Abstract
We present a new formalization of the algebraic hierarchy in Coq, exploiting its new type class mechanism to make practical a solution formerly thought infeasible. Our approach addresses both traditional challenges as well as new ones resulting from our ambition to build upon this development a library of constructive analysis in which abstraction penalties inhibiting efficient computation are reduced to a bare minimum. To support mathematically sound abstract interfaces for ℕ, ℤ, and ℚ, our formalization includes portions of category theory and multisorted universal algebra.
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References
Asperti, A., Ricciotti, W., Coen, C.S., Tassi, E.: Hints in unification. In: Proceedings of the 22nd International Conference on Theorem Proving in Higher Order Logics, p. 98. Springer, Heidelberg (2009)
Berardi, S., Damiani, F., de’Liguoro, U. (eds.): TYPES 2008. LNCS, vol. 5497. Springer, Heidelberg (2009)
Capretta, V.: Universal algebra in type theory. In: Bertot, Y., Dowek, G., Hirschowitz, A., Paulin, C., Théry, L. (eds.) TPHOLs 1999. LNCS, vol. 1690, pp. 131–148. Springer, Heidelberg (1999)
Cruz-Filipe, L., Geuvers, H., Wiedijk, F.: C-coRN, the constructive coq repository at nijmegen. In: Asperti, A., Bancerek, G., Trybulec, A. (eds.) MKM 2004. LNCS, vol. 3119, pp. 88–103. Springer, Heidelberg (2004)
Domínguez, C.: Formalizing in Coq Hidden Algebras to Specify Symbolic Computation Systems. 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. 270–284. Springer, Heidelberg (2008)
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)
Geuvers, H., Pollack, R., Wiedijk, F., Zwanenburg, J.: A constructive algebraic hierarchy in Coq. J. Symb. Comput. 34(4), 271–286 (2002)
Haftmann, F., Wenzel, M.: Local theory specifications in Isabelle/Isar. In: Berardi, et al. (eds.) [2], pp. 153–168
Luo, Z.: Manifest fields and module mechanisms in intensional type theory. In: Berardi, et al. (eds.) [2], pp. 237–255
Sozeau, M.: A new look at generalized rewriting in type theory. Journal of Formalized Reasoning 2(1), 41–62 (2009)
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Spitters, B., van der Weegen, E. (2010). Developing the Algebraic Hierarchy with Type Classes in Coq. In: Kaufmann, M., Paulson, L.C. (eds) Interactive Theorem Proving. ITP 2010. Lecture Notes in Computer Science, vol 6172. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14052-5_35
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DOI: https://doi.org/10.1007/978-3-642-14052-5_35
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14051-8
Online ISBN: 978-3-642-14052-5
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