Proof using.
intros.
find_apply_lem_hyp refl_trans_1n_n1_trace.
prep_induction H.
induction H; intros; subst.
- auto using ZOOOPST_nil.
- repeat concludes.
apply ZOOOPST_app; auto.
invc H1; simpl in *.
+ find_apply_lem_hyp PB_net_defn.
intuition; subst; auto using ZOOOPST_singleton_nil.
break_exists. break_and. break_match.
intuition; subst; auto using ZOOOPST_singleton_singleton.
+ rewrite cons_cons_app.
apply ZOOOPST_app.
* auto using ZOOOPST_singleton_inl.
* find_apply_lem_hyp PB_input_handler_defn; intuition; subst;
auto using ZOOOPST_singleton_nil, ZOOOPST_singleton_singleton.
Qed.

Lemma inputs_1_m_revert :
forall net tr,
step_async_star (params := PB_multi_params) step_async_init net tr ->
inputs_m tr = inputs_1 (revert_trace tr) ++ queue (nwState net Primary).