Canonical Equational Proofs PDF

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.