Internalising modified realisability in constructive type theory

A modified realisability interpretation of infinitary logic is formalised and proved sound in constructive type theory (CTT). The logic considered subsumes first order logic. The interpretation makes it possible to extract programs with simplified types and to incorporate and reason about them in CTT. 1 Modified realisability Modified realisability interpretation is a well-known method for giving constructive interpretation of some intuitionistic logical system into a simple type structure (Troelstra 1973). The method is used, for instance, in Minlog and Coq for extracting programs from proofs (cf. Schwichtenberg 2004 and Letouzey 2004). These programs are to a large extent free from the computationally irrelevant parts that might be present in programs arising from direct interpretations into constructive type theory. The realisability interpretation requires a separate proof of correctness, which is usually left unformalised. In this note we present a completely formalised modified realisability interpretation carried out in the proof support system Agda (Coquand 2000). We shall here use what is calledmodified realisability with truthwhich has the property that anything realised is also true in the system (Theorem 1.2). One difference from usual interpretations as in Minlog is that the logic interpreted goes beyond first order logic: it is a (constructively) infinitary logic, which arises naturally from the type-theoretic notion of universe. Our extension to infinitary logic seems to be a novel result. Agda is based on Martin-Löf constructive type theory (1972) with an infinite hierarchy of universes #0 = Set, #1 = Type, #2 = Kind, #3, . . . . Each of these