Justifying Algorithms for βη-Conversion

Deciding the typing judgement of type theories with dependent types such as the Logical Framework relies on deciding the equality judgement for the same theory. Implementing the conversion algorithm for βη-equality and justifying this algorithm is therefore an important problem for applications such as proof assistants and modules systems. This article gives a proof of decidability, correctness and completeness of the conversion algorithms for βη-equality defined by Coquand [3] and Harper and Pfenning [8] for the Logical Framework, relying on established metatheoretic results for the type theory. Proofs are also given of the same properties for a typed algorithm for conversion for System F, a new result.