Abstract
This paper proposes generic design patterns to define and combine algebraic structures, using dependent records, coercions and type inference, inside the Coq system. This alternative to telescopes in particular supports multiple inheritance, maximal sharing of notations and theories, and automated structure inference. Our methodology is robust enough to handle a hierarchy comprising a broad variety of algebraic structures, from types with a choice operator to algebraically closed fields. Interfaces for the structures enjoy the convenience of a classical setting, without requiring any axiom. Finally, we present two applications of our proof techniques: a key lemma for characterising the discrete logarithm, and a matrix decomposition problem.
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
Lang, S.: Algebra. Springer, Heidelberg (2002)
Jackson, P.: Enhancing the Nuprl proof-development system and applying it to computational abstract algebra. PhD thesis, Cornell University (1995)
Betarte, G., Tasistro, A.: Formalisation of systems of algebras using dependent record types and subtyping: An example. In: Proc. 7th Nordic workshop on Programming Theory (1995)
Pottier, L.: User contributions in Coq, Algebra (1999), http://coq.inria.fr/contribs/Algebra.html
Geuvers, H., Pollack, R., Wiedijk, F., Zwanenburg, J.: A constructive algebraic hierarchy in Coq. Journal of Symbolic Computation 34(4), 271–286 (2002)
Haftmann, F., Wenzel, M.: Local theory specifications in Isabelle/Isar. In: Berardi, S., Damiani, F., de’Liguoro, U. (eds.) TYPES 2008. LNCS, vol. 5497, pp. 153–168. Springer, Heidelberg (2009)
Rudnicki, P., Schwarzweller, C., Trybulec, A.: Commutative algebra in the Mizar system. J. Symb. Comput. 32(1), 143–169 (2001)
Gonthier, G., Mahboubi, A., Rideau, L., Tassi, E., Théry, L.: A Modular Formalisation of Finite Group Theory. In: Schneider, K., Brandt, J. (eds.) TPHOLs 2007. LNCS, vol. 4732, pp. 86–101. Springer, Heidelberg (2007)
Bertot, Y., Gonthier, G., Ould Biha, S., Pasca, I.: Canonical big operators. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 86–101. Springer, Heidelberg (2008)
Gonthier, G., Mahboubi, A.: A small scale reflection extension for the Coq system. INRIA Technical report, http://hal.inria.fr/inria-00258384
Paulin-Mohring, C.: Définitions Inductives en Théorie des Types d’Ordre Supérieur. Habilitation à diriger les recherches, Université Claude Bernard Lyon I (1996)
Sozeau, M., Oury, N.: First-Class Type Classes. In: Mohamed, O.A., Muñoz, C., Tahar, S. (eds.) TPHOLs 2008. LNCS, vol. 5170, pp. 278–293. Springer, Heidelberg (2008)
Pollack, R.: Dependently typed records in type theory. Formal Aspects of Computing 13, 386–402 (2002)
Bruijn, N.G.D.: Telescopic mappings in typed lambda calculus. Information and Computation 91, 189–204 (1991)
Paulson, L.C.: Defining Functions on Equivalence Classes. ACM Transactions on Computational Logic 7(4), 658–675 (2006)
Barthe, G., Capretta, V., Pons, O.: Setoids in type theory. Journal of Functional Programming 13(2), 261–293 (2003)
Altenkirch, T., McBride, C., Swierstra, W.: Observational equality, now! In: Proceedings of the PLPV 2007 workshop, pp. 57–68. ACM, New York (2007)
Olteanu, G.: Computing the Wedderburn decomposition of group algebras by the Brauer-Witt theorem. Mathematics of Computation 76(258), 1073–1087 (2007)
Rotman, J.J.: An Introduction to the Theory of Groups. Springer, Heidelberg (1994)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. McGraw-Hill, New York (2003)
Sacerdoti Coen, C., Tassi, E.: Working with Mathematical Structures in Type Theory. In: Miculan, M., Scagnetto, I., Honsell, F. (eds.) TYPES 2007. LNCS, vol. 4941, pp. 157–172. Springer, Heidelberg (2008)
Sacerdoti Coen, C., Tassi, E.: A constructive and formal proof of Lebesgue Dominated Convergence Theorem in the interactive theorem prover Matita. Journal of Formalized Reasoning 1, 51–89 (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Garillot, F., Gonthier, G., Mahboubi, A., Rideau, L. (2009). Packaging Mathematical Structures. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds) Theorem Proving in Higher Order Logics. TPHOLs 2009. Lecture Notes in Computer Science, vol 5674. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03359-9_23
Download citation
DOI: https://doi.org/10.1007/978-3-642-03359-9_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03358-2
Online ISBN: 978-3-642-03359-9
eBook Packages: Computer ScienceComputer Science (R0)