A Proof of Strong Normalisation using Domain Theory

U. Berger, (2005) significantly simplified Tait´s normalisation proof for bar recursion, replacing Tait´s introduction of infinite terms by the construction of a domain having the property that a term, is strongly normalizing if its semantics is neperp. The goal of this paper is to show that, using ideas from the theory of intersection types and Martin-Lof´s domain interpretation of type theory, 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-Lof dependent type theory extended with bar recursion, and with some form of proof-irrelevance