now rewrite Char.compare_same_is_eq; rewrite IHx.
Qed.
(** Test if two strings are equal. *)
Definition eqb (x y : t) : bool :=
match compare x y with
| Eq => true
| _ => false
end.

Lemma eqb_implies_eq : forall (x y : t), eqb x y = true -> x = y.
intros x y; unfold eqb; case_eq (compare x y); try congruence.