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).