Corollary step_ordered_pt_mapped_simulation_star_1 :
  forall net tr,
    @step_ordered_star _ multi_fst step_ordered_init net tr ->
    @step_ordered_star _ multi_snd step_ordered_init (pt_map_onet net) (filterMap pt_map_trace_ev tr).

Lemma pt_map_map_pair_eq : forall m m' h failed, pt_map_msg m = Some m' -> Permutation (filterMap pt_map_name_msg (map2snd m (filter_rel adjacent_to_dec h (remove_all name_eq_dec failed nodes)))) (map2snd m' (filter_rel adjacent_to_dec (tot_map_name h) (remove_all name_eq_dec (map tot_map_name failed) nodes))).