Proof using.
intros a p; elim p; simpl in |- *; auto.
intros a0 H; inversion H.
exists (nil (A:=bool)); auto.
intros p0 H p1 H0 H1; inversion H1.
case H; auto.
intros x Hx; exists (false :: x).
apply in_or_app; left; auto.
change
(In ((fun v => match v with
| (a1, b1) => (a1, false :: b1)
end) (a, x))
(map (fun v => match v with
| (a1, b1) => (a1, false :: b1)
end) (compute_code p0))) in |- *;
apply in_map; auto.
case H0; auto.
intros x Hx; exists (true :: x).
apply in_or_app; right; auto.
change
(In ((fun v => match v with
| (a1, b1) => (a1, true :: b1)
end) (a, x))
(map (fun v => match v with
| (a1, b1) => (a1, true :: b1)
end) (compute_code p1))) in |- *;
apply in_map; auto.
Qed.

(* The computed code is in the alphabet of its leaves *)
Theorem in_alphabet_compute_code :
forall m t,
(forall a : A, In a m -> inb (leaf a) t) -> in_alphabet m (compute_code t).