Proof using.
unfold PB_net. intros.
pb_unfold. repeat (first [break_let | break_match]); repeat find_inversion; auto.
- right. left. intuition. right. eexists. eexists. intuition eauto.
find_rewrite. auto.
- right. left. intuition. right. eexists. eexists. intuition eauto.
find_rewrite. auto.
- right. right. intuition. eexists. eauto.
Qed.

Lemma PB_net_defn' :
forall dst src m d os d' ms,
PB_net dst src m d = (os, d', ms) ->
(os = [] /\ d' = d /\ ms = [] /\
((dst = Primary /\ exists i, m = BackItUp i) \/
(dst = Backup /\ m = Ack))) \/
(dst = Primary /\ m = Ack /\ (
(queue d = [] /\ os = [] /\ ms = [] /\ d' = d) \/
(exists h t, queue d = h :: t /\
queue d' = t /\
((t = [] /\ ms = []) \/
(exists y ys,
t = y :: ys /\
ms = [(Backup, BackItUp y)])) /\
let (us, st') := handler h (state d) in
os = [RequestResponse h us] /\
state d' = st'))) \/
(dst = Backup /\ ms = [(Primary, Ack)] /\ queue d' = queue d /\ os = [] /\
(exists i, m = BackItUp i /\
state d' = snd (handler i (state d)))).