(** Even if [positive] can be seen as an ordered type with respect to the usual order (see above), we can also use a lexicographic order over bits (lower bits are considered first). This is more natural when using [positive] as indexes for sets or maps (see MSetPositive). *)
Local Open Scope positive.
Module PositiveOrderedTypeBits <: UsualOrderedType.
Definition t:=positive.
Include HasUsualEq <+ UsualIsEq.
Definition eqb := Pos.eqb.
Definition eqb_eq := Pos.eqb_eq.
Include HasEqBool2Dec.
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.