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From informal to formal proofs in Euclidean geometry

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Abstract

In this paper, we propose a new approach for automated verification of informal proofs in Euclidean geometry using a fragment of first-order logic called coherent logic and a corresponding proof representation. We use a TPTP inspired language to write a semi-formal proof of a theorem, that fairly accurately depicts a proof that can be found in mathematical textbooks. The semi-formal proof is verified by generating more detailed proof objects expressed in the coherent logic vernacular. Those proof objects can be easily transformed to Isabelle and Coq proof objects, and also in natural language proofs written in English and Serbian. This approach is tested on two sets of theorem proofs using classical axiomatic system for Euclidean geometry created by David Hilbert, and a modern axiomatic system E created by Jeremy Avigad, Edward Dean, and John Mumma.

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References

  1. Avigad, J.: Understanding proofs. In: Mancosu, P. (ed.) The Philosophy of Mathematical Practice. Oxford University Press, Oxford (2008)

  2. Avigad, J., Dean, E., Mumma, J.: A formal system for Euclid’s elements. The Review of Symbolic Logic (2009)

  3. Beeson, M., Narboux, J., Wiedijk, F.: Proof-checking Euclid. CoRR (2017). arXiv:1710.00787

  4. Bezem, M., Coquand, T.: Automating coherent logic. In: Sutcliffe, G., Voronkov, A. (eds.) 12th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning — LPAR 2005, Lecture Notes in Computer Science, vol. 3835. Springer (2005)

  5. Bezem, M., Hendriks, D.: On the mechanization of the proof of Hessenberg’s theorem in coherent logic. J. Autom. Reas., 40(1) (2008)

  6. Blanchette, J.C.: Automatic Proofs and Refutations for Higher-Order Logic. Ph.D. thesis. Department of Informatics, Technische Universität München (2012)

  7. Blanchette, J.C.: Redirecting proofs by contradiction. In: Blanchette, J.C., Urban, J. (eds.) Third International Workshop on Proof Exchange for Theorem Proving, PxTP 2013, Lake Placid, NY, USA, June 9-10, 2013, EPiC Series, vol. 14, pp 11–26. EasyChair (2013)

  8. Blanchette, J.C., Böhme, S., Paulson, L.C.: Extending sledgehammer with SMT solvers. J. Autom. Reason. 51(1), 109–128 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Blanchette, J.C., Bulwahn, L., Nipkow, T.: Automatic proof and disproof in Isabelle/HOL. In: Tinelli, C., Sofronie-Stokkermans, V. (eds.) Frontiers of Combining Systems, 8th International Symposium, Proceedings, Lecture Notes in Computer Science, vol. 6989, pp 12–27. Springer (2011)

  10. Braun, G., Narboux, J.: From Tarski to Hilbert. In: Ida, T., Fleuriot, J. (eds.) Automated Deduction in Geometry 2012. http://hal.inria.fr/hal-00727117, Edinburgh (2012)

  11. Braun, G., Narboux, J.: A synthetic proof of Pappus’ theorem in Tarski’s geometry. J. Autom. Reason., 1–22 (2016)

  12. Dehlinger, C., Dufourd, J.F., Schreck, P.: Higher-order intuitionistic formalization and proofs in Hilbert’s elementary geometry. In: Automated Deduction in Geometry, Lecture Notes in Computer Science, vol. 2061. Springer (2001)

  13. Despotović, R., Tošić, R., Šešelja, B.: Matematika za I razred srednje škole. Zavod za udžbenike i nastavna sredstva, Beograd (2000)

    Google Scholar 

  14. Ekici, B., Katz, G., Keller, C., Mebsout, A., Reynolds, A.J., Tinelli, C.: Extending smtcoq, a certified checker for smt (extended abstract). In: Blanchette, J.C., Kaliszyk, C. (eds.) Proceedings First International Workshop on Hammers for Type Theories, Coimbra, Portugal, July 1, 2016, Electronic Proceedings in Theoretical Computer Science, vol. 210, pp 21–29. Open Publishing Association (2016)

  15. Euclid: The Thirteen Books of the Elements, 2nd edn. Dover Publications, New York (1956). Translated with introduction and commentary by Sir Thomas L. Heath, from the text of Heiberg

    MATH  Google Scholar 

  16. Fisher, J., Bezem, M.: Skolem machines and geometric logic. In: Jones, C.B., Liu, Z., Woodcock, J. (eds.) 4th International Colloquium on Theoretical Aspects of Computing — ICTAC 2007, Lecture Notes in Computer Science, vol. 4711. Springer (2007)

  17. Ganesalingam, M., Gowers, W.T.: A fully automatic theorem prover with human-style output. J. Autom. Reas., 1–39 (2016). https://doi.org/10.1007/s10817-016-9377-1

  18. Gonthier, G., Asperti, A., Avigad, J., Bertot, Y., Cohen, C., Garillot, F., Le Roux, S., Mahboubi, A., O’Connor, R., Ould Biha, S., Pasca, I., Rideau, L., Solovyev, A., Tassi, E., Théry, L.: A machine-checked proof of the odd order theorem. In: Blazy, S., Paulin, C., Pichardie, D. (eds.) ITP 2013, 4th Conference on Interactive Theorem Proving, LNCS, vol. 7998, pp 163–179. Springer, Rennes (2013). https://doi.org/10.1007/978-3-642-39634-2_14

    Google Scholar 

  19. Hales, T.C.: Introduction to the Flyspeck Project, Mathematics, Algorithms, Proofs, Dagstuhl Seminar Proceedings, vol. 05021. Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany (2006)

    Google Scholar 

  20. Harrison, J.: HOL light: A tutorial introduction. In: Srivas, M.K., Camilleri, A.J. (eds.) Formal Methods in Computer-Aided Design, Lecture Notes in Computer Science, vol. 1166. Springer (1996)

  21. Hilbert, D.: Foundations of Geometry. Open Court Classics, 10th edn (1971)

  22. Janicić, P.: GCLC – A tool for constructive Euclidean geometry and more than that. In: Takayama, N., Iglesias, A., Gutierrez, J. (eds.) Proceedings of International Congress of Mathematical Software (ICMS 2006), Lecture Notes in Computer Science, vol. 4151, pp 58–73. Springer (2006)

  23. Kaliszyk, C., Krauss, A.: Scalable LCF-style proof translation. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) Interactive Theorem Proving - 4th International Conference, ITP 2013, Rennes, France, July 22-26, 2013. Proceedings, Lecture Notes in Computer Science, vol. 7998, pp 51–66. Springer (2013)

  24. Kaliszyk, C., Urban, J.: Mizar 40 for mizar 40. J. Autom. Reason. 55(3), 245–256 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kaliszyk, C., Urban, J., Vyskočil, J: Learning to Parse on Aligned Corpora (Rough Diamond), pp 227–233. Springer International Publishing, Cham (2015)

    MATH  Google Scholar 

  26. Kaliszyk, C., Urban, J., Vyskočil, J.: Automating Formalization by Statistical and Semantic Parsing of Mathematics, pp 12–27. Springer International Publishing, Cham (2017)

    MATH  Google Scholar 

  27. Kaliszyk, C., Urban, J., Vyskočil, J., Geuvers, H.: Developing Corpus-Based Translation Methods between Informal and Formal Mathematics: Project Description, pp 435–439. Springer International Publishing, Cham (2014)

    MATH  Google Scholar 

  28. Kinyon, M., Veroff, R., Vojtěchovský, P: Loops with Abelian Inner Mapping Groups: An Application of Automated Deduction, pp 151–164. Springer, Berlin (2013)

    MATH  Google Scholar 

  29. McCune, W.: Solution of the robbins problem. J. Autom. Reason. 19(3), 263–276 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. Meikle, L., Fleuriot, J.: Formalizing Hilbert’s Grundlagen in Isabelle/Isar. In: Proceedings of TPHOLs, Lecture Notes in Computer Science, vol. 2758. Springer (2003)

  31. Meng, J., Paulson, L.C.: Translating higher-order clauses to first-order clauses. J. Autom. Reason. 40(1), 35–60 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Meng, J., Quigley, C., Paulson, L.C.: Automation for interactive proof: first prototype. Inf. Comput. Special Issue: Comb. Logical Syst. 204(10), 1575–1596 (2006)

    MathSciNet  MATH  Google Scholar 

  33. Mitrović, M., Ognjanović, S., Veljković, M., Petković, L., Lazarević, N.: Geometrija za prvi razred Matematičke gimnazije. KRUG Beograd (1998)

  34. de Moura, L., Bjørner, N.: Z3: An Efficient SMT Solver, pp 337–340. Springer, Berlin (2008)

    Google Scholar 

  35. Narboux, J.: Mechanical theorem proving in Tarski’s geometry. In: Proceedings of Automated Deduction in Geometry 2006, Lecture Notes in Computer Science, vol. 4869. Springer (2007)

  36. Nikoliċ, M., Janičiċ, P.: CDCL-Based Abstract State Transition System for Coherent Logic, pp 264–279. Springer, Berlin (2012)

    MATH  Google Scholar 

  37. Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL - A Proof Assistant for Higher-Order Logic, Lecture Notes in Computer Science, vol. 2283. Springer (2002)

  38. Northrop, B.: Automated Diagrammatic Reasoning: A proof checker for the language of E. Master’s thesis, Department of Philosophy Carnegie Mellon University (2011)

  39. Phillips, J.D., Stanovský, D.: Automated theorem proving in quasigroup and loop theory. AI Commun. 23(2-3), 267–283 (2010)

    MathSciNet  MATH  Google Scholar 

  40. Polonsky, A.: Proofs, Types and Lambda Calculus. Ph.D. thesis, University of Bergen (2011)

  41. Riazanov, A., Voronkov, A.: The design and implementation of VAMPIRE. AI Commun. 15(2-3), 91–110 (2002)

    MATH  Google Scholar 

  42. Rojas, K.: EuclidZ3 – a Computational Proof-Checker for the Language E: Possible Backend for Interactive Geometric Proof Environments. Master’s thesis, Department of Philosophy Carnegie Mellon University (2015)

  43. Schulz, S.: E - a brainiac theorem prover. AI Commun. 15(2-3), 111–126 (2002)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  45. Scott, P., Fleuriot, J.: An investigation of Hilbert’s implicit reasoning through proof discovery in idle-time. In: Schreck, P., Narboux, J., Richter-Gebert, J. (eds.) Automated Deduction in Geometry, pp 182–200. Springer, Berlin (2011)

    Chapter  Google Scholar 

  46. Stojanović, S.: Formalization and automation of Euclidean geometry (in serbian). Ph.D. thesis, University of Belgrade. Advisor: Predrag Janicić (2016)

  47. Stojanović, S., Narboux, J., Bezem, M., Janičić, P.: A vernacular for coherent logic. In: S.W., other (eds.) Intelligent Computer Mathematics - CICM 2014, Lecture Notes in Computer Science, vol. 8543. Springer (2014)

  48. Stojanović, S., Pavlović, V., Janicić, P.: A coherent logic based geometry theorem prover capable of producing formal and readable proofs. In: Schreck, P., Narboux, J., Richter-Gebert, J. (eds.) Automated Deduction in Geometry, Lecture Notes in Computer Science, vol. 6877. Springer (2011)

  49. Stojanović, V.: Matematiskop 3 - Odabrani zadaci (sa rešenjima) za učenike srednjih škola. Beograd, Naučna knjiga (1988)

    Google Scholar 

  50. Stojanović-Ðurđević, S.: Automated verification of informal proofs from high school geometry (in Serbian) InfoM (2016)

  51. Stojanović-Ðurđević, S., Narboux, J., Janičić, P.: Automated generation of machine verifiable and readable proofs: A case study of Tarski’s geometry. Annals of Mathematics and Artificial Intelligence (2015)

  52. Sutcliffe, G.: The TPTP problem library and associated infrastructure: The FOF and CNF parts, v3.5.0. J. Autom. Reason. 43(4), 337–362 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  53. The Coq development team: The Coq proof assistant reference manual, Version 8.2. TypiCal Project (2009). http://coq.inria.fr

  54. Weidenbach, C., Dimova, D., Fietzke, A., Kumar, R., Suda, M., Wischnewski, P.: Spass version 3.5. In: Automated Deduction - CADE-22 Proceedings, Lecture Notes in Computer Science, vol. 5663, pp. 140–145. Springer (2009)

  55. Wenzel, M.: Isar - a generic interpretative approach to readable formal proof documents. In: Bertot, Y., Dowek, G., Hirschowitz, A., Paulin, C., Théry, L. (eds.) Theorem Proving in Higher Order Logics (TPHOLs’99), Lecture Notes in Computer Science, vol. 1690, pp 167–184. Springer (1999)

  56. Wiedijk, F.: A synthesis of the procedural and declarative styles of interactive theorem proving. Logical Methods Comput. Sci., 8(1) (2012)

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Acknowledgements

We are grateful to the anonymous reviewers for the extremely helpful feedback on the first version of the paper.

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Authors and Affiliations

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

    Sana Stojanović-Ðurđević

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  1. Sana Stojanović-Ðurđević
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Correspondence to Sana Stojanović-Ðurđević.

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The author is partly supported by the grant ON174021 of the Ministry of Science of Serbia.

Appendix: Invocation of the system ArgoGeoChecker

Appendix: Invocation of the system ArgoGeoChecker

  • Detailed README.txt file can be found on the system’s web-page. Here we will give just one example of the invocation of the system. Input file should be named th_name_proof.txt and located in the folder theorems_and_proofs

  • System invocation: ./ArgoGeoChecker axiom_system th_name_proof.txt

  • The shortcut has been added for Hilbert’s axiomatic system:./ArgoGeoChecker I th_1_proof.txt The option “I” can be used when we want to use just the first group of Hilbert’s axioms for verification of the proof, and the option “II” can be used when we want to use the first and the second group of Hilbert’s axioms together (definitions are always used).

  • System will automatically generate (in the folder theorem_and_proofs) files for all target languages (with extensions *.thy, *.v, *.tex, *.pdf).

  • An example:


    ./ArgoGeoChecker I th_1_proof.txt th_1_01.p| vampire| 0.03| 0.13| ax_I3a | 0.00|| th_1_02.p| vampire| 0.27| 0.21| ax_sym_ncol ax_D1a | 0.01|| th_1_03.p| e| 0.28| 0.13| ax_I4a | 0.02|| th_1_04.p| vampire| 0.16| 0.41| ax_I6 | 0.00|| Seconds: 6

    Here we can see: the name of the auxiliary lemma that is proved — resolution theorem prover that found the smallest axiom set that the lemma can be proved with — Vampire time — E time — set of axioms that was found — ArgoCLP time.

  • List of semi-formal proofs in Hilbert’s axiomatic system (with the shorcut for the specific set of Hilbert’s axioms):


    ./ArgoGeoChecker I th_1_proof.txt ./ArgoGeoChecker I th_2_proof.txt ./ArgoGeoChecker I th_3_proof.txt ./ArgoGeoChecker I th_4_proof.txt ./ArgoGeoChecker I th_5_proof.txt ./ArgoGeoChecker I th_6_proof.txt ./ArgoGeoChecker II th_7_proof.txt ./ArgoGeoChecker II th_8_proof.txt ./ArgoGeoChecker I th_9_proof.txt ./ArgoGeoChecker I th_10_proof.txt ./ArgoGeoChecker I th_11_proof.txt ./ArgoGeoChecker I th_12_proof.txt ./ArgoGeoChecker I th_13_proof.txt ./ArgoGeoChecker I th_14_proof.txt ./ArgoGeoChecker II th_15_proof.txt ./ArgoGeoChecker II th_16_proof.txt ./ArgoGeoChecker II th_17_proof.txt ./ArgoGeoChecker II th_18_proof.txt

  • List of semi-formal proofs in the axiomatic system E:


    ./ArgoGeoChecker axiomatic_system/avigad_axioms.p th_prop1_proof.txt ./ArgoGeoChecker axiomatic_system/avigad_axioms.p th_prop1aux1_proof.txt ./ArgoGeoChecker axiomatic_system/avigad_axioms.p th_prop1aux2_proof.txt ./ArgoGeoChecker axiomatic_system/avigad_axioms.p th_prop2_proof.txt

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Stojanović-Ðurđević, S. From informal to formal proofs in Euclidean geometry. Ann Math Artif Intell 85, 89–117 (2019). https://doi.org/10.1007/s10472-018-9597-7

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