International Conference on Interactive Theorem Proving

ITP 2015: Interactive Theorem Proving pp 234-252 | Cite as

A Consistent Foundation for Isabelle/HOL

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9236)

Abstract

The interactive theorem prover Isabelle/HOL is based on the well understood Higher-Order Logic (HOL), which is widely believed to be consistent (and provably consistent in set theory by a standard semantic argument). However, Isabelle/HOL brings its own personal touch to HOL: overloaded constant definitions, used to achieve Haskell-like type classes in the user space. These features are a delight for the users, but unfortunately are not easy to get right as an extension of HOL—they have a history of inconsistent behavior. It has been an open question under which criteria overloaded constant definitions and type definitions can be combined together while still guaranteeing consistency. This paper presents a solution to this problem: non-overlapping definitions and termination of the definition-dependency relation (tracked not only through constants but also through types) ensures relative consistency of Isabelle/HOL.

References

  1. 1.
  2. 2.
    The HOL4 Theorem Prover. http://hol.sourceforge.net/
  3. 3.
    Adams, M.: Introducing HOL Zero. In: Fukuda, K., Hoeven, J., Joswig, M., Takayama, N. (eds.) ICMS 2010. LNCS, vol. 6327, pp. 142–143. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  4. 4.
    Anand, A., Rahli, V.: Towards a formally verified proof assistant. In: Klein, G., Gamboa, R. (eds.) ITP 2014. LNCS, vol. 8558, pp. 27–44. Springer, Heidelberg (2014) Google Scholar
  5. 5.
    Arthan, R.D.: Some mathematical case studies in ProofPower-HOL. In: TPHOLs 2004 (2004)Google Scholar
  6. 6.
    Barras, B.: Coq en Coq. Technical report 3026, INRIA (1996)Google Scholar
  7. 7.
    Barras, B.: Sets in Coq, Coq in Sets. J. Formalized Reasoning 3(1), 29–48 (2010)MATHMathSciNetGoogle Scholar
  8. 8.
    Bertot, Y., Casteran, P.: Interactive Theorem Proving and Program Development: Coq’Art: The Calculus of Inductive Constructions. Springer, Heidelberg (2004) CrossRefGoogle Scholar
  9. 9.
    Blanchette, J.C., Popescu, A., Traytel, D.: Foundational extensible corecursion. In: ICFP 2015. ACM (2015)Google Scholar
  10. 10.
    Bove, A., Dybjer, P., Norell, U.: A brief overview of Agda – a functional language with dependent types. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 73–78. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  11. 11.
    Dénès, M.: [Coq-Club] Propositional extensionality is inconsistent in Coq, archived at https://sympa.inria.fr/sympa/arc/coq-club/2013-12/msg00119.html
  12. 12.
    Gordon, M.J.C., Melham, T.F. (eds.): Introduction to HOL: A Theorem Proving Environment for Higher Order Logic. Cambridge University Press, New York (1993)MATHGoogle Scholar
  13. 13.
    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
  14. 14.
    Harrison, J.: HOL Light: a tutorial introduction. In: Srivas, M., Camilleri, A. (eds.) FMCAD 1996. LNCS, vol. 1166, pp. 265–269. Springer, Heidelberg (1996) CrossRefGoogle Scholar
  15. 15.
    Harrison, J.: Towards self-verification of HOL Light. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 177–191. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  16. 16.
    Hölzl, J., Immler, F., Huffman, B.: Type classes and filters for mathematical analysis in Isabelle/HOL. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 279–294. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  17. 17.
    Huffman, B., Urban, C.: Proof pearl: a new foundation for Nominal Isabelle. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 35–50. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  18. 18.
    Kumar, R., Arthan, R., Myreen, M.O., Owens, S.: HOL with definitions: semantics, soundness, and a verified implementation. In: Klein, G., Gamboa, R. (eds.) ITP 2014. LNCS, vol. 8558, pp. 308–324. Springer, Heidelberg (2014) Google Scholar
  19. 19.
    Kunčar, O.: Correctness of Isabelle’s cyclicity checker: implementability of overloading in proof assistants. In: CPP 2015. ACM (2015)Google Scholar
  20. 20.
    Kunčar, O., Popescu, A.: A consistent foundation for Isabelle/HOL. Technical report (2015). www.eis.mdx.ac.uk/staffpages/andreipopescu/pdf/IsabelleHOL.pdf
  21. 21.
    Leino, K.R.M., Moskal, M.: Co-induction simply–automatic co-inductive proofs in a program verifier. In: FM 2014 (2014)Google Scholar
  22. 22.
    Lochbihler, A.: Light-Weight Containers for Isabelle: Efficient, Extensible, Nestable. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 116–132. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  23. 23.
    McBride, C., et al.: [HoTT] Newbie questions about homotopy theory and advantage of UF/Coq, archived at http://article.gmane.org/gmane.comp.lang.agda/6106
  24. 24.
    Müller, O., Nipkow, T., von Oheimb, D., Slotosch, O.: HOLCF = HOL + LCF. J. Funct. Program. 9, 191–223 (1999)MATHCrossRefGoogle Scholar
  25. 25.
    Myreen, M.O., Davis, J.: The reflective Milawa theorem prover is sound. In: Klein, G., Gamboa, R. (eds.) ITP 2014. LNCS, vol. 8558, pp. 421–436. Springer, Heidelberg (2014) Google Scholar
  26. 26.
    Nipkow, T., Klein, G.: Concrete Semantics - With Isabelle/HOL. Springer, New York (2014)MATHGoogle Scholar
  27. 27.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002) Google Scholar
  28. 28.
    Kang, J., Adibi, S.: Type classes and overloading resolution via order-sorted unification. In: Doss, R., Piramuthu, S., ZHOU, W. (eds.) Functional Programming Languages and Computer Architecture. LNCS, vol. 523, pp. 1–14. Springer, Heidelberg (1991) CrossRefGoogle Scholar
  29. 29.
    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
  30. 30.
    Pitts, A.: Introduction to HOL: a theorem proving environment for higher order logic. Chapter The HOL Logic, pp. 191–232. In: Gordon and Melham [12] (1993)Google Scholar
  31. 31.
    Shankar, N., Owre, S., Rushby, J.M.: PVS Tutorial. Computer Science Laboratory, SRI International (1993)Google Scholar
  32. 32.
    Sozeau, M., Oury, N.: First-class type classes. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 278–293. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  33. 33.
    Urban, C.: Nominal techniques in Isabelle/HOL. J. Autom. Reason. 40(4), 327–356 (2008)MATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Wadler, P., Blott, S.: How to make ad-hoc polymorphism less ad-hoc. In: POPL (1989)Google Scholar
  35. 35.
    Wenzel, M.: Type classes and overloading in higher-order logic. In: Gunter, E.L., Felty, A.P. (eds.) TPHOLs 1997. LNCS, vol. 1275, pp. 307–322. Springer, Heidelberg (1997) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Fakultät für InformatikTechnische Universität MünchenMunichGermany
  2. 2.Department of Computer Science, School of Science and TechnologyMiddlesex UniversityLondonUK

Personalised recommendations