Proof using.
intros m; elim m; simpl in |- *; auto.
intros a l H t H0; cut (inb (leaf a) t); auto; intros H1.
case inb_compute_ex with (1 := H1).
intros l1 Hl1; apply in_alphabet_cons with (ca := l1); auto.
Qed.

(* Two associations in the list with same prefix have same key *)
Theorem btree_unique_prefix1 :
forall (t : btree) (a1 a2 : A) (lb1 lb2 : list bool),
In (a1, lb1) (compute_code t) ->
In (a2, lb2) (compute_code t) -> is_prefix lb1 lb2 -> a1 = a2.