We describe the application of proof orderings---a technique for reasoning about inference systems---to various rewrite-based theorem-proving methods, including re#nements of the standard Knuth-Bendix completion procedure based on critical pair criteria; Huet's procedure for rewriting modulo a congruence; ordered completion #a refutationally complete extension of standard completion#; and a proof by consistency procedure for proving inductive theorems. # This is a substantially revised version of the paper, #Orderings for equational proofs," co-authored with J. Hsiang and presented at the Symp. on Logic in Computer Science #Boston, Massachusetts, June 1986#. It includes material from the paper #Proof by consistency in equational theories," by the #rst author, presented at the ThirdAnnual Symp. on Logic in Computer Science #Edinburgh, Scotland, July 1988#. This researchwas supported in part by the National Science Foundation under grants CCR-89-01322, CCR-90-07195, and CCR-90-24271. 1 ...

. Zhang, Kapur and Krishnamoorthy introduced a cover set method for designing induction schemes for automating proofs by induction from specifications expressed as equations and conditional equations. This method has been implemented in the theorem prover Rewrite Rule Laboratory (RRL) and a proof management system Tecton built on top of RRL, and it has been used to prove many nontrivial theorems and reason about sequential as well as parallel programs. The cover set method is based on the assumption that a function symbol is defined using a finite set of terminating (conditional or unconditional) rewrite rules. The termination ordering employed in orienting the rules is used to perform proofs by well-founded induction. The left side of the rules are used to design different cases of an induction scheme, and recursive calls to the function made in the right side can be used to design appropriate instantiations for generating induction hypotheses. A weakness of this method is that it rel...

The inductionless induction (also called proof by consistency) approach for proving equations by induction from an equational theory, requires a consistency check for equational theories. A new method using test sets for checking consistency of an equational theory is proposed. Using this method, a variation of the Knuth-Bendix completion procedure can be used for automatically proving equations by induction. The method does not suffer from limitations imposed by the methods proposed by Musser as well as by Huet and Hullot, and is as powerful as Jouannaud and Kounalis' method based on ground-reducibility. A theoretical comparison of the test set method with Jouannaud and Kounalis' method is given showing that the test set method is generally much better. Both the methods have been implemented in RRL, Rewrite Rule Laboratory, a theorem proving environment based on rewriting techniques and completion. In practice also, the test set method is faster than Jouannaud and Kounalis' ...

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Formal software development techniques facilitate the design and implementation of more reliable computer systems, which is particularly important for the development of safety-critical systems. In particular, formal specification languages provide a means for precisely characterizing the behavior of a computer system and its components, and facilitates the determination of correct implementation using automated reasoning techniques. While formal specifications can be created using a word processor or typesetter, the process is neither easy nor suitable for large scale software specification. Tools supporting the application of formal methods are needed to make these activities easier, and thus more practical to use. This paper discusses Spectacle, an object-oriented library of software components designed for constructing formal specification editing tools; prototype specification editors built from this library are presented. 1 Introduction The number of applications to which the r...

Introduction Theorem provers and automated reasoning tools are not as widely used in speci cation analysis, debugging and verication of hardware and software as one would hope. A major reason is perhaps that these tools are often found dicult to use. The learning curve for a typical theorem prover is quite high; it takes a great deal of eort by typical users to eectively use such a tool for their application domain. Even then, considerable resources must be expended by building a large knowledge base and library of useful properties in a representation suitable for the prover, to bring it to a level so that it can start playing a useful role. For theorem provers to be acceptable to application experts, the key requirement is that a theorem prover should be able to easily perform reasoning steps considered routine in an application domain; performing such reasoning should denitely not become a burden on the expert. We have called this reasoning in the large