On Proving Inductive Properties of Abstract Data Types

Abstract

The equational axioms of an algebraic specification of a data type (such as finite sequences) often can be formed into a convergent set of rewrite rules; i.e. such that all sequences of rewrites are finite and uniquely terminating. If one adds a rewrite rule corresponding to a data type property whose proof requires induction (such as associativity of sequence concatenation), convergence may be destroyed, but often can be restored by using the Knuth-Bendix algorithm to generate additional rules. A convergent set of rules thus obtained can be used as a decision procedure for the equational theory for the axioms plus the property added. This fact, combined with a "full specification" property of axiomatizations, leads to a new method of proof of inductive properties--not requiring the explicit invocation of an inductive rule of inference.