Section Restrict.
Variable A : Type.
Variable empty : A.
Variable eqA_dec : forall a b : A, {a = b} + {a <> b}.
Variable m : list A.
(* Restrict the code putting only codes of element in the frequency list *)
Definition restrict_code (m : list A) (c : code A) :
code A :=
map (fun x => (fst x, find_code eqA_dec (fst x) c))
(frequency_list eqA_dec m).
Proof using.
intros c; apply ulist_unique_key.
unfold restrict_code in |- *.
replace
(map (fst (B:=_))
(map (fun x : A * nat => (fst x, find_code eqA_dec (fst x) c))
(frequency_list eqA_dec m))) with
(map (fst (B:=_)) (frequency_list eqA_dec m)).
apply unique_key_ulist; auto.
elim (frequency_list eqA_dec m); simpl in |- *; auto with datatypes.
intros a l H; apply f_equal2 with (f := cons (A:=A)); auto.
Qed.
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.
Proof using.
intros m1 c; elim m1; simpl in |- *; auto.
intros a l H H0.
apply f_equal2 with (f := app (A:=bool)); auto with datatypes.
apply restrict_code_in; auto with datatypes.
apply H; apply incl_tran with (2 := H0); auto with datatypes.
Qed.
Proof using.
intros c; apply restrict_code_encode_incl; auto with datatypes.
Qed.

(* The restriction does not change the unique prefix property if the message is in the alphabet *)
Theorem restrict_unique_prefix :
forall c : code A,
not_null c ->
in_alphabet m c -> unique_prefix c -> unique_prefix (restrict_code m c).