# Proof-checking Euclid

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## Abstract

We used computer proof-checking methods to verify the correctness of our proofs of the propositions in Euclid Book I. We used axioms as close as possible to those of Euclid, in a language closely related to that used in Tarski’s formal geometry. We used proofs as close as possible to those given by Euclid, but filling Euclid’s gaps and correcting errors. Euclid Book I has 48 propositions; we proved 235 theorems. The extras were partly “Book Zero”, preliminaries of a very fundamental nature, partly propositions that Euclid omitted but were used implicitly, partly advanced theorems that we found necessary to fill Euclid’s gaps, and partly just variants of Euclid’s propositions. We wrote these proofs in a simple fragment of first-order logic corresponding to Euclid’s logic, debugged them using a custom software tool, and then checked them in the well-known and trusted proof checkers HOL Light and Coq.

## Keywords

Euclid Proof-checking Euclidean geometry HOL light Coq## Mathematics Subject Classification (2010)

03B30 51M05## Notes

## References

- 1.Avigad, J., Dean, E., Mumma, J.: A formal system for Euclid’s Elements. Rev. Symb. Log.
**2**, 700–768 (2009)MathSciNetCrossRefGoogle Scholar - 2.Beeson, M.: A constructive version of Tarski’s geometry. Ann. Pure Appl. Logic
**166**, 1199–1273 (2015)MathSciNetCrossRefGoogle Scholar - 3.Beeson, M.: Constructive geometry and the parallel postulate. Bull. Symb. Log.
**22**, 1–104 (2016)MathSciNetCrossRefGoogle Scholar - 4.Bledsoe, W. W., Loveland, D.: Automated Theorem Proving: After 25 Years. American Mathematical Society, Providence (1983)Google Scholar
- 5.Boutry, P.: On the Formalization of Foundations of Geometry. PhD thesis, University of Strasbourg (2018)Google Scholar
- 6.Boutry, P., Braun, G., Narboux, J.: From Tarski to Descartes: formalization of the arithmetization of Euclidean geometry. In: Davenport, J.H., Ghourabi, F. (eds.) SCSS 2016, the 7th International Symposium on Symbolic Computation in Software Science, vol. 39 of EPiC Series in Computing, Tokyo, Japan, EasyChair, pp. 14–28 (2016)Google Scholar
- 7.Boutry, P., Braun, G., Narboux, J.: Formalization of the arithmetization of Euclidean plane geometry and applications. J. Symb. Comput.
**90**, 149–168 (2019). https://doi.org/10.1016/j.jsc.2018.04.007 MathSciNetCrossRefzbMATHGoogle Scholar - 8.Boutry, P., Gries, C., Narboux, J., Schreck, P.: Parallel postulates and continuity axioms: a mechanized study in intuitionistic logic using Coq. J. Autom. Reason., p. 68. https://link.springer.com/article/10.1007%2Fs10817-017-9422-8 (2017)
- 9.Braun, G., Boutry, P., Narboux, J.: From Hilbert to Tarski. In: 11th International Workshop on Automated Deduction in Geometry, Proceedings of ADG 2016, Strasbourg, France, p. 19 (2016)Google Scholar
- 10.Braun, G., Narboux, J.: From Tarski to Hilbert. In: Ida, T., Fleuriot, J. (eds.) Automated Deduction in Geometry 2012 vol. 7993, Edinburgh, United Kingdom, Jacques Fleuriot, Springer, pp. 89–109 (2012)Google Scholar
- 11.Braun, G., Narboux, J.: A synthetic proof of Pappus’ theorem in Tarski’s geometry. J. Autom. Reason.
**58**, 23 (2017)MathSciNetCrossRefGoogle Scholar - 12.Chou, S. C.: Mechanical Geometry Theorem Proving. Kluwer Academic Publishers, Norwell (1987)CrossRefGoogle Scholar
- 13.Chou, S. C.: An introduction to Wu’s method for mechanical theorem proving in geometry. J. Autom. Reason.
**4**, 237–267 (1988)MathSciNetCrossRefGoogle Scholar - 14.Chou, S.-C., Gao, X.-S., Zhang, J.-Z.: Machine Proofs in Geometry: Automated Production of Readable Proofs for Geometry Theorems. World Scientific (1994)Google Scholar
- 15.Chou, S. C., Schelter, W. F.: Proving geometry theorems with rewrite rules. J. Autom. Reason.
**2**, 253–273 (1986)CrossRefGoogle Scholar - 16.Corbineau, P.: A declarative language for the Coq proof assistant. In: Types for Proofs and Programs, vol. 4941 of LNCS, Springer (2008)Google Scholar
- 17.Euclid: The Elements of Euclid, viz. the first six books, together with the eleventh and the twelfth, Nourse and Balfous, Edinburgh, 7th ed., 1787. Translated by Robert Simson. Available from Bibliotheque Nationale at http://gallica.bnf.fr/ark:/12148/bpt6k1163221v
- 18.Euclid: The Thirteen Books of the Elements. Dover, New York (1956). Three volumes. Includes commentary by the translator, Sir Thomas L. HeathzbMATHGoogle Scholar
- 19.Gelernter, H.: Realization of a geometry theorem-proving machine. In: Feigenbaum, E., Feldman, J. (eds.) Computers and Thought, pp 134–152. McGraw-Hill, New York (1963)Google Scholar
- 20.Gelernter, H., Hansen, J.R., Loveland, D.W.: Empirical explorations of a geometry-theorem proving machine. In: Feigenbaum, E., Feldman, J. (eds.) Computers and Thought, pp 153–167. McGraw-Hill, New York (1963)Google Scholar
- 21.Greenberg, M. J.: Euclidean and non-Euclidean Geometries: Development and History, 4th edn. W. H. Freeman, New York (2008)zbMATHGoogle Scholar
- 22.Gries, C.: Axiomes de continuité en géométrie neutre : Une étude méchanisée en Coq. University of Strasbourg, internship report (2017)Google Scholar
- 23.Gupta, H. N.: Contributions to the Axiomatic Foundations of Geometry. PhD Thesis, University of California, Berkeley (1965)Google Scholar
- 24.Hartshorne, R.: Geometry: Euclid and Beyond. Springer, New York (2000)CrossRefGoogle Scholar
- 25.Hilbert, D.: Foundations of Geometry (Grundlagen der Geometrie). Open Court, La Salle, Illinois Second English edition, translated from the tenth German edition by Leo Unger. Original publication year (1899)Google Scholar
- 26.Kapur, D.: A refutational approach to geometry theorem proving. J. Artif. Intell.
**37**, 61–93 (1988)MathSciNetCrossRefGoogle Scholar - 27.Kutzler, B., Stifter, S.: Automated geometry theorem proving using Buchberger’s algorithm. In: Proceedings of the International Symposium on Symbolic and Algebraic Computation (SYMSAC 86), Waterloo, 1986, B. Char, ed., pp. 209–214 (1986)Google Scholar
- 28.Manders, K.: The Euclidean diagram (1995). In: The Philosophy of Mathematical Practice, Paolo Mancosu. Oxford University Press ed. (2011)Google Scholar
- 29.Miller, N.: A Diagrammatic Formal System for Euclidean Geometry. PhD thesis, Cornell University (2001)Google Scholar
- 30.Miller, N.: Euclid and his twentieth century rivals: diagrams in the logic of Euclidean geometry. Studies in the theory and applications of diagrams, CSLI Publications, Stanford, Calif, 2007. OCLC: ocm71947628Google Scholar
- 31.Miller, N.: On the inconsistency of Mumma’s Eu. Notre Dame Journal of Formal Logic
**53**, 27–52 (2012)MathSciNetCrossRefGoogle Scholar - 32.Mollerup, J.: Die Beweise der ebenen Geometrie ohne Benutzung der Gleichheit und Ungleichheit der Winkel. Math. Ann.
**58**, 479–496 (1904)MathSciNetCrossRefGoogle Scholar - 33.Mumma, J.: Proofs, pictures, and Euclid. Synthese
**175**, 255–287 (2010)MathSciNetCrossRefGoogle Scholar - 34.Narboux, J., Janičić, P., Fleuriot, J.: Computer-assisted theorem proving in synthetic geometry. In: Sitharam, M., John, A.S., Sidman, J. (eds.) Handbook of Geometric Constraint Systems Principles, Chapman and Hall/CRC, ch. 2. in press (2018)Google Scholar
- 35.Pasch, M.: Vorlesung über Neuere Geometrie. Teubner, Leipzig (1882)zbMATHGoogle Scholar
- 36.Peano, G.: Principii de Geometria. Fratelli Bocca, Torino (1889)zbMATHGoogle Scholar
- 37.Potts, R.: Euclid’s Elements of Geometry [Books 1-6, 11,12] with Explanatory Notes; Together with a Selection of Geometrical Exercises. To which is Prefixed an Intr., Containing a Brief Outline of the History of Geometry. Oxford University Press, Oxford (1845)Google Scholar
- 38.Proclus: A Commentary on the First Book of Euclid. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
- 39.Quaife, A.: Automated development of Tarski’s geometry. J. Autom. Reason.
**5**, 97–118 (1989)MathSciNetCrossRefGoogle Scholar - 40.Schwabhäuser, W., Szmielew, W., Tarski, A.: Metamathematische Methoden in der Geometrie: Teil I: Ein axiomatischer Aufbau der euklidischen Geometrie. Teil II: Metamathematische Betrachtungen (Hochschultext). Springer–Verlag, 1983. Reprinted 2011 by Ishi Press, with a new foreword by Michael BeesonGoogle Scholar
- 41.Stojanovic Durdevic, S., Narboux, J., Janicic, P.: Automated generation of machine verifiable and readable proofs: a case study of Tarski’s geometry. Ann. Math. Artif. Intell.
**73**, 25 (2015)MathSciNetzbMATHGoogle Scholar - 42.Strommer, J.: Über die Kreisaxiome. Period. Math. Hung.
**4**, 3–16 (1973)CrossRefGoogle Scholar - 43.Tarski, A., Givant, S.: Tarski’s system of geometry. Bull. Symb. Log.
**5**, 175–214 (1999)MathSciNetCrossRefGoogle Scholar - 44.Veronese, G.: Fondamenti di geometria a più dimensioni e a più specie di unità rettilinee esposti in forma elementare. Lezioni per la scuola di magistero in matematica, Padova: Tipografia del Seminario, 1891 (1891)Google Scholar
- 45.Wu, W.-T.: On the decision problem and the mechanization of theorem-proving in elementary geometry. Sci. Sinica
**21**, reprinted in [4] (1978)MathSciNetGoogle Scholar - 46.Wu, W.-T.: Mechanical Theorem Proving in Geometries: Basic Principles. Springer, Wien/ New York (1994)CrossRefGoogle Scholar

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