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.

(* Specifications: *)
(* Searching in a list with a common first element is equivalent to the search skipping the element *)
Theorem not_in_find_map :
forall p a b,
(forall l, ~ In (a, l) p) ->
find_code a
(map
(fun v : A * list bool => match v with
| (a1, b1) => (a1, b :: b1)
end) p) = nil.