Automatic Generation of Epsilon-Delta Proofs of Continuity

As part of a project on automatic generation of proofs involving both logic and computation, we have automated the production of some proofs involving epsilon-delta arguments. These proofs involve two or three quantifiers on the logical side, and on the computational side, they involve algebra, trigonometry, and some calculus. At the border of logic and computation, they involve several types of arguments involving inequalities, including transitivity chaining and several types of bounding arguments, in which bounds are sought that do not depend on certain variables. Control mechanisms have been developed for intermixing logical deduction steps with computational steps and with inequality reasoning. Problems discussed here as examples involve the continuity and uniform continuity of various specific functions. 1 Context of this Research Mathematics consists of logic and computation, interwoven in tapestries of proofs. “Logic” is represented by the manipulation of phrases (or symbols) such as for all x, there exists an x, implies, etc. “Computation” refers to chains of formulas progressing towards an “answer”, such as one makes when evaluating an integral or solving an equation. Typically computational steps move “forwards” (from the known facts further facts are derived) and logical steps move “backwards” (from the goal towards the hypothesis, as in it would suffice to prove. The mixture of logic and computation gives mathematics a rich structure that has not yet been captured, either in the formal systems of logic, or in computer programs. The research reported on here is part of a larger research program to do just that: capture and computerize mathematics. At present, there exist computer programs that can do mathematical computations, such as Mathematica, Maple, and Macsyma. These programs, however, do not keep track of the logical conditions needed to make computations legal, and can easily be made to produce incorrect results. 1 This research partially supported by NSF Grant Number CCR-9528913. 2 Just to give one example: Start with the equation a = 0. Divide both sides by a. In all the three systems mentioned, you can get 1 = 0 since the system thinks a/a = 1 and 0/a = 0. Many other examples have been given in the literature [1],[15]. Jacques Calmet and Jan Plaza (Eds.): AISC’98, LNAI 1476, pp. 67–83, 1998. c © Springer-Verlag Berlin Heidelberg 1998