GADTs and Exhaustiveness: Looking for the Impossible

Sound exhaustiveness checking of pattern-matching is an essential feature of GADTs, and OCaml has supported it from day one, by showing that the remaining cases could never be typed [1]. Not only does it allow the programmer to be confident in the soundness of his code, but it also permits optimizations which make GADTs more efficient. However, while this approach is sound and can prune some simple uses of GADTs, some other uses caused superfluous warnings. In this talk we describe the original approach and how we ensure its soundness, and show that one can do better by turning the type-checking of extra cases into a backtracking proof search algorithm. We also show that the exhaustiveness problem is undecidable for GADTs, so that this proof search must be kept partial.