Definition tot_map_trace_occ (e : @name _ multi_fst * (@input base_fst + list (@output base_fst))) :=
match e with
| (n, inl io) => (tot_map_name n, inl (tot_map_input io))
| (n, inr lo) => (tot_map_name n, inr (map tot_map_output lo))
end.Definition tot_map_packet (p : @packet base_fst multi_fst)  :=
match p with
| mkPacket src dst m =>
  mkPacket (tot_map_name src) (tot_map_name dst) (tot_map_msg m)
end.Definition tot_map_net (net : @network _ multi_fst) : @network _ multi_snd :=
  {| nwPackets := map tot_map_packet net.(nwPackets) ;
     nwState := fun n => tot_map_data (net.(nwState) (tot_map_name_inv n)) |}.

Lemma tot_map_update_eq : forall f d h, (fun n : name => tot_map_data (update name_eq_dec f h d (tot_map_name_inv n))) = update name_eq_dec (fun n : name => tot_map_data (f (tot_map_name_inv n))) (tot_map_name h) (tot_map_data d).