opens.grothendieckTopology ↥V) (Opens.grothendieckTopology ↥↑U)
this✝¹ :
  ((Opens.overEquivalence V).symm.functor ⋙ Over.map f).IsContinuous (Opens.grothendieckTopology ↥V)
    ((Opens.grothendieckTopology ↥X).over U)
this✝ :
  (Opens.overEquivalence U).symm.functor.IsContinuous (Opens.grothendieckTopology ↥↑U)
    ((Opens.grothendieckTopology ↥X).over U)
this :
  (Hom.opensFunctor (X.homOfLE ⋯) ⋙ (Opens.overEquivalence U).symm.functor).IsContinuous (Opens.grothendieckTopology ↥V)
    ((Opens.grothendieckTopology ↥X).over U)
W : (TopologicalSpace.Opens ↥V)ᵒᵖ
x : ↑(((Opens.sheafRestrict V).obj X.ringCatSheaf).obj.obj W)
⊢ (ConcreteCategory.hom (X.presheaf.map (eqToHom ⋯))) x = (ConcreteCategory.hom (X.presheaf.map (homOfLE ⋯).op)) x

Tactic:
rfl

Goal state after: