In [6] we have proposed a general higher-order unification method using a theory of explicit substitutions and we have proved its completeness. In this paper, we investigate the case of higher-order patterns as introduced by Miller. We show that our general algorithm specializes in a very convenient way to patterns. We also sketch an efficient implementation of the abstract algorithm and its generalization to constraint simplification, which has yielded good experimental results at the core of a higher-order constraint logic programming language.

Abstract not found

Here we present a first-order formalization of set theory that has a finite number of axioms and which syntax is similar to the one often used in books: it provides an encoding of the comprehension symbol. Other formalizations of set theory exist: Zermelo theory with existence axioms is first-order but has no comprehension symbol and has an infinite number of axioms.

This abstract has a pragmatic intent: it explains the use of an explicit substitution notation in an implementation of the higher-order logic programming language λProlog. The particular aspects of this language that are of interest here are its provision of typed lambda terms as a means for representing objects and of higher-order unification as a tool for probing the structures of these objects. There are many uses for these facilities originating from the fact that they lead to direct and declarative support for a higher-order abstract syntax view of objects such as formulas and programs [MN87, PE88]. Detailed discussions of applications can be found in the literature, e.g. see [Fel93, HM92, NM94, Per91, Pfe88]. Success encountered in these various experiments has driven an effort on our part towards developing a good implementation of the language. An important ingredient of such an implementation is, of course, a sensible treatment of lambda terms. The use that is made of...