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 : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
a : A
n : ℕ
hn : a ∈ 𝒜 n
⊢ f ^ n ∈ 𝒜 (m * n)

Tactic:
mul_comm

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 : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
a : A
n : ℕ
hn : a ∈ 𝒜 n
⊢ f ^ n ∈ 𝒜 (n * 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
a : A
n : ℕ
hn : a ∈ 𝒜 n
⊢ f ^ n ∈ 𝒜 (m * n)

Tactic:
mul_comm

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
a : A
n : ℕ
hn : a ∈ 𝒜 n
⊢ f ^ n ∈ 𝒜 (n * 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 : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
a : A
n : ℕ
hn : a ∈ 𝒜 (n * m)
⊢ f ^ n ∈ 𝒜 (m * n)

Tactic:
mul_comm

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 : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
a : A
n : ℕ
hn : a ∈ 𝒜 (n * m)
⊢ f ^ n ∈ 𝒜 (n * m)
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.map' (fun x ↦ { deg := x.deg, num := ⟨g ↑x.num, ⋯⟩, den := ⟨g ↑x.den, ⋯⟩, den_mem := ⋯ }) ⋯
      (Quotient.mk'' x * Quotient.mk'' y) =
    Quotient.map' (fun x ↦ { deg := x.deg, num := ⟨g ↑x.num, ⋯⟩, den := ⟨g ↑x.den, ⋯⟩, den_mem := ⋯ }) ⋯
        (Quotient.mk'' x) *
      Quotient.map' (fun x ↦ { deg := x.deg, num := ⟨g ↑x.num, ⋯⟩, den := ⟨g ↑x.den, ⋯⟩, den_mem := ⋯ }) ⋯
        (Quotient.mk'' y)

Tactic:
simp only [← mk_mul, Quotient.map'_mk'', num_mul, map_mul, den_mul]; 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 j : ℕ
h2 : ¬m + m < j
h1 : j ≤ m
⊢ f ^ i ∈ 𝒜 (m * i)

Tactic:
mul_comm

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 j : ℕ
h2 : ¬m + m < j
h1 : j ≤ m
⊢ f ^ i ∈ 𝒜 (i * 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
q : ↑↑(Spec A⁰_ f).toPresheafedSpace
a b : A
ha : a ∈ carrier f_deg q
hb : b ∈ carrier f_deg q
i j : ℕ
h2 : ¬m + m < j
h1 : ¬j ≤ m
⊢ f ^ i ∈ 𝒜 (m * i)

Tactic:
mul_comm

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 j : ℕ
h2 : ¬m + m < j
h1 : ¬j ≤ m
⊢ f ^ i ∈ 𝒜 (i * m)
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f g : A
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this :
  ((Scheme.ΓSpecIso (CommRingCat.of (Away 𝒜 f))).inv ≫ Scheme.Hom.app (basicOpenToSpec 𝒜 f) ⊤) ≫
      (basicOpen 𝒜 f).topIso.hom =
    awayToSection 𝒜 f
⊢ IsIso
    (((Scheme.ΓSpecIso (CommRingCat.of (Away 𝒜 f))).inv ≫ Scheme.Hom.app (basicOpenIsoSpec 𝒜 f f_deg hm).hom ⊤) ≫
      (basicOpen 𝒜 f).topIso.hom)

Tactic:
infer_instance

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

Goal state before:
pp: σ : Type u_1
A : Type u
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
m' : ℕ
g : A
g_deg : g ∈ 𝒜 m'
hm' : 0 < m'
this : Algebra (Away 𝒜 f) (Away 𝒜 (f * g)) := (awayMap 𝒜 g_deg ⋯).toAlgebra
⊢ Set.range ⇑(Spec.map (CommRingCat.ofHom (awayMap 𝒜 g_deg ⋯))) =
    ↑(PrimeSpectrum.basicOpen (Away.isLocalizationElem f_deg g_deg))

Tactic:
let := HomogeneousLocalization.Away.isLocalization_mul f_deg g_deg rfl hm.ne'

Goal state after:
σ : Type u_1
A : Type u
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
m' : ℕ
g : A
g_deg : g ∈ 𝒜 m'
hm' : 0 < m'
this✝ : Algebra (Away 𝒜 f) (Away 𝒜 (f * g)) := (awayMap 𝒜 g_deg ⋯).toAlgebra
this : IsLocalization.Away (Away.isLocalizationElem f_deg g_deg) (Away 𝒜 (f * g)) :=
  Away.isLocalization_mul f_deg g_deg rfl (LT.lt.ne' hm)
⊢ Set.range ⇑(Spec.map (CommRingCat.ofHom (awayMap 𝒜 g_deg ⋯))) =
    ↑(PrimeSpectrum.basicOpen (Away.isLocalizationElem f_deg g_deg))
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f : A →+* ↑Γ(X, ⊤)
x x' : ↑Γ(X, ⊤)
t t' : A
d d' : ℕ
H : f t = x
h0d : 0 < d
hd : t ∈ 𝒜 d
H' : f t' = x'
h0d' : 0 < d'
hd' : t' ∈ 𝒜 d'
s : A
hts : t * s = t'
n : ℕ
hn : d + n = d'
hs : s ∈ 𝒜 n
hx'x : PrimeSpectrum.basicOpen x' ≤ PrimeSpectrum.basicOpen x
⊢ Scheme.Hom.resLE X.toSpecΓ (PrimeSpectrum.basicOpen x) (X.basicOpen x') ⋯ ≫
      (basicOpenIsoSpecAway x).hom ≫
        Spec.map (CommRingCat.ofHom (IsLocalization.map (Localization.Away x) f ⋯)) ≫
          Spec.map (CommRingCat.ofHom (algebraMap (Away 𝒜 t) (Localization.Away t))) ≫ awayι 𝒜 t hd h0d =
    Scheme.Hom.resLE X.toSpecΓ (PrimeSpectrum.basicOpen x') (X.basicOpen x') ⋯ ≫
      (basicOpenIsoSpecAway x').hom ≫
        Spec.map (CommRingCat.ofHom (IsLocalization.map (Localization.Away x') f ⋯)) ≫
          Spec.map (CommRingCat.ofHom (algebraMap (Away 𝒜 t') (Localization.Away t'))) ≫ awayι 𝒜 t' hd' h0d'

Tactic:
← Scheme.Hom.resLE_map_assoc _ (by simp [X.toSpecΓ_preimage_basicOpen]) hx'x

Goal state after:
σ : Type u_1
A : Type u
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
X : Scheme
f : A →+* ↑Γ(X, ⊤)
x x' : ↑Γ(X, ⊤)
t t' : A
d d' : ℕ
H : f t = x
h0d : 0 < d
hd : t ∈ 𝒜 d
H' : f t' = x'
h0d' : 0 < d'
hd' : t' ∈ 𝒜 d'
s : A
hts : t * s = t'
n : ℕ
hn : d + n = d'
hs : s ∈ 𝒜 n
hx'x : PrimeSpectrum.basicOpen x' ≤ PrimeSpectrum.basicOpen x
⊢ Scheme.Hom.resLE X.toSpecΓ (PrimeSpectrum.basicOpen x') (X.basicOpen x') ⋯ ≫
      (Spec Γ(X, ⊤)).homOfLE hx'x ≫
        (basicOpenIsoSpecAway x).hom ≫
          Spec.map (CommRingCat.ofHom (IsLocalization.map (Localization.Away x) f ⋯)) ≫
            Spec.map (CommRingCat.ofHom (algebraMap (Away 𝒜 t) (Localization.Away t))) ≫ awayι 𝒜 t hd h0d =
    Scheme.Hom.resLE X.toSpecΓ (PrimeSpectrum.basicOpen x') (X.basicOpen x') ⋯ ≫
      (basicOpenIsoSpecAway x').hom ≫
        Spec.map (CommRingCat.ofHom (IsLocalization.map (Localization.Away x') f ⋯)) ≫
          Spec.map (CommRingCat.ofHom (algebraMap (Away 𝒜 t') (Localization.Away t'))) ≫ awayι 𝒜 t' hd' h0d'
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: case exists_of_eq
A : Type u_2
σ : Type u_3
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
e d : ℕ
f : A
hf : f ∈ 𝒜 d
g : A
hg : g ∈ 𝒜 e
x : A
hx : x = f * g
hd : d ≠ 0
this : Algebra (Away 𝒜 f) (Away 𝒜 x) := ⋯
n : ℕ
a : A
ha : a ∈ 𝒜 (n • d)
m : ℕ
b : A
hb : b ∈ 𝒜 (m • d)
k : ℕ
hc : ↑⟨(fun x_1 ↦ x ^ x_1) k, ⋯⟩ * (x ^ m * (a * g ^ n)) = ↑⟨(fun x_1 ↦ x ^ x_1) k, ⋯⟩ * (x ^ n * (b * g ^ m))
⊢ val (↑⟨(fun x ↦ isLocalizationElem hf hg ^ x) (k + m + n), ⋯⟩ * Away.mk 𝒜 hf n a ha) =
    val (↑⟨(fun x ↦ isLocalizationElem hf hg ^ x) (k + m + n), ⋯⟩ * Away.mk 𝒜 hf m b hb)

Tactic:
simp only [val_mul, val_pow, val_mk, Localization.mk_pow,
      Localization.mk_eq_mk_iff, Localization.r_iff_exists, Submonoid.coe_mul, Localization.mk_mul,
      SubmonoidClass.coe_pow, Subtype.exists, exists_prop]

Goal state after:
case exists_of_eq
A : Type u_2
σ : Type u_3
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
e d : ℕ
f : A
hf : f ∈ 𝒜 d
g : A
hg : g ∈ 𝒜 e
x : A
hx : x = f * g
hd : d ≠ 0
this : Algebra (Away 𝒜 f) (Away 𝒜 x) := (awayMap 𝒜 hg hx).toAlgebra
n : ℕ
a : A
ha : a ∈ 𝒜 (n • d)
m : ℕ
b : A
hb : b ∈ 𝒜 (m • d)
k : ℕ
hc : ↑⟨(fun x_1 ↦ x ^ x_1) k, ⋯⟩ * (x ^ m * (a * g ^ n)) = ↑⟨(fun x_1 ↦ x ^ x_1) k, ⋯⟩ * (x ^ n * (b * g ^ m))
⊢ ∃ a_1 ∈ Submonoid.powers f,
    a_1 * ((f ^ e) ^ (k + m + n) * f ^ m * ((g ^ d) ^ (k + m + n) * a)) =
      a_1 * ((f ^ e) ^ (k + m + n) * f ^ n * ((g ^ d) ^ (k + m + n) * b))
Module: Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Goal state before:
pp: case neg.smul
ι : Type u_1
A : Type u_2
σ : Type u_3
inst✝⁶ : CommRing A
inst✝⁵ : SetLike σ A
inst✝⁴ : AddSubgroupClass σ A
inst✝³ : AddCommMonoid ι
inst✝² : DecidableEq ι
𝒜 : ι → σ
inst✝¹ : GradedRing 𝒜
f : A
d : ι
hf : f ∈ 𝒜 d
ι' : Type u_4
inst✝ : Fintype ι'
v : ι' → A
hx✝ : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤
dv : ι' → ι
hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)
HH : ¬Subsingleton (Away 𝒜 f)
n : ι
a : A
j : ℕ
hb' : (fun x ↦ f ^ x) j ∈ 𝒜 n
hfj : ¬f ^ j = 0
r : ↥(𝒜 0)
x : A
hx : x ∈ Submodule.span ↥(𝒜 0) ↑(Submonoid.closure (Set.range v))
hx' :
  mk { deg := n, num := ((decompose 𝒜) x) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ } ∈
    Submodule.span ↥(𝒜 0) {x | ∃ a ai, ∃ (hai : ∑ i, ai i • dv i = a • d), Away.mk 𝒜 hf a (∏ i, v i ^ ai i) ⋯ = x}
⊢ mk { deg := n, num := ((decompose 𝒜) (r • x)) n, den := ⟨f ^ j, hb'⟩, den_mem := ⋯ } ∈
    Submodule.span ↥(𝒜 0) {x | ∃ a ai, ∃ (hai : ∑ i, ai i • dv i = a • d), Away.mk 𝒜 hf a (∏ i, v i ^ ai i) ⋯ = x}

Tactic:
| smul r x hx hx' =>

Goal state after:
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:
by
    simp_rw [continuous_iff_isClosed, isClosed_iff_zeroLocus, exists_imp, forall_eq_apply_imp_iff]
    exact fun s ↦ ⟨f '' s, by ext; simp⟩

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

Goal state before:
pp: R : CommRingCat
M : ModuleCat ↑R
⊢ IsIso tilde.toTildeΓNatIso.hom

Tactic:
infer_instance

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

Goal state before:
pp: A B C σ τ ψ : Type u
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₊)
⊢ map (g.comp f) ⋯ = map g hg ≫ map f hf

Tactic:
refine (mapAffineOpenCover _ <| irrelevant_le_map_comp hf hg).openCover.hom_ext _ _ fun s ↦ ?_

Goal state after:
A B C σ τ ψ : Type u
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₊)
s : (mapAffineOpenCover (g.comp f) ⋯).openCover.I₀
⊢ (mapAffineOpenCover (g.comp f) ⋯).openCover.f s ≫ map (g.comp f) ⋯ =
    (mapAffineOpenCover (g.comp f) ⋯).openCover.f s ≫ map g hg ≫ map f hf
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:
infer_instance

Goal state after:
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 (Functor.whiskerRight α (forget₂ CommRingCat RingCat))

Tactic:
infer_instance

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

Goal state before:
pp: X : Scheme
M : X.Modules
inst✝ : SheafOfModules.IsQuasicoherent M
⊢ ∃ 𝒰, ∀ (i : 𝒰.I₀), Nonempty (SheafOfModules.Presentation (M.restrict (𝒰.f i)))

Tactic:
obtain ⟨ι, U, pres, hU, hU'⟩ := M.exists_isOpenCover_presentation
  refine ⟨Scheme.AffineOpenCover.ofIsOpenCover _ hU hU', fun i ↦ ⟨?_⟩⟩
  exact SheafOfModules.Presentation.ofIsIso.{u, u, u} ((restrictFunctorComp _ _).app M).inv <|
    (presentationRestrict (hU' i).isoSpec.inv (pres i))

Goal state after:
Module: Mathlib.RingTheory.Valuation.LocalSubring

Goal state before:
pp: R : Type u_1
S : Type u_2
K : Type u_3
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Field K
V : ValuationSubring K
⊢ IsIntegrallyClosed ↥V.valuation.integer

Tactic:
infer_instance

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
a : A
i : ℕ
⊢ f ^ i ∈ 𝒜 (m * i)

Tactic:
mul_comm

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
a : A
i : ℕ
⊢ f ^ i ∈ 𝒜 (i * m)
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper

Goal state before:
pp: case hxj.convert_8
σ : Type u_1
A : Type u_2
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
d e : ℕ
f : A
hf : f ∈ 𝒜 d
g : A
hg : g ∈ 𝒜 e
x : A
hx : x = f * g
hd : 0 < d
n : ℕ
a : A
ha : a ∈ 𝒜 n
j : ℕ
hb' : (fun x_1 ↦ x ^ x_1) j ∈ 𝒜 n
hfg : ¬(f * g) ^ j = 0
⊢ SetLike.GradedMonoid 𝒜

Tactic:
infer_instance

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
this : IsIso (toSpec 𝒜 f).base
x : ↑(Proj.restrict ⋯).toTopCat
⊢ IsIso
    ((specStalkEquiv 𝒜 f x f_deg hm).hom ≫
      («Proj».stalkIso' 𝒜 ↑x).toCommRingCatIso.inv ≫ (Proj.restrictStalkIso ⋯ x).inv)

Tactic:
infer_instance

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

Goal state before:
pp: σ : Type u_1
A : Type u_2
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
⊢ IsSeparated (toSpecZero 𝒜)

Tactic:
refine ⟨IsZariskiLocalAtTarget.of_openCover (Pullback.openCoverOfLeftRight
    (affineOpenCover 𝒜).openCover (affineOpenCover 𝒜).openCover _ _) ?_⟩

Goal state after:
σ : Type u_1
A : Type u_2
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
⊢ ∀
    (i :
      (Pullback.openCoverOfLeftRight (affineOpenCover 𝒜).openCover (affineOpenCover 𝒜).openCover (toSpecZero 𝒜)
            (toSpecZero 𝒜)).toPreZeroHypercover.1),
    IsClosedImmersion
      (Cover.pullbackHom
        (Pullback.openCoverOfLeftRight (affineOpenCover 𝒜).openCover (affineOpenCover 𝒜).openCover (toSpecZero 𝒜)
          (toSpecZero 𝒜))
        (diagonal (toSpecZero 𝒜)) i)
Module: Mathlib.CategoryTheory.Sites.Point.Basic

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
Φ : J.Point
A : Type u'
inst✝⁵ : Category.{v', u'} A
B : Type u''
inst✝⁴ : Category.{v'', u''} B
inst✝³ : HasColimitsOfSize.{w, w, v', u'} A
inst✝² : HasColimitsOfSize.{w, w, v'', u''} B
FC : A → A → Type u_1
CC : A → Type w'
inst✝¹ : (X Y : A) → FunLike (FC X Y) (CC X) (CC Y)
inst✝ : ConcreteCategory A FC
⊢ ∀ {J_1 : Type w} [inst : Category.{w, w} J_1],
    PreservesColimitsOfShape J_1
      ((Functor.whiskeringLeft Φ.fiber.Elementsᵒᵖ Cᵒᵖ A).obj (CategoryOfElements.π Φ.fiber).op ⋙ colim)

Tactic:
infer_instance

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Point.Basic

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
Φ : J.Point
inst✝¹ : LocallySmall.{w, v, u} C
U T : C
f : U ⟶ T
inst✝ : Mono f
⊢ Mono (Φ.fiber.map f)

Tactic:
infer_instance

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

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
hf : IsOpenImmersion f
⊢ Etale f

Tactic:
infer_instance

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

Goal state before:
pp: R : CommRingCat
M : (Spec (CommRingCat.of ↑R)).Modules
N : ModuleCat ↑R
e : (tilde.functor R).obj N ≅ M
⊢ SheafOfModules.isQuasicoherent (Spec R).ringCatSheaf (tilde N)

Tactic:
infer_instance

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

Goal state before:
pp: R : CommRingCat
M x✝ : ModuleCat ↑R
⊢ SheafOfModules.IsQuasicoherent (tilde x✝)

Tactic:
infer_instance

Goal state after:
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense

Goal state before:
pp: C₀ : Type u₀
C : Type u
inst✝³ : Category.{v₀, u₀} C₀
inst✝² : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝¹ : Category.{v', u'} A
inst✝ : IsDenseSubsite J₀ J F
data : (X : C) → F.OneHypercoverDenseData J₀ J X
G : Cᵒᵖ ⥤ A
⊢ Presheaf.IsSheaf J G ↔
    Presheaf.IsSheaf J₀ (F.op ⋙ G) ∧ ∀ (X : C), Nonempty (IsLimit ((data X).toOneHypercover.multifork G))

Tactic:
refine ⟨fun hG ↦ ⟨op_comp_isSheaf F J₀ J ⟨_, hG⟩,
    fun X ↦ ⟨(data X).toOneHypercover.isLimitMultifork ⟨G, hG⟩⟩⟩, fun ⟨hG₀, hG⟩ ↦ ?_⟩

Goal state after:
C₀ : Type u₀
C : Type u
inst✝³ : Category.{v₀, u₀} C₀
inst✝² : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝¹ : Category.{v', u'} A
inst✝ : IsDenseSubsite J₀ J F
data : (X : C) → F.OneHypercoverDenseData J₀ J X
G : Cᵒᵖ ⥤ A
x✝ : Presheaf.IsSheaf J₀ (F.op ⋙ G) ∧ ∀ (X : C), Nonempty (IsLimit ((data X).toOneHypercover.multifork G))
hG₀ : Presheaf.IsSheaf J₀ (F.op ⋙ G)
hG : ∀ (X : C), Nonempty (IsLimit ((data X).toOneHypercover.multifork G))
⊢ Presheaf.IsSheaf J G
Module: Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Sheaf

Goal state before:
pp: C : Type u
A : Type u₁
K✝ : Type u₂
inst✝⁴ : Category.{v, u} C
inst✝³ : Category.{v₁, u₁} A
inst✝² : Category.{v₂, u₂} K✝
J : GrothendieckTopology C
inst✝¹ : HasSheafify J A
inst✝ : HasFilteredColimitsOfSize.{v₂, u₂, v₁, u₁} A
K : Type u₂
⊢ ∀ [inst : Category.{v₂, u₂} K] [IsFiltered K], HasColimitsOfShape K (Sheaf J A)

Tactic:
infer_instance

Goal state after:
Module: Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Sheaf

Goal state before:
pp: C : Type u
A : Type u₁
K✝ : Type u₂
inst✝⁶ : Category.{v, u} C
inst✝⁵ : Category.{v₁, u₁} A
inst✝⁴ : Category.{v₂, u₂} K✝
J : GrothendieckTopology C
inst✝³ : HasFiniteLimits A
inst✝² : HasSheafify J A
inst✝¹ : HasFilteredColimitsOfSize.{v₂, u₂, v₁, u₁} A
inst✝ : AB5OfSize.{v₂, u₂, v₁, u₁} A
K : Type u₂
x✝¹ : Category.{v₂, u₂} K
x✝ : IsFiltered K
⊢ HasExactColimitsOfShape K (Sheaf J A)

Tactic:
infer_instance

Goal state after:
Module: Mathlib.CategoryTheory.Sites.ConstantSheaf

Goal state before:
pp: C : Type u_1
inst✝¹² : Category.{v_1, u_1} C
J : GrothendieckTopology C
D : Type u_2
inst✝¹¹ : Category.{v_2, u_2} D
inst✝¹⁰ : HasWeakSheafify J D
B : Type u_3
inst✝⁹ : Category.{v_3, u_3} B
U : D ⥤ B
inst✝⁸ : HasWeakSheafify J B
inst✝⁷ : J.PreservesSheafification U
inst✝⁶ : J.HasSheafCompose U
F : Sheaf J D
inst✝⁵ : (constantSheaf J D).Faithful
inst✝⁴ : (constantSheaf J D).Full
inst✝³ : (constantSheaf J B).Faithful
inst✝² : (constantSheaf J B).Full
inst✝¹ : (sheafCompose J U).ReflectsIsomorphisms
inst✝ : IsConstant J ((sheafCompose J U).obj F)
T : C
hT : IsTerminal T
⊢ IsIso
    ((constantCommuteCompose J U).hom.app (F.obj.obj (Opposite.op T)) ≫
      (constantSheafAdj J B hT).counit.app ((sheafCompose J U).obj F))

Tactic:
infer_instance

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

Goal state before:
pp: S : Scheme
X Y : Scheme
f : X ⟶ Y
hf : IsIso f
⊢ QuasiCompactCover (PreZeroHypercover.singleton f)

Tactic:
infer_instance

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

Goal state before:
pp: case refine_2
S : Scheme
X Y : Scheme
f : X ⟶ Y
E : PreZeroHypercover Y
h : ∀ (i : E.I₀), HasPullback f (E.f i)
hE : QuasiCompactCover E
⊢ QuasiCompactCover (PreZeroHypercover.pullback₁ f E)

Tactic:
infer_instance

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

Goal state before:
pp: S : Scheme
X : Scheme
E : PreZeroHypercover X
F : (i : E.I₀) → PreZeroHypercover (E.X i)
hE : QuasiCompactCover E
hF : ∀ (i : E.I₀), QuasiCompactCover (F i)
⊢ QuasiCompactCover (E.bind F)

Tactic:
infer_instance

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

Goal state before:
pp: S : Scheme
X : Scheme
E F : PreZeroHypercover X
hE : QuasiCompactCover E
hF : QuasiCompactCover F
⊢ QuasiCompactCover (E.sum F)

Tactic:
infer_instance

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

Goal state before:
pp: X : Scheme
inst✝ : IsEmpty ↥X
⊢ QuasiCompactCover (PreZeroHypercover.empty X)

Tactic:
infer_instance

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

Goal state before:
pp: S : Scheme
⊢ Functor.IsDenseSubsite ((AffineEtale.Spec S).inducedTopology S.smallEtaleTopology) S.smallEtaleTopology
    (AffineEtale.Spec S)

Tactic:
infer_instance

Goal state after: