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A MODULAR TYPE-CHECKING ALGORITHM FOR TYPE THEORY WITH SINGLETON TYPES AND
 

Summary: A MODULAR TYPE-CHECKING ALGORITHM FOR
TYPE THEORY WITH SINGLETON TYPES AND
PROOF IRRELEVANCE
ANDREAS ABEL, THIERRY COQUAND, AND MIGUEL PAGANO
Ludwig-Maximilians-Universit¨at M¨unchen
e-mail address: andreas.abel@ifi.lmu.de
G¨oteborg University
e-mail address: coquand@chalmers.se
Universidad Nacional de C´ordoba
e-mail 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-
ation-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.
1. Introduction and Related Work
One of the raisons d'^etre of proof-checkers 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