mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify

Goal state before:
pp: C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj
inst✝⁶ : Presheaf.IsLocallyInjective J α
inst✝⁵ : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrpCat
φ : M₀.presheaf ⟶ A.obj
inst✝⁴ : Presheaf.IsLocallyInjective J φ
inst✝³ : Presheaf.IsLocallySurjective J φ
inst✝² : J.WEqualsLocallyBijective AddCommGrpCat
M₀' : PresheafOfModules R₀
A' : Sheaf J AddCommGrpCat
φ' : M₀'.presheaf ⟶ A'.obj
inst✝¹ : Presheaf.IsLocallyInjective J φ'
inst✝ : Presheaf.IsLocallySurjective J φ'
τ₀ : M₀ ⟶ M₀'
τ : A ⟶ A'
fac : (toPresheaf R₀).map τ₀ ≫ φ' = φ ≫ τ.hom
X : Cᵒᵖ
r : ↑(R.obj.obj X)
m : ↑((sheafify α φ).val.obj X)
f : (sheafify α φ).val ⟶ (sheafify α φ').val := (sheafifyHomEquiv' α φ ⋯).symm (τ₀ ≫ toSheafify α φ')
⊢ φ ≫ τ.hom = (toPresheaf R₀).map (τ₀ ≫ toSheafify α φ')

Tactic:
]

Goal state after:
C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj
inst✝⁶ : Presheaf.IsLocallyInjective J α
inst✝⁵ : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrpCat
φ : M₀.presheaf ⟶ A.obj
inst✝⁴ : Presheaf.IsLocallyInjective J φ
inst✝³ : Presheaf.IsLocallySurjective J φ
inst✝² : J.WEqualsLocallyBijective AddCommGrpCat
M₀' : PresheafOfModules R₀
A' : Sheaf J AddCommGrpCat
φ' : M₀'.presheaf ⟶ A'.obj
inst✝¹ : Presheaf.IsLocallyInjective J φ'
inst✝ : Presheaf.IsLocallySurjective J φ'
τ₀ : M₀ ⟶ M₀'
τ : A ⟶ A'
fac : (toPresheaf R₀).map τ₀ ≫ φ' = φ ≫ τ.hom
X : Cᵒᵖ
r : ↑(R.obj.obj X)
m : ↑((sheafify α φ).val.obj X)
f : (sheafify α φ).val ⟶ (sheafify α φ').val := (sheafifyHomEquiv' α φ ⋯).symm (τ₀ ≫ toSheafify α φ')
⊢ φ ≫ τ.hom = (toPresheaf R₀).map (τ₀ ≫ toSheafify α φ')
Module: Mathlib.Algebra.Category.ModuleCat.Presheaf.Sheafify

Goal state before:
pp: C : Type u₁
inst✝⁷ : Category.{v₁, u₁} C
J : GrothendieckTopology C
R₀ : Cᵒᵖ ⥤ RingCat
R : Sheaf J RingCat
α : R₀ ⟶ R.obj
inst✝⁶ : Presheaf.IsLocallyInjective J α
inst✝⁵ : Presheaf.IsLocallySurjective J α
M₀ : PresheafOfModules R₀
A : Sheaf J AddCommGrpCat
φ : M₀.presheaf ⟶ A.obj
inst✝⁴ : Presheaf.IsLocallyInjective J φ
inst✝³ : Presheaf.IsLocallySurjective J φ
inst✝² : J.WEqualsLocallyBijective AddCommGrpCat
M₀' : PresheafOfModules R₀
A' : Sheaf J AddCommGrpCat
φ' : M₀'.presheaf ⟶ A'.obj
inst✝¹ : Presheaf.IsLocallyInjective J φ'
inst✝ : Presheaf.IsLocallySurjective J φ'
τ₀ : M₀ ⟶ M₀'
τ : A ⟶ A'
fac : (toPresheaf R₀).map τ₀ ≫ φ' = φ ≫ τ.hom
X : Cᵒᵖ
r : ↑(R.obj.obj X)
m : ↑((sheafify α φ).val.obj X)
f : (sheafify α φ).val ⟶ (sheafify α φ').val := (sheafifyHomEquiv' α φ ⋯).symm (τ₀ ≫ toSheafify α φ')
⊢ (toPresheaf R₀).map τ₀ ≫ φ' = (toPresheaf R₀).map τ₀ ≫ φ'

Tactic:
]

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

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X Y : C
f : Y ⟶ X
S : Sieve X
hS : J₁.IsClosed S
Z : C
g : Z ⟶ Y
hg : J₁.Covers (Sieve.pullback f S) g
⊢ Sieve.pullback g (Sieve.pullback f S) ∈ J₁ Z

Tactic:
]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X Y : C
f : Y ⟶ X
S : Sieve X
hS : J₁.IsClosed S
Z : C
g : Z ⟶ Y
hg : J₁.Covers (Sieve.pullback f S) g
⊢ Sieve.pullback g (Sieve.pullback f S) ∈ J₁ Z
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: case a
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X Y : C
f : Y ⟶ X
S : Sieve X
Z : C
g : Z ⟶ Y
hg : (Sieve.pullback f (J₁.close S)).arrows g
⊢ Sieve.pullback (g ≫ f) S ∈ J₁ Z

Tactic:
]

Goal state after:
case a
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X Y : C
f : Y ⟶ X
S : Sieve X
Z : C
g : Z ⟶ Y
hg : (Sieve.pullback f (J₁.close S)).arrows g
⊢ Sieve.pullback (g ≫ f) S ∈ J₁ Z
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: case mp
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
h : J₁.close S = ⊤
Y : C
f : Y ⟶ X
hf : ⊤.arrows f
⊢ ⊤.arrows f

Tactic:
]

Goal state after:
case mp
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
h : J₁.close S = ⊤
Y : C
f : Y ⟶ X
hf : ⊤.arrows f
⊢ ⊤.arrows f
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: case mpr
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
⊢ ⊤ ≤ J₁.close S

Tactic:
]

Goal state after:
case mpr
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
⊢ ⊤ ≤ J₁.close S
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
⊢ Presieve.IsSeparatedFor (Functor.closedSieves J₁).toFunctor S.arrows ∧
    ∀ (x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows),
      x.Compatible → ∃ t, x.IsAmalgamation t

Tactic:
]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
⊢ Presieve.IsSeparatedFor (Functor.closedSieves J₁).toFunctor S.arrows ∧
    ∀ (x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows),
      x.Compatible → ∃ t, x.IsAmalgamation t
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: case refine_1
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
M : Sieve X
hM : M ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
N : Sieve X
hN : N ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
hM₂ : x.IsAmalgamation ⟨M, hM⟩
hN₂ : x.IsAmalgamation ⟨N, hN⟩
Y : C
f : Y ⟶ X
⊢ J₁.Covers M f ↔ J₁.Covers N f

Tactic:
]

Goal state after:
case refine_1
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
M : Sieve X
hM : M ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
N : Sieve X
hN : N ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
hM₂ : x.IsAmalgamation ⟨M, hM⟩
hN₂ : x.IsAmalgamation ⟨N, hN⟩
Y : C
f : Y ⟶ X
⊢ J₁.Covers M f ↔ J₁.Covers N f
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: case refine_1.mp
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
M : Sieve X
hM : M ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
N : Sieve X
hN : N ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
hM₂ : x.IsAmalgamation ⟨M, hM⟩
hN₂ : x.IsAmalgamation ⟨N, hN⟩
Y : C
f : Y ⟶ X
q : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N
MSNS : M ⊓ S = N ⊓ S
hf : J₁.Covers M f
⊢ Sieve.pullback f N ∈ J₁ Y

Tactic:
]

Goal state after:
case refine_1.mp
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
M : Sieve X
hM : M ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
N : Sieve X
hN : N ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
hM₂ : x.IsAmalgamation ⟨M, hM⟩
hN₂ : x.IsAmalgamation ⟨N, hN⟩
Y : C
f : Y ⟶ X
q : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N
MSNS : M ⊓ S = N ⊓ S
hf : J₁.Covers M f
⊢ Sieve.pullback f N ∈ J₁ Y
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: case refine_1.mp
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
M : Sieve X
hM : M ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
N : Sieve X
hN : N ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
hM₂ : x.IsAmalgamation ⟨M, hM⟩
hN₂ : x.IsAmalgamation ⟨N, hN⟩
Y : C
f : Y ⟶ X
q : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N
MSNS : M ⊓ S = N ⊓ S
hf : J₁.Covers M f
⊢ Sieve.pullback f (M ⊓ S) ∈ J₁ Y

Tactic:
]

Goal state after:
case refine_1.mp
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
M : Sieve X
hM : M ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
N : Sieve X
hN : N ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
hM₂ : x.IsAmalgamation ⟨M, hM⟩
hN₂ : x.IsAmalgamation ⟨N, hN⟩
Y : C
f : Y ⟶ X
q : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N
MSNS : M ⊓ S = N ⊓ S
hf : J₁.Covers M f
⊢ Sieve.pullback f (M ⊓ S) ∈ J₁ Y
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: case refine_1.mpr
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
M : Sieve X
hM : M ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
N : Sieve X
hN : N ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
hM₂ : x.IsAmalgamation ⟨M, hM⟩
hN₂ : x.IsAmalgamation ⟨N, hN⟩
Y : C
f : Y ⟶ X
q : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N
MSNS : M ⊓ S = N ⊓ S
hf : J₁.Covers N f
⊢ Sieve.pullback f M ∈ J₁ Y

Tactic:
]

Goal state after:
case refine_1.mpr
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
M : Sieve X
hM : M ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
N : Sieve X
hN : N ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
hM₂ : x.IsAmalgamation ⟨M, hM⟩
hN₂ : x.IsAmalgamation ⟨N, hN⟩
Y : C
f : Y ⟶ X
q : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N
MSNS : M ⊓ S = N ⊓ S
hf : J₁.Covers N f
⊢ Sieve.pullback f M ∈ J₁ Y
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: case refine_1.mpr
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
M : Sieve X
hM : M ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
N : Sieve X
hN : N ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
hM₂ : x.IsAmalgamation ⟨M, hM⟩
hN₂ : x.IsAmalgamation ⟨N, hN⟩
Y : C
f : Y ⟶ X
q : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N
MSNS : M ⊓ S = N ⊓ S
hf : J₁.Covers N f
⊢ Sieve.pullback f (N ⊓ S) ∈ J₁ Y

Tactic:
]

Goal state after:
case refine_1.mpr
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
M : Sieve X
hM : M ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
N : Sieve X
hN : N ∈ (Functor.closedSieves J₁).obj (Opposite.op X)
hM₂ : x.IsAmalgamation ⟨M, hM⟩
hN₂ : x.IsAmalgamation ⟨N, hN⟩
Y : C
f : Y ⟶ X
q : ∀ ⦃Z : C⦄ (g : Z ⟶ X), S.arrows g → Sieve.pullback g M = Sieve.pullback g N
MSNS : M ⊓ S = N ⊓ S
hf : J₁.Covers N f
⊢ Sieve.pullback f (N ⊓ S) ∈ J₁ Y
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: case refine_2
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
hx : x.SieveCompatible
⊢ ∃ t, x.IsAmalgamation t

Tactic:
]

Goal state after:
case refine_2
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
hx : x.SieveCompatible
⊢ ∃ t, x.IsAmalgamation t
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: case a
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
hx : x.SieveCompatible
M : Sieve X := Sieve.bind S.arrows fun Y f hf ↦ ↑(x f hf)
Y : C
f : Y ⟶ X
hf : S.arrows f
Z : C
u : Z ⟶ Y
W : C
g : Z ⟶ W
f' : W ⟶ X
hf' : S.arrows f'
hg : (↑(x f' hf')).arrows g
c : g ≫ f' = u ≫ f
⊢ ↑(x (u ≫ f) ⋯) = ⊤

Tactic:
]

Goal state after:
case a
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
hx : x.SieveCompatible
M : Sieve X := Sieve.bind S.arrows fun Y f hf ↦ ↑(x f hf)
Y : C
f : Y ⟶ X
hf : S.arrows f
Z : C
u : Z ⟶ Y
W : C
g : Z ⟶ W
f' : W ⟶ X
hf' : S.arrows f'
hg : (↑(x f' hf')).arrows g
c : g ≫ f' = u ≫ f
⊢ ↑(x (u ≫ f) ⋯) = ⊤
Module: Mathlib.CategoryTheory.Functor.Flat

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
J : Type v₁
inst✝² : SmallCategory J
inst✝¹ : FinCategory J
F : C ⥤ D
inst✝ : RepresentablyFlat F
K : J ⥤ C
c : Cone K
hc : IsLimit c
s : Cone (K ⋙ F)
f₁ f₂ : s.pt ⟶ F.obj c.pt
h₁ : ∀ (j : J), f₁ ≫ (F.mapCone c).π.app j = s.π.app j
h₂ : ∀ (j : J), f₂ ≫ (F.mapCone c).π.app j = s.π.app j
α₁ : (F.mapCone c).toStructuredArrow ⋙ map f₁ ⟶ s.toStructuredArrow := { app := fun X ↦ eqToHom ⋯, naturality := ⋯ }
α₂ : (F.mapCone c).toStructuredArrow ⋙ map f₂ ⟶ s.toStructuredArrow := { app := fun X ↦ eqToHom ⋯, naturality := ⋯ }
c₁ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.whiskerRight α₁ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₁)
c₂ : Cone (s.toStructuredArrow ⋙ pre s.pt K F) :=
  (Cone.postcompose (Functor.whiskerRight α₂ (pre s.pt K F))).obj (c.toStructuredArrowCone F f₂)
c₀ : Cone (biconeMk J c₁ c₂) := IsCofiltered.cone (biconeMk J c₁ c₂)
g₁ : c₀.pt ⟶ c₁.pt := c₀.π.app Bicone.left
g₂ : c₀.pt ⟶ c₂.pt := c₀.π.app Bicone.right
this✝¹ : ∀ (j : J), Hom.right g₁ ≫ c.π.app j = Hom.right g₂ ≫ c.π.app j
this✝ : c.extend (Hom.right g₁) = c.extend (Hom.right g₂)
this : Hom.right g₁ = Hom.right g₂
⊢ c₀.pt.hom ≫ F.map (Hom.right g₂) = c₀.pt.hom ≫ F.map (Hom.right g₂)

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.Zero

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreZeroHypercover S
ι : Type u_1
f : ι → E.I₀
⊢ ∀ (i : (E.restrictIndex f).I₀), E.presieve₀ ((E.restrictIndex f).f i)

Tactic:
]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreZeroHypercover S
ι : Type u_1
f : ι → E.I₀
⊢ ∀ (i : (E.restrictIndex f).I₀), E.presieve₀ ((E.restrictIndex f).f i)
Module: Mathlib.CategoryTheory.Sites.Hypercover.Zero

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreZeroHypercover S
ι : Type u_1
f : ι → E.I₀
⊢ (E.restrictIndex f).presieve₀ ≤ E.presieve₀

Tactic:
rw [Presieve.ofArrows_le_iff]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreZeroHypercover S
ι : Type u_1
f : ι → E.I₀
⊢ ∀ (i : (E.restrictIndex f).I₀), E.presieve₀ ((E.restrictIndex f).f i)
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: case a
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
hx : x.SieveCompatible
M : Sieve X := Sieve.bind S.arrows fun Y f hf ↦ ↑(x f hf)
Y : C
f : Y ⟶ X
hf : S.arrows f
Z : C
u : Z ⟶ Y
W : C
g : Z ⟶ W
f' : W ⟶ X
hf' : S.arrows f'
hg : (↑(x f' hf')).arrows g
c : g ≫ f' = u ≫ f
⊢ ↑((ConcreteCategory.hom ((Functor.closedSieves J₁).toFunctor.map g.op)) (x f' hf')) = ⊤

Tactic:
]

Goal state after:
case a
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
hx : x.SieveCompatible
M : Sieve X := Sieve.bind S.arrows fun Y f hf ↦ ↑(x f hf)
Y : C
f : Y ⟶ X
hf : S.arrows f
Z : C
u : Z ⟶ Y
W : C
g : Z ⟶ W
f' : W ⟶ X
hf' : S.arrows f'
hg : (↑(x f' hf')).arrows g
c : g ≫ f' = u ≫ f
⊢ ↑((ConcreteCategory.hom ((Functor.closedSieves J₁).toFunctor.map g.op)) (x f' hf')) = ⊤
Module: Mathlib.CategoryTheory.Sites.Hypercover.Zero

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreZeroHypercover S
F : PreZeroHypercover S
⊢ Presieve.ofArrows (E.sum F).X (E.sum F).f = Presieve.ofArrows E.X E.f ⊔ Presieve.ofArrows F.X F.f

Tactic:
]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
S : C
E : PreZeroHypercover S
F : PreZeroHypercover S
⊢ Presieve.ofArrows (E.sum F).X (E.sum F).f = Presieve.ofArrows E.X E.f ⊔ Presieve.ofArrows F.X F.f
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: case refine_2
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
hx : x.SieveCompatible
M : Sieve X := Sieve.bind S.arrows fun Y f hf ↦ ↑(x f hf)
this : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = ↑(x f hf)
Y : C
f : Y ⟶ X
hf : S.arrows f
⊢ J₁.close ↑(x f hf) = ↑(x f hf)

Tactic:
]

Goal state after:
case refine_2
C : Type u
inst✝ : Category.{v, u} C
J₁ : GrothendieckTopology C
X : C
S : Sieve X
hS : S ∈ J₁ X
x : Presieve.FamilyOfElements (Functor.closedSieves J₁).toFunctor S.arrows
hx : x.SieveCompatible
M : Sieve X := Sieve.bind S.arrows fun Y f hf ↦ ↑(x f hf)
this : ∀ ⦃Y : C⦄ (f : Y ⟶ X) (hf : S.arrows f), Sieve.pullback f M = ↑(x f hf)
Y : C
f : Y ⟶ X
hf : S.arrows f
⊢ J₁.close ↑(x f hf) = ↑(x f hf)
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X : C
S : Sieve X
H : Presieve.IsSheafFor (Functor.closedSieves J).toFunctor S.arrows
⊢ J.close S = ⊤

Tactic:
]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X : C
S : Sieve X
H : Presieve.IsSheafFor (Functor.closedSieves J).toFunctor S.arrows
⊢ J.close S = ⊤
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X : C
S : Sieve X
H : Presieve.IsSheafFor (Functor.closedSieves J).toFunctor S.arrows
this✝ : J.IsClosed ⊤
this : ↑⟨J.close S, ⋯⟩ = ↑⟨⊤, this✝⟩
⊢ J.close S = ⊤

Tactic:
]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X : C
S : Sieve X
H : Presieve.IsSheafFor (Functor.closedSieves J).toFunctor S.arrows
this✝ : J.IsClosed ⊤
this : ↑⟨J.close S, ⋯⟩ = ↑⟨⊤, this✝⟩
⊢ J.close S = ⊤
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X : C
S : Sieve X
H : Presieve.IsSheafFor (Functor.closedSieves J).toFunctor S.arrows
this : J.IsClosed ⊤
Y : C
f : Y ⟶ X
hf : S.arrows f
⊢ J.close (Sieve.pullback f S) = ⊤

Tactic:
S.pullback_eq_top_of_mem hf,

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X : C
S : Sieve X
H : Presieve.IsSheafFor (Functor.closedSieves J).toFunctor S.arrows
this : J.IsClosed ⊤
Y : C
f : Y ⟶ X
hf : S.arrows f
⊢ J.close ⊤ = ⊤
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X : C
S : Sieve X
H : Presieve.IsSheafFor (Functor.closedSieves J).toFunctor S.arrows
this : J.IsClosed ⊤
Y : C
f : Y ⟶ X
hf : S.arrows f
⊢ ⊤ ∈ J Y

Tactic:
]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X : C
S : Sieve X
H : Presieve.IsSheafFor (Functor.closedSieves J).toFunctor S.arrows
this : J.IsClosed ⊤
Y : C
f : Y ⟶ X
hf : S.arrows f
⊢ ⊤ ∈ J Y
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : Presieve.IsSheaf J₁ (Functor.closedSieves J₂).toFunctor
X : C
S : Sieve X
hS : S ∈ J₁ X
⊢ Presieve.IsSheafFor (Functor.closedSieves J₂).toFunctor S.arrows

Tactic:
]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : Presieve.IsSheaf J₁ (Functor.closedSieves J₂).toFunctor
X : C
S : Sieve X
hS : S ∈ J₁ X
⊢ Presieve.IsSheafFor (Functor.closedSieves J₂).toFunctor S.arrows
Module: Mathlib.CategoryTheory.Sites.Hypercover.Zero

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S T : C
E✝¹ : PreZeroHypercover S
F✝¹ : PreZeroHypercover S
E✝ : PreZeroHypercover S
F✝ : PreZeroHypercover S
G : PreZeroHypercover S
E : PreZeroHypercover S
F : PreZeroHypercover S
f g : E.Hom F
hs : f.s₀ = g.s₀
i : E.I₀
⊢ F.X (g.s₀ i) = F.X (g.s₀ i)

Tactic:
]

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

Goal state before:
pp: case mpr.a
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ Presieve.IsSheaf J₂ (Functor.closedSieves J₂).toFunctor

Tactic:
]

Goal state after:
case mpr.a
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ Presieve.IsSheaf J₂ (Functor.closedSieves J₂).toFunctor
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: case a
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ Presieve.IsSheaf J₁ (Functor.closedSieves J₁).toFunctor

Tactic:
]

Goal state after:
case a
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
h : ∀ (P : Cᵒᵖ ⥤ Type (max v u)), Presieve.IsSheaf J₁ P ↔ Presieve.IsSheaf J₂ P
⊢ Presieve.IsSheaf J₁ (Functor.closedSieves J₁).toFunctor
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
c : (X : C) → ClosureOperator (Sieve X)
hc : ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve X), (c Y) (Sieve.pullback f S) = Sieve.pullback f ((c X) S)
X Y : C
S : Sieve X
f : Y ⟶ X
hS : (c X) S = ⊤
⊢ Sieve.pullback f S ∈ {S | (c Y) S = ⊤}

Tactic:
]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
c : (X : C) → ClosureOperator (Sieve X)
hc : ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve X), (c Y) (Sieve.pullback f S) = Sieve.pullback f ((c X) S)
X Y : C
S : Sieve X
f : Y ⟶ X
hS : (c X) S = ⊤
⊢ Sieve.pullback f S ∈ {S | (c Y) S = ⊤}
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
c : (X : C) → ClosureOperator (Sieve X)
hc : ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve X), (c Y) (Sieve.pullback f S) = Sieve.pullback f ((c X) S)
X Y : C
S : Sieve X
f : Y ⟶ X
hS : (c X) S = ⊤
⊢ ⊤ = ⊤

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.Sites.Hypercover.Zero

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
S✝ T : C
E✝¹ : PreZeroHypercover S✝
F✝¹ : PreZeroHypercover S✝
E✝ : PreZeroHypercover S✝
F✝ : PreZeroHypercover S✝
G : PreZeroHypercover S✝
S : C
E F : PreZeroHypercover S
s₀ : E.I₀ ≃ F.I₀
h₀ : (i : E.I₀) → E.X i ≅ F.X (s₀ i)
w₀ : ∀ (i : E.I₀), (h₀ i).hom ≫ F.f (s₀ i) = E.f i
i : E.I₀
⊢ E.f i ≫ eqToHom ⋯ = E.f i

Tactic:
]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
S✝ T : C
E✝¹ : PreZeroHypercover S✝
F✝¹ : PreZeroHypercover S✝
E✝ : PreZeroHypercover S✝
F✝ : PreZeroHypercover S✝
G : PreZeroHypercover S✝
S : C
E F : PreZeroHypercover S
s₀ : E.I₀ ≃ F.I₀
h₀ : (i : E.I₀) → E.X i ≅ F.X (s₀ i)
w₀ : ∀ (i : E.I₀), (h₀ i).hom ≫ F.f (s₀ i) = E.f i
i : E.I₀
⊢ E.f i ≫ eqToHom ⋯ = E.f i
Module: Mathlib.CategoryTheory.Sites.Closed

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
c : (X : C) → ClosureOperator (Sieve X)
hc : ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve X), (c Y) (Sieve.pullback f S) = Sieve.pullback f ((c X) S)
X : C
S : Sieve X
hS : (c X) S = ⊤
R : Sieve X
hR : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Sieve.pullback f R ∈ {S | (c Y) S = ⊤}
⊢ R ∈ {S | (c X) S = ⊤}

Tactic:
]

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
J₁ J₂ : GrothendieckTopology C
c : (X : C) → ClosureOperator (Sieve X)
hc : ∀ ⦃X Y : C⦄ (f : Y ⟶ X) (S : Sieve X), (c Y) (Sieve.pullback f S) = Sieve.pullback f ((c X) S)
X : C
S : Sieve X
hS : (c X) S = ⊤
R : Sieve X
hR : ∀ ⦃Y : C⦄ ⦃f : Y ⟶ X⦄, S.arrows f → Sieve.pullback f R ∈ {S | (c Y) S = ⊤}
⊢ R ∈ {S | (c X) S = ⊤}