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. Our work includes the specification of the language via a bidirectional type system, which works "up-to-congruence,'' and an algorithm for elaborating expressions in this language to an explicitly typed core language. We prove that our elaboration algorithm is complete with respect to the source type system, and always produces well typed terms in the core language. This algorithm has been implemented in the Zombie language, which includes general recursion, irrelevant arguments, heterogeneous equality and datatypes.