Programming up to congruence

This paper presents a dependently-typed programming language that uses an adaptation of a congruence closure algorithm for proof and type inference. While most dependently-typed languages auto-matically use equalities that follow from β-reductions but do not automatically use known assumptions from the context, our lan-guage does the opposite. It uses assumptions but does not automat-ically reduce expressions. 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 explicitly typed core language. We prove that our elaboration algorithm is complete with respect to the source type system, and always pro-duces well typed terms in the core language. This algorithm has been implemented in the ZOMBIE language, which includes gen-eral recursion, irrelevant arguments, heterogeneous equality and datatypes