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

Lemma listd_pure_atomd_pureu atms s0 s :
listd (fun atms : list pure_atom => listd pure_atom_denote un FF atms)
inter TT atms s0 ->
join_sub s0 s ->
listd (fun atms : list pure_atom => listd pure_atom_denote un FF atms)
inter TT atms s.