Xu, Chuangjie
(2020):
A SYNTACTIC APPROACH TO CONTINUITY OF TDEFINABLE FUNCTIONALS.
In: Logical Methods in Computer Science, Vol. 16, No. 1, 22

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Abstract
We give a new proof of the wellknown fact that all functions (N > N) > N which are definable in Godel's System T are continuous via a syntactic approach. Differing from the usual syntactic method, we firstly perform a translation of System T into itself in which natural numbers are translated to functions (N > N) > N. Then we inductively define a continuity predicate on the translated elements and show that the translation of any term in System T satisfies the continuity predicate. We obtain the desired result by relating terms and their translations via a parametrized logical relation. Our constructions and proofs have been formalized in the Agda proof assistant. Because Agda is also a programming language, we can execute our proof to compute moduli of continuity of Tdefinable functions.