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

Lemma InputHandler_trace_preserved :
forall h i d u o d' l,
InputHandler h i d = (u, o, d', l) ->
trace_inputs [(h, inl i)] =
trace_outputs [(h, inr o)] +
inc_in_flight_to_backup (send_packets h l) +
ack_in_flight_to_primary (send_packets h l).