Theorem Proving and Algebra

Abstract

This book can be seen either as a text on theorem proving that uses techniques from general algebra, or else as a text on general algebra illustrated and made concrete by practical exercises in theorem proving. The book considers several different logical systems, including first-order logic, Horn clause logic, equational logic, and first-order logic with equality. Similarly, several different proof paradigms are considered. However, we do emphasize equational logic, and for simplicity we use only the OBJ3 software system, though it is used in a rather flexible manner. We do not pursue the lofty goal of mechanizing proofs like those of which mathematicians are justly so proud; instead, we seek to take steps towards providing mechanical assistance for proofs that are useful for computer scientists in developing software and hardware. This more modest goal has the advantage of both being achievable and having practical benefits. The following topics are covered: many-sorted signature, algebra and homomorphism; term algebra and substitution; equation and satisfaction; conditional equations; equational deduction and its completeness; deduction for conditional equations; the theorem of constants; interpretation and equivalence of theories; term rewriting, termination, confluence and normal form; abstract rewrite systems; standard models, abstract data types, initiality, and induction; rewriting and deduction modulo equations; first-order logic, models, and proof planning; second-order algebra; order-sorted algebra and rewriting; modules; unification and completion; and hidden algebra. In parallel with these are a gradual introduction to OBJ3, applications to group theory, various abstract data types (such as number systems, lists, and stacks), propositional calculus, hardware verification, the λ-calculus, correctness of functional programs, and other topics.