Careful algebraic translations of geometry theorems

Abstract

Modernapplication areas like computer-aided design and robotics have revived interest in geometry.

Computer Aided Proofs in Analysis

“…Our specification language from (KUTZLER 1988(KUTZLER ,1989) has been designed under this aspect. Answering the question whether there exists an extension by definitions of, let say PEg, together with suitable algebraic interpretations, such that a given geometry theorem is of type (a) certainly is an interesting future research goal.…”

“…(,8) has been completely investigated in (KUTZLER 1988(KUTZLER ,1989) and can, in particular, be fully mechanized. (a) is the step from the informal description to a formal description, hence, no formal argument about its correctness is possible.…”

“…It certainly stimulated most of the research on this topic including our work on applying Buchberger's algorithm. But recent investigations in (KUTZLER 1988(KUTZLER ,1989) demonstrated a severe flaw in Wu's method, namely the inadequacy of his way of translating geometry theorems into algebraic problems. From this new point of view, almost all results obtained with Wu's algorithm can not be regarded as proofs but only as "near-proofs" in a sense fully explained in the above reference.…”

“…Since the languages of usual axiomatizations of PEg by Hilbert, Tarski, and others are not very convenient to use, we have established a suitable specification language by extending PEg by definitions l with various useful predicates like collinear, online, etc. (see (Kutzler 1988(Kutzler ,1989). The formulation of (Ex) in this extended theory is:…”