Proof using.
intros.
simpl in *.
break_match; auto.
Qed.

Lemma correspond_reachable :
forall net tr_m,
step_async_star step_async_init net tr_m ->
forall st tr_1,
step_1_star init st tr_1 ->
inputs_1 tr_1 = inputs_m tr_m ->
correspond st (nwState net) tr_1 tr_m.