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_in_adj_map_pair : forall (m : msg) ns failed n h, In n ns -> ~ In n (map tot_map_name failed) -> adjacent_to h n -> In (n, m) (map2snd m (filter_rel adjacent_to_dec h (remove_all name_eq_dec (map tot_map_name failed) ns))).