Computer Proofs of Limit Theorems

Some re lat ive ly simple concepts have been developed which, when incorporated into existing automatic theorem proving programs (including those using resolut ion), enable them to prove e f f i c ien t l y a number of the l im i t theorems of elementary calculus, including the theorem that d i f f e r e n t i a t e functions are con­ tinuous. These concepts include: (1) A l imited theory of types, to designate whether a given variable belongs to a certain interval on the real l i ne , (2) An algebraic s impl i f icat ion routine, (3) A routine for solving linear i n ­ equal i t ies, applicable to a l l areas of analysis, and (4) A " l im i t heur is t ic" , designed especially for the l im i t theorems of calculus. jL Introduction. In this paper we describe some re lat ive ly simple changes that have been made to an existing automatic theorem proving program to enable it to prove e f f i c ien t l y a num­ ber of the l im i t theorems of elementary calculus. These changes include subroutines of a general nature which apply to a l l areas of analysis, and a special " l im i t -heur is t ic " designed for the l im i t theorems of calculus. These concepts have been incorporated into an existing LISP program and run on the PDP-10 at the A . I . Laboratory, M.I.T., to obtain com­ puter proofs of many of the l im i t theorems, including the theorem that the l im i t of the sum of two real functions is the sum of their l im i t s , and a similar theorem about products. Also computer proofs have been obtained (or are easily obtainable) of the theorems that a continuous function of a continuous function is continuous, and that a function having a derivative at a point is continuous there, as well as l im i t results for polynomial functions. The l im i t theorems of calculus present a surprisingly d i f f i c u l t challenge for general purpose automatic theorem provers. One reason for this is that calculus is a branch of analysis, and proofs in analysis require manipulation of algebraic expressions, solutions of inequal i t ies, and other operations which depend upon the axioms of an ordered f i e l d . It is in applying these f i e l d axioms that automatic provers are usually forced into long and d i f f i c u l t searches. On the other hand, a human mathematician is often able to easily perform the necessary operations of analysis without being aware of the exp l i c i t use of the f i e l d axioms. One purpose of this paper is to describe ways in which automatic provers can also avoid the use of the f i e l d axioms and and speed up proofs in analysis. Section 2 ex­ plains how this is done using a l imited theory of types and routines for algebraic s impl i f ica­ t ion and solving l inear inequal i t ies. In Section 3 we present the l im i t -heur is t i c , give examples of i t s use, and discuss i t s "forcing" nature which enables it to cur ta i l combinatorial searches. The reader interested only in resolution based programs should skip Sections 4 and 5 and go d i rect ly to Section 6, where we explain how resolution programs can be altered to make use of the l im i t heuristic and other concepts. In Section 5 we give a detailed description of a computer proof of the theorem that the l im i t of the product of two functions is the product of their l im i t s . This proof was made by a program which is the same as that described in in n LI J, except that the subroutine, RESOLUTION, n [1] has been replaced by a new subroutine called IMPLY. We have thus eliminated resolution altogether from our program,replacing it by an "implication method" which we believe is faster and easier to use (though not complete). This implication method is described br ie f l y in Section 4, and excerpts from actual computer proofs using it are given there and in Section 5. It appears that some of these ideas may have wider implications than the l imited scope in which they were used here. This is discussed in the comments of Section 7 and throughout the paper. Session No. 14 Theorem Proving 587 as a solution of (2) the substi tut ion [ b / x ] , 2 and require (0 < b -> b < b) in (3 ) , which is impossible. Of course (1) is unprovable with out further hypotheses (or axioms) but it can be easily handled by the use of types (which im­ p l i c i t l y assumes certain axioms). Our approach in proving (1) is to assign type <0 »> to b, and then t ry to prove by assigning the type <0 b^ to x. The result ing type of x, <0 b>, was derived as the in ter­ section of i t s i n i t i a l type <0 <*> gotten from (5) , and the interval <-« b-, which would have been the type gotten from (6) alone. Since this intersection is not empty (because b has type <0 <*>), it is assigned as the result ing type of x. Even though the variable x had already been "solved for" in (5) (typed), it remains a va r i ­ able in the solution of (6) (though l imited in scope) and therefore could be "solved for" again (retyped). In the examples of Section 5 some of the variables are retyped two or three times, and this greatly simpli f ies the proofs. Types are used by the routines SOLVE< and SET-TYPE which are described below.