Section inner.
Variable client : clientId.
Variable id : nat.
Proof using.
intros. unfold key_in_output_trace in *.
break_exists_exists.
intuition. do_in_app; intuition.
exfalso. eauto.
Qed.
Proof using.
intros.
unfold key_in_output_list in *.
break_exists; do_in_app; intuition eauto.
Qed.

Lemma key_in_output_list_empty :
~ key_in_output_list client id [].