mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Bicategory.Adjunction.Mate

Goal state before:
pp: B : Type u
inst✝ : Bicategory B
a b c d e f x y z : B
g₁ : a ⟶ d
h₁ : b ⟶ e
k₁ : c ⟶ f
g₂ : d ⟶ x
h₂ : e ⟶ y
k₂ : f ⟶ z
l₁ : a ⟶ b
l₂ : b ⟶ c
l₃ : d ⟶ e
l₄ : e ⟶ f
l₅ : x ⟶ y
l₆ : y ⟶ z
α : g₁ ≫ l₃ ⟶ l₁ ≫ h₁
β : h₁ ≫ l₄ ⟶ l₂ ≫ k₁
γ : g₂ ≫ l₅ ⟶ l₃ ≫ h₂
δ : h₂ ≫ l₆ ⟶ l₄ ≫ k₂
⊢ comp α β γ δ = hcomp (vcomp α γ) (vcomp β δ)

Tactic:
dsimp only [comp, vcomp, hcomp]
  calc
    _ = 𝟙 _ ⊗≫ g₁ ◁ γ ▷ l₆ ⊗≫ ((g₁ ≫ l₃) ◁ δ ≫ α ▷ (l₄ ≫ k₂)) ⊗≫ l₁ ◁ β ▷ k₂ ⊗≫ 𝟙 _ := by
      bicategory
    _ = _ := by
      rw [whisker_exchange]
      bicategory

Goal state after:
Module: Mathlib.AlgebraicGeometry.PointsPi

Goal state before:
pp: ι : Type u
R : ι → CommRingCat
V : Scheme
f : (∐ fun i ↦ Spec (R i)) ⟶ V
g : V ⟶ Spec (CommRingCat.of ((i : ι) → ↑(R i)))
inst✝¹ : IsImmersion g
inst✝ : CompactSpace ↥V
hU' : f ≫ g = sigmaSpec R
this✝ : Scheme.Hom.coborderRange g = ⊤
this : IsIso (Scheme.Hom.coborderRange g).ι
⊢ IsClosedImmersion g

Tactic:
rw [← g.liftCoborder_ι]

Goal state after:
ι : Type u
R : ι → CommRingCat
V : Scheme
f : (∐ fun i ↦ Spec (R i)) ⟶ V
g : V ⟶ Spec (CommRingCat.of ((i : ι) → ↑(R i)))
inst✝¹ : IsImmersion g
inst✝ : CompactSpace ↥V
hU' : f ≫ g = sigmaSpec R
this✝ : Scheme.Hom.coborderRange g = ⊤
this : IsIso (Scheme.Hom.coborderRange g).ι
⊢ IsClosedImmersion (Scheme.Hom.liftCoborder g ≫ (Scheme.Hom.coborderRange g).ι)
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) ↔ x ∈ (zeroLocus 𝒜 {r})ᶜ

Tactic:
simp only [Set.mem_compl_iff, mem_zeroLocus, Set.singleton_subset_iff]

Goal state after:
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
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 g : A
⊢ basicOpen 𝒜 (f * g) ≤ basicOpen 𝒜 f

Tactic:
rw [basicOpen_mul 𝒜 f g]

Goal state after:
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 g : A
⊢ basicOpen 𝒜 f ⊓ basicOpen 𝒜 g ≤ basicOpen 𝒜 f
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 g : A
⊢ basicOpen 𝒜 (f * g) ≤ basicOpen 𝒜 g

Tactic:
rw [basicOpen_mul 𝒜 f g]

Goal state after:
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 g : A
⊢ basicOpen 𝒜 f ⊓ basicOpen 𝒜 g ≤ basicOpen 𝒜 g
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

Goal state before:
pp: case mp
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
h :
  HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal
⊢ Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩ ∈ ⇑(algebraMap (A⁰_ f) (Localization.Away f)) '' {s | s ∈ q.asIdeal}

pp: case mpr
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
h : Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩ ∈ ⇑(algebraMap (A⁰_ f) (Localization.Away f)) '' {s | s ∈ q.asIdeal}
⊢ HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal

Tactic:
· rw [Set.mem_image]; refine ⟨_, h, rfl⟩

Goal state after:
case mpr
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
h : Localization.mk ((proj 𝒜 i) a ^ m) ⟨f ^ i, ⋯⟩ ∈ ⇑(algebraMap (A⁰_ f) (Localization.Away f)) '' {s | s ∈ q.asIdeal}
⊢ HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ } ∈ q.asIdeal
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
ι : Type u_2
f : ι → A
hfn : ∀ (i : ι), ∃ n, f i ∈ 𝒜 n
hf : Algebra.adjoin (↥(𝒜 0)) (Set.range f) = ⊤
⊢ ⨆ i, basicOpen 𝒜 (f i) = ⊤

Tactic:
apply Proj.iSup_basicOpen_eq_top

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
ι : Type u_2
f : ι → A
hfn : ∀ (i : ι), ∃ n, f i ∈ 𝒜 n
hf : Algebra.adjoin (↥(𝒜 0)) (Set.range f) = ⊤
⊢ (HomogeneousIdeal.irrelevant 𝒜).toIdeal ≤ Ideal.span (Set.range f)
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'
x : A
hx : x = f * g
⊢ Spec.map (CommRingCat.ofHom (awayMap 𝒜 g_deg hx)) ≫ awayι 𝒜 f f_deg hm = awayι 𝒜 x ⋯ ⋯

Tactic:
awayι,

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'
x : A
hx : x = f * g
⊢ Spec.map (CommRingCat.ofHom (awayMap 𝒜 g_deg hx)) ≫ (basicOpenIsoSpec 𝒜 f f_deg hm).inv ≫ (basicOpen 𝒜 f).ι =
    awayι 𝒜 x ⋯ ⋯
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'
x : A
hx : x = f * g
⊢ Spec.map (CommRingCat.ofHom (awayMap 𝒜 g_deg hx)) ≫ (basicOpenIsoSpec 𝒜 f f_deg hm).inv ≫ (basicOpen 𝒜 f).ι =
    awayι 𝒜 x ⋯ ⋯

Tactic:
awayι,

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'
x : A
hx : x = f * g
⊢ Spec.map (CommRingCat.ofHom (awayMap 𝒜 g_deg hx)) ≫ (basicOpenIsoSpec 𝒜 f f_deg hm).inv ≫ (basicOpen 𝒜 f).ι =
    (basicOpenIsoSpec 𝒜 x ⋯ ⋯).inv ≫ (basicOpen 𝒜 x).ι
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'
⊢ ⇑(awayι 𝒜 f f_deg hm) ⁻¹' ↑(basicOpen 𝒜 g) =
    ⇑(TopCat.Hom.hom (Limits.pullback.fst (awayι 𝒜 f f_deg hm) (awayι 𝒜 g g_deg hm')).base) ''
      Set.range ⇑(TopCat.Hom.hom (pullbackAwayιIso 𝒜 f_deg hm g_deg hm' ⋯).inv.base)

Tactic:
rw [Set.range_eq_univ.mpr (by exact
      (pullbackAwayιIso 𝒜 f_deg hm g_deg hm' rfl).inv.homeomorph.surjective),
      ← opensRange_awayι _ _ g_deg hm']

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'
⊢ ⇑(awayι 𝒜 f f_deg hm) ⁻¹' ↑(Scheme.Hom.opensRange (awayι 𝒜 g g_deg hm')) =
    ⇑(TopCat.Hom.hom (Limits.pullback.fst (awayι 𝒜 f f_deg hm) (awayι 𝒜 g g_deg hm')).base) '' Set.univ
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic

Goal state before:
pp: case a
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, ⊤)
hf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤
r : A
n : ℕ
hn : 0 < n
hr : r ∈ 𝒜 n
x : ↥X
hx : x ∈ fromOfGlobalSections 𝒜 f hf ⁻¹ᵁ basicOpen 𝒜 r
⊢ x ∈ X.basicOpen (f r)

Tactic:
obtain ⟨i, x, rfl⟩ := (openCoverOfMapIrrelevantEqTop 𝒜 f hf).exists_eq x

Goal state after:
case a
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, ⊤)
hf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤
r : A
n : ℕ
hn : 0 < n
hr : r ∈ 𝒜 n
i : (openCoverOfMapIrrelevantEqTop 𝒜 f hf).I₀
x : ↥((openCoverOfMapIrrelevantEqTop 𝒜 f hf).X i)
hx : ((openCoverOfMapIrrelevantEqTop 𝒜 f hf).f i) x ∈ fromOfGlobalSections 𝒜 f hf ⁻¹ᵁ basicOpen 𝒜 r
⊢ ((openCoverOfMapIrrelevantEqTop 𝒜 f hf).f i) x ∈ X.basicOpen (f r)
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
⊢ ∑ x, ai x % d * dv x + d * ∑ i, ai i / d * dv i - (∑ i, dv i * (ai i / d)) * d = ∑ x, ai x % d * dv x

Tactic:
mul_comm d,

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
⊢ ∑ x, ai x % d * dv x + (∑ i, ai i / d * dv i) * d - (∑ i, dv i * (ai i / d)) * d = ∑ x, ai x % d * dv x
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor

Goal state before:
pp: 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
⊢ awayι ℬ (f s) ⋯ hi ≫ map f hf = Spec.map (CommRingCat.ofHom (Away.map f s)) ≫ awayι 𝒜 s hs hi

Tactic:
awayι,

Goal state after:
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
⊢ ((basicOpenIsoSpec ℬ (f s) ⋯ hi).inv ≫ (basicOpen ℬ (f s)).ι) ≫ map f hf =
    Spec.map (CommRingCat.ofHom (Away.map f s)) ≫ awayι 𝒜 s hs hi
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor

Goal state before:
pp: 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
⊢ ((basicOpenIsoSpec ℬ (f s) ⋯ hi).inv ≫ (basicOpen ℬ (f s)).ι) ≫ map f hf =
    Spec.map (CommRingCat.ofHom (Away.map f s)) ≫ awayι 𝒜 s hs hi

Tactic:
awayι,

Goal state after:
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
⊢ ((basicOpenIsoSpec ℬ (f s) ⋯ hi).inv ≫ (basicOpen ℬ (f s)).ι) ≫ map f hf =
    Spec.map (CommRingCat.ofHom (Away.map f s)) ≫ (basicOpenIsoSpec 𝒜 s hs hi).inv ≫ (basicOpen 𝒜 s).ι
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 : (Opens.grothendieckTopology ↥X).CoversTop 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:
Opens.coversTop_iff,

Goal state after:
case refine_2
X : Scheme
M : X.Modules
inst✝ : SheafOfModules.IsQuasicoherent M
I : Type u
W : I → TopologicalSpace.Opens ↥X
cov : IsOpenCover 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.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 ⋯
a : ℕ
hai : ∑ i, 0 i • dx i = a • d
hai' : ∀ (i : ↥s), 0 i ≤ d
⊢ Away.mk 𝒜 hf a (∏ i, ↑i ^ 0 i) ⋯ ∈ ↑(↥(𝒜 0))[f']

Tactic:
simp only [Finset.univ_eq_attach, Pi.zero_apply, pow_zero, Finset.prod_const_one, mem_coe]

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 ⋯
a : ℕ
hai : ∑ i, 0 i • dx i = a • d
hai' : ∀ (i : ↥s), 0 i ≤ d
⊢ Away.mk 𝒜 hf a 1 ⋯ ∈ (↥(𝒜 0))[f']
Module: Mathlib.RingTheory.Valuation.LocalSubring

Goal state before:
pp: K : Type u_3
inst✝ : Field K
R : ValuationSubring K
S : LocalSubring K
hS : R.toLocalSubring ≤ S
x : K
hx : x ∈ S.toSubring
h : x⁻¹ ∈ R.carrier
h' : x ∉ R.toLocalSubring.toSubring
hx0 : x ≠ 0
this : IsUnit ((Subring.inclusion ⋯) ⟨x⁻¹, h⟩)
x' : ↥R.toLocalSubring.toSubring
hx' : ⟨x⁻¹, h⟩ * x' = 1
⊢ ↑x' = x

Tactic:
simpa [Subtype.ext_iff, inv_mul_eq_iff_eq_mul₀ hx0] using hx'

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
⊢ Subtype.val ⁻¹' ↑(pbo ↑(((decompose 𝒜) a) i)) =
    ⇑(ConcreteCategory.hom (toSpec 𝒜 f)) ⁻¹'
      (sbo
          HomogeneousLocalization.mk
            { deg := m * i, num := ⟨↑(((decompose 𝒜) a) i) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ }).carrier

Tactic:
Opens.carrier_eq_coe,

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
⊢ Subtype.val ⁻¹' ↑(pbo ↑(((decompose 𝒜) a) i)) =
    ⇑(ConcreteCategory.hom (toSpec 𝒜 f)) ⁻¹'
      ↑(sbo
          HomogeneousLocalization.mk
            { deg := m * i, num := ⟨↑(((decompose 𝒜) a) i) ^ m, ⋯⟩, den := ⟨f ^ i, ⋯⟩, den_mem := ⋯ })
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
⊢ (Scheme.ΓSpecIso (CommRingCat.of (A⁰_ f))).inv ≫
      (Spec A⁰_ f).presheaf.germ ⊤ ((ConcreteCategory.hom (toSpec 𝒜 f).base) x) ⋯ ≫
        LocallyRingedSpace.Hom.stalkMap (toSpec 𝒜 f) x =
    awayToΓ 𝒜 f ≫ (Proj.restrict ⋯).presheaf.Γgerm x

Tactic:
dsimp
  erw [LocallyRingedSpace.stalkMap_germ (toSpec 𝒜 f) ⊤ x (by simp)]
  erw [toOpen_toSpec_val_c_app_assoc]
  rfl

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

Goal state before:
pp: case mp
P Q : MorphismProperty Scheme
S : Scheme
inst✝⁵ : P.IsStableUnderBaseChange
inst✝⁴ : P.IsMultiplicative
inst✝³ : P.RespectsIso
inst✝² : Q.IsStableUnderComposition
inst✝¹ : Q.IsStableUnderBaseChange
inst✝ : Q.HasOfPostcompProperty Q
hPQ : P ≤ Q
X : Q.Over ⊤ S
R : Sieve X
this : (MorphismProperty.Over.forget Q ⊤ S).LocallyCoverDense (overGrothendieckTopology P S)
⊢ (∃ 𝒰 x, 𝒰.toPresieveOver ≤ (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R).arrows) →
    ∃ R_1 ∈ (smallPretopology P Q).coverings X, R_1 ≤ R.arrows

Tactic:
intro ⟨𝒰, h, le⟩

Goal state after:
case mp
P Q : MorphismProperty Scheme
S : Scheme
inst✝⁵ : P.IsStableUnderBaseChange
inst✝⁴ : P.IsMultiplicative
inst✝³ : P.RespectsIso
inst✝² : Q.IsStableUnderComposition
inst✝¹ : Q.IsStableUnderBaseChange
inst✝ : Q.HasOfPostcompProperty Q
hPQ : P ≤ Q
X : Q.Over ⊤ S
R : Sieve X
this : (MorphismProperty.Over.forget Q ⊤ S).LocallyCoverDense (overGrothendieckTopology P S)
𝒰 : Cover (precoverage P) ((MorphismProperty.Over.forget Q ⊤ S).obj X).left
h : Cover.Over S 𝒰
le : 𝒰.toPresieveOver ≤ (Sieve.functorPushforward (MorphismProperty.Over.forget Q ⊤ S) R).arrows
⊢ ∃ R_1 ∈ (smallPretopology P Q).coverings X, R_1 ≤ R.arrows
Module: Mathlib.AlgebraicGeometry.ValuativeCriterion

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
inst✝² : QuasiCompact f
inst✝¹ : QuasiSeparated f
inst✝ : LocallyOfFiniteType f
H : ValuativeCriterion f
⊢ IsProper f

Tactic:
by
  rw [eq_valuativeCriterion]
  exact ⟨⟨⟨‹_›, ‹_›⟩, ‹_›⟩, ‹_›⟩

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

Goal state before:
pp: case right
σ : Type u_1
A : Type u_2
inst✝¹⁰ : CommRing A
inst✝⁹ : SetLike σ A
inst✝⁸ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝⁷ : GradedRing 𝒜
O : Type u_3
inst✝⁶ : CommRing O
inst✝⁵ : IsDomain O
inst✝⁴ : ValuationRing O
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra O K
inst✝¹ : IsFractionRing O K
φ₀ : ↥(𝒜 0) →+* O
ι : Type u_5
inst✝ : Finite ι
x : ι → A
h2 : Algebra.adjoin (↥(𝒜 0)) (Set.range x) = ⊤
j : ι
φ : Away 𝒜 (x j) →+* K
hcomm : (algebraMap O K).comp φ₀ = φ.comp (fromZeroRingHom 𝒜 (Submonoid.powers (x j)))
d : ι → ℕ
hdi : ∀ (i : ι), 0 < d i
hxdi : ∀ (i : ι), x i ∈ 𝒜 (d i)
val✝ : Fintype ι
ψ : ι → ValueGroup O K := ⋯
this✝¹ : Nonempty ι
Kmax : ValueGroup O K := ⋯
i₀ : ι
hi1 : ψ i₀ = Kmax
hi₀ : ∀ (j : ι), ψ j ≤ ψ i₀
hKmax : 0 < Kmax
this✝ : Algebra (Away 𝒜 (x j)) (Away 𝒜 (x j * x i₀)) := ⋯
this : IsLocalization.Away (Away.isLocalizationElem ⋯ ⋯) (Away 𝒜 (x j * x i₀))
hunit : IsUnit (φ (Away.isLocalizationElem ⋯ ⋯))
φ' : Away 𝒜 (x j * x i₀) →+* K := ⋯
hφ'1 : ∀ (s : Away 𝒜 (x j)), φ' ((awayMap 𝒜 ⋯ ⋯) s) = φ s
⊢ (φ'.comp (awayMap 𝒜 ⋯ ⋯)).range ≤ (algebraMap O K).range

Tactic:
rintro _ ⟨sx, rfl⟩

Goal state after:
case right
σ : Type u_1
A : Type u_2
inst✝¹⁰ : CommRing A
inst✝⁹ : SetLike σ A
inst✝⁸ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝⁷ : GradedRing 𝒜
O : Type u_3
inst✝⁶ : CommRing O
inst✝⁵ : IsDomain O
inst✝⁴ : ValuationRing O
K : Type u_4
inst✝³ : Field K
inst✝² : Algebra O K
inst✝¹ : IsFractionRing O K
φ₀ : ↥(𝒜 0) →+* O
ι : Type u_5
inst✝ : Finite ι
x : ι → A
h2 : Algebra.adjoin (↥(𝒜 0)) (Set.range x) = ⊤
j : ι
φ : Away 𝒜 (x j) →+* K
hcomm : (algebraMap O K).comp φ₀ = φ.comp (fromZeroRingHom 𝒜 (Submonoid.powers (x j)))
d : ι → ℕ
hdi : ∀ (i : ι), 0 < d i
hxdi : ∀ (i : ι), x i ∈ 𝒜 (d i)
val✝ : Fintype ι
ψ : ι → ValueGroup O K := fun i ↦ (valuation O K) (φ (Away.isLocalizationElem ⋯ ⋯) ^ ∏ k ∈ Finset.univ.erase i, d k)
this✝¹ : Nonempty ι
Kmax : ValueGroup O K := (Finset.image ψ Finset.univ).max' ⋯
i₀ : ι
hi1 : ψ i₀ = Kmax
hi₀ : ∀ (j : ι), ψ j ≤ ψ i₀
hKmax : 0 < Kmax
this✝ : Algebra (Away 𝒜 (x j)) (Away 𝒜 (x j * x i₀)) := (awayMap 𝒜 ⋯ ⋯).toAlgebra
this : IsLocalization.Away (Away.isLocalizationElem ⋯ ⋯) (Away 𝒜 (x j * x i₀))
hunit : IsUnit (φ (Away.isLocalizationElem ⋯ ⋯))
φ' : Away 𝒜 (x j * x i₀) →+* K := IsLocalization.Away.lift (Away.isLocalizationElem ⋯ ⋯) hunit
hφ'1 : ∀ (s : Away 𝒜 (x j)), φ' ((awayMap 𝒜 ⋯ ⋯) s) = φ s
sx : Away 𝒜 (x i₀)
⊢ (φ'.comp (awayMap 𝒜 ⋯ ⋯)) sx ∈ (algebraMap O K).range
Module: Mathlib.CategoryTheory.Sites.Point.Basic

Goal state before:
pp: C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
Φ : J.Point
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : HasColimitsOfSize.{w, w, v', u'} A
FC : A → A → Type u_1
CC : A → Type w'
inst✝⁴ : (X Y : A) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory A FC
P Q : Cᵒᵖ ⥤ A
inst✝² : PreservesFilteredColimitsOfSize.{w, w, v', w', u', w' + 1} (forget A)
inst✝¹ : LocallySmall.{w, v, u} C
f : P ⟶ Q
inst✝ : Presheaf.IsLocallyInjective J f
this :
  ∀ (X : C) (x : Φ.fiber.obj X) (p₁ p₂ : ToType (P.obj (op X))),
    (ConcreteCategory.hom (f.app (op X))) p₁ = (ConcreteCategory.hom (f.app (op X))) p₂ →
      (ConcreteCategory.hom (Φ.toPresheafFiber X x P)) p₁ = (ConcreteCategory.hom (Φ.toPresheafFiber X x P)) p₂
q₁ q₂ : CC (Φ.presheafFiber.obj P)
h : (ConcreteCategory.hom (Φ.presheafFiber.map f)) q₁ = (ConcreteCategory.hom (Φ.presheafFiber.map f)) q₂
⊢ q₁ = q₂

Tactic:
obtain ⟨X, x, p₁, p₂, rfl, rfl⟩ := Φ.toPresheafFiber_jointly_surjective₂ q₁ q₂

Goal state after:
C : Type u
inst✝⁷ : Category.{v, u} C
J : GrothendieckTopology C
Φ : J.Point
A : Type u'
inst✝⁶ : Category.{v', u'} A
inst✝⁵ : HasColimitsOfSize.{w, w, v', u'} A
FC : A → A → Type u_1
CC : A → Type w'
inst✝⁴ : (X Y : A) → FunLike (FC X Y) (CC X) (CC Y)
inst✝³ : ConcreteCategory A FC
P Q : Cᵒᵖ ⥤ A
inst✝² : PreservesFilteredColimitsOfSize.{w, w, v', w', u', w' + 1} (forget A)
inst✝¹ : LocallySmall.{w, v, u} C
f : P ⟶ Q
inst✝ : Presheaf.IsLocallyInjective J f
this :
  ∀ (X : C) (x : Φ.fiber.obj X) (p₁ p₂ : ToType (P.obj (op X))),
    (ConcreteCategory.hom (f.app (op X))) p₁ = (ConcreteCategory.hom (f.app (op X))) p₂ →
      (ConcreteCategory.hom (Φ.toPresheafFiber X x P)) p₁ = (ConcreteCategory.hom (Φ.toPresheafFiber X x P)) p₂
X : C
x : Φ.fiber.obj X
p₁ p₂ : ToType (P.obj (op X))
h :
  (ConcreteCategory.hom (Φ.presheafFiber.map f)) ((ConcreteCategory.hom (Φ.toPresheafFiber X x P)) p₁) =
    (ConcreteCategory.hom (Φ.presheafFiber.map f)) ((ConcreteCategory.hom (Φ.toPresheafFiber X x P)) p₂)
⊢ (ConcreteCategory.hom (Φ.toPresheafFiber X x P)) p₁ = (ConcreteCategory.hom (Φ.toPresheafFiber X x P)) p₂
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
hG₀ : Presheaf.IsSheaf J₀ (F.op ⋙ G)
hG : (X : C) → IsLimit ((data X).toOneHypercover.multifork G)
X : C
S : J.Cover X
s : Multifork (S.index G)
a : S.Arrow
i : (data a.Y).I₀
X₀ : C₀
b : X₀ ⟶ (data a.Y).X i
Y : C
j : (data X).I₀
c : F.obj X₀ ⟶ F.obj ((data X).X j)
fac₁ : c ≫ (data X).f j = F.map b ≫ (data a.Y).f i ≫ a.f
Y₀ : C₀
d : Y₀ ⟶ X₀
e : Y₀ ⟶ (data X).X j
fac₂ : F.map e = F.map d ≫ c
⊢ (↑S).arrows (F.map e ≫ (data X).f j)

Tactic:
fac₂,

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
hG₀ : Presheaf.IsSheaf J₀ (F.op ⋙ G)
hG : (X : C) → IsLimit ((data X).toOneHypercover.multifork G)
X : C
S : J.Cover X
s : Multifork (S.index G)
a : S.Arrow
i : (data a.Y).I₀
X₀ : C₀
b : X₀ ⟶ (data a.Y).X i
Y : C
j : (data X).I₀
c : F.obj X₀ ⟶ F.obj ((data X).X j)
fac₁ : c ≫ (data X).f j = F.map b ≫ (data a.Y).f i ≫ a.f
Y₀ : C₀
d : Y₀ ⟶ X₀
e : Y₀ ⟶ (data X).X j
fac₂ : F.map e = F.map d ≫ c
⊢ (↑S).arrows ((F.map d ≫ c) ≫ (data X).f j)