Proof using.
intros c a b H H0 H1.
unfold not_null in |- *; intros a1; red in |- *; intros Ha1.
case (H0 a); simpl in |- *; auto; intros l1 Hl1.
case (H0 b); simpl in |- *; auto; intros l2 Hl2.
case H; apply trans_equal with a1.
apply sym_equal; apply unique_prefix1 with (3 := Hl1) (2 := Ha1); auto.
apply unique_prefix1 with (3 := Hl2) (2 := Ha1); auto.
Qed.

(* Unique prefix is preserved by permutation *)
Theorem unique_prefix_permutation :
forall c1 c2 : code,
permutation c1 c2 -> unique_prefix c1 -> unique_prefix c2.