Proof with simpl; auto.
intros A B; destruct c as [gamma delta| |]; destruct d as [gamma' delta'| |]...
2: simpl; destruct gamma, delta... 2: destruct p... 2: destruct e...
2: destruct p0... 2: destruct e...
2: simpl; destruct gamma, delta... 2: destruct p... 2: destruct e...
2: destruct p0... 2: destruct e...
destruct delta... destruct p... destruct e...
destruct delta'... destruct p... destruct e...
remember (expr_eq (Var v) (Var v0)) as b; destruct b... if_tac...
rewrite <-(expr_eq_eq' _ _ Heqb) in *. intros H. split... intros H1.
simpl in A, B. apply listd_merge_inter' in H1. destruct H1 as [H2 H3].
2: intros; rewrite pure_atom_cmp_eq...
specialize (A H2); specialize (B H3); rewrite (@listd_insert_uniq_un _ state).
2: intros x y; rewrite <-pure_atom_cmp_eq... destruct A, B.
assert (C: (e0 === e1) s) by (eapply var_eq_trans; apply var_eq_sym in H0; eauto).
left. if_tac... apply var_eq_sym...
right; apply listd_merge_un2... solve[apply pure_atom_cmp_eq; auto].
right; apply listd_merge_un1... solve[apply pure_atom_cmp_eq; auto].
right; apply listd_merge_un1... solve[apply pure_atom_cmp_eq; auto].
Qed.

Lemma negative_superposition_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 (negative_superposition c d l0) s.