Skip to main content
SpringerLink
Log in

Nominal Techniques in Isabelle/HOL

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

This paper describes a formalisation of the lambda-calculus in a HOL-based theorem prover using nominal techniques. Central to the formalisation is an inductive set that is bijective with the alpha-equated lambda-terms. Unlike de-Bruijn indices, however, this inductive set includes names and reasoning about it is very similar to informal reasoning with “pencil and paper”. To show this we provide a structural induction principle that requires to prove the lambda-case for fresh binders only. Furthermore, we adapt work by Pitts providing a recursion combinator for the inductive set. The main technical novelty of this work is that it is compatible with the axiom of choice (unlike earlier nominal logic work by Pitts et al); thus we were able to implement all results in Isabelle/HOL and use them to formalise the standard proofs for Church-Rosser, strong-normalisation of beta-reduction, the correctness of the type-inference algorithm W, typical proofs from SOS and much more.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ambler, S.J., Crole, R.L., Momigliano, A.: Combining higher order abstract syntax with tactical theorem proving and (co)induction. In: Proceedings of the 15th International Conference on Theorem Proving in Higher Order Logics (TPHOLs). LNCS, vol. 2410, pp. 13–30. Hampton, 20–23 August 2002

  2. Aydemir, B., Bohannon, A., Weihrich, S.: Nominal reasoning techniques in Coq (work in progress). In: Proceedings of the International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice (LFMTP). ENTCS, pp. 60–68. Seattle, 16 August 2006

  3. Aydemir, B., Charguéraud, A., Pierce, B.C., Pollack, R., Weirich, S.: Engineering formal metatheory. In: Proceedings of the 35rd Symposium on Principles of Programming Languages (POPL), pp. 3–15. ACM, New York (2008)

  4. Aydemir, B.E., Bohannon, A., Fairbairn, M., Foster, J.N., Pierce, B.C., Sewell, P., Vytiniotis, D., Washburn, G., Weirich, S., Zdancewic, S.: Mechanized metatheory for the masses: the Poplmark challenge. In: Proceedings of the 18th International Conference on Theorem Proving in Higher-Order Logics (TPHOLs). LNCS, vol. 3603, pp. 50–65. Oxford, 22–25 August 2005

  5. Barendregt, H.: The Lambda Calculus: Its Syntax and Semantics. Studies in Logic and the Foundations of Mathematics, vol. 103. North-Holland, Amsterdam (1981)

    Google Scholar 

  6. Bengtson, J., Parrow, J.: Formalising the pi-Calculus using nominal logic. In: Proceedings of the 10th International Conference on Foundations of Software Science and Computation Structures (FOSSACS). LNCS, vol. 4423, pp. 63–77. Braga, March 2007

  7. Crary, K.: Logical relations and a case study in equivalence checking. In: Pierce, B.C. (ed.) Advanced Topics in Types and Programming Languages, pp. 223–244. MIT, Cambridge (2005)

    Google Scholar 

  8. Despeyroux, J., Felty, A., Hirschowitz, A.: Higher-order abstract syntax in Coq. In: Proceedings of the 2nd International Conference on Typed Lambda Calculi and Applications (TLCA). LNCS, vol. 902, pp. 124–138. Springer, New York (1995)

    Google Scholar 

  9. Dowek, G., Hardin, T., Kirchner, C.: Higher-order unification via explicit substitutions. Inf. Comput. 157, 183–235 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gabbay, M.J.: A theory of inductive definitions with α-equivalence. PhD thesis, University of Cambridge (2001)

  11. Gabbay, M.J., Pitts, A.M.: A new approach to abstract syntax with variable binding. Form. Asp. Comput. 13, 341–363 (2001)

    Article  Google Scholar 

  12. Girard, J.-Y., Lafont, Y., Taylor, P.: Proofs and Types. Cambridge Tracts in Theoretical Computer Science, vol. 7. Cambridge University Press, Cambridge (1989)

    Google Scholar 

  13. Gordon, A.D.: A mechanisation of name-carrying syntax up to alpha-conversion. In: Proceedings of the 6th International Workshop on Higher-order Logic Theorem Proving and its Applications (HUG). LNCS, vol. 780, pp. 414–426. Vancouver, 11–13 August 1994

  14. Gordon, A.D., Melham, T.: Five axioms of alpha conversion. In: Proceedings of the 9th International Conference on Theorem Proving in Higher Order Logics (TPHOLs). LNCS, vol. 1125, pp. 173–190. Turku, 26–30 August 1996

  15. Harper, R., Pfenning, F.: On equivalence and canonical forms in the LF type theory. ACM Trans. Comput. Log. 6(1), 61–101 (2005)

    Article  MathSciNet  Google Scholar 

  16. Homeier, P.: A design structure for higher order quotients. In: Proceedings of the 18th International Conference on Theorem Proving in Higher Order Logics (TPHOLs). LNCS, vol. 3603, pp. 130–146. Oxford, 22–25 August 2005

  17. Kleene, S.C.: Disjunction and existence under implication in elementary intuitionistic formalisms. J. Symb. Log. 27(1), 11–18 (1962)

    Article  MathSciNet  Google Scholar 

  18. Leroy, X.: Polymorphic typing of an algorithmic language. Ph.D. thesis, University Paris 7, INRIA Research Report, No 1778 (1992)

  19. Melham, T.: Automating recursive type definitions in higher order logic. Technical Report 146, Computer Laboratory, University of Cambridge, September (1988)

  20. Melham, T.: Automating recursive type definitions in higher order logic. In: Current Trends in Hardware Verification and Automated Theorem Proving, pp. 341–386. Springer, New York (1989)

    Google Scholar 

  21. Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle HOL: A proof assistant for higher-order logic. LNCS, vol. 2283. Springer, New York (2002)

    Google Scholar 

  22. Norrish, M.: Recursive function definition for types with binders. In: Proceedings of the 17th International Conference Theorem Proving in Higher Order Logics (TPHOLs). LNCS, vol. 3223, pp. 241–256. Park City, 14–17 September 2004

  23. Norrish, M.: Mechanising λ-calculus using a classical first order theory of terms with permutation. High. Order Symb. Comput. 19, 169–195 (2006)

    Article  MATH  Google Scholar 

  24. Paulson, L.: Defining functions on equivalence classes. ACM Trans. Comput. Log. 7(4), 658–675 (2006)

    Article  MathSciNet  Google Scholar 

  25. Pfenning, F., Elliott, C.: Higher-order abstract syntax. In: Proceedings of the 10th Conference on Conference on Programming Language Design and Implementation (PLDI), pp. 199–208. ACM, New York (1989)

    Google Scholar 

  26. Pitts, A.M.: Nominal logic, a first order theory of names and binding. Inf. Comput. 186, 165–193 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  27. Pitts, A.M.: Alpha-structural recursion and induction. J. ACM 53, 459–506 (2006)

    Article  MathSciNet  Google Scholar 

  28. Slind, K.: Wellfounded schematic definitions. In: Proceedings of the 17th International Conference on Automated Deduction (CADE). LNCS, vol. 1831, pp. 45–63. Pittsburgh, 17–20 June 2000

  29. Takahashi, M.: Parallel reductions in lambda-calculus. Inf. Comput. 118(1), 120–127 (1995)

    Article  MATH  Google Scholar 

  30. Tobin-Hochstadt, S., Felleisen, M.: The design and implementation of typed scheme. In: Proceedings of the 35rd Symposium on Principles of Programming Languages (POPL), pp. 395–406. ACM, New York (2008)

    Google Scholar 

  31. Urban, C.: Classical logic and computation. Ph.D. thesis, Cambridge University, October (2000)

  32. Urban, C., Berghofer, S.: A recursion combinator for nominal datatypes implemented in Isabelle/HOL. In: Proceedings of the 3rd International Joint Conference on Automated Reasoning (IJCAR). LNAI, vol. 4130, pp. 498–512. Seattle, 17–20 August 2006

  33. Urban, C., Berghofer, S., Norrish, M.: Barendregt’s variable convention in rule inductions. In: Proceedings of the 21st International Conference on Automated Deduction (CADE). LNAI, vol. 4603, pp. 35–50. Bremen, 17–20 July 2007

  34. Urban, C., Norrish, M.: A formal treatment of the Barendregt variable convention in rule inductions. In: Proceedings of the 3rd International ACM Workshop on Mechanized Reasoning about Languages with Variable Binding and Names, pp. 25–32. ACM, New York (2005)

    Chapter  Google Scholar 

  35. Urban, C., Pitts, A.M., Gabbay, M.J.: Nominal unification. Theor. Comp. Sci. 323(1–2), 473–497 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  36. Urban, C., Tasson, C.: Nominal techniques in Isabelle/HOL. In: Proceedings of the 20th International Conference on Automated Deduction (CADE). LNCS, vol. 3632, pp. 38–53. Tallinn, 22–27 July 2005

  37. Wenzel, M.: Using axiomatic type classes in Isabelle. Manual in the Isabelle distribution. http://isabelle.in.tum.de/doc/axclass.pdf (2000)

  38. Wenzel, M.: Structured induction proofs in Isabelle/Isar. In: Proceedings of the 5th International Conference on Mathematical Knowledge Management (MKM). LNAI, vol. 4108, pp. 17–30. Wokingham, 11–12 August 2006

Download references

Author information

Authors and Affiliations

  1. Technical University Munich, Munich, Germany

    Christian Urban

Authors
  1. Christian Urban
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Urban.

Additional information

This paper is a revised and much extended version of Urban and Berghofer [32], and Urban and Tasson [36].

Rights and permissions

Reprints and permissions

About this article

Cite this article

Urban, C. Nominal Techniques in Isabelle/HOL. J Autom Reasoning 40, 327–356 (2008). https://doi.org/10.1007/s10817-008-9097-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10817-008-9097-2

Keywords

Search

Navigation