Hostname: page-component-5db58dd55d-m58mf Total loading time: 0 Render date: 2026-06-16T19:19:28.644Z Has data issue: false hasContentIssue false
  • English
  • Français

Formalization of real analysis: a survey of proof assistants and libraries†

Published online by Cambridge University Press:  02 January 2015

SYLVIE BOLDO ,
CATHERINE LELAY  and
GUILLAUME MELQUIOND
SYLVIE BOLDO
Affiliation:
Inria, LRI, bâtiment 650, Université Paris-Sud, F-91405 Orsay Cedex, France Email: Sylvie.Boldo@inria.fr
CATHERINE LELAY
Affiliation:
Inria, LRI, bâtiment 650, Université Paris-Sud, F-91405 Orsay Cedex, France Email: Sylvie.Boldo@inria.fr
GUILLAUME MELQUIOND
Affiliation:
Inria, LRI, bâtiment 650, Université Paris-Sud, F-91405 Orsay Cedex, France Email: Sylvie.Boldo@inria.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

In the recent years, numerous proof systems have improved enough to be used for formally verifying non-trivial mathematical results. They, however, have different purposes and it is not always easy to choose which one is adapted to undertake a formalization effort. In this survey, we focus on properties related to real analysis: real numbers, arithmetic operators, limits, differentiability, integrability and so on. We have chosen to look into the formalizations provided in standard by the following systems: Coq, HOL4, HOL Light, Isabelle/HOL, Mizar, ProofPower-HOL, and PVS. We have also accounted for large developments that play a similar role or extend standard libraries: ACL2(r) for ACL2, C-CoRN/MathClasses for Coq, and the NASA PVS library. This survey presents how real numbers have been defined in these various provers and how the notions of real analysis described above have been formalized. We also look at the methods of automation these systems provide for real analysis.

Information

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 26 , Issue 7 , October 2016 , pp. 1196 - 1233
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

†

This work was supported by Project Coquelicot from RTRA Digiteo and Région Île-de-France.

References

Arthan, R. D. (2001) An irrational construction of ℝ from ℤ. In: Boulton, and Jackson, P. B. (eds.) Proceedings of the 14th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2001), Edinburgh, UK. Lecture Notes in Computer Science 2152 43–58.Google Scholar
Arthan, R. D. (2012) Mathematical Case Studies: Basic Analysis, Lemma 1 Ltd.Google Scholar
Arthan, R. D. and King, D. (1996) Development of practical verification tools. ICL Systems Journal 11 (1) 106–122.Google Scholar
Avigad, J. and Donnelly, K. (2004) Formalizing O notation in Isabelle/HOL. In: Basin, D. and Rusinowitch, M. (eds.) Proceedings of the 2nd International Joint Conference on Automated Reasoning (IJCAR 2004). Lecture Notes in Computer Science 3097 357–371.Google Scholar
Bertot, Y. and Castéran, P. (2004) Interactive Theorem Proving and Program Development. Coq'Art: The Calculus of Inductive Constructions, Texts in Theoretical Computer Science, Springer Verlag, Berlin Heidelberg.CrossRefGoogle Scholar
Besson, F. (2006) Fast reflexive arithmetic tactics: The linear case and beyond. In: Proceedings of the International Conference on Types for proofs and programs (TYPES'06), Nottingham, UK. Lecture Notes in Computer Science 4502 48–62.Google Scholar
Bickford, M. (2008) Formalizing Constructive Analysis in Nuprl, Computing Science Department, Cornell University. Available at: http://www.nuprl.org/MathLibrary/ConstructiveAnalysis/ Google Scholar
Boldo, S., Clément, F., Filliâtre, J.-C., Mayero, M., Melquiond, G. and Weis, P. (2013) Wave equation numerical resolution: A comprehensive mechanized proof of a C program. Journal of Automated Reasoning 50 (4) 423–456.Google Scholar
Boldo, S., Lelay, C. and Melquiond, G. (2012) Improving real analysis in Coq: A user-friendly approach to integrals and derivatives. In: Hawblitzel, C. and Miller, D. (eds.) Proceedings of the 2nd International Conference on Certified Programs and Proofs (CPP 2012), Kyoto, Japan. Lecture Notes in Computer Science 7679 289–304.Google Scholar
Boldo, S. and Melquiond, G. (2011) Flocq: A unified library for proving floating-point algorithms in Coq. In: Antelo, E., Hough, D. and Ienne, P. (eds.) Proceedings of the 20th IEEE Symposium on Computer Arithmetic (ARITH 20), Tübingen, 243–252.Google Scholar
Butler, R. W. (2009) Formalization of the integral calculus in the PVS theorem prover. Journal of Formalized Reasoning 2 (1) 1–26.Google Scholar
Byliński, C. (1990) Functions and their basic properties. Journal of Formalized Mathematics 1 (1) 55–65.Google Scholar
Chaieb, A. (2006) Verifying mixed real-integer quantifier elimination. In: Furbach, U. and Shankar, N. (eds.) Proceedings of the 3rd International Joint on Automated Reasoning (IJCAR 2006), Seattle, WA, USA. Lecture Notes in Computer Science 4130 528–540.Google Scholar
Chaieb, A. (2008) Parametric linear arithmetic over ordered fields in Isabelle/HOL. In: Autexier, S., Campbell, J., Rubio, J., Sorge, V., Suzuki, M. and Wiedijk, F. (eds.) Intelligent Computer Mathematics. Lecture Notes in Artificial Intelligence 5144 246–260. (Birmingham, UK)Google Scholar
Chaieb, A. and Wenzel, M. (2007) Context aware calculation and deduction – ring equalities via Gröbner bases in Isabelle. In: Kauers, M., Kerber, M., Miner, R. and Windsteiger, W. (eds.) Proceedings of the 14th Symposium Calculemus 2007, Hagenberg, Austria, Towards Mechanized Mathematical Assistants. Lecture Notes in Artificial Intelligence 4573 27–39.Google Scholar
Ciaffaglione, A. and Di-Gianantonio, P. (2006) A certified, corecursive implementation of exact real numbers. Theoretical Computer Science 351 (1) 39–51.Google Scholar
Cohen, C. (2012) Formalized Algebraic Numbers: Construction and First Order Theory, Ph.D. thesis, École Polytechnique.Google Scholar
Constable, R. L., Allen, S. F., Allen, S. F., Bromley, H. M., Cleaveland, W. R., Cremer, J. F., Harper, R. W., Howe, D. J., Knoblock, T. B., Mendler, N. P., Panangaden, P., Smith, S. F., Sasaki, J. T. and Smith, S. F. (1986) Implementing Mathematics with the Nuprl Proof Development System, Computing Science Department, Cornell University, Ithaca, NY.Google Scholar
Cowles, J. R. and Gamboa, R. (2010) Using a first order logic to verify that some set of reals has no Lesbegue measure. In: Kaufmann, M. and Paulson, L. C. (eds.) Proceeding of the 1st International Conference of Interactive Theorem Proving (ITP 2010), Edinburgh, UK. Lecture Notes in Computer Science 6172 25–34.Google Scholar
Cruz-Filipe, L. (2003) A constructive formalization of the fundamental theorem of calculus. In: Proceedings of the International Conference on Types for Proofs and Programs (TYPES'02), Berg en Dal, The Netherlands. Lecture Notes in Computer Science 2646 108–126.Google Scholar
Cruz-Filipe, L. (2004) Constructive Real Analysis: A Type-Theoretical Formalization and Applications, Ph.D. thesis, University of Nijmegen.Google Scholar
Cruz-Filipe, L., Geuvers, H. and Wiedijk, F. (2004) C-CoRN: The constructive Coq repository at Nijmegen. In: Asperti, A., Bancerek, G. and Trybulec, A. (eds.) Proceedings of the 3rd International Conference of Mathematical Knowledge Management (MKM 2004). Lecture Notes in Computer Science 3119 88–103.Google Scholar
Daumas, M., Lester, D. and Muñoz, C. (2009) Verified real number calculations: A library for interval arithmetic. IEEE Transactions on Computers 58 (2) 226–237.Google Scholar
Delahaye, D. and Mayero, M. (2001) Field, une procédure de décision pour les nombres réels en Coq. In: Castéran, P. (ed.) Journées Francophones des Langages Applicatifs (JFLA'01) 33–48.Google Scholar
Desmettre, O. (2002) Coq standard library – RiemannInt.Google Scholar
Dutertre, B. (1996) Elements of mathematical analysis in PVS. In: von Wright, J., Grundy, J. and Harrison, J. (eds.) Proceedings of the 9th International Conference Theorem Proving in Higher Order Logics (TPHOLs 1996), Turku, Finland. Lecture Notes in Computer Science 1125 141–156.Google Scholar
Endou, N. and Korniłowicz, A. (1999) The definition of the Riemann definite integral and some related lemmas. Journal of Formalized Mathematics 8 (1) 93–102.Google Scholar
Endou, N., Wasaki, K. and Shidama, Y. (2001) Definition of integrability for partial functions from ℝ to ℝ and integrability for continuous functions. Formalized Mathematics 9 (2) 281–284.Google Scholar
Fleuriot, J. (2000) On the mechanization of real analysis in Isabelle/HOL. In: Aagaard, M. and Harrison, J. (eds.) Proceeding of the 13th International Conference of Theorem Proving in Higher Order Logics (TPHOLs 2000), Portland, OR, USA. Lecture Notes in Computer Science 1869 145–161.Google Scholar
Gamboa, R. (2000) Continuity and differentiability in ACL2. In: Kaufmann, M., Manolios, P. and Moore, J. S. (eds.) Computer-Aided Reasoning: ACL2 Case Studies. Advances in Formal Methods, Springer, US 301–316.Google Scholar
Gamboa, R. and Cowles, J. R. (2009) Inverse functions in ACL2(r) In: Proceedings of the 8th International Workshop on the ACL2 Theorem Prover and its Applications 57–61.Google Scholar
Gamboa, R., Cowles, J. R. and Kuzmina, N. (2004) Axiomatic events in ACL2(r): A story of defun, defun-std, and encapsulate. In: Proceedings of the 5th International Workshop on the ACL2 Theorem Prover and its Applications, Austin, TX, USA 1–18.Google Scholar
Gamboa, R. and Kaufmann, M. (2001) Nonstandard analysis in ACL2. Journal of Automated Reasoning 27 (4) 323–351.Google Scholar
Garillot, F., Gonthier, G., Mahboubi, A. and Rideau, L. (2009) Packaging mathematical structures. In: Berghofer, S., Nipkow, T., Urban, C. and Wenzel, M. (eds.) Proceedings of the 22nd International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2009), München, Germany. Lecture Notes in Computer Science 5674 327–342.Google Scholar
Geuvers, H. and Niqui, M. (2002) Constructive reals in Coq: Axioms and categoricity. In: Callaghan, P., Luo, Z., McKinna, J. and Pollack, R. (eds.) Selected Papers of the International Workshop on Types for Proofs and Programs (TYPES 2000), Durham, UK. Lecture Notes in Computer Science 2277 79–95.Google Scholar
Gordon, M. J. C. and Melham, T. F. (1993) Introduction to HOL: A Theorem Proving Environment for Higher Order Logic, Cambridge University Press, Cambridge, England.Google Scholar
Gottliebsen, H. (2000) Transcendental functions and continuity checking in PVS. In: Aagaard, M. and Harrison, J. (eds.) Proceedings of the 13th International Conference Theorem Proving in Higher Order Logics (TPHOLs 2000), Portland, OR, USA. Lecture Notes in Computer Science 1869 197–214.Google Scholar
Grégoire, B. and Mahboubi, A. (2005) Proving equalities in a commutative ring done right in Coq. In: Hurd, J. and Melham, T. (eds.) Proceedings of the 18th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2005), Oxford, UK. Lecture Notes in Computer Science 3603 98–113.Google Scholar
Guard, J. R., Oglesby, F. C., Bennett, J. H. and Settle, L. G. (1969) Semi-automated mathematics. Journal of the ACM 16 49–62.Google Scholar
Hales, T. (2012) Dense Sphere Packings: A Blueprint for Formal Proofs, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, England.Google Scholar
Harrison, J. (1994) Constructing the real numbers in HOL. Formal Methods in System Design 5 (1–2) 35–59.Google Scholar
Harrison, J. (1996) Proof style. In: Proceedings of the International Workshop on Types for Proofs and Programs (TYPES'96), Aussois, France. Lecture Notes in Computer Science 1512 154–172.Google Scholar
Harrison, J. (1997) Floating point verification in HOL Light: The exponential function, Technical Report 428, University of Cambridge Computer Laboratory.Google Scholar
Harrison, J. (1998) Theorem Proving with the Real Numbers, Springer-Verlag, London.Google Scholar
Harrison, J. (2005) A HOL theory of Euclidean space. In: Hurd, J. and Melham, T. (eds.) Proceedings of the 18th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2005), Oxford, UK. Lecture Notes in Computer Science 3603 114–129.Google Scholar
Harrison, J. (2006) Towards self-verification of HOL Light. In: Furbach, U. and Shankar, N. (eds.) Proceedings of the 3rd International Joint Conference (IJCAR 2006), Seattle, WA, USA. Lecture Notes in Computer Science 4130 177–191.Google Scholar
Harrison, J. (2007) Verifying nonlinear real formulas via sums of squares. In: Schneider, K. and Brandt, J. (eds.) Proceedings of the 20th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2007), Kaiserslautern, Germany. Lecture Notes in Computer Science 4732 102–118.Google Scholar
Harrison, J. (2009) HOL Light: An overview. In: Berghofer, S., Nipkow, T., Urban, C. and Wenzel, M. (eds.) Proceedings of the 22nd International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2009), München, Germany. Lecture Notes in Computer Science 5674 60–66.Google Scholar
Harrison, J. (2013) The HOL Light theory of Euclidean space. Journal of Automated Reasoning 50 173–190.Google Scholar
HOL4 development team (2012) The HOL system LOGIC, Technical Report, University of Cambridge, NICTA and University of Utah.Google Scholar
Hölzl, J., Immler, F. and Huffman, B. (2013) Type classes and filters for mathematical analysis in Isabelle/HOL. In: Blazy, S., Paulin-Mohring, C. and Pichardie, D. (eds.) Proceedings of the 4th International Conference on Interactive Theorem Proving (ITP 2013), Rennes, France. Lecture Notes in Computer Science 7998 279–294.Google Scholar
Hunt, W. A. Jr., Krug, R. B. and Moore, J. (2003) Linear and nonlinear arithmetic in ACL2. In: Geist, D. and Tronci, E. (eds.) Proceedings of the 12th IFIP WG 10.5 Conference on Correct Hardware Design and Verification Methods (CHARME 2003), L'Aquila, Italy. Lecture Notes in Computer Science 2860 319–333.Google Scholar
Hurd, J. (2011) The OpenTheory standard theory library. In: Bobaru, M., Havelund, K., Holzmann, G. J. and Joshi, R. (eds.) Proceedings of the 3rd International Symposium on NASA Formal Methods (NFM 2011), Pasadena, CA, USA. Lecture Notes in Computer Science 6617 177–191.CrossRefGoogle Scholar
Hölzl, J. and Heller, A. (2011) Three chapters of measure theory in Isabelle/HOL. In: Eekelen, M., Geuvers, H., Schmaltz, J. and Wiedijk, F. (eds.) Proceedings of the 2nd International Conference of Interactive Theorem Proving (ITP 2011), Berg en Dal, The Netherlands. Lecture Notes in Computer Science 6898 135–151.Google Scholar
Immler, F. and Hölzl, J. (2012) Numerical analysis of ordinary differential equations in Isabelle/HOL. In: Beringer, L. and Felty, A. (eds.) Proceedings of the 3rd International Conference on Interactive Theorem Proving (ITP 2012), Princeton, NJ, USA. Lecture Notes in Computer Science 7406 377–392.Google Scholar
Jones, C. (1993) Completing the rationals and metric spaces in LEGO. In: Huet, G. and Plotkin, G. (eds.) Papers Presented at the 2nd Annual Workshop on Logical Environments, Cambridge University Press, Edinburgh, UK 297–316.Google Scholar
Julien, N. and Paşca, I. (2009) Formal verification of exact computations using Newton's method. In: Berghofer, S., Nipkow, T., Urban, C. and Wenzel, M. (eds.) Proceedings of the 22nd International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2009), München, Germany. Lecture Notes in Computer Science 5674 408–423.Google Scholar
Jutting, L. S. v. B. (1977) Checking Landau's ‘Grundlagen’ in the AUTOMATH System, Ph.D. thesis, Eindhoven University of Technology.Google Scholar
Kaliszyk, C. and O'Connor, R. (2009) Computing with classical real numbers. Journal of Automated Reasoning 2 (1) 27–29.Google Scholar
Kaufmann, M., Moore, J. S. and Manolios, P. (2000) Computer-Aided Reasoning: An Approach, Kluwer Academic Publishers, Norwell, MA, USA.Google Scholar
Kotowicz, J. (1990a) Convergent sequences and the limit of sequences. Journal of Formalized Mathematics 1 (2) 273–275.Google Scholar
Kotowicz, J. (1990b) Real sequences and basic operations on them. Journal of Formalized Mathematics 1 (2) 269–272.Google Scholar
Kotowicz, J. (1991a) The limit of a real function at a point. Journal of Formalized Mathematics 2 (1) 71–80.Google Scholar
Kotowicz, J. (1991b) The limit of a real function at infinity. Journal of Formalized Mathematics 2, 17–28.Google Scholar
Kotowicz, J. (1991c) One-side limits of a real function at a point. Journal of Formalized Mathematics 2 (1) 29–40.Google Scholar
Krebbers, R. and Spitters, B. (2013) Type classes for efficient exact real arithmetic in Coq. Logical Methods in Computer Science 9 (1:1) 1–27.Google Scholar
Lester, D. R. (2007) Topology in PVS: Continuous mathematics with applications. In: Proceedings of the 2nd Workshop on Automated Formal Methods (AFM '07), Atlanta, GA, USA, 11–20.Google Scholar
Lester, D. R. (2008) Real number calculations and theorem proving. In: Mohamed, O. A., Muñoz, C. A. and Tahar, S. (eds.) Proceedings of the 21st International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2008). Lecture Notes in Computer Science 5170 215–229.Google Scholar
Makarov, E. and Spitters, B. (2013) The Picard algorithm for ordinary differential equations in Coq. In: Blazy, S., Paulin-Mohring, C. and Pichardie, D. (eds.) Proceeding of the 4th International Conference of Interactive Theorem Proving (ITP 2013), Rennes, France. Lecture Notes in Computer Science 7998 463–468.Google Scholar
Manolios, P. and Moore, J. S. (2003) Partial functions in ACL2. Journal of Automated Reasoning 31 (2) 107–127.Google Scholar
Mayero, M. (2001) Formalisation et automatisation de preuves en analyses réelle et numérique, Ph.D. thesis, Université Paris VI.Google Scholar
Mayero, M. (2012) Problèmes critiques et preuves formelles, Habilitation à diriger des recherches, Université Paris 13.Google Scholar
McLaughlin, S. and Harrison, J. (2005) A proof-producing decision procedure for real arithmetic. In: Nieuwenhuis, R. (ed.) Proceedings of the 20th International Conference on Automated Deduction (CADE 20), Tallinn, Estonia. Lecture Notes in Computer Science 3632 295–314.Google Scholar
Mhamdi, T., Hasan, O. and Tahar, S. (2010) On the formalization of the Lebesgue integration theory in HOL. In: Kaufmann, M. and Paulson, L. C. (eds.) Proceedings of the 1st International Conference of Interactive Theorem Proving (ITP 2010), Edinburgh, UK. Lecture Notes in Computer Science 6172 387–402.Google Scholar
Monniaux, D. and Corbineau, P. (2011) On the generation of Positivstellensatz witnesses in degenerate cases. In: Van Eekelen, M., Geuvers, H., Schmaltz, J. and Wiedijk, F. (eds.) Proceedings of the 2nd International Conference on Interactive Theorem Proving (ITP 2011), Berg en Dal, The Netherlands. Lecture Notes in Computer Science 6898 249–264.Google Scholar
Moore, J. S., Lynch, T. and Kaufmann, M. (1998) A mechanically checked proof of the correctness of the kernel of the AMD5K86 floating point division algorithm. IEEE Transactions on Computers 47 (9) 913–926.Google Scholar
Muñoz, C. and Mayero, M. (2001) Real automation in the field. Technical Report NASA/CR-2001-211271, ICASE.Google Scholar
Muñoz, C. and Narkawicz, A. (2013) Formalization of a Bernstein polynomials and applications to global optimization. Journal of Automated Reasoning 51 (2) 151–196.Google Scholar
Narkawicz, A., Muñoz, C. and Dowek, G. (2012) Provably correct conflict prevention bands algorithms. Science of Computer Programming 77 (10–11) 1039–1057.Google Scholar
Naumowicz, A. and Korniłowicz, A. (2009) A brief overview of Mizar. In: Berghofer, S., Nipkow, T., Urban, C. and Wenzel, M. (eds.) Proceedings of the 22th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2009), München, Germany. Lecture Notes in Computer Science 5674 67–72.Google Scholar
Nipkow, T., Paulson, L. C. and Wenzel, M. (2002) Isabelle/HOL – A Proof Assistant for Higher-Order Logic, Lecture Notes in Computer Science volume 2283, Springer, Berlin Heidelberg.Google Scholar
O'Connor, R. (2007) A monadic, functional implementation of real numbers. Mathematical Structures in Computer Science 17 129–159.Google Scholar
O'Connor, R. (2008) Certified exact transcendental real number computation in Coq. In: Mohamed, O. A., Muñoz, C. and Tahar, S. (eds.) Proceedings of the 21th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2008), Montreal, Canada. Lecture Notes in Computer Science 5170 246–261.Google Scholar
O'Connor, R. and Spitters, B. (2010) A computer-verified monadic functional implementation of the integral. Theoretical Computer Science 411 (37) 3386–3402.Google Scholar
Owre, S., Rushby, J. M. and Shankar, N. (1992) PVS: A prototype verification system. In: Kapur, D. (ed.) Proceedings of the 11th International Conference on Automated Deduction (CADE 11), Saratoga, NY. Lecture Notes in Artificial Intelligence 607 748–752.Google Scholar
Owre, S. and Shankar, N. (2003) The PVS Prelude Library. Technical Report, CSL Technical Report SRI-CSL-03-01.Google Scholar
Perkowska, B. (1992) Functional sequence from a domain to a domain. Journal of Formalized Mathematics 3 (1) 17–21.Google Scholar
Pottier, L. (2008) Connecting Gröbner bases programs with Coq to do proofs in algebra, geometry and arithmetics. In: Sutcliffe, G., Rudnicki, P., Schmidt, R., Konev, B. and Schulz, S. (eds.) Knowledge Exchange: Automated Provers and Proof Assistants, CEUR Workshop Proceedings, Doha, Qatar 67–76.Google Scholar
ProofPower development team (2006) Proofpower - HOL reference manual, Technical Report, Lemma 1 Ltd.Google Scholar
Raczkowski, K. (1991) Integer and rational exponents. Journal of Formalized Mathematics 2 (1) 125–130.Google Scholar
Raczkowski, K. and Nedzusiak, A. (1991a) Real exponents and logarithms. Journal of Formalized Mathematics 2 (2) 213–216.Google Scholar
Raczkowski, K. and Nedzusiak, A. (1991b) Series. Journal of Formalized Mathematics 2 (4) 449–452.Google Scholar
Raczkowski, K. and Sadowski, P. (1990a) Real function continuity. Journal of Formalized Mathematics 1 (4) 787–791.Google Scholar
Raczkowski, K. and Sadowski, P. (1990b) Real function differentiability. Journal of Formalized Mathematics 1 (4) 797–801.Google Scholar
Reid, P. and Gamboa, R. (2011) Automatic differentiation in ACL2. In: Eekelen, M., Geuvers, H., Schmaltz, J. and Wiedijk, F. (eds.) Proceeding of the 2nd International Conference of Interactive Theorem Proving (ITP 2011), Berg en Dal, The Netherlands. Lecture Notes in Computer Science 6898 312–324.Google Scholar
Richter, S. (2004) Formalizing integration theory with an application to probabilistic algorithms. In: Slind, K., Bunker, A. and Gopalakrishnan, G. (eds.) Proceedings of the 17th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2004), Park City, UT, USA. Lecture Notes in Computer Science 3223 271–286.Google Scholar
Rushby, J., Owre, S. and Shankar, N. (1998) Subtypes for specifications: Predicate subtyping in PVS. IEEE Transactions on Software Engineering 24 (9) 709–720.Google Scholar
Shi, Z., Gu, W., Li, X., Guan, Y., Ye, S., Zhang, J. and Wei, H. (2013) The gauge integral theory in HOL4. Journal of Applied Mathematics. Available at http://dx.doi.org/10.1155/2013/160875 Google Scholar
Solovyev, A. and Hales, T. C. (2013) Formal verification of nonlinear inequalities with taylor interval approximations. In: Brat, G., Rungta, N. and Venet, A. (eds.) Proceedings of the 5th NASA Formal Methods Symposium (NFM 2013). Lecture Notes in Computer Science 7871 383–397.Google Scholar
Spitters, B. and van der Weegen, E. (2011) Type classes for mathematics in type theory. Mathematical Structures of Computer Science 21 1–31.Google Scholar
Trybulec, A. (1993) Some features of the Mizar language. In: Proceedings of the ESPRIT Workshop, Torino, Italy 1–17.Google Scholar
Trybulec, A. (1998) Non negative real numbers. Part I. Journal of Formalized Mathematics Addenda.Google Scholar
Vito, B. L. D. (2003) Strategy-enhanced interactive proving and arithmetic simplification for PVS. In: Proceedings of STRATA 2003, 1st International Workshop on Design and Application of Strategies/Tactics in Higher Order Logics, number CP-2003-212448, NASA, Rome, Italy.Google Scholar
Wenzel, M. (2002) Isabelle/Isar – A Versatile Environment for Human-Readable Formal Proof Documents, Ph.D. thesis, Institut für Informatik, Technische Universität München.Google Scholar
Wenzel, M. and Wiedijk, F. (2002) A comparison of the mathematical proof languages Mizar and Isar. Journal of Automated Reasoning 29 389–411.Google Scholar
Wiedijk, F. (2006) The Seventeen Provers of the World, Lecture Notes in Artificial Intelligence volume 3600, Springer, Berlin Heidelberg. Foreword by Dana S. Scott.Google Scholar
Wiedijk, F. (2009) Statistics on digital libraries of mathematics. Studies in Logic, Grammar and Rhetoric 18 (31) 137–151. (Computer Reconstruction of the Body of Mathematics.)Google Scholar