mathlib-initiative/mathlib-tactics huggingface dataset:
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
      ↑{ deg := m * i, num := ⟨(a * ↑(((decompose 𝒜) x) (i - n))) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }.num
      ⟨↑{ deg := m * i, num := ⟨(a * ↑(((decompose 𝒜) x) (i - n))) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }.den, ⋯⟩ =
    Localization.mk ↑{ deg := m * n, num := ⟨a ^ m, ⋯⟩, den := ⟨f ^ n, ?refine_2.refine_1⟩, den_mem := ⋯ }.num
        ⟨↑{ deg := m * n, num := ⟨a ^ m, ⋯⟩, den := ⟨f ^ n, ?refine_2.refine_1⟩, den_mem := ⋯ }.den, ⋯⟩ *
      Localization.mk
        ↑{ deg := m * (i - n), num := ⟨↑(((decompose 𝒜) x) (i - n)) ^ m, ⋯⟩, den := ⟨f ^ (i - n), ?refine_2.refine_2⟩,
              den_mem := ⋯ }.num
        ⟨↑{ deg := m * (i - n), num := ⟨↑(((decompose 𝒜) x) (i - n)) ^ m, ⋯⟩, den := ⟨f ^ (i - n), ?refine_2.refine_2⟩,
                den_mem := ⋯ }.den,
          ⋯⟩

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
      ↑{ deg := m * i, num := ⟨(a * ↑(((decompose 𝒜) x) (i - n))) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }.num
      ⟨↑{ deg := m * i, num := ⟨(a * ↑(((decompose 𝒜) x) (i - n))) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }.den, ⋯⟩ =
    Localization.mk ↑{ deg := m * n, num := ⟨a ^ m, ⋯⟩, den := ⟨f ^ n, ?refine_2.refine_1⟩, den_mem := ⋯ }.num
        ⟨↑{ deg := m * n, num := ⟨a ^ m, ⋯⟩, den := ⟨f ^ n, ?refine_2.refine_1⟩, den_mem := ⋯ }.den, ⋯⟩ *
      Localization.mk
        ↑{ deg := m * (i - n), num := ⟨↑(((decompose 𝒜) x) (i - n)) ^ m, ⋯⟩, den := ⟨f ^ (i - n), ?refine_2.refine_2⟩,
              den_mem := ⋯ }.num
        ⟨↑{ deg := m * (i - n), num := ⟨↑(((decompose 𝒜) x) (i - n)) ^ m, ⋯⟩, den := ⟨f ^ (i - n), ?refine_2.refine_2⟩,
                den_mem := ⋯ }.den,
          ⋯⟩
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: R : CommRingCat
M : ModuleCat ↑R
s : Set ↑M
hs : Submodule.span (↑R) s = ⊤
t : Set (↑s →₀ ↑R)
ht : Submodule.span (↑R) t = (Finsupp.linearCombination (↑R) Subtype.val).ker
H₁ :
  Function.Exact ⇑(ConcreteCategory.hom (ModuleCat.ofHom (Finsupp.linearCombination (↑R) Subtype.val)))
    ⇑(ConcreteCategory.hom (ModuleCat.ofHom (Finsupp.linearCombination (↑R) Subtype.val)))
⊢ tilde.map
      (ModuleCat.ofHom (Finsupp.linearCombination (↑R) Subtype.val ∘ₗ Finsupp.linearCombination (↑R) Subtype.val)) =
    0

Tactic:
]

Goal state after:
R : CommRingCat
M : ModuleCat ↑R
s : Set ↑M
hs : Submodule.span (↑R) s = ⊤
t : Set (↑s →₀ ↑R)
ht : Submodule.span (↑R) t = (Finsupp.linearCombination (↑R) Subtype.val).ker
H₁ :
  Function.Exact ⇑(ConcreteCategory.hom (ModuleCat.ofHom (Finsupp.linearCombination (↑R) Subtype.val)))
    ⇑(ConcreteCategory.hom (ModuleCat.ofHom (Finsupp.linearCombination (↑R) Subtype.val)))
⊢ tilde.map
      (ModuleCat.ofHom (Finsupp.linearCombination (↑R) Subtype.val ∘ₗ Finsupp.linearCombination (↑R) Subtype.val)) =
    0
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: R : CommRingCat
M : (Spec R).Modules
P : SheafOfModules.Presentation M
⊢ (tilde.functor R).essImage M

Tactic:
]

Goal state after:
R : CommRingCat
M : (Spec R).Modules
P : SheafOfModules.Presentation M
⊢ (tilde.functor R).essImage M
Module: Mathlib.RingTheory.LocalRing.LocalSubring

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
K : Type u_3
inst✝¹ : Field K
A : Subring K
P : Ideal ↥A
inst✝ : P.IsPrime
x : ↥A
s : ↥P.primeCompl
y : ↥A
t : ↥P.primeCompl
e :
  (IsLocalization.liftAlgHom ⋯) (IsLocalization.mk' (Localization.AtPrime P) x s) =
    (IsLocalization.liftAlgHom ⋯) (IsLocalization.mk' (Localization.AtPrime P) y t)
H : ∀ (x : ↥P.primeCompl), ↑x ≠ 0
this : ↑x = ↑y * (↑↑t)⁻¹ * ↑↑s
⊢ (algebraMap (↥A) (Localization.AtPrime P)) (↑t * x) = (algebraMap (↥A) (Localization.AtPrime P)) (↑s * y)

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing S
K : Type u_3
inst✝¹ : Field K
A : Subring K
P : Ideal ↥A
inst✝ : P.IsPrime
x : ↥A
s : ↥P.primeCompl
y : ↥A
t : ↥P.primeCompl
e :
  (IsLocalization.liftAlgHom ⋯) (IsLocalization.mk' (Localization.AtPrime P) x s) =
    (IsLocalization.liftAlgHom ⋯) (IsLocalization.mk' (Localization.AtPrime P) y t)
H : ∀ (x : ↥P.primeCompl), ↑x ≠ 0
this : ↑x = ↑y * (↑↑t)⁻¹ * ↑↑s
⊢ (algebraMap (↥A) (Localization.AtPrime P)) (↑t * x) = (algebraMap (↥A) (Localization.AtPrime P)) (↑s * y)
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 ^ m * ↑(((decompose 𝒜) x) (i - n)) ^ m) ⟨f ^ i, ⋯⟩ =
    Localization.mk (a ^ m * ↑(((decompose 𝒜) x) (i - n)) ^ m) (⟨f ^ n, ⋯⟩ * ⟨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 ^ m * ↑(((decompose 𝒜) x) (i - n)) ^ m) ⟨f ^ i, ⋯⟩ =
    Localization.mk (a ^ m * ↑(((decompose 𝒜) x) (i - n)) ^ m) (⟨f ^ n, ⋯⟩ * ⟨f ^ (i - n), ⋯⟩)
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: case refine_1
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)
s : Finset ι' := {x | 0 < dv x}
⊢ ∀ (y : ι'), v y ∈ ↑(Algebra.adjoin (↥(𝒜 0)) (Set.range (v ∘ Subtype.val)))

Tactic:
]

Goal state after:
case refine_1
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)
s : Finset ι' := {x | 0 < dv x}
⊢ ∀ (y : ι'), v y ∈ ↑(Algebra.adjoin (↥(𝒜 0)) (Set.range (v ∘ Subtype.val)))
Module: Mathlib.RingTheory.Valuation.Integral

Goal state before:
pp: case pos
R : Type u
Γ₀ : Type v
inst✝³ : CommRing R
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O R
hv : v.Integers O
x : R
a✝ : Nontrivial R
this : Nontrivial O
f : O[X]
hm : f.Monic
hf : eval₂ (algebraMap O R) x f = 0
hn : ∃ x, C x = f
⊢ v x ≤ 1

Tactic:
]

Goal state after:
case pos
R : Type u
Γ₀ : Type v
inst✝³ : CommRing R
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O R
hv : v.Integers O
x : R
a✝ : Nontrivial R
this : Nontrivial O
f : O[X]
hm : f.Monic
hf : eval₂ (algebraMap O R) x f = 0
hn : ∃ x, C x = f
⊢ v x ≤ 1
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
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
⊢ f ^ i = f ^ i

Tactic:
]

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

Goal state before:
pp: R : CommRingCat
M N : TopCat.Sheaf (ModuleCat ↑R) ↑(Spec R).toPresheafedSpace
φ : M ≅ N
hM : IsLocalizing M
f : ↑R
⊢ IsLocalizedModule (Submonoid.powers f) (ModuleCat.Hom.hom (M.obj.map (basicOpen f).leTop.op))

Tactic:
]

Goal state after:
R : CommRingCat
M N : TopCat.Sheaf (ModuleCat ↑R) ↑(Spec R).toPresheafedSpace
φ : M ≅ N
hM : IsLocalizing M
f : ↑R
⊢ IsLocalizedModule (Submonoid.powers f) (ModuleCat.Hom.hom (M.obj.map (basicOpen f).leTop.op))
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: case refine_2
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)
s : Finset ι' := {x | 0 < dv x}
this :
  Algebra.adjoin ↥(𝒜 0)
      {x |
        ∃ a ai,
          ∃ (hai : ∑ i, ai i • (dv ∘ Subtype.val) i = a • d) (_ : ∀ (i : ↥s), ai i ≤ d),
            Away.mk 𝒜 hf a (∏ i, (v ∘ Subtype.val) i ^ ai i) ⋯ = x} =
    ⊤
⊢ Algebra.adjoin ↥(𝒜 0)
      {x |
        ∃ a ai,
          ∃ (hai : ∑ i, ai i • (dv ∘ Subtype.val) i = a • d) (_ : ∀ (i : ↥s), ai i ≤ d),
            Away.mk 𝒜 hf a (∏ i, (v ∘ Subtype.val) i ^ ai i) ⋯ = x} ≤
    Algebra.adjoin ↥(𝒜 0)
      {x |
        ∃ a ai, ∃ (hai : ∑ i, ai i • dv i = a • d) (_ : ∀ (i : ι'), ai i ≤ d), Away.mk 𝒜 hf a (∏ i, v i ^ ai i) ⋯ = x}

Tactic:
]

Goal state after:
case refine_2
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)
s : Finset ι' := {x | 0 < dv x}
this :
  Algebra.adjoin ↥(𝒜 0)
      {x |
        ∃ a ai,
          ∃ (hai : ∑ i, ai i • (dv ∘ Subtype.val) i = a • d) (_ : ∀ (i : ↥s), ai i ≤ d),
            Away.mk 𝒜 hf a (∏ i, (v ∘ Subtype.val) i ^ ai i) ⋯ = x} =
    ⊤
⊢ Algebra.adjoin ↥(𝒜 0)
      {x |
        ∃ a ai,
          ∃ (hai : ∑ i, ai i • (dv ∘ Subtype.val) i = a • d) (_ : ∀ (i : ↥s), ai i ≤ d),
            Away.mk 𝒜 hf a (∏ i, (v ∘ Subtype.val) i ^ ai i) ⋯ = x} ≤
    Algebra.adjoin ↥(𝒜 0)
      {x |
        ∃ a ai, ∃ (hai : ∑ i, ai i • dv i = a • d) (_ : ∀ (i : ι'), ai i ≤ d), Away.mk 𝒜 hf a (∏ i, v i ^ ai i) ⋯ = x}
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: R : CommRingCat
M N : TopCat.Sheaf (ModuleCat ↑R) ↑(Spec R).toPresheafedSpace
φ : M ⟶ N
h : IsIso (φ.hom.app (op ⊤))
hM : IsLocalizing M
hN : IsLocalizing N
f : ↑R
⊢ IsLocalizedModule (Submonoid.powers f) (ModuleCat.Hom.hom (φ.hom.app (op ⊤) ≫ N.obj.map (basicOpen f).leTop.op))

Tactic:
]

Goal state after:
R : CommRingCat
M N : TopCat.Sheaf (ModuleCat ↑R) ↑(Spec R).toPresheafedSpace
φ : M ⟶ N
h : IsIso (φ.hom.app (op ⊤))
hM : IsLocalizing M
hN : IsLocalizing N
f : ↑R
⊢ IsLocalizedModule (Submonoid.powers f) (ModuleCat.Hom.hom (φ.hom.app (op ⊤) ≫ N.obj.map (basicOpen f).leTop.op))
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: R : CommRingCat
M : ModuleCat ↑R
f : ↑R
heq :
  tilde.toOpen M ⊤ ≫ (modulesSpecToSheaf.obj (tilde M)).obj.map (basicOpen f).leTop.op = tilde.toOpen M (basicOpen f)
⊢ IsLocalizedModule (Submonoid.powers f) (ModuleCat.Hom.hom (tilde.toOpen M (basicOpen f)))

Tactic:
]

Goal state after:
R : CommRingCat
M : ModuleCat ↑R
f : ↑R
heq :
  tilde.toOpen M ⊤ ≫ (modulesSpecToSheaf.obj (tilde M)).obj.map (basicOpen f).leTop.op = tilde.toOpen M (basicOpen f)
⊢ IsLocalizedModule (Submonoid.powers f) (ModuleCat.Hom.hom (tilde.toOpen M (basicOpen f)))
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: case refine_1
R : CommRingCat
M : (Spec R).Modules
h : IsIso M.fromTildeΓ
⊢ IsLocalizing (modulesSpecToSheaf.obj (tilde ((modulesSpecToSheaf.obj M).presheaf.obj (op ⊤))))

Tactic:
]

Goal state after:
case refine_1
R : CommRingCat
M : (Spec R).Modules
h : IsIso M.fromTildeΓ
⊢ IsLocalizing (modulesSpecToSheaf.obj (tilde ((modulesSpecToSheaf.obj M).presheaf.obj (op ⊤))))
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: case refine_2
R : CommRingCat
M : (Spec R).Modules
h : IsLocalizing (modulesSpecToSheaf.obj M)
⊢ IsIso (modulesSpecToSheaf.map M.fromTildeΓ)

Tactic:
]

Goal state after:
case refine_2
R : CommRingCat
M : (Spec R).Modules
h : IsLocalizing (modulesSpecToSheaf.obj M)
⊢ IsIso (modulesSpecToSheaf.map M.fromTildeΓ)
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: case refine_2
R : CommRingCat
M : (Spec R).Modules
h : IsLocalizing (modulesSpecToSheaf.obj M)
⊢ IsIso ((modulesSpecToSheaf.obj M).obj.map (homOfLE ⋯).op)

Tactic:
]

Goal state after:
case refine_2
R : CommRingCat
M : (Spec R).Modules
h : IsLocalizing (modulesSpecToSheaf.obj M)
⊢ IsIso ((modulesSpecToSheaf.obj M).obj.map (homOfLE ⋯).op)
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: R S : CommRingCat
φ : R ⟶ S
M : (Spec S).Modules
h : IsLocalizing (modulesSpecToSheaf.obj M)
⊢ IsLocalizing
    ((modulesSpecToSheaf ⋙
          Sheaf.pushforward (ModuleCat ↑S) (Spec.map φ).base ⋙
            sheafCompose (Opens.grothendieckTopology ↥(Spec R)) (ModuleCat.restrictScalars (CommRingCat.Hom.hom φ))).obj
      M)

Tactic:
]

Goal state after:
R S : CommRingCat
φ : R ⟶ S
M : (Spec S).Modules
h : IsLocalizing (modulesSpecToSheaf.obj M)
⊢ IsLocalizing
    ((modulesSpecToSheaf ⋙
          Sheaf.pushforward (ModuleCat ↑S) (Spec.map φ).base ⋙
            sheafCompose (Opens.grothendieckTopology ↥(Spec R)) (ModuleCat.restrictScalars (CommRingCat.Hom.hom φ))).obj
      M)
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

Goal state before:
pp: case refine_2.refine_1
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
⊢ f ^ n ∈ 𝒜 ?m.422

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
⊢ m • n = ?m.422

Tactic:
]

Goal state after:
case refine_2.refine_1
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
⊢ f ^ n ∈ 𝒜 ?m.422

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
⊢ m • n = ?m.422
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: R : CommRingCat
M✝ : ModuleCat ↑R
X Y : Scheme
f : X ⟶ Y
inst✝¹ : IsOpenImmersion f
M : Y.Modules
inst✝ : SheafOfModules.IsQuasicoherent M
α : X.presheaf ⟶ (Hom.opensFunctor f).op ⋙ Y.presheaf := { app := fun U ↦ (Hom.appIso f (unop U)).inv, naturality := ⋯ }
hα : IsIso α
φ : X.ringCatSheaf ⟶
  ((Hom.opensFunctor f).sheafPushforwardContinuous RingCat (Opens.grothendieckTopology ↥X)
        (Opens.grothendieckTopology ↥Y)).obj
    Y.ringCatSheaf :=
  { hom := Functor.whiskerRight α (forget₂ CommRingCat RingCat) }
⊢ IsIso ((ObjectProperty.ι (CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥X))).map φ)

Tactic:
]

Goal state after:
R : CommRingCat
M✝ : ModuleCat ↑R
X Y : Scheme
f : X ⟶ Y
inst✝¹ : IsOpenImmersion f
M : Y.Modules
inst✝ : SheafOfModules.IsQuasicoherent M
α : X.presheaf ⟶ (Hom.opensFunctor f).op ⋙ Y.presheaf := { app := fun U ↦ (Hom.appIso f (unop U)).inv, naturality := ⋯ }
hα : IsIso α
φ : X.ringCatSheaf ⟶
  ((Hom.opensFunctor f).sheafPushforwardContinuous RingCat (Opens.grothendieckTopology ↥X)
        (Opens.grothendieckTopology ↥Y)).obj
    Y.ringCatSheaf :=
  { hom := Functor.whiskerRight α (forget₂ CommRingCat RingCat) }
⊢ IsIso ((ObjectProperty.ι (CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↥X))).map φ)
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

Goal state before:
pp: case refine_2.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
⊢ f ^ (i - n) ∈ 𝒜 ?m.461

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
⊢ m • (i - n) = ?m.461

Tactic:
]

Goal state after:
case refine_2.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
⊢ f ^ (i - n) ∈ 𝒜 ?m.461

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
⊢ m • (i - n) = ?m.461
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: case refine_2
X : Scheme
M : X.Modules
inst✝ : SheafOfModules.IsQuasicoherent M
I : Type u
W : I → TopologicalSpace.Opens ↥X
cov : iSup W = ⊤
pres : (i : I) → (SheafOfModules.over M (W i)).Presentation
κ : I → Set (TopologicalSpace.Opens ↥X)
hsub : ∀ (i : I), κ i ⊆ X.affineOpens
heq : ∀ (i : I), W i = sSup (κ i)
⊢ IsOpenCover fun j ↦ ↑j.snd

Tactic:
]

Goal state after:
case refine_2
X : Scheme
M : X.Modules
inst✝ : SheafOfModules.IsQuasicoherent M
I : Type u
W : I → TopologicalSpace.Opens ↥X
cov : iSup W = ⊤
pres : (i : I) → (SheafOfModules.over M (W i)).Presentation
κ : I → Set (TopologicalSpace.Opens ↥X)
hsub : ∀ (i : I), κ i ⊆ X.affineOpens
heq : ∀ (i : I), W i = sSup (κ i)
⊢ IsOpenCover fun j ↦ ↑j.snd
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: case refine_2
X : Scheme
M : X.Modules
inst✝ : SheafOfModules.IsQuasicoherent M
I : Type u
W : I → TopologicalSpace.Opens ↥X
cov : iSup W = ⊤
pres : (i : I) → (SheafOfModules.over M (W i)).Presentation
κ : I → Set (TopologicalSpace.Opens ↥X)
hsub : ∀ (i : I), κ i ⊆ X.affineOpens
heq : ∀ (i : I), W i = sSup (κ i)
⊢ ⨆ i, ⨆ j, ↑⟨i, j⟩.snd = iSup W

Tactic:
]

Goal state after:
case refine_2
X : Scheme
M : X.Modules
inst✝ : SheafOfModules.IsQuasicoherent M
I : Type u
W : I → TopologicalSpace.Opens ↥X
cov : iSup W = ⊤
pres : (i : I) → (SheafOfModules.over M (W i)).Presentation
κ : I → Set (TopologicalSpace.Opens ↥X)
hsub : ∀ (i : I), κ i ⊆ X.affineOpens
heq : ∀ (i : I), W i = sSup (κ i)
⊢ ⨆ i, ⨆ j, ↑⟨i, j⟩.snd = iSup W
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: case refine_2
X : Scheme
M : X.Modules
inst✝ : SheafOfModules.IsQuasicoherent M
I : Type u
W : I → TopologicalSpace.Opens ↥X
cov : iSup W = ⊤
pres : (i : I) → (SheafOfModules.over M (W i)).Presentation
κ : I → Set (TopologicalSpace.Opens ↥X)
hsub : ∀ (i : I), κ i ⊆ X.affineOpens
heq : ∀ (i : I), W i = sSup (κ i)
i : I
⊢ ⨆ j, ↑⟨i, j⟩.snd = ⨆ a, ↑a

Tactic:
]

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
hm : 0 < m
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
rid : f ∈ asIdeal f_deg hm q
⊢ 1 =
    Localization.mk ↑{ deg := m * m, num := ⟨(proj 𝒜 m) f ^ m, ⋯⟩, den := ⟨f ^ m, ⋯⟩, den_mem := ⋯ }.num
      ⟨↑{ deg := m * m, num := ⟨(proj 𝒜 m) f ^ m, ⋯⟩, den := ⟨f ^ m, ⋯⟩, den_mem := ⋯ }.den, ⋯⟩

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 ↑{ deg := m * m, num := ⟨(proj 𝒜 m) f ^ m, ⋯⟩, den := ⟨f ^ m, ⋯⟩, den_mem := ⋯ }.num
      ⟨↑{ deg := m * m, num := ⟨(proj 𝒜 m) f ^ m, ⋯⟩, den := ⟨f ^ m, ⋯⟩, den_mem := ⋯ }.den, ⋯⟩
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: R : CommRingCat
M : (Spec R).Modules
V : (Spec R).Opens
g : ↑R
hg : basicOpen g ≤ V
hV : Aux M V
f : ↑R
hfg : basicOpen f ≤ basicOpen g
t : ↑Γ(M, basicOpen g)
ht : (ConcreteCategory.hom (M.presheaf.map (homOfLE hfg).op)) t = 0
n : ℕ
t' : ↑Γ(M, V)
ht' : (ConcreteCategory.hom (M.presheaf.map (homOfLE hg).op)) t' = g ^ n • t
⊢ g ^ n • 0 = 0

Tactic:
]

Goal state after:
R : CommRingCat
M : (Spec R).Modules
V : (Spec R).Opens
g : ↑R
hg : basicOpen g ≤ V
hV : Aux M V
f : ↑R
hfg : basicOpen f ≤ basicOpen g
t : ↑Γ(M, basicOpen g)
ht : (ConcreteCategory.hom (M.presheaf.map (homOfLE hfg).op)) t = 0
n : ℕ
t' : ↑Γ(M, V)
ht' : (ConcreteCategory.hom (M.presheaf.map (homOfLE hg).op)) t' = g ^ n • t
⊢ g ^ n • 0 = 0
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: case inl
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)
hd : d = 0
f' : Away 𝒜 f := Away.mk 𝒜 hf 1 1 ⋯
⊢ {x | ∃ a ai, ∃ (hai : ∑ i, ai i • dx i = a • d) (_ : ∀ (i : ↥s), ai i ≤ d), Away.mk 𝒜 hf a (∏ i, ↑i ^ ai i) ⋯ = x} ⊆
    ↑(↥(𝒜 0))[f']

Tactic:
]

Goal state after:
case inl
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)
hd : d = 0
f' : Away 𝒜 f := Away.mk 𝒜 hf 1 1 ⋯
⊢ {x | ∃ a ai, ∃ (hai : ∑ i, ai i • dx i = a • d) (_ : ∀ (i : ↥s), ai i ≤ d), Away.mk 𝒜 hf a (∏ i, ↑i ^ ai i) ⋯ = x} ⊆
    ↑(↥(𝒜 0))[f']