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. 
Andreas Abel (Ludwig-Maximilians-Universit\"at M\"unchen), Thierry   Coquand (G\"oteborg University), Miguel Pagano (Univers...
cs.LO