Corollary step_async_tot_mapped_simulation_star_1 :
  forall net tr,
    @step_async_star _ multi_fst step_async_init net tr ->
    @step_async_star _ multi_snd step_async_init (tot_map_net net) (map tot_map_trace_occ tr).

Theorem step_async_tot_mapped_simulation_2 : forall net net' out mnet mout, @step_async _ multi_snd net net' out -> tot_map_net mnet = net -> map tot_map_trace_occ mout = out -> exists mnet', @step_async _ multi_fst mnet mnet' mout /\ tot_map_net mnet' = net'.