Abstract
We present in this paper the design of a graphical user interface to deal with proofs in geometry. The software developed combines three tools: a dynamic geometry software to explore, measure, and invent conjectures; an automatic theorem prover to check facts; and an interactive proof system (Coq) to mechanically check proofs built interactively by the user.
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Ag-Almouloud, S.: L’ordinateur, outil d’aide à l’apprentissage de la démonstration et de traitement de données didactiques. Ph.D. thesis, Université de Rennes (1992)
Amerkad, A., Bertot, Y., Pottier, L., Rideau, L.: Mathematics and proof presentation in Pcoq. In: Workshop Proof Transformation and Presentation and Proof Complexities in connection with IJCAR 2001 (2001)
Anderson, J.R., Boyle, C.F., Yost, G.: The geometry tutor. In: IJCAI Proceedings, pp. 1–7 (1985)
Aspinall, D., Lüth, C., Winterstein, D.: A framework for interactive proof. Technical report (2004)
Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, New York (1998)
Balacheff, N., Pesty, S., Occello, M., Caffera, R., Webber, C., Peltier, N.: Baghera. http://www-baghera.imag.fr (1999–2002)
Barwise, J., Allwein, G. (eds.): Logical Reasoning with Diagrams. Oxford University Press (1996)
Bernat, P.: CHYPRE: Un logiciel d’aide au raisonnement. Technical report 10, IREM (1993)
Bertot, Y., Guilhot, F., Pottier, L.: Visualizing geometrical statements with GeoView. Electronic Notes in Theoretical Computer Science 103, 49–65 (2003)
Bertot, Y., Thery, L.: A generic approach to building user interfaces for theorem provers. J. Symb. Comput. 25, 161–194 (1998)
Chou, S.-C.: Mechanical geometry theorem proving. Reidel (1988)
Chou, S.-C., Gao, X.-S.: A class of geometry statements of constructive type and geometry theorem proving. In: Proceedings of CADE’92 (1992)
Chou, S.-C., Gao, X.-S., Zhang, J.-Z.: Machine Proofs in Geometry. Singapore: World Scientific. (1994)
Chou, S.-C., Gao, X.-S., Zhang, J.-Z.: Automated generation of readable proofs with geometric invariants, theorem proving with full angle. J. Autom. Reason. 17, 325–347 (1996)
Coq development team: The Coq proof assistant reference manual. Version 8.0. LogiCal Project (2004)
Furinghetti, F., Domingo, P.: To produce conjectures and to prove them within a dynamic geometry environment: a case study. In: Proceedings of Psychology of Mathematics 27th International Conference, pp. 397–404 (2003)
Gao, X.-S.: Geometry expert, software package. http://www.mmrc.iss.ac.cn/gex (2000)
Gao, X.-S., Lin, Q.: MMP/Geometer – a software package for automated geometry reasoning. In: Winkler, F. (ed.) Proceedings of ADG 2002, pp. 44–46 (2002)
Gressier, J.: Geometrix. http://perso.orange.fr/jgressier/ENGLISH/english.html (1988–1998)
Guilhot, F.: Formalisation en Coq et visualisation d’un cours de géométrie pour le lycée. Revue des Sciences et Technologies de l’Information, Technique et Science Informatiques, Langages applicatifs 24, 1113–1138 (2005) Lavoisier.
Harrison, J.: Automatic theorem proving examples. http://www.cl.cam.ac.uk/users/jrh/atp/index.html (2003)
Jackiw, N.: The geometer’s sketchpad. http://www.keypress.com/ (1990)
Jamnik, M.: Mathematical Reasoning with Diagrams: From Intuition to Automation. CSLI Press. (2001)
Kortenkamp, U.: Foundations of dynamic geometry. Ph.D. thesis, ETH Zürich (1999)
Kortenkamp, U., Richter-Gebert, J.: Using automatic theorem proving to improve the usability of geometry software. In: Mathematical User Interface (2004)
Laborde, J.-M., Bellemain, F.: Cabri-geometry II. http://www.cabri.net (1993–1998)
Luengo, V.: Cabri-Euclide: Un micromonde de Preuve intégrant la réfutation. Ph.D. thesis, Université Joseph Fourier (1997)
Meikle, L., Fleuriot, J.: Formalizing Hilbert’s Grundlagen in Isabelle/Isar. In: Theorem Proving in Higher Order Logics, pp. 319–334 (2003)
Miller, N.: A diagrammatic formal system for Euclidean geometry. Ph.D. thesis, Cornell University (2001)
Narboux, J.: A decision procedure for geometry in Coq. In: Konrad, S., Annett, B., Ganesh, G. (eds.) Proceedings of TPHOLs’ 2004, vol. 3223 of Lecture Notes in Computer Science (2004)
Narboux, J.: Formalisation et automatisation du raisonnement géométrique en Coq’. Ph.D. thesis, Université Paris Sud (2006a)
Narboux, J.: A formalization of diagrammatic proofs in abstract rewriting (2006b)
Narboux, J.: Mechanical theorem proving in Tarski’s geometry. In: Proceedings of Automatic Deduction in Geometry 06 (2006c)
Narboux, J.: The user manual of GeoProof. http://home.gna.org/geoproof/documentation.html (2006d)
Py, D.: Reconnaissance de plan pour l’aide à la démonstration dans un tuteur intelligent de la géométrie. Ph.D. thesis, Université de Rennes (1990)
Richter-Gebert, J., Kortenkamp, U.: Die interaktive Geometrie software Cinderella Book and CD-ROM. German school-edition of the Cinderella software. http://cinderella.de (1999)
Schwartz, J. T.: Probabilistic algorithms for verification of polynomial identities. In: Symbolic and Algebraic Computation, vol. 72. Lecture Notes in Computer Science. Marseille, pp. 200–215 (1979)
Wilson, S., Fleuriot, J.: Combining dynamic geometry, automated geometry theorem proving and diagrammatic proofs. In: ETAPS Satellite Workshop on User Interfaces for Theorem Provers (UITP). Edinburgh (2005)
Winterstein, D.: Dr. Doodle: A diagrammatic theorem prover. In: Proceedings of IJCAR 2004 (2004a)
Winterstein, D.: Using diagrammatic reasoning for theorem proving in continous domain. Ph.D. thesis, The University of Edinburgh (2004b)
Winterstein, D., Aspinall, D., Lüth, C.: PG/Eclipse: A generic interface for interactive proof. Technical report (2004)
Wu, W.-T.: On the decision problem and the mechanization of theorem proving in elementary geometry. In: Scientia Sinica, vol. 21. pp. 157–179 (1978)
Yevdokimov, O.: About a constructivist approach for stimulating students’ thinking to produce conjectures and their proving in active learning of geometry. In: Fourth Congress of the European Society for Research in Mathematics Education (2004)
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Narboux, J. A Graphical User Interface for Formal Proofs in Geometry. J Autom Reasoning 39, 161–180 (2007). https://doi.org/10.1007/s10817-007-9071-4
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DOI: https://doi.org/10.1007/s10817-007-9071-4