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.