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 in_pt_map_map_pair : forall m m' l n, pt_map_msg m = Some m' -> (forall nm, In nm l -> snd nm = m) -> In (tot_map_name_inv n, m) l -> In (n, m') (filterMap pt_map_name_msg l).