On the algebraic formulation of certain geometry statements and mechanical geometry theorem proving

In this paper we analyze the algebraic formulations of certain geometry statements appearing in recent literature related to mechanical geometry theorem proving and give several examples to show that one of these formulations can cause serious problems. We clarify a formulation which is essentially due to W. T. Wu and, in our opinion, is the most satisfactory.

[1]  Shang-Ching Chou,et al.  Proving geometry theorems with rewrite rules , 2006, Journal of Automated Reasoning.

[2]  R. Hartshorne.  Foundations of projective geometry , 1967 .

[3]  Joseph R. Shoenfield,et al.  Mathematical logic , 1967 .

[4]  Wen-tsün Wu,et al.  TOWARD MECHANIZATION OF GEOMETRY — SOME COMMENTS ON HILBERT'S GRUNDLAGEN DER GEOMETRIE” , 1982 .

[5]  Philip J. Davis,et al.  Formac Meets Pappus Some Observations on Elementary Analytic Geometry by Computer , 1969 .

[6]  Shang-Ching Chou,et al.  An algorithm for constructing gröbner bases from characteristic sets and its application to geometry , 1990, Algorithmica.

[7]  Wanda Szmielew.  Some Metamathematical Problems Concerning Elementary Hyperbolic Geometry , 1959 .

[8]  Marvin J. Greenberg,et al.  Euclidean and non-euclidean geometries : development and history / Marvin Jay Greenberg , 1975 .

[9]  Philip J. Davis,et al.  Proof, Completeness, Transcendentals, and Sampling , 1977, JACM.

[10]  Sabine Stifter,et al.  Automated geometry theorem proving using Buchberger's algorithm , 1986, SYMSAC '86.

[11]  S. Chou.  Mechanical Geometry Theorem Proving , 1987 .

[12]  A. Tarski.  A Decision Method for Elementary Algebra and Geometry , 2023 .

[13]  Deepak Kapur,et al.  Geometry theorem proving using Hilbert's Nullstellensatz , 1986, SYMSAC '86.

[14]  Marvin J. Greenberg,et al.  Euclidean and non-Euclidean geometries , 1982 .

[15]  K. Reidemeister,et al.  Die metrische Form in der absoluten und der elliptischen Geometrie , 1937 .

[16]  Shang-Ching Chou,et al.  An introduction to Wu's method for mechanical theorem proving in geometry , 1988, Journal of Automated Reasoning.

[17]  David Hilbert,et al.  The Foundations of Geometry , 1903, The Mathematical Gazette.

[18]  P. Szasz.  Direct Introduction of Weierstrass Homogeneous Coordinates in the Hyperbolic Plane, on the Basis of the Endcalculus of Hilbert1 , 1959 .

[19]  A. Tarski.  What is Elementary Geometry , 1959 .