Proof with auto.
intro sh.
destruct (eq_dec sh top); subst.
left. apply fullshare_full.
right. intro. apply n.
generalize (Share.top_correct sh);intro.
apply leq_join_sub in H0.
destruct H0.
specialize ( H x). spec H. exists top...
specialize ( H sh top (join_comm H0))...
Qed.

Lemma nonemp_split_neq1: forall sh sh1 sh2, nonidentity sh -> split sh = (sh1, sh2) -> sh1 <> sh.
Proof with auto.