A Modular Type-checking algorithm for Type Theory with Singleton Types and Proof Irrelevance

We define a logical framework with singleton types and one universe of small types. We give the semantics using a PER model; it is used for constructing a normalisation-by-evaluation algorithm. We prove completeness and soundness of the algorithm; and get as a corollary the injectivity of type constructors. Then we give the definition of a correct and complete type-checking algorithm for terms in normal form. We extend the results to proof-irrelevant propositions.

• Type: Journal
• Year: 2011
• Where: CORR
• Authors: Andreas Abel, Thierry Coquand, Miguel Pagano
• Keywords: Algorithm, Complete Type-checking Algorithm, CORR 2011, Education, Normalisation-by-evaluation Algorithm