mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
hP :
  ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [inst : HasPullback π π],
    Function.Bijective ⇑(ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯))
⊢ EqualizerCondition P

Tactic:
intro X

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
hP :
  ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [inst : HasPullback π π],
    Function.Bijective ⇑(ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯))
X : C
⊢ ∀ ⦃B : C⦄ (π : X ⟶ B) [EffectiveEpi π], SingleEqualizerCondition P π
Module: Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hc : IsLimit c
this : HasPullback π π
hP :
  ∀
    (b :
      (fun X ↦ X)
        ↑{x |
            (ConcreteCategory.hom (P.map (pullback.fst π π).op)) x =
              (ConcreteCategory.hom (P.map (pullback.snd π π).op)) x}),
    ∃! a, (ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)) a = b
b : P.obj (op X)
⊢ (ConcreteCategory.hom (P.map c.fst.op)) b = (ConcreteCategory.hom (P.map c.snd.op)) b →
    ∃! x, (ConcreteCategory.hom (P.map π.op)) x = b

Tactic:
hb

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hc : IsLimit c
this : HasPullback π π
hP :
  ∀
    (b :
      (fun X ↦ X)
        ↑{x |
            (ConcreteCategory.hom (P.map (pullback.fst π π).op)) x =
              (ConcreteCategory.hom (P.map (pullback.snd π π).op)) x}),
    ∃! a, (ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)) a = b
b : P.obj (op X)
hb : (ConcreteCategory.hom (P.map c.fst.op)) b = (ConcreteCategory.hom (P.map c.snd.op)) b
⊢ ∃! x, (ConcreteCategory.hom (P.map π.op)) x = b
Module: Mathlib.CategoryTheory.Sites.EpiMono

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁵ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w
inst✝⁴ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝³ : ConcreteCategory A FA
inst✝² : HasFunctorialSurjectiveInjectiveFactorization A
inst✝¹ : J.WEqualsLocallyBijective A
data : FunctorialSurjectiveInjectiveFactorizationData A
inst✝ : HasWeakSheafify J A
x✝ : Arrow (Sheaf J A)
⊢ Presheaf.IsLocallyInjective J
    ((MorphismProperty.FunctorialFactorizationData.functorCategory data Cᵒᵖ).p.app (Arrow.mk x✝.hom.hom))

Tactic:
apply Presheaf.isLocallyInjective_of_injective

Goal state after:
C : Type u
inst✝⁶ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝⁵ : Category.{v', u'} A
FA : A → A → Type u_1
CA : A → Type w
inst✝⁴ : (X Y : A) → FunLike (FA X Y) (CA X) (CA Y)
inst✝³ : ConcreteCategory A FA
inst✝² : HasFunctorialSurjectiveInjectiveFactorization A
inst✝¹ : J.WEqualsLocallyBijective A
data : FunctorialSurjectiveInjectiveFactorizationData A
inst✝ : HasWeakSheafify J A
x✝ : Arrow (Sheaf J A)
⊢ ∀ (X : Cᵒᵖ),
    Function.Injective
      ⇑(ConcreteCategory.hom
          (((MorphismProperty.FunctorialFactorizationData.functorCategory data Cᵒᵖ).p.app (Arrow.mk x✝.hom.hom)).app X))
Module: Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hc : IsLimit c
this : HasPullback π π
hP :
  ∀
    (b :
      (fun X ↦ X)
        ↑{x |
            (ConcreteCategory.hom (P.map (pullback.fst π π).op)) x =
              (ConcreteCategory.hom (P.map (pullback.snd π π).op)) x}),
    ∃! a, (ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)) a = b
b : P.obj (op X)
hb : (ConcreteCategory.hom (P.map c.fst.op)) b = (ConcreteCategory.hom (P.map c.snd.op)) b
h₁ : ((pullbackIsPullback π π).conePointUniqueUpToIso hc).hom ≫ c.fst = pullback.fst π π
⊢ (ConcreteCategory.hom (P.map ((pullbackIsPullback π π).conePointUniqueUpToIso hc).hom.op))
      ((ConcreteCategory.hom (P.map c.fst.op)) b) =
    (ConcreteCategory.hom (P.map (pullback.snd π π).op)) b

Tactic:
hb

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
X B : C
π : X ⟶ B
inst✝ : EffectiveEpi π
c : PullbackCone π π
hc : IsLimit c
this : HasPullback π π
hP :
  ∀
    (b :
      (fun X ↦ X)
        ↑{x |
            (ConcreteCategory.hom (P.map (pullback.fst π π).op)) x =
              (ConcreteCategory.hom (P.map (pullback.snd π π).op)) x}),
    ∃! a, (ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯)) a = b
b : P.obj (op X)
hb : (ConcreteCategory.hom (P.map c.fst.op)) b = (ConcreteCategory.hom (P.map c.snd.op)) b
h₁ : ((pullbackIsPullback π π).conePointUniqueUpToIso hc).hom ≫ c.fst = pullback.fst π π
⊢ (ConcreteCategory.hom (P.map ((pullbackIsPullback π π).conePointUniqueUpToIso hc).hom.op))
      ((ConcreteCategory.hom (P.map c.snd.op)) b) =
    (ConcreteCategory.hom (P.map (pullback.snd π π).op)) b
Module: Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves

Goal state before:
pp: case mpr
C : Type u_1
inst✝ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
hP : ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [inst : HasPullback π π], IsIso (equalizer.lift (P.map π.op) ⋯)
⊢ ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [inst : HasPullback π π],
    Function.Bijective ⇑(ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯))

Tactic:
intro X

Goal state after:
case mpr
C : Type u_1
inst✝ : Category.{v_1, u_1} C
P : Cᵒᵖ ⥤ Type u_4
hP : ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [inst : HasPullback π π], IsIso (equalizer.lift (P.map π.op) ⋯)
X : C
⊢ ∀ (B : C) (π : X ⟶ B) [EffectiveEpi π] [inst : HasPullback π π],
    Function.Bijective ⇑(ConcreteCategory.hom (mapToEqualizer P π (pullback.fst π π) (pullback.snd π π) ⋯))
Module: Mathlib.CategoryTheory.Sites.Coherent.SequentialLimit

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preregular C
inst✝¹ : FinitaryExtensive C
F : ℕᵒᵖ ⥤ Sheaf (coherentTopology C) (Type v)
c : Cone F
hc : IsLimit c
hF : ∀ (n : ℕ), Sheaf.IsLocallySurjective (F.map (homOfLE ⋯).op)
inst✝ : HasLimitsOfShape ℕᵒᵖ C
h : ∀ (G : ℕᵒᵖ ⥤ C), (∀ (n : ℕ), EffectiveEpi (G.map (homOfLE ⋯).op)) → EffectiveEpi (limit.π G (op 0))
⊢ ∀ (X : C) (y : ToType ((F.obj (op 0)).obj.obj (op X))),
    ∃ X' φ,
      ∃ (_ : EffectiveEpi φ),
        ∃ x,
          (ConcreteCategory.hom ((c.π.app (op 0)).hom.app (op X'))) x =
            (ConcreteCategory.hom ((F.obj (op 0)).obj.map (op φ))) y

Tactic:
intro X

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Preregular C
inst✝¹ : FinitaryExtensive C
F : ℕᵒᵖ ⥤ Sheaf (coherentTopology C) (Type v)
c : Cone F
hc : IsLimit c
hF : ∀ (n : ℕ), Sheaf.IsLocallySurjective (F.map (homOfLE ⋯).op)
inst✝ : HasLimitsOfShape ℕᵒᵖ C
h : ∀ (G : ℕᵒᵖ ⥤ C), (∀ (n : ℕ), EffectiveEpi (G.map (homOfLE ⋯).op)) → EffectiveEpi (limit.π G (op 0))
X : C
⊢ ∀ (y : ToType ((F.obj (op 0)).obj.obj (op X))),
    ∃ X' φ,
      ∃ (_ : EffectiveEpi φ),
        ∃ x,
          (ConcreteCategory.hom ((c.π.app (op 0)).hom.app (op X'))) x =
            (ConcreteCategory.hom ((F.obj (op 0)).obj.map (op φ))) y
Module: Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves

Goal state before:
pp: C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Category.{v_3, u_3} E
P : Cᵒᵖ ⥤ D
X B : C
π : X ⟶ B
c : PullbackCone π π
hc : IsLimit c
S : Presieve B := (Sieve.ofArrows (fun x ↦ X) fun x ↦ π).arrows
X' : S.category := S.categoryMk π ⋯
P' : S.category := S.categoryMk (c.fst ≫ π) ⋯
fst : P' ⟶ X' := ObjectProperty.homMk (Over.homMk c.fst ⋯)
snd : P' ⟶ X' := ObjectProperty.homMk (Over.homMk c.snd ⋯)
⊢ IsLimit (Fork.ofι (P.map π.op) ⋯) ≃ IsLimit (P.mapCone (Sieve.ofArrows (fun x ↦ X) fun x ↦ π).arrows.cocone.op)

Tactic:
let F : S.categoryᵒᵖ ⥤ D := S.diagram.op ⋙ P

Goal state after:
C : Type u_1
D : Type u_2
E : Type u_3
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Category.{v_2, u_2} D
inst✝ : Category.{v_3, u_3} E
P : Cᵒᵖ ⥤ D
X B : C
π : X ⟶ B
c : PullbackCone π π
hc : IsLimit c
S : Presieve B := (Sieve.ofArrows (fun x ↦ X) fun x ↦ π).arrows
X' : S.category := S.categoryMk π ⋯
P' : S.category := S.categoryMk (c.fst ≫ π) ⋯
fst : P' ⟶ X' := ObjectProperty.homMk (Over.homMk c.fst ⋯)
snd : P' ⟶ X' := ObjectProperty.homMk (Over.homMk c.snd ⋯)
F : S.categoryᵒᵖ ⥤ D := S.diagram.op ⋙ P
⊢ IsLimit (Fork.ofι (P.map π.op) ⋯) ≃ IsLimit (P.mapCone (Sieve.ofArrows (fun x ↦ X) fun x ↦ π).arrows.cocone.op)
Module: Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preregular C
W : C
⊢ ∀ {X : C}, ∀ R ∈ (regularCoverage C).coverings X, IsSheafFor (yoneda.obj W) R

Tactic:
intro X

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preregular C
W X : C
⊢ ∀ R ∈ (regularCoverage C).coverings X, IsSheafFor (yoneda.obj W) R
Module: Mathlib.CategoryTheory.Sites.Descent.Precoverage

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι
Z : C
q : Z ⟶ X i
j₁ : ι'
a₁ : Z ⟶ X' j₁
j₂ : ι'
a₂ : Z ⟶ X' j₂
fac₁ : a₁ ≫ f' j₁ = q ≫ f i
fac₂ : a₂ ≫ f' j₂ = q ≫ f i
this :
  pullHom (φ.hom j₁) a₁ (a₁ ≫ p' j₁) (a₁ ≫ p' j₁) ⋯ ⋯ ≫
      D₂.hom (q ≫ f i) (a₁ ≫ p' j₁) (a₂ ≫ p' j₂) ⋯ ⋯ ≫
        (F.mapComp' (p' j₂).op.toLoc a₂.op.toLoc ((p' j₂).op.toLoc ≫ a₂.op.toLoc) ⋯).hom.toNatTrans.app
          (D₂.obj (α j₂)) =
    D₁.hom (q ≫ f i) (a₁ ≫ p' j₁) (a₂ ≫ p' j₂) ⋯ ⋯ ≫
      pullHom (φ.hom j₂) a₂ (a₂ ≫ p' j₂) (a₂ ≫ p' j₂) ⋯ ⋯ ≫
        (F.mapComp' (p' j₂).op.toLoc a₂.op.toLoc ((p' j₂).op.toLoc ≫ a₂.op.toLoc) ⋯).hom.toNatTrans.app (D₂.obj (α j₂))
⊢ mor w φ q a₁ fac₁ = mor w φ q a₂ fac₂

Tactic:
simp only [← Category.assoc, cancel_mono] at this

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι
Z : C
q : Z ⟶ X i
j₁ : ι'
a₁ : Z ⟶ X' j₁
j₂ : ι'
a₂ : Z ⟶ X' j₂
fac₁ : a₁ ≫ f' j₁ = q ≫ f i
fac₂ : a₂ ≫ f' j₂ = q ≫ f i
this :
  pullHom (φ.hom j₁) a₁ (a₁ ≫ p' j₁) (a₁ ≫ p' j₁) ⋯ ⋯ ≫ D₂.hom (q ≫ f i) (a₁ ≫ p' j₁) (a₂ ≫ p' j₂) ⋯ ⋯ =
    D₁.hom (q ≫ f i) (a₁ ≫ p' j₁) (a₂ ≫ p' j₂) ⋯ ⋯ ≫ pullHom (φ.hom j₂) a₂ (a₂ ≫ p' j₂) (a₂ ≫ p' j₂) ⋯ ⋯
⊢ mor w φ q a₁ fac₁ = mor w φ q a₂ fac₂
Module: Mathlib.CategoryTheory.Sites.Descent.Precoverage

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι
Z : C
q : Z ⟶ X i
j₁ : ι'
a₁ : Z ⟶ X' j₁
j₂ : ι'
a₂ : Z ⟶ X' j₂
fac₁ : a₁ ≫ f' j₁ = q ≫ f i
fac₂ : a₂ ≫ f' j₂ = q ≫ f i
this :
  pullHom (φ.hom j₁) a₁ (a₁ ≫ p' j₁) (a₁ ≫ p' j₁) ⋯ ⋯ ≫ D₂.hom (q ≫ f i) (a₁ ≫ p' j₁) q ⋯ ⋯ =
    (D₁.hom (q ≫ f i) (a₁ ≫ p' j₁) (a₂ ≫ p' j₂) ⋯ ⋯ ≫ pullHom (φ.hom j₂) a₂ (a₂ ≫ p' j₂) (a₂ ≫ p' j₂) ⋯ ⋯) ≫
      D₂.hom (q ≫ f i) (a₂ ≫ p' j₂) q ⋯ ⋯
⊢ mor w φ q a₁ fac₁ = mor w φ q a₂ fac₂

Tactic:
rw [mor_eq _ _ _ _ _ _ _ rfl rfl, mor_eq _ _ _ _ _ _ _ rfl rfl, this]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
ι : Type t
S : C
X : ι → C
f : (i : ι) → X i ⟶ S
ι' : Type t'
X' : ι' → C
f' : (j : ι') → X' j ⟶ S
α : ι' → ι
p' : (j : ι') → X' j ⟶ X (α j)
w : ∀ (j : ι'), p' j ≫ f (α j) = f' j
D₁ D₂ : F.DescentData f
φ : (pullFunctor F ⋯).obj D₁ ⟶ (pullFunctor F ⋯).obj D₂
i : ι
Z : C
q : Z ⟶ X i
j₁ : ι'
a₁ : Z ⟶ X' j₁
j₂ : ι'
a₂ : Z ⟶ X' j₂
fac₁ : a₁ ≫ f' j₁ = q ≫ f i
fac₂ : a₂ ≫ f' j₂ = q ≫ f i
this :
  pullHom (φ.hom j₁) a₁ (a₁ ≫ p' j₁) (a₁ ≫ p' j₁) ⋯ ⋯ ≫ D₂.hom (q ≫ f i) (a₁ ≫ p' j₁) q ⋯ ⋯ =
    (D₁.hom (q ≫ f i) (a₁ ≫ p' j₁) (a₂ ≫ p' j₂) ⋯ ⋯ ≫ pullHom (φ.hom j₂) a₂ (a₂ ≫ p' j₂) (a₂ ≫ p' j₂) ⋯ ⋯) ≫
      D₂.hom (q ≫ f i) (a₂ ≫ p' j₂) q ⋯ ⋯
⊢ D₁.hom (q ≫ f i) q (a₁ ≫ p' j₁) ⋯ ⋯ ≫
      (D₁.hom (q ≫ f i) (a₁ ≫ p' j₁) (a₂ ≫ p' j₂) ⋯ ⋯ ≫ pullHom (φ.hom j₂) a₂ (a₂ ≫ p' j₂) (a₂ ≫ p' j₂) ⋯ ⋯) ≫
        D₂.hom (q ≫ f i) (a₂ ≫ p' j₂) q ⋯ ⋯ =
    D₁.hom (q ≫ f i) q (a₂ ≫ p' j₂) ⋯ ⋯ ≫
      pullHom (φ.hom j₂) a₂ (a₂ ≫ p' j₂) (a₂ ≫ p' j₂) ⋯ ⋯ ≫ D₂.hom (q ≫ f i) (a₂ ≫ p' j₂) q ⋯ ⋯
Module: Mathlib.CategoryTheory.Sites.Hypercover.Homotopy

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
S : C
E : PreOneHypercover S
F : PreOneHypercover S
inst✝ : HasPullbacks C
f g : E.Hom F
p : (cylinder f g).I₀
⊢ pullback.snd (pullback.lift (f.h₀ p.fst) (g.h₀ p.fst) ⋯) (F.toPullback p.snd) ≫ F.p₁ p.snd =
    ((cylinderHom f g).comp f).h₀ p

Tactic:
by
    have : F.p₁ p.snd = pullback.lift _ _ (F.w p.2) ≫ pullback.fst _ _ := by simp
    nth_rw 1 [this]
    rw [← pullback.condition_assoc]
    simp

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

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
inst✝ : HasWeakSheafify J (Type v)
F : Sheaf J (Type v)
sq : Square C
⊢ (sq.op.map ((yoneda ⋙ presheafToSheaf J (Type v)).op ⋙ yoneda.obj F)).IsPullback ↔ (sq.op.map F.obj).IsPullback

Tactic:
by
  refine Square.IsPullback.iff_of_equiv _ _
    (((sheafificationAdjunction J (Type v)).homEquiv _ _).trans yonedaEquiv)
    (((sheafificationAdjunction J (Type v)).homEquiv _ _).trans yonedaEquiv)
    (((sheafificationAdjunction J (Type v)).homEquiv _ _).trans yonedaEquiv)
    (((sheafificationAdjunction J (Type v)).homEquiv _ _).trans yonedaEquiv) ?_ ?_ ?_ ?_
  all_goals
    ext x
    simp [Adjunction.homEquiv, yonedaEquiv_naturality]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Descent.Precoverage

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
inst✝³ : HasPullbacks C
J : Precoverage C
inst✝² : J.HasIsos
inst✝¹ : J.IsStableUnderBaseChange
inst✝ : J.IsStableUnderComposition
hF : ∀ (S : C), ∀ R ∈ J.coverings S, F.IsPrestackFor R
S : C
M N : ↑(F.obj { as := op S })
⊢ ∀ {X : Over S},
    ∀ R ∈ (Precoverage.toPretopology (Over S) (Precoverage.comap (Over.forget S) J)).coverings X,
      Presieve.IsSheafFor (F.presheafHom M N) R

Tactic:
intro X

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
inst✝³ : HasPullbacks C
J : Precoverage C
inst✝² : J.HasIsos
inst✝¹ : J.IsStableUnderBaseChange
inst✝ : J.IsStableUnderComposition
hF : ∀ (S : C), ∀ R ∈ J.coverings S, F.IsPrestackFor R
S : C
M N : ↑(F.obj { as := op S })
X : Over S
⊢ ∀ R ∈ (Precoverage.toPretopology (Over S) (Precoverage.comap (Over.forget S) J)).coverings X,
    Presieve.IsSheafFor (F.presheafHom M N) R
Module: Mathlib.CategoryTheory.Sites.Descent.Precoverage

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
inst✝³ : HasPullbacks C
J : Precoverage C
inst✝² : J.HasIsos
inst✝¹ : J.IsStableUnderBaseChange
inst✝ : J.IsStableUnderComposition
hF : ∀ (S : C), ∀ R ∈ J.coverings S, F.IsPrestackFor R
S : C
M N : ↑(F.obj { as := op S })
X : Over S
R : Presieve X
⊢ R ∈ (Precoverage.toPretopology (Over S) (Precoverage.comap (Over.forget S) J)).coverings X →
    Presieve.IsSheafFor (F.presheafHom M N) R

Tactic:
hR

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
F : LocallyDiscrete Cᵒᵖ ⥤ᵖ Cat
inst✝³ : HasPullbacks C
J : Precoverage C
inst✝² : J.HasIsos
inst✝¹ : J.IsStableUnderBaseChange
inst✝ : J.IsStableUnderComposition
hF : ∀ (S : C), ∀ R ∈ J.coverings S, F.IsPrestackFor R
S : C
M N : ↑(F.obj { as := op S })
X : Over S
R : Presieve X
hR : R ∈ (Precoverage.toPretopology (Over S) (Precoverage.comap (Over.forget S) J)).coverings X
⊢ Presieve.IsSheafFor (F.presheafHom M N) R
Module: Mathlib.CategoryTheory.Sites.MayerVietorisSquare

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
inst✝¹ : HasWeakSheafify J (Type v)
S : J.MayerVietorisSquare
A : Type u'
inst✝ : Category.{v, u'} A
F : Sheaf J A
⊢ ∀ (X : Aᵒᵖ), S.SheafCondition (F.obj ⋙ coyoneda.obj X)

Tactic:
intro X

Goal state after:
C : Type u
inst✝² : Category.{v, u} C
J : GrothendieckTopology C
inst✝¹ : HasWeakSheafify J (Type v)
S : J.MayerVietorisSquare
A : Type u'
inst✝ : Category.{v, u'} A
F : Sheaf J A
X : Aᵒᵖ
⊢ S.SheafCondition (F.obj ⋙ coyoneda.obj X)
Module: Mathlib.CategoryTheory.Sites.SheafHom

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
hG : IsSheaf J G
⊢ Presieve.IsSheaf J (presheafHom F G)

Tactic:
intro X

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
A : Type u'
inst✝ : Category.{v', u'} A
F G : Cᵒᵖ ⥤ A
hG : IsSheaf J G
X : C
⊢ ∀ S ∈ J X, Presieve.IsSheafFor (presheafHom F G) S.arrows
Module: Mathlib.CategoryTheory.Sites.Precoverage.Generates

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
K : Precoverage C
J : GrothendieckTopology C
h : K.Generates J
F : Cᵒᵖ ⥤ Type w
H : ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, Presieve.IsSheafFor F R
inst✝ : ∀ (Z : C), _root_.Small.{max u v, w} (F.obj (Opposite.op Z))
⊢ Presieve.IsSheaf J F

Tactic:
intro X

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
K : Precoverage C
J : GrothendieckTopology C
h : K.Generates J
F : Cᵒᵖ ⥤ Type w
H : ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, Presieve.IsSheafFor F R
inst✝ : ∀ (Z : C), _root_.Small.{max u v, w} (F.obj (Opposite.op Z))
X : C
⊢ ∀ S ∈ J X, Presieve.IsSheafFor F S.arrows
Module: Mathlib.CategoryTheory.Sites.Precoverage.Generates

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
K : Precoverage C
J : GrothendieckTopology C
h : K.Generates J
F : Cᵒᵖ ⥤ Type w
H : ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, Presieve.IsSheafFor F R
⊢ Presieve.IsSheaf J F

Tactic:
intro X

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
K : Precoverage C
J : GrothendieckTopology C
h : K.Generates J
F : Cᵒᵖ ⥤ Type w
H : ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, Presieve.IsSheafFor F R
X : C
⊢ ∀ S ∈ J X, Presieve.IsSheafFor F S.arrows
Module: Mathlib.CategoryTheory.Sites.PreservesLimits

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : J.Subcanonical
K : Type u_1
inst✝¹ : Category.{v_1, u_1} K
F : K ⥤ C
inst✝ : ∀ (X : Sheaf J (Type v)), PreservesLimit F.op X.obj
c : Cocone F
hc : IsColimit c
⊢ (k : Sheaf J (Type v)) →
    IsLimit
      (((evaluation (Sheaf J (Type v)) (Type (max u v))).obj k).mapCone (coyoneda.mapCone (J.yoneda.mapCocone c).op))

Tactic:
intro X

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : J.Subcanonical
K : Type u_1
inst✝¹ : Category.{v_1, u_1} K
F : K ⥤ C
inst✝ : ∀ (X : Sheaf J (Type v)), PreservesLimit F.op X.obj
c : Cocone F
hc : IsColimit c
X : Sheaf J (Type v)
⊢ IsLimit
    (((evaluation (Sheaf J (Type v)) (Type (max u v))).obj X).mapCone (coyoneda.mapCone (J.yoneda.mapCocone c).op))
Module: Mathlib.CategoryTheory.Sites.PreservesLimits

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : J.Subcanonical
K : Type u_1
inst✝¹ : Category.{v_1, u_1} K
F : K ⥤ C
inst✝ : ∀ (X : Sheaf J (Type (max v v'))), PreservesLimit F.op X.obj
c : Cocone F
hc : IsColimit c
⊢ (k : Sheaf J (Type (max v v'))) →
    IsLimit
      (((evaluation (Sheaf J (Type (max v v'))) (Type (max (max u v) v'))).obj k).mapCone
        (coyoneda.mapCone ((GrothendieckTopology.uliftYoneda.{v', v, u} J).mapCocone c).op))

Tactic:
intro X

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : J.Subcanonical
K : Type u_1
inst✝¹ : Category.{v_1, u_1} K
F : K ⥤ C
inst✝ : ∀ (X : Sheaf J (Type (max v v'))), PreservesLimit F.op X.obj
c : Cocone F
hc : IsColimit c
X : Sheaf J (Type (max v v'))
⊢ IsLimit
    (((evaluation (Sheaf J (Type (max v v'))) (Type (max (max u v) v'))).obj X).mapCone
      (coyoneda.mapCone ((GrothendieckTopology.uliftYoneda.{v', v, u} J).mapCocone c).op))
Module: Mathlib.CategoryTheory.Sites.Precoverage.Generates

Goal state before:
pp: case refine_2
C : Type u
inst✝ : Category.{v, u} C
K : Precoverage C
J : GrothendieckTopology C
H : K.Generates J
⊢ ∀ ⦃X : C⦄, ∀ R ∈ K.coverings X, Presieve.IsSheafFor (Functor.closedSieves K.toGrothendieck).toFunctor R

Tactic:
intro X

Goal state after:
case refine_2
C : Type u
inst✝ : Category.{v, u} C
K : Precoverage C
J : GrothendieckTopology C
H : K.Generates J
X : C
⊢ ∀ R ∈ K.coverings X, Presieve.IsSheafFor (Functor.closedSieves K.toGrothendieck).toFunctor R
Module: Mathlib.CategoryTheory.Sites.Precoverage.Generates

Goal state before:
pp: case refine_2
C : Type u
inst✝ : Category.{v, u} C
K : Precoverage C
J : GrothendieckTopology C
H : K.Generates J
X : C
R : Presieve X
⊢ R ∈ K.coverings X → Presieve.IsSheafFor (Functor.closedSieves K.toGrothendieck).toFunctor R

Tactic:
hR

Goal state after:
case refine_2
C : Type u
inst✝ : Category.{v, u} C
K : Precoverage C
J : GrothendieckTopology C
H : K.Generates J
X : C
R : Presieve X
hR : R ∈ K.coverings X
⊢ Presieve.IsSheafFor (Functor.closedSieves K.toGrothendieck).toFunctor R
Module: Mathlib.CategoryTheory.Subfunctor.SubmonoidFunctor

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
M✝ : C ⥤ MonCat
S✝ : SubmonoidFunctor M✝
M M' M'' : C ⥤ MonCat
S : SubmonoidFunctor M
S' : SubmonoidFunctor M'
⊢ ∀ (X : C), Mono (S.ι.app X)

Tactic:
intro X

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
M✝ : C ⥤ MonCat
S✝ : SubmonoidFunctor M✝
M M' M'' : C ⥤ MonCat
S : SubmonoidFunctor M
S' : SubmonoidFunctor M'
X : C
⊢ Mono (S.ι.app X)
Module: Mathlib.CategoryTheory.Sites.SheafCohomology.MayerVietoris

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
inst✝¹ : HasSheafify J AddCommGrpCat
inst✝ : HasExt (Sheaf J AddCommGrpCat)
S : J.MayerVietorisSquare
F : Sheaf J AddCommGrpCat
n : ℕ
x₂ : ↑(F.H' n S.X₂)
x₃ : ↑(F.H' n S.X₃)
⊢ (Ext.mk₀ S.shortComplex.f).comp
      (Ext.biprodAddEquiv.symm
        ((ConcreteCategory.hom ((F.H' n S.X₂).biprodIsoProd (F.H' n S.X₃)).hom)
          (((F.H' n S.X₂).biprodIsoProd (F.H' n S.X₃)).addCommGroupIsoToAddEquiv.symm (x₂, x₃))))
      ⋯ =
    (ConcreteCategory.hom (S.fromBiprod F n))
      (((F.H' n S.X₂).biprodIsoProd (F.H' n S.X₃)).addCommGroupIsoToAddEquiv.symm (x₂, x₃))

Tactic:
Ext.biprodAddEquiv_symm_apply,

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
J : GrothendieckTopology C
inst✝² : HasWeakSheafify J (Type v)
inst✝¹ : HasSheafify J AddCommGrpCat
inst✝ : HasExt (Sheaf J AddCommGrpCat)
S : J.MayerVietorisSquare
F : Sheaf J AddCommGrpCat
n : ℕ
x₂ : ↑(F.H' n S.X₂)
x₃ : ↑(F.H' n S.X₃)
⊢ (Ext.mk₀ S.shortComplex.f).comp
      ((Ext.mk₀ biprod.fst).comp
          ((ConcreteCategory.hom ((F.H' n S.X₂).biprodIsoProd (F.H' n S.X₃)).hom)
              (((F.H' n S.X₂).biprodIsoProd (F.H' n S.X₃)).addCommGroupIsoToAddEquiv.symm (x₂, x₃))).1
          ⋯ +
        (Ext.mk₀ biprod.snd).comp
          ((ConcreteCategory.hom ((F.H' n S.X₂).biprodIsoProd (F.H' n S.X₃)).hom)
              (((F.H' n S.X₂).biprodIsoProd (F.H' n S.X₃)).addCommGroupIsoToAddEquiv.symm (x₂, x₃))).2
          ⋯)
      ⋯ =
    (ConcreteCategory.hom (S.fromBiprod F n))
      (((F.H' n S.X₂).biprodIsoProd (F.H' n S.X₃)).addCommGroupIsoToAddEquiv.symm (x₂, x₃))
Module: Mathlib.CategoryTheory.Topos.Sheaf

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type (max u v))
m : F ⟶ G
inst✝ : Mono m
⊢ Subfunctor.range (Presheaf.χ m.hom) ≤ closedSieves J

Tactic:
intro X

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
J : GrothendieckTopology C
F G : Sheaf J (Type (max u v))
m : F ⟶ G
inst✝ : Mono m
X : Cᵒᵖ
⊢ (Subfunctor.range (Presheaf.χ m.hom)).obj X ⊆ (closedSieves J).obj X