Corollary tot_map_update_packet_eq :
forall f p d,
  (fun n : name => tot_map_data (update name_eq_dec f (pDst p) d (tot_map_name_inv n))) =
  (update name_eq_dec (fun n : name => tot_map_data (f (tot_map_name_inv n))) (pDst (tot_map_packet p)) (tot_map_data d)).

Theorem step_async_tot_mapped_simulation_1 : forall net net' tr, @step_async _ multi_fst net net' tr -> @step_async _ multi_snd (tot_map_net net) (tot_map_net net') (map tot_map_trace_occ tr).