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.

Lemma eq_scIMPLYle_sc : forall a b : SecClass, eq_sc a b -> le_sc a b.
unfold eq_sc, le_sc in |- *; intros.