The use of machines to assist in rigorous proof

Abstract

A methodology for computer assisted proof is presented with an example. A central ingredient in the method is the presentation of tactics (or strategies) in an algorithmic metalanguage. Further, the same language is also used to express combinators, by which simple elementary tactics - which often correspond to the inference rules of the logic employed - are combined into more complex tactics, which may even be strategies complete for a class of problems. However, the emphasis is not upon completeness but upon providing a metalogical framework within which a user may express his insight into proof methods and may delegate routine (but error-prone) work to the computer. This method of tactic composition is presented at the start of the paper in the form of an elementary theory of goal-seeking. A second ingredient of the methodology is the stratification of machine-assisted proof by an ancestry graph of applied theories, and the example illustrates this stratification. In the final section, some recent developments and applications of the method are cited.