 
Summary: Irrelevance in Type Theory with a
Heterogeneous Equality Judgement
Andreas Abel
Department of Computer Science
LudwigMaximiliansUniversity Munich
andreas.abel@ifi.lmu.de
Abstract. Dependently typed programs contain an excessive amount of static
terms which are necessary to please the type checker but irrelevant for computa
tion. To obtain reasonable performance of not only the compiled program but also
the type checker such static terms need to be erased as early as possible, prefer
ably immediately after type checking. To this end, Pfenning's type theory with
irrelevant quantification, that models a distinction between static and dynamic
code, is extended to universes and large eliminations. Novel is a heterogeneously
typed implementation of equality which allows the smooth construction of a uni
versal Kripke model that proves normalization, consistency and decidability.
Keywords: dependent types, proof irrelevance, heterogeneously typed equality,
algorithmic equality, logical relation, universal Kripke model.
1 Introduction and Related Work
Dependently typed programming languages such as Agda [10], Coq [14], and Epigram
[16] allow the programmer to express in one language programs, their types, rich in
