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A DIRECT PROOF OF SCHWICHTENBERG'S BAR RECURSION CLOSURE THEOREM
In 1979 Schwichtenberg showed that the System definable
functionals are closed under a rule-like version Spector's bar recursion of
lowest type levels and . More precisely, if the functional which
controls the stopping condition of Spector's bar recursor is
-definable, then the corresponding bar recursion of type levels
and is already -definable. Schwichtenberg's original proof,
however, relies on a detour through Tait's infinitary terms and the
correspondence between ordinal recursion for and
primitive recursion over finite types. This detour makes it hard to calculate
on given concrete system input, what the corresponding system
output would look like. In this paper we present an alternative
(more direct) proof based on an explicit construction which we prove correct
via a suitably defined logical relation. We show through an example how this
gives a straightforward mechanism for converting bar recursive definitions into
-definitions under the conditions of Schwichtenberg's theorem.
Finally, with the explicit construction we can also easily state a sharper
result: if is in the fragment then terms built from
for this particular are definable in the
fragment .Comment: 12 page