Section rmdirIsSecure.
StartPSS.
inversion H.
unfold SecureState in |- *; BreakSS; split.
unfold DACSecureState in |- *; simpl in |- *; intros.
elim (OBJeq_dec o o0); intros.
rewrite <- a.
replace (fsecmat (secmat s) o) with (None (A:=ReadersWriters)).
trivial.
symmetry  in |- *; unfold fsecmat in |- *; auto.
unfold PreDACRead, PreDACWrite in |- *; simpl in |- *.
replace (facl (rmdir_acl s o) o0) with (facl (acl s) o0).
unfold DACSecureState, PreDACRead, PreDACWrite in DAC; apply DAC.
auto.
unfold MACSecureState in |- *; simpl in |- *.
intros.
elim (OBJeq_dec o o0).
intro; rewrite <- a.
replace (fsecmat (secmat s) o) with (None (A:=ReadersWriters)).
elim (fOSC (rmdir_oSC s o) o); elim (fSSC (subjectSC s) u0); trivial.
symmetry  in |- *; unfold fsecmat in |- *; auto.
intro EQ.
replace (fOSC (rmdir_oSC s o) o0) with (fOSC (objectSC s) o0).
unfold MACSecureState in MAC; apply MAC.
auto.
Qed.
StartPSP.
inversion H.
unfold StarProperty in |- *; simpl in |- *; intros.
elim (OBJeq_dec o o1); elim (OBJeq_dec o o2); intros EQ2 EQ1.
replace (fsecmat (secmat s) o1) with (None (A:=ReadersWriters)).
elim (fOSC (objectSC s) o2); elim (fOSC (objectSC s) o1);
elim (fOSC (rmdir_oSC s o) o2); elim (fOSC (rmdir_oSC s o) o1);
elim (fsecmat (secmat s) o2); trivial.
rewrite <- EQ1; symmetry  in |- *; unfold fsecmat in |- *; auto.
replace (fsecmat (secmat s) o1) with (None (A:=ReadersWriters)).
elim (fsecmat (secmat s) o2); elim (fOSC (objectSC s) o2);
elim (fOSC (objectSC s) o1); elim (fOSC (rmdir_oSC s o) o2);
elim (fOSC (rmdir_oSC s o) o1); trivial.
rewrite <- EQ1; symmetry  in |- *; unfold fsecmat in |- *; auto.
replace (fsecmat (secmat s) o2) with (None (A:=ReadersWriters)).
elim (fsecmat (secmat s) o1); elim (fOSC (objectSC s) o2);
elim (fOSC (objectSC s) o1); elim (fOSC (rmdir_oSC s o) o2);
elim (fOSC (rmdir_oSC s o) o1); trivial.
rewrite <- EQ2; symmetry  in |- *; unfold fsecmat in |- *; auto.
replace (fOSC (rmdir_oSC s o) o2) with (fOSC (objectSC s) o2).
replace (fOSC (rmdir_oSC s o) o1) with (fOSC (objectSC s) o1).
unfold StarProperty in SP; apply SP.
auto.
auto.
Qed.

Lemma RmdirPCP :
forall s t : SFSstate,
FuncPre4 s -> FuncPre6 s -> PreservesControlProp s Rmdir t.