Abstract
When faced with the question of how to represent properties in a formal proof system any user has to make design decisions. We have proved three of the theorems from Maskin’s 2004 survey article on Auction Theory using the Isabelle/HOL system, and we have verified software code that implements combinatorial Vickrey auctions. A fundamental question in this was how to represent some basic concepts: since set theory is available inside Isabelle/HOL, when introducing new definitions there is often the issue of balancing the amount of set-theoretical objects and of objects expressed using entities which are more typical of higher order logic such as functions or lists. Likewise, a user has often to answer the question whether to use a constructive or a non-constructive definition. Such decisions have consequences for the proof development and the usability of the formalization. For instance, sets are usually closer to the representation that economists would use and recognize, while the other objects are closer to the extraction of computational content. We have studied the advantages and disadvantages of these approaches, and their relationship, in the concrete application setting of auction theory. In addition, we present the corresponding Isabelle library of definitions and theorems, most prominently those dealing with relations and quotients.
This work has been supported by EPSRC grant EP/J007498/1 and an LMS Computer Science Small Grant.
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References
Bergman, C.: Universal Algebra: Fundamentals and Selected Topics. Chapman & Hall Pure and Applied Mathematics. Taylor & Francis (2011)
Blanchette, J.C., Paulson, L.C.: Hammering Away. A User’s Guide to Sledgehammer for Isabelle/HOL (December 5, 2013), http://isabelle.in.tum.de/dist/Isabelle2013-2/doc/sledgehammer.pdf
Bowen, J., Gordon, M.: Z and HOL. In: Z User Workshop, Cambridge 1994, pp. 141–167. Springer (1994)
Caminati, M.B., et al.: Proving soundness of combinatorial Vickrey auctions and generating verified executable code, arXiv:1308.1779 [cs.GT] (2013)
Haftmann, F., Nipkow, T.: Code generation via higher-order rewrite systems. In: Blume, M., Kobayashi, N., Vidal, G. (eds.) FLOPS 2010. LNCS, vol. 6009, pp. 103–117. Springer, Heidelberg (2010)
Klein, G., et al. (eds.): Archive of Formal Proofs (2014), http://afp.sf.net/ (visited on March 14, 2014)
Lange, C., Caminati, M.B., Kerber, M., Mossakowski, T., Rowat, C., Wenzel, M., Windsteiger, W.: A qualitative comparison of the suitability of four theorem provers for basic auction theory. In: Carette, J., Aspinall, D., Lange, C., Sojka, P., Windsteiger, W. (eds.) CICM 2013. LNCS (LNAI), vol. 7961, pp. 200–215. Springer, Heidelberg (2013)
Lange, C., Rowat, C., Kerber, M.: The ForMaRE Project – Formal Mathematical Reasoning in Economics. In: Carette, J., Aspinall, D., Lange, C., Sojka, P., Windsteiger, W. (eds.) CICM 2013. LNCS (LNAI), vol. 7961, pp. 330–334. Springer, Heidelberg (2013), arXiv:1303.4194[cs.CE]
Leinster, T.: Rethinking set theory. arXiv preprint arXiv:1212.6543 (2012)
Maskin, E.: The unity of auction theory: Milgrom’s master class. Journal of Economic Literature 42(4), 1102–1115 (2004), http://scholar.harvard.edu/files/maskin/files/unity_of_auction_theory.pdf
Nipkow, T., Paulson, L.C.: Proof pearl: Defining functions over finite sets. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 385–396. Springer, Heidelberg (2005)
Paulson, L.C.: Defining functions on equivalence classes. ACM Transactions on Computational Logic (TOCL) 7(4), 658–675 (2006)
Paulson, L.C.: Set Theory for Verification: I. From Foundations to Functions. Journal of Automated Reasoning 11, 353–389 (1993)
Paulson, L.C., et al.: Isabelle/HOL. A Proof Assistant for Higher-Order Logic (2013)
Simpson, S.G.: Subsystems of second order arithmetic, vol. 1. Cambridge University Press (2009)
Valentine, S.H., et al.: AZ Patterns Catalogue II-definitions and laws, v0.1 (2004)
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Caminati, M.B., Kerber, M., Lange, C., Rowat, C. (2014). Set Theory or Higher Order Logic to Represent Auction Concepts in Isabelle?. In: Watt, S.M., Davenport, J.H., Sexton, A.P., Sojka, P., Urban, J. (eds) Intelligent Computer Mathematics. CICM 2014. Lecture Notes in Computer Science(), vol 8543. Springer, Cham. https://doi.org/10.1007/978-3-319-08434-3_18
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DOI: https://doi.org/10.1007/978-3-319-08434-3_18
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