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		Case 2: Some stage s starts but fails to terminate. For each we have by the construction that and is total; hence . Note that for each . We show that M fails to Aexa-identify at least one of these  j p(i)For each , let . Clearly, the Ai's are disjoint. Hence, by Lemma 7.2(a), , which is . Now, if it were the case that , then we would have for each ; but then , a contradiction. Therefore, for at least one , .

		From the above cases it follows that .

		The Aex criteria is thus seen to be a hierarchy that sits above the Exa criteria and which is incomparable to the Bca hierarchy, except for the obvious degenerative case of Bc* = Aex1.

		We conclude this section by brief consideration of the text analogs of the Aex criteria.

		7.13 Definition Suppose . We say that M TxtAexa-identifies L (written: ) if and only if, for each text T for L, and . Define "M TxtAexa-identifies S" and "TxtAexa" in the usual way.

		The results for the Aex criteria all carry over to this new setting. For example, we have:

		7.14 Proposition If , then .

		In proving these results one cannot employ the trick of Section 6.2 used to convert Ex-based results to analogous TxtEx-based results. The difficulty is that the conversion plays havoc with densities. On the other hand, Lemma 7.3—the foundation for the arguments of this section—works equally well in the text setting. Hence, one can fairly directly translate Aex arguments to TxtAex arguments. Exercise 7-9 illustrates this method by working through a proof of Proposition 7.14. Since these translations are all straightforward, we leave TxtAex versions of the other results of this section as exercises for the reader.

		§7.4 Uniform Approximate Explanatory Identification

		The Aex criterion allows infinite many errors provided that the density of these errors is suitably bounded. Notice, however, that Aex permits approximations that —although