A DIRECT PROOF OF SCHWICHTENBERG'S BAR RECURSION CLOSURE THEOREM

In [12], Schwichtenberg showed that the System T definable functionals are closed under a rule-like version Spector's bar recursion of lowest type levels 0 and 1. More precisely, if the functional Y which controls the stopping condition of Spector's bar recursor is T-definable, then the corresponding bar recursion of type levels 0 and 1 is already T-definable. Schwichtenberg's original proof, however, relies on a detour through Tait's infinitary terms and the correspondence between ordinal recursion for alpha < epsilon(0) and primitive recursion over finite types. This detour makes it hard to calculate on given concrete system T input, what the corresponding system T 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 T-definitions under the conditions of Schwichtenberg's theorem. Finally, with the explicit construction we can also easily state a sharper result: if Y is in the fragment T-i then terms built from BRN,sigma for this particular Y are definable in the fragment T-i+max{1,T-level(sigma)}+2.