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

Lemma do_well1_2_sound sigma s :
space_denote sigma s ->
listd (fun atms => listd pure_atom_denote un FF atms) inter TT
(do_well1_2 sigma) s.