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_some : forall net d p p', pt_map_packet p = Some p' -> (fun n : name => pt_ext_map_data (update name_eq_dec (nwState net) (pDst p) d (tot_map_name_inv n)) (tot_map_name_inv n)) = update name_eq_dec (fun n : name => pt_ext_map_data (nwState net (tot_map_name_inv n)) (tot_map_name_inv n)) (pDst p') (pt_ext_map_data d (pDst p)).