intros s1 s2 H_le.
case H_le; intro H_le'.
- induction H_le'.
* apply lex_order_char_lt.
assumption.
* apply lex_order_char_eq.
apply IHH_le'.
left.
assumption.
* apply lex_order_empty_string.
- rewrite <- H_le'.
apply lex_order_refl.
Qed.

Lemma nat_of_ascii_injective:
forall c1 c2, nat_of_ascii c1 = nat_of_ascii c2 -> c1 = c2.