Programming up to Congruence

Abstract

This paper presents the design of Zombie, a dependently-typed programming language that uses an adaptation of a congruence closure algorithm for proof and type inference. This algorithm allows the type checker to automatically use equality assumptions from the context when reasoning about equality. Most dependently-typed languages automatically use equalities that follow from beta-reduction during type checking; however, such reasoning is incompatible with congruence closure. In contrast, Zombie does not use automatic beta-reduction because types may contain potentially diverging terms. Therefore Zombie provides a unique opportunity to explore an alternative definition of equivalence in dependently-typed language design.