Corollary step_failure_tot_mapped_simulation_star_1 :
  forall net failed tr,
    @step_failure_star _ _ fail_fst step_failure_init (failed, net) tr ->
    @step_failure_star _ _ fail_snd step_failure_init (map tot_map_name failed, tot_map_net net) (map tot_map_trace_occ tr).

Lemma not_in_failed_in_exclude : forall ns n failed, ~ In (tot_map_name_inv n) failed -> In (tot_map_name_inv n) ns -> In (tot_map_name_inv n) (remove_all name_eq_dec failed ns).