Summary: A MODULAR TYPE­CHECKING ALGORITHM FOR
TYPE THEORY WITH SINGLETON TYPES AND
PROOF IRRELEVANCE
ANDREAS ABEL, THIERRY COQUAND, AND MIGUEL PAGANO
Ludwig­Maximilians­Universit˜at M˜unchen
e­mail address: andreas.abel@ifi.lmu.de
G˜oteborg University
e­mail address: coquand@chalmers.se
Universidad Nacional de C’ordoba
e­mail address: pagano@famaf.unc.edu.ar
Abstract. 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 normalis­
ation­by­evaluation 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 type­checking algorithm for terms in normal form. We extend the
results to proof­irrelevant propositions.
1. Introduction and Related Work
One of the raisons d'“etre of proof­checkers like Agda [46], Coq [33], and Epigram [40] is
to decide if a given term has some type (either checking for a given type or inferring one);
i.e., if a term corresponds to a proof of a proposition [32]. Hence, the convenience of such

Source: Abel, Andreas - Theoretische Informatik, Ludwig-Maximilians-Universität München

Collections: Computer Technologies and Information Sciences