Third-Order Matching in the Presence of Type Constructors

We show that it is decidable whether a third-order matching problem in ! (an extension of the simply typed lambda calculus with type constructors) has a solution or not. We present an algorithm which, given such a problem, returns a solution for this problem if the problem has a solution and returns fail otherwise. We also show that it is undecidable whether a third-order matching problem in ! has a closed solution or not. 1 Introduction It is well-known that type theory is a good basis for the implementation of proof checkers. Although there are various ways to use type theory for proof checking, they all exploit the fact that type theory provides a uniform way to represent and manipulate proofs, formulas and data types. The man-machine interaction of proof checking can be considerably improved if some kind of matching algorithm can be implemented for the terms of the underlying type theory. For if one wants to prove OE(t) for a certain formula OE and term t, and one already has a pr..