Proof with auto.
intros H1 H2. unfold ef. destruct ce... destruct c... destruct delta...
destruct p... destruct e... if_tac...
apply ef_aux_sound... simpl; auto. apply var_eq_refl.
Qed.

Lemma remove_trivial_atoms_sound b atms s :
listd pure_atom_denote inter b atms s ->
listd pure_atom_denote inter b (remove_trivial_atoms atms) s.