Definition compare : forall x y : t, Compare lt eq x y.Definition eq_dec : forall x y : t, {eq x y} + {~ eq x y}.Definition t:=positive.Definition eq:=@eq positive.Definition eq_refl := @eq_refl t.Definition eq_sym := @eq_sym t.Definition eq_trans := @eq_trans t.Fixpoint bits_lt (p q:positive) : Prop :=
   match p, q with
   | xH, xI _ => True
   | xH, _ => False
   | xO p, xO q => bits_lt p q
   | xO _, _ => True
   | xI p, xI q => bits_lt p q
   | xI _, _ => False
   end.Definition lt:=bits_lt.

Lemma lt_not_eq : forall x y : t, lt x y -> ~ eq x y.