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 in_failed_exclude : forall ns failed n, In (tot_map_name_inv n) (remove_all name_eq_dec failed ns) -> ~ In (tot_map_name_inv n) failed /\ In (tot_map_name_inv n) ns.