Section Code.
(* Arbitrary set *)
Variable A : Type.
(* Equality is decidable *)
Variable eqA_dec : forall a b : A, {a = b} + {a <> b}.
(* A code is an association list *)
Definition code := list (A * list bool).
(* Predicate that tells that a message (a list of message) is in the alphabet of the code *)
Definition in_alphabet (m : list A) (c : code) :=
forall a : A, In a m -> exists l : list bool, In (a, l) c.
Proof using.
intros m1 m2 c H H0; red in |- *.
intros a H1; case (H0 a); auto.
Qed.
Proof using.
intros c a H; inversion H.
Qed.
Hint Resolve in_alphabet_nil : core.
Proof using.
intros m c a ca H H0; red in |- *; simpl in |- *.
intros a1 [H1| H1].
exists ca; rewrite <- H1; auto.
case (H0 a1); auto.
Qed.
Hint Resolve in_alphabet_cons : core.
Proof using.
intros c a l H; red in |- *.
intros a1 H1; apply H; simpl in |- *; auto.
Qed.
Definition code_dec :
forall (c : code) a,
{(exists l : list bool, In (a, l) c)} +
{(forall l : list bool, ~ In (a, l) c)}.
intros c a; elim c; simpl in |- *; auto.
intros (a1, l1) l H; case (eqA_dec a1 a); intros H1.
left; exists l1; rewrite H1; auto.
case H.
intros e; left; case e; intros l2 H2; exists l2; auto.
intros n; right; intros l0; red in |- *; intros [H0| H0];
[ case H1 | case (n l0) ]; auto.
injection H0; auto.
Defined.
(* Checking if a code is in an alphabet is decidable *)
Definition in_alphabet_dec :
forall m c, {in_alphabet m c} + {~ in_alphabet m c}.
intros m; elim m; simpl in |- *; auto.
intros a l H c; case (H c); intros H1.
case (code_dec c a); intros H2.
left; red in |- *; simpl in |- *.
intros a1 [H3| H3];
[ case H2; intros l1 Hl1; exists l1; rewrite <- H3 | idtac ];
auto.
right; red in |- *; intros H3; case (H3 a); simpl in |- *; auto.
right; Contradict H1; auto; red in |- *.
intros a0 H0; case (H1 a0); simpl in |- *; auto.
intros x H2; exists x; auto.
Defined.
(* The empty list is associated to none of element of A in the code *)
Definition not_null (c : code) := forall a : A, ~ In (a, nil) c.
Proof using.
intros a l H; red in |- *.
intros a0; red in |- *; intros H0; case (H a0); simpl in |- *; auto.
Qed.
Proof using.
intros a b l H H0; red in |- *.
intros a1; simpl in |- *; red in |- *; intros [H1| H1]; auto.
case H; injection H1; auto.
case (H0 a1); simpl in |- *; auto.
Qed.
Hint Resolve not_null_cons : core.
Proof using.
intros l1; elim l1; simpl in |- *; auto.
intros (a2, l2); case l2; auto.
intros l H l0 H0 H1; case (H0 a2); simpl in |- *; auto.
intros b l l0 H l3 H0 H1; apply not_null_cons; auto.
red in |- *; intros H2; discriminate.
apply H; auto.
apply not_null_inv with (1 := H0).
Qed.
Hint Resolve not_null_app : core.
Proof using.
intros l; elim l; simpl in |- *; auto.
intros b; red in |- *; intros a; red in |- *; intros H; inversion H.
intros (a1, l1) l0 H b; apply not_null_cons; auto.
red in |- *; intros; discriminate.
Qed.
Hint Resolve not_null_map : core.
(* Define the property of list of booleans to be a prefix of another *)
Inductive is_prefix : list bool -> list bool -> Prop :=
| prefixNull : forall l, is_prefix nil l
| prefixCons :
forall (b : bool) l1 l2,
is_prefix l1 l2 -> is_prefix (b :: l1) (b :: l2).
Hint Constructors is_prefix : core.
Proof using.
intros l; elim l; simpl in |- *; auto.
Qed.
Hint Resolve is_prefix_refl : core.
(* A code is unique_prefix, if there is no encoding that is prefix of another *)
Definition unique_prefix (l : code) :=
(forall (a1 a2 : A) (lb1 lb2 : list bool),
In (a1, lb1) l -> In (a2, lb2) l -> is_prefix lb1 lb2 -> a1 = a2) /\
unique_key l.
Proof using.
split; auto.
intros a1 a2 lb1 lb2 H; inversion H; auto.
Qed.
Hint Resolve unique_prefix_nil : core.
Proof using.
intros c a1 a2 lb1 lb2 (H1, H2); apply (H1 a1 a2 lb1 lb2); auto.
Qed.
Proof using.
intros c (H1, H2); auto.
Qed.
Proof using.
intros c a l (H1, H2); split.
intros a1 a2 lb1 lb2 H H0 H3; apply (H1 a1 a2 lb1 lb2); simpl in |- *; auto.
apply unique_key_inv with (1 := H2); auto.
Qed.
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.
Proof using.
intros c1 c2 H (H1, H2).
cut (permutation c2 c1); [ intros HP; split | apply permutation_sym; auto ].
intros a1 a2 lb1 lb2 H0 H3 H4.
apply (H1 a1 a2 lb1 lb2); auto.
apply permutation_in with (2 := H0); auto.
apply permutation_in with (2 := H3); auto.
apply unique_key_perm with (2 := H2); auto.
Qed.
(* Find the list of booleans associated with an element, return the empty list if the element does not exists *)
Fixpoint find_code (a : A) (l : code) {struct l} : list bool :=
match l with
| nil => nil
| (b, c) :: l1 =>
match eqA_dec a b with
| left _ => c
| right _ => find_code a l1
end
end.
Proof using.
intros c a b l; elim c; simpl in |- *; auto.
intros; discriminate.
intros a0; case a0.
intros a1; case (eqA_dec a a1); simpl in |- *; auto.
intros e l0 l1 H H0; left;
apply f_equal2 with (f := pair (A:=A) (B:=list bool));
auto.
Qed.
Proof using.
intros c a l; generalize a l; elim c; clear a l c; simpl in |- *; auto.
intros a l H H0; case H0.
intros a; case a.
intros a0 l l0 H a1 l1 H0 H1; case (eqA_dec a1 a0).
intros H2; case H1; intros H3.
injection H3; auto.
apply unique_key_in_inv with (l := (a0, l) :: l0) (a := a0); simpl in |- *;
auto.
rewrite <- H2; auto.
intros H2; case H1; auto.
intros H3; case H2; injection H3; auto.
intros; apply H; auto.
apply unique_key_inv with (1 := H0); auto.
Qed.
Proof using.
intros a l; elim l; simpl in |- *; auto.
intros (a1, l1) l0 H H0.
case (eqA_dec a a1); auto.
intros H1; case (H0 l1); rewrite H1; auto.
intros H1; apply H; intros p; red in |- *; intros Hp; case (H0 p); auto.
Qed.
Proof using.
intros a l1; generalize a; elim l1; simpl in |- *; auto; clear a l1.
intros (a1, l1) l H a l2 H1; case (eqA_dec a a1); auto.
generalize H1; case l1; auto.
intros H0; case (H0 a1); simpl in |- *; auto.
cut (not_null l); simpl in |- *; auto.
apply not_null_inv with (1 := H1); auto.
Qed.
Proof using.
intros a c1 c2 H; elim H; simpl in |- *; auto.
intros a0; case a0.
intros a1; case (eqA_dec a a1); auto.
intros n l L1 L2 H0 H1 H2; apply H1.
apply unique_prefix_inv with (1 := H2).
intros a0; case a0.
intros a1 l1.
case (eqA_dec a a1).
intros Ha1 b; case b; auto.
intros a2 l2 L HL; case (eqA_dec a a2); auto.
intros e.
case unique_key_in with (1 := unique_prefix2 _ HL) (b2 := l2); auto.
rewrite <- Ha1; rewrite e; simpl in |- *; auto.
intros Ha1 b; case b; auto.
intros L1 L2 L3 H0 H1 H2 H3 H4; apply trans_equal with (find_code a L2); auto.
apply H3.
apply (unique_prefix_permutation _ _ H0); auto.
Qed.
Proof using.
intros p; elim p; simpl in |- *; auto.
intros a l b H; case H.
intros (a1, l1) l H a0 l0 b [H0| H0]; auto.
case (eqA_dec a0 a1); auto.
intros HH; case HH; injection H0; auto.
case (eqA_dec a0 a1); auto.
intros n; apply (H a0 l0 b); auto.
Qed.
Proof using.
intros p; elim p; simpl in |- *; auto.
intros (a1, l1) l H a0 b H0; case (eqA_dec a0 a1); auto.
intros e; case (H0 l1); rewrite e; auto.
intros n; apply (H a0 b); auto.
intros l0; red in |- *; intros H1; case (H0 l0); auto.
Qed.
(* Find the element associated with a list of booleans, return None if it does not exist in the code *)
Fixpoint find_val (a : list bool) (l : code) {struct l} : option A :=
match l with
| nil => None
| (b, c) :: l1 =>
match list_eq_dec eq_bool_dec a c with
| left _ => Some b
| right _ => find_val a l1
end
end.
Proof using.
intros c a l; elim c; simpl in |- *; auto.
intros; discriminate.
intros a0; case a0.
intros a1 l0 l1; case (list_eq_dec eq_bool_dec l l0); auto.
intros e H H0; injection H0.
intros e1; left; apply f_equal2 with (f := pair (A:=A) (B:=list bool)); auto.
Qed.
Proof using.
intros c a l; generalize a l; elim c; clear a l c; simpl in |- *; auto.
intros a l H H0; case H0.
intros a; case a.
intros a0 l l0 H a1 l1 H0 H1; case (list_eq_dec eq_bool_dec l1 l).
intros H2; apply f_equal with (f := Some (A:=A)).
case H1; intros H3.
injection H3; auto.
apply unique_prefix1 with (1 := H0) (lb1 := l) (lb2 := l1); simpl in |- *;
auto.
rewrite H2; auto.
intros H2; case H1; auto.
intros H3; case H2; injection H3; auto.
intros; apply H; auto.
apply unique_prefix_inv with (1 := H0); auto.
Qed.
Opaque list_eq_dec.
Proof using.
intros c; elim c; simpl; auto.
intros a; case a.
intros a1 l; case (list_eq_dec eq_bool_dec nil l); auto.
intros e l0 H H0; case (H0 a1); rewrite e; simpl in |- *; auto.
intros n l0 H H0; apply H.
unfold not_null in |- *; intros a2; red in |- *; intros H1.
case (H0 a2); simpl in |- *; auto.
Qed.
(* Encoding is done by iteratively applying find_code *)
Fixpoint encode (c : code) (m : list A) {struct m} : list bool :=
match m with
| nil => nil
| a :: b => find_code a c ++ encode c b
end.
Proof using.
intros l1; elim l1; simpl in |- *; auto.
intros a l H l2 c; rewrite H; auto.
rewrite app_ass; auto.
Qed.
Proof using.
intros a l1 l m1; generalize a l1 l; elim m1; simpl in |- *; auto;
clear a l1 l m1.
intros a l H a0 l1 l0 H0; case (eqA_dec a a0); simpl in |- *; auto.
intros H1; case H0; auto.
intros e; rewrite H; auto.
Qed.
Proof using.
intros m; elim m; simpl in |- *; auto.
intros a l H c1 c2 H0 H1.
apply f_equal2 with (f := app (A:=bool)); auto.
apply find_code_permutation; auto.
Qed.
Proof using.
intros m1 m2 c H; elim H; simpl in |- *; auto; clear H m1 m2.
intros; repeat rewrite <- app_ass; auto.
intros l1 l2 l3 H H0 H1 H2; apply permutation_trans with (1 := H0); auto.
Qed.
(* To decode, first find the initial section of booleans that maps an encoding, this initial section is accumulated in the head *)
Fixpoint decode_aux (c : code) (head m : list bool) {struct m} : list A :=
match m with
| nil => match find_val head c with
| Some a => a :: nil
| None => nil
end
| a :: m1 =>
match find_val head c with
| Some a1 => a1 :: decode_aux c (a :: nil) m1
| None => decode_aux c (head ++ a :: nil) m1
end
end.
Proof using.
intros c Hc1 Hc2 m1; elim m1; simpl in |- *; auto.
intros m2; case m2; simpl in |- *; auto.
intros head a; rewrite <- app_nil_end.
intros H1; rewrite H1; auto.
rewrite not_null_find_val; auto.
intros b l head a; rewrite <- app_nil_end.
intros H1; rewrite H1; auto.
rewrite not_null_find_val; auto.
intros a l Rec m2 head a1 H2.
rewrite <- Rec with (head := head ++ a :: nil).
generalize (fun a => find_val_correct1 c a head).
case (find_val head c); auto.
intros a0 H; (cut (In (a0, head) c); [ intros Hin1 | auto ]).
cut (In (a1, head ++ a :: l) c);
[ intros Hin2 | apply find_val_correct1; auto ].
cut (a1 = a0); [ intros Heq | idtac ].
absurd (head ++ a :: l = head); auto.
elim head; simpl in |- *; auto.
red in |- *; intros; discriminate.
intros a2 l0 H0; red in |- *; intros H1; case H0; auto.
injection H1; auto.
apply unique_key_in_inv with (l := c) (a := a1); auto.
apply unique_prefix2; auto.
rewrite Heq; auto.
apply sym_equal; apply (unique_prefix1 c a0 a1 head (head ++ a :: l)); auto.
elim head; simpl in |- *; auto.
rewrite app_ass; auto.
Qed.
(* Decode is just decode_aux with an empty head *)
Definition decode (c : list (A * list bool)) (m : list bool) :=
decode_aux c nil m.
Proof using.
intros c H H0 m1 m2 a H1; unfold decode in |- *; apply decode_aux_correct;
auto.
Qed.
Proof using.
intros c H1 H2 m; elim m; simpl in |- *; auto.
intros H3; unfold decode in |- *; simpl in |- *.
rewrite not_null_find_val; auto.
intros a l H H0; rewrite decode_correct with (a := a); auto.
apply f_equal2 with (f := cons (A:=A)); auto.
apply H; apply in_alphabet_inv with (1 := H0); auto.
case (H0 a); simpl in |- *; auto.
intros l1 Hl1.
apply find_val_correct2; auto.
case (H0 a); simpl in |- *; auto.
intros x1; case x1.
intros H3; case (H2 a); auto.
intros b l0 H3; cut (find_code a c = b :: l0).
intros H4; rewrite H4; apply find_code_correct1; auto.
apply find_code_correct2 with (2 := H3); auto.
case H1; auto.
Qed.

(* A unique prefix code in an alphabet of at least 2 elements is non-empty *)
Theorem frequency_not_null :
forall m (c : code),
1 < length (frequency_list eqA_dec m) ->
unique_prefix c -> in_alphabet m c -> not_null c.