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_failure_tot_mapped_simulation_1 : forall net net' failed failed' tr, @step_failure _ _ fail_fst (failed, net) (failed', net') tr -> @step_failure _ _ fail_snd (map tot_map_name failed, tot_map_net net) (map tot_map_name failed', tot_map_net net') (map tot_map_trace_occ tr).