1 - Introduction

Abstract

This chapter discusses the use of logic and the theorem prover. The logic is a quantifier-free, first-order logic resembling Pure Lisp. Its axioms and rules of inference are obtained from the propositional calculus with equality and function symbols by adding (1) axioms characterizing certain basic function symbols, (2) two extension principles, with which one can add new axioms to the logic to introduce new inductively defined data types and recursive functions, and (3) mathematical induction on the ordinals up to ɛ0 as a rule of inference. The logic is mechanized by a collection of Lisp programs that permit the user to axiomatize inductively constructed data types, define recursive functions, and prove theorems about them. This collection of programs is frequently referred to as the Boyer–Moore theorem prover, although the program that implements the theorem prover is only one of many programs provided in the system. When new theorems are submitted the currently enabled rules determine how certain parts of the theorem prover behave. In this way, the user can lead the machine to the proofs of exceedingly deep theorems by presenting it with an appropriate graduated sequence of lemmas.