Letters in Mathematical Physics

, Volume 69, Issue 1, pp 287–315 | Cite as

Computer Theorem Proving in Mathematics

  • Carlos Simpson


We give an overview of issues surrounding computer-verified theorem proving in the standard pure-mathematical context. This includes the basic reasons why it should be interesting to pure mathematicians, some history, natural desiderata for a useful system, viewpoints on what kind of logic to use, a short explanation of how things work, an overview of different options for encoding sets, and perspectives on future developments.


Lambda calculus Type theory Theorem proving Verification Set theory 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aczel, P. The type theoretic interpretation of constructive set theory, In: Macintyre etal (eds), Logic Colloquium 77, Springer, New York, 1977Google Scholar
  2. Agerholm S., Gordon M.:(1995) Experiments with ZF set theory in HOL and Isabelle. In: Higher Order Logic Theorem Proving and its Applications (Aspen Grove, UT, 1995), Lecture Notes in Comput. Sci. 971, Springer, New York, . 32–45.Google Scholar
  3. Alexandre G.: An axiomatisation of intuitionistic Zermelo–Fraenkel set theory. See Scholar
  4. Alonso Tarrí o, L., Jeremí as López A.., Lipman J.: (1999). Studies in Duality on Noetherian Formal Schemes and Non-Noetherian Ordinary Schemes. Contemp. Math. 244 .Amer. Math. Soc. Providence.Google Scholar
  5. Altenkirch T., McBride C. Generic programming within dependently typed programming, To appear, WCGP 2002. Scholar
  6. Barras B., Coquand T., Werner B.: Paradoxes in set theory and type theory.User-contribution of Coq. INRIA. Rocquencourt.Google Scholar
  7. Bertot Y. Castéran P.: Coq’Art. Book in press.Google Scholar
  8. Boom H.: Message on [101], Feb. 16, 2001.Google Scholar
  9. Boyer, R., Moore, J.S. 1979A Computational LogicACM Monogr Ser Academic PressNew YorkGoogle Scholar
  10. Bledsoe W.W., Boyer R., Henneman W. (1971). Computer proofs of limit theorems. IJCAI . 586–600Google Scholar
  11. Boyer, R. eds. 1991Automated Reasoning: Essays in Honor of Woody Bledsoe Automated Reasoning Ser.Kluwer Acad. PublDordrechtGoogle Scholar
  12. Bundy A..:(1999) A survey of automated deduction, In: Michael J. Wooldridge and Manuela Veloso (ed.), Artificial Intelligence Today. Recent Trends and Developments, Lecture Notes in Comput. Sci. 1600, Springer, New York, . 153–174Google Scholar
  13. Capretta V.: Universal Algebra in Coq. universal_algebra.html.Google Scholar
  14. Chicli, L.: Sur la formalisation des mathématiques dans le calcul des constructions Inductives, Thesis, Université de Nice-Sophia Antipolis (Nov. 2003). http://www-sop.inria. fr/lemme/Laurent.Chicli/ Scholar
  15. Chicli, L., Pottier, L., Simpson, C. 2003Mathematical quotients and quotient types in CoqGeuvers, H.Wiedijk, F. eds. Types for Proofs and Programs, Lecture Notes in Comput. Sci. 2646SpringerNew York95107Google Scholar
  16. Coquand, T.: An analysis of Girard’s paradox, In: Proc. LICS, IEEE Press, 1985.Google Scholar
  17. Courant, J. 2002Explicit universes for the calculus of constructionsCarreño, V.Muñoz, C.Tahar, S. eds. Theorem Proving in Higher Order Logics 2002, Lecture Notes in Comput. Sci. 2410SpringerNew York115130Google Scholar
  18. Cuihtlauac, A.: Reflexion pour la reecriture dans le calcul de constructions inductives, Thesis, 18 Dec. 2002.Google Scholar
  19. Cruz-Filipe, L. 2003A constructive formalization of the fundamental theorem of calculusGeuvers, H.Wiedijk, F. eds. Types for Proofs and Programs, Lecture Notes in Comput. Sci. 2646SpringerNew York108126Google Scholar
  20. Deligne, P., Milne, J. S., Ogus, A., Shih, K. 1982Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Comput. Sci. 900SpringerNew YorkGoogle Scholar
  21. Feferman, S.: Typical ambiguity: trying to have your cake and eat it too, to appear in the proceedings of the conference Russell 2001, Munich, 2–5 June 2001.Google Scholar
  22. Fiore, M., Rosolini, G. 1997Two models of synthetic domain theoryJ. Pure Appl. Algebra.116151162Google Scholar
  23. Fitelson, B., Ulrich, D. and Wos, L.: XCB, the last of the shortest single axioms for the classical equivalential calculus, cs.LO/0211015, and Vanquishing the XCB question: the methodology discovery of the last shortest single axiom for the equivalential calculus, cs.LO/0211014.Google Scholar
  24. Friedman, H. 1998Finite functions and the necessary use of large cardinalsAnn Math148803893Google Scholar
  25. Gabai, D., Meyerhoff, G.R., Thurston, N. 2003Homotopy hyperbolic 3-manifolds are hyperbolicAnn. Math.157335431Google Scholar
  26. Geuvers, H.: Inconsistency of classical logic in type theory. herman/ See also other publications at pubs.html.Google Scholar
  27. Geuvers, H. Barendregt, H. Proof assistants using dependent type systems, Chapter 18 of A. Robinson and A. Voronkov (eds), Handbook of Automated Reasoning, Vol 2, Elsevier, Amsterdam, 2001, pp. 1149–1238.Google Scholar
  28. Geuvers, H., Wiedijk, F., Zwanenburg, J., Pollack, R. and Barendregt, H.: A formalized proof of the fundamental theorem of algebra in the theorem prover Coq, Contributions to Coq V.7, April 2001, Scholar
  29. Girard, J.-Y.: Interpretation fonctionnelle et élimination des coupures de l’arithmétique d’ordre supérieure, Thèse d’Etat, Université Paris 7 (1972).Google Scholar
  30. Gordon, M. From LCF to HOL: a short history. Proof, Language, and Interaction, Found. Comput. Ser., MIT Press, 2000, pp. 169–185.Google Scholar
  31. Hales, T.: The Flyspeck Project Fact Sheet. index.html.Google Scholar
  32. Harrison, J.: Formalized mathematics. (1996). form-math3.html, see also an html version at jrh0100.htm.Google Scholar
  33. Herbelin, H.: A program from an A-translated impredicative proof of Higman’s Lemma. User-contribution in [82], see Scholar
  34. Holmes, R.: Undefined terms, and the thread of messages following it (in particular J. Harrison’s reply), Volume 3, May 1995.Google Scholar
  35. Hohti, A.: Recursive synthesis and the foundations of mathematics. HO/0208184Google Scholar
  36. Huet, G.: Unification in typed lambda calculus, In: λ- Calculus and Computer Science Theory (Proc. Sympos., Rome, 1975), Lecture Notes in Comput. Sci. 37, Springer, New York, 1975, pp. 192–212.Google Scholar
  37. Huet, G. and Saï bi, A.: Constructive category theory, In: Proof, Language, and Interaction, Found. Comput. Ser., MIT Press, 2000, pp. 239–275Google Scholar
  38. Jutting, L. and van Bentham, S.: Checking Landau’s Grundlagen in the AUTOMATH system. Thesis, Eindhoven University of Technology, 1977.Google Scholar
  39. Kaufmann, M., Moore, J. S. 2002A computational logic for applicative common LISPJacquette, D. eds. A Companion to Philosophical LogicBlackwellOxford724741Google Scholar
  40. Kitoda, H. Is mathematics consistent? math.GM/0306007.Google Scholar
  41. Kunen, K., Ramsey, A. 1995Theorem in Boyer–Moore logicJ. Automat. Reason.15217235Google Scholar
  42. Lam, C., Thiel, L., Swiercz, S. 1989The nonexistence of finite projective planes of order 10Canad. J. Math.4111171123Google Scholar
  43. Lam, C., Thiel, L., Swiercz, S., McKay, J. 1983The nonexistence of ovals in a projective plane of order 10Discrete Math.45319321Google Scholar
  44. Lamport L. Types considered harmful, or Types are not harmless. This appeared under the first title in a posting by P. Rudnicki on [100] Volume 2, Aug. 1994. A revised version with the second title appeared as a technical report. A balanced discussion presenting both points of view is in the next reference.Google Scholar
  45. Lamport, L. Paulson, L. Should your specification language Be typed? ACM Trans. Programming Languages and Systems 21 (3) (May 1999), 502–526. See Scholar
  46. Luo, Z. An extended calculus of constructions, Thesis, University of Edinburgh, 1990.Google Scholar
  47. Maggesi, M. Proof of JMeq_eq, see posting on [101], Oct. 17th 2002, Scholar
  48. Martin-Löf, P. Intuitionistic Type Theory, Studies in Proof Theory, Bibliopolis, 1984.Google Scholar
  49. McCarthy, J. 1960Recursive functions of symbolic expressions and their computation by machine (Part I)CACM3184195Google Scholar
  50. McCarthy, J. Towards a mathematical science of computation. Proc. Information Processing Congn. 62: North-Holland, Amsterdam, 1962, pp. 21–28.Google Scholar
  51. McCarthy, J. 1963A basis for a mathematical theory of computationComputer Programming and Formal Systems. North-HollandAmsterdamGoogle Scholar
  52. McCune. W. and Veroff. R. A short Sheffer axiom for Boolean algebra., Scholar
  53. McKinna, J. Reply to thread How to prove two constructors are different, on [101] 6 Oct. 2003.Google Scholar
  54. Milner, R. LCF: a way of doing proofs with a machine, In: Mathematical Foundations of Computer Science (Proc. Eighth Sympos., Olomouc, 1979), Lecture Notes in Comput. Sci. 74, Springer New York 1979, pp. 146–159.Google Scholar
  55. Moerdijk, I. and MacLane, S. Sheaves in Geometry and Logic, Springer, New York, 1992.Google Scholar
  56. Moore, J.S. 1979A mechanical proof of the termination of Takeuchi’s functionInform. Process. Lett.9176181Google Scholar
  57. Neeman, A. 2002A counterexample to a 1961 theorem in homological algebraInvent. Math.148397420Google Scholar
  58. Nowak, D. Ensembles and the axiom of choice, on [101], 25 Nov. 1998.Google Scholar
  59. O’Connor, R. Proof of Gödel’s first incompleteness theorem http://math.berkeley. edu/roconnor/godel.html.Google Scholar
  60. Melville, D. J. Sumerian metrological numeration systems*. The relevance of ancient numbering systems as one of the origins of formalized mathematics was mentionned in [32] . dmelvill/mesomath/sumerian.html.Google Scholar
  61. Plotkin, B. Algebraic geometry in first order logic, math.GM/0312485.Google Scholar
  62. QED manifesto, Scholar
  63. Robinson, J. 1963A. Theorem-proving on the computerJ. Assoc. Comput. Mach.10163174Google Scholar
  64. Robinson, J.A. 1965A machine-oriented logic based on the resolution principleJ. Assoc. Comput. Mach.122341Google Scholar
  65. Russinoff, D. 1992A mechanical proof of quadratic reciprocityJ. Automat. Reason.8321Google Scholar
  66. Schmidhuber, C. Strings from logic. CERN-TH/2000-316, hep-th/0011065.Google Scholar
  67. Scott, D. Domains for denotational semantics, In: Automata, Languages and Programming (Aarhus, 1982), Lecture Notes in Comput. Sci. 140, Springer, New York, 1982, pp. 577–613.Google Scholar
  68. Shankar, N. 1988A mechanical proof of the Church-Rosser theoremJ. Assoc. Comput. Mach.35475522Google Scholar
  69. Shankar N. Metamathematics, Machines, and Gödel’s Proof, Cambridge Tracts in Theoretical Comput. Sci. 38, Cambridge University Press, 1994.Google Scholar
  70. Shimada, I. Vanishing cycles, the generalized Hodge conjecture, and Gröbner bases, math.AG/0311180.Google Scholar
  71. Simpson, A. Computational adequacy in an elementary topos, Proceedings CSL ‘98, Leucture Notes in Comput. Sci. 1584, Springer, New York, 1998, pp. 323–342.Google Scholar
  72. Simpson, C. Set-theoretical mathematics in Coq, Preprint with attached proof files, math.LO/0402336.Google Scholar
  73. Simpson, S. (ed): Reverse Mathematics 2001, to appear.Google Scholar
  74. Streicher, T. Lifting Grothendieck universes (with M. Hofmann); and Universes in toposes. Preprints available at Scholar
  75. Trybulec A. (1978). The Mizar-QC/6000 Logic Information Language. ALLC Bull. 6(2)Google Scholar
  76. Wenzel, M., Wiedijk, F. 2002A comparison of Mizar and Isar. J. AutomatReason.29389411Google Scholar
  77. Werner, B. Sets in types, types in sets, In: Theoretical Aspects of Computer Software (Sendai 1997), Lecture Notes in Comput. Sci. 1281, Springer, New York, 1999, pp. 530–546. Scholar
  78. Werner, B. An encoding of Zermolo-Fraenkel set theory in Coq:see http://coq.inria. fr/contribs-eng.html.Google Scholar
  79. Werner, B. Une théorie des constructions inductives, Thèse d’Etat, Univ. Paris 7 (1994).Google Scholar
  80. ACL2 system: Scholar
  81. Alfa system (formerly ALF): Scholar
  82. Coq system:, especially the reference manual: main.html.Google Scholar
  83. Ghilbert system: Scholar
  84. Helm Coq-on-line library (University of Bologna): library.html.Google Scholar
  85. HOL system: Scholar
  86. HOL-Light: Scholar
  87. IMPS system, see particularly the theory library: Scholar
  88. Metamath system: Scholar
  89. Isabelle system: Scholar
  90. The Elf meta-language. Scholar
  91. LEGO system: Scholar
  92. Mizar system: Scholar
  93. Nuprl system: Scholar
  94. The PhoX proof assistant: RAFFALLI/af2.html.Google Scholar
  95. PVS (Proof Verification System): Scholar
  96. TPS (Theorem Proving System): Scholar
  97. Z/EVES system, see particularly the Mathematical Toolkit: z-eves/.Google Scholar
  98. Formal Methods web page at Oxford (this has a very complete listing of items on the web concerning formal methods): Scholar
  99. Pfenning, F. Bibliography on logical frameworks (449 entries!) http://www-2.cs. Scholar
  100. The QED project (see particularly the archives of the QED mailing list, volumes 1–3): Scholar
  101. Coq-club mailing list archives: Scholar
  102. Jo. Formalized Math.: Scholar
  103. arXiv e-Print archive Scholar
  104. MathSci Net. (by subscription).Google Scholar
  105. The Google search engine. Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  1. 1.CNRS, Laboratoire J. A. DieudonnéUniversité de Nice-Sophia AntipolisNice Cedex 2France

Personalised recommendations