Corollary step_ordered_failure_pt_mapped_simulation_star_1 :
  forall net failed tr,
    @step_ordered_failure_star _ _ overlay_fst fail_msg_fst step_ordered_failure_init (failed, net) tr ->
    @step_ordered_failure_star _ _ overlay_snd fail_msg_snd step_ordered_failure_init (map tot_map_name failed, pt_map_onet net) (filterMap pt_map_trace_ev tr).Definition pt_map_odnet (net : @ordered_dynamic_network _ multi_fst) : @ordered_dynamic_network _ multi_snd :=
  {| odnwNodes := map tot_map_name net.(odnwNodes) ;
     odnwPackets := fun src dst => filterMap pt_map_msg (net.(odnwPackets) (tot_map_name_inv src) (tot_map_name_inv dst)) ;
     odnwState := fun n => match net.(odnwState) (tot_map_name_inv n) with
                         | None => None
                         | Some d => Some (pt_map_data d)
                         end |}.

(* Specifications: *)
Lemma collate_ls_pt_map_eq : forall ns f h m m', pt_map_msg m = Some m' -> (fun src dst => filterMap pt_map_msg (collate_ls name_eq_dec ns f h m (tot_map_name_inv src) (tot_map_name_inv dst))) = collate_ls name_eq_dec (map tot_map_name ns) (fun src dst => filterMap pt_map_msg (f (tot_map_name_inv src) (tot_map_name_inv dst))) (tot_map_name h) m'.