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

  • SYLVIE BOLDO (a1), CATHERINE LELAY (a1) and GUILLAUME MELQUIOND (a1)

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.

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

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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 4358.
Arthan, R. D. (2012) Mathematical Case Studies: Basic Analysis, Lemma 1 Ltd.
Arthan, R. D. and King, D. (1996) Development of practical verification tools. ICL Systems Journal 11 (1) 106122.
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 357371.
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.
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 4862.
Bickford, M. (2008) Formalizing Constructive Analysis in Nuprl, Computing Science Department, Cornell University. Available at: http://www.nuprl.org/MathLibrary/ConstructiveAnalysis/
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) 423456.
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 289304.
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.
Butler, R. W. (2009) Formalization of the integral calculus in the PVS theorem prover. Journal of Formalized Reasoning 2 (1) 126.
Byliński, C. (1990) Functions and their basic properties. Journal of Formalized Mathematics 1 (1) 5565.
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 528540.
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 246260. (Birmingham, UK)
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 2739.
Ciaffaglione, A. and Di-Gianantonio, P. (2006) A certified, corecursive implementation of exact real numbers. Theoretical Computer Science 351 (1) 3951.
Cohen, C. (2012) Formalized Algebraic Numbers: Construction and First Order Theory, Ph.D. thesis, École Polytechnique.
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.
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 2534.
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 108126.
Cruz-Filipe, L. (2004) Constructive Real Analysis: A Type-Theoretical Formalization and Applications, Ph.D. thesis, University of Nijmegen.
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 88103.
Daumas, M., Lester, D. and Muñoz, C. (2009) Verified real number calculations: A library for interval arithmetic. IEEE Transactions on Computers 58 (2) 226237.
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.
Desmettre, O. (2002) Coq standard library – RiemannInt.
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 141156.
Endou, N. and Korniłowicz, A. (1999) The definition of the Riemann definite integral and some related lemmas. Journal of Formalized Mathematics 8 (1) 93102.
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) 281284.
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 145161.
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 301316.
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.
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.
Gamboa, R. and Kaufmann, M. (2001) Nonstandard analysis in ACL2. Journal of Automated Reasoning 27 (4) 323351.
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 327342.
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 7995.
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.
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 197214.
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 98113.
Guard, J. R., Oglesby, F. C., Bennett, J. H. and Settle, L. G. (1969) Semi-automated mathematics. Journal of the ACM 16 4962.
Hales, T. (2012) Dense Sphere Packings: A Blueprint for Formal Proofs, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, England.
Harrison, J. (1994) Constructing the real numbers in HOL. Formal Methods in System Design 5 (1–2) 3559.
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 154172.
Harrison, J. (1997) Floating point verification in HOL Light: The exponential function, Technical Report 428, University of Cambridge Computer Laboratory.
Harrison, J. (1998) Theorem Proving with the Real Numbers, Springer-Verlag, London.
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 114129.
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 177191.
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 102118.
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 6066.
Harrison, J. (2013) The HOL Light theory of Euclidean space. Journal of Automated Reasoning 50 173190.
HOL4 development team (2012) The HOL system LOGIC, Technical Report, University of Cambridge, NICTA and University of Utah.
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 279294.
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 319333.
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 177191.
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 135151.
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 377392.
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 297316.
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 408423.
Jutting, L. S. v. B. (1977) Checking Landau's ‘Grundlagen’ in the AUTOMATH System, Ph.D. thesis, Eindhoven University of Technology.
Kaliszyk, C. and O'Connor, R. (2009) Computing with classical real numbers. Journal of Automated Reasoning 2 (1) 2729.
Kaufmann, M., Moore, J. S. and Manolios, P. (2000) Computer-Aided Reasoning: An Approach, Kluwer Academic Publishers, Norwell, MA, USA.
Kotowicz, J. (1990a) Convergent sequences and the limit of sequences. Journal of Formalized Mathematics 1 (2) 273275.
Kotowicz, J. (1990b) Real sequences and basic operations on them. Journal of Formalized Mathematics 1 (2) 269272.
Kotowicz, J. (1991a) The limit of a real function at a point. Journal of Formalized Mathematics 2 (1) 7180.
Kotowicz, J. (1991b) The limit of a real function at infinity. Journal of Formalized Mathematics 2, 1728.
Kotowicz, J. (1991c) One-side limits of a real function at a point. Journal of Formalized Mathematics 2 (1) 2940.
Krebbers, R. and Spitters, B. (2013) Type classes for efficient exact real arithmetic in Coq. Logical Methods in Computer Science 9 (1:1) 127.
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.
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 215229.
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 463468.
Manolios, P. and Moore, J. S. (2003) Partial functions in ACL2. Journal of Automated Reasoning 31 (2) 107127.
Mayero, M. (2001) Formalisation et automatisation de preuves en analyses réelle et numérique, Ph.D. thesis, Université Paris VI.
Mayero, M. (2012) Problèmes critiques et preuves formelles, Habilitation à diriger des recherches, Université Paris 13.
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 295314.
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 387402.
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 249264.
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) 913926.
Muñoz, C. and Mayero, M. (2001) Real automation in the field. Technical Report NASA/CR-2001-211271, ICASE.
Muñoz, C. and Narkawicz, A. (2013) Formalization of a Bernstein polynomials and applications to global optimization. Journal of Automated Reasoning 51 (2) 151196.
Narkawicz, A., Muñoz, C. and Dowek, G. (2012) Provably correct conflict prevention bands algorithms. Science of Computer Programming 77 (10–11) 10391057.
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 6772.
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.
O'Connor, R. (2007) A monadic, functional implementation of real numbers. Mathematical Structures in Computer Science 17 129159.
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 246261.
O'Connor, R. and Spitters, B. (2010) A computer-verified monadic functional implementation of the integral. Theoretical Computer Science 411 (37) 33863402.
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 748752.
Owre, S. and Shankar, N. (2003) The PVS Prelude Library. Technical Report, CSL Technical Report SRI-CSL-03-01.
Perkowska, B. (1992) Functional sequence from a domain to a domain. Journal of Formalized Mathematics 3 (1) 1721.
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.
ProofPower development team (2006) Proofpower - HOL reference manual, Technical Report, Lemma 1 Ltd.
Raczkowski, K. (1991) Integer and rational exponents. Journal of Formalized Mathematics 2 (1) 125130.
Raczkowski, K. and Nedzusiak, A. (1991a) Real exponents and logarithms. Journal of Formalized Mathematics 2 (2) 213216.
Raczkowski, K. and Nedzusiak, A. (1991b) Series. Journal of Formalized Mathematics 2 (4) 449452.
Raczkowski, K. and Sadowski, P. (1990a) Real function continuity. Journal of Formalized Mathematics 1 (4) 787791.
Raczkowski, K. and Sadowski, P. (1990b) Real function differentiability. Journal of Formalized Mathematics 1 (4) 797801.
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 312324.
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 271286.
Rushby, J., Owre, S. and Shankar, N. (1998) Subtypes for specifications: Predicate subtyping in PVS. IEEE Transactions on Software Engineering 24 (9) 709720.
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
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 383397.
Spitters, B. and van der Weegen, E. (2011) Type classes for mathematics in type theory. Mathematical Structures of Computer Science 21 131.
Trybulec, A. (1993) Some features of the Mizar language. In: Proceedings of the ESPRIT Workshop, Torino, Italy 1–17.
Trybulec, A. (1998) Non negative real numbers. Part I. Journal of Formalized Mathematics Addenda.
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.
Wenzel, M. (2002) Isabelle/Isar – A Versatile Environment for Human-Readable Formal Proof Documents, Ph.D. thesis, Institut für Informatik, Technische Universität München.
Wenzel, M. and Wiedijk, F. (2002) A comparison of the mathematical proof languages Mizar and Isar. Journal of Automated Reasoning 29 389411.
Wiedijk, F. (2006) The Seventeen Provers of the World, Lecture Notes in Artificial Intelligence volume 3600, Springer, Berlin Heidelberg. Foreword by Dana S. Scott.
Wiedijk, F. (2009) Statistics on digital libraries of mathematics. Studies in Logic, Grammar and Rhetoric 18 (31) 137151. (Computer Reconstruction of the Body of Mathematics.)

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