A Theorem Prover in Mathematica

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Analytica- A Theorem Prover in Mathematica Edmund Clarke Xudong Zhao School of Computer Science Carnegie Mellon Pittsburgh, PA 15213 Current automatic theorem provers, particularly those based on

1 Analytica- A Theorem Prover in Mathematica Edmund Clarke Xudong Zhao School of Computer Science Carnegie Mellon Pittsburgh, PA Current automatic theorem provers, particularly those based on some variant of resolution, have concentrated on obtaining ever higher inference rates by using clever programming techniques, parallelism, etc. We believe that this approach is unlikely to lead to a useful system for actually doing mathematics. The main problem is the large amount of domain knowledge that is required for even the simplest proofs. In this paper, we describe an alternative approach that involves combining an automatic theorem prover with a symbolic computation system. The theorem prover, which we call Analytica, is able to exploit the mathematical knowledge that is built into this symbolic computation system. In addition, it can guarantee the correctness of certain steps that are made by the symbolic computation system and, therefore, prevent common errors like division by an expression that may be zero. Analytica is written in the Mathematica programming language and runs in the interactive environment provided by this system [4]. Since we wanted to generate proofs that were similar to proofs constructed by humans, we have used a variant of the sequent calculus in the inference phase of our theorem prover. However, quantifiers are handled by skolemization instead of explicit quantifier intro.duction and elimination rules. Although inequalities play a key role in all of analysis, Mathematica is only able to handle very simple numeric inequalities. We have developed a technique that is complete for linear inequalities and is able to handle a large class of non-linear inequalities as well. This technique is more closely related to the BOUNDER system developed at MIT [3] than to the traditional SUP-INF method. Another important component of Analytica deals with expressions involving summation and product operators. A large number of rules are devoted to the basic properties of these operators. We have also integrated Gosper's algorithm [2] for hypergeometric sums with the other summation rules, since it can be used to find closed form representations for a wide class of summations that occur in practice. In a related project that we plan to describe in a forthcoming paper, we have managed to prove all of the theorems and examples in Chapter 2 of Ramanujan's Collected Works[i] completely automatically. The techniques that we use are similar to those described in this paper. We believe that the examples that we have been able to prove provide convincing justification for combining powerful symbolic computation techniques with theorem provers. Analytica consists of four different phases: skolemization, simplification, inference, and rewriting. When a new formula is submitted to Analytica for proof, it This research was sponsored in part by the National Science Foundation under Contract Number CCR , and also by the Defense Advanced Research Projects Agency (DOD), ARPA Order No. 4976, Amendment 20, under Contract Number F C-1499.

2 762 is first skolemized to a quantifier free form. Then it is simplified using a collection of algebraic and logical reduction rules. A formula is simplified with respect to its proof context. The proof context consists of the formulas that may be assumed true when the formula is encountered in the proof. If the formula reduces to true, the current branch of the inference tree terminates with success. If not, the theorem prover checks to see if the formula matches the conclusion of some inference rule. If a match is found, Analytica will try to establish the hypothesis of the rule. If the hypothesis consists of a single formula, then it will try to prove that formula. If the hypothesis consists of a series of formulas, then Analytica will attempt to prove each of the formulas in sequential order. Special tactics are also included in the inference phase for handling inequalities and constructing inductive proofs. The inequality tactic is complete for linear inequalities and can handle many nonlinear inequalities as well. The induction tactic enables Analytica to select a suitable induction scheme for the formula to be proved and attempts to establish the basis and induction steps. If no inference rule is applicable, then various rewrite rules are used attempting to convert the formula to another equivalent form. If the rewriting phase is unsuccessful, the search terminates in failure; otherwise the simplification, inference and rewriting phases will repeat with the new formula. Backtracking will cause the entire inference tree to be searched before the proof of the original goal formula terminates with failure. In the appendix, we give a few examples that Analytica can prove. Due to lack of space, we omit the proofs in all but the first example. Mathematica automatically generates I.$TEXcommands to typeset formulas involving algebraic expressions. In addition to these examples, Analytica has also proved some other non-trivial theorems including the Bernstein approximation theorem for continuous functions and main parts of the Fundamental Theorem of Algebra. References [1] B.C.Berndt, Ramanujan's Notebooks, Part I, Springer-Verlag, 1985, pp [2] R.W.Gosper, Indefinite IIypergeometric sums in MA CSYMA, [3] E.Sacks, Hierarchical Inequality Reasoning, Technique Report, MIT Laboratory for Computer Science, [4] S. Wolfram. Mathematica: A System for Doing Mathematics by Computer, \Volfram Research Inc., 1988.

3 m 763 Appendix. Examples proved by Analytica A Closed form for a summation Theorem : Proof : 2 k 1 2 ~+1 (integer(n) AOSnAmr 1~ l+m ~ - m m 2"+') k=o prove use induction on n 2 k ~ 1 + m 2~ -1 + m 1 - rn -~2~ base case with n = 0 reduces to m = l V l+m -l+m 1 -m 2 True induction step 2 k ~ A 0 n integer(n) <_ A 1 + rn 2~ m n.z2.2 ~ 1-]-n 2 k ~ /71 1 V 1 + ra 2k -1 + m 1 -- m 4"2" calculate summations substitute 2 k '~ g () A 0 _ < n A > inte_ern 1 + rn 2k -1 + m 1-2'2'~ m= 1V 1+m22" + l+m 2k -l+m 1-m 42" t k--0 ) using equation reduces to [] 2 k '~ _ q - :=:::> integer(n) A 0 < n A 1+ m 2k -1 + m 1 - rn 22~ '~ '~ 1 4 ' 2 ~ m=lv rn 22" -1 + m 1 - m >2" -1 + m 1 - m 42" True

4 764 B Stereographical projection Consider the function that maps each point (xi, x2, x3) in 3-space to the complex plane C: 8p(Xl, X2, X3) -- zl Jr- ix, 1 -za We will use Analytica to prove that this mapping is bijection between the unit sphere S whose equation is x 2 i x~-t-x3 9 2 = 1 and the complex plane. We will also prove that this mapping is a projection, i.e. if sp(xi, x2, x3) = a+bi, then the north pole(0,0,1), (x~, x2, x3) and (a, b, 0) are all collinear. With the definitions that: unil(xi, x~, x3) = T if and only if Xl ~ + x.~ + x 3 = 1 collinear({xi, x2, x3}, {Yi, Y2, Y3}, {Zl, z2, z3}) = T if and only if xl x2 1 I xl z3 1 x2 x3 1 Yl Y2 1 = 0 A I Yl Y3 1 = 0A yo Y3 1 Zl z2 1 Zl Z3 1 z2 Z3 1 =0 The following theorems show the properties of the Stereographical projection: (UTter (Xl, X2, X3) /k unil(yl, Y2, Y3) A sp(xl, x2, x3) = sp(yl, y2, Y3) =:~ x3 : Y3 A Xl : Yl A x2 -: Y2) 3{Xl, x2, x3}[sp(xl, x~, ~3) = Zl + z~i A ~nit(xl, ~2, ~)] z = sp(xl, x2, x3) ==~ collinear({r(z), I(z), 0}, {xl, x2, x3}, {0, 0, 1}) All three of these theorems are proved completely automatically. In second theorem, Analytica is able to find the appropriate instantiation for the existentially quantified variables. Analytica avoids division by zero in this example by making sure that x3 in sp(zl, x2, x3) is not equal to 1.

5 765 C Weierstrass' everywhere continuous but nowhere differentiable function Weierstrass's non-differentiable function is defined be the series f(~) = ~ oo rt-~-0 b- cos(a~) where 0 < b < 1, and a is a odd positive integer. When ab > 1 + ~, of the function does not exist for any value of x. With 3 the derivative It is sufficient to prove that diff(h,f,x) = -f(x) + f(h + x) h h = 1 - arnx -F round(amx) a m Continuous(f (x), {x, z0}) Ve[(e > 0 =~ m-.~lim lhl < e)] VM[lirn [diff(h, f, z)[ > M] The first two theorems can be proved without help from the user, however, the last one cannot be proved fully automatically. Three lemmas are added: where diff(h, f, x) = R(m) + S(m) 2arab m [~(m)l >_ (a m b m IS(~)l < ab----yt- 1 S(~) = a m R(m) = (~7~-1-t -mb n (-- cos(tranx) + cos(tra -m+n (1 + round(a'~x))))) \z-..,~=0 1 - (arnx - round(amx)) (--1)ro~nd(om')am(E.~m b~ (1 + cos(~a-'~+~d.~))) ) 1 - (amx - round(amx)) All three lemmas are proved automatically. Using these lemmas, Analytica can prove the remaining theorem without further help.

Keywords: Theorem Prover, Symbolic computation, Mathemmatica, Analytica

Keywords: Theorem Prover, Symbolic computation, Mathemmatica, Analytica AD-A258 656 Analytica - An Experiment in Combining Theorem Proving and Symbolic Computation Edmund Clarke Xudong Zhao October 1992 CMU-CS-92-147 School of Computer Science Carnegie Mellon University Pittsburgh,