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_in_snd : forall m m' h ns nm, pt_map_msg m' = Some m -> In nm (filterMap pt_map_name_msg (map2snd m' (filter_rel adjacent_to_dec h ns))) -> snd nm = m.