Programming Up to Congruence (Extended version)

This paper presents the design of ZOMBIE, a dependently-typedprogramming language that uses an adaptation of a congruenceclosure algorithm for proof and type inference. This algorithm al-lows the type checker to automatically use equality assumptionsfrom the context when reasoning about equality. Most dependently-typed languages automatically use equalities that follow from β-reduction during type checking; however, such reasoning is incom-patible with congruence closure. In contrast, ZOMBIE does not useautomatic β-reduction because types may contain potentially di-verging terms. Therefore ZOMBIE provides a unique opportunityto explore an alternative definition of equivalence in dependently-typed language design. Our work includes the specification of the language via a bidirec-tional type system, which works “up-to-congruence,” and an algo-rithm for elaborating expressions in this language to an explicitlytyped core language. We prove that our elaboration algorithm iscomplete with respect to the source type system, and always pro-duces well typed terms in the core language. This algorithm hasbeen implemented in the ZOMBIE language, which includes gen-eral recursion, irrelevant arguments, heterogeneous equality anddatatypes.