Abstract
Pitts et al introduced a beautiful theory about names and binding based on the notions of permutation and support. The engineering challenge is to smoothly adapt this theory to a theorem prover environment, in our case Isabelle/HOL. We present a formalisation of this work that differs from our earlier approach in two important respects: First, instead of representing permutations as lists of pairs of atoms, we now use a more abstract representation based on functions. Second, whereas the earlier work modeled different sorts of atoms using different types, we now introduce a unified atom type that includes all sorts of atoms. Interestingly, we allow swappings, that is permutations build from two atoms, to be ill-sorted. As a result of these design changes, we can iron out inconveniences for the user, considerably simplify proofs and also drastically reduce the amount of custom ML-code. Furthermore we can extend the capabilities of Nominal Isabelle to deal with variables that carry additional information. We end up with a pleasing and formalised theory of permutations and support, on which we can build an improved and more powerful version of Nominal Isabelle.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bengtson, J., Parrow, J.: Formalising the pi-Calculus using Nominal Logic. In: Seidl, H. (ed.) FOSSACS 2007. LNCS, vol. 4423, pp. 63–77. Springer, Heidelberg (2007)
Benzmüller, C., Paulson, L.C.: Quantified Multimodal Logics in Simple Type Theory. SEKI Report SR–2009–02 (ISSN 1437-4447). SEKI Publications (2009), http://arxiv.org/abs/0905.2435
Cheney, J.: Completeness and Herbrand Theorems for Nominal Logic. Journal of Symbolic Logic 71(1), 299–320 (2006)
Church, A.: A Formulation of the Simple Theory of Types. Journal of Symbolic Logic 5(2), 56–68 (1940)
Gabbay, M.J., Pitts, A.M.: A New Approach to Abstract Syntax with Variable Binding. Formal Aspects of Computing 13, 341–363 (2002)
Gunter, E., Osborn, C., Popescu, A.: Theory Support for Weak Higher Order Abstract Syntax in Isabelle/HOL. In: Proc. of the 4th International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice (LFMTP). ENTCS, pp. 12–20 (2009)
Pitts, A.M.: Syntax and Semantics. Part of the documentation for the HOL4 system
Pitts, A.M.: Nominal Logic, A First Order Theory of Names and Binding. Information and Computation 183, 165–193 (2003)
Sato, M., Pollack, R.: External and Internal Syntax of the Lambda-Calculus. To appear in Journal of Symbolic Computation
Tobin-Hochstadt, S., Felleisen, M.: The Design and Implementation of Typed Scheme. In: Proc. of the 35rd Symposium on Principles of Programming Languages (POPL), pp. 395–406. ACM, New York (2008)
Urban, C., Cheney, J., Berghofer, S.: Mechanizing the Metatheory of LF. In: Proc. of the 23rd IEEE Symposium on Logic in Computer Science (LICS), pp. 45–56 (2008)
Urban, C., Pitts, A., Gabbay, M.: Nominal Unification. Theoretical Computer Science 323(1-3), 473–497 (2004)
Urban, C., Zhu, B.: Revisiting Cut-Elimination: One Difficult Proof is Really a Proof. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 409–424. Springer, Heidelberg (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Huffman, B., Urban, C. (2010). A New Foundation for Nominal Isabelle. In: Kaufmann, M., Paulson, L.C. (eds) Interactive Theorem Proving. ITP 2010. Lecture Notes in Computer Science, vol 6172. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14052-5_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-14052-5_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14051-8
Online ISBN: 978-3-642-14052-5
eBook Packages: Computer ScienceComputer Science (R0)