l3lab/ntp-mathlib huggingface dataset:
@[simp]
theorem div2_bit (b n) : div2 (bit b n) = n := by
  

Proof state:
a b✝ : ℤ
n✝ : ℕ
b : Bool
n : ℤ
⊢ div2 (bit b n) = n

Next tactic:
rw [bit_val, div2_val, add_comm, Int.add_mul_ediv_left, (_ : (_ / 2 : ℤ) = 0), zero_add]
@[simp]
theorem div2_bit (b n) : div2 (bit b n) = n := by
  rw [bit_val, div2_val, add_comm, Int.add_mul_ediv_left, (_ : (_ / 2 : ℤ) = 0), zero_add]
  

Proof state:
a b✝ : ℤ
n✝ : ℕ
b : Bool
n : ℤ
⊢ (bif b then 1 else 0) / 2 = 0
case H a b✝ : ℤ n✝ : ℕ b : Bool n : ℤ ⊢ 2 ≠ 0

Next tactic:
cases b
@[simp]
theorem div2_bit (b n) : div2 (bit b n) = n := by
  rw [bit_val, div2_val, add_comm, Int.add_mul_ediv_left, (_ : (_ / 2 : ℤ) = 0), zero_add]
  cases b
  · 

Proof state:
case false
a b : ℤ
n✝ : ℕ
n : ℤ
⊢ (bif false then 1 else 0) / 2 = 0

Next tactic:
decide
@[simp]
theorem div2_bit (b n) : div2 (bit b n) = n := by
  rw [bit_val, div2_val, add_comm, Int.add_mul_ediv_left, (_ : (_ / 2 : ℤ) = 0), zero_add]
  cases b
  · decide
  · 

Proof state:
case true
a b : ℤ
n✝ : ℕ
n : ℤ
⊢ (bif true then 1 else 0) / 2 = 0

Next tactic:
show ofNat _ = _
@[simp]
theorem div2_bit (b n) : div2 (bit b n) = n := by
  rw [bit_val, div2_val, add_comm, Int.add_mul_ediv_left, (_ : (_ / 2 : ℤ) = 0), zero_add]
  cases b
  · decide
  · show ofNat _ = _
    

Proof state:
case true
a b : ℤ
n✝ : ℕ
n : ℤ
⊢ ofNat (1 / 2) = 0

Next tactic:
rw [Nat.div_eq_of_lt]
@[simp]
theorem div2_bit (b n) : div2 (bit b n) = n := by
  rw [bit_val, div2_val, add_comm, Int.add_mul_ediv_left, (_ : (_ / 2 : ℤ) = 0), zero_add]
  cases b
  · decide
  · show ofNat _ = _
    rw [Nat.div_eq_of_lt] <;> 

Proof state:
case true
a b : ℤ
n✝ : ℕ
n : ℤ
⊢ ofNat 0 = 0

Next tactic:
simp
@[simp]
theorem div2_bit (b n) : div2 (bit b n) = n := by
  rw [bit_val, div2_val, add_comm, Int.add_mul_ediv_left, (_ : (_ / 2 : ℤ) = 0), zero_add]
  cases b
  · decide
  · show ofNat _ = _
    rw [Nat.div_eq_of_lt] <;> 

Proof state:
case true
a b : ℤ
n✝ : ℕ
n : ℤ
⊢ 1 < 2

Next tactic:
simp
@[simp]
theorem div2_bit (b n) : div2 (bit b n) = n := by
  rw [bit_val, div2_val, add_comm, Int.add_mul_ediv_left, (_ : (_ / 2 : ℤ) = 0), zero_add]
  cases b
  · decide
  · show ofNat _ = _
    rw [Nat.div_eq_of_lt] <;> simp
  · 

Proof state:
case H
a b✝ : ℤ
n✝ : ℕ
b : Bool
n : ℤ
⊢ 2 ≠ 0

Next tactic:
decide
/-- 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_inverse' :
    OpensLeCover U ⥤
    (FullSubcategory fun f : Over Y =>
      (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom) where
  obj := fun V => ⟨⟨V.obj, ⟨⟨⟩⟩, homOfLE <| hY ▸ (V.2.choose_spec.trans (le_iSup U (V.2.choose)))⟩,
    ⟨U V.2.choose, V.2.choose_spec.hom, homOfLE <| hY ▸ le_iSup U V.2.choose,
      ⟨V.2.choose, rfl⟩, rfl⟩⟩
  map {_ _} g := Over.homMk g
  map_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
x✝ : OpensLeCover U
⊢ {
          obj := fun V =>
            {
              obj :=
                { left := V.obj, right := { as := PUnit.unit },
                  hom := homOfLE (_ : (𝟭 (Opens ↑X)).obj V.obj ≤ (Functor.fromPUnit Y).obj { as := PUnit.unit }) },
              property :=
                (_ :
                  ∃ Y_1 h g,
                    presieveOfCoveringAux U Y g ∧
                      h ≫ g =
                        { left := V.obj, right := { as := PUnit.unit },
                            hom :=
                              homOfLE
                                (_ :
                                  (𝟭 (Opens ↑X)).obj V.obj ≤ (Functor.fromPUnit Y).obj { as := PUnit.unit }) }.hom) },
          map := fun {x x_1} g => Over.homMk g }.map
      (𝟙 x✝) =
    𝟙
      ({
            obj := fun V =>
              {
                obj :=
                  { left := V.obj, right := { as := PUnit.unit },
                    hom := homOfLE (_ : (𝟭 (Opens ↑X)).obj V.obj ≤ (Functor.fromPUnit Y).obj { as := PUnit.unit }) },
                property :=
                  (_ :
                    ∃ Y_1 h g,
                      presieveOfCoveringAux U Y g ∧
                        h ≫ g =
                          { left := V.obj, right := { as := PUnit.unit },
                              hom :=
                                homOfLE
                                  (_ :
                                    (𝟭 (Opens ↑X)).obj V.obj ≤ (Functor.fromPUnit Y).obj { as := PUnit.unit }) }.hom) },
            map := fun {x x_1} g => Over.homMk g }.obj
        x✝)

Next tactic:
refine Over.OverMorphism.ext ?_
/-- 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_inverse' :
    OpensLeCover U ⥤
    (FullSubcategory fun f : Over Y =>
      (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom) where
  obj := fun V => ⟨⟨V.obj, ⟨⟨⟩⟩, homOfLE <| hY ▸ (V.2.choose_spec.trans (le_iSup U (V.2.choose)))⟩,
    ⟨U V.2.choose, V.2.choose_spec.hom, homOfLE <| hY ▸ le_iSup U V.2.choose,
      ⟨V.2.choose, rfl⟩, rfl⟩⟩
  map {_ _} g := Over.homMk g
  map_id _ := by
    refine Over.OverMorphism.ext ?_
    

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✝ : OpensLeCover U
⊢ ({
            obj := fun V =>
              {
                obj :=
                  { left := V.obj, right := { as := PUnit.unit },
                    hom := homOfLE (_ : (𝟭 (Opens ↑X)).obj V.obj ≤ (Functor.fromPUnit Y).obj { as := PUnit.unit }) },
                property :=
                  (_ :
                    ∃ Y_1 h g,
                      presieveOfCoveringAux U Y g ∧
                        h ≫ g =
                          { left := V.obj, right := { as := PUnit.unit },
                              hom :=
                                homOfLE
                                  (_ :
                                    (𝟭 (Opens ↑X)).obj V.obj ≤ (Functor.fromPUnit Y).obj { as := PUnit.unit }) }.hom) },
            map := fun {x x_1} g => Over.homMk g }.map
        (𝟙 x✝)).left =
    (𝟙
        ({
              obj := fun V =>
                {
                  obj :=
                    { left := V.obj, right := { as := PUnit.unit },
                      hom := homOfLE (_ : (𝟭 (Opens ↑X)).obj V.obj ≤ (Functor.fromPUnit Y).obj { as := PUnit.unit }) },
                  property :=
                    (_ :
                      ∃ Y_1 h g,
                        presieveOfCoveringAux U Y g ∧
                          h ≫ g =
                            { left := V.obj, right := { as := PUnit.unit },
                                hom :=
                                  homOfLE
                                    (_ :
                                      (𝟭 (Opens ↑X)).obj V.obj ≤
                                        (Functor.fromPUnit Y).obj { as := PUnit.unit }) }.hom) },
              map := fun {x x_1} g => Over.homMk g }.obj
          x✝)).left

Next tactic:
simp only [Functor.id_obj, Sieve.generate_apply, Functor.const_obj_obj, Over.homMk_left,
      eq_iff_true_of_subsingleton]
/-- 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 

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 : FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom),
    (𝟭 (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).obj X_1 =
      (generateEquivalenceOpensLe_functor' U ?m.48360 ⋙ generateEquivalenceOpensLe_inverse' U ?m.48407).obj X_1

Next tactic:
rintro ⟨⟨_, _⟩, _⟩
/-- 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 ⟨⟨_, _⟩, _⟩; 

Proof state:
case mk.mk
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
left✝ : Opens ↑X
right✝ : Discrete PUnit.{1}
hom✝ : (𝟭 (Opens ↑X)).obj left✝ ⟶ (Functor.fromPUnit Y).obj right✝
property✝ : (Sieve.generate (presieveOfCoveringAux U Y)).arrows { left := left✝, right := right✝, hom := hom✝ }.hom
⊢ (𝟭 (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).obj
      { obj := { left := left✝, right := right✝, hom := hom✝ }, property := property✝ } =
    (generateEquivalenceOpensLe_functor' U ?m.48360 ⋙ generateEquivalenceOpensLe_inverse' U ?m.48407).obj
      { obj := { left := left✝, right := right✝, hom := hom✝ }, property := property✝ }

Next tactic:
dsimp
/-- 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; 

Proof state:
case mk.mk
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
left✝ : Opens ↑X
right✝ : Discrete PUnit.{1}
hom✝ : (𝟭 (Opens ↑X)).obj left✝ ⟶ (Functor.fromPUnit Y).obj right✝
property✝ : (Sieve.generate (presieveOfCoveringAux U Y)).arrows { left := left✝, right := right✝, hom := hom✝ }.hom
⊢ { obj := { left := left✝, right := right✝, hom := hom✝ }, property := property✝ } =
    (generateEquivalenceOpensLe_inverse' U ?m.48407).obj
      ((generateEquivalenceOpensLe_functor' U ?m.48360).obj
        { obj := { left := left✝, right := right✝, hom := hom✝ }, property := property✝ })

Next tactic:
congr
/-- 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 

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 Y_1 : FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom) (f : X_1 ⟶ Y_1),
    (𝟭 (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).map f =
      eqToHom
          (_ :
            (𝟭 (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).obj X_1 =
              (generateEquivalenceOpensLe_functor' U ?m.48360 ⋙ generateEquivalenceOpensLe_inverse' U ?m.48407).obj
                X_1) ≫
        (generateEquivalenceOpensLe_functor' U ?m.48360 ⋙ generateEquivalenceOpensLe_inverse' U ?m.48407).map f ≫
          eqToHom
            (_ :
              (generateEquivalenceOpensLe_functor' U ?m.48360 ⋙ generateEquivalenceOpensLe_inverse' U ?m.48407).obj
                  Y_1 =
                (𝟭 (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).obj Y_1)

Next tactic:
intros
/-- 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; 

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✝ : FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom
f✝ : X✝ ⟶ Y✝
⊢ (𝟭 (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).map f✝ =
    eqToHom
        (_ :
          (𝟭 (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).obj X✝ =
            (generateEquivalenceOpensLe_functor' U ?m.48360 ⋙ generateEquivalenceOpensLe_inverse' U ?m.48407).obj X✝) ≫
      (generateEquivalenceOpensLe_functor' U ?m.48360 ⋙ generateEquivalenceOpensLe_inverse' U ?m.48407).map f✝ ≫
        eqToHom
          (_ :
            (generateEquivalenceOpensLe_functor' U ?m.48360 ⋙ generateEquivalenceOpensLe_inverse' U ?m.48407).obj Y✝ =
              (𝟭 (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).obj Y✝)

Next tactic:
refine Over.OverMorphism.ext ?_
/-- 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 ?_; 

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✝ : FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom
f✝ : X✝ ⟶ Y✝
⊢ ((𝟭 (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).map f✝).left =
    (eqToHom
          (_ :
            (𝟭 (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).obj X✝ =
              (generateEquivalenceOpensLe_functor' U ?m.48360 ⋙ generateEquivalenceOpensLe_inverse' U ?m.48407).obj
                X✝) ≫
        (generateEquivalenceOpensLe_functor' U ?m.48360 ⋙ generateEquivalenceOpensLe_inverse' U ?m.48407).map f✝ ≫
          eqToHom
            (_ :
              (generateEquivalenceOpensLe_functor' U ?m.48360 ⋙ generateEquivalenceOpensLe_inverse' U ?m.48407).obj Y✝ =
                (𝟭 (FullSubcategory fun f => (Sieve.generate (presieveOfCoveringAux U Y)).arrows f.hom)).obj Y✝)).left

Next tactic:
aesop_cat
/-- 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 

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 : OpensLeCover U),
    (generateEquivalenceOpensLe_inverse' U (_ : Y = iSup U) ⋙
            generateEquivalenceOpensLe_functor' U (_ : Y = iSup U)).obj
        X_1 =
      (𝟭 (OpensLeCover U)).obj X_1

Next tactic:
intro
/-- 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; 

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✝ : OpensLeCover U
⊢ (generateEquivalenceOpensLe_inverse' U (_ : Y = iSup U) ⋙ generateEquivalenceOpensLe_functor' U (_ : Y = iSup U)).obj
      X✝ =
    (𝟭 (OpensLeCover U)).obj X✝

Next tactic:
refine FullSubcategory.ext _ _ ?_
/-- 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 _ _ ?_; 

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✝ : OpensLeCover U
⊢ ((generateEquivalenceOpensLe_inverse' U (_ : Y = iSup U) ⋙ generateEquivalenceOpensLe_functor' U (_ : Y = iSup U)).obj
        X✝).obj =
    ((𝟭 (OpensLeCover U)).obj X✝).obj

Next tactic:
rfl
/-- 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 

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 Y_1 : OpensLeCover U) (f : X_1 ⟶ Y_1),
    HEq
      ((generateEquivalenceOpensLe_inverse' U (_ : Y = iSup U) ⋙
            generateEquivalenceOpensLe_functor' U (_ : Y = iSup U)).map
        f)
      ((𝟭 (OpensLeCover U)).map f)

Next tactic:
intros