Proof with simpl; auto.
intros H1 H2 H3.
generalize (negative_superposition_sound c d l0 s) as H4; intro.
generalize (positive_superposition_sound c d l0 s) as H5; intro.
generalize (equality_factoring_sound c l0 s) as H6; intro.
generalize (ordered_factoring_sound c l0 s) as H7; intro.
specialize (H4 H1 H2 H3); specialize (H5 H1 H2 H3).
specialize (H6 H1 H3); specialize (H7 H1 H3).
rewrite (@listd_unfold_inter _ state). split...
destruct c, d... simpl in H4, H5, H6.
destruct gamma, delta... destruct p...
destruct gamma0, delta0... destruct p... if_tac... split... intros _.
destruct H5 as [H5 _].
rewrite (@listd_unfold_un _ state). left. apply H5...
Qed.

Lemma superpose_sound c d l0 s :
clause_denote c s -> clause_denote d s -> listd clause_denote inter TT l0 s ->
listd clause_denote inter TT (superpose c d l0) s.