mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ MorphismProperty.CostructuredArrow.homMk (𝟙 ((𝒰.X i).asOverProp S ⋯).left) True.intro ⋯ ≫
      MorphismProperty.Over.homMk (𝒰.f i) ⋯ trivial =
    Hom.asOverProp (𝒰.f i) S

Tactic:
by

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
⊢ Sieve.coverByImage (MorphismProperty.CostructuredArrow.toOver P Scheme.Spec S) U ∈ (smallGrothendieckTopology P S) U

Tactic:
by

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✝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))
rid : φ (Away.isLocalizationElem ⋯ ⋯) = 0
⊢ 0 = Kmax

Tactic:
by

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:
by

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

Goal state before:
pp: 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
⊢ ⨆ i, n i = (⨆ i, n i) - n i + n i

Tactic:
by

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
S : C
ι : C → Type w
U : (S : C) → ι S → C₀
f : (S : C) → (i : ι S) → F.obj (U S i) ⟶ S
hf : ∀ (S : C), Sieve.ofArrows (fun i ↦ F.obj (U S i)) (f S) ∈ J S
i j : ι S
W₀ : C₀
p₁ : W₀ ⟶ U S i
p₂ : W₀ ⟶ U S j
hp : F.map p₁ ≫ f S i = F.map p₂ ≫ f S j
this : F.IsCoverDense J
T : C
Z : C₀
q : Z ⟶ W₀
r : T ⟶ F.obj Z
Y : C
k : ι (pullback (f S i) (f S j))
s : F.obj Z ⟶ F.obj (U (pullback (f S i) (f S j)) k)
fac : s ≫ f (pullback (f S i) (f S j)) k = F.map q ≫ pullback.lift (F.map p₁) (F.map p₂) hp
fac₁ : s ≫ f (pullback (f S i) (f S j)) k ≫ pullback.fst (f S i) (f S j) = F.map q ≫ F.map p₁
fac₂ : s ≫ f (pullback (f S i) (f S j)) k ≫ pullback.snd (f S i) (f S j) = F.map q ≫ F.map p₂
⊢ F.map (q ≫ p₁) =
    F.map
      (F.preimage s ≫
        { I₀ := ι S, X := U S, f := f S, I₁ := fun i j ↦ ι (pullback (f S i) (f S j)),
              Y := fun i j ↦ U (pullback (f S i) (f S j)),
              p₁ := fun i j k ↦ F.preimage (f (pullback (f S i) (f S j)) k ≫ pullback.fst (f S i) (f S j)),
              p₂ := fun i j k ↦ F.preimage (f (pullback (f S i) (f S j)) k ≫ pullback.snd (f S i) (f S j)), w := ⋯ }.p₁
          k)

Tactic:
by

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
S : C
ι : C → Type w
U : (S : C) → ι S → C₀
f : (S : C) → (i : ι S) → F.obj (U S i) ⟶ S
hf : ∀ (S : C), Sieve.ofArrows (fun i ↦ F.obj (U S i)) (f S) ∈ J S
i j : ι S
W₀ : C₀
p₁ : W₀ ⟶ U S i
p₂ : W₀ ⟶ U S j
hp : F.map p₁ ≫ f S i = F.map p₂ ≫ f S j
this : F.IsCoverDense J
T : C
Z : C₀
q : Z ⟶ W₀
r : T ⟶ F.obj Z
Y : C
k : ι (pullback (f S i) (f S j))
s : F.obj Z ⟶ F.obj (U (pullback (f S i) (f S j)) k)
fac : s ≫ f (pullback (f S i) (f S j)) k = F.map q ≫ pullback.lift (F.map p₁) (F.map p₂) hp
fac₁ : s ≫ f (pullback (f S i) (f S j)) k ≫ pullback.fst (f S i) (f S j) = F.map q ≫ F.map p₁
fac₂ : s ≫ f (pullback (f S i) (f S j)) k ≫ pullback.snd (f S i) (f S j) = F.map q ≫ F.map p₂
⊢ F.map (q ≫ p₂) =
    F.map
      (F.preimage s ≫
        { I₀ := ι S, X := U S, f := f S, I₁ := fun i j ↦ ι (pullback (f S i) (f S j)),
              Y := fun i j ↦ U (pullback (f S i) (f S j)),
              p₁ := fun i j k ↦ F.preimage (f (pullback (f S i) (f S j)) k ≫ pullback.fst (f S i) (f S j)),
              p₂ := fun i j k ↦ F.preimage (f (pullback (f S i) (f S j)) k ≫ pullback.snd (f S i) (f S j)), w := ⋯ }.p₂
          k)

Tactic:
by

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✝³ : (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 := ⋯ }
x✝² : P.Over ⊤ S
x✝¹ : x✝² ⟶ X
x✝ : 𝒱.toPresieveOverProp ⋯ x✝¹
i : 𝒱.I₀
⊢ Hom.asOverProp (𝒱.f i) S = 𝟙 ((𝒱.X i).asOverProp S ⋯) ≫ MorphismProperty.CostructuredArrow.homMk (𝒰.f i) ⋯ ⋯

Tactic:
by

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
S : C
ι : C → Type w
U : (S : C) → ι S → C₀
f : (S : C) → (i : ι S) → F.obj (U S i) ⟶ S
hf : ∀ (S : C), Sieve.ofArrows (fun i ↦ F.obj (U S i)) (f S) ∈ J S
i j : ι S
W₀ : C₀
p₁ : W₀ ⟶ U S i
p₂ : W₀ ⟶ U S j
hp : F.map p₁ ≫ f S i = F.map p₂ ≫ f S j
⊢ { I₀ := ι S, X := U S, f := f S, I₁ := fun i j ↦ ι (pullback (f S i) (f S j)),
          Y := fun i j ↦ U (pullback (f S i) (f S j)),
          p₁ := fun i j k ↦ F.preimage (f (pullback (f S i) (f S j)) k ≫ pullback.fst (f S i) (f S j)),
          p₂ := fun i j k ↦ F.preimage (f (pullback (f S i) (f S j)) k ≫ pullback.snd (f S i) (f S j)), w := ⋯ }.sieve₁₀
      p₁ p₂ ∈
    J₀ W₀

Tactic:
by

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
⊢ ∃ ι U f,
    Sieve.ofArrows (fun i ↦ (MorphismProperty.CostructuredArrow.toOver P Scheme.Spec S).obj (U i)) f ∈
      (smallGrothendieckTopology P S) X

Tactic:
by

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:
by

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:
by

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
c : Cone F
⊢ (hom (A.map (limit.lift (F ⋙ π A) ((π A).mapCone c)))) c.pt.snd = liftedConeElement F

Tactic:
by

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
i : I
⊢ (hom (A.map (limit.π (F ⋙ π A) i))) (liftedConeElement F) = (F.obj i).snd

Tactic:
by

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
i : I
⊢ (hom (A.map (limit.π (F ⋙ π A) i))) (((Functor.const I).obj ⟨limit (F ⋙ π A), liftedConeElement F⟩).obj i).snd =
    (F.obj i).snd

Tactic:
by

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
i i' : I
f : i ⟶ i'
⊢ ((Functor.const I).obj ⟨limit (F ⋙ π A), liftedConeElement F⟩).map f ≫ ⟨limit.π (F ⋙ π A) i', ⋯⟩ =
    ⟨limit.π (F ⋙ π A) i, ⋯⟩ ≫ F.map f

Tactic:
by

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

Goal state before:
pp: 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)))
Z : C
g : Z ⟶ W
hg : S.arrows g
⊢ Sieve Z

Tactic:
by

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:
by

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
m : s.pt ⟶ (liftedCone F).pt
h : ∀ (j : I), m ≫ (liftedCone F).π.app j = s.π.app j
i : I
⊢ ↑m ≫ limit.π (F ⋙ π A) i = ↑⟨limit.lift (F ⋙ π A) ((π A).mapCone s), ⋯⟩ ≫ limit.π (F ⋙ π A) i

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
E : F.Elements
he : IsInitial E
X✝ : C
f : X✝ ⟶ unop E.fst
⊢ (fun y ↦ (↑(he.to ⟨op X✝, y⟩)).unop) ((fun f ↦ (ConcreteCategory.hom (F.map f.op)) E.snd) f) = f

Tactic:
by

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

Goal state before:
pp: 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))
⊢ Aux M V

Tactic:
by

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

Goal state before:
pp: 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 ⋯)))
⊢ S ∈ J W

Tactic:
by

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

Goal state before:
pp: 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 ⋯)))
hS : S ∈ J W
Z : C
g : Z ⟶ W
hg : S.arrows g
⊢ Sieve.pullback (Presieve.getFunctorPushforwardStructure ⋯).lift
      (Sieve.functorPushforward F (data.sieve₁₀ (Exists.choose ⋯) (Exists.choose ⋯))) ∈
    J Z

Tactic:
by

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

Goal state before:
pp: R : CommRingCat
M : (Spec R).Modules
⊢ IsLocalizing (modulesSpecToSheaf.obj M) ↔ Aux M ⊤

Tactic:
by

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✝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)
a : ℕ
ai : ι → ℕ
hai✝ : ∑ i, ai i • d i = a • d i₀
hai : ∑ x, ai x * d x = a * d i₀
⊢ (∏ i, x i ^ ai i) * x i₀ ^ (a * (d j - 1)) ∈ 𝒜 ((a * d i₀) • d j)

Tactic:
by

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

Goal state before:
pp: R : CommRingCat
M : (Spec R).Modules
g : ↑R
a : R ⟶ CommRingCat.of (Localization.Away g) := CommRingCat.ofHom (algebraMap (↑R) (Localization.Away g))
ψ : Spec (CommRingCat.of (Localization.Away g)) ⟶ Spec (CommRingCat.of ↑R) := Spec.map a
M' : (Spec (CommRingCat.of (Localization.Away g))).Modules := M.restrict ψ
h : Aux M' ⊤
f : ↑R
hf : basicOpen f ≤ basicOpen g
⊢ basicOpen f = ψ ''ᵁ Spec.map a ⁻¹ᵁ basicOpen f

Tactic:
Scheme.Hom.image_preimage_eq_opensRange_inf

Goal state after:
R : CommRingCat
M : (Spec R).Modules
g : ↑R
a : R ⟶ CommRingCat.of (Localization.Away g) := CommRingCat.ofHom (algebraMap (↑R) (Localization.Away g))
ψ : Spec (CommRingCat.of (Localization.Away g)) ⟶ Spec (CommRingCat.of ↑R) := Spec.map a
M' : (Spec (CommRingCat.of (Localization.Away g))).Modules := M.restrict ψ
h : Aux M' ⊤
f : ↑R
hf : basicOpen f ≤ basicOpen g
⊢ basicOpen f = Scheme.Hom.opensRange ψ ⊓ basicOpen f
Module: Mathlib.AlgebraicGeometry.Modules.Tilde

Goal state before:
pp: R : CommRingCat
M : (Spec R).Modules
g : ↑R
a : R ⟶ CommRingCat.of (Localization.Away g) := CommRingCat.ofHom (algebraMap (↑R) (Localization.Away g))
ψ : Spec (CommRingCat.of (Localization.Away g)) ⟶ Spec (CommRingCat.of ↑R) := Spec.map a
M' : (Spec (CommRingCat.of (Localization.Away g))).Modules := M.restrict ψ
h : Aux M' ⊤
f : ↑R
hf : basicOpen f ≤ basicOpen g
⊢ basicOpen f = ψ ''ᵁ basicOpen ((ConcreteCategory.hom a) f)

Tactic:
by

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

Goal state before:
pp: R : CommRingCat
M : (Spec R).Modules
g : ↑R
a : R ⟶ CommRingCat.of (Localization.Away g) := CommRingCat.ofHom (algebraMap (↑R) (Localization.Away g))
ψ : Spec (CommRingCat.of (Localization.Away g)) ⟶ Spec (CommRingCat.of ↑R) := Spec.map a
M' : (Spec (CommRingCat.of (Localization.Away g))).Modules := M.restrict ψ
h : Aux M' ⊤
heq : ∀ (f : ↑R), basicOpen f ≤ basicOpen g → basicOpen f = ψ ''ᵁ basicOpen ((ConcreteCategory.hom a) f)
⊢ basicOpen g = ψ ''ᵁ ⊤

Tactic:
by

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
A : Type u'
inst✝¹ : Category.{v', u'} A
inst✝ : HasColimitsOfSize.{w, w, v', u'} A
P : Cᵒᵖ ⥤ A
T : A
f g : Φ.presheafFiber.obj P ⟶ T
h : ∀ (X : C) (x : Φ.fiber.obj X), Φ.toPresheafFiber X x P ≫ f = Φ.toPresheafFiber X x P ≫ g
⊢ ∀ (j : Φ.fiber.Elementsᵒᵖ),
    colimit.ι (((Functor.whiskeringLeft Φ.fiber.Elementsᵒᵖ Cᵒᵖ A).obj (CategoryOfElements.π Φ.fiber).op).obj P) j ≫ f =
      colimit.ι (((Functor.whiskeringLeft Φ.fiber.Elementsᵒᵖ Cᵒᵖ A).obj (CategoryOfElements.π Φ.fiber).op).obj P) j ≫ g

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
J : GrothendieckTopology C
Φ : J.Point
A : Type u'
inst✝³ : Category.{v', u'} A
B : Type u''
inst✝² : Category.{v'', u''} B
inst✝¹ : HasColimitsOfSize.{w, w, v', u'} A
inst✝ : HasColimitsOfSize.{w, w, v'', u''} B
X : C
x : Φ.fiber.obj X
x✝¹ x✝ : Cᵒᵖ ⥤ A
f : x✝¹ ⟶ x✝
⊢ ((evaluation Cᵒᵖ A).obj (op X)).map f ≫ Φ.toPresheafFiber X x x✝ = Φ.toPresheafFiber X x x✝¹ ≫ Φ.presheafFiber.map f

Tactic:
by

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
Φ : J.Point
inst✝ : LocallySmall.{w, v, u} C
X Y : C
f : X ⟶ Y
x : Φ.fiber.obj X
⊢ (ConcreteCategory.hom (Φ.presheafFiber.map (shrinkYoneda.{w, v, u}.map f)))
      ((ConcreteCategory.hom (Φ.shrinkYonedaCompPresheafFiberIso.inv.app X)) x) =
    (ConcreteCategory.hom (Φ.toPresheafFiber X x (shrinkYoneda.{w, v, u}.obj Y))) (shrinkYonedaObjObjEquiv.symm f)

Tactic:
by

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

Goal state before:
pp: R : CommRingCat
M : (Spec R).Modules
g : ↑R
a : R ⟶ CommRingCat.of (Localization.Away g) := CommRingCat.ofHom (algebraMap (↑R) (Localization.Away g))
ψ : Spec (CommRingCat.of (Localization.Away g)) ⟶ Spec (CommRingCat.of ↑R) := Spec.map a
M' : (Spec (CommRingCat.of (Localization.Away g))).Modules := M.restrict ψ
h : Aux M' ⊤
heq : ∀ (f : ↑R), basicOpen f ≤ basicOpen g → basicOpen f = ψ ''ᵁ basicOpen ((ConcreteCategory.hom a) f)
iso : Γ(M.restrict ψ, ⊤) ≅ Γ(M, basicOpen g) :=
  Scheme.Modules.restrictAppIso ψ M ⊤ ≪≫ Functor.mapIso M.presheaf (eqToIso ⋯).op
e : (f : ↑R) → basicOpen f ≤ basicOpen g → (Γ(M', basicOpen ((ConcreteCategory.hom a) f)) ≅ Γ(M, basicOpen f)) :=
  fun f hf ↦
    Scheme.Modules.restrictAppIso ψ M (basicOpen ((ConcreteCategory.hom a) f)) ≪≫
      Functor.mapIso M.presheaf (eqToIso ⋯).op
f : ↑R
hf : basicOpen f ≤ basicOpen g
t : ↑Γ(M, basicOpen g)
ht : (ConcreteCategory.hom (M.presheaf.map (homOfLE hf).op)) t = 0
⊢ (ConcreteCategory.hom (M'.presheaf.map (homOfLE ⋯).op)) ((ConcreteCategory.hom iso.inv) t) = 0

Tactic:
by

Goal state after:
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
inst✝⁴ : HasColimitsOfSize.{w, w, v', u'} A
FC : A → A → Type u_1
CC : A → Type w'
inst✝³ : (X Y : A) → FunLike (FC X Y) (CC X) (CC Y)
inst✝² : ConcreteCategory A FC
P : Cᵒᵖ ⥤ A
inst✝¹ : PreservesFilteredColimitsOfSize.{w, w, v', w', u', w' + 1} (forget A)
inst✝ : LocallySmall.{w, v, u} C
p : ToType (Φ.presheafFiber.obj P)
⊢ ∃ X x z, (ConcreteCategory.hom (Φ.toPresheafFiber X x P)) z = p

Tactic:
by

Goal state after:
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
inst✝⁴ : HasColimitsOfSize.{w, w, v', u'} A
FC : A → A → Type u_1
CC : A → Type w'
inst✝³ : (X Y : A) → FunLike (FC X Y) (CC X) (CC Y)
inst✝² : ConcreteCategory A FC
P : Cᵒᵖ ⥤ A
inst✝¹ : PreservesFilteredColimitsOfSize.{w, w, v', w', u', w' + 1} (forget A)
inst✝ : LocallySmall.{w, v, u} C
p₁ p₂ : ToType (Φ.presheafFiber.obj P)
⊢ ∃ X x z₁ z₂,
    (ConcreteCategory.hom (Φ.toPresheafFiber X x P)) z₁ = p₁ ∧ (ConcreteCategory.hom (Φ.toPresheafFiber X x P)) z₂ = p₂

Tactic:
by

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

Goal state before:
pp: 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
⊢ data.toPreOneHypercover.sieve₁ p₁ p₂ ∈ J W

Tactic:
by

Goal state after: