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

Goal state before:
pp: case refine_1
R : CommRingCat
M : (Spec R).Modules
V : (Spec R).Opens
ι : Type u_1
inst✝ : Finite ι
g : ι → ↑R
hg : V = ⨆ i, basicOpen (g i)
h₁ : ∀ (i : ι), Aux M (basicOpen (g i))
h₂ : ∀ (i j : ι), Aux M (basicOpen (g i * g j))
hgle : ∀ (i : ι), basicOpen (g i) ≤ V
hug : ∀ (i : ι) (m : ℕ), IsUnit ((algebraMap (↑R) (Module.End ↑R ↑Γ(M, basicOpen (g i)))) (g i ^ m))
f : ↑R
hf : basicOpen f ≤ V
s : ↑Γ(M, basicOpen f)
hfgi : ∀ (i : ι), basicOpen (f * g i) ≤ basicOpen (g i)
s' : (i : ι) → ↑Γ(M, basicOpen (f * g i)) := fun i ↦ (ConcreteCategory.hom (M.presheaf.map (homOfLE ⋯).op)) s
N : ℕ
t : (i : ι) → ↑Γ(M, basicOpen (g i))
ht : ∀ (i : ι), f ^ N • s' i = (ConcreteCategory.hom (M.presheaf.map (homOfLE ⋯).op)) (t i)
i j : ι
hfgigi : basicOpen (f * (g i * g j)) ≤ basicOpen (f * g i)
hfgigj : basicOpen (f * (g i * g j)) ≤ basicOpen (f * g j)
⊢ f ^ N • (ConcreteCategory.hom (M.presheaf.map (homOfLE ⋯).op)) s -
      f ^ N • (ConcreteCategory.hom (M.presheaf.map (homOfLE ⋯).op)) s =
    0

Tactic:
simp

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

Goal state before:
pp: A : Type u_1
σ : Type u_2
inst✝³ : CommRing A
inst✝² : SetLike σ A
inst✝¹ : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst✝ : GradedRing 𝒜
f : A
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAlgebra
y z : NumDenSameDeg 𝒜 (Submonoid.powers f)
e :
  HomogeneousLocalization.val
      ((algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal)) (HomogeneousLocalization.mk y)) =
    HomogeneousLocalization.val
      ((algebraMap (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal)) (HomogeneousLocalization.mk z))
⊢ ∃ i, ∃ c ∈ 𝒜 i, c ∉ (↑x).asHomogeneousIdeal ∧ c * (↑z.den * ↑y.num) = c * (↑y.den * ↑z.num)

Tactic:
simp only [RingHom.algebraMap_toAlgebra, map_mk, GradedRingHom.id_apply, val_mk, mk_eq_mk',
        IsLocalization.mk'_eq_iff_eq] at e

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
x : ↥(pbo f)
m : ℕ
f_deg : f ∈ 𝒜 m
hm : 0 < m
this : Algebra (A⁰_ f) (AtPrime 𝒜 (↑x).asHomogeneousIdeal.toIdeal) := (mapId 𝒜 ⋯).toAlgebra
y z : NumDenSameDeg 𝒜 (Submonoid.powers f)
e :
  (algebraMap A (Localization (↑x).asHomogeneousIdeal.toIdeal.primeCompl)) (↑z.den * ↑y.num) =
    (algebraMap A (Localization (↑x).asHomogeneousIdeal.toIdeal.primeCompl)) (↑y.den * ↑z.num)
⊢ ∃ i, ∃ c ∈ 𝒜 i, c ∉ (↑x).asHomogeneousIdeal ∧ c * (↑z.den * ↑y.num) = c * (↑y.den * ↑z.num)
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper

Goal state before:
pp: case e'_4.succ
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
e : ℕ
f g : A
hg : g ∈ 𝒜 e
x : A
hx : x = f * g
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
n✝ : ℕ
hf : f ∈ 𝒜 (n✝ + 1)
hd : 0 < n✝ + 1
this : n = j * (n✝ + 1 + e)
⊢ j * ((n✝ + 1) * (e + 1)) = j * (n✝ + 1 + e) + (j * (n✝ + 1 - 1)) • e

Tactic:
simp

Goal state after:
case e'_4.succ
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
e : ℕ
f g : A
hg : g ∈ 𝒜 e
x : A
hx : x = f * g
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
n✝ : ℕ
hf : f ∈ 𝒜 (n✝ + 1)
hd : 0 < n✝ + 1
this : n = j * (n✝ + 1 + e)
⊢ j * ((n✝ + 1) * (e + 1)) = j * (n✝ + 1 + e) + j * n✝ * e
Module: Mathlib.AlgebraicGeometry.ValuativeCriterion

Goal state before:
pp: X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
Y' X X' Y : Scheme
Y'_to_Y : Y' ⟶ Y
f : X ⟶ Y
X'_to_X : X' ⟶ X
f' : X' ⟶ Y'
hP : IsPullback X'_to_X f' f Y'_to_Y
⊢ Existence f → Existence f'

Tactic:
hf

Goal state after:
X✝ Y✝ : Scheme
f✝ : X✝ ⟶ Y✝
Y' X X' Y : Scheme
Y'_to_Y : Y' ⟶ Y
f : X ⟶ Y
X'_to_X : X' ⟶ X
f' : X' ⟶ Y'
hP : IsPullback X'_to_X f' f Y'_to_Y
hf : Existence f
⊢ Existence f'
Module: Mathlib.AlgebraicGeometry.ValuativeCriterion

Goal state before:
pp: case refine_2.h₀
X Y : Scheme
f : X ⟶ Y
inst✝ : QuasiSeparated f
hf : ValuativeCriterion.Uniqueness f
S : ValuativeCommSq (pullback.diagonal f)
hc : CommSq S.i₁ (Spec.map (CommRingCat.ofHom (algebraMap S.R S.K))) f (S.i₂ ≫ pullback.fst f f ≫ f)
S' : ValuativeCommSq f := ⋯
this : Subsingleton ⋯.LiftStruct
S'l₁ : ⋯.LiftStruct := ⋯
S'l₂ : ⋯.LiftStruct := ⋯
h₁₂ : S'l₁ = S'l₂
⊢ (S.i₂ ≫ pullback.fst f f ≫ pullback.diagonal f) ≫ pullback.fst f f = S.i₂ ≫ pullback.fst f f

Tactic:
simp

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

Goal state before:
pp: case refine_1
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
Y X : Q.Over ⊤ S
f : X ⟶ Y
𝒰 : Cover (precoverage P) Y.left
h : Cover.Over S 𝒰
p : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
⊢ ∀ (j : (Cover.pullbackCoverOverProp' S 𝒰 f.left ⋯ ⋯ p).I₀),
    Q ((Cover.pullbackCoverOverProp' S 𝒰 f.left ⋯ ⋯ p).X j ↘ S)

Tactic:
intro j

Goal state after:
case refine_1
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
Y X : Q.Over ⊤ S
f : X ⟶ Y
𝒰 : Cover (precoverage P) Y.left
h : Cover.Over S 𝒰
p : ∀ (j : 𝒰.I₀), Q (𝒰.X j ↘ S)
j : (Cover.pullbackCoverOverProp' S 𝒰 f.left ⋯ ⋯ p).I₀
⊢ Q ((Cover.pullbackCoverOverProp' S 𝒰 f.left ⋯ ⋯ p).X j ↘ S)
Module: Mathlib.AlgebraicGeometry.ValuativeCriterion

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
inst✝ : IsSeparated f
S : ValuativeCommSq f
l₁ : Spec (CommRingCat.of S.R) ⟶ X
hl₁ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₁ = S.i₁
hl₁' : l₁ ≫ f = S.i₂
l₂ : Spec (CommRingCat.of S.R) ⟶ X
hl₂ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₂ = S.i₁
hl₂' : l₂ ≫ f = S.i₂
h : l₁ ≫ f = l₂ ≫ f
Z : Scheme := pullback (pullback.diagonal f) (pullback.lift l₁ l₂ h)
g : Z ⟶ Spec (CommRingCat.of S.R) := pullback.snd (pullback.diagonal f) (pullback.lift l₁ l₂ h)
this✝ : IsClosedImmersion g
hZ : IsAffine Z
this : IsIso g
| g ≫ l₁

Tactic:
← pullback.lift_fst l₁ l₂ h,

Goal state after:
X Y : Scheme
f : X ⟶ Y
inst✝ : IsSeparated f
S : ValuativeCommSq f
l₁ : Spec (CommRingCat.of S.R) ⟶ X
hl₁ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₁ = S.i₁
hl₁' : l₁ ≫ f = S.i₂
l₂ : Spec (CommRingCat.of S.R) ⟶ X
hl₂ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₂ = S.i₁
hl₂' : l₂ ≫ f = S.i₂
h : l₁ ≫ f = l₂ ≫ f
Z : Scheme := pullback (pullback.diagonal f) (pullback.lift l₁ l₂ h)
g : Z ⟶ Spec (CommRingCat.of S.R) := pullback.snd (pullback.diagonal f) (pullback.lift l₁ l₂ h)
this✝ : IsClosedImmersion g
hZ : IsAffine Z
this : IsIso g
| g ≫ pullback.lift l₁ l₂ h ≫ pullback.fst f f
Module: Mathlib.AlgebraicGeometry.ValuativeCriterion

Goal state before:
pp: X Y : Scheme
f : X ⟶ Y
inst✝ : IsSeparated f
S : ValuativeCommSq f
l₁ : Spec (CommRingCat.of S.R) ⟶ X
hl₁ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₁ = S.i₁
hl₁' : l₁ ≫ f = S.i₂
l₂ : Spec (CommRingCat.of S.R) ⟶ X
hl₂ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₂ = S.i₁
hl₂' : l₂ ≫ f = S.i₂
h : l₁ ≫ f = l₂ ≫ f
Z : Scheme := pullback (pullback.diagonal f) (pullback.lift l₁ l₂ h)
g : Z ⟶ Spec (CommRingCat.of S.R) := pullback.snd (pullback.diagonal f) (pullback.lift l₁ l₂ h)
this✝ : IsClosedImmersion g
hZ : IsAffine Z
this : IsIso g
⊢ pullback.fst (pullback.diagonal f) (pullback.lift l₁ l₂ h) ≫ pullback.diagonal f ≫ pullback.fst f f =
    pullback.fst (pullback.diagonal f) (pullback.lift l₁ l₂ h) ≫ pullback.diagonal f ≫ pullback.snd f f

Tactic:
simp

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

Goal state before:
pp: case h₁
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 j : (affineOpenCover 𝒜).openCover.I₀
e₁ : pullback ((affineOpenCover 𝒜).f i ≫ toSpecZero 𝒜) ((affineOpenCover 𝒜).f j ≫ toSpecZero 𝒜) ≅
  Spec (CommRingCat.of (Away 𝒜 ↑i.snd ⊗[↥(𝒜 0)] Away 𝒜 ↑j.snd)) :=
  congrHom ⋯ ⋯ ≪≫ pullbackSpecIso (↥(𝒜 0)) (Away 𝒜 ↑i.snd) (Away 𝒜 ↑j.snd)
e₂ : pullback ((affineOpenCover 𝒜).f i) ((affineOpenCover 𝒜).f j) ≅ Spec (CommRingCat.of (Away 𝒜 (↑i.snd * ↑j.snd))) :=
  pullbackAwayιIso 𝒜 ⋯ ⋯ ⋯ ⋯ ⋯
F : Away 𝒜 ↑i.snd ⊗[↥(𝒜 0)] Away 𝒜 ↑j.snd →+* Away 𝒜 (↑i.snd * ↑j.snd) :=
  (Algebra.TensorProduct.lift (awayMapₐ 𝒜 ⋯ ⋯) (awayMapₐ 𝒜 ⋯ ⋯) ⋯).toRingHom
this : Function.Surjective ⇑F
⊢ (pullbackAwayιIso 𝒜 ⋯ ⋯ ⋯ ⋯ ⋯).inv ≫
      (pullbackDiagonalMapIdIso ((affineOpenCover 𝒜).openCover.f i) ((affineOpenCover 𝒜).openCover.f j)
            (toSpecZero 𝒜)).inv ≫
        snd (diagonal (toSpecZero 𝒜))
            ((Pullback.openCoverOfLeftRight (affineOpenCover 𝒜).openCover (affineOpenCover 𝒜).openCover (toSpecZero 𝒜)
                  (toSpecZero 𝒜)).f
              (i, j)) ≫
          snd ((affineOpenCover 𝒜).f i ≫ toSpecZero 𝒜) ((affineOpenCover 𝒜).f j ≫ toSpecZero 𝒜) =
    Spec.map (CommRingCat.ofHom ↑Algebra.TensorProduct.includeRight ≫ CommRingCat.ofHom F)

Tactic:
pullbackDiagonalMapIdIso_inv_snd_snd

Goal state after:
case h₁
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 j : (affineOpenCover 𝒜).openCover.I₀
e₁ : pullback ((affineOpenCover 𝒜).f i ≫ toSpecZero 𝒜) ((affineOpenCover 𝒜).f j ≫ toSpecZero 𝒜) ≅
  Spec (CommRingCat.of (Away 𝒜 ↑i.snd ⊗[↥(𝒜 0)] Away 𝒜 ↑j.snd)) :=
  congrHom ⋯ ⋯ ≪≫ pullbackSpecIso (↥(𝒜 0)) (Away 𝒜 ↑i.snd) (Away 𝒜 ↑j.snd)
e₂ : pullback ((affineOpenCover 𝒜).f i) ((affineOpenCover 𝒜).f j) ≅ Spec (CommRingCat.of (Away 𝒜 (↑i.snd * ↑j.snd))) :=
  pullbackAwayιIso 𝒜 ⋯ ⋯ ⋯ ⋯ ⋯
F : Away 𝒜 ↑i.snd ⊗[↥(𝒜 0)] Away 𝒜 ↑j.snd →+* Away 𝒜 (↑i.snd * ↑j.snd) :=
  (Algebra.TensorProduct.lift (awayMapₐ 𝒜 ⋯ ⋯) (awayMapₐ 𝒜 ⋯ ⋯) ⋯).toRingHom
this : Function.Surjective ⇑F
⊢ (pullbackAwayιIso 𝒜 ⋯ ⋯ ⋯ ⋯ ⋯).inv ≫ snd ((affineOpenCover 𝒜).openCover.f i) ((affineOpenCover 𝒜).openCover.f j) =
    Spec.map (CommRingCat.ofHom ↑Algebra.TensorProduct.includeRight ≫ CommRingCat.ofHom F)
Module: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper

Goal state before:
pp: σ : Type u_1
A : Type u_2
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
x : Finset A
hx : Algebra.adjoin ↥(𝒜 0) ↑x = ⊤
d : (i : A) → i ∈ x → ℕ
hd : ∀ (i : A) (a : i ∈ x), d i a ≠ 0
hxd : ∀ (i : A) (a : i ∈ x), i ∈ 𝒜 (d i a)
i : ↥x
⊢ IsCompact (Hom.opensRange (awayι 𝒜 ↑i ⋯ ⋯)).carrier

Tactic:
exact isCompact_range (Proj.awayι _ _ (hxd _ i.2) (hd _ i.2).bot_lt).continuous

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

Goal state before:
pp: 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 : Nonempty ι
⊢ (Finset.image ψ Finset.univ).Nonempty

Tactic:
simp

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

Goal state before:
pp: 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 : Nonempty ι
Kmax : ValueGroup O K := (Finset.image ψ Finset.univ).max' ⋯
i₀ : ι
hi1 : ψ i₀ = Kmax
j : ι
⊢ ψ j ∈ Finset.image ψ Finset.univ

Tactic:
simp

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

Goal state before:
pp: 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 : 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 : Kmax = 0
i : ι
⊢ ψ i ∈ Finset.image ψ Finset.univ

Tactic:
simp

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

Goal state before:
pp: 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✝ : Nonempty ι
Kmax : ValueGroup O K := ⋯
i₀ : ι
hi1 : ψ i₀ = Kmax
hi₀ : ∀ (j : ι), ψ j ≤ ψ i₀
hKmax : Kmax = 0
this : ∀ (i : ι), φ (Away.isLocalizationElem ⋯ ⋯) = 0 ∧ ¬∏ k ∈ Finset.univ.erase i, d k = 0
⊢ val 1 = val (Away.isLocalizationElem ⋯ ⋯)

Tactic:
simp

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

Goal state before:
pp: S : Scheme
P : MorphismProperty Scheme
inst✝³ : P.IsMultiplicative
inst✝² : IsZariskiLocalAtSource P
inst✝¹ : P.IsStableUnderBaseChange
inst✝ : P.HasOfPostcompProperty P
U : P.Over ⊤ S
𝒰 : Cover (precoverage P) U.left := Cover.changeProp U.left.affineCover ⋯
x✝² : (i : 𝒰.I₀) → (𝒰.X i).Over S := fun i ↦ { hom := 𝒰.f i ≫ U.hom }
x✝¹ : Cover.Over S 𝒰 := { over := inferInstance, isOver_map := ⋯ }
X : P.Over ⊤ S
f : X ⟶ U
x✝ : 𝒰.toPresieveOverProp ⋯ f
i : 𝒰.I₀
⊢ 𝒰.f i ≫ U.hom = ((MorphismProperty.CostructuredArrow.toOver P Scheme.Spec S).obj (affineOverMk (𝒰.f i ≫ U.hom) ⋯)).hom

Tactic:
simp

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

Goal state before:
pp: S : Scheme
P : MorphismProperty Scheme
inst✝³ : P.IsMultiplicative
inst✝² : IsZariskiLocalAtSource P
inst✝¹ : P.IsStableUnderBaseChange
inst✝ : P.HasOfPostcompProperty P
X : P.Over ⊤ S
𝒰 : X.left.AffineOpenCover := X.left.affineOpenCover
𝒱 : Cover (precoverage P) X.left := Cover.changeProp 𝒰.openCover ⋯
x✝ : (i : 𝒱.I₀) → (𝒱.X i).Over S := fun i ↦ { hom := 𝒰.f i ≫ X.hom }
this : Cover.Over S 𝒱 := { over := inferInstance, isOver_map := ⋯ }
i : 𝒰.I₀
⊢ ⊤ (𝒰.f i)

Tactic:
simp

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

Goal state before:
pp: C₀ : Type u₀
C : Type u
inst✝⁵ : Category.{v₀, u₀} C₀
inst✝⁴ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
inst✝³ : IsDenseSubsite J₀ J F
inst✝² : HasPullbacks C
inst✝¹ : F.Full
inst✝ : F.Faithful
hF : ∀ (S : C), ∃ ι U f, Sieve.ofArrows (fun i ↦ F.obj (U i)) f ∈ J S
S : C
⊢ Nonempty (F.OneHypercoverDenseData J₀ J S)

Tactic:
choose ι U f hf using hF
    exact ⟨{
      I₀ := ι S
      X := U S
      f := f S
      I₁ i j := ι (pullback (f _ i) (f _ j))
      Y i j := U (pullback (f _ i) (f _ j))
      p₁ i j k := F.preimage (f _ k ≫ pullback.fst _ _)
      p₂ i j k := F.preimage (f _ k ≫ pullback.snd _ _)
      w i j k := by simp [pullback.condition]
      mem₀ := hf S
      mem₁₀ i j W₀ p₁ p₂ hp := by
        have := IsDenseSubsite.isCoverDense J₀ J F
        rw [← functorPushforward_mem_iff J₀ J F]
        refine J.superset_covering ?_
          (IsCoverDense.functorPullback_pushforward_covering
            ⟨_, J.pullback_stable (pullback.lift _ _ hp) (hf (pullback (f _ i) (f _ j)))⟩)
        rintro T _ ⟨Z, q, r, ⟨_, s, _, ⟨k⟩, fac⟩, rfl⟩
        have fac₁ := fac =≫ pullback.fst _ _
        have fac₂ := fac =≫ pullback.snd _ _
        simp only [Category.assoc, pullback.lift_fst, pullback.lift_snd] at fac₁ fac₂
        exact ⟨Z, q, r, ⟨k, F.preimage s, F.map_injective (by simp [fac₁]),
          F.map_injective (by simp [fac₂])⟩, rfl⟩ }⟩

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

Goal state before:
pp: case right.zero
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)
⊢ (valuation O K) ((φ'.comp (awayMap 𝒜 ⋯ ⋯)) 0) ≤ 1

Tactic:
simp

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

Goal state before:
pp: case h.h
R : CommRingCat
M : (Spec R).Modules
V : (Spec R).Opens
ι : Type u_1
inst✝ : Finite ι
g : ι → ↑R
hg : V = ⨆ i, basicOpen (g i)
h₁ : ∀ (i : ι), Aux M (basicOpen (g i))
h₂ : ∀ (i j : ι), Aux M (basicOpen (g i * g j))
hgle : ∀ (i : ι), basicOpen (g i) ≤ V
hug : ∀ (i : ι) (m : ℕ), IsUnit ((algebraMap (↑R) (Module.End ↑R ↑Γ(M, basicOpen (g i)))) (g i ^ m))
f : ↑R
hf : basicOpen f ≤ V
t : ↑Γ(M, V)
hs : (ConcreteCategory.hom (M.presheaf.map (homOfLE hf).op)) t = 0
n : ι → ℕ
hn : ∀ (i : ι), (ConcreteCategory.hom (M.presheaf.map (homOfLE ⋯).op)) (f ^ n i • t) = 0
i : ι
this✝ : n i ≤ ⨆ i, n i
this : ⨆ i, n i = (⨆ i, n i) - n i + n i
⊢ f ^ ((⨆ i, n i) - n i) • 0 = (ConcreteCategory.hom ({ obj := M.presheaf, property := ⋯ }.obj.map (homOfLE ⋯).op)) 0

Tactic:
simp

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Elements

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
A : C ⥤ Type w
I : Type u₁
inst✝¹ : Category.{v₁, u₁} I
inst✝ : Small.{w, u₁} I
F : I ⥤ A.Elements
⊢ ∀ (j j' : I) (f : j ⟶ j'), (hom (((F ⋙ π A) ⋙ A).map f)) (F.obj j).snd = (F.obj j').snd

Tactic:
simp

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Elements

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
A : C ⥤ Type w
I : Type u₁
inst✝³ : Category.{v₁, u₁} I
inst✝² : Small.{w, u₁} I
F : I ⥤ A.Elements
inst✝¹ : HasLimitsOfShape I C
inst✝ : PreservesLimitsOfShape I A
s : Cone F
⊢ (hom (A.map (limit.lift (F ⋙ π A) ((π A).mapCone s)))) s.pt.snd = (liftedCone F).pt.snd

Tactic:
simp

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

Goal state before:
pp: case hR
C₀ : Type u₀
C : Type u
inst✝² : Category.{v₀, u₀} C₀
inst✝¹ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
inst✝ : IsDenseSubsite J₀ J F
X : C
data : F.OneHypercoverDenseData J₀ J X
i₁ i₂ : data.I₀
W : C
p₁ : W ⟶ F.obj (data.X i₁)
p₂ : W ⟶ F.obj (data.X i₂)
w : p₁ ≫ data.f i₁ = p₂ ≫ data.f i₂
this : F.IsCoverDense J
S : Sieve W :=
  Sieve.bind (Sieve.coverByImage F W).arrows fun Y f hf ↦
    Sieve.pullback (Nonempty.some hf).lift
      (Sieve.functorPushforward F
        (F.imageSieve ((Nonempty.some hf).map ≫ p₁) ⊓ F.imageSieve ((Nonempty.some hf).map ≫ p₂)))
T : Sieve W :=
  Sieve.bind S.arrows fun Z g hg ↦
    Sieve.pullback (Presieve.getFunctorPushforwardStructure ⋯).lift
      (Sieve.functorPushforward F (data.sieve₁₀ (Exists.choose ⋯) (Exists.choose ⋯)))
Y : C
f : Y ⟶ W
⊢ ∀ (H : (Sieve.coverByImage F W).arrows f),
    Sieve.pullback (Nonempty.some H).lift
        (Sieve.functorPushforward F
          (F.imageSieve ((Nonempty.some H).map ≫ p₁) ⊓ F.imageSieve ((Nonempty.some H).map ≫ p₂))) ∈
      J Y

Tactic:
hf

Goal state after:
case hR
C₀ : Type u₀
C : Type u
inst✝² : Category.{v₀, u₀} C₀
inst✝¹ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
inst✝ : IsDenseSubsite J₀ J F
X : C
data : F.OneHypercoverDenseData J₀ J X
i₁ i₂ : data.I₀
W : C
p₁ : W ⟶ F.obj (data.X i₁)
p₂ : W ⟶ F.obj (data.X i₂)
w : p₁ ≫ data.f i₁ = p₂ ≫ data.f i₂
this : F.IsCoverDense J
S : Sieve W :=
  Sieve.bind (Sieve.coverByImage F W).arrows fun Y f hf ↦
    Sieve.pullback (Nonempty.some hf).lift
      (Sieve.functorPushforward F
        (F.imageSieve ((Nonempty.some hf).map ≫ p₁) ⊓ F.imageSieve ((Nonempty.some hf).map ≫ p₂)))
T : Sieve W :=
  Sieve.bind S.arrows fun Z g hg ↦
    Sieve.pullback (Presieve.getFunctorPushforwardStructure ⋯).lift
      (Sieve.functorPushforward F (data.sieve₁₀ (Exists.choose ⋯) (Exists.choose ⋯)))
Y : C
f : Y ⟶ W
hf : (Sieve.coverByImage F W).arrows f
⊢ Sieve.pullback (Nonempty.some hf).lift
      (Sieve.functorPushforward F
        (F.imageSieve ((Nonempty.some hf).map ≫ p₁) ⊓ F.imageSieve ((Nonempty.some hf).map ≫ p₂))) ∈
    J Y