Proof with simpl; auto.
induction atms... intros [A B] C. split... solve [eapply pure_atomd_pure; eauto].
Qed.

Lemma listd_pure_atomdu atms s0 s :
listd pure_atom_denote un FF atms s0 -> join_sub s0 s ->
listd pure_atom_denote un FF atms s.