Higher-Order Proof by Consistency Henrik Linnestad 1 , Christian Prehofer  2 and Olav Lysne  1 1 Department of Informatics, University of Oslo, PB 1080 Blindern, 0316 Oslo, Norway. 2 Institut f¨ ur Informatik, Technische Universit¨ at M¨ unchen, 80290 M¨ unchen, Germany. Abstract. We investigate an integration of the first-order method of proof by consistency (PBC), also known as term rewriting induction, into theorem proving in higher-order specifications. PBC may be seen as well-founded in- duction over an ordering which contains the rewrite relation, and in this paper we extend this method to the higher-order rewrite relation due to Nipkow. This yields a proof procedure which has several advantages over conventional induction. First, it is less control demanding; second, it is more flexible in the sense that it does not instantiate variables precisely with every constructor, but instantiates according to the rewrite rules. We show how a number of technical problems can be solved in order for this integration to work, and point out some desirable refinements that involve challenging problems. 1 Introduction The field of term rewriting has attracted much attention over the last twenty-odd years, largely triggered by seminal work of Knuth and Bendix on completion [14]. From the late seventies we have seen an ever increasing body of research on methods for the analysis of first-order rewrite systems. For a survey of this part of the field we refer to [4]. Due to their expressive power, higher-order logics are widely used for specification and verification. For the extension of term rewriting in this direction, there exist several different formalisms which integrate typed lambda calculus and term rewrite systems, including Klop [13], Breazu-Tannen [3] and Nipkow [22]. We follow the approach given in the latter work, where a rewriting relation modulo α-, β- and η-conversion is considered. In this paper we adapt the first-order proof method called inductionless induc- tion , or proof by consistency , to the higher-order setting. The rationale behind this method, which was first described in a paper by Musser [21], is that the Knuth and Bendix completion process can be used to prove or disprove properties of a rewrite system. This is roughly done by studying the new equations that emerge in the com- pletion process wrt. a notion of consistency. Since 1980 we have seen a lot of work on this first-order method, removing some of its limitations [6, 8], relaxing its close connection with the full completion process [5, 1], and extending the set of rewrite- based specifications that the method applies to [2, 16, 17]. In [26] it was pointed out  Email: henrikl@ifi.uio.no. Partly supported by the Norwegian Research Council.  Email: prehofer@informatik.tu-muenchen.de  Email: olavly@ifi.uio.no. Supported by Esprit projects OMI-MACRAME and OMI- ARCHES.