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).

(* Specifications: *)
Lemma pt_msg_in_map : forall m l n m', (forall nm, In nm l -> snd nm = m) -> In (tot_map_name n, m') (filterMap pt_map_name_msg l) -> pt_map_msg m = Some m'.
Lemma pt_map_in_in : forall m m0 n l, (forall nm, In nm l -> snd nm = m) -> ~ In (n, m) l -> ~ In (tot_map_name n, m0) (filterMap pt_map_name_msg l).
Lemma nodup_pt_map : forall m nms, (forall nm, In nm nms -> snd nm = m) -> NoDup nms -> NoDup (filterMap pt_map_name_msg nms).
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.
Lemma pt_in_tot_map_name : forall m m' l n, pt_map_msg m = Some m' -> (forall nm, In nm l -> snd nm = m) -> In (n, m') (filterMap pt_map_name_msg l) -> In (tot_map_name_inv n, m) l.
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).
Lemma pt_nodup_perm_map_map_pair_perm : forall m m' h failed ns ns', NoDup ns -> Permutation (map tot_map_name ns) ns' -> 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 ns)))) (map2snd m' (filter_rel adjacent_to_dec (tot_map_name h) (remove_all name_eq_dec (map tot_map_name failed) ns'))).
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))).