l3lab/ntp-mathlib huggingface dataset:
/-- Given a family of opens `U` and an open `Y` equal to the union of opens in `U`, we may
    take the presieve on `Y` associated to `U` and the sieve generated by it, and form the
    full subcategory (subposet) of opens contained in `Y` (`over Y`) consisting of arrows
    in the sieve. This full subcategory is equivalent to `OpensLeCover U`, the (poset)
    category of opens contained in some `U i`. -/
@[simps]
def generateEquivalenceOpensLe :
    (FullSubcategory fun f : Over Y => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom) ≌
    OpensLeCover U where
  -- Porting note : split it out to prevent timeout
  functor := generateEquivalenceOpensLe_functor' _ _
  inverse := generateEquivalenceOpensLe_inverse' _ _
  unitIso := eqToIso <| CategoryTheory.Functor.ext
    (by rintro ⟨⟨_, _⟩, _⟩; dsimp; congr)
    (by intros; refine Over.OverMorphism.ext ?_; aesop_cat)
  counitIso := eqToIso <| CategoryTheory.Functor.hext
    (by intro; refine FullSubcategory.ext _ _ ?_; rfl) (by intros; 

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι : Type w
U : ι → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U
X✝ Y✝ : OpensLeCover U
f✝ : X✝ ⟶ Y✝
⊢ HEq
    ((generateEquivalenceOpensLe_inverse' U (_ : Y = iSup U) ⋙
          generateEquivalenceOpensLe_functor' U (_ : Y = iSup U)).map
      f✝)
    ((𝟭 (OpensLeCover U)).map f✝)

Next tactic:
rfl
/-- Given a family of opens `opensLeCoverCocone U` is essentially the natural cocone
    associated to the sieve generated by the presieve associated to `U` with indexing
    category changed using the above equivalence. -/
@[simps]
def whiskerIsoMapGenerateCocone :
    (F.mapCone (opensLeCoverCocone U).op).whisker (generateEquivalenceOpensLe U hY).op.functor ≅
      F.mapCone (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op where
  hom :=
    { hom := F.map (eqToHom (congr_arg op hY.symm))
      w := fun j => by
        

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι : Type w
U : ι → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U
j : (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)ᵒᵖ
⊢ F.map
        (eqToHom
          (_ :
            op (opensLeCoverCocone U).pt =
              op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt)) ≫
      (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows))).π.app j =
    (Cone.whisker (Equivalence.op (generateEquivalenceOpensLe U hY)).functor
            (F.mapCone (Cocone.op (opensLeCoverCocone U)))).π.app
      j

Next tactic:
erw [← F.map_comp]
/-- Given a family of opens `opensLeCoverCocone U` is essentially the natural cocone
    associated to the sieve generated by the presieve associated to `U` with indexing
    category changed using the above equivalence. -/
@[simps]
def whiskerIsoMapGenerateCocone :
    (F.mapCone (opensLeCoverCocone U).op).whisker (generateEquivalenceOpensLe U hY).op.functor ≅
      F.mapCone (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op where
  hom :=
    { hom := F.map (eqToHom (congr_arg op hY.symm))
      w := fun j => by
        erw [← F.map_comp]
        

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι : Type w
U : ι → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U
j : (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)ᵒᵖ
⊢ F.map
      (eqToHom
          (_ :
            op (opensLeCoverCocone U).pt =
              op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt) ≫
        (Cocone.op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows)).π.app j) =
    (Cone.whisker (Equivalence.op (generateEquivalenceOpensLe U hY)).functor
            (F.mapCone (Cocone.op (opensLeCoverCocone U)))).π.app
      j

Next tactic:
dsimp
/-- Given a family of opens `opensLeCoverCocone U` is essentially the natural cocone
    associated to the sieve generated by the presieve associated to `U` with indexing
    category changed using the above equivalence. -/
@[simps]
def whiskerIsoMapGenerateCocone :
    (F.mapCone (opensLeCoverCocone U).op).whisker (generateEquivalenceOpensLe U hY).op.functor ≅
      F.mapCone (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op where
  hom :=
    { hom := F.map (eqToHom (congr_arg op hY.symm))
      w := fun j => by
        erw [← F.map_comp]
        dsimp
        

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι : Type w
U : ι → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U
j : (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)ᵒᵖ
⊢ F.map (eqToHom (_ : op (opensLeCoverCocone U).pt = op Y) ≫ j.unop.obj.hom.op) =
    F.map ((opensLeCoverCocone U).ι.app ((generateEquivalenceOpensLe_functor' U (_ : Y = iSup U)).obj j.unop)).op

Next tactic:
congr 1
/-- Given a family of opens `opensLeCoverCocone U` is essentially the natural cocone
    associated to the sieve generated by the presieve associated to `U` with indexing
    category changed using the above equivalence. -/
@[simps]
def whiskerIsoMapGenerateCocone :
    (F.mapCone (opensLeCoverCocone U).op).whisker (generateEquivalenceOpensLe U hY).op.functor ≅
      F.mapCone (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op where
  hom :=
    { hom := F.map (eqToHom (congr_arg op hY.symm))
      w := fun j => by
        erw [← F.map_comp]
        dsimp
        congr 1 }
  inv :=
    { hom := F.map (eqToHom (congr_arg op hY))
      w := fun j => by
        

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι : Type w
U : ι → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U
j : (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)ᵒᵖ
⊢ F.map
        (eqToHom
          (_ :
            op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt =
              op (opensLeCoverCocone U).pt)) ≫
      (Cone.whisker (Equivalence.op (generateEquivalenceOpensLe U hY)).functor
              (F.mapCone (Cocone.op (opensLeCoverCocone U)))).π.app
        j =
    (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows))).π.app j

Next tactic:
erw [← F.map_comp]
/-- Given a family of opens `opensLeCoverCocone U` is essentially the natural cocone
    associated to the sieve generated by the presieve associated to `U` with indexing
    category changed using the above equivalence. -/
@[simps]
def whiskerIsoMapGenerateCocone :
    (F.mapCone (opensLeCoverCocone U).op).whisker (generateEquivalenceOpensLe U hY).op.functor ≅
      F.mapCone (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op where
  hom :=
    { hom := F.map (eqToHom (congr_arg op hY.symm))
      w := fun j => by
        erw [← F.map_comp]
        dsimp
        congr 1 }
  inv :=
    { hom := F.map (eqToHom (congr_arg op hY))
      w := fun j => by
        erw [← F.map_comp]
        

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι : Type w
U : ι → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U
j : (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)ᵒᵖ
⊢ F.map
      (eqToHom
          (_ :
            op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt =
              op (opensLeCoverCocone U).pt) ≫
        (Cocone.op (opensLeCoverCocone U)).π.app ((Equivalence.op (generateEquivalenceOpensLe U hY)).functor.obj j)) =
    (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows))).π.app j

Next tactic:
dsimp
/-- Given a family of opens `opensLeCoverCocone U` is essentially the natural cocone
    associated to the sieve generated by the presieve associated to `U` with indexing
    category changed using the above equivalence. -/
@[simps]
def whiskerIsoMapGenerateCocone :
    (F.mapCone (opensLeCoverCocone U).op).whisker (generateEquivalenceOpensLe U hY).op.functor ≅
      F.mapCone (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op where
  hom :=
    { hom := F.map (eqToHom (congr_arg op hY.symm))
      w := fun j => by
        erw [← F.map_comp]
        dsimp
        congr 1 }
  inv :=
    { hom := F.map (eqToHom (congr_arg op hY))
      w := fun j => by
        erw [← F.map_comp]
        dsimp
        

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι : Type w
U : ι → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U
j : (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)ᵒᵖ
⊢ F.map
      (eqToHom (_ : op Y = op (opensLeCoverCocone U).pt) ≫
        ((opensLeCoverCocone U).ι.app ((generateEquivalenceOpensLe_functor' U (_ : Y = iSup U)).obj j.unop)).op) =
    F.map j.unop.obj.hom.op

Next tactic:
congr 1
/-- Given a family of opens `opensLeCoverCocone U` is essentially the natural cocone
    associated to the sieve generated by the presieve associated to `U` with indexing
    category changed using the above equivalence. -/
@[simps]
def whiskerIsoMapGenerateCocone :
    (F.mapCone (opensLeCoverCocone U).op).whisker (generateEquivalenceOpensLe U hY).op.functor ≅
      F.mapCone (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op where
  hom :=
    { hom := F.map (eqToHom (congr_arg op hY.symm))
      w := fun j => by
        erw [← F.map_comp]
        dsimp
        congr 1 }
  inv :=
    { hom := F.map (eqToHom (congr_arg op hY))
      w := fun j => by
        erw [← F.map_comp]
        dsimp
        congr 1 }
  hom_inv_id := by
    

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι : Type w
U : ι → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U
⊢ ConeMorphism.mk
        (F.map
          (eqToHom
            (_ :
              op (opensLeCoverCocone U).pt =
                op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt))) ≫
      ConeMorphism.mk
        (F.map
          (eqToHom
            (_ :
              op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt =
                op (opensLeCoverCocone U).pt))) =
    𝟙
      (Cone.whisker (Equivalence.op (generateEquivalenceOpensLe U hY)).functor
        (F.mapCone (Cocone.op (opensLeCoverCocone U))))

Next tactic:
ext
/-- Given a family of opens `opensLeCoverCocone U` is essentially the natural cocone
    associated to the sieve generated by the presieve associated to `U` with indexing
    category changed using the above equivalence. -/
@[simps]
def whiskerIsoMapGenerateCocone :
    (F.mapCone (opensLeCoverCocone U).op).whisker (generateEquivalenceOpensLe U hY).op.functor ≅
      F.mapCone (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op where
  hom :=
    { hom := F.map (eqToHom (congr_arg op hY.symm))
      w := fun j => by
        erw [← F.map_comp]
        dsimp
        congr 1 }
  inv :=
    { hom := F.map (eqToHom (congr_arg op hY))
      w := fun j => by
        erw [← F.map_comp]
        dsimp
        congr 1 }
  hom_inv_id := by
    ext
    

Proof state:
case w
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι : Type w
U : ι → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U
⊢ (ConeMorphism.mk
          (F.map
            (eqToHom
              (_ :
                op (opensLeCoverCocone U).pt =
                  op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt))) ≫
        ConeMorphism.mk
          (F.map
            (eqToHom
              (_ :
                op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt =
                  op (opensLeCoverCocone U).pt)))).hom =
    (𝟙
        (Cone.whisker (Equivalence.op (generateEquivalenceOpensLe U hY)).functor
          (F.mapCone (Cocone.op (opensLeCoverCocone U))))).hom

Next tactic:
simp [eqToHom_map]
/-- Given a family of opens `opensLeCoverCocone U` is essentially the natural cocone
    associated to the sieve generated by the presieve associated to `U` with indexing
    category changed using the above equivalence. -/
@[simps]
def whiskerIsoMapGenerateCocone :
    (F.mapCone (opensLeCoverCocone U).op).whisker (generateEquivalenceOpensLe U hY).op.functor ≅
      F.mapCone (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op where
  hom :=
    { hom := F.map (eqToHom (congr_arg op hY.symm))
      w := fun j => by
        erw [← F.map_comp]
        dsimp
        congr 1 }
  inv :=
    { hom := F.map (eqToHom (congr_arg op hY))
      w := fun j => by
        erw [← F.map_comp]
        dsimp
        congr 1 }
  hom_inv_id := by
    ext
    simp [eqToHom_map]
  inv_hom_id := by
    

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι : Type w
U : ι → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U
⊢ ConeMorphism.mk
        (F.map
          (eqToHom
            (_ :
              op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt =
                op (opensLeCoverCocone U).pt))) ≫
      ConeMorphism.mk
        (F.map
          (eqToHom
            (_ :
              op (opensLeCoverCocone U).pt =
                op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt))) =
    𝟙 (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows)))

Next tactic:
ext
/-- Given a family of opens `opensLeCoverCocone U` is essentially the natural cocone
    associated to the sieve generated by the presieve associated to `U` with indexing
    category changed using the above equivalence. -/
@[simps]
def whiskerIsoMapGenerateCocone :
    (F.mapCone (opensLeCoverCocone U).op).whisker (generateEquivalenceOpensLe U hY).op.functor ≅
      F.mapCone (Sieve.generate (presieveOfCoveringAux U Y)).arrows.cocone.op where
  hom :=
    { hom := F.map (eqToHom (congr_arg op hY.symm))
      w := fun j => by
        erw [← F.map_comp]
        dsimp
        congr 1 }
  inv :=
    { hom := F.map (eqToHom (congr_arg op hY))
      w := fun j => by
        erw [← F.map_comp]
        dsimp
        congr 1 }
  hom_inv_id := by
    ext
    simp [eqToHom_map]
  inv_hom_id := by
    ext
    

Proof state:
case w
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι : Type w
U : ι → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U
⊢ (ConeMorphism.mk
          (F.map
            (eqToHom
              (_ :
                op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt =
                  op (opensLeCoverCocone U).pt))) ≫
        ConeMorphism.mk
          (F.map
            (eqToHom
              (_ :
                op (opensLeCoverCocone U).pt =
                  op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows).pt)))).hom =
    (𝟙 (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U Y)).arrows)))).hom

Next tactic:
simp [eqToHom_map]
/-- Given a presheaf `F` on the topological space `X` and a presieve `R` whose generated sieve
    is covering for the associated Grothendieck topology (equivalently, the presieve is covering
    for the associated pretopology), the natural cone associated to `F` and the family of opens
    associated to `R` is a limit cone iff the natural cone associated to `F` and the generated
    sieve is a limit cone.
    Since only the existence of a 1-1 correspondence will be used, the exact definition does
    not matter, so tactics are used liberally. -/
def isLimitOpensLeEquivGenerate₂ (R : Presieve Y)
    (hR : Sieve.generate R ∈ Opens.grothendieckTopology X Y) :
    IsLimit (F.mapCone (opensLeCoverCocone (coveringOfPresieve Y R)).op) ≃
      IsLimit (F.mapCone (Sieve.generate R).arrows.cocone.op) := by
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι : Type w
U : ι → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U
R : Presieve Y
hR : Sieve.generate R ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) Y
⊢ IsLimit (F.mapCone (Cocone.op (opensLeCoverCocone (coveringOfPresieve Y R)))) ≃
    IsLimit (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate R).arrows)))

Next tactic:
convert isLimitOpensLeEquivGenerate₁ F (coveringOfPresieve Y R)
      (coveringOfPresieve.iSup_eq_of_mem_grothendieck Y R hR).symm using 1
/-- Given a presheaf `F` on the topological space `X` and a presieve `R` whose generated sieve
    is covering for the associated Grothendieck topology (equivalently, the presieve is covering
    for the associated pretopology), the natural cone associated to `F` and the family of opens
    associated to `R` is a limit cone iff the natural cone associated to `F` and the generated
    sieve is a limit cone.
    Since only the existence of a 1-1 correspondence will be used, the exact definition does
    not matter, so tactics are used liberally. -/
def isLimitOpensLeEquivGenerate₂ (R : Presieve Y)
    (hR : Sieve.generate R ∈ Opens.grothendieckTopology X Y) :
    IsLimit (F.mapCone (opensLeCoverCocone (coveringOfPresieve Y R)).op) ≃
      IsLimit (F.mapCone (Sieve.generate R).arrows.cocone.op) := by
  convert isLimitOpensLeEquivGenerate₁ F (coveringOfPresieve Y R)
      (coveringOfPresieve.iSup_eq_of_mem_grothendieck Y R hR).symm using 1
  

Proof state:
case h.e'_2
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι : Type w
U : ι → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U
R : Presieve Y
hR : Sieve.generate R ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) Y
⊢ IsLimit (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate R).arrows))) =
    IsLimit
      (F.mapCone
        (Cocone.op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux (coveringOfPresieve Y R) Y)).arrows)))

Next tactic:
rw [covering_presieve_eq_self R]
/-- A presheaf `(opens X)ᵒᵖ ⥤ C` on a topological space `X` is a sheaf on the site `opens X` iff
    it satisfies the `IsSheafOpensLeCover` sheaf condition. The latter is not the
    official definition of sheaves on spaces, but has the advantage that it does not
    require `has_products C`. -/
theorem isSheaf_iff_isSheafOpensLeCover : F.IsSheaf ↔ F.IsSheafOpensLeCover := by
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι : Type w
U : ι → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U
⊢ IsSheaf F ↔ IsSheafOpensLeCover F

Next tactic:
refine' (Presheaf.isSheaf_iff_isLimit _ _).trans _
/-- A presheaf `(opens X)ᵒᵖ ⥤ C` on a topological space `X` is a sheaf on the site `opens X` iff
    it satisfies the `IsSheafOpensLeCover` sheaf condition. The latter is not the
    official definition of sheaves on spaces, but has the advantage that it does not
    require `has_products C`. -/
theorem isSheaf_iff_isSheafOpensLeCover : F.IsSheaf ↔ F.IsSheafOpensLeCover := by
  refine' (Presheaf.isSheaf_iff_isLimit _ _).trans _
  

Proof state:
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι : Type w
U : ι → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U
⊢ (∀ ⦃X_1 : Opens ↑X⦄,
      ∀ S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) X_1,
        Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows))))) ↔
    IsSheafOpensLeCover F

Next tactic:
constructor
/-- A presheaf `(opens X)ᵒᵖ ⥤ C` on a topological space `X` is a sheaf on the site `opens X` iff
    it satisfies the `IsSheafOpensLeCover` sheaf condition. The latter is not the
    official definition of sheaves on spaces, but has the advantage that it does not
    require `has_products C`. -/
theorem isSheaf_iff_isSheafOpensLeCover : F.IsSheaf ↔ F.IsSheafOpensLeCover := by
  refine' (Presheaf.isSheaf_iff_isLimit _ _).trans _
  constructor
  · 

Proof state:
case mp
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι : Type w
U : ι → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U
⊢ (∀ ⦃X_1 : Opens ↑X⦄,
      ∀ S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) X_1,
        Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows))))) →
    IsSheafOpensLeCover F

Next tactic:
intro h ι U
/-- A presheaf `(opens X)ᵒᵖ ⥤ C` on a topological space `X` is a sheaf on the site `opens X` iff
    it satisfies the `IsSheafOpensLeCover` sheaf condition. The latter is not the
    official definition of sheaves on spaces, but has the advantage that it does not
    require `has_products C`. -/
theorem isSheaf_iff_isSheafOpensLeCover : F.IsSheaf ↔ F.IsSheafOpensLeCover := by
  refine' (Presheaf.isSheaf_iff_isLimit _ _).trans _
  constructor
  · intro h ι U
    

Proof state:
case mp
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι✝ : Type w
U✝ : ι✝ → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U✝
h :
  ∀ ⦃X_1 : Opens ↑X⦄,
    ∀ S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) X_1,
      Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows))))
ι : Type w
U : ι → Opens ↑X
⊢ Nonempty (IsLimit (F.mapCone (Cocone.op (opensLeCoverCocone U))))

Next tactic:
rw [(isLimitOpensLeEquivGenerate₁ F U rfl).nonempty_congr]
/-- A presheaf `(opens X)ᵒᵖ ⥤ C` on a topological space `X` is a sheaf on the site `opens X` iff
    it satisfies the `IsSheafOpensLeCover` sheaf condition. The latter is not the
    official definition of sheaves on spaces, but has the advantage that it does not
    require `has_products C`. -/
theorem isSheaf_iff_isSheafOpensLeCover : F.IsSheaf ↔ F.IsSheafOpensLeCover := by
  refine' (Presheaf.isSheaf_iff_isLimit _ _).trans _
  constructor
  · intro h ι U
    rw [(isLimitOpensLeEquivGenerate₁ F U rfl).nonempty_congr]
    

Proof state:
case mp
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι✝ : Type w
U✝ : ι✝ → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U✝
h :
  ∀ ⦃X_1 : Opens ↑X⦄,
    ∀ S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) X_1,
      Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows))))
ι : Type w
U : ι → Opens ↑X
⊢ Nonempty
    (IsLimit (F.mapCone (Cocone.op (Presieve.cocone (Sieve.generate (presieveOfCoveringAux U (iSup U))).arrows))))

Next tactic:
apply h
/-- A presheaf `(opens X)ᵒᵖ ⥤ C` on a topological space `X` is a sheaf on the site `opens X` iff
    it satisfies the `IsSheafOpensLeCover` sheaf condition. The latter is not the
    official definition of sheaves on spaces, but has the advantage that it does not
    require `has_products C`. -/
theorem isSheaf_iff_isSheafOpensLeCover : F.IsSheaf ↔ F.IsSheafOpensLeCover := by
  refine' (Presheaf.isSheaf_iff_isLimit _ _).trans _
  constructor
  · intro h ι U
    rw [(isLimitOpensLeEquivGenerate₁ F U rfl).nonempty_congr]
    apply h
    

Proof state:
case mp.a
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι✝ : Type w
U✝ : ι✝ → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U✝
h :
  ∀ ⦃X_1 : Opens ↑X⦄,
    ∀ S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) X_1,
      Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows))))
ι : Type w
U : ι → Opens ↑X
⊢ Sieve.generate (presieveOfCoveringAux U (iSup U)) ∈
    GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) (iSup U)

Next tactic:
apply presieveOfCovering.mem_grothendieckTopology
/-- A presheaf `(opens X)ᵒᵖ ⥤ C` on a topological space `X` is a sheaf on the site `opens X` iff
    it satisfies the `IsSheafOpensLeCover` sheaf condition. The latter is not the
    official definition of sheaves on spaces, but has the advantage that it does not
    require `has_products C`. -/
theorem isSheaf_iff_isSheafOpensLeCover : F.IsSheaf ↔ F.IsSheafOpensLeCover := by
  refine' (Presheaf.isSheaf_iff_isLimit _ _).trans _
  constructor
  · intro h ι U
    rw [(isLimitOpensLeEquivGenerate₁ F U rfl).nonempty_congr]
    apply h
    apply presieveOfCovering.mem_grothendieckTopology
  · 

Proof state:
case mpr
C : Type u
inst✝ : Category.{v, u} C
X : TopCat
F : Presheaf C X
ι : Type w
U : ι → Opens ↑X
Y : Opens ↑X
hY : Y = iSup U
⊢ IsSheafOpensLeCover F →
    ∀ ⦃X_1 : Opens ↑X⦄,
      ∀ S ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) X_1,
        Nonempty (IsLimit (F.mapCone (Cocone.op (Presieve.cocone S.arrows))))

Next tactic:
intro h Y S