(*SUBJECT, Property and TransFunc*)
Ltac OpDontChangeStPSS :=
intros s t Sub SS TF; inversion TF;
match goal with
| id:(s = t) |- _ => rewrite <- id; auto
end.
Ltac OpDontChangeStPSP :=
intros s t Sub SP TF;  (* SUBJECT, Property and TransFunc *)
inversion TF; match goal with
| id:(s = t) |- _ => rewrite <- id; auto
end.
Ltac StartPSS := intros s t Sub SS TF; inversion TF.
Ltac BreakSS :=
match goal with
| SS:(SecureState _) |- _ =>
unfold SecureState in SS; elim SS; intros DAC MAC
end.
Ltac StartPSP := intros s t Sub SP TF; inversion TF.
unfold DACSecureState in |- *.
intros.
cut
match fsecmat (secmat s) o with
| Some y =>
(set_In u (ActReaders y) -> PreDACRead s u o) /\
(set_In u (ActWriters y) -> PreDACWrite s u o)
| None => True
end.
elim (fsecmat (secmat s) o).
intros.
elim H0; intros.
auto.
auto.
apply H.
Qed.
unfold DACSecureState in |- *.
intros.
cut
match fsecmat (secmat s) o with
| Some y =>
(set_In u (ActReaders y) -> PreDACRead s u o) /\
(set_In u (ActWriters y) -> PreDACWrite s u o)
| None => True
end.
elim (fsecmat (secmat s) o).
intros.
elim H0; intros.
auto.
auto.
apply H.
Qed.
intros.
cut (~ set_In u (ActReaders rw) /\ ~ set_In u (ActWriters rw)).
tauto.
auto.
Qed.
intros.
cut (~ set_In u (ActReaders rw) /\ ~ set_In u (ActWriters rw)).
tauto.
auto.
Qed.
intro; intro; unfold FuncPre1 in |- *; elim o; simpl in |- *; intros.
cut
((forall o : OBJECT,
set_In o (DOM OBJeq_dec (directories s)) -> ObjType o = Directory) /\
(forall o : OBJECT, set_In o (DOM OBJeq_dec (files s)) -> ObjType o = File) /\
DOM OBJeq_dec (acl s) =
set_union OBJeq_dec (DOM OBJeq_dec (files s))
(DOM OBJeq_dec (directories s))).
intro CUT; elim CUT; intros.
elim H3; intros.
rewrite H5; intro.
cut
(set_In (a, File) (domf (files s)) \/
set_In (a, Directory) (domd (directories s))).
intro H7; elim H7; intro.
auto.
auto.
generalize H6; elim b; intro.
left; cut (~ set_In (a, File) (DOM OBJeq_dec (directories s))).
intro; unfold domf in |- *.
cut
(set_In (a, File)
(set_union OBJeq_dec (DOM OBJeq_dec (directories s))
(DOM OBJeq_dec (files s)))).
intro; eauto.
auto.
intro; absurd (ObjType (a, File) = Directory).
simpl in |- *.
intro H11; discriminate H11.
auto.
auto.
right.
cut (~ set_In (a, Directory) (DOM OBJeq_dec (files s))).
intro; unfold domd in |- *; eauto.
intro; absurd (ObjType (a, Directory) = File).
simpl in |- *.
intro H11; discriminate H11.
auto.
auto.
Qed.
Hint Resolve UniqNames.
unfold eq_sc, le_sc in |- *; intros.
elim H; intros.
rewrite H0; rewrite H1.
auto.
Qed.

Lemma NotInDOMIsUndef2 :
forall (s : SFSstate) (o1 o2 : OBJECT),
~ set_In o1 (domsecmat (secmat s)) ->
o1 = o2 -> None = fsecmat (secmat s) o2.