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: simpl; destruct gamma, delta... 2: destruct p...
destruct gamma, delta as [|atm delta]...
destruct atm as [e1 e2], delta'... (*destruct p...*) simpl in A; spec A...
destruct gamma'... destruct gamma'... destruct p...
remember (expr_eq e1 e) as b; destruct b. 2: simpl; auto.
if_tac... split... intro H1. specialize (B H1). simpl in B.
rewrite (@listd_insert_uniq_un _ state)... 2: apply pure_atom_cmp_eq.
destruct A as [A|A];destruct B as [B|B]... rewrite <-(expr_eq_eq' _ _ Heqb) in B.
assert (C: (e0 === e2) s) by (eapply var_eq_trans; apply var_eq_sym in B; eauto).
left; if_tac... solve[apply var_eq_sym; auto].
right; apply listd_merge_un2... solve[apply pure_atom_cmp_eq].
right; apply listd_merge_un1... solve[apply pure_atom_cmp_eq].
right; apply listd_merge_un2... solve[apply pure_atom_cmp_eq].
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.