Fixpoint RoseS_spec (x y z : bs) :=
match z with
| nil => nil
| _ :: z' =>
if leb (length z) (length x - length y)
then false :: RoseS_spec x y z'
else hd false (skipn (length x - length z) x) :: RoseS_spec x y z'
end.

Lemma RoseS_cons : forall x y z b1 b2,
length z <= length x ->
RoseS_spec (b1 :: x) (b2 :: y) z = RoseS_spec x y z.