A Geometric Procedure with Prover9

Here we give an automated proof of the fact that a cubic curve admits at most one group law. This is achieved by proving the tight connection between the chord-tangent law of composition and any potential group law (as a morphism) on the curve. An automated proof of this is accomplished by implementing the rigidity lemma and the CayleyBacharach theorem of algebraic geometry as formal inference rules in Prover9, a first-order theorem prover developed by Dr. William McCune. 1 In Memory of Bill McCune This book is dedicated to the memory of our friend and colleague, Bill McCune. Both authors of this article collaborated with Bill for many years before his untimely passing in 2011. R. P. has included comments about Bill in the article, Group Embedding of the Projective Plane PG(2, 3), that also appears in the book. We preface this article with the following personal reflections from Bob. After Bill moved to New Mexico in 2006, we would get together regularly to discuss specific math applications, new features for his programs and implementation strategies for some of the more complex features. We preferred to meet at my house rather than at the university, and when we did meet, Bill would stay for dinner with my family and an evening of shooting pool and playing games. He enjoyed cooking, and sometimes he would bring bread or a dessert that he had made. He especially liked cooking with New Mexico green chiles; during chile harvest we’d buy a year’s supply of fresh roasted chiles and spend an afternoon peeling and bagging them for freezing. Bill enjoyed taking hikes on the many mountain trails in the Albuquerque area. He often hiked alone, but I would join him occasionally. We had our share of adventures—bushwhacking to find our way back to hard-to-follow trails and hiking through unexpected long stretches of calf-deep snow—but the hikes always were great fun. Bill was very reserved in public, but he appreciated the humor of situations such as our hiking adventures or realizing that something we Partially supported by a University of Manitoba research leave grant. M.P. Bonacina and M.E. Stickel (Eds.): McCune Festschrift, LNAI 7788, pp. 139–150, 2013. c © Springer-Verlag Berlin Heidelberg 2013 140 R. Padmanabhan and R. Veroff were working on was a really bad idea. He’d smile skeptically at some of my suggestions for new features for his programs, but sometimes he would implement them anyway, just to humor me. 2 A First-Order Property for Cubic Curves Lying at the crossroads of algebra, arithmetic and geometry, the modern theory of elliptic curves has many fascinating applications. For example, AndrewWiles’s celebrated 1995 proof of Fermat’s Last Theorem was accomplished using the theory of elliptic curves. In this paper, we show how the modern technology of automated deduction can be successfully employed in understanding some of the inference rules enjoyed by elliptic curves. An elliptic curve is simply the solution set of a non-singular cubic equation defined in a projective plane over an algebraically closed field. A remarkable fact from algebraic geometry is that elliptic curves admit a group law (see Figure 1) and such a group law is unique and commutative. This is a simple consequence of the so-called “rigidity lemma”, a deep theorem in classical algebraic geometry [4, p. 43]. Also, the associativity of the group law on an elliptic curve—the most non-trivial of the group properties—is a simple consequence of an intersection theorem known as the Cayley-Bacharach theorem. This is a far-reaching generalization of the Pappus-Pascal theorem of classical projective geometry. During the late 1980s, the rigidity lemma and the Cayley-Bacharach theorem were formalized as inference rules in the language of first-order logic with equality [5]. The resulting formal rules were successfully implemented by the late Dr. William McCune in his theorem prover Otter [2]. Now there are new powerful theorem provers, like Prover9 [3], also created by McCune. In this paper, we show how one can use Prover9 to prove incidence theorems on non-singular cubic curves within the framework of first-order logic with equality. We use a scheme of inference rules derived from the rigidity lemma and the Cayley-Bacharach theorem, incorporate these rules in Prover9 and then obtain first-order proofs of several well-known incidence theorems including the associativity and the uniqueness of the group law on cubic curves. 3 Implementing gL in Prover9 In [6], Padmanabhan and McCune describe the implementation of a first-order property for cubic curves, called gL, as an inference rule in McCune’s theorem prover Otter. Although gL is not supported directly as an inference rule in Prover9, we can implement applications of gL using hyperresolution with appropriate hyperresolution nuclei included as additional input clauses. For example, consider the gL clause (z * x0) * x1 != (z * y0) * y1 | (w * x0) * x1 = (w * y0) * y1. A Geometric Procedure with Prover9 141 This clause is an implication which simply says that if (z ∗x0)∗x1 = (z ∗ y0)∗ y1 for some z (which occurs in the same location on both sides of the equation), then the common element z can be replaced with an arbitrary variable w (see Figure 2, the basic gL). In other words, resolving with the negative literal of this clause has the effect of replacing the subterm that matches the z in the first literal with a newly introduced variable w. Since gL operates at the term level, there needs to be a separate gL clause for every combination of relevant functions and every possible term position. As a matter of practicality, when working on a specific problem, we limit the nesting level of the gL clauses that are included in the input file. To facilitate these studies, we have written a gL clause generator [9] that takes as arguments a signature and a maximum nesting level and generates all of the relevant gL clauses up to the specified nesting-level. As an example, here is the output of the gL clause generator for a signature with two functions, f(x, y) and g(x), and a maximum nesting level of 1. % ----------------------------------------------------% The following is output from the gL clause generator. % ----------------------------------------------------formulas(assumptions). % Signature: [[’f’, 2], [’g’, 1]] % Clauses to implement gL to nesting level 1 f(z,x0) != f(z,y0) | f(w,x0) = f(w,y0). f(f(z,x0),x1) != f(f(z,y0),y1) | f(f(w,x0),x1) = f(f(w,y0),y1). f(x0,f(z,x1)) != f(y0,f(z,y1)) | f(x0,f(w,x1)) = f(y0,f(w,y1)). g(f(z,x0)) != g(f(z,y0)) | g(f(w,x0)) = g(f(w,y0)). f(x0,z) != f(y0,z) | f(x0,w) = f(y0,w). f(f(x0,z),x1) != f(f(y0,z),y1) | f(f(x0,w),x1) = f(f(y0,w),y1). f(x0,f(x1,z)) != f(y0,f(y1,z)) | f(x0,f(x1,w)) = f(y0,f(y1,w)). g(f(x0,z)) != g(f(y0,z)) | g(f(x0,w)) = g(f(y0,w)). g(z) != g(z) | g(w) = g(w). f(g(z),x0) != f(g(z),y0) | f(g(w),x0) = f(g(w),y0). f(x0,g(z)) != f(y0,g(z)) | f(x0,g(w)) = f(y0,g(w)). g(g(z)) != g(g(z)) | g(g(w)) = g(g(w)).