Corollary step_ordered_failure_tot_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, tot_map_onet net) (map tot_map_trace tr).Definition tot_map_odnet (net : @ordered_dynamic_network _ multi_fst) : @ordered_dynamic_network _ multi_snd :=
  {| odnwNodes := map tot_map_name net.(odnwNodes) ;
     odnwPackets := fun src dst => map tot_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 (tot_map_data d)
                   end |}.

Lemma collate_ls_tot_map_eq : forall ns f h m, (fun src dst => map tot_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 => map tot_map_msg (f (tot_map_name_inv src) (tot_map_name_inv dst))) (tot_map_name h) (tot_map_msg m).