Equations occur in many computer applications, such as symbolic compu- tation, functional programming, abstract data type specifications, program verification, program synthesis, and automated theorem proving.
Equations occur in many computer applications, such as symbolic compu- tation, functional programming, abstract data type specifications, program verification, program synthesis, and automated theorem proving. Rewrite systems are directed equations used to compute by replacing subterms in a given formula by equal terms until a simplest form possible, called a normal form, is obtained. The theory of rewriting is concerned with the compu- tation of normal forms. We shall study the use of rewrite techniques for reasoning about equations. Reasoning about equations may, for instance, involve deciding whether an equation is a logical consequence of a given set of equational axioms. Convergent rewrite systems are those for which the rewriting process de- fines unique normal forms. They can be thought of as non-deterministic functional programs and provide reasonably efficient decision procedures for the underlying equational theories. The Knuth-Bendix completion method provides a means of testing for convergence and can often be used to con- struct convergent rewrite systems from non-convergent ones. We develop a proof-theoretic framework for studying completion and related rewrite- based proof procedures. We shall view theorem provers as proof transformation procedures, so as to express their essential properties as proof normalization theorems.
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1 Equational Proofs.- 1.1. Introduction.- 1.2. Terms.- 1.3. Equations.- 1.4. Orderings.- 1.5. Proofs.- 2 Standard Completion.- 2.1. Basic Completion.- 2.2. Proof Transformation.- 2.3. Proof Simplification.- 2.4. Fairness and Correctness.- 2.5. Standard Completion.- 2.6. Critical Pair Criteria.- 3 Extended Completion.- 3.1. Rewriting Modulo a Congruence.- 3.2. The Left-Linear Rule Method.- 3.3. Church-Rosser Systems.- 3.4. Extended Completion.- 3.5. The Extended Rule Method.- 3.6. Associative-Commutative Completion.- 3.7. The Protected Rule Method.- 3.8. Extended Critical Pair Criteria.- 4 Ordered Completion.- 4.1. Ordered Completion.- 4.2. Construction of Convergent Rewrite Systems.- 4.3. Refutational Theorem Proving.- 4.4. Horn Clauses with Equality.- 5 Proof by Consistency.- 5.1. Consistency and Ground Reducibility.- 5.2. Proof by Consistency.- 5.3. Refutation Completeness.- 5.4. Covering Sets.
Canonical Equational Proofs by Leo Bachmair
Progress in Theoretical Computer Science
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Birkhauser Boston Inc
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