Formalising foundations of mathematics

Mathematical Structures in Computer Science

Volumn 21, Issue 4, 2011, Pages 883-911

  Iancu, Mihnea ^a   Rabe, Florian ^a  

^a JACOBS UNIVERSITY BREMEN   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

FORMALISATION; ISABELLE/HOL; MATHEMATICAL FOUNDATIONS; SOPHISTICATED SYSTEM; SYSTEMATIC STUDY; TRANSLATION FUNCTIONS;

MATHEMATICAL TECHNIQUES;

FOUNDATIONS;

EID: 80053069075     PISSN: 09601295     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0960129511000144     Document Type: Article

References (56)

1. Aczel, P. (1998) On Relating Type Theories and Set Theories. In: Altenkirch, T., Naraschewski, W. and Reus, B. (eds.) Types for Proofs and Programs 1-18.
2. Bancerek, G. (2003) On the structure of Mizar types. Electronic Notes in Theoretical Computer Science 85 69-85.
3. Bourbaki, N. (1964) Univers. In: Séminaire de Géomé trie Algébrique du Bois Marie -Théorie des topos et cohomologie étale des schémas, Springer-Verlag 185-217.
4. Brown, C. (2006) Combining Type Theory and Untyped Set Theory. In: Furbach, U. and Shankar, N. (eds.) International Joint Conference on Automated Reasoning. Springer-Verlag Lecture Notes in Computer Science 4130 205-219. (Pubitemid 44532675)
5. Church, A. (1940) A Formulation of the Simple Theory of Types. Journal of Symbolic Logic 5 (1) 56-68.
6. Constable, R. et al. (1986) Implementing Mathematics with the Nuprl Development System, Prentice-Hall.
7. Coquand, T. and Huet, G. (1988) The Calculus of Constructions. Information and Computation 76 (2/3) 95-120.
8. de Bruijn, N. (1970) The Mathematical Language AUTOMATH. In: Laudet, M. (ed.) Proceedings of the Symposium on Automated Demonstration. Springer-Verlag Lecture Notes in Computer Science 25 29-61.
9. de Nivelle, H. (2010) Classical Logic with Partial Functions. In: Giesl, J. and Hähnle, R. (eds.) Automated Reasoning. Springer-Verlag Lecture Notes in Computer Science 6173 203-217.
10. Dumbrava, S. and Rabe, F. (2010) Structuring Theories with Partial Morphisms, Workshop on Abstract Development Techniques.
11. Farmer, W., Guttman, J. and Thayer, F. (1993) IMPS: An Interactive Mathematical Proof System. Journal of Automated Reasoning 11 (2) 213-248.
12. Fraenkel, A. (1922) The notion of 'definite' and the independence of the axiom of choice.
13. Gordon, M. (1988) HOL: A Proof Generating System for Higher-Order Logic. In: Birtwistle, G. and Subrahmanyam, P. (eds.) VLSI Specification, Verification and Synthesis, Kluwer-Academic Publishers 73-128.
14. Gordon, M., Milner, R. and Wadsworth, C. (1979) Edinburgh LCF: A Mechanized Logic of Computation. Springer-Verlag Lecture Notes in Computer Science 78.
15. Gordon, M. and Pitts, A. (1993) The HOL Logic. In: Gordon, M. and Melham, T. (eds.) Introduction to HOL, Part III, Cambridge University Press 191-232.
16. Hales, T. (2003) The flyspeck project. (Available at http://code.google. com/p/flyspeck/.)
17. Harper, R., Honsell, F. and Plotkin, G. (1993) A framework for defining logics. Journal of the Association for Computing Machinery 40 (1) 143-184.
18. Harper, R., Sannella, D. and Tarlecki, A. (1994) Structured presentations and logic representations. Annals of Pure and Applied Logic 67 113-160.
19. Harrison, J. (1996) HOL Light: A tutorial introduction. In: Srivas, M. and Camilleri, A. (eds.) Proceedings of the First International Conference on Formal Methods in Computer-Aided Design (FMCAD'96). Springer-Verlag Lecture Notes in Computer Science 1166 265-269. (Pubitemid 126149763)
20. Hurd, J. (2009) OpenTheory: Package Management for Higher Order Logic Theories. In: Reis, G. D. and Théry, L. (eds.) Programming Languages for Mechanized Mathematics Systems, ACM 31-37.
21. Iancu, M. and Rabe, F. (2010) Formalizing Foundations of Mathematics, LF Encodings. (Available at https://latin.omdoc.org/wiki/FormalizingFoundations.)
22. Jaśkowski, S. (1934) On the rules of suppositions in formal logic. Studia Logica 1 5-32.
23. Keller, C. and Werner, B. (2010) Importing HOL Light into Coq. In: Kaufmann, M. and Paulson, L. (eds.) Proceedings Interactive Theorem Proving, ITP 2010. Springer-Verlag Lecture Notes in Computer Science 6172 307-322.
24. Klein, G., Nipkow, T. and Paulson, L. (2004) Archive of Formal Proofs. (Available at http: //afp.sourceforge.net/.)
25. Kohlhase, M., Mossakowski, T. and Rabe, F. (2009) The LATIN Project. (Available at https: //trac.omdoc.org/LATIN/.)
26. Krauss, A. and Schropp, A. (2010) A Mechanized Translation from Higher-Order Logic to Set Theory. In: Kaufmann, M. and Paulson, L. (eds.) Proceedings Interactive Theorem Proving, ITP 2010. Springer-Verlag Lecture Notes in Computer Science 6172 323-338.
27. Landau, E. (1930) Grundlagen der Analysis, Akademische Verlagsgesellschaft.
28. Lovas, W. and Pfenning, F. (2009) Refinement Types as Proof Irrelevance. In: Curien, P. (ed.) Typed Lambda Calculi and Applications. Springer-Verlag Lecture Notes in Computer Science 5608 157-171.
29. Matuszewski, R. (1990) Formalized Mathematics. (Available at http://mizar.uwb.edu.pl/fm/.)
30. McLaughlin, S. (2006) An Interpretation of Isabelle/HOL in HOL Light. In: Shankar, N. and Furbach, U. (eds.) International Joint Conference on Automated Reasoning. Springer-Verlag Lecture Notes in Computer Science 4130 192-204. (Pubitemid 44532674)
31. Mizar (2009) Grammar, version 7.11.02. (Available at http://mizar.org/ language/mizar-grammar.xml.)
32. Naumov, P., Stehr, M. and Meseguer, J. (2001) The HOL/NuPRL proof translator -a practical approach to formal interoperability. In: Boulton, R. and Jackson, P. (eds.) 14th International Conference on Theorem Proving in Higher Order Logics. Springer-Verlag Lecture Notes in Computer Science 2152 329-345. (Pubitemid 33329009)
33. Nipkow, T., Paulson, L. and Wenzel, M. (2002) Isabelle/HOL -A Proof Assistant for Higher-Order Logic. Springer-Verlag Lecture Notes in Computer Science 2283.
34. Norell, U. (2005) The Agda WiKi. (Available at \url{http://wiki.portal. chalmers.se/agda}.)
35. Obua, S. and Skalberg, S. (2006) Importing HOL into Isabelle/HOL. In: Shankar, N. and Furbach, U. (eds.) International Joint Conference on Automated Reasoning. Springer-Verlag Lecture Notes in Computer Science 4130 298-302. (Pubitemid 44532683)
36. Owre, S., Rushby, J. and Shankar, N. (1992) PVS: A Prototype Verification System. In: Kapur, D. (ed.) 11th International Conference on Automated Deduction (CADE). Springer-Verlag Lecture Notes in Computer Science 607 748-752.
37. Paulson, L. (1994) Isabelle: A Generic Theorem Prover. Springer-Verlag Lecture Notes in Computer Science 828.
38. Paulson, L. and Coen, M. (1993) Zermelo-Fraenkel Set Theory. Isabelle distribution, ZF/ZF.thy.
39. Pfenning, F. and Schürmann, C. (1999) System description: Twelf -a meta-logical framework for deductive systems. Lecture Notes in Computer Science 1632 202-206.
40. Pfenning, F., Schürmann, C., Kohlhase, M., Shankar, N. and Owre, S. (2003) The Logosphere Project. (Available at http://www.logosphere.org/.)
41. Poswolsky, A. and Schürmann, C. (2008) System Description: Delphin -A Functional Programming Language for Deductive Systems. In: Abel, A. and Urban, C. (eds.) International Workshop on Logical Frameworks and Metalanguages: Theory and Practice. Electronic Notes in Theoretical Computer Science 228 135-141.
42. Rabe, F. (2010) Representing Isabelle in LF. In: Crary, K. and Miculan, M. (eds.) Logical Frameworks and Meta-languages: Theory and Practice. EPTCS 34 85-99.
43. Rabe, F. and Schürmann, C. (2009) A Practical Module System for LF. In: Cheney, J. and Felty, A. (eds.) Proceedings of the Workshop on Logical Frameworks: Meta-Theory and Practice (LFMTP), ACM Press 40-48.
44. Schürmann, C. and Stehr, M. (2004) An Executable Formalization of the HOL/Nuprl Connection in the Metalogical Framework Twelf. In: Hermann, M. and Voronkov, A. (eds.) 11th International Conference on Logic for Programming Artificial Intelligence and Reasoning. Springer-Verlag Lecture Notes in Computer Science 4246 150-166. (Pubitemid 44850675)
45. Hales, T., Gonthier, G., Harrison, J. and Wiedijk, F. (2008) A Special Issue on Formal Proof. Notices of the AMS 55 (11).
46. Tarski, A. (1938) Über Unerreichbare Kardinalzahlen. Fundamenta Mathematicae 30 176-183.
47. Trybulec, A. (1989) Tarski Grothendieck Set Theory. Journal of Formalized Mathematics, Axiomatics 1 (1) 9-11.
48. Trybulec, A. and Blair, H. (1985) Computer Assisted Reasoning with MIZAR. In: Joshi, A. (ed.) Proceedings of the 9th International Joint Conference on Artificial Intelligence, Morgan Kaufmann Publishers 26-28.
49. Urban, J. (2003) Translating Mizar for First Order Theorem Provers. In: Asperti, A., Buchberger, B. and Davenport, J. (eds.) Mathematical Knowledge Management: Second International Conference, MKM 2003. Springer-Verlag Lecture Notes in Computer Science 2594 203-215.
50. van Benthem Jutting, L. (1977) Checking Landau's 'Grundlagen' in the AUTOMATH system, Ph.D. thesis, Eindhoven University of Technology.
51. Wenzel, M. (2009) The Isabelle/Isar Reference Manual. (Available at http://isabelle.in.tum. de/documentation.html.)
52. Whitehead, A. and Russell, B. (1913) Principia Mathematica, Cambridge University Press.
53. Wiedijk, F. (2006) Is ZF a hack? Comparing the complexity of some (formalist interpretations of) foundational systems for mathematics. Journal of Applied Logic 4 (4) 622-645. (Pubitemid 44647986)
54. Wiedijk, F. (2007) Mizar's Soft Type System. In: Schneider, K. and Brandt, J. (eds.) Theorem Proving in Higher Order Logics. Springer-Verlag Lecture Notes in Computer Science 4732 383-399.
55. Wiener, N. (1967) A Simplification of the Logic of Relations. In: van Heijenoort, J. (ed.) From Frege to Gödel, Harvard University Press 224-227.
56. Zermelo, E. (1908) Untersuchungen über die Grundlagen der Mengenlehre I. Mathematische Annalen 65 261-281. (English title: Investigations in the foundations of set theory I.)