Abstract
We motivate and give semantics to theory presentation combinators as the foundational building blocks for a scalable library of theories. The key observation is that the category of contexts and fibered categories are the ideal theoretical tools for this purpose.
This research was supported by NSERC.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Asperti, A., Sacerdoti Coen, C.: Some Considerations on the Usability of Interactive Provers. In: Autexier, S., Calmet, J., Delahaye, D., Ion, P.D.F., Rideau, L., Rioboo, R., Sexton, A.P. (eds.) AISC 2010. LNCS, vol. 6167, pp. 147–156. Springer, Heidelberg (2010), http://dl.acm.org/citation.cfm?id=1894483.1894498
Burstall, R.M., Goguen, J.A.: Putting theories together to make specifications. In: IJCAI, pp. 1045–1058 (1977)
Burstall, R.M., Goguen, J.A.: The Semantics of Clear, a Specification Language. In: Bjorner, D. (ed.) Abstract Software Specifications. LNCS, vol. 86, pp. 292–332. Springer, Heidelberg (1980)
Carette, J., Farmer, W.M.: High-Level Theories. In: Autexier, S., Campbell, J., Rubio, J., Sorge, V., Suzuki, M., Wiedijk, F. (eds.) AISC 2008, Calculemus 2008, and MKM 2008. LNCS (LNAI), vol. 5144, pp. 232–245. Springer, Heidelberg (2008)
Carette, J., Farmer, W.M., Jeremic, F., Maccio, V., O’Connor, R., Tran, Q.: The mathscheme library: Some preliminary experiments. Tech. rep., University of Bologna, Italy (2011), uBLCS-2011-04
Carette, J., Kiselyov, O.: Multi-stage programming with functors and monads: Eliminating abstraction overhead from generic code. Sci. Comput. Program. 76(5), 349–375 (2011)
Carette, J., O’Connor, R.: Theory Presentation Combinators (2012), http://arxiv.org/abs/1204.0053v2
Cartmell, J.: Generalised algebraic theories and contextual categories. Annals of Pure and Applied Logic 32, 209–243 (1986), http://www.sciencedirect.com/science/article/pii/0168007286900539
CoFI (The Common Framework Initiative): Casl Reference Manual. LNCS, IFIP Series, vol. 2960. Springer (2004)
Farmer, W.M., Guttman, J.D., Thayer, F.J.: Little Theories. In: Kapur, D. (ed.) CADE 1992. LNCS, vol. 607, pp. 567–581. Springer, Heidelberg (1992)
Garillot, F., Gonthier, G., Mahboubi, A., Rideau, L.: Packaging Mathematical Structures. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 327–342. Springer, Heidelberg (2009), http://dx.doi.org/10.1007/978-3-642-03359-9_23
Geuvers, H., Wiedijk, F., Zwanenburg, J.: A Constructive Proof of the Fundamental Theorem of Algebra without Using the Rationals. In: Callaghan, P., Luo, Z., McKinna, J., Pollack, R. (eds.) TYPES 2000. LNCS, vol. 2277, pp. 96–111. Springer, Heidelberg (2002)
Grabowski, A., Schwarzweller, C.: On Duplication in Mathematical Repositories. In: Autexier, S., Calmet, J., Delahaye, D., Ion, P., Rideau, L., Rioboo, R., Sexton, A. (eds.) AISC 2010. LNCS, vol. 6167, pp. 300–314. Springer, Heidelberg (2010)
Jacobs, B.: Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics, vol. 141. North Holland, Amsterdam (1999)
Oriat, C.: Detecting equivalence of modular specifications with categorical diagrams. Theor. Comput. Sci. 247(1-2), 141–190 (2000)
Rabe, F., Kohlhase, M.: A Scalable Module System, http://kwarc.info/frabe/Research/mmt.pdf
Sacerdoti Coen, C., Tassi, E.: Nonuniform coercions via unification hints. In: Hirschowitz, T. (ed.) TYPES. EPTCS, vol. 53, pp. 16–29 (2009)
Smith, D.R.: Constructing specification morphisms. Journal of Symbolic Computation 15, 5–6 (1993)
Smith, D.R.: Mechanizing the development of software. In: Broy, M., Steinbrueggen, R. (eds.) Calculational System Design, Proceedings of the NATO Advanced Study Institute, pp. 251–292. IOS Press, Amsterdam (1999)
Spitters, B., van der Weegen, E.: Type classes for mathematics in type theory. Mathematical Structures in Computer Science 21(4), 795–825 (2011)
Wiedijk, F.: Estimating the cost of a standard library for a mathematical proof checker (2001), http://www.cs.ru.nl/~freek/notes/mathstdlib2.pdf
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Carette, J., O’Connor, R. (2012). Theory Presentation Combinators. In: Jeuring, J., et al. Intelligent Computer Mathematics. CICM 2012. Lecture Notes in Computer Science(), vol 7362. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31374-5_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-31374-5_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31373-8
Online ISBN: 978-3-642-31374-5
eBook Packages: Computer ScienceComputer Science (R0)