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 adjacent_in_in_msg : forall (m : @msg base_fst multi_fst) ns n h, adjacent_to h (tot_map_name_inv n) -> In (tot_map_name_inv n) ns -> In (tot_map_name_inv n, m) (map2snd m (filter_rel adjacent_to_dec h ns)).