Proof using name_map_bijective multi_map_congr.
move => net net' tr.
case => {net net' tr}.
- move => net net' p ms ms' out d l H_eq H_hnd H_eq'.
rewrite /=.
case H_m: (pt_map_packet p) => [p'|].
have ->: tot_map_name (pDst p) = pDst p'.
case: p H_eq H_hnd H_eq' H_m => /= src dst m H_eq H_hnd H_eq'.
case (pt_map_msg m) => //= m' H_m.
by inversion H_m.
left.
rewrite H_eq' /=.
apply (@StepAsync_deliver _ _ _ _ _ (filterMap pt_map_packet ms) (filterMap pt_map_packet ms') (filterMap pt_map_output out) (pt_map_data d) (filterMap pt_map_name_msg l)).
* rewrite /= H_eq filterMap_app /=.
case H_p: (pt_map_packet _) => [p0|].
rewrite H_p in H_m.
by injection H_m => H_eq_p; rewrite H_eq_p.
by rewrite H_p in H_m.
* rewrite /=.
case: p H_eq H_hnd H_eq' H_m => /= src dst m H_eq H_hnd H_eq'.
case H_m: (pt_map_msg m) => [m'|] //.
case: p' H_eq' => src' dst' m0 H_eq' H_eq_p.
inversion H_eq_p; subst.
rewrite /= {H_eq_p}.
have H_q := pt_net_handlers_some dst src m (nwState net dst) H_m.
rewrite /pt_mapped_net_handlers in H_q.
rewrite H_hnd in H_q.
rewrite H_q.
by rewrite tot_map_name_inv_inverse.
* rewrite /= /pt_map_net /= 2!filterMap_app.
by rewrite (filterMap_pt_map_packet_map_eq_some _ _ H_m) (pt_map_update_eq_some _ _ _ H_m).
right.
split.
* rewrite H_eq' {H_eq'}.
rewrite /pt_map_net /=.
case: p H_eq H_hnd H_m => /= src dst m H_eq H_hnd.
case H_m: (pt_map_msg _) => [m'|] // H_eq'.
rewrite 2!filterMap_app H_eq filterMap_app /=.
case H_m': (pt_map_msg _) => [m'|]; first by rewrite H_m' in H_m.
have [H_d [H_l H_o]] := pt_net_handlers_none _ _ _ _ H_m H_hnd.
rewrite (filterMap_pt_map_name_msg_empty_eq _ dst H_l) /=.
set nwS1 := fun _ => _.
set nwS2 := fun _ => _.
have H_eq_s: nwS1 = nwS2.
rewrite /nwS1 /nwS2 /=.
apply functional_extensionality => n.
rewrite /update /=.
case (name_eq_dec _ _) => H_dec //.
by rewrite H_dec H_d.
by rewrite H_eq_s.
* move {H_eq'}.
case: p H_eq H_hnd H_m => /= src dst m H_eq H_hnd.
case H_m: (pt_map_msg _) => [m'|] // H_eq'.
have [H_d [H_l H_o]] := pt_net_handlers_none _ _ _ _ H_m H_hnd.
by rewrite H_o.
- move => h net net' out inp d l H_hnd H_eq.
rewrite /=.
case H_i: (pt_map_input inp) => [inp'|].
left.
apply (@StepAsync_input _ _ _ _ _ _ _ (pt_map_data d) (filterMap pt_map_name_msg l)).
rewrite /=.
have H_q := pt_input_handlers_some h inp (nwState net h) H_i.
rewrite /pt_mapped_input_handlers /= in H_q.
rewrite H_hnd in H_q.
rewrite H_q.
by rewrite tot_map_name_inv_inverse.
rewrite /= H_eq /= /pt_map_net /=.
rewrite -filterMap_pt_map_packet_map_eq filterMap_app.
by rewrite -pt_map_update_eq.
right.
split.
* rewrite H_eq /pt_map_net /=.
have [H_d [H_l H_o]] := pt_input_handlers_none _ _ _ H_i H_hnd.
rewrite filterMap_app.
rewrite (filterMap_pt_map_name_msg_empty_eq _ h H_l) /=.
set nwS1 := fun _ => _.
set nwS2 := fun _ => _.
have H_eq_s: nwS1 = nwS2.
rewrite /nwS1 /nwS2 /=.
apply functional_extensionality => n.
rewrite /update /=.
case (name_eq_dec _ _) => H_dec //.
by rewrite H_dec H_d.
by rewrite H_eq_s.
* rewrite /=.
have [H_d [H_l H_o]] := pt_input_handlers_none _ _ _ H_i H_hnd.
by rewrite H_o.
Qed.

Corollary step_async_pt_mapped_simulation_star_1 :
forall net tr,
@step_async_star _ multi_fst step_async_init net tr ->
exists tr', @step_async_star _ multi_snd step_async_init (pt_map_net net) tr' /\
filterMap trace_non_empty_out (filterMap pt_map_trace_occ tr) = filterMap trace_non_empty_out tr'.