Corollary step_failure_tot_mapped_simulation_star_1 :
  forall net failed tr,
    @step_failure_star _ _ fail_fst step_failure_init (failed, net) tr ->
    @step_failure_star _ _ fail_snd step_failure_init (map tot_map_name failed, tot_map_net net) (map tot_map_trace_occ tr).

Lemma tot_map_in_in : forall n m l, (forall nm, In nm l -> snd nm = m) -> ~ In (n, m) l -> ~ In (tot_map_name n, tot_map_msg m) (map (fun nm : name * msg => (tot_map_name (fst nm), tot_map_msg (snd nm))) l).