Proof using.
intros.
unfold key_in_output_list in *.
break_exists; do_in_app; intuition eauto.
Qed.

(* Specifications: *)
Lemma key_in_output_list_empty :
forall client id,
~ key_in_output_list client id [].
Lemma key_in_output_list_empty :
~ key_in_output_list client id [].