Higher-Order Proof by Consistency

We investigate an integration of the rst order method of proof by consistency PBC also known as term rewriting induction into theorem proving in higher order speci cations 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 exible 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 re nements that involve challenging problems