Proof using.
intros l; elim l; simpl in |- *; auto.
intros a l0 H; apply add_frequency_list_unique_key; auto.
Qed.
Hint Resolve frequency_list_unique : core.

(* Elements of the message are keys of the frequency list *)
Theorem in_frequency_map :
forall l a, In a l -> In a (map fst (frequency_list l)).