Definition ltb (s1 s2 : string) : bool :=
if compare s1 s2 is Lt then true else false.
Definition leb (s1 s2 : string) : bool :=
if compare s1 s2 is Gt then false else true.

Lemma leb_total (s1 s2 : string) : leb s1 s2 = true \/ leb s2 s1 = true.
Infix "?="  := compare : string_scope.