The Relative Complement Problem for Higher-Order Patterns

We address the problem of complementing higher-order patterns without repetitions of free variables. Di erently from the rst-order case, the complement of a pattern cannot, in general, be described by a pattern, or even by a nite set of patterns. We therefore generalize the simply-typed -calculus to include an internal notion of strict function so that we can directly express that a term must depend on a given variable. We show that, in this more expressive calculus, nite sets of patterns without repeated variables are closed under complement and uni cation. Our principal application is the transformational approach to negation in higher-order logic programs.