Proof using.
apply cross_relation; intros.
- intuition.
+ red. red. auto.
+ unfold last_holder in *. simpl in *. discriminate.
+ unfold last_holder in *. simpl in *. discriminate.
- match goal with
| [ H : step_async _ _ _ |- _ ] => invcs H
end; monad_unfold; unlabeled_unfold;
unfold lb_net_handlers, lb_input_handlers in *; simpl in *;
repeat break_let; repeat find_inversion.
+ unfold NetHandler in *. break_match.
* find_apply_lem_hyp ClientNetHandler_cases; eauto.
break_and.
{ break_or_hyp.
- intuition; subst.
+ apply trace_mutex'_no_out_extend; auto.
+ rewrite update_nop_ext.
find_apply_lem_hyp last_holder'_no_out_inv.
auto.
+ match goal with
| [ H : _ |- _ ] => rewrite update_nop in H
end.
find_apply_hyp_hyp.
apply last_holder'_no_out_extend. auto.
- intuition; subst.
+ apply trace_mutex'_locked_extend. auto.
destruct (last_holder' None tr) eqn:?; auto.
find_apply_hyp_hyp.
erewrite locked_in_flight_all_clients_false in * by
eauto using locks_correct_locked_invariant, reachable_intro,
locks_correct_invariant.
discriminate.
+ my_update_destruct; try find_inversion; rewrite_update; auto.
find_apply_lem_hyp last_holder'_locked_some_eq. congruence.
+ my_update_destruct; try find_inversion; rewrite_update.
* apply last_holder'_locked_extend.
* erewrite locked_in_flight_all_clients_false in * by
eauto using locks_correct_locked_invariant, reachable_intro,
locks_correct_invariant.
discriminate.
}
* { find_apply_lem_hyp ServerNetHandler_cases. break_and. subst.
repeat split.
- apply trace_mutex'_no_out_extend. auto.
- intros. my_update_destruct; try discriminate.
rewrite_update.
find_apply_lem_hyp last_holder'_no_out_inv.
auto.
- intros. my_update_destruct; try discriminate; rewrite_update.
apply last_holder'_no_out_extend. auto.
}
+ unfold InputHandler in *. break_match.
* unfold ClientIOHandler in *.
{ monad_unfold.
repeat break_match; repeat find_inversion; intuition;
repeat rewrite snoc_assoc in *;
try apply trace_mutex'_no_out_extend;
try find_apply_lem_hyp last_holder'_no_out_inv;
try (apply last_holder'_no_out_extend; auto).
- apply trace_mutual_exclusion'_extend_input; auto. congruence.
- rewrite update_nop_ext.
find_apply_lem_hyp last_holder'_input_inv; try congruence.
auto.
- match goal with
| [ H : _ |- _ ] => rewrite update_nop in H
end.
apply last_holder'_input_extend; auto. congruence.
- apply trace_mutex'_unlock_extend; auto.
- rewrite last_holder_unlock_none in *; auto. discriminate.
- my_update_destruct; try find_inversion; rewrite_update.
+ discriminate.
+ assert (mutual_exclusion (nwState st))
by eauto using mutual_exclusion_invariant, reachable_intro.
unfold mutual_exclusion in *.
assert (c = n) by eauto. congruence.
- apply trace_mutex'_unlock_extend. auto.
- rewrite update_nop.
find_apply_lem_hyp last_holder_some_unlock_inv.
auto.
- match goal with
| [ H : _ |- _ ] => rewrite update_nop in H
end.
assert (n <> c) by congruence.
find_apply_hyp_hyp.
apply last_holder'_neq_unlock_extend; auto.
- apply trace_mutual_exclusion'_extend_input; auto. congruence.
- rewrite update_nop_ext. find_apply_lem_hyp last_holder'_input_inv; try congruence.
auto.
- match goal with
| [ H : _ |- _ ] => rewrite update_nop in H
end.
apply last_holder'_input_extend; auto. congruence.
}
* unfold ServerIOHandler in *.
monad_unfold. find_inversion.
{ intuition;
repeat rewrite snoc_assoc in *.
- apply trace_mutex'_no_out_extend.
apply trace_mutual_exclusion'_extend_input_server. auto.
- find_apply_lem_hyp last_holder'_no_out_inv.
rewrite update_nop. find_apply_lem_hyp last_holder'_input_inv_server. auto.
- apply last_holder'_no_out_extend; auto.
rewrite_update. unfold last_holder. rewrite last_holder'_server_extend.
auto.
}
Qed.

Lemma head_grant_state_unlock :
forall st tr c t,
step_async_star step_async_init st tr ->
queue (nwState st Server) = c :: t ->
(In (mkPacket Server (Client c) Locked) (nwPackets st)) \/
(held (nwState st (Client c)) = true) \/
(In (mkPacket (Client c) Server Unlock) (nwPackets st)).