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_adjacent_exclude_in_exlude : forall ns failed n h, In (tot_map_name_inv n) (filter_rel adjacent_to_dec h (remove_all name_eq_dec failed ns)) -> In (tot_map_name_inv n) (remove_all name_eq_dec failed ns) /\ adjacent_to h (tot_map_name_inv n).