mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: R : CommRingCat
M : (Spec R).Modules
S : CommRingCat
f : R ⟶ S
inst✝ : IsOpenImmersion (Spec.map f)
U : (Spec S).Opens
r : ↑R
x : ↑Γ(M.restrict (Spec.map f), U)
⊢ (ConcreteCategory.hom (restrictAppIso (Spec.map f) M U).hom)
      ((ConcreteCategory.hom ((Spec S).presheaf.map (Opens.leTop U).op))
          ((ConcreteCategory.hom (ΓSpecIso S).inv) ((ConcreteCategory.hom f) r)) •
        x) =
    (ConcreteCategory.hom ((Spec R).presheaf.map (Opens.leTop (Spec.map f ''ᵁ U)).op))
        ((ConcreteCategory.hom (ΓSpecIso R).inv) r) •
      (ConcreteCategory.hom (restrictAppIso (Spec.map f) M U).hom) x

Tactic:
[

Goal state after:
R : CommRingCat
M : (Spec R).Modules
S : CommRingCat
f : R ⟶ S
inst✝ : IsOpenImmersion (Spec.map f)
U : (Spec S).Opens
r : ↑R
x : ↑Γ(M.restrict (Spec.map f), U)
⊢ (ConcreteCategory.hom (restrictAppIso (Spec.map f) M U).hom)
      ((ConcreteCategory.hom ((Spec S).presheaf.map (Opens.leTop U).op))
          ((ConcreteCategory.hom (ΓSpecIso S).inv) ((ConcreteCategory.hom f) r)) •
        x) =
    (ConcreteCategory.hom ((Spec R).presheaf.map (Opens.leTop (Spec.map f ''ᵁ U)).op))
        ((ConcreteCategory.hom (ΓSpecIso R).inv) r) •
      (ConcreteCategory.hom (restrictAppIso (Spec.map f) M U).hom) x
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: R : CommRingCat
M : (Spec R).Modules
S : CommRingCat
f : R ⟶ S
inst✝ : IsOpenImmersion (Spec.map f)
U : (Spec S).Opens
r : ↑R
x : ↑Γ(M.restrict (Spec.map f), U)
this :
  f ≫ (ΓSpecIso S).inv ≫ (Spec S).presheaf.map (Opens.leTop U).op ≫ (Hom.appIso (Spec.map f) U).inv =
    (ΓSpecIso R).inv ≫ (Spec R).presheaf.map (Opens.leTop (Spec.map f ''ᵁ U)).op
⊢ (ConcreteCategory.hom
          (f ≫ (ΓSpecIso S).inv ≫ (Spec S).presheaf.map (Opens.leTop U).op ≫ (Hom.appIso (Spec.map f) U).inv))
        r •
      (ConcreteCategory.hom (restrictAppIso (Spec.map f) M U).hom) x =
    (ConcreteCategory.hom ((ΓSpecIso R).inv ≫ (Spec R).presheaf.map (Opens.leTop (Spec.map f ''ᵁ U)).op)) r •
      (ConcreteCategory.hom (restrictAppIso (Spec.map f) M U).hom) x

Tactic:
this

Goal state after:
R : CommRingCat
M : (Spec R).Modules
S : CommRingCat
f : R ⟶ S
inst✝ : IsOpenImmersion (Spec.map f)
U : (Spec S).Opens
r : ↑R
x : ↑Γ(M.restrict (Spec.map f), U)
this :
  f ≫ (ΓSpecIso S).inv ≫ (Spec S).presheaf.map (Opens.leTop U).op ≫ (Hom.appIso (Spec.map f) U).inv =
    (ΓSpecIso R).inv ≫ (Spec R).presheaf.map (Opens.leTop (Spec.map f ''ᵁ U)).op
⊢ (ConcreteCategory.hom ((ΓSpecIso R).inv ≫ (Spec R).presheaf.map (Opens.leTop (Spec.map f ''ᵁ U)).op)) r •
      (ConcreteCategory.hom (restrictAppIso (Spec.map f) M U).hom) x =
    (ConcreteCategory.hom ((ΓSpecIso R).inv ≫ (Spec R).presheaf.map (Opens.leTop (Spec.map f ''ᵁ U)).op)) r •
      (ConcreteCategory.hom (restrictAppIso (Spec.map f) M U).hom) x
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

Goal state before:
pp: case e_y
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
g : ℕ → A⁰_ f :=
  fun j ↦
    (m + m).choose j •
      if h2 : m + m < j then 0
      else
        if h1 : j ≤ m then
          HomogeneousLocalization.mk
              { deg := m * i, num := ⟨(proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
            HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }
        else
          HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
            HomogeneousLocalization.mk
              { deg := m * i, num := ⟨(proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩,
                den_mem := ⋯ }
j : ℕ
hj : j ∈ range (m + m + 1)
h2 : ¬m + m < j
h1 : j ≤ m
⊢ ⟨f ^ (i + i), ⋯⟩ = ⟨f ^ i, ⋯⟩ * ⟨f ^ i, ⋯⟩

Tactic:
[

Goal state after:
case e_y
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
g : ℕ → A⁰_ f :=
  fun j ↦
    (m + m).choose j •
      if h2 : m + m < j then 0
      else
        if h1 : j ≤ m then
          HomogeneousLocalization.mk
              { deg := m * i, num := ⟨(proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
            HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }
        else
          HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
            HomogeneousLocalization.mk
              { deg := m * i, num := ⟨(proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩,
                den_mem := ⋯ }
j : ℕ
hj : j ∈ range (m + m + 1)
h2 : ¬m + m < j
h1 : j ≤ m
⊢ ⟨f ^ (i + i), ⋯⟩ = ⟨f ^ i, ⋯⟩ * ⟨f ^ i, ⋯⟩
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: A : Type u_2
σ : Type u_3
inst✝⁴ : CommRing A
inst✝³ : SetLike σ A
inst✝² : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ℕ
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ℕ
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
hxd' : ∀ (i : ι'), 0 < dv i
a : ℕ
ai : ι' → ℕ
hai : ∑ i, ai i • dv i = a • d
⊢ ∑ i, (ai i % d + d * (ai i / d)) • dv i - (∑ i, dv i * (ai i / d)) * d = ∑ k, (ai k % d) • dv k

Tactic:
[

Goal state after:
A : Type u_2
σ : Type u_3
inst✝⁴ : CommRing A
inst✝³ : SetLike σ A
inst✝² : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ℕ
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ℕ
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
hxd' : ∀ (i : ι'), 0 < dv i
a : ℕ
ai : ι' → ℕ
hai : ∑ i, ai i • dv i = a • d
⊢ ∑ i, (ai i % d + d * (ai i / d)) • dv i - (∑ i, dv i * (ai i / d)) * d = ∑ k, (ai k % d) • dv k
Module: Mathlib.RingTheory.GradedAlgebra.Homogeneous.Maps

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
σ : Type u_4
τ : Type u_5
ω : Type u_6
ι : Type u_7
F : Type u_8
G : Type u_9
inst✝¹³ : Semiring A
inst✝¹² : Semiring B
inst✝¹¹ : Semiring C
inst✝¹⁰ : SetLike σ A
inst✝⁹ : SetLike τ B
inst✝⁸ : SetLike ω C
inst✝⁷ : AddSubmonoidClass σ A
inst✝⁶ : AddSubmonoidClass τ B
inst✝⁵ : AddSubmonoidClass ω C
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
𝒜 : ι → σ
ℬ : ι → τ
𝒞 : ι → ω
inst✝² : GradedRing 𝒜
inst✝¹ : GradedRing ℬ
inst✝ : GradedRing 𝒞
f : 𝒜 →+*ᵍ ℬ
g : ℬ →+*ᵍ 𝒞
I : HomogeneousIdeal ℬ
n : ι
a : A
ha : a ∈ Ideal.comap f I.toIdeal
⊢ f ↑(((DirectSum.decompose 𝒜) a) n) ∈ I

Tactic:
f.map_directSumDecompose

Goal state after:
A : Type u_1
B : Type u_2
C : Type u_3
σ : Type u_4
τ : Type u_5
ω : Type u_6
ι : Type u_7
F : Type u_8
G : Type u_9
inst✝¹³ : Semiring A
inst✝¹² : Semiring B
inst✝¹¹ : Semiring C
inst✝¹⁰ : SetLike σ A
inst✝⁹ : SetLike τ B
inst✝⁸ : SetLike ω C
inst✝⁷ : AddSubmonoidClass σ A
inst✝⁶ : AddSubmonoidClass τ B
inst✝⁵ : AddSubmonoidClass ω C
inst✝⁴ : DecidableEq ι
inst✝³ : AddMonoid ι
𝒜 : ι → σ
ℬ : ι → τ
𝒞 : ι → ω
inst✝² : GradedRing 𝒜
inst✝¹ : GradedRing ℬ
inst✝ : GradedRing 𝒞
f : 𝒜 →+*ᵍ ℬ
g : ℬ →+*ᵍ 𝒞
I : HomogeneousIdeal ℬ
n : ι
a : A
ha : a ∈ Ideal.comap f I.toIdeal
⊢ ↑(((DirectSum.decompose ℬ) (f a)) n) ∈ I
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor

Goal state before:
pp: A : Type u_1
B : Type u_2
C : Type u_3
σ : Type u_4
τ : Type u_5
ψ : Type u_6
inst✝¹¹ : CommRing A
inst✝¹⁰ : SetLike σ A
inst✝⁹ : AddSubgroupClass σ A
inst✝⁸ : CommRing B
inst✝⁷ : SetLike τ B
inst✝⁶ : AddSubgroupClass τ B
inst✝⁵ : CommRing C
inst✝⁴ : SetLike ψ C
inst✝³ : AddSubgroupClass ψ C
𝒜 : ℕ → σ
ℬ : ℕ → τ
𝒞 : ℕ → ψ
inst✝² : GradedRing 𝒜
inst✝¹ : GradedRing ℬ
inst✝ : GradedRing 𝒞
f : 𝒜 →+*ᵍ ℬ
g : ℬ →+*ᵍ 𝒞
hf : ℬ₊ ≤ HomogeneousIdeal.map f (𝒜₊)
hg : 𝒞₊ ≤ HomogeneousIdeal.map g (ℬ₊)
⊢ Continuous (comapFun f hf)

Tactic:
[

Goal state after:
A : Type u_1
B : Type u_2
C : Type u_3
σ : Type u_4
τ : Type u_5
ψ : Type u_6
inst✝¹¹ : CommRing A
inst✝¹⁰ : SetLike σ A
inst✝⁹ : AddSubgroupClass σ A
inst✝⁸ : CommRing B
inst✝⁷ : SetLike τ B
inst✝⁶ : AddSubgroupClass τ B
inst✝⁵ : CommRing C
inst✝⁴ : SetLike ψ C
inst✝³ : AddSubgroupClass ψ C
𝒜 : ℕ → σ
ℬ : ℕ → τ
𝒞 : ℕ → ψ
inst✝² : GradedRing 𝒜
inst✝¹ : GradedRing ℬ
inst✝ : GradedRing 𝒞
f : 𝒜 →+*ᵍ ℬ
g : ℬ →+*ᵍ 𝒞
hf : ℬ₊ ≤ HomogeneousIdeal.map f (𝒜₊)
hg : 𝒞₊ ≤ HomogeneousIdeal.map g (ℬ₊)
⊢ Continuous (comapFun f hf)
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor

Goal state before:
pp: A : Type u_1
B : Type u_2
σ : Type u_4
τ : Type u_5
inst✝⁷ : CommRing A
inst✝⁶ : SetLike σ A
inst✝⁵ : AddSubgroupClass σ A
inst✝⁴ : CommRing B
inst✝³ : SetLike τ B
inst✝² : AddSubgroupClass τ B
𝒜 : ℕ → σ
ℬ : ℕ → τ
inst✝¹ : GradedRing 𝒜
inst✝ : GradedRing ℬ
f : 𝒜 →+*ᵍ ℬ
hf : ℬ₊ ≤ HomogeneousIdeal.map f (𝒜₊)
U : Opens (ProjectiveSpectrum 𝒜)
V : Opens (ProjectiveSpectrum ℬ)
hUV : V.carrier ⊆ ⇑(ProjectiveSpectrum.comap f hf) ⁻¹' U.carrier
s : (x : ↥U) → AtPrime 𝒜 (↑x).asHomogeneousIdeal.toSubmodule
hs : (ProjectiveSpectrum.StructureSheaf.isLocallyFraction 𝒜).pred s
p : ↑(top ℬ)
hpV : p ∈ V
W : Opens ↑(top 𝒜)
m : ↑⟨(ProjectiveSpectrum.comap f hf) p, ⋯⟩ ∈ W
iWU : W ⟶ U
i : ℕ
a b : ↥(𝒜 i)
hb : ∀ (x : ↥W), ↑b ∉ (↑x).asHomogeneousIdeal
h_frac : ∀ (x : ↥W), (fun x ↦ s (iWU x)) x = HomogeneousLocalization.mk { deg := i, num := a, den := b, den_mem := ⋯ }
x✝ : ↥((Opens.comap (ProjectiveSpectrum.comap f hf)) W ⊓ V)
q : ↑(top ℬ)
hqW : q ∈ ↑((Opens.comap (ProjectiveSpectrum.comap f hf)) W)
hqV : q ∈ ↑V
⊢ val
      ((fun x ↦
          comapStructureSheafFun f hf U V hUV s ((((Opens.comap (ProjectiveSpectrum.comap f hf)) W).infLERight V) x))
        ⟨q, ⋯⟩) =
    (HomogeneousLocalization.mk
        { deg := i, num := (f.gradedAddHom i) a, den := (f.gradedAddHom i) b, den_mem := ⋯ }).val

Tactic:
specialize h_frac ⟨_, hqW⟩

Goal state after:
A : Type u_1
B : Type u_2
σ : Type u_4
τ : Type u_5
inst✝⁷ : CommRing A
inst✝⁶ : SetLike σ A
inst✝⁵ : AddSubgroupClass σ A
inst✝⁴ : CommRing B
inst✝³ : SetLike τ B
inst✝² : AddSubgroupClass τ B
𝒜 : ℕ → σ
ℬ : ℕ → τ
inst✝¹ : GradedRing 𝒜
inst✝ : GradedRing ℬ
f : 𝒜 →+*ᵍ ℬ
hf : ℬ₊ ≤ HomogeneousIdeal.map f (𝒜₊)
U : Opens (ProjectiveSpectrum 𝒜)
V : Opens (ProjectiveSpectrum ℬ)
hUV : V.carrier ⊆ ⇑(ProjectiveSpectrum.comap f hf) ⁻¹' U.carrier
s : (x : ↥U) → AtPrime 𝒜 (↑x).asHomogeneousIdeal.toSubmodule
hs : (ProjectiveSpectrum.StructureSheaf.isLocallyFraction 𝒜).pred s
p : ↑(top ℬ)
hpV : p ∈ V
W : Opens ↑(top 𝒜)
m : ↑⟨(ProjectiveSpectrum.comap f hf) p, ⋯⟩ ∈ W
iWU : W ⟶ U
i : ℕ
a b : ↥(𝒜 i)
hb : ∀ (x : ↥W), ↑b ∉ (↑x).asHomogeneousIdeal
x✝ : ↥((Opens.comap (ProjectiveSpectrum.comap f hf)) W ⊓ V)
q : ↑(top ℬ)
hqW : q ∈ ↑((Opens.comap (ProjectiveSpectrum.comap f hf)) W)
h_frac :
  (fun x ↦ s (iWU x)) ⟨(ProjectiveSpectrum.comap f hf) q, hqW⟩ =
    HomogeneousLocalization.mk { deg := i, num := a, den := b, den_mem := ⋯ }
hqV : q ∈ ↑V
⊢ val
      ((fun x ↦
          comapStructureSheafFun f hf U V hUV s ((((Opens.comap (ProjectiveSpectrum.comap f hf)) W).infLERight V) x))
        ⟨q, ⋯⟩) =
    (HomogeneousLocalization.mk
        { deg := i, num := (f.gradedAddHom i) a, den := (f.gradedAddHom i) b, den_mem := ⋯ }).val
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

Goal state before:
pp: case e_y
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
g : ℕ → A⁰_ f :=
  fun j ↦
    (m + m).choose j •
      if h2 : m + m < j then 0
      else
        if h1 : j ≤ m then
          HomogeneousLocalization.mk
              { deg := m * i, num := ⟨(proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
            HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }
        else
          HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
            HomogeneousLocalization.mk
              { deg := m * i, num := ⟨(proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩,
                den_mem := ⋯ }
j : ℕ
hj : j ∈ range (m + m + 1)
h2 : ¬m + m < j
h1 : ¬j ≤ m
⊢ ⟨f ^ (i + i), ⋯⟩ = ⟨f ^ i, ⋯⟩ * ⟨f ^ i, ⋯⟩

Tactic:
[

Goal state after:
case e_y
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
g : ℕ → A⁰_ f :=
  fun j ↦
    (m + m).choose j •
      if h2 : m + m < j then 0
      else
        if h1 : j ≤ m then
          HomogeneousLocalization.mk
              { deg := m * i, num := ⟨(proj 𝒜 i) a ^ j * (proj 𝒜 i) b ^ (m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
            HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) b ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }
        else
          HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } *
            HomogeneousLocalization.mk
              { deg := m * i, num := ⟨(proj 𝒜 i) a ^ (j - m) * (proj 𝒜 i) b ^ (m + m - j), ⋯⟩, den := ⟨f ^ i, ⋯⟩,
                den_mem := ⋯ }
j : ℕ
hj : j ∈ range (m + m + 1)
h2 : ¬m + m < j
h1 : ¬j ≤ m
⊢ ⟨f ^ (i + i), ⋯⟩ = ⟨f ^ i, ⋯⟩ * ⟨f ^ i, ⋯⟩
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: case e_y.inr
A : Type u_2
σ : Type u_3
inst✝⁴ : CommRing A
inst✝³ : SetLike σ A
inst✝² : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ℕ
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ℕ
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
hxd' : ∀ (i : ι'), 0 < dv i
a : ℕ
ai : ι' → ℕ
hai : ∑ i, ai i • dv i = a • d
H₀ : (a - ∑ i, dv i * (ai i / d)) • d = ∑ k, (ai k % d) • dv k
hd : d > 0
⊢ ∑ i, dv i * (ai i / d) * d ≤ ∑ i, ai i • dv i

Tactic:
[

Goal state after:
case e_y.inr
A : Type u_2
σ : Type u_3
inst✝⁴ : CommRing A
inst✝³ : SetLike σ A
inst✝² : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ℕ
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ℕ
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
hxd' : ∀ (i : ι'), 0 < dv i
a : ℕ
ai : ι' → ℕ
hai : ∑ i, ai i • dv i = a • d
H₀ : (a - ∑ i, dv i * (ai i / d)) • d = ∑ k, (ai k % d) • dv k
hd : d > 0
⊢ ∑ i, dv i * (ai i / d) * d ≤ ∑ i, ai i • dv i
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
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
i : ℕ
⊢ Localization.mk ((proj 𝒜 i) 0 ^ m) ⟨f ^ i, ⋯⟩ = 0

Tactic:
[

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
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
i : ℕ
⊢ Localization.mk ((proj 𝒜 i) 0 ^ m) ⟨f ^ i, ⋯⟩ = 0
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

Goal state before:
pp: case refine_2
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
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
⊢ HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) (a • x) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈
    q.asIdeal

Tactic:
[

Goal state after:
case refine_2
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
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
⊢ HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) (a • x) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈
    q.asIdeal
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor

Goal state before:
pp: case e_a
A B σ τ : Type u
inst✝⁷ : CommRing A
inst✝⁶ : SetLike σ A
inst✝⁵ : AddSubgroupClass σ A
inst✝⁴ : CommRing B
inst✝³ : SetLike τ B
inst✝² : AddSubgroupClass τ B
𝒜 : ℕ → σ
ℬ : ℕ → τ
inst✝¹ : GradedRing 𝒜
inst✝ : GradedRing ℬ
f : 𝒜 →+*ᵍ ℬ
hf : ℬ₊ ≤ HomogeneousIdeal.map f (𝒜₊)
i : ℕ
hi : 0 < i
s : A
hs : s ∈ 𝒜 i
⊢ Scheme.Hom.resLE (map f hf) (basicOpen 𝒜 s) (map f hf ⁻¹ᵁ basicOpen 𝒜 s) ⋯ =
    ((basicOpenIsoSpec ℬ (f s) ⋯ hi).hom ≫ Spec.map (CommRingCat.ofHom (Away.map f s))) ≫
      (basicOpenIsoSpec 𝒜 s hs hi).inv

Tactic:
Iso.eq_comp_inv

Goal state after:
case e_a
A B σ τ : Type u
inst✝⁷ : CommRing A
inst✝⁶ : SetLike σ A
inst✝⁵ : AddSubgroupClass σ A
inst✝⁴ : CommRing B
inst✝³ : SetLike τ B
inst✝² : AddSubgroupClass τ B
𝒜 : ℕ → σ
ℬ : ℕ → τ
inst✝¹ : GradedRing 𝒜
inst✝ : GradedRing ℬ
f : 𝒜 →+*ᵍ ℬ
hf : ℬ₊ ≤ HomogeneousIdeal.map f (𝒜₊)
i : ℕ
hi : 0 < i
s : A
hs : s ∈ 𝒜 i
⊢ Scheme.Hom.resLE (map f hf) (basicOpen 𝒜 s) (map f hf ⁻¹ᵁ basicOpen 𝒜 s) ⋯ ≫ (basicOpenIsoSpec 𝒜 s hs hi).hom =
    (basicOpenIsoSpec ℬ (f s) ⋯ hi).hom ≫ Spec.map (CommRingCat.ofHom (Away.map f s))
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor

Goal state before:
pp: A σ : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
⊢ map (GradedRingHom.id 𝒜) ⋯ = 𝟙 (Proj 𝒜)

Tactic:
refine (affineOpenCover _).openCover.hom_ext _ _ fun s ↦ ?_

Goal state after:
A σ : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
s : (affineOpenCover 𝒜).openCover.I₀
⊢ (affineOpenCover 𝒜).openCover.f s ≫ map (GradedRingHom.id 𝒜) ⋯ = (affineOpenCover 𝒜).openCover.f s ≫ 𝟙 (Proj 𝒜)
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
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
product : A⁰_ f :=
  HomogeneousLocalization.mk { deg := m • n, num := ⟨a ^ m, ⋯⟩, den := ⟨f ^ n, ?refine_2.refine_1⟩, den_mem := ⋯ } *
    HomogeneousLocalization.mk
      { deg := m • (i - n), num := ⟨(proj 𝒜 (i - n)) x ^ m, ⋯⟩, den := ⟨f ^ (i - n), ?refine_2.refine_2⟩, den_mem := ⋯ }
h : n ≤ i
⊢ Localization.mk ((a * ↑(((decompose 𝒜) x) (i - n))) ^ m) ⟨f ^ i, ⋯⟩ =
    Localization.mk (a ^ m) ⟨f ^ n, ⋯⟩ * Localization.mk (↑(((decompose 𝒜) x) (i - n)) ^ m) ⟨f ^ (i - n), ⋯⟩

Tactic:
[

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
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x : A
hx : x ∈ carrier f_deg q
n : ℕ
a : A
ha : a ∈ 𝒜 n
i : ℕ
product : A⁰_ f :=
  HomogeneousLocalization.mk { deg := m • n, num := ⟨a ^ m, ⋯⟩, den := ⟨f ^ n, ?refine_2.refine_1⟩, den_mem := ⋯ } *
    HomogeneousLocalization.mk
      { deg := m • (i - n), num := ⟨(proj 𝒜 (i - n)) x ^ m, ⋯⟩, den := ⟨f ^ (i - n), ?refine_2.refine_2⟩, den_mem := ⋯ }
h : n ≤ i
⊢ Localization.mk ((a * ↑(((decompose 𝒜) x) (i - n))) ^ m) ⟨f ^ i, ⋯⟩ =
    Localization.mk (a ^ m) ⟨f ^ n, ⋯⟩ * Localization.mk (↑(((decompose 𝒜) x) (i - n)) ^ m) ⟨f ^ (i - n), ⋯⟩
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: A : Type u_2
σ : Type u_3
inst✝⁴ : CommRing A
inst✝³ : SetLike σ A
inst✝² : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝¹ : GradedRing 𝒜
inst✝ : Algebra.FiniteType (↥(𝒜 0)) A
f : A
d : ℕ
hf : f ∈ 𝒜 d
s : Finset A
hs : Algebra.adjoin ↥(𝒜 0) ↑s = ⊤
hs' : ∀ i ∈ s, ∃ n, n ≠ 0 ∧ i ∈ 𝒜 n
dx : ↥s → ℕ
hdx : ∀ (x : ↥s), dx x ≠ 0
hxd : ∀ (x : ↥s), ↑x ∈ 𝒜 (dx x)
⊢ ⊤.FG

Tactic:
[

Goal state after:
A : Type u_2
σ : Type u_3
inst✝⁴ : CommRing A
inst✝³ : SetLike σ A
inst✝² : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝¹ : GradedRing 𝒜
inst✝ : Algebra.FiniteType (↥(𝒜 0)) A
f : A
d : ℕ
hf : f ∈ 𝒜 d
s : Finset A
hs : Algebra.adjoin ↥(𝒜 0) ↑s = ⊤
hs' : ∀ i ∈ s, ∃ n, n ≠ 0 ∧ i ∈ 𝒜 n
dx : ↥s → ℕ
hdx : ∀ (x : ↥s), dx x ≠ 0
hxd : ∀ (x : ↥s), ↑x ∈ 𝒜 (dx x)
⊢ ⊤.FG
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

Goal state before:
pp: case refine_3
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
x : A
hx : x ∈ carrier f_deg q
⊢ ∀ (m_1 m' : A), m_1 • x ∈ carrier f_deg q → m' • x ∈ carrier f_deg q → (m_1 + m') • x ∈ carrier f_deg q

Tactic:
[

Goal state after:
case refine_3
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
x : A
hx : x ∈ carrier f_deg q
⊢ ∀ (m_1 m' : A), m_1 • x ∈ carrier f_deg q → m' • x ∈ carrier f_deg q → (m_1 + m') • x ∈ carrier f_deg q
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
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
rid : f ∈ asIdeal f_deg hm q
⊢ 1 = Localization.mk (↑(((decompose 𝒜) f) m) ^ m) ⟨f ^ m, ⋯⟩

Tactic:
[

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
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
rid : f ∈ asIdeal f_deg hm q
⊢ 1 = Localization.mk (↑(((decompose 𝒜) f) m) ^ m) ⟨f ^ m, ⋯⟩
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
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : IsHomogeneousElem 𝒜 x
x✝ : IsHomogeneousElem 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
⊢ HomogeneousLocalization.mk
        { deg := m * nx, num := ⟨↑(((decompose 𝒜) x) nx) ^ m, ⋯⟩, den := ⟨f ^ nx, ⋯⟩, den_mem := ⋯ } *
      HomogeneousLocalization.mk
        { deg := m * ny, num := ⟨↑(((decompose 𝒜) y) ny) ^ m, ⋯⟩, den := ⟨f ^ ny, ⋯⟩, den_mem := ⋯ } =
    HomogeneousLocalization.mk
      { deg := m * (nx + ny), num := ⟨↑(((decompose 𝒜) (x * y)) (nx + ny)) ^ m, ⋯⟩, den := ⟨f ^ (nx + ny), ⋯⟩,
        den_mem := ⋯ }

Tactic:
[

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
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
x y : A
x✝¹ : IsHomogeneousElem 𝒜 x
x✝ : IsHomogeneousElem 𝒜 y
hxy : x * y ∈ asIdeal f_deg hm q
nx : ℕ
hnx : x ∈ 𝒜 nx
ny : ℕ
hny : y ∈ 𝒜 ny
⊢ HomogeneousLocalization.mk
        { deg := m * nx, num := ⟨↑(((decompose 𝒜) x) nx) ^ m, ⋯⟩, den := ⟨f ^ nx, ⋯⟩, den_mem := ⋯ } *
      HomogeneousLocalization.mk
        { deg := m * ny, num := ⟨↑(((decompose 𝒜) y) ny) ^ m, ⋯⟩, den := ⟨f ^ ny, ⋯⟩, den_mem := ⋯ } =
    HomogeneousLocalization.mk
      { deg := m * (nx + ny), num := ⟨↑(((decompose 𝒜) (x * y)) (nx + ny)) ^ m, ⋯⟩, den := ⟨f ^ (nx + ny), ⋯⟩,
        den_mem := ⋯ }