Abstract
Representation determines how we can reason about a specific problem. Sometimes one representation helps us to find a proof more easily than others. Most current automated reasoning tools focus on reasoning within one representation. There is, therefore, a need for the development of better tools to mechanise and automate formal and logically sound changes of representation. In this paper we look at examples of representational transformations in discrete mathematics, and show how we have used tools from Isabelle’s Transfer package to automate the use of these transformations in proofs. We give an overview of a general theory of transformations that we consider appropriate for thinking about the matter, and we explain how it relates to the Transfer package. We show a few reasoning tactics we developed in Isabelle to improve the use of transformations, including the automation of search in the space of representations. We present and analyse some results of the use of these tactics.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ballarin, C.: Interpretation of locales in Isabelle: theories and proof contexts. In: International Conference on Mathematical Knowledge Management, pp. 31–43. Springer, Berlin, Heidelberg (2006)
Blanchette, J.C., Nipkow, T.: Nitpick: a counterexample generator for higher-order logic based on a relational model finder. In: International conference on interactive theorem proving, pp. 131–146. Springer, Berlin, Heidelberg (2010)
Böhme, S., Nipkow, T.: Sledgehammer: judgement day. In: International Joint Conference on Automated Reasoning, pp. 107–121. Springer, Berlin, Heidelberg (2010)
Bóna, M.: A walk through combinatorics: an introduction to enumeration and graph theory. World scientific, Singapore (2011)
Bulwahn, L.: The new Quickcheck for Isabelle. In: International Conference on Certified Programs and Proofs, pp. 92–108. Springer, Berlin, Heidelberg (2012)
Farmer, W.M., Guttman, J.D., Thayer, F.T.: Little theories. In: International Conference on Automated Deduction, pp. 567–581. Springer, Berlin, Heidelberg (1992)
Farmer, W.M., Guttman, J.D., Thayer, F.T.: IMPS: an interactive mathematical proof system. J. Autom. Reasoning 9(11), 213–248 (1993)
Goguen, J.A., Burstall, R.M.: Institutions: abstract model theory for specication and programming. J. ACM (JACM) 39(1), 95–146 (1992)
Haftmann, F., Krauss, A., Kunčar, O., Nipkow, T.: Data refinement in Isabelle/HOL. In: International Conference on Interactive Theorem Proving, pp. 100–115. Springer, Berlin, Heidelberg (2013)
Haftmann, F., Wenzel, M,: Constructive type classes in Isabelle. In: International Workshop on Types for Proofs and Programs, pp. 160–174. Springer, Berlin, Heidelberg (2006)
Huffman, B., Kunčar, O.: Lifting and transfer: a modular design for quotients in Isabelle/HOL. In: International Conference on Certified Programs and Proofs, pp. 131–146. Springer, Switzerland (2013)
Hurd, J.: System description: the Metis proof tactic. In: Benzmüller, C., Harrison, J., Schürmann, C. (eds.) Empirically Successful Automated Reasoning in Higher-Order Logic (ESHOL), pp. 103–104, Montego Bay (2005)
Kerber, M., Präcklein, A.: Using tactics to reformulate formulae for resolution theorem proving. Ann. Math. Artif. Intell. 18(2), 221–241 (1996)
Kumar, R., Arthan, R., Myreen, M.O., Owens, S.: HOL with definitions: semantics, soundness, and a verified implementation. In: International Conference on Interactive Theorem Proving, pp. 308–324. Springer, Switzerland (2014)
Kunčar, O., Popescu, A.: A consistent foundation for Isabelle/HOL. In: International Conference on Interactive Theorem Proving, pp. 234–252. Springer, Switzerland (2015)
Meng, J., Paulson, L.C.: Translating higher-order clauses to first-order clauses. J. Autom. Reasoning 40(1), 35–60 (2008)
Mossakowski, T., Maeder, C., Lüttich, K.: The heterogeneous tool set, HETS. In: International Conference on Tools and Algorithms for the Construction and Analysis of Systems, pp, 519–522. Springer, Berlin, Heidelberg (2007)
Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: a proof assistant for higher-order logic, vol. 2283. Springer, Berlin, Heidelberg (2002)
Paulson, L.C., Blanchette, J.C.: Three years of experience with Sledgehammer, a practical link between automatic and interactive theorem provers. In: 8th International Workshop on the Implementation of Logics, 2010. Invited talk
Sozeau, M.: A new look at generalized rewriting in type theory. J. Formalized Reasoning 2(1), 41–62 (2009)
van Lint, J.H., Wilson, R.M.: A course in combinatorics. Cambridge University Press, Cambridge (2001)
Weber, T.: SMT solvers: new oracles for the HOL theorem prover. Int. J. Softw. Tools Technol. Transfer 13(5), 419–429 (2011)
Zimmermann, T., Herbelin, H.: Automatic and transparent transfer of theorems along isomorphisms in the Coq proof assistant. arXiv preprint, arXiv:1505.05028 (2015)
Author information
Authors and Affiliations
Corresponding author
Additional information
D. Raggi: This work has been supported by the Mexican Council of Science and Technology (CONACYT), with scholarship no. 214095.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Raggi, D., Bundy, A., Grov, G. et al. Automating Change of Representation for Proofs in Discrete Mathematics (Extended Version). Math.Comput.Sci. 10, 429–457 (2016). https://doi.org/10.1007/s11786-016-0275-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11786-016-0275-z