On the Values of Reducibility Candidates
TLCA '09 Proceedings of the 9th International Conference on Typed Lambda Calculi and Applications
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U. Bergel; [ I I ] signijicantly simplijied Tait's normalisation proof for bar recursion [27], see also [9], replacing Tait's introduction of injinite terms by the construction of a domain having the property that a term is strongly normalizing if its semantics is \ne. The goal of this paper is to show that, using ideas from the theory of intersection types [2, 6, 7, 211 and Martin-Liif's domain interpretation of type theory [18], we can in turn simplify U. Berger's argument in the construction of such a domain model. We think that our domain model can be used to give modular proofs of strong normalization for various type theory. As an example, we show in some details how it can be used to prove strong normalization for Martin-Liif dependent type theory extended with bar recursion, and with some form ofproof-irrelevance.