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 p...
2: simpl; destruct gamma, delta... 2: destruct p... 2: destruct p...
destruct gamma... destruct p... destruct delta'... destruct p...
destruct e1... remember (expr_eq (Var v) e) as b; destruct b...
if_tac; [simpl|simpl; auto]; rewrite interS; simpl. intro H3. split...
rewrite (@listd_insert_uniq_inter _ state)... 2: apply pure_atom_cmp_eq.
intros [H1 H2].
assert ((e2 === e0) s). if_tac in H1; simpl in H1... apply var_eq_sym...
apply listd_merge_inter' in H2. destruct H2 as [H4 H5]. simpl in B; spec B...
2: intros x y; rewrite <-pure_atom_cmp_eq...
destruct B.
assert ((e === e0) s). apply var_eq_trans with (y:=e2)...
solve[apply expr_eq_eq' in Heqb; rewrite Heqb in H0; auto].
simpl in A; spec A... split...
apply listd_merge_un2... solve[apply pure_atom_cmp_eq].
apply listd_merge_un1... solve[apply pure_atom_cmp_eq].
Qed.

Lemma reflexivity_resolution_sound c l0 s :
clause_denote c s -> listd clause_denote inter TT l0 s ->
listd clause_denote inter TT (reflexivity_resolution c l0) s.