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 inc_in_flight_to_backup_app : forall xs ys, inc_in_flight_to_backup (xs ++ ys) = inc_in_flight_to_backup xs + inc_in_flight_to_backup ys.