intros s t Sub FP5 SP TF; inversion TF.
inversion H.
unfold StarProperty in |- *; simpl in |- *; intros.
cut
match
fsecmat (secmat s) o1, fsecmat (secmat s) o2, fOSC (objectSC s) o2,
fOSC (objectSC s) o1
with
| None, _, _, _ => True
| _, None, _, _ => True
| _, _, None, _ => True
| _, _, _, None => True
| Some w, Some x, Some y, Some z =>
set_In u0 (ActWriters w) -> set_In u0 (ActReaders x) -> le_sc y z
end.
cut
match fsecmat (close_sm s Sub o) o, fsecmat (secmat s) o with
| _, None => False
| None, Some z => True
| Some y, Some z =>
(set_In u0 (ActReaders y) -> set_In u0 (ActReaders z)) /\
(set_In u0 (ActWriters y) -> set_In u0 (ActWriters z))
end.
elim (OBJeq_dec o o1); elim (OBJeq_dec o o2); intros EQ2 EQ1.
rewrite <- EQ1; rewrite <- EQ2.
elim (fsecmat (secmat s) o); elim (fOSC (objectSC s) o);
elim (fsecmat (close_sm s Sub o) o); intros; auto.
replace (fsecmat (close_sm s Sub o) o2) with (fsecmat (secmat s) o2).
rewrite <- EQ1.
elim (fsecmat (secmat s) o2); elim (fOSC (objectSC s) o);
elim (fOSC (objectSC s) o2); elim (fsecmat (secmat s) o);
elim (fsecmat (close_sm s Sub o) o); intros; contradiction || auto.
tauto.
auto.
replace (fsecmat (close_sm s Sub o) o1) with (fsecmat (secmat s) o1).
rewrite <- EQ2.
elim (fsecmat (secmat s) o1); elim (fOSC (objectSC s) o);
elim (fOSC (objectSC s) o1); elim (fsecmat (secmat s) o);
elim (fsecmat (close_sm s Sub o) o); intros; contradiction || auto.
tauto.
auto.
replace (fsecmat (close_sm s Sub o) o2) with (fsecmat (secmat s) o2).
replace (fsecmat (close_sm s Sub o) o1) with (fsecmat (secmat s) o1).
auto.
auto.
auto.
apply Close_smCorr2; auto.
unfold StarProperty in SP; apply SP.
Qed.

Lemma ClosePCP : forall s t : SFSstate, PreservesControlProp s Close t.
intros; unfold PreservesControlProp in |- *; intros Sub TF; inversion TF;
unfold ControlProperty in |- *.