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		12.24 Claim For all i:

		(a) If i = g(j) for some j, then .

		(b) If g(j) < i < g(j) + j + 3 for some j and is infinite, then .

		(c) If for some j, then .

		The above claim follows immediately from the definition of  y . We also have:

		12.25 Claim Suppose L is an infinite r.e. set with MinGram y (L) > 0. Let j + 1 = MinGram y (L) and let A be the set of h-minimal  y -grammars for L. Then:

		(a) .

		(b) For each i e A and each  i  < j, we have that .

		Proof: Let S = { g(j) + 1, . . ., g(j) + j + 2}. Suppose and there is no  i  < j such that . Then it follows from the definition of that as s goes to infinity, so does the s' of clause (ii). Hence, . Since there are only j + 1 many and j + 2 many , there is at least one  y -grammar for L in S. Let i* be the least  y -grammar for L in S. Since L is infinite, it follows from our choice of j + 1 and Claim 12.24 that i* = MinGram y (L). Since h is strictly increasing, we have that . Hence, by Claim 12.24(c), all the h-minimal  y -grammars must be . Part (a) of the claim thus follows. Part (b) follows using part (a) and the construction of .

		We leave the rest of the proof to the reader.

		Jain and Sharma [89] discuss other properties of the h-TxtMex paradigm.

		§12.7 Bibliographic Notes

		The subject of bounding the number of mind changes was first considered by Barzdins * and Freivalds [15].

		Case and Smith [35] were the first to study the interaction of mind changes and anomalies in the final program. For other results relating to mind changes, see Case, Jain, and Ngo-Manguelle [28]. Case, Jain, and Sharma [31] investigate mind changes in the context of vacillatory identification of functions. Mind change complexity of indexed families of computable languages has been investigated by Mukouchi [135] and by Lange and Zeugmann [118]. Recently, Freivalds and Smith [66] proposed a generalized notion