Proof using.
intros.
simpl in *.
unfold revertNetwork.
simpl in *.
repeat find_rewrite.
match goal with
[ |- context [ dedup ?eq (map ?f ?l ++ ?xs ++ ?ys) ] ] =>
assert (dedup eq (map f l ++ xs ++ ys) = (map f l) ++ (dedup eq (xs ++ ys)))
by (find_eapply_lem_hyp processPackets_dedup;
eauto; repeat find_rewrite; in_crush)
end.
find_rewrite.
rewrite filter_app.
rewrite map_app.
assert (In p (nwPackets st)) by (find_rewrite; in_crush).
match goal with
| H : In _ _ |- _ =>
apply dedup_In with (A_eq_dec := pkt_eq_dec) in H
end.
find_apply_lem_hyp in_split. break_exists. repeat find_rewrite.
find_copy_apply_lem_hyp dedup_partition.
unfold seq_num_packet in *.
repeat find_rewrite.
assert (NoDup (x ++ p :: x0)) by (repeat find_reverse_rewrite; apply NoDup_dedup).
match goal with
| [ |- context [ filter ?f (?xs ++ ?p :: ?ys) ] ] =>
assert (In p (filter f (xs ++ p :: ys)))
by (apply filter_In; repeat break_if; intuition)
end. find_apply_lem_hyp in_split. break_exists.
repeat find_rewrite.
exists (map revertPacket x1), (map revertPacket x2).
intuition; map_crush; auto.
simpl in *.
f_equal. f_equal.
- rewrite filter_true_id.
+ match goal with
| _ : processPackets ?n ?l = _ |- _ =>
rewrite <- (processPackets_preserves_messages n l)
end.
find_rewrite. simpl in *.
unfold revertPacket.
repeat rewrite map_map. auto.
+ intros.
in_crush.
find_eapply_lem_hyp processPackets_ge_start; [|eauto].
match goal with
p : packet |- _ =>
assert (tmNum (pBody p) < tdNum (nwState st (pSrc p))) by
(eapply_prop sequence_sane; find_rewrite; in_crush)
end.
repeat (break_if; simpl in *; repeat find_rewrite; intuition);
unfold sequence_seen in *.
-- case (name_eq_dec (pSrc p) (pDst p)); intro.
++ rewrite <- e0 in i.
rewrite get_set_same_default in i.
rewrite e0 in i.
case i; intro.
** rewrite <- H12 in H11.
rewrite <- e0 in H11.
omega.
** pose proof (H0 _ _ _ H12).
omega.
++ match goal with
| [H: In _ (assoc_default _ (assoc_set _ _ _ _) _ _) |- _ ] =>
rewrite get_set_diff_default in H
end; auto.
pose proof (H0 _ _ _ i).
omega.
-- pose proof (H0 _ _ _ i).
omega.
- rewrite <- filter_except_one
with (f := (fun p0 : seq_num_packet =>
if member (tmNum (pBody p0)) (assoc_default name_eq_dec (tdSeen (nwState st (pDst p0))) (pSrc p0) [])
then false
else true)) (x := p) (A_eq_dec := pkt_eq_dec).
+ unfold seq_num_packet in *. repeat find_rewrite.
rewrite filter_app.
find_apply_lem_hyp filter_partition; intuition.
repeat find_rewrite. eauto using map_app.
+ intros. repeat (break_if; simpl in *; repeat find_rewrite; intuition).
-- contradict n0.
case (name_eq_dec (pSrc p) (pSrc y)); intro.
** rewrite <- e0.
rewrite get_set_same_default.
right.
rewrite e0.
auto.
** rewrite get_set_diff_default; auto.
-- exfalso.
match goal with H : _ = _ -> False |- _ => apply H end.
eapply_prop sequence_equality; repeat find_rewrite; in_crush.
** find_apply_lem_hyp in_dedup_was_in. in_crush.
** case (name_eq_dec (pSrc p) (pSrc y)); intro.
++ rewrite e0 in i.
rewrite get_set_same_default in i.
case i; intro; intuition.
++ rewrite get_set_diff_default in i; intuition.
** case (name_eq_dec (pSrc p) (pSrc y)); intro; auto.
rewrite get_set_diff_default in i; intuition.
+ repeat (break_if; simpl in *; intuition).
contradict n0.
rewrite get_set_same_default.
left; auto.
- apply functional_extensionality. intros; repeat break_if; simpl in *; intuition.
Qed.

Theorem revertNetwork_input :
forall (st : seq_num_network) h (d : @data orig_base_params) l l' n,
sequence_sane st ->
sequence_seen st ->
sequence_equality st ->
processPackets (tdNum (nwState st h)) l = (n, l') ->
revertNetwork
(@mkNetwork _ multi_params
(send_packets h l' ++ nwPackets st)
(update name_eq_dec (nwState st) h
(@mkseq_num_data _ orig_multi_params
n
(tdSeen (nwState st h))
d)))
= {|
nwPackets := (@send_packets _ orig_multi_params h l) ++ (nwPackets (revertNetwork st));
nwState := update name_eq_dec (nwState (revertNetwork st)) h d
|}.