Abstract
In earlier work presented at CICM, four theorem provers (Isabelle, Mizar, Hets/CASL/TPTP, and Theorema) were compared based on a case study in theoretical economics, the formalization of the landmark Theorem of Vickrey in auction theory. At the time of this comparison the Theorema system was in a state of transition: The original Theorema system (Theorema 1) had been shut down by the Theorema group and the successor system Theorema 2.0 was just about to be launched. Theorema 2.0 participated in the competition, but only parts of the system were ready for use. In particular, the new reasoning engines had not been set up, so that some of the results in the system comparison had to be extrapolated from experience we had with Theorema 1. In this paper, we now want to compare a complete formalization of Vickrey’s Theorem in Theorema 2.0 with the original formalization in Isabelle. On the one hand, we compare the mathematical setup of the two theories and, on the other hand, we also give an overview on statistical indicators, such as number of auxiliary lemmas and the total number of proof steps needed for all proofs in the theory. Last but not least, we present a shorter version of proof of the main theorem in Isabelle.
A. Maletzky—The research was funded by the Austrian Science Fund (FWF): P 29498-N31.
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
Notes
- 1.
It should be noted that not all methods that were implemented in Theorema 1 are already available in Theorema 2.0. The standard method available, which was used also in the current formalization, is a natural-deduction-like prover for first-order predicate logic with certain enhancements for Theorema-specific language constructs.
- 2.
Theorema knowledge archives, which will be an efficient way of storing Theorema formalizations for later use in a structured hierarchical build-up of theories, are not yet available in the current release of Theorema 2.0. Currently, the formalization of Vickrey’s Theorem is written in one Mathematica/Theorema notebook containing the statement of all pieces of formalized maths (definitions, lemmas, and theorems). A proof in Theorema 2.0 is represented in a data-structure called proof object, which contains the information about all logical steps the proof consists of. Proof objects and additional statistics about the proof run and user and system settings are stored in separate files. The Theorema formalization thus consists of the Theorema notebook together with its accompanying files, the current formalization being available for download in zip-format from the Theorema homepage at www.risc.jku.at/research/theorema/software/Vickrey.zip.
- 3.
This lemma could be omitted also, the four steps of its proof could as well be done in an extra branch of the proof of the main theorem. We rather see it as a means to structure the theory.
- 4.
- 5.
The proofs of the important theorems are still fine, though.
- 6.
The number of steps is no measure for the effort needed to generate the proof, since the steps are generated automatically.
- 7.
To get a feeling, how big a 41-step proof is: Theorema’s proof display with natural-language proof explanation consumes ample space because the explanation of every proof step starts in a new line and every formula is printed nicely 2D-formatted in a separate line (see a sample screen-shot of a proof in Fig. 4). In this format the proof is approximately 5 pages. Its fully automated generation took 110 s on a standard laptop (4 cores 2.20 GHz each), 106 s for proof generation plus 4 s for subsequent simplification of the generated proof.
- 8.
Note that the authors of the formalization in Isabelle introduced a type-synonym ‘vectors’ for such function types, which corresponds exactly to what tuples are in Theorema. We want to emphasize, however, that the choice of tuples in Theorema is not system-enforced, it is more a matter of taste. The mathematical representation of the objects to be studied in Theorema can be chosen freely. In general, using built-in structures (like tuples) has the advantage of getting system support in computation and proving. Using non-built structures (like functions as done in Isabelle) would require to prove auxiliary knowledge about these entities. This knowledge would then be part of the formalization, like the knowledge about ‘maximum’ in the Isabelle formalization.
- 9.
The universal quantifier for n is not visible locally, neither in the theorem nor in the definitions, because we use a ‘global universal quantifier’ \(\mathop \forall \limits _{n\in \mathbb {N}}\) at the beginning of the document. This mechanism in Theorema 2.0 is explained in detail in [2]. The Theorema language is untyped. Quantifiers range over all objects that can be expressed in the Theorema language, i.e. sets, tuples, and various kinds of numbers available in Mathematica. There are special ranges with limited domain, such as \(i=1,\dots ,n\) for finite integer fragments. The domain can also be restricted using conditions. All what is needed in the current formalization is essentially first-order.
- 10.
The Isabelle proof is spelled out in more detail than necessary, in order to be easily comparable to the proof in Theorema; a single application of fastforce would suffice. The aforementioned 185 steps refer to the short version, which is not shown here.
- 11.
- 12.
Definitions, lemmas, and theorems have user-defined labels. Formulas that are generated automatically during a proof have system-generated labels, where A#... and G#... refer to assumptions and goals, respectively.
References
Bancerek, G., Byliński, C., Grabowski, A., Korniłowicz, A., Matuszewski, R., Naumowicz, A., Pa̧k, K., Urban, J.: Mizar: state-of-the-art and beyond. In: Kerber, M., Carette, J., Kaliszyk, C., Rabe, F., Sorge, V. (eds.) CICM 2015. LNCS, vol. 9150, pp. 261–279. Springer, Cham (2015). doi:10.1007/978-3-319-20615-8_17
Buchberger, B., Jebelean, T., Kutsia, T., Maletzky, A., Windsteiger, W.: Theorema 2.0: computer-assisted natural-style mathematics. JFR 9(1), 149–185 (2016)
Caminati, M.B., Kerber, M., Lange, C., Rowat, C.: VCG - Combinatorial Vickrey-Clarke-Groves Auctions. Archive of Formal Proofs, April 2015
Grabowski, A., Korniłowicz, A., Naumowicz, A.: Mizar in a nutshell. J. Formal. Reason. 3(2), 153–245 (2010)
Kerber, M., Lange, C., Rowat, C., Windsteiger, W.: Developing an auction theory toolbox. In: Kerber, M., Lange, C., Rowat, C. (eds.) AISB 2013, pp. 1–4 (2013). Proceedings available online. http://www.cs.bham.ac.uk/research/projects/formare/events/aisb2013/proceedings.php
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, vol. 7961, pp. 200–215. Springer, Heidelberg (2013). doi:10.1007/978-3-642-39320-4_13
Maskin, E.: The unity of auction theory. J. Econ. Lit. 42(4), 1102–1115 (2004)
Milgrom, P.: Putting Auction Theory to Work. Cambridge University Press, Cambridge (2004)
Mossakowski, T., Haxthausen, A.E., Sannella, D., Tarlecki, A.: CASL - the common algebraic specification language. In: Bjørner, D., Henson, M.C. (eds.) Logics of Specification Languages. Monographs in Theoretical Computer Science, pp. 241–298. Springer, Heidelberg (2008)
Mossakowski, T., Maeder, C., Codescu, M.: Hets user guide. Technical report. version 0.98, DFKI Bremen (2013)
Paulson, L.C.: Isabelle: the next 700 theorem provers. In: Odifreddi, P. (ed.) Logic and Computer Science, pp. 361–386. Academic Press (1990)
Sutcliffe, G.: The TPTP problem library and associated infrastructure: the FOF and CNF parts, v3.5.0. J. Autom. Reason. 43(4), 337–362 (2009)
Vickrey, W.: Counterspeculation, auctions, and competitive sealed tenders. J. Financ. XVI, 8–37 (1961)
Wenzel, M.: Isabelle/Isar Reference Manual (2017)
Windsteiger, W.: Theorema 2.0: a graphical user interface for a mathematical assistant system. In: Kaliszyk, C., Lueth, C. (eds.) Proceedings of the 10th International Workshop UITP. EPTCS, vol. 118, pp. 72–82. Open Publishing Association (2012). doi10.4204/EPTCS.118.5. http://arxiv.org/abs/1307.1945v1
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Maletzky, A., Windsteiger, W. (2017). The Formalization of Vickrey Auctions: A Comparison of Two Approaches in Isabelle and Theorema. In: Geuvers, H., England, M., Hasan, O., Rabe, F., Teschke, O. (eds) Intelligent Computer Mathematics. CICM 2017. Lecture Notes in Computer Science(), vol 10383. Springer, Cham. https://doi.org/10.1007/978-3-319-62075-6_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-62075-6_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-62074-9
Online ISBN: 978-3-319-62075-6
eBook Packages: Computer ScienceComputer Science (R0)