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).

Lemma remove_tot_map_eq : forall h failed, map tot_map_name (remove name_eq_dec h failed) = remove name_eq_dec (tot_map_name h) (map tot_map_name failed).