Normalization by Evaluation for Martin-Löf Type Theory with One Universe

Authors:
Andreas Abel;Klaus Aehlig;Peter Dybjer
Affiliations:
Institut für Informatik, Ludwig-Maximilians-Universität, Oettingenstr. 67, D-80538 München;Department of Computer Science, University of Wales Swansea, Singleton Park, Swansea SA2 8PP;Department of Computer Science, Chalmers University of Technology, Rännvägen 6, S-41296 Göteborg
Venue:
Electronic Notes in Theoretical Computer Science (ENTCS)
Year:
2007

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Abstract

We present an algorithm for computing normal terms and types in Martin-Lof type theory with one universe and eta-conversion. We prove that two terms or types are equal in the theory iff the normal forms are identical (as de Bruijn terms). It thus follows that our algorithm can be used for deciding equality in Martin-Lof type theory. The algorithm uses the technique of normalization by evaluation; normal forms are computed by first evaluating terms and types in a suitable model. The normal forms are then extracted from the semantic elements. We prove its completeness by a PER model and its soundness by a Kripke logical relation.