Proof with simpl; auto.
induction atms... intros [A | A] B.
left; eapply pure_atomd_pure; eauto. right...
Qed.

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