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Anba h_ b,a arbitrary one-place function assumed functions based on index classes complete sequence completes the proof computation based constant function Corollaries course-of-values induction course-of-values recursion Define the function DOCTOR OF PHILOSOPHY exponent fin(y fin(yn follows by induction function 0 otherwise function Aba function defined function h function primitive recursive function value Furthermore given general recursive h b,a h is primitive h-types hypothesis of induction index of h k-recursive Lemma limit notation majorizes the one-place monotone increasing monotone non-decreasing one-place primitive recursive ordinal notations ordinals less pr b,a predicate or set primitive recursive degree primitive recursive function recursive in Anba recursive in h recursive in pr(b,a recursively as follows representing function schemata segment of notations sequences of notations suffices to take uniqueness property fails uniqueness property holds University of Wisconsin y-index