Proof using name_map_bijective multi_map_congr.
move => net net' tr.
case => {net net' tr}.
- move => net net' tr m ms out d l from to H_eq H_hnd H_eq' H_eq_tr.
destruct (pt_map_msg m) eqn:?.
left.
rewrite H_eq' /= /pt_map_onet /=.
apply (@StepOrdered_deliver _ _ _ _ _ m0 (filterMap pt_map_msg ms) (filterMap pt_map_output out) (pt_map_data d) (filterMap pt_map_name_msg l) (tot_map_name from) (tot_map_name to)).
* rewrite /= 2!tot_map_name_inv_inverse H_eq /=.
destruct (pt_map_msg _) eqn:?; last by congruence.
by find_injection.
* rewrite /= tot_map_name_inv_inverse.
rewrite -(pt_net_handlers_some _ _ _ _ Heqo).
rewrite /pt_mapped_net_handlers /=.
repeat break_let.
by inversion H_hnd.
* by rewrite /= pt_map_update_eq collate_pt_map_update2_eq.
* by rewrite H_eq_tr /= filterMap_pt_map_trace_ev_outputs_eq.
right.
rewrite /=.
have [H_eq_d [H_ms H_out]] := pt_net_handlers_none _ _ _ _ Heqo H_hnd.
rewrite H_eq_tr filterMap_pt_map_trace_ev_outputs_eq H_out /=.
split => //.
rewrite H_eq' /pt_map_onet /=.
rewrite pt_map_update_eq /= H_eq_d.
rewrite collate_pt_map_eq H_ms /=.
set nwS1 := update _ _ _ _.
set nwS2 := fun n => pt_map_data _.
set nwP1 := fun _ _ => _.
set nwP2 := fun _ _ => _.
have H_eq_s: nwS1 = nwS2.
rewrite /nwS1 /nwS2 {nwS1 nwS2}.
apply functional_extensionality => n.
rewrite /update /=.
case (name_eq_dec _ _) => H_dec //.
by rewrite H_dec tot_map_name_inv_inverse.
have H_eq_p: nwP1 = nwP2.
rewrite /nwP1 /nwP2 /=.
apply functional_extensionality => src.
apply functional_extensionality => dst.
rewrite /update2 /=.
case (sumbool_and _ _ _ _) => H_dec //.
move: H_dec => [H_eq_from H_eq_to].
rewrite -H_eq_from -H_eq_to H_eq /=.
case H_m': (pt_map_msg _) => [m'|] //.
by rewrite H_m' in Heqo.
by rewrite H_eq_s H_eq_p.
- move => h net net' tr out inp d l H_hnd H_eq H_eq_tr.
destruct (pt_map_input inp) eqn:?.
left.
apply (@StepOrdered_input _ _ (tot_map_name h) _ _ _ (filterMap pt_map_output out) i (pt_map_data d) (filterMap pt_map_name_msg l)).
* rewrite /=.
have H_q := pt_input_handlers_some h inp (onwState net h) Heqo.
rewrite /pt_mapped_input_handlers /= in H_q.
rewrite H_hnd in H_q.
rewrite H_q.
by rewrite tot_map_name_inv_inverse.
* by rewrite H_eq /pt_map_onet /= pt_map_update_eq collate_pt_map_eq.
* by rewrite H_eq_tr /= Heqo filterMap_pt_map_trace_ev_outputs_eq.
right.
rewrite /=.
have [H_d [H_l H_o]] := pt_input_handlers_none h inp (onwState net h) Heqo H_hnd.
rewrite H_eq_tr /= filterMap_pt_map_trace_ev_outputs_eq Heqo H_o /=.
split => //.
rewrite H_eq /= /pt_map_onet /=.
rewrite pt_map_update_eq /= H_d.
rewrite collate_pt_map_eq H_l /=.
set nwS1 := update _ _ _ _.
set nwS2 := fun n => pt_map_data _.
have H_eq_n: nwS1 = nwS2.
rewrite /nwS1 /nwS2 /=.
apply functional_extensionality => n.
rewrite /update /=.
case (name_eq_dec _ _) => H_dec //.
by rewrite H_dec tot_map_name_inv_inverse.
by rewrite H_eq_n.
Qed.

Corollary step_ordered_pt_mapped_simulation_star_1 :
forall net tr,
@step_ordered_star _ multi_fst step_ordered_init net tr ->
@step_ordered_star _ multi_snd step_ordered_init (pt_map_onet net) (filterMap pt_map_trace_ev tr).