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. Subcase, b) is safe. We must now prove that the run is maximal. None of the previous cases of the algorithm apply, so we must have b = a = ?, o = a = ?, and b = o. Since a = ? and b = ?, we have a ? b. Moreover, we have b ? o and b = ?. Hence the predicate localconflict(o, a, b) is satisfied. As before, by safety condition (3), the only safe run is, hence it is maximal

R. Case, Since previous cases of the algorithm do not apply, we have a = b, o = X , a = ?, b = ?, and a ? b. By Lemma 11, each of the schemas S(k) are path consistent for k ? dom(a) ? dom(b) By the definition of schema projection, for each k we have