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 InputHandler_inc_in_flight_to_backup_preserved :
forall h i d u o d' l,
InputHandler h i d = (u, o, d', l) ->
d' = d + inc_in_flight_to_backup (send_packets h l).