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_map_pair_adjacent_to : forall (m : @msg _ multi_fst) ns failed h n, In (tot_map_name_inv n, m) (map2snd m (filter_rel adjacent_to_dec h (remove_all name_eq_dec failed ns))) -> In (tot_map_name_inv n) (filter_rel adjacent_to_dec h (remove_all name_eq_dec failed ns)).