Lemma join_symmetry: forall S D a b, joins S D a b -> joins S D b a.
Proof.

++ clear IHmultipar. left. destruct H1 as [a1' K]. exists a1'. inversion K. clear K. inversion H2. clear H2. repeat split; eauto. ++ clear IHmultipar. right. destruct H1 as [a1' K]. exists a1'. inversion K. clear K. repeat split; eauto.