Definition inc_in_flight_to_backup (l : list packet) : nat :=
  length (filterMap
            (fun p => if msg_eq_dec (pBody p) inc
                   then if name_eq_dec (pDst p) backup
                        then Some tt else None
                   else None)
            l).

Lemma backup_plus_network_eq_primary : forall net tr, step_async_star (params := Counter_MultiParams) step_async_init net tr -> nwState net backup + inc_in_flight_to_backup (nwPackets net) = nwState net primary.