Proof using.
intros. unfold key_in_output_trace in *.
break_exists_exists.
intuition. do_in_app; intuition.
exfalso. eauto.
Qed.

Lemma key_in_output_list_split :
forall l l',
key_in_output_list client id (l ++ l') ->
key_in_output_list client id l \/ key_in_output_list client id l'.