Proof using.
intros a c H.
apply sym_equal; apply find_code_correct2; auto.
apply restrict_code_unique_key.
generalize (in_frequency_map _ eqA_dec m a H).
unfold restrict_code in |- *; elim (frequency_list eqA_dec m); simpl in |- *;
auto with datatypes.
intros a0; case a0; simpl in |- *; auto with datatypes.
intros a1 n l H0 [H1| H1]; try rewrite H1; auto.
Qed.

(* Specifications: *)
(* The restriction does not change the encoding for messages in the same alphabet *)
Theorem restrict_code_encode_incl :
forall (m1 : list A) (c : code A),
incl m1 m -> encode eqA_dec c m1 = encode eqA_dec (restrict_code m c) m1.