Definition Pos_compare_cont_antisym :
  forall (p q : positive) (c : comparison),
       eq (CompOpp (Pos.compare_cont c p q))
         (Pos.compare_cont (CompOpp c) q p ) :=
fun (p q : positive) (c : comparison) =>
positive_ind
  (fun p0 : positive =>
   forall (q0 : positive) (c0 : comparison),
   CompOpp (Pos.compare_cont c0 p0 q0) =
   Pos.compare_cont (CompOpp c0) q0 p0)
  (fun (p0 : positive)
     (IHp : forall (q0 : positive) (c0 : comparison),
            CompOpp (Pos.compare_cont c0 p0 q0) =
            Pos.compare_cont (CompOpp c0) q0 p0)
     (q0 : positive) =>
   match
     q0 as p1
     return
       (forall c0 : comparison,
        CompOpp (Pos.compare_cont c0 p0~1 p1) =
        Pos.compare_cont (CompOpp c0) p1 p0~1)
   with
   | (q1~1)%positive =>
       fun c0 : comparison => IHp q1 c0
   | (q1~0)%positive =>
       fun c0 : comparison => IHp q1 Gt
   | 1%positive => fun c0 : comparison => eq_refl
   end)
  (fun (p0 : positive)
     (IHp : forall (q0 : positive) (c0 : comparison),
            CompOpp (Pos.compare_cont c0 p0 q0) =
            Pos.compare_cont (CompOpp c0) q0 p0)
     (q0 : positive) =>
   match
     q0 as p1
     return
       (forall c0 : comparison,
        CompOpp (Pos.compare_cont c0 p0~0 p1) =
        Pos.compare_cont (CompOpp c0) p1 p0~0)
   with
   | (q1~1)%positive =>
       fun c0 : comparison => IHp q1 Lt
   | (q1~0)%positive =>
       fun c0 : comparison => IHp q1 c0
   | 1%positive => fun c0 : comparison => eq_refl
   end)
  (fun q0 : positive =>
   match
     q0 as p0
     return
       (forall c0 : comparison,
        CompOpp (Pos.compare_cont c0 1 p0) =
        Pos.compare_cont (CompOpp c0) p0 1)
   with
   | (q1~1)%positive =>
       fun c0 : comparison => eq_refl
   | (q1~0)%positive =>
       fun c0 : comparison => eq_refl
   | 1%positive => fun c0 : comparison => eq_refl
   end) p q c.Definition Pos_compare_antisym:
  forall p q : positive,
       eq (Pos.compare q p) (CompOpp (Pos.compare p q)) :=
  fun p q : positive =>
eq_ind_r (fun c : comparison => eq (Pos.compare_cont Eq q p) c) eq_refl
  (Pos_compare_cont_antisym p q Eq).

Lemma Pos_compare_eq: forall x y, eq (Pos.compare x y) Eq -> eq x y.