A Direct Proof of Schwichtenberg's Bar Recursion Closure Theorem

Abstract

In 1979 Schwichtenberg showed that the System $\text{T}$ 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 $\text{T}$-definable, then the corresponding bar recursion of type levels $0$ and $1$ is already $\text{T}$-definable. Schwichtenberg's original proof, however, relies on a detour through Tait's infinitary terms and the correspondence between ordinal recursion for $\alpha < \varepsilon_0$ and primitive recursion over finite types. This detour makes it hard to calculate on given concrete system $\text{T}$ input, what the corresponding system $\text{T}$ 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 $\text{T}$-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 $\text{T}_i$ then terms built from $\text{BR}^{\mathbb{N}, \sigma}$ for this particular $Y$ are definable in the fragment $\text{T}_{i + \max \{ 1, \text{level}{\sigma} \} + 2}$.