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.

(* Inversion theorem *)
Theorem not_null_inv :
forall (a : A * list bool) l, not_null (a :: l) -> not_null l.