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Mechanising λ-calculus using a classical first order theory of terms with permutations

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Higher-Order and Symbolic Computation

Abstract

This paper describes the mechanisation in HOL of some basic λ-calculus theory. The proofs are taken from standard sources (books by Hankin and Barendregt), and cover: equational theory, reduction theory, residuals, finiteness of developments, and the standardisation theorem. The issues in mechanising pen-and-paper proofs are discussed; in particular, those difficulties arising from the sources’ use of the Barendregt Variable Convention.

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References

  1. Ambler, S.J., Crole, R.L., Momigliano, A.: Combining Higher Order Abstract Syntax with Tactical Theorem Proving and (Co)Induction. In: Carreño, V.A., Muñoz, C.A., Tahar, S. (eds.), Theorem Proving in Higher Order Logics, 15th International Conference, vol. 2410 of Lecture Notes in Computer Science, pp. 13–30 (2002)

  2. Barendregt, H.P.: The Lambda Calculus: its Syntax and Semantics, vol. 103 of Studies in Logic and the Foundations of Mathematics. Elsevier, Amsterdam: revised edition (1984)

    Google Scholar 

  3. Ford, J.M., Mason, I.A.: Operational Techniques in PVS: A Preliminary Evaluation. In: Fidge, C. (ed.), Computing: The Australasian Theory Symposium (CATS 2001), vol.42 of Electronic Notes in Theoretical Computer Science (2001)

  4. Gordon, A.D.: A Mechanisation of Name-Carrying Syntax up to Alpha-Conversion. In: Joyce, J.J., Seger, C.J.H. (eds.), In: Proceedings of Higher-Order Logic Theorem Proving and its Applications (HUG’93), vol. 780 of Lecture Notes in Computer Science, pp. 413–425 (1994)

  5. Gordon, A.D., Melham, T.: Five Axioms of Alpha Conversion. In: von Wright, J., Grundy, J., Harrison, J. (eds.), Theorem Proving in Higher Order Logics: 9th International Conference, TPHOLs’96, vol. 1125 of Lecture Notes in Computer Science, pp. 173–190 (1996)

  6. Hankin, C.: Lambda Calculi: A Guide for Computer Scientists, vol.3 of Graduate Texts in Computer Science, Clarendon Press, Oxford (1994)

    Google Scholar 

  7. Homeier, P.: A Proof of the Church-Rosser Theorem for the Lambda Calculus in Higher Order Logic. In: Boulton, R.J., Jackson, P.B. (eds.), TPHOLs 2001: Supplemental Proceedings. Available as Informatics Research Report EDI-INF-RR-0046 pp. 207–222 (2001)

  8. Homeier, P.: A Design Structure for Higher Order Quotients in tphols2005, pp.130–146 (2005)

  9. Huet, G.P.: Residual Theory in Lambda-Calculus: A Formal Development. Journal of Functional Programming 4(3), 371–394 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  10. Hurd, J., Melham, T. (eds.): Theorem Proving in Higher Order Logics 18th International Conference, Lecture Notes in Computer Science, Springer, vol.3603 (2005)

  11. McKinna, J., Pollack, R.: Some Lambda Calculus and Type Theory Formalized. Journal of Automated Reasoning 23(3–4), 373–409 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Norrish, M.: Mechanising Hankin and Barendregt Using the Gordon-Melham Axioms. In: Honsell, F., Miculan, M., Momigliano, A. (eds.), Merlin 2003, Proceedings of the Second ACM SIGPLAN Workshop on Mechanized Reasoning about Languages with Variable Binding. Available at http://doi.acm.org/10.1145/976571.976577 (2003)

  13. Norrish, M.: Recursive Function Definition for Types with Binders. In: Slind, K., Bunker, A., Gopalakrishnan, G. (eds.) Theorem Proving in Higher Order Logics, 17th International Conference, vol. 3223 of LNCS. pp. 241–256 (2004)

  14. Paulson, L.: Defining Functions on Equivalence Classes. ACM Transactions on Computational Logic. To appear.

  15. Pitts, A.M.: Nominal Logic, A First Order Theory of Names and Binding. Information and Computation 186, 165–193 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Pitts, A.M.: Alpha-Structural Recursion and Induction (Extended Abstract) in phols2005, pp.17–34 (2005)

  17. Urban, C., Tasson, C.: Nominal Techniques in Isabelle/HOL. In: Nieuwenhuis, R. (ed.), Proceedings of the 20th International Conference on Automated Deduction (CADE), vol. 3632 of Lecture Notes in Computer Science, pp. 38–53 (2005)

  18. Vestergaard, R.: The Primitive Proof Theory of the λ-Calculus. Ph.D. thesis, School of Mathematical and Computer Sciences, Heriot-Watt University (2003)

  19. Vestergaard, R., Brotherston, J.: A Formalised First-Order Confluence Proof for the λ-Calculus using One-Sorted Variable Names (Barendregt was right after all almost). In: Middeldorp, A. (ed.), Proceedings of RTA-12, vol. 2051 of LNCS (2001a)

  20. Vestergaard, R., Brotherston, J.: The Mechanisation of Barendregt-Style Equational Proofs (The Residual Perspective). Electronic Notes in Theoretical Computer Science 58(1) (2001b)

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  1. Canberra Research Laboratory, National ICT Australia, Research School of Information Science and Engineering, Australian National University, Acton 0200, Australia

    Michael Norrish

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  1. Michael Norrish
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Norrish, M. Mechanising λ-calculus using a classical first order theory of terms with permutations. Higher-Order Symb Comput 19, 169–195 (2006). https://doi.org/10.1007/s10990-006-8745-7

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  • DOI: https://doi.org/10.1007/s10990-006-8745-7

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