Proof using.
intros.
apply refl_trans_1n_n1_trace in H.
prep_induction H.
induction H; intros; subst.
- unfold step_async_init, revert_state. constructor.
- repeat concludes. rewrite revert_trace_app.
unfold step_1_star.
find_apply_lem_hyp inductive_simulation.
simpl in *.
unfold step_1_star in *.
eauto using refl_trans_1n_trace_trans.
Qed.

Theorem transformer :
forall (P : list (input * output) -> Prop),
(forall st tr,
step_1_star init st tr ->
P tr) ->
(forall net tr,
step_async_star step_async_init net tr ->
P (revert_trace tr)).