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 map_map_pair_eq : forall h m failed, Permutation (map (fun nm : name * msg => (tot_map_name (fst nm), tot_map_msg (snd nm))) (map2snd m (filter_rel adjacent_to_dec h (remove_all name_eq_dec failed nodes)))) (map2snd (tot_map_msg m) (filter_rel adjacent_to_dec (tot_map_name h) (remove_all name_eq_dec (map tot_map_name failed) nodes))).