mathlib-initiative/mathlib-tactics huggingface dataset:
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
O : Type u_2
commRing✝ : CommRing O
domain✝ : IsDomain O
valuationRing✝ : ValuationRing O
K : Type u_2
field✝ : Field K
algebra✝ : Algebra O K
isFractionRing✝ : IsFractionRing O K
i₁ : Spec (CommRingCat.of K) ⟶ Proj 𝒜
i₂ : Spec (CommRingCat.of O) ⟶ Spec (CommRingCat.of ↥(𝒜 0))
w : CommSq i₁ (Spec.map (CommRingCat.ofHom (algebraMap O K))) (toSpecZero 𝒜) i₂
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
hi : i₁ (IsLocalRing.closedPoint K) ∈ basicOpen 𝒜 ↑i
this : Set.range ⇑i₁ ⊆ ↑(Hom.opensRange (awayι 𝒜 ↑i ⋯ ⋯))
φ : Spec (CommRingCat.of K) ⟶ Spec (CommRingCat.of (Away 𝒜 ↑i)) := IsOpenImmersion.lift (awayι 𝒜 ↑i ⋯ ⋯) i₁ this
⊢ i₁ ≫ toSpecZero 𝒜 = i₁ ≫ toSpecZero 𝒜

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
O : Type u_2
commRing✝ : CommRing O
domain✝ : IsDomain O
valuationRing✝ : ValuationRing O
K : Type u_2
field✝ : Field K
algebra✝ : Algebra O K
isFractionRing✝ : IsFractionRing O K
i₁ : Spec (CommRingCat.of K) ⟶ Proj 𝒜
i₂ : Spec (CommRingCat.of O) ⟶ Spec (CommRingCat.of ↥(𝒜 0))
w : CommSq i₁ (Spec.map (CommRingCat.ofHom (algebraMap O K))) (toSpecZero 𝒜) i₂
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
hi : i₁ (IsLocalRing.closedPoint K) ∈ basicOpen 𝒜 ↑i
this : Set.range ⇑i₁ ⊆ ↑(Hom.opensRange (awayι 𝒜 ↑i ⋯ ⋯))
φ : Spec (CommRingCat.of K) ⟶ Spec (CommRingCat.of (Away 𝒜 ↑i)) := IsOpenImmersion.lift (awayι 𝒜 ↑i ⋯ ⋯) i₁ this
⊢ 0 < d ↑i ⋯

Tactic:
]

Goal state after:
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
O : Type u_2
commRing✝ : CommRing O
domain✝ : IsDomain O
valuationRing✝ : ValuationRing O
K : Type u_2
field✝ : Field K
algebra✝ : Algebra O K
isFractionRing✝ : IsFractionRing O K
i₁ : Spec (CommRingCat.of K) ⟶ Proj 𝒜
i₂ : Spec (CommRingCat.of O) ⟶ Spec (CommRingCat.of ↥(𝒜 0))
w : CommSq i₁ (Spec.map (CommRingCat.ofHom (algebraMap O K))) (toSpecZero 𝒜) i₂
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
hi : i₁ (IsLocalRing.closedPoint K) ∈ basicOpen 𝒜 ↑i
this : Set.range ⇑i₁ ⊆ ↑(Hom.opensRange (awayι 𝒜 ↑i ⋯ ⋯))
φ : Spec (CommRingCat.of K) ⟶ Spec (CommRingCat.of (Away 𝒜 ↑i)) := IsOpenImmersion.lift (awayι 𝒜 ↑i ⋯ ⋯) i₁ this
⊢ 0 < d ↑i ⋯
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense

Goal state before:
pp: C₀ : Type u₀
C : Type u
inst✝⁴ : Category.{v₀, u₀} C₀
inst✝³ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : IsDenseSubsite J₀ J F
data : (X : C) → F.OneHypercoverDenseData J₀ J X
inst✝ : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf J₀ A
X Y : C
X₀ : C₀
f : F.obj X₀ ⟶ X
g : X ⟶ Y
U₀ Y₀ : C₀
c : U₀ ⟶ Y₀
d : Y₀ ⟶ X₀
hd : Presieve.functorPullback F (Sieve.pullback f (data X).toPreOneHypercover.sieve₀).arrows d
hc :
  Presieve.functorPullback F
    (Sieve.pullback (Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) ≫ g)
        (data Y).toPreOneHypercover.sieve₀).arrows
    c
hc' :
  Sieve.ofArrows.h hc ≫ (data Y).f (Sieve.ofArrows.i hc) =
    F.map c ≫ Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) ≫ g
hd' : Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) = F.map d ≫ f
this : F.imageSieve (Sieve.ofArrows.h hc) ⊓ Sieve.pullback c (F.imageSieve (Sieve.ofArrows.h hd)) ∈ J₀ U₀
V₀ : C₀
a : V₀ ⟶ U₀
x₁ : V₀ ⟶ (data Y).X (Sieve.ofArrows.i hc)
fac₁ : F.map x₁ = F.map a ≫ Sieve.ofArrows.h hc
x₂ : V₀ ⟶ (data X).X (Sieve.ofArrows.i hd)
fac₂ : F.map x₂ = F.map (a ≫ c) ≫ Sieve.ofArrows.h hd
⊢ restriction data G₀ ((data X).f (Sieve.ofArrows.i hd) ≫ g) ≫ G₀.obj.map x₂.op =
    presheafObjπ data G₀ Y (Sieve.ofArrows.i hc) ≫ G₀.obj.map x₁.op

Tactic:
]

Goal state after:
C₀ : Type u₀
C : Type u
inst✝⁴ : Category.{v₀, u₀} C₀
inst✝³ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : IsDenseSubsite J₀ J F
data : (X : C) → F.OneHypercoverDenseData J₀ J X
inst✝ : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf J₀ A
X Y : C
X₀ : C₀
f : F.obj X₀ ⟶ X
g : X ⟶ Y
U₀ Y₀ : C₀
c : U₀ ⟶ Y₀
d : Y₀ ⟶ X₀
hd : Presieve.functorPullback F (Sieve.pullback f (data X).toPreOneHypercover.sieve₀).arrows d
hc :
  Presieve.functorPullback F
    (Sieve.pullback (Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) ≫ g)
        (data Y).toPreOneHypercover.sieve₀).arrows
    c
hc' :
  Sieve.ofArrows.h hc ≫ (data Y).f (Sieve.ofArrows.i hc) =
    F.map c ≫ Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) ≫ g
hd' : Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) = F.map d ≫ f
this : F.imageSieve (Sieve.ofArrows.h hc) ⊓ Sieve.pullback c (F.imageSieve (Sieve.ofArrows.h hd)) ∈ J₀ U₀
V₀ : C₀
a : V₀ ⟶ U₀
x₁ : V₀ ⟶ (data Y).X (Sieve.ofArrows.i hc)
fac₁ : F.map x₁ = F.map a ≫ Sieve.ofArrows.h hc
x₂ : V₀ ⟶ (data X).X (Sieve.ofArrows.i hd)
fac₂ : F.map x₂ = F.map (a ≫ c) ≫ Sieve.ofArrows.h hd
⊢ restriction data G₀ ((data X).f (Sieve.ofArrows.i hd) ≫ g) ≫ G₀.obj.map x₂.op =
    presheafObjπ data G₀ Y (Sieve.ofArrows.i hc) ≫ G₀.obj.map x₁.op
Module: Mathlib.AlgebraicGeometry.Sites.SheafQuasiCompact

Goal state before:
pp: case refine_2
P : MorphismProperty Scheme
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : P.IsMultiplicative
F : Schemeᵒᵖ ⥤ Type u_1
inst✝ : IsZariskiLocalAtSource P
x✝ :
  Presieve.IsSheaf zariskiTopology F ∧
    ∀ {R S : CommRingCat} (f : R ⟶ S),
      P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
hzar : Presieve.IsSheaf zariskiTopology F
hff :
  ∀ {R S : CommRingCat} (f : R ⟶ S),
    P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
R : CommRingCat
𝒰 : Cover (propQCPrecoverage P) (Spec R)
this :
  ∀ (R : CommRingCat) (𝒰 : Cover (propQCPrecoverage P) (Spec R)),
    (∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)) ∧ Finite 𝒰.I₀ → Presieve.IsSheafFor F 𝒰.presieve₀
h𝒰 : ¬((∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)) ∧ Finite 𝒰.I₀)
𝒱 : AffineCover P (Spec R)
f✝ : 𝒱.cover ⟶ 𝒰.forgetQc
hfin : Finite 𝒱.I₀
ho : ∀ (j : 𝒱.cover.I₀), IsOpenImmersion (f✝.h₀ j)
V : Scheme
f : V ⟶ Spec R
i j : V.affineOneHypercover.I₀
k : V.affineOneHypercover.I₁ i j
⊢ Presieve.IsSeparatedFor F
    (Presieve.ofArrows
      (fun i_1 ↦
        pullback ((Precoverage.ZeroHypercover.pullback₂ f 𝒱.cover).f i_1)
          (V.affineOneHypercover.p₁ k ≫ V.affineOneHypercover.f i))
      fun x ↦
      pullback.snd ((Precoverage.ZeroHypercover.pullback₂ f 𝒱.cover).f x)
        (V.affineOneHypercover.p₁ k ≫ V.affineOneHypercover.f i))

Tactic:
]

Goal state after:
case refine_2
P : MorphismProperty Scheme
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : P.IsMultiplicative
F : Schemeᵒᵖ ⥤ Type u_1
inst✝ : IsZariskiLocalAtSource P
x✝ :
  Presieve.IsSheaf zariskiTopology F ∧
    ∀ {R S : CommRingCat} (f : R ⟶ S),
      P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
hzar : Presieve.IsSheaf zariskiTopology F
hff :
  ∀ {R S : CommRingCat} (f : R ⟶ S),
    P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
R : CommRingCat
𝒰 : Cover (propQCPrecoverage P) (Spec R)
this :
  ∀ (R : CommRingCat) (𝒰 : Cover (propQCPrecoverage P) (Spec R)),
    (∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)) ∧ Finite 𝒰.I₀ → Presieve.IsSheafFor F 𝒰.presieve₀
h𝒰 : ¬((∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)) ∧ Finite 𝒰.I₀)
𝒱 : AffineCover P (Spec R)
f✝ : 𝒱.cover ⟶ 𝒰.forgetQc
hfin : Finite 𝒱.I₀
ho : ∀ (j : 𝒱.cover.I₀), IsOpenImmersion (f✝.h₀ j)
V : Scheme
f : V ⟶ Spec R
i j : V.affineOneHypercover.I₀
k : V.affineOneHypercover.I₁ i j
⊢ Presieve.IsSeparatedFor F
    (Presieve.ofArrows
      (fun i_1 ↦
        pullback ((Precoverage.ZeroHypercover.pullback₂ f 𝒱.cover).f i_1)
          (V.affineOneHypercover.p₁ k ≫ V.affineOneHypercover.f i))
      fun x ↦
      pullback.snd ((Precoverage.ZeroHypercover.pullback₂ f 𝒱.cover).f x)
        (V.affineOneHypercover.p₁ k ≫ V.affineOneHypercover.f i))
Module: Mathlib.CategoryTheory.Sites.EffectiveEpimorphic

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
B✝ : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B✝
H : EffectiveEpiFamilyStruct X π
D : Type (max u v) := ObjectProperty.FullSubcategory fun T ↦ (Sieve.generateFamily X π).arrows T.hom
F : D ⥤ C := (Sieve.generateFamily X π).arrows.diagram
S : Cocone (Sieve.generateFamily X π).arrows.diagram
T : Over B✝
a : α
g : T.left ⟶ X a
hT : g ≫ π a = T.hom
h₁ : ∀ (a : α), (Sieve.generateFamily X π).arrows (Over.mk (π a)).hom
h₂ :
  ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂),
    g₁ ≫ π a₁ = g₂ ≫ π a₂ →
      g₁ ≫ S.ι.app { obj := Over.mk (π a₁), property := ⋯ } = g₂ ≫ S.ι.app { obj := Over.mk (π a₂), property := ⋯ }
h₃ : (Sieve.generateFamily X π).arrows T.hom
A : D := { obj := Over.mk (π a), property := ⋯ }
B : D := { obj := Over.mk T.hom, property := ⋯ }
i : B ⟶ A := ObjectProperty.homMk (Over.homMk g ⋯)
⊢ S.ι.app B = S.ι.app { obj := T, property := h₃ }

Tactic:
]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
B✝ : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B✝
H : EffectiveEpiFamilyStruct X π
D : Type (max u v) := ObjectProperty.FullSubcategory fun T ↦ (Sieve.generateFamily X π).arrows T.hom
F : D ⥤ C := (Sieve.generateFamily X π).arrows.diagram
S : Cocone (Sieve.generateFamily X π).arrows.diagram
T : Over B✝
a : α
g : T.left ⟶ X a
hT : g ≫ π a = T.hom
h₁ : ∀ (a : α), (Sieve.generateFamily X π).arrows (Over.mk (π a)).hom
h₂ :
  ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂),
    g₁ ≫ π a₁ = g₂ ≫ π a₂ →
      g₁ ≫ S.ι.app { obj := Over.mk (π a₁), property := ⋯ } = g₂ ≫ S.ι.app { obj := Over.mk (π a₂), property := ⋯ }
h₃ : (Sieve.generateFamily X π).arrows T.hom
A : D := { obj := Over.mk (π a), property := ⋯ }
B : D := { obj := Over.mk T.hom, property := ⋯ }
i : B ⟶ A := ObjectProperty.homMk (Over.homMk g ⋯)
⊢ S.ι.app B = S.ι.app { obj := T, property := h₃ }
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense

Goal state before:
pp: C₀ : Type u₀
C : Type u
inst✝⁴ : Category.{v₀, u₀} C₀
inst✝³ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : IsDenseSubsite J₀ J F
data : (X : C) → F.OneHypercoverDenseData J₀ J X
inst✝ : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf J₀ A
X Y : C
X₀ : C₀
f : F.obj X₀ ⟶ X
g : X ⟶ Y
U₀ Y₀ : C₀
c : U₀ ⟶ Y₀
d : Y₀ ⟶ X₀
hd : Presieve.functorPullback F (Sieve.pullback f (data X).toPreOneHypercover.sieve₀).arrows d
hc :
  Presieve.functorPullback F
    (Sieve.pullback (Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) ≫ g)
        (data Y).toPreOneHypercover.sieve₀).arrows
    c
hc' :
  Sieve.ofArrows.h hc ≫ (data Y).f (Sieve.ofArrows.i hc) =
    F.map c ≫ Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) ≫ g
hd' : Sieve.ofArrows.h hd ≫ (data X).f (Sieve.ofArrows.i hd) = F.map d ≫ f
this : F.imageSieve (Sieve.ofArrows.h hc) ⊓ Sieve.pullback c (F.imageSieve (Sieve.ofArrows.h hd)) ∈ J₀ U₀
V₀ : C₀
a : V₀ ⟶ U₀
x₁ : V₀ ⟶ (data Y).X (Sieve.ofArrows.i hc)
fac₁ : F.map x₁ = F.map a ≫ Sieve.ofArrows.h hc
x₂ : V₀ ⟶ (data X).X (Sieve.ofArrows.i hd)
fac₂ : F.map x₂ = F.map (a ≫ c) ≫ Sieve.ofArrows.h hd
⊢ presheafObjπ data G₀ Y (Sieve.ofArrows.i hc) ≫ G₀.obj.map x₁.op =
    presheafObjπ data G₀ Y (Sieve.ofArrows.i hc) ≫ G₀.obj.map x₁.op

Tactic:
]

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

Goal state before:
pp: C₀ : Type u₀
C : Type u
inst✝⁴ : Category.{v₀, u₀} C₀
inst✝³ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : IsDenseSubsite J₀ J F
data : (X : C) → F.OneHypercoverDenseData J₀ J X
inst✝ : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf J₀ A
X : C
i : (data X).I₀
⊢ presheafObjπ data G₀ X i = presheafObjπ data G₀ X i

Tactic:
]

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

Goal state before:
pp: P : MorphismProperty Scheme
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : P.IsMultiplicative
F : Schemeᵒᵖ ⥤ Type u_1
inst✝ : IsZariskiLocalAtSource P
x✝ :
  Presieve.IsSheaf zariskiTopology F ∧
    ∀ {R S : CommRingCat} (f : R ⟶ S),
      P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
hzar : Presieve.IsSheaf zariskiTopology F
hff :
  ∀ {R S : CommRingCat} (f : R ⟶ S),
    P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
R : CommRingCat
𝒰 : Cover (propQCPrecoverage P) (Spec R)
left✝ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)
right✝ : Finite 𝒰.I₀
𝒰' : Cover (precoverage P) (Spec R) := 𝒰.forgetQc.sigma
⊢ Presieve.IsSheafFor F (Presieve.singleton (𝒰.forgetQc.sigma.f default))

Tactic:
]

Goal state after:
P : MorphismProperty Scheme
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : P.IsMultiplicative
F : Schemeᵒᵖ ⥤ Type u_1
inst✝ : IsZariskiLocalAtSource P
x✝ :
  Presieve.IsSheaf zariskiTopology F ∧
    ∀ {R S : CommRingCat} (f : R ⟶ S),
      P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
hzar : Presieve.IsSheaf zariskiTopology F
hff :
  ∀ {R S : CommRingCat} (f : R ⟶ S),
    P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
R : CommRingCat
𝒰 : Cover (propQCPrecoverage P) (Spec R)
left✝ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)
right✝ : Finite 𝒰.I₀
𝒰' : Cover (precoverage P) (Spec R) := 𝒰.forgetQc.sigma
⊢ Presieve.IsSheafFor F (Presieve.singleton (𝒰.forgetQc.sigma.f default))
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense

Goal state before:
pp: C₀ : Type u₀
C : Type u
inst✝⁴ : Category.{v₀, u₀} C₀
inst✝³ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : IsDenseSubsite J₀ J F
data : (X : C) → F.OneHypercoverDenseData J₀ J X
inst✝ : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf J₀ A
X Y Z : C
f : X ⟶ Y
g : Y ⟶ Z
i : (data X).I₀
⊢ restriction data G₀ ((data X).f i ≫ f ≫ g) = restriction data G₀ ((data X).f i ≫ f ≫ g)

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
B✝ : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B✝
H : IsColimit (Sieve.generateFamily X π).arrows.cocone
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
A : Over B✝
a : α
g₁ : A.left ⟶ X a
ha : g₁ ≫ π a = A.hom
B : Over B✝
b : α
g₂ : B.left ⟶ X b
hb : g₂ ≫ π b = B.hom
q : A ⟶ B
⊢ (Over.Hom.left q ≫ ⋯.choose) ≫ e ⋯.choose = ⋯.choose ≫ e ⋯.choose

Tactic:
]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
B✝ : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B✝
H : IsColimit (Sieve.generateFamily X π).arrows.cocone
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
A : Over B✝
a : α
g₁ : A.left ⟶ X a
ha : g₁ ≫ π a = A.hom
B : Over B✝
b : α
g₂ : B.left ⟶ X b
hb : g₂ ≫ π b = B.hom
q : A ⟶ B
⊢ (Over.Hom.left q ≫ ⋯.choose) ≫ e ⋯.choose = ⋯.choose ≫ e ⋯.choose
Module: Mathlib.CategoryTheory.Sites.EffectiveEpimorphic

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
B✝ : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B✝
H : IsColimit (Sieve.generateFamily X π).arrows.cocone
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
A : Over B✝
a : α
g₁ : A.left ⟶ X a
ha : g₁ ≫ π a = A.hom
B : Over B✝
b : α
g₂ : B.left ⟶ X b
hb : g₂ ≫ π b = B.hom
q : A ⟶ B
⊢ Over.Hom.left q ≫ ⋯.choose ≫ π ⋯.choose = ⋯.choose ≫ π ⋯.choose

Tactic:
]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
B✝ : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B✝
H : IsColimit (Sieve.generateFamily X π).arrows.cocone
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
A : Over B✝
a : α
g₁ : A.left ⟶ X a
ha : g₁ ≫ π a = A.hom
B : Over B✝
b : α
g₂ : B.left ⟶ X b
hb : g₂ ≫ π b = B.hom
q : A ⟶ B
⊢ Over.Hom.left q ≫ ⋯.choose ≫ π ⋯.choose = ⋯.choose ≫ π ⋯.choose
Module: Mathlib.AlgebraicGeometry.Sites.SheafQuasiCompact

Goal state before:
pp: P : MorphismProperty Scheme
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : P.IsMultiplicative
F : Schemeᵒᵖ ⥤ Type u_1
inst✝ : IsZariskiLocalAtSource P
x✝ :
  Presieve.IsSheaf zariskiTopology F ∧
    ∀ {R S : CommRingCat} (f : R ⟶ S),
      P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
hzar : Presieve.IsSheaf zariskiTopology F
hff :
  ∀ {R S : CommRingCat} (f : R ⟶ S),
    P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
R : CommRingCat
𝒰 : Cover (propQCPrecoverage P) (Spec R)
left✝ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)
right✝ : Finite 𝒰.I₀
𝒰' : Cover (precoverage P) (Spec R) := 𝒰.forgetQc.sigma
this : IsAffine (𝒰.forgetQc.sigma.X default)
f : Spec Γ(𝒰.forgetQc.sigma.X default, ⊤) ⟶ Spec R :=
  (𝒰.forgetQc.sigma.X default).isoSpec.inv ≫ 𝒰.forgetQc.sigma.f default
φ : R ⟶ Γ(𝒰.forgetQc.sigma.X default, ⊤)
hφ : Spec.map φ = f
⊢ Presieve.IsSheafFor F (Presieve.singleton (Spec.map φ))

Tactic:
]

Goal state after:
P : MorphismProperty Scheme
inst✝² : P.IsStableUnderBaseChange
inst✝¹ : P.IsMultiplicative
F : Schemeᵒᵖ ⥤ Type u_1
inst✝ : IsZariskiLocalAtSource P
x✝ :
  Presieve.IsSheaf zariskiTopology F ∧
    ∀ {R S : CommRingCat} (f : R ⟶ S),
      P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
hzar : Presieve.IsSheaf zariskiTopology F
hff :
  ∀ {R S : CommRingCat} (f : R ⟶ S),
    P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Presieve.singleton (Spec.map f))
R : CommRingCat
𝒰 : Cover (propQCPrecoverage P) (Spec R)
left✝ : ∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)
right✝ : Finite 𝒰.I₀
𝒰' : Cover (precoverage P) (Spec R) := 𝒰.forgetQc.sigma
this : IsAffine (𝒰.forgetQc.sigma.X default)
f : Spec Γ(𝒰.forgetQc.sigma.X default, ⊤) ⟶ Spec R :=
  (𝒰.forgetQc.sigma.X default).isoSpec.inv ≫ 𝒰.forgetQc.sigma.f default
φ : R ⟶ Γ(𝒰.forgetQc.sigma.X default, ⊤)
hφ : Spec.map φ = f
⊢ Presieve.IsSheafFor F (Presieve.singleton (Spec.map φ))
Module: Mathlib.CategoryTheory.Sites.EffectiveEpimorphic

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
B✝ : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B✝
H : IsColimit (Sieve.generateFamily X π).arrows.cocone
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
A : Over B✝
a : α
g₁ : A.left ⟶ X a
ha : g₁ ≫ π a = A.hom
B : Over B✝
b : α
g₂ : B.left ⟶ X b
hb : g₂ ≫ π b = B.hom
q : A ⟶ B
h1 : ∃ a g, g ≫ π a = B.hom
h2 : ∃ g, g ≫ π h1.choose = B.hom
h3 : ∃ a g, g ≫ π a = A.hom
h4 : ∃ g, g ≫ π h3.choose = A.hom
⊢ A.hom = A.hom

Tactic:
]

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

Goal state before:
pp: C₀ : Type u₀
C : Type u
inst✝⁴ : Category.{v₀, u₀} C₀
inst✝³ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : IsDenseSubsite J₀ J F
data : (X : C) → F.OneHypercoverDenseData J₀ J X
inst✝ : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf J₀ A
X₀ W₀ : C₀
a : W₀ ⟶ X₀
ha : Presieve.functorPullback F (data (F.obj X₀)).toPreOneHypercover.sieve₀.arrows a
T₀ : C₀
b : T₀ ⟶ X₀
hb : Presieve.functorPullback F (data (F.obj X₀)).toPreOneHypercover.sieve₀.arrows b
U₀ : C₀
p₁ : U₀ ⟶ W₀
p₂ : U₀ ⟶ T₀
fac : p₁ ≫ a = p₂ ≫ b
ha' : Sieve.ofArrows.h ha ≫ (data (F.obj X₀)).f (Sieve.ofArrows.i ha) = F.map a
hb' : Sieve.ofArrows.h hb ≫ (data (F.obj X₀)).f (Sieve.ofArrows.i hb) = F.map b
⊢ F.map p₁ ≫ F.map a = F.map p₁ ≫ F.map a

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
B : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B
H : IsColimit (Sieve.generateFamily X π).arrows.cocone
aux : {W : C} →
  (e : (a : α) → X a ⟶ W) →
    (∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂) →
      Cocone (Sieve.generateFamily X π).arrows.diagram :=
  fun {W} e h ↦
    { pt := W,
      ι :=
        {
          app := fun x ↦
            match x with
            | { obj := obj, property := hT } => ⋯.choose ≫ e (Exists.choose hT),
          naturality := ⋯ } }
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
a : α
this :
  π a ≫ H.desc { pt := W, ι := { app := fun x ↦ ⋯.choose ≫ e (Exists.choose ⋯), naturality := ⋯ } } =
    ⋯.choose ≫ e (Exists.choose ⋯)
⊢ ⋯.choose ≫ e (Exists.choose ⋯) = e a

Tactic:
]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
B : C
α : Type u_1
X : α → C
π : (a : α) → X a ⟶ B
H : IsColimit (Sieve.generateFamily X π).arrows.cocone
aux : {W : C} →
  (e : (a : α) → X a ⟶ W) →
    (∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂) →
      Cocone (Sieve.generateFamily X π).arrows.diagram :=
  fun {W} e h ↦
    { pt := W,
      ι :=
        {
          app := fun x ↦
            match x with
            | { obj := obj, property := hT } => ⋯.choose ≫ e (Exists.choose hT),
          naturality := ⋯ } }
W : C
e : (a : α) → X a ⟶ W
h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂), g₁ ≫ π a₁ = g₂ ≫ π a₂ → g₁ ≫ e a₁ = g₂ ≫ e a₂
a : α
this :
  π a ≫ H.desc { pt := W, ι := { app := fun x ↦ ⋯.choose ≫ e (Exists.choose ⋯), naturality := ⋯ } } =
    ⋯.choose ≫ e (Exists.choose ⋯)
⊢ ⋯.choose ≫ e (Exists.choose ⋯) = e a
Module: Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense

Goal state before:
pp: C₀ : Type u₀
C : Type u
inst✝⁴ : Category.{v₀, u₀} C₀
inst✝³ : Category.{v, u} C
F : C₀ ⥤ C
J₀ : GrothendieckTopology C₀
J : GrothendieckTopology C
A : Type u'
inst✝² : Category.{v', u'} A
inst✝¹ : IsDenseSubsite J₀ J F
data : (X : C) → F.OneHypercoverDenseData J₀ J X
inst✝ : HasLimitsOfSize.{w, w, v', u'} A
G₀ : Sheaf J₀ A
X₀ W₀ : C₀
a : W₀ ⟶ X₀
ha : Presieve.functorPullback F (data (F.obj X₀)).toPreOneHypercover.sieve₀.arrows a
T₀ : C₀
b : T₀ ⟶ X₀
hb : Presieve.functorPullback F (data (F.obj X₀)).toPreOneHypercover.sieve₀.arrows b
U₀ : C₀
p₁ : U₀ ⟶ W₀
p₂ : U₀ ⟶ T₀
fac : p₁ ≫ a = p₂ ≫ b
ha' : Sieve.ofArrows.h ha ≫ (data (F.obj X₀)).f (Sieve.ofArrows.i ha) = F.map a
hb' : Sieve.ofArrows.h hb ≫ (data (F.obj X₀)).f (Sieve.ofArrows.i hb) = F.map b
⊢ F.map p₂ ≫ F.map b = F.map p₂ ≫ F.map b

Tactic:
]

Goal state after: