Third-Order Matching in the Polymorphic Lambda Calculus

We show that it is decidable whether a third-order matching problem in the polymorphic lambda calculus has a solution. The proof is constructive in the sense that an algorithm can be extracted from it that, given such a problem, returns a substitution if it has a solution and fail otherwise. 1 Introduction This paper is a contribution to the theory of (pattern) matching in higher order type theory. The starting point is the fact that third-order matching is decidable in the simply typed lambda calculus with constant types (see [5]). The question we would like to answer is: what happens if we extend this calculus with the type features that are characteristic for the Calculus of Constructions [2]: dependent types, type constructors and polymorphism. In [3], Dowek showed that in lambda calculi with dependent types third-order matching is undecidable. In contrast, we showed in [15] that the presence of type constructors is not sufficient to make third-order matching undecidable. In this ..