Proof with try tauto.
intros. rewrite Share.lub_rel_recompose in H.
generalize (Share.decompose_recompose (Share.lub a1 a2, Share.lub b1 b2));intro.
rewrite H in H0.
rewrite Share.lub_rel_recompose in H0.
rewrite Share.decompose_recompose in H0.
split;congruence.
Qed.

Lemma join_splice2_aux1:
forall a1 a2 b1 b2,
Share.lub (Share.rel Share.Lsh (Share.glb a1 a2)) (Share.rel Share.Rsh (Share.glb b1 b2)) = Share.bot ->
Share.glb a1 a2 = Share.bot /\ Share.glb b1 b2 = Share.bot.