intros s t Sub FP5 SS TF; inversion TF.
inversion H.
unfold SecureState in |- *.
BreakSS.
split.
unfold DACSecureState in |- *; intros; simpl in |- *.
elim (OBJeq_dec o o0); intro y0.
rewrite <- y0.
elim (SUBeq_dec Sub u0); intro y1.
rewrite <- y1.
cut
match fsecmat (close_sm s Sub o) o with
| None => True
| Some y => ~ set_In Sub (ActReaders y) /\ ~ set_In Sub (ActWriters y)
end.
elim (fsecmat (close_sm s Sub o) o).
intros; split; intro; tauto.
auto.
apply Close_smCorr; auto.
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.
cut
match fsecmat (secmat s) o with
| None => True
| Some y =>
(set_In u0 (ActReaders y) -> PreDACRead s u0 o) /\
(set_In u0 (ActWriters y) -> PreDACWrite s u0 o)
end.
unfold close_sm at 3 4, PreDACRead, PreDACWrite in |- *; simpl in |- *;
elim (fsecmat (secmat s) o); elim (fsecmat (close_sm s Sub o) o).
intros.
elim H6; elim H5; intros.
split; intro.
apply H7; apply H9; auto.
apply H8; apply H10; auto.
auto.
tauto.
auto.
unfold DACSecureState in DAC; apply DAC.
apply Close_smCorr2; auto.
replace (fsecmat (close_sm s Sub o) o0) with (fsecmat (secmat s) o0).
unfold close_sm, PreDACRead, PreDACWrite in |- *; simpl in |- *;
unfold DACSecureState, PreDACRead, PreDACWrite in DAC;
apply DAC.
auto.
unfold MACSecureState in |- *; intros; simpl in |- *.
elim (OBJeq_dec o o0); intro y0.
rewrite <- y0.
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.
cut
match fsecmat (secmat s) o, fOSC (objectSC s) o, fSSC (subjectSC s) u0 with
| None, _, _ => True
| _, None, _ => True
| _, _, None => True
| Some x, Some y, Some z =>
set_In u0 (ActReaders x) \/ set_In u0 (ActWriters x) -> le_sc y z
end.
elim (fsecmat (secmat s) o); elim (fsecmat (close_sm s Sub o) o);
elim (fOSC (objectSC s) o); elim (fSSC (subjectSC s) u0);
contradiction || trivial.
tauto.
unfold MACSecureState in MAC; apply MAC.
apply Close_smCorr2; auto.
replace (fsecmat (close_sm s Sub o) o0) with (fsecmat (secmat s) o0).
unfold MACSecureState in MAC; apply MAC.
auto.
Qed.

(* Specifications: *)
Lemma ClosePSP :
forall (s t : SFSstate) (u : SUBJECT),
FuncPre5 s -> StarProperty s -> TransFunc u s Close t -> StarProperty t.