Definition tot_map_net_event e :=
{| evt_a := (List.map tot_map_name (fst e.(evt_a)), tot_map_net (snd e.(evt_a))) ;
   evt_l := tot_map_label e.(evt_l) ;
   evt_trace := List.map tot_map_trace_occ e.(evt_trace) |}.

Lemma tot_map_net_hd_step_failure_star_always : forall s, event_step_star step_failure step_failure_init (hd s) -> lb_step_execution lb_step_failure s -> always (now (event_step_star step_failure step_failure_init)) (map tot_map_net_event s).