Proof using name_map_bijective.
move => net d p p'.
case: p => src dst m.
case: p' => src' dst' m' /=.
case H_eq: (pt_map_msg _) => [m0|] // H_eq'.
inversion H_eq'; subst.
move {H_eq H_eq'}.
exact: pt_map_update_eq.
Qed.

Theorem step_async_pt_mapped_simulation_1 :
forall net net' tr,
@step_async _ multi_fst net net' tr ->
@step_async _ multi_snd (pt_map_net net) (pt_map_net net') (filterMap pt_map_trace_occ tr) \/
(pt_map_net net' = pt_map_net net /\ filterMap trace_non_empty_out (filterMap pt_map_trace_occ tr) = []).