On definition trees of ordinal recursive functionals: Reduction of the recursion orders by means of type level raising

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Abstract
It is known that every < ε0-recursive function is also a primitive recursive functional. Kreisel has proved this by means of Gödel's functional-interpretation, using that every < ε0-recursive function is provably recursive in Heyting's arithmetic [2, §3.4]. Parsons obtained a refinement of Kreisel's result by a further examination of Gödel's interpretation with regard to type levels [3, Theorem 5], [4, §4]. A quite different proof is provided by the research into extensions of the Grzegorczyk hierarchy as done by Schwichtenberg and Wainer: this yields another characterization of the < ε0-recursive functions from which easily appears that these are primitive recursive functionals (see [5] in combination with [6, Chapter II]).However, these proofs are indirect and do not show how, in general, given a definition tree of an ordinal recursive functional, transfinite recursions can be replaced (in a straightforward way) by recursions over wellorderings of lower order types. The argument given by Tait in [9, pp. 189–191] seems to be an improvement in this respect, but the crucial step in it is (at least in my opinion) not very clear.

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