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 tot_map_in_snd : forall h m ns nm, In nm (map (fun nm0 : name * msg => (tot_map_name (fst nm0), tot_map_msg (snd nm0))) (map2snd m (filter_rel adjacent_to_dec h ns))) -> snd nm = tot_map_msg m.