A synthesis of the procedural and declarative styles of interactive theorem proving

Logical Methods in Computer Science

Volumn 8, Issue 1, 2012, Pages

  Wiedijk, Freek ^a  

^a RADBOUD UNIVERSITY NIJMEGEN   (Netherlands)

Author keywords

Declarative proof style; Formal mathematics; HOL; Interactive theorem proving; Mizar; Natural deduction; Procedural proof style; Proof assistants; Tactics

Indexed keywords

EID: 84859966332     PISSN: None     EISSN: 18605974     Source Type: Journal    
DOI: 10.2168/LMCS-8(1:30)2012     Document Type: Article

References (55)

1. Jean-Raymond Abrial. The B-Book: Assigning Programs to Meanings. Cambridge University Press, 1996.
2. Agda Development Team. The Agda Wiki. http://wiki.portal.chalmers.se/agda/pmwiki.php.
3. Andrea Asperti, Claudio Sacerdoti Coen, Enrico Tassi, and Stefano Zacchiroli. Crafting a proof assistant. In Proceedings of the 2006 international conference on Types for proofs and programs, TYPES'06, pages 18-32, Berlin, Heidelberg, 2007. Springer-Verlag.
4. Andrea Asperti et al. Mathematical Knowledge Management in HELM. Annals of Mathematics and Artificial Intelligence, 38(1-3):27-46, 2003.
5. Andrea Asperti and Bernd Wegner. MOWGLI-A New Approach for the Content Description in Digital Documents. In Proc. of the 9th Intl. Conference on Electronic Resources and the Social Role of Libraries in the Future, volume 1, Autonomous Republic of Crimea, Ukraine, 2002.
6. David Aspinall. Proof General: A generic tool for proof development. In European Joint Conferences on Theory and Practice of Software (ETAPS), 2000.
7. Henk Barendregt. Towards an Interactive Mathematical Proof Language. In F. Kamareddine, editor, Thirty Five Years of Automath, pages 25-36. Kluwer, 2003.
8. Henk Barendregt and Freek Wiedijk. The Challenge of Computer Mathematics. Transactions A of the Royal Society, 363(1835):2351-2375, 2006.
9. Douglas Beaver et al. Amazon. com Interview: Larry Wall. The father of Perl talks about XML, Unicode, the Win32 port, and the philosophy behind the language, http://www.amazon.com/gp/feature.html?docId=7137, 1998.
10. B. Buchberger, T. Jebelean, F. Kriftner, M. Marin, and D. Vasaru. An Overview on the Theorema project. In W. Kuechlin, editor, Proceedings of ISSAC'97 (International Symposium on Symbolic and Algebraic Computation), Maui, Hawaii, 1997. ACM Press.
11. Paul Cairns and Jeremy Gow. Using and Parsing the Mizar Language. Electronic Notes in Theoretical Computer Science, 93:60-69, 2004.
12. Robert L. Constable, Stuart F. Allen, H.M. Bromley, W.R. Cleaveland, J.F. Cremer, R.W. Harper, Douglas J. Howe, T.B. Knoblock, N.P. Mendler, P. Panangaden, James T. Sasaki, and Scott F. Smith. Implementing Mathematics with the Nuprl Development System. Prentice-Hall, NJ, 1986.
13. Coq Development Team. The Coq Proof Assistant Reference Manual, 2010.
14. Pierre Corbineau. A declarative proof language for the Coq proof assistant. In TYPES 2007: Types for Proofs and Programs, number 4941 in LNCS. Springer-Verlag, 2007.
15. N.G. de Bruijn. The Mathematical Vernacular, a language for mathematics with typed sets. In P. Dybjer et al., editors, Proceedings of the Workshop on Programming Languages, Marstrand, Sweden, 1987.
16. Anthony Fox. Formal Specification and Verification of ARM6. In D.A. Basin and B. Wolff, editors, TPHOLs 2003, volume 2758 of LNCS. Springer, 2003.
17. Georges Gonthier. A computer-checked proof of the Four Colour Theorem. http://research.microsoft.com/~gonthier/4colproof.pdf, 2006.
18. Georges Gonthier, Assia Mahboubi, and Enrico Tassi. A Small Scale Reflection Extension for the Coq system. Rapport de recherche RR-6455, INRIA, 2008.
19. Mike Gordon and Tom Melham, editors. Introduction to HOL. Cambridge University Press, Cambridge, 1993.
20. Adam Grabowski, Artur Kornilowicz, and Adam Naumowicz. Mizar in a Nutshell. Journal of Formalized Reasoning, 3:153-245, 2010.
21. Frédérique Guilhot. Proof explanations: using natural language and graph view, 2003. presentation for the MoWGLI project.
22. Tom Hales. Mathematics in the Age of the Turing Machine, 2011. To appear in 2012 in a special volume of Lecture Notes in Logic (CUP) in commemoration of the centennial of Turing's birth.
23. John Harrison. The HOL Light theorem prover. http://www.cl.cam.ac.uk/~jrh13/hol-light/.
24. John Harrison. A Mizar Mode for HOL. In J. von Wright, J. Grundy, and J. Harrison, editors, TPHOLs '96, volume 1125 of LNCS, 1996.
25. John Harrison. Optimizing Proof Search in Model Elimination. In M. A. McRobbie and J. K. Slaney, editors, 13th International Conference on Automated Deduction, volume 1104 of LNCS, pages 313-327, New Brunswick, NJ, 1996. Springer-Verlag.
26. John Harrison. Proof Style. In Eduardo Giménez and Christine Paulin-Möhring, editors, Types for Proofs and Programs: International Workshop TYPES'96, volume 1512 of LNCS, pages 154-172, Aus-sois, France, 1996. Springer-Verlag.
27. John Harrison. The HOL Light manual (1.1), 2000.
28. John Harrison. HOL Light Tutorial (for version 2.20), 2007.
29. John Harrison. Formalizing an Analytic Proof of the Prime Number Theorem. In R. Boulton, J. Hurd, and K. Slind, editors, Tools and Techniques for Verification of System Infrastructure, pages 243-261. The Royal Society, 2008.
30. Michael Huth and Mark Ryan. Logic in Computer Science: Modelling and Reasoning about Systems. Cambridge University Press, 2nd edition, 2004.
31. Cezary Kaliszyk and Freek Wiedijk. Merging procedural and declarative proof. In S. Berardi, F. Dami-ani, and U. De'Liquoro, editors, TYPES 2008, 2009.
32. Matt Kaufmann, Panagiotis Manolios, and J Strother Moore. Computer-Aided Reasoning: An Approach. Kluwer Academic Publishers, Boston, 2000.
33. Gerwin Klein et al. seL4: formal verification of an OS kernel. In J.N. Matthews and Th. Anderson, editors, Proceedings of the ACM SIGOPS 22nd symposium on Operating systems principles, pages 207-220, 2009.
34. Xavier Leroy. Formal Certification of a Compiler Back-end, or: Programming a Compiler with a Proof Assistant. In POPL'06, 2006.
35. Lemma 1 Ltd. ProofPower-Description. Lemma 1 Ltd., 2000.
36. Conor McBride and James McKinna. The View from the Left. Journal of Functional Programming, 14(1), 2004.
37. Norman D. Megill. Metamath, A Computer Language for Pure Mathematics. http://us.metamath.org/downloads/metamath.pdf, 1997.
38. Tobias Nipkow, Larry Paulson, and Markus Wenzel. Isabelle/HOL-A Proof Assistant for Higher-Order Logic, volume 2283 of LNCS. Springer, 2002.
39. Sam Owre, John M. Rushby, and Natarajan Shankar. PVS: A Prototype Verification System. In D. Kapur, editor, 11th International Conference on Automated Deduction (CADE), volume 607 of LNAI, pages 748-752, Berlin, Heidelberg, New York, 1992. Springer.
40. Andrei Paskevich. The syntax and semantics of the ForTheL language. http://nevidal.org/download/forthel.pdf.
41. Jeff Pelletier. A History of Natural Deduction and Elementary Logic Textbooks. In J. Woods and B. Brown, editors, Logical Consequence: Rival Approaches and New Studies, Vol. 1. Hermes Science Pubs., 2000.
42. Frank Pfenning and Carsten Schuermann. Twelf User's Guide, Version 1.4, 2002. http://www-2.cs.cmu.edu/~twelf/guide-1-4/twelf.ps.
43. Randy Pollack. How to Believe a Machine-Checked Proof. In G. Sambin and J. Smith, editors, Twenty-Five Years of Constructive Type Theory. Oxford University Press, Oxford, 1998.
44. Christophe Raffalli. The PhoX Proof Assistant. http://www.lama.univ-savoie. fr/~RAFFALLI/phox.html.
45. Claudio Sacerdoti Coen. Declarative Representation of Proof Terms. J. Autom. Reason., 44:25-52, 2010.
46. Raymond Smullyan. What Is the Name of this Book? Penguin Books Ltd, 1990.
47. Josef Urban. MizarMode-an integrated proof assistance tool for the Mizar way of formalizing mathematics. J. Applied Logic, 4(4):414-427, 2006.
48. B.L. van der Waerden. Algebra I. Springer-Verlag, 1971. Achte Auflage.
49. Markus Wenzel. Isabelle/Isar-a versatile environment for human-readable formal proof documents. PhD thesis, Institut fur Informatik, Technische Universitat München, 2002. http://tumb1.biblio.tu-muenchen.de/publ/diss/in/2002/wenzel.html.
50. Markus Wenzel. The Isabelle/Isar Reference Manual. TU München, 2002.
51. Markus Wenzel and Freek Wiedijk. A comparison of the mathematical proof languages Mizar and Isar. Journal of Automated Reasoning, 29:389-411, 2002.
52. Freek Wiedijk. Mizar Light for HOL Light. In R. J. Boulton and P. B. Jackson, editors, TPHOLs 2001, volume 2152 of LNCS, 2001.
53. Freek Wiedijk. Formal Proof Sketches. In S. Berardi, M. Coppo, and F. Damiani, editors, TYPES 2003, volume 3085 of LNCS, pages 378-393, 2004.
54. Freek Wiedijk, editor. The Seventeen Provers of the World, volume 3600 of LNCS. Springer, 2006. With a foreword by Dana S. Scott.
55. Freek Wiedijk. Mizar's Soft Type System. In K. Schneider and J. Brandt, editors, Theorem Proving in Higher Order Logics 2007, volume 4732 of LNCS, pages 383-399, 2007.