Module String_as_OT <: UsualOrderedType.
Definition t := string.
Definition eq := @eq string.
Definition eq_refl := @eq_refl t.
Definition eq_sym := @eq_sym t.
Definition eq_trans := @eq_trans t.
Inductive lts : string -> string -> Prop :=
| lts_empty : forall a s, lts EmptyString (String a s)
| lts_tail : forall a s1 s2, lts s1 s2 -> lts (String a s1) (String a s2)
| lts_head : forall (a b : ascii) s1 s2,
lt (nat_of_ascii a) (nat_of_ascii b) ->
lts (String a s1) (String b s2).
Definition lt := lts.

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