A case of automatic theorem proving in Euclidean geometry: the Maclane 8₃ theorem

Abstract

In this paper we consider a semi-automatic proof for a theorem on a configuration of points that is true over the reals but false over the complex field. The method, that can be completely automated, consists in Gröbner basis computations, factorizations and determination of singular loci. The difference with the usual automatic proof methods over the complex field consists in a step that checks if an irreducible component has smooth real points, and otherwise substitutes this component with its singular locus.

This research was performed with the contribution of C.N.R., M.U.R.S.T, and CEC contract ESPRIT B.R.A. n.6846 POSSO

After the submission of this paper, we have been informed that the same theorem has been considered in the technical report [2], using characteristic sets instead of Gröbner basis, and Collin's CAD instead of analysis of critical points. The conclusions have been the same. A fair comparison of the performance of the two methods is difficult, since both are only partly automated, the implementations are different, the computers are different, and 5 years of computer technology advances makes a big difference in timings. However we may remark that with the methods of [2] the theorem is one of the most difficult, and the proof took more than 4 hours on a Symbolics; the timings of the automated parts of our proof are much smaller, totaling less than 4 seconds on a Sun Sparc2 (using the Gröbner of the POSSO library).

One would conjecture that for theorems of this kind the Gröbner-based provers are more efficient that the characteristic-set based provers.