A Mechanized Translation from Higher-Order Logic to Set Theory

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6172)


In order to make existing formalizations available for set-theoretic developments, we present an automated translation of theories from Isabelle/HOL to Isabelle/ZF. This covers all fundamental primitives, particularly type classes. The translation produces LCF-style theorems that are checked by Isabelle’s inference kernel. Type checking is replaced by explicit reasoning about set membership.


Type Variable Theorem Prove Type Class Type Check Mechanized Translation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Bortin, M., Broch Johnsen, E., Lüth, C.: Structured formal development in Isabelle. Nordic Journal of Computing 13, 1–20 (2006)MathSciNetGoogle Scholar
  2. 2.
    Coquand, T., Huet, G.: The calculus of constructions. Information and Computation 76(2-3), 95–120 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Furbach, U., Shankar, N. (eds.): IJCAR 2006. LNCS (LNAI), vol. 4130. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  4. 4.
    Gaifman, H.: Global and local choice functions. Israel Journal of Mathematics 22(3-4), 257–265 (1975)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Gordon, M.J.C.: Set theory, higher order logic or both? In: von Wright, J., Harrison, J., Grundy, J. (eds.) TPHOLs 1996. LNCS, vol. 1125, pp. 191–201. Springer, Heidelberg (1996)Google Scholar
  6. 6.
    Gordon, M.J.C.: Twenty years of theorem proving for HOLs: Past, present and future. In: Ait Mohamed, O., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 1–5. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Haftmann, F., Wenzel, M.: Constructive type classes in Isabelle. In: Altenkirch, T., McBride, C. (eds.) TYPES 2006. LNCS, vol. 4502, pp. 160–174. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Homeier, P.V.: The HOL-omega logic. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 244–259. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Lamport, L., Paulson, L.C.: Should your specification language be typed? ACM Transactions on Programming Languages and Systems 21(3), 502–526 (1999)CrossRefGoogle Scholar
  10. 10.
    McLaughlin, S.: An interpration of Isabelle/HOL in HOL Light. In: Furbach, Shankar: [3], pp. 192–204Google Scholar
  11. 11.
    Moschovakis, Y.N.: Notes on Set Theory. Springer, Heidelberg (1994)zbMATHGoogle Scholar
  12. 12.
    Müller, O., Nipkow, T., von Oheimb, D., Slotosch, O.: HOLCF=HOL+LCF. Journal of Functional Programming 9(2), 191–223 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Obua, S.: Checking conservativity of overloaded definitions in higher-order logic. In: Pfenning, F. (ed.) RTA 2006. LNCS, vol. 4098, pp. 212–226. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Obua, S., Skalberg, S.: Importing HOL into Isabelle/HOL. In: Furbach, Shankar: [3], pp. 298–302Google Scholar
  15. 15.
    Paulson, L.C.: The foundation of a generic theorem prover. Journal of Automated Reasoning 5, 363–397 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Paulson, L.C.: Set theory for verification: I. From foundations to functions. Journal of Automated Reasoning 11, 353–389 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Pitts, A.: The HOL logic. In: Gordon, M., Melham, T. (eds.) Introduction to HOL: A theorem proving environment for Higher Order Logic, pp. 191–232. Cambridge University Press, Cambridge (1993)Google Scholar
  18. 18.
    Schmidt-Schauß, M. (ed.): Computational Aspects of an Order-Sorted Logic with Term Declarations. LNCS, vol. 395. Springer, Heidelberg (1989)zbMATHGoogle Scholar
  19. 19.
    Wenzel, M.: Type classes and overloading in higher-order logic. In: Gunter, E.L., Felty, A.P. (eds.) TPHOLs 1997. LNCS, vol. 1275. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  20. 20.
    Wiedijk, F.: The QED manifesto revisited. In: Matuszewski, R., Zalewska, A. (eds.) From Insight To Proof – Festschrift in Honour of Andrzej Trybulec, pp. 121–133. University of Białystok (2007)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.Institut für InformatikTechnische Universität München 

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