l3lab/ntp-mathlib huggingface dataset:
theorem le_iff_sign {x y : EReal} :
    x ≤ y ↔ sign x < sign y ∨
      sign x = SignType.neg ∧ sign y = SignType.neg ∧ y.abs ≤ x.abs ∨
        sign x = SignType.zero ∧ sign y = SignType.zero ∨
          sign x = SignType.pos ∧ sign y = SignType.pos ∧ x.abs ≤ y.abs := by
  constructor
  · intro h
    refine (sign.monotone h).lt_or_eq.imp_right (fun hs => ?_)
    rw [← x.sign_mul_abs, ← y.sign_mul_abs] at h
    cases hy : sign y <;> rw [hs, hy] at h ⊢
    · simp
    · left; simpa using h
    · 

Proof state:
case mp.pos
x y : EReal
h : ↑SignType.pos * ↑(EReal.abs x) ≤ ↑SignType.pos * ↑(EReal.abs y)
hs : sign x = sign y
hy : sign y = SignType.pos
⊢ SignType.pos = SignType.neg ∧ SignType.pos = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨
    SignType.pos = SignType.zero ∧ SignType.pos = SignType.zero ∨
      SignType.pos = SignType.pos ∧ SignType.pos = SignType.pos ∧ EReal.abs x ≤ EReal.abs y

Next tactic:
right
theorem le_iff_sign {x y : EReal} :
    x ≤ y ↔ sign x < sign y ∨
      sign x = SignType.neg ∧ sign y = SignType.neg ∧ y.abs ≤ x.abs ∨
        sign x = SignType.zero ∧ sign y = SignType.zero ∨
          sign x = SignType.pos ∧ sign y = SignType.pos ∧ x.abs ≤ y.abs := by
  constructor
  · intro h
    refine (sign.monotone h).lt_or_eq.imp_right (fun hs => ?_)
    rw [← x.sign_mul_abs, ← y.sign_mul_abs] at h
    cases hy : sign y <;> rw [hs, hy] at h ⊢
    · simp
    · left; simpa using h
    · right; 

Proof state:
case mp.pos.h
x y : EReal
h : ↑SignType.pos * ↑(EReal.abs x) ≤ ↑SignType.pos * ↑(EReal.abs y)
hs : sign x = sign y
hy : sign y = SignType.pos
⊢ SignType.pos = SignType.zero ∧ SignType.pos = SignType.zero ∨
    SignType.pos = SignType.pos ∧ SignType.pos = SignType.pos ∧ EReal.abs x ≤ EReal.abs y

Next tactic:
right
theorem le_iff_sign {x y : EReal} :
    x ≤ y ↔ sign x < sign y ∨
      sign x = SignType.neg ∧ sign y = SignType.neg ∧ y.abs ≤ x.abs ∨
        sign x = SignType.zero ∧ sign y = SignType.zero ∨
          sign x = SignType.pos ∧ sign y = SignType.pos ∧ x.abs ≤ y.abs := by
  constructor
  · intro h
    refine (sign.monotone h).lt_or_eq.imp_right (fun hs => ?_)
    rw [← x.sign_mul_abs, ← y.sign_mul_abs] at h
    cases hy : sign y <;> rw [hs, hy] at h ⊢
    · simp
    · left; simpa using h
    · right; right; 

Proof state:
case mp.pos.h.h
x y : EReal
h : ↑SignType.pos * ↑(EReal.abs x) ≤ ↑SignType.pos * ↑(EReal.abs y)
hs : sign x = sign y
hy : sign y = SignType.pos
⊢ SignType.pos = SignType.pos ∧ SignType.pos = SignType.pos ∧ EReal.abs x ≤ EReal.abs y

Next tactic:
simpa using h
theorem le_iff_sign {x y : EReal} :
    x ≤ y ↔ sign x < sign y ∨
      sign x = SignType.neg ∧ sign y = SignType.neg ∧ y.abs ≤ x.abs ∨
        sign x = SignType.zero ∧ sign y = SignType.zero ∨
          sign x = SignType.pos ∧ sign y = SignType.pos ∧ x.abs ≤ y.abs := by
  constructor
  · intro h
    refine (sign.monotone h).lt_or_eq.imp_right (fun hs => ?_)
    rw [← x.sign_mul_abs, ← y.sign_mul_abs] at h
    cases hy : sign y <;> rw [hs, hy] at h ⊢
    · simp
    · left; simpa using h
    · right; right; simpa using h
  · 

Proof state:
case mpr
x y : EReal
⊢ sign x < sign y ∨
      sign x = SignType.neg ∧ sign y = SignType.neg ∧ EReal.abs y ≤ EReal.abs x ∨
        sign x = SignType.zero ∧ sign y = SignType.zero ∨
          sign x = SignType.pos ∧ sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y →
    x ≤ y

Next tactic:
rintro (h | h | h | h)
theorem le_iff_sign {x y : EReal} :
    x ≤ y ↔ sign x < sign y ∨
      sign x = SignType.neg ∧ sign y = SignType.neg ∧ y.abs ≤ x.abs ∨
        sign x = SignType.zero ∧ sign y = SignType.zero ∨
          sign x = SignType.pos ∧ sign y = SignType.pos ∧ x.abs ≤ y.abs := by
  constructor
  · intro h
    refine (sign.monotone h).lt_or_eq.imp_right (fun hs => ?_)
    rw [← x.sign_mul_abs, ← y.sign_mul_abs] at h
    cases hy : sign y <;> rw [hs, hy] at h ⊢
    · simp
    · left; simpa using h
    · right; right; simpa using h
  · rintro (h | h | h | h)
    · 

Proof state:
case mpr.inl
x y : EReal
h : sign x < sign y
⊢ x ≤ y

Next tactic:
exact (sign.monotone.reflect_lt h).le
theorem le_iff_sign {x y : EReal} :
    x ≤ y ↔ sign x < sign y ∨
      sign x = SignType.neg ∧ sign y = SignType.neg ∧ y.abs ≤ x.abs ∨
        sign x = SignType.zero ∧ sign y = SignType.zero ∨
          sign x = SignType.pos ∧ sign y = SignType.pos ∧ x.abs ≤ y.abs := by
  constructor
  · intro h
    refine (sign.monotone h).lt_or_eq.imp_right (fun hs => ?_)
    rw [← x.sign_mul_abs, ← y.sign_mul_abs] at h
    cases hy : sign y <;> rw [hs, hy] at h ⊢
    · simp
    · left; simpa using h
    · right; right; simpa using h
  · rintro (h | h | h | h)
    · exact (sign.monotone.reflect_lt h).le
    

Proof state:
case mpr.inr.inl
x y : EReal
h : sign x = SignType.neg ∧ sign y = SignType.neg ∧ EReal.abs y ≤ EReal.abs x
⊢ x ≤ y
case mpr.inr.inr.inl
x y : EReal
h : sign x = SignType.zero ∧ sign y = SignType.zero
⊢ x ≤ y
case mpr.inr.inr.inr x y : EReal h : sign x = SignType.pos ∧ sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y ⊢ x ≤ y

Next tactic:
all_goals rw [← x.sign_mul_abs, ← y.sign_mul_abs]; simp [h]
theorem le_iff_sign {x y : EReal} :
    x ≤ y ↔ sign x < sign y ∨
      sign x = SignType.neg ∧ sign y = SignType.neg ∧ y.abs ≤ x.abs ∨
        sign x = SignType.zero ∧ sign y = SignType.zero ∨
          sign x = SignType.pos ∧ sign y = SignType.pos ∧ x.abs ≤ y.abs := by
  constructor
  · intro h
    refine (sign.monotone h).lt_or_eq.imp_right (fun hs => ?_)
    rw [← x.sign_mul_abs, ← y.sign_mul_abs] at h
    cases hy : sign y <;> rw [hs, hy] at h ⊢
    · simp
    · left; simpa using h
    · right; right; simpa using h
  · rintro (h | h | h | h)
    · exact (sign.monotone.reflect_lt h).le
    all_goals 

Proof state:
case mpr.inr.inl
x y : EReal
h : sign x = SignType.neg ∧ sign y = SignType.neg ∧ EReal.abs y ≤ EReal.abs x
⊢ x ≤ y

Next tactic:
rw [← x.sign_mul_abs, ← y.sign_mul_abs]
theorem le_iff_sign {x y : EReal} :
    x ≤ y ↔ sign x < sign y ∨
      sign x = SignType.neg ∧ sign y = SignType.neg ∧ y.abs ≤ x.abs ∨
        sign x = SignType.zero ∧ sign y = SignType.zero ∨
          sign x = SignType.pos ∧ sign y = SignType.pos ∧ x.abs ≤ y.abs := by
  constructor
  · intro h
    refine (sign.monotone h).lt_or_eq.imp_right (fun hs => ?_)
    rw [← x.sign_mul_abs, ← y.sign_mul_abs] at h
    cases hy : sign y <;> rw [hs, hy] at h ⊢
    · simp
    · left; simpa using h
    · right; right; simpa using h
  · rintro (h | h | h | h)
    · exact (sign.monotone.reflect_lt h).le
    all_goals rw [← x.sign_mul_abs, ← y.sign_mul_abs]; 

Proof state:
case mpr.inr.inl
x y : EReal
h : sign x = SignType.neg ∧ sign y = SignType.neg ∧ EReal.abs y ≤ EReal.abs x
⊢ ↑(sign x) * ↑(EReal.abs x) ≤ ↑(sign y) * ↑(EReal.abs y)

Next tactic:
simp [h]
theorem le_iff_sign {x y : EReal} :
    x ≤ y ↔ sign x < sign y ∨
      sign x = SignType.neg ∧ sign y = SignType.neg ∧ y.abs ≤ x.abs ∨
        sign x = SignType.zero ∧ sign y = SignType.zero ∨
          sign x = SignType.pos ∧ sign y = SignType.pos ∧ x.abs ≤ y.abs := by
  constructor
  · intro h
    refine (sign.monotone h).lt_or_eq.imp_right (fun hs => ?_)
    rw [← x.sign_mul_abs, ← y.sign_mul_abs] at h
    cases hy : sign y <;> rw [hs, hy] at h ⊢
    · simp
    · left; simpa using h
    · right; right; simpa using h
  · rintro (h | h | h | h)
    · exact (sign.monotone.reflect_lt h).le
    all_goals 

Proof state:
case mpr.inr.inr.inl
x y : EReal
h : sign x = SignType.zero ∧ sign y = SignType.zero
⊢ x ≤ y

Next tactic:
rw [← x.sign_mul_abs, ← y.sign_mul_abs]
theorem le_iff_sign {x y : EReal} :
    x ≤ y ↔ sign x < sign y ∨
      sign x = SignType.neg ∧ sign y = SignType.neg ∧ y.abs ≤ x.abs ∨
        sign x = SignType.zero ∧ sign y = SignType.zero ∨
          sign x = SignType.pos ∧ sign y = SignType.pos ∧ x.abs ≤ y.abs := by
  constructor
  · intro h
    refine (sign.monotone h).lt_or_eq.imp_right (fun hs => ?_)
    rw [← x.sign_mul_abs, ← y.sign_mul_abs] at h
    cases hy : sign y <;> rw [hs, hy] at h ⊢
    · simp
    · left; simpa using h
    · right; right; simpa using h
  · rintro (h | h | h | h)
    · exact (sign.monotone.reflect_lt h).le
    all_goals rw [← x.sign_mul_abs, ← y.sign_mul_abs]; 

Proof state:
case mpr.inr.inr.inl
x y : EReal
h : sign x = SignType.zero ∧ sign y = SignType.zero
⊢ ↑(sign x) * ↑(EReal.abs x) ≤ ↑(sign y) * ↑(EReal.abs y)

Next tactic:
simp [h]
theorem le_iff_sign {x y : EReal} :
    x ≤ y ↔ sign x < sign y ∨
      sign x = SignType.neg ∧ sign y = SignType.neg ∧ y.abs ≤ x.abs ∨
        sign x = SignType.zero ∧ sign y = SignType.zero ∨
          sign x = SignType.pos ∧ sign y = SignType.pos ∧ x.abs ≤ y.abs := by
  constructor
  · intro h
    refine (sign.monotone h).lt_or_eq.imp_right (fun hs => ?_)
    rw [← x.sign_mul_abs, ← y.sign_mul_abs] at h
    cases hy : sign y <;> rw [hs, hy] at h ⊢
    · simp
    · left; simpa using h
    · right; right; simpa using h
  · rintro (h | h | h | h)
    · exact (sign.monotone.reflect_lt h).le
    all_goals 

Proof state:
case mpr.inr.inr.inr
x y : EReal
h : sign x = SignType.pos ∧ sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y
⊢ x ≤ y

Next tactic:
rw [← x.sign_mul_abs, ← y.sign_mul_abs]
theorem le_iff_sign {x y : EReal} :
    x ≤ y ↔ sign x < sign y ∨
      sign x = SignType.neg ∧ sign y = SignType.neg ∧ y.abs ≤ x.abs ∨
        sign x = SignType.zero ∧ sign y = SignType.zero ∨
          sign x = SignType.pos ∧ sign y = SignType.pos ∧ x.abs ≤ y.abs := by
  constructor
  · intro h
    refine (sign.monotone h).lt_or_eq.imp_right (fun hs => ?_)
    rw [← x.sign_mul_abs, ← y.sign_mul_abs] at h
    cases hy : sign y <;> rw [hs, hy] at h ⊢
    · simp
    · left; simpa using h
    · right; right; simpa using h
  · rintro (h | h | h | h)
    · exact (sign.monotone.reflect_lt h).le
    all_goals rw [← x.sign_mul_abs, ← y.sign_mul_abs]; 

Proof state:
case mpr.inr.inr.inr
x y : EReal
h : sign x = SignType.pos ∧ sign y = SignType.pos ∧ EReal.abs x ≤ EReal.abs y
⊢ ↑(sign x) * ↑(EReal.abs x) ≤ ↑(sign y) * ↑(EReal.abs y)

Next tactic:
simp [h]
instance : CommMonoidWithZero EReal :=
  { inferInstanceAs (MulZeroOneClass EReal) with
    mul_assoc := fun x y z => by
      

Proof state:
src✝ : MulZeroOneClass EReal := inferInstanceAs (MulZeroOneClass EReal)
x y z : EReal
⊢ x * y * z = x * (y * z)

Next tactic:
rw [← sign_eq_and_abs_eq_iff_eq]
instance : CommMonoidWithZero EReal :=
  { inferInstanceAs (MulZeroOneClass EReal) with
    mul_assoc := fun x y z => by
      rw [← sign_eq_and_abs_eq_iff_eq]
      

Proof state:
src✝ : MulZeroOneClass EReal := inferInstanceAs (MulZeroOneClass EReal)
x y z : EReal
⊢ EReal.abs (x * y * z) = EReal.abs (x * (y * z)) ∧ sign (x * y * z) = sign (x * (y * z))

Next tactic:
simp only [mul_assoc, abs_mul, eq_self_iff_true, sign_mul, and_self_iff]
instance : PosMulMono EReal := posMulMono_iff_covariant_pos.2 <| .mk <| by
  

Proof state:
⊢ Covariant { x // 0 < x } EReal (fun x y => ↑x * y) fun x x_1 => x ≤ x_1

Next tactic:
rintro ⟨x, x0⟩ a b h
instance : PosMulMono EReal := posMulMono_iff_covariant_pos.2 <| .mk <| by
  rintro ⟨x, x0⟩ a b h
  

Proof state:
case mk
x : EReal
x0 : 0 < x
a b : EReal
h : a ≤ b
⊢ (fun x y => ↑x * y) { val := x, property := x0 } a ≤ (fun x y => ↑x * y) { val := x, property := x0 } b

Next tactic:
simp only [le_iff_sign, EReal.sign_mul, sign_pos x0, one_mul, EReal.abs_mul] at h ⊢
instance : PosMulMono EReal := posMulMono_iff_covariant_pos.2 <| .mk <| by
  rintro ⟨x, x0⟩ a b h
  simp only [le_iff_sign, EReal.sign_mul, sign_pos x0, one_mul, EReal.abs_mul] at h ⊢
  

Proof state:
case mk
x : EReal
x0 : 0 < x
a b : EReal
h :
  sign a < sign b ∨
    sign a = SignType.neg ∧ sign b = SignType.neg ∧ EReal.abs b ≤ EReal.abs a ∨
      sign a = SignType.zero ∧ sign b = SignType.zero ∨
        sign a = SignType.pos ∧ sign b = SignType.pos ∧ EReal.abs a ≤ EReal.abs b
⊢ sign a < sign b ∨
    sign a = SignType.neg ∧ sign b = SignType.neg ∧ EReal.abs x * EReal.abs b ≤ EReal.abs x * EReal.abs a ∨
      sign a = SignType.zero ∧ sign b = SignType.zero ∨
        sign a = SignType.pos ∧ sign b = SignType.pos ∧ EReal.abs x * EReal.abs a ≤ EReal.abs x * EReal.abs b

Next tactic:
exact h.imp_right <| Or.imp (And.imp_right <| And.imp_right (mul_le_mul_left' · _)) <|
    Or.imp_right <| And.imp_right <| And.imp_right (mul_le_mul_left' · _)
/-- Given a sieve `S` on `X : C`, a presheaf `P : Cᵒᵖ ⥤ A`, and an object `E` of `A`,
    the cones over the natural diagram `S.arrows.diagram.op ⋙ P` associated to `S` and `P`
    with cone point `E` are in 1-1 correspondence with sieve_compatible family of elements
    for the sieve `S` and the presheaf of types `Hom (E, P -)`. -/
@[simps]
def conesEquivSieveCompatibleFamily :
    (S.arrows.diagram.op ⋙ P).cones.obj E ≃
      { x : FamilyOfElements (P ⋙ coyoneda.obj E) (S : Presieve X) // x.SieveCompatible } where
  toFun π :=
    ⟨fun Y f h => π.app (op ⟨Over.mk f, h⟩), fun X Y f g hf => by
      

Proof state:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X✝ : C
S : Sieve X✝
R : Presieve X✝
E : Aᵒᵖ
π : (Functor.cones ((diagram S.arrows).op ⋙ P)).obj E
X Y : C
f : X ⟶ X✝
g : Y ⟶ X
hf : S.arrows f
⊢ (fun Y f h => π.app (op { obj := Over.mk f, property := h })) Y (g ≫ f) (_ : S.arrows (g ≫ f)) =
    (P ⋙ coyoneda.obj E).map g.op ((fun Y f h => π.app (op { obj := Over.mk f, property := h })) X f hf)

Next tactic:
apply (id_comp _).symm.trans
/-- Given a sieve `S` on `X : C`, a presheaf `P : Cᵒᵖ ⥤ A`, and an object `E` of `A`,
    the cones over the natural diagram `S.arrows.diagram.op ⋙ P` associated to `S` and `P`
    with cone point `E` are in 1-1 correspondence with sieve_compatible family of elements
    for the sieve `S` and the presheaf of types `Hom (E, P -)`. -/
@[simps]
def conesEquivSieveCompatibleFamily :
    (S.arrows.diagram.op ⋙ P).cones.obj E ≃
      { x : FamilyOfElements (P ⋙ coyoneda.obj E) (S : Presieve X) // x.SieveCompatible } where
  toFun π :=
    ⟨fun Y f h => π.app (op ⟨Over.mk f, h⟩), fun X Y f g hf => by
      apply (id_comp _).symm.trans
      

Proof state:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X✝ : C
S : Sieve X✝
R : Presieve X✝
E : Aᵒᵖ
π : (Functor.cones ((diagram S.arrows).op ⋙ P)).obj E
X Y : C
f : X ⟶ X✝
g : Y ⟶ X
hf : S.arrows f
⊢ 𝟙
        (((Functor.const (FullSubcategory fun f => S.arrows f.hom)ᵒᵖ).op.obj E).unop.obj
          (op { obj := Over.mk (g ≫ f), property := (_ : S.arrows (g ≫ f)) })) ≫
      (fun Y f h => π.app (op { obj := Over.mk f, property := h })) Y (g ≫ f) (_ : S.arrows (g ≫ f)) =
    (P ⋙ coyoneda.obj E).map g.op ((fun Y f h => π.app (op { obj := Over.mk f, property := h })) X f hf)

Next tactic:
dsimp
/-- Given a sieve `S` on `X : C`, a presheaf `P : Cᵒᵖ ⥤ A`, and an object `E` of `A`,
    the cones over the natural diagram `S.arrows.diagram.op ⋙ P` associated to `S` and `P`
    with cone point `E` are in 1-1 correspondence with sieve_compatible family of elements
    for the sieve `S` and the presheaf of types `Hom (E, P -)`. -/
@[simps]
def conesEquivSieveCompatibleFamily :
    (S.arrows.diagram.op ⋙ P).cones.obj E ≃
      { x : FamilyOfElements (P ⋙ coyoneda.obj E) (S : Presieve X) // x.SieveCompatible } where
  toFun π :=
    ⟨fun Y f h => π.app (op ⟨Over.mk f, h⟩), fun X Y f g hf => by
      apply (id_comp _).symm.trans
      dsimp
      

Proof state:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X✝ : C
S : Sieve X✝
R : Presieve X✝
E : Aᵒᵖ
π : (Functor.cones ((diagram S.arrows).op ⋙ P)).obj E
X Y : C
f : X ⟶ X✝
g : Y ⟶ X
hf : S.arrows f
⊢ 𝟙 E.unop ≫ π.app (op { obj := Over.mk (g ≫ f), property := (_ : S.arrows (g ≫ f)) }) =
    π.app (op { obj := Over.mk f, property := hf }) ≫ P.map g.op

Next tactic:
exact π.naturality (Quiver.Hom.op (Over.homMk _ (by rfl)))
/-- Given a sieve `S` on `X : C`, a presheaf `P : Cᵒᵖ ⥤ A`, and an object `E` of `A`,
    the cones over the natural diagram `S.arrows.diagram.op ⋙ P` associated to `S` and `P`
    with cone point `E` are in 1-1 correspondence with sieve_compatible family of elements
    for the sieve `S` and the presheaf of types `Hom (E, P -)`. -/
@[simps]
def conesEquivSieveCompatibleFamily :
    (S.arrows.diagram.op ⋙ P).cones.obj E ≃
      { x : FamilyOfElements (P ⋙ coyoneda.obj E) (S : Presieve X) // x.SieveCompatible } where
  toFun π :=
    ⟨fun Y f h => π.app (op ⟨Over.mk f, h⟩), fun X Y f g hf => by
      apply (id_comp _).symm.trans
      dsimp
      exact π.naturality (Quiver.Hom.op (Over.homMk _ (by 

Proof state:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X✝ : C
S : Sieve X✝
R : Presieve X✝
E : Aᵒᵖ
π : (Functor.cones ((diagram S.arrows).op ⋙ P)).obj E
X Y : C
f : X ⟶ X✝
g : Y ⟶ X
hf : S.arrows f
⊢ g ≫ { obj := Over.mk f, property := hf }.obj.hom =
    { obj := Over.mk (g ≫ f), property := (_ : S.arrows (g ≫ f)) }.obj.hom

Next tactic:
rfl
/-- Given a sieve `S` on `X : C`, a presheaf `P : Cᵒᵖ ⥤ A`, and an object `E` of `A`,
    the cones over the natural diagram `S.arrows.diagram.op ⋙ P` associated to `S` and `P`
    with cone point `E` are in 1-1 correspondence with sieve_compatible family of elements
    for the sieve `S` and the presheaf of types `Hom (E, P -)`. -/
@[simps]
def conesEquivSieveCompatibleFamily :
    (S.arrows.diagram.op ⋙ P).cones.obj E ≃
      { x : FamilyOfElements (P ⋙ coyoneda.obj E) (S : Presieve X) // x.SieveCompatible } where
  toFun π :=
    ⟨fun Y f h => π.app (op ⟨Over.mk f, h⟩), fun X Y f g hf => by
      apply (id_comp _).symm.trans
      dsimp
      exact π.naturality (Quiver.Hom.op (Over.homMk _ (by rfl)))⟩
  invFun x :=
    { app := fun f => x.1 f.unop.1.hom f.unop.2
      naturality := fun f f' g => by
        

Proof state:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E : Aᵒᵖ
x : { x // SieveCompatible x }
f f' : (FullSubcategory fun f => S.arrows f.hom)ᵒᵖ
g : f ⟶ f'
⊢ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)ᵒᵖ).op.obj E).unop.map g ≫
      (fun f => ↑x f.unop.obj.hom (_ : S.arrows f.unop.obj.hom)) f' =
    (fun f => ↑x f.unop.obj.hom (_ : S.arrows f.unop.obj.hom)) f ≫ ((diagram S.arrows).op ⋙ P).map g

Next tactic:
refine' Eq.trans _ (x.2 f.unop.1.hom g.unop.left f.unop.2)
/-- Given a sieve `S` on `X : C`, a presheaf `P : Cᵒᵖ ⥤ A`, and an object `E` of `A`,
    the cones over the natural diagram `S.arrows.diagram.op ⋙ P` associated to `S` and `P`
    with cone point `E` are in 1-1 correspondence with sieve_compatible family of elements
    for the sieve `S` and the presheaf of types `Hom (E, P -)`. -/
@[simps]
def conesEquivSieveCompatibleFamily :
    (S.arrows.diagram.op ⋙ P).cones.obj E ≃
      { x : FamilyOfElements (P ⋙ coyoneda.obj E) (S : Presieve X) // x.SieveCompatible } where
  toFun π :=
    ⟨fun Y f h => π.app (op ⟨Over.mk f, h⟩), fun X Y f g hf => by
      apply (id_comp _).symm.trans
      dsimp
      exact π.naturality (Quiver.Hom.op (Over.homMk _ (by rfl)))⟩
  invFun x :=
    { app := fun f => x.1 f.unop.1.hom f.unop.2
      naturality := fun f f' g => by
        refine' Eq.trans _ (x.2 f.unop.1.hom g.unop.left f.unop.2)
        

Proof state:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E : Aᵒᵖ
x : { x // SieveCompatible x }
f f' : (FullSubcategory fun f => S.arrows f.hom)ᵒᵖ
g : f ⟶ f'
⊢ ((Functor.const (FullSubcategory fun f => S.arrows f.hom)ᵒᵖ).op.obj E).unop.map g ≫
      (fun f => ↑x f.unop.obj.hom (_ : S.arrows f.unop.obj.hom)) f' =
    ↑x (g.unop.left ≫ f.unop.obj.hom) (_ : S.arrows (g.unop.left ≫ f.unop.obj.hom))

Next tactic:
dsimp
/-- Given a sieve `S` on `X : C`, a presheaf `P : Cᵒᵖ ⥤ A`, and an object `E` of `A`,
    the cones over the natural diagram `S.arrows.diagram.op ⋙ P` associated to `S` and `P`
    with cone point `E` are in 1-1 correspondence with sieve_compatible family of elements
    for the sieve `S` and the presheaf of types `Hom (E, P -)`. -/
@[simps]
def conesEquivSieveCompatibleFamily :
    (S.arrows.diagram.op ⋙ P).cones.obj E ≃
      { x : FamilyOfElements (P ⋙ coyoneda.obj E) (S : Presieve X) // x.SieveCompatible } where
  toFun π :=
    ⟨fun Y f h => π.app (op ⟨Over.mk f, h⟩), fun X Y f g hf => by
      apply (id_comp _).symm.trans
      dsimp
      exact π.naturality (Quiver.Hom.op (Over.homMk _ (by rfl)))⟩
  invFun x :=
    { app := fun f => x.1 f.unop.1.hom f.unop.2
      naturality := fun f f' g => by
        refine' Eq.trans _ (x.2 f.unop.1.hom g.unop.left f.unop.2)
        dsimp
        

Proof state:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E : Aᵒᵖ
x : { x // SieveCompatible x }
f f' : (FullSubcategory fun f => S.arrows f.hom)ᵒᵖ
g : f ⟶ f'
⊢ 𝟙 E.unop ≫ ↑x f'.unop.obj.hom (_ : S.arrows f'.unop.obj.hom) =
    ↑x (g.unop.left ≫ f.unop.obj.hom) (_ : S.arrows (g.unop.left ≫ f.unop.obj.hom))

Next tactic:
rw [id_comp]
/-- Given a sieve `S` on `X : C`, a presheaf `P : Cᵒᵖ ⥤ A`, and an object `E` of `A`,
    the cones over the natural diagram `S.arrows.diagram.op ⋙ P` associated to `S` and `P`
    with cone point `E` are in 1-1 correspondence with sieve_compatible family of elements
    for the sieve `S` and the presheaf of types `Hom (E, P -)`. -/
@[simps]
def conesEquivSieveCompatibleFamily :
    (S.arrows.diagram.op ⋙ P).cones.obj E ≃
      { x : FamilyOfElements (P ⋙ coyoneda.obj E) (S : Presieve X) // x.SieveCompatible } where
  toFun π :=
    ⟨fun Y f h => π.app (op ⟨Over.mk f, h⟩), fun X Y f g hf => by
      apply (id_comp _).symm.trans
      dsimp
      exact π.naturality (Quiver.Hom.op (Over.homMk _ (by rfl)))⟩
  invFun x :=
    { app := fun f => x.1 f.unop.1.hom f.unop.2
      naturality := fun f f' g => by
        refine' Eq.trans _ (x.2 f.unop.1.hom g.unop.left f.unop.2)
        dsimp
        rw [id_comp]
        

Proof state:
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E : Aᵒᵖ
x : { x // SieveCompatible x }
f f' : (FullSubcategory fun f => S.arrows f.hom)ᵒᵖ
g : f ⟶ f'
⊢ ↑x f'.unop.obj.hom (_ : S.arrows f'.unop.obj.hom) =
    ↑x (g.unop.left ≫ f.unop.obj.hom) (_ : S.arrows (g.unop.left ≫ f.unop.obj.hom))

Next tactic:
convert rfl
/-- Given a sieve `S` on `X : C`, a presheaf `P : Cᵒᵖ ⥤ A`, and an object `E` of `A`,
    the cones over the natural diagram `S.arrows.diagram.op ⋙ P` associated to `S` and `P`
    with cone point `E` are in 1-1 correspondence with sieve_compatible family of elements
    for the sieve `S` and the presheaf of types `Hom (E, P -)`. -/
@[simps]
def conesEquivSieveCompatibleFamily :
    (S.arrows.diagram.op ⋙ P).cones.obj E ≃
      { x : FamilyOfElements (P ⋙ coyoneda.obj E) (S : Presieve X) // x.SieveCompatible } where
  toFun π :=
    ⟨fun Y f h => π.app (op ⟨Over.mk f, h⟩), fun X Y f g hf => by
      apply (id_comp _).symm.trans
      dsimp
      exact π.naturality (Quiver.Hom.op (Over.homMk _ (by rfl)))⟩
  invFun x :=
    { app := fun f => x.1 f.unop.1.hom f.unop.2
      naturality := fun f f' g => by
        refine' Eq.trans _ (x.2 f.unop.1.hom g.unop.left f.unop.2)
        dsimp
        rw [id_comp]
        convert rfl
        

Proof state:
case h.e'_3.h.e'_4
C : Type u₁
inst✝¹ : Category.{v₁, u₁} C
A : Type u₂
inst✝ : Category.{v₂, u₂} A
J : GrothendieckTopology C
P : Cᵒᵖ ⥤ A
X : C
S : Sieve X
R : Presieve X
E : Aᵒᵖ
x : { x // SieveCompatible x }
f f' : (FullSubcategory fun f => S.arrows f.hom)ᵒᵖ
g : f ⟶ f'
⊢ g.unop.left ≫ f.unop.obj.hom = f'.unop.obj.hom

Next tactic:
rw [Over.w]