Definition pt_ext_map_net (net : @network  _ multi_fst) : @network _ multi_snd :=
  {| nwPackets := filterMap pt_map_packet net.(nwPackets) ;
     nwState := fun n => pt_ext_map_data (net.(nwState) (tot_map_name_inv n)) (tot_map_name_inv n) |}.

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