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Automating Proofs in Category Theory

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Automated Reasoning (IJCAR 2006)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4130))

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Abstract

We introduce a semi-automated proof system for basic category-theoretic reasoning. It is based on a first-order sequent calculus that captures the basic properties of categories, functors and natural transformations as well as a small set of proof tactics that automate proof search in this calculus. We demonstrate our approach by automating the proof that the functor categories Fun[ C× D, E] and Fun[ C, Fun[ D, E] ] are naturally isomorphic.

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References

  1. Allen, S., et al.: Innovations in computational type theory using Nuprl. Journal of Applied Logic (to appear, 2006)

    Google Scholar 

  2. Allen, S., Constable, R., Eaton, R., Kreitz, C., Lorigo, L.: The Nuprl open logical environment. In: McAllester, D. (ed.) CADE 2000. LNCS, vol. 1831, pp. 170–176. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  3. Bancerek, G.: Concrete categories. J. formalized mathematics 13 (2001)

    Google Scholar 

  4. Bancerek, G.: Miscellaneous facts about functors. J. form. math. 13 (2001)

    Google Scholar 

  5. Bancerek, G.: Categorial background for duality theory. J. form. math. 13 (2001)

    Google Scholar 

  6. Barr, M., Wells, C.: Category Theory for Computing Science. Prentice-Hall, Englewood Cliffs (1990)

    MATH  Google Scholar 

  7. Bird, R.: A Calculus of Functions for Program Derivation, Research Topics in Functional Programming, pp. 287–307. Addison-Wesley, Reading (1990)

    Google Scholar 

  8. Bundy, A.: The Use of Explicit Plans to Guide Inductive Proofs. In: Lusk, E.‘., Overbeek, R. (eds.) CADE 1988. LNCS, vol. 310, pp. 111–120. Springer, Heidelberg (1988)

    Chapter  Google Scholar 

  9. Constable, R., et al.: Implementing Mathematics with the Nuprl proof development system. Prentice-Hall, Englewood Cliffs (1986)

    Google Scholar 

  10. Cáccamo, M.J., Winskel, G.: A higher-order calculus for categories. Technical Report RS-01-27, BRICS, University of Aarhus (2001)

    Google Scholar 

  11. Eklund, P., et al.: A graphical approach to monad compositions. Electronic Notes in Theoretical Computer Science 40 (2002)

    Google Scholar 

  12. Eilenberg, S., MacLane, S.: General theory of natural equivalences. Trans. Amer. Math. Soc. 58, 231–244 (1945)

    MATH  MathSciNet  Google Scholar 

  13. Glimming, J.: Logic and automation for algebra of programming. Master thesis, University of Oxford (2001)

    Google Scholar 

  14. Goguen, J.: A categorical manifesto. Mathematical Structures in Computer Science 1(1), 49–67 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  15. Harrison, J.: Formalized mathematics. Technical report of Turku Centre for Computer Science 36 (1996)

    Google Scholar 

  16. Huet, G., Saïbi, A.: Constructive category theory. In: Joint CLICS-TYPES Workshop on Categories and Type Theory, MIT Press, Cambridge (1995)

    Google Scholar 

  17. Knuth, D., Bendix, P.: Simple word problems in universal algebra. In: Computational Problems in Abstract Algebra, pp. 263–297. Pergamon Press, Oxford (1970)

    Google Scholar 

  18. Kozen, D.: Toward the automation of category theory. Technical Report 2004-1964, Computer Science Department, Cornell University (2004)

    Google Scholar 

  19. Kreitz, C.: The Nuprl Proof Development System, V5: Reference Manual and User’s Guide. Computer Science Department, Cornell University (2002)

    Google Scholar 

  20. MacLane, S.: Categories for the Working Mathematician. Springer, Heidelberg (1971)

    Google Scholar 

  21. Moggi, E.: Notions of computation and monads. Inf. and Comp. 93 (1991)

    Google Scholar 

  22. O’Keefe, G.: Towards a readable formalisation of category theory. Electronic Notes in Theoretical Computer Science, vol. 91, pp. 212–228. Elsevier, Amsterdam (2004)

    Google Scholar 

  23. Paulson, L.C.: Isabelle. LNCS, vol. 828. Springer, Heidelberg (1994)

    MATH  Google Scholar 

  24. Rydeheard, D., Burstall, R.: Computational Category Theory. Prentice-Hall, Englewood Cliffs (1988)

    MATH  Google Scholar 

  25. Reed, G.M., Roscoe, A.W., Wachter, R.F.: Topology and Category Theory in Computer Science. Oxford University Press, Oxford (1991)

    MATH  Google Scholar 

  26. Saïbi, A.: Constructive category theory (1995), http://coq.inria.fr/contribs/category.tar.gz

  27. Simpson, C.: Category theory in ZFC (2004), coq.inria.fr/contribs/CatsInZFC.tar.gz

  28. Takeyama, M.: Universal Structure and a Categorical Framework for Type Theory. PhD thesis, University of Edinburgh (1995)

    Google Scholar 

  29. Trybulec, A.: Some isomorphisms between functor categories. J. formalized mathematics 4 (1992)

    Google Scholar 

  30. Wadler, P.: Monads for functional programming. In: Jeuring, J., Meijer, E. (eds.) AFP 1995. LNCS, vol. 925, pp. 24–52. Springer, Heidelberg (1995)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Computer Science Department, Cornell University, Ithaca, NY, 14853-7501, USA

    Dexter Kozen & Christoph Kreitz

  2. Institut für Informatik, Universität Potsdam, 14482, Potsdam, Germany

    Christoph Kreitz & Eva Richter

Authors
  1. Dexter Kozen
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  2. Christoph Kreitz
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  3. Eva Richter
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Artificial Intelligence Research Group, University of Koblenz-Landau, Universitätsstr. 1, 56070, Koblenz

    Ulrich Furbach

  2. SRI International, MS EL256,, 333 Ravenswood Avenue, 94025-3493, Menlo Park, CA, USA

    Natarajan Shankar

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© 2006 Springer-Verlag Berlin Heidelberg

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Kozen, D., Kreitz, C., Richter, E. (2006). Automating Proofs in Category Theory. In: Furbach, U., Shankar, N. (eds) Automated Reasoning. IJCAR 2006. Lecture Notes in Computer Science(), vol 4130. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11814771_34

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  • DOI: https://doi.org/10.1007/11814771_34

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-37187-8

  • Online ISBN: 978-3-540-37188-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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