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

doi:10.1016/j.entcs.2007.02.025

Electronic Notes in Theoretical Computer Science

Volume 173, 2 April 2007, Pages 17-39

https://doi.org/10.1016/j.entcs.2007.02.025 Get rights and content

Under a Creative Commons license

open access

Abstract

We present an algorithm for computing normal terms and types in Martin-Löf 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-Löf 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.

Keywords

Dependent Types

Domain Semantics

Normalization by Evaluation

Type Theory

Universe

Cited by (0)

¹: Partially supported by the EU coordination action TYPES (510996).

³: Partially supported by grant EP/D03809X/1 of the British Engineering and Physical Sciences Research Council (EPSRC). Currently visiting University of Toronto, supported by grant Ae 102/1-1 of the Deutsche Forschungsgemeinschaft (DFG).

⁵: Partially supported by project TLCA, Vetenskapsrådet.