Skip to main content
SpringerLink
Log in

Formalization of the Poincaré Disc Model of Hyperbolic Geometry

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

We describe formalization of the Poincaré disc model of hyperbolic geometry within the Isabelle/HOL proof assistant. The model is defined within the complex projective line \(\mathbb {C}{}P^1\)and is shown to satisfy Tarski’s axioms except for Euclid’s axiom—it is shown to satisfy it’s negation, and, moreover, to satisfy the existence of limiting parallels axiom.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Notes

  1. We use the numbering of theorems as of the tenth edition.

  2. http://afp.sourceforge.net/.

  3. To highlight the fact that all of the axioms except for Euclid’s axiom indeed defines a neutral geometry we provide figures both in the Euclidean model and a non-Euclidean model, namely the Poincaré disc model. The figure on the left hand side illustrates the validity of the axiom in Euclidean geometry. The figure on the right hand side either depicts the validity of the statement in the Poincaré disc model or exhibits a counter-example.

  4. From this example it can be seen that the lifting must be done in two stages (one for the subtype and the other for the quotient type). However, to simplify the presentation, in the rest of the paper we shall show only the initial and the final definition.

  5. Note that such claim is slightly imprecise. In a strictly typed setting the value of the cross ratio is a number in the extended complex plane \(\mathbb {C}{}P^1\). If different from \(\infty _h\), then it can be converted to an ordinary complex number and we claim that its imaginary part is equal to 0. Formally we claim that is_real (to_complex (cross_ratio u \(i_1\) v \(i_2\))), where is_real x \(\equiv \) Im x = 0. For simplicity, we shall sometimes make such simplifications.

  6. There already exists tools that translate proofs from one proof assistant to another [1].

  7. there is no development about real closed fields in Isabelle/HOL. This is the reason that motivated us to avoid the restriction of predicates f and g to FOL predicates.

References

  1. Assaf, A.: A framework for defining computational higher-order logics. Theses, École polytechnique (2015)

  2. Ballarin, C.: Interpretation of locales in Isabelle: theories and proof contexts. In: Mathematical Knowledge Management, MKM, Proceedings, pp. 31–43 (2006)

  3. Beeson, M.: Proving Hilbert’s axioms in Tarski geometry (2014). http://www.michaelbeeson.com/research/papers/TarskiProvesHilbert.pdf

  4. Beeson, M.: A constructive version of Tarski’s geometry. Ann. Pure Appl. Log. 166(11), 1199–1273 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Beeson, M., Boutry, P., Braun, G., Gries, C., Narboux, J.: GeoCoq (2018). https://hal.inria.fr/hal-01912024/

  6. Beeson, M., Wos, L.: Finding proofs in Tarskian geometry. J. Autom. Reason. 58(1), 181–207 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Beltrami, E.: Saggio di interpretazione della geometria Non-Euclidea. s.n. (1868)

  8. Bolyai, J.: Appendix, Scientiam Spatii absolute veram exhibens: a veritate aut falsitate Axiomatis XI. Euclidei (a priori haud unquam decidenda) independentem; adjecta ad casum falsitatis, quadratura circuli geometrica. Auctore Johanne Bolyai de eadem, Geometrarum in Exercitu Caesareo Regio Austriaco Castrensium Capitaneo. Coll. Ref. (1832)

  9. Borsuk, K., Szmielew, W.: Foundations of Geometry. North-Holland, New York (1960)

    MATH  Google Scholar 

  10. Boutry, P., Braun, G., Narboux, J.: Formalization of the arithmetization of Euclidean plane geometry and applications. J. Symbol. Comput. 90, 149–168 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Boutry, P., Gries, C., Narboux, J., et al.: Parallel postulates and continuity axioms: a mechanized study in intuitionistic logic using Coq. J Autom. Reason. 62, 1–68 (2019). https://doi.org/10.1007/s10817-017-9422-8

    Article  MathSciNet  MATH  Google Scholar 

  12. Boutry, P., Narboux, J., Schreck, P., Braun, G.: A short note about case distinctions in Tarski’s geometry. In: Botana, F., Quaresma, P. (eds.) Proceedings of the Tenth International Workshop on Automated Deduction in Geometry, Proceedings of ADG 2014, pp. 51–65. Coimbra, Portugal (2014)

  13. Braun, D., Magaud, N., Schreck, P.: An equivalence proof between rank theory and incidence projective geometry. Proce. ADG 2016, 62–77 (2016)

    Google Scholar 

  14. Braun, G., Boutry, P., Narboux, J.: From Hilbert to Tarski. In: Narboux, J., Schreck, P., Streinu, I. (eds.) Proceedings of the Eleventh International Workshop on Automated Deduction in Geometry, Proceedings of ADG 2016, pp. 78–96. Strasbourg, France (2016)

  15. Braun, G., Narboux, J.: From Tarski to Hilbert. In: Ida, T., Fleuriot, J. (eds.) Automated Deduction in Geometry (ADG 2012). Lecture Notes in Computer Science, vol. 7993, pp. 89–109. Springer, Edinburgh (2012)

    Chapter  Google Scholar 

  16. Braun, G., Narboux, J.: A synthetic proof of Pappus’ theorem in Tarski’s geometry. J. Autom. Reason. 58(2), 23 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Brun, C., Dufourd, J.-F., Magaud, N.: Formal proof in Coq and derivation of a program in C++ to compute convex hulls. In: Ida, T., Fleuriot, J. (eds.) Automated Deduction in Geometry (ADG 2012), volume 7993 of Lecture Notes in Computer Science, pp. 71–88. Springer Verlag, Berlin (2012)

  18. Coghetto, R.: Klein-Beltrami model. Part I. Formaliz. Math. 26(1), 21–32 (2018)

    Article  MATH  Google Scholar 

  19. Coghetto, R.: Klein-Beltrami model. Part II. Formaliz. Math. 26(1), 33–48 (2018)

    Article  MATH  Google Scholar 

  20. Coxeter, H.S.M.: Projective Geometry. Springer, Berlin (2003)

    MATH  Google Scholar 

  21. De Risi, V.: The development of Euclidean axiomatics. Arch. Hist. Exact Sci. 70(6), 591–676 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Dehlinger, C., Dufourd, J.-F.: Formalizing generalized maps in Coq. Theor. Comput. Sci. 323(1), 351–397 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Dehlinger, C., Dufourd, J.-F., Schreck, P.: Higher-order intuitionistic formalization and proofs in Hilbert’s elementary geometry. In: Automated Deduction in Geometry (ADG 2000), volume 2061 of Lecture Notes in Computer Science, pp. 306–324 (2001)

  24. Dufourd, J.-F.: A hypermap framework for computer-aided proofs in surface subdivisions: genus theorem and Euler’s formula. In: Proceedings of the 2007 ACM Symposium on Applied Computing, SAC ’07, pp. 757–761. ACM, New York, NY, USA (2007)

  25. Dufourd, J.-F., Bertot, Y.: Formal study of plane Delaunay triangulation. In: Kaufmann, M., Paulson, L.C. (eds.) Interactive Theorem Proving’2010 (In Federative Logic Conference: FLoC’2010), number 6172 in Lecture Notes in Computer Science, pp. 211–226. Springer (2010)

  26. Fleuriot, J.: Nonstandard geometric proofs. In: Richter-Gebert, J., Wang, D. (eds.) Automated Deduction in Geometry (ADG 2000), pp. 246–267. Springer, Berlin Heidelberg (2001)

    Chapter  Google Scholar 

  27. Fleuriot, J.: Theorem proving in infinitesimal geometry. Log. J. IGPL 9(3), 447–474 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Fleuriot, J.: Exploring the foundations of discrete analytical geometry in Isabelle/HOL. In: Schreck, P., Narboux, J., Richter-Gebert, J. (eds.) Automated Deduction in Geometry (ADG 2010), volume 6877 of Lecture Notes in Computer Science, pp. 34–50. Springer (2010)

  29. Freudenthal, H., et al.: K. Borsuk and Wanda Szmielew, foundations of geometry, Euclidean and Bolyai-Lobachevskian geometry, projective geometry. Bull. Am. Math. Soc. 67(4), 342–344 (1961)

    Article  Google Scholar 

  30. Gauss, C.F.: Disquisitiones generales circa superficies curvas. [Mathematical tracts. Typis Dieterichianis (1828)

  31. Gries, C., Narboux, J., Boutry, P.: Axiomes de continuité en géométrie neutre : une étude mécanisée en Coq. In: Magaud, N., Dargaye, Z. (eds.) Journées Francophones des Langages Applicatifs 2019, Acte des Journées Francophones des Langages Applicatifs (JFLA 2019). Les Rousses, France (2019)

  32. Gupta, H.N.: Contributions to the Axiomatic Foundations of Geometry. PhD thesis, University of California, Berkley (1965)

  33. Harrison, J.: Without loss of generality. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) Theorem Proving in Higher Order Logics, pp. 43–59. Springer, Berlin Heidelberg (2009)

    Chapter  Google Scholar 

  34. Huffman, B., Kunčar, O.: Lifting and transfer: a modular design for quotients in Isabelle/HOL. In: Gonthier, G., Norrish, M. (eds.) Certified Programs and Proofs, pp. 131–146. Springer, Cham (2013)

    Chapter  MATH  Google Scholar 

  35. Kahn, G.: Constructive geometry according to Jan von Plato (1995)

  36. Lobatschewsky, N.: Geometrische Untersuchungen zur Theorie der Parallellinien, pp. 159–223. Springer, Vienna (1985)

  37. Lorenz, J.F.: Grundriss der Reinen und Angewandten Mathematik. Fleckeisen, Wolfenbuttel(1791)

  38. Magaud, N., Chollet, A., Fuchs, L.: Formalizing a discrete model of the continuum in Coq from a discrete geometry perspective. Ann. Math. Artif. Intell. 74(3–4), 309–332 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  39. Magaud, N., Narboux, J., Schreck, P.: A case study in formalizing projective geometry in Coq: Desargues theorem. Comput. Geom. 45(8), 406–424 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Makarios, T.J.M.: A mechanical verification of the independence of Tarski’s Euclidean axiom. Master’s thesis, Victoria University of Wellington (2012)

  41. Marić, F., Simić, D.: Formalizing complex plane geometry. Ann. Math. Artif. Intell. 74(3–4), 271–308 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  42. Marić, F., Simić, D.: Complex geometry. Archive of Formal Proofs (2019). http://isa-afp.org/entries/Complex_Geometry.html, Formal proof development

  43. McFarland, A., McFarland, J., Smith, J. (eds.): Alfred Tarski: Early Work in Poland-Geometry and Teaching. Springer, New York (2014)

    MATH  Google Scholar 

  44. Meikle, L., Fleuriot, J.: Formalizing Hilbert’s Grundlagen in Isabelle/Isar. In: Theorem Proving in Higher Order Logics, pp. 319–334 (2003)

  45. Meikle, L., Fleuriot, J.: Mechanical theorem proving in computational geometry. In: Hong, H., Wang, D. (eds.) Automated Deduction in Geometry (ADG 2004), pp. 1–18. Springer, Berlin Heidelberg (2006)

  46. Narboux, J.: Mechanical theorem proving in Tarski’s geometry. In: Eugenio, F., Roanes, L. (eds) Automated Deduction in Geometry (ADG 2006), volume 4869 of Lecture Notes in Computer Science, pp. 139–156. Springer, Pontevedra, Spain (2007)

  47. Narboux, J., Janičić, P., Fleuriot, J.: Computer-assisted theorem proving in synthetic geometry. In: Sitharam, M., St. John, A., Sidman, J. (eds.) Handbook of Geometric Constraint Systems Principles, chapter 2, pp. 25–73. Chapman and Hall/CRC (2018)

  48. Needham, T.: Visual Complex Analysis. Oxford University Press, Oxford (1998)

    MATH  Google Scholar 

  49. Nipkow, T.: Programming and Proving in Isabelle/HOL (2013)

  50. Nipkow, T., Paulson, L., Wenzel, M.: Isabelle/HOL: A Proof Assistant for Higher-Order Logic, vol. 2283. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  51. Pasch, M.: Vorlesung über Neuere Geometrie. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  52. Paulson, L.C.: Natural deduction as higher-order resolution. J Log Program 3(3), 237–258 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  53. Paulson, L.C.: The foundation of a generic theorem prover. J Autom Reason 5(3), 363–397 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  54. Pichardie, D., Bertot, Y.: Formalizing convex hull algorithms. In: Boulton, R., Jackson, P. (eds.) Theorem Proving in Higher Order Logics, pp. 346–361. Springer, Berlin Heidelberg (2001)

  55. Puitg, F., Dufourd, J.-F.: Formal specification and theorem proving breakthroughs in geometric modeling. In: Grundy, J., Newey, M. (eds.) Theorem Proving in Higher Order Logics, pp. 401–422. Springer, Berlin Heidelberg (1998)

  56. Richter, W.: Formalizing Rigorous Hilbert axiomatic geometry proofs in the proof assistant Hol light. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.363.2373&rep=rep1&type=pdf

  57. Richter, W., Grabowski, A., Alama, J.: Tarski geometry axioms. Formaliz. Math. 22(2), 167–176 (2014)

    Article  MATH  Google Scholar 

  58. Riemann, B., Weyl, Hermann: Über die Hypothesen, welche der Geometrie zu Grunde liegen. Springer, Berlin (1921)

    Book  MATH  Google Scholar 

  59. Saccheri, G.G.: Euclides ab omni naevo vindicatus. Mediolani: Ex typographia Pauli Antonio Montani (1733)

  60. Schwabhäuser, W., Szmielew, W., Tarski, A.: Metamathematische Methoden in der Geometrie. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  61. Schwerdtfeger, H.: Geometry of Complex Numbers: Circle Geometry, Moebius Transformation, Non-euclidean Geometry. North Chelmsford, North Chelmsford (1979)

    MATH  Google Scholar 

  62. Scott, P.: Mechanising Hilbert’s foundations of geometry in Isabelle. Master’s thesis, University of Edinburgh (2008)

  63. Scott, P., Fleuriot, J.: An investigation of Hilbert’s implicit reasoning through proof discovery in idle-time. In: Proceedings of the Seventh International Workshop on Automated Deduction in Geometry, pp. 182–200 (2010)

  64. Simić, D., Marić, F., Boutry, P.: Poincaré disc model. Archive of Formal Proofs, (2019). http://isa-afp.org/entries/Poincare_Disc.html, Formal proof development

  65. Stojanović Đurđević, S., Narboux, J., Janičić, P.: Automated generation of machine verifiable and readable proofs: a case study of Tarski’s geometry. Ann. Math. Artific. Intell. pp. 249–269 (2015)

  66. Tarski, A., Givant, S.: Tarski’s system of geometry. Bull. Symbol. Log. 5(2), 175–214 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  67. von Plato, J.: The axioms of constructive geometry. Ann. Pure Appl. Log. 76(2), 169–200 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  68. Wenzel, M.: Isabelle/Isar—a versatile environment for human-readable formal proof documents. PhD thesis, Technische Universität München (2002)

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics, University of Belgrade, Studentski trg 16, Belgrade, 11000, Serbia

    Danijela Simić & Filip Marić

  2. ICube, UMR 7357 CNRS, University of Strasbourg, Pôle API, Bd Sébastien Brant, BP 10413, 67412, Illkirch, France

    Pierre Boutry

Authors
  1. Danijela Simić
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Filip Marić
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pierre Boutry
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Danijela Simić.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is partially supported by the Serbian Ministry of Education and Science Grant 174021.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Simić, D., Marić, F. & Boutry, P. Formalization of the Poincaré Disc Model of Hyperbolic Geometry. J Autom Reasoning 65, 31–73 (2021). https://doi.org/10.1007/s10817-020-09551-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10817-020-09551-2

Keywords

Search

Navigation