Abstract. Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. 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, preferably 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 universal Kripke model that proves normalization, consistency and decidability.

This paper introduces a new recursion principle for inductive data modulo

We extend the Logical Framework by proof irrelevant equality types and present an algorithm that computes unique long normal forms. The algorithm is inspired by normalization-by-evaluation. Equality proofs which are not reflexivity are erased to a single object ∗. The algorithm decides judgmental equality, its completeness is established by a PER model. 1.

Abstract. We consider the Calculus of Constructions with typed beta-eta equality and an algorithm which computes long normal forms. The normalization algorithm evaluates terms into a semantic domain, and reifies the values back to terms in normal form. To show termination, we interpret types as partial equivalence relations between values and type constructors as operators on PERs. This models also yields consistency of the beta-eta-Calculus of Constructions. The model construction can be carried out directly in impredicative type theory, enabling a formalization in Coq. 1

Abstract. Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. 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, preferably 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. Normalization, consistency, and decidability are obtained via a universal Kripke model based on algorithmic equality. 1. Introduction and Related