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

Lemma in_output_list_split :
forall l l',
in_output_list client id out (l ++ l') ->
in_output_list client id out l \/ in_output_list client id out l'.