Definition t:=positive.Definition eqb := Pos.eqb.Definition eqb_eq := Pos.eqb_eq.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 bits_lt_trans : forall x y z : positive, bits_lt x y -> bits_lt y z -> bits_lt x z.