Abstract
Datatypes freely generated by their constructors are well supported in mainstream proof assistants. Algebraic specification languages offer more expressive datatypes on axiomatic means: nonfree datatypes generated from constructors modulo equations. We have implemented an Isabelle/HOL package for nonfree datatypes, without compromising foundations. The use of the package, and its nonfree iterator in particular, is illustrated with examples: bags, polynomials and λ-terms modulo α-equivalence. The many-sorted metatheory of nonfree datatypes is formalized as an ordinary Isabelle theory and is animated by the package into user-specified instances. HOL lacks a type of types, so we employ an ad hoc construction of a universe embedding the relevant parameter types.
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.
Similar content being viewed by others
References
The Coq Proof Assistant (2013), http://coq.inria.fr
Maude ITP (2013), http://maude.cs.uiuc.edu/tools/itp
Altenkirch, T., McBride, C., Morris, P.: Generic programming with dependent types. In: Backhouse, R., Gibbons, J., Hinze, R., Jeuring, J. (eds.) SSDGP 2006. LNCS, vol. 4719, pp. 209–257. Springer, Heidelberg (2007)
Blanqui, F., Hardin, T., Weis, P.: On the implementation of construction functions for non-free concrete data types. In: De Nicola, R. (ed.) ESOP 2007. LNCS, vol. 4421, pp. 95–109. Springer, Heidelberg (2007)
Bove, A., Dybjer, P.: Dependent types at work. In: Bove, A., Barbosa, L.S., Pardo, A., Pinto, J.S. (eds.) LerNet ALFA Summer School 2008. LNCS, vol. 5520, pp. 57–99. Springer, Heidelberg (2009)
Clavel, M., Durán, F., Eker, S., Lincoln, P., Martí-Oliet, N., Meseguer, J., Quesada, J.F.: The Maude system. In: Narendran, P., Rusinowitch, M. (eds.) RTA 1999. LNCS, vol. 1631, pp. 240–243. Springer, Heidelberg (1999)
CoFI task group on semantics, CASL — The Common Algebraic Specification Language, Semantics (1999), http://www.informatik.uni-bremen.de/cofi/wiki/index.php/CASL
Coquand, T., Danielsson, N.A.: Isomorphism is equality. Draft (2013)
Haftmann, F., Wenzel, M.: Constructive type classes in Isabelle. In: Altenkirch, T., McBride, C. (eds.) TYPES 2006. LNCS, vol. 4502, pp. 160–174. Springer, Heidelberg (2007)
Harrison, J.: HOL Light: A tutorial introduction. In: Srivas, M., Camilleri, A. (eds.) FMCAD 1996. LNCS, vol. 1166, pp. 265–269. Springer, Heidelberg (1996)
Homeier, P.V.: A design structure for higher order quotients. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 130–146. Springer, Heidelberg (2005)
Huffman, B., Kuncar, O.: Lifting and transfer: A modular design for quotients in Isabelle/HOL. In: Isabelle Users Workshop (2012)
Huffman, B., Urban, C.: A new foundation for Nominal Isabelle. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 35–50. Springer, Heidelberg (2010)
Kaliszyk, C., Urban, C.: Quotients revisited for Isabelle/HOL. In: SAC, pp. 1639–1644 (2011)
Kammüller, F., Wenzel, M., Paulson, L.C.: Locales - A sectioning concept for Isabelle. In: Bertot, Y., Dowek, G., Hirschowitz, A., Paulin, C., Théry, L. (eds.) TPHOLs 1999. LNCS, vol. 1690, pp. 149–166. Springer, Heidelberg (1999)
Nadathur, G., Miller, D.: An overview of Lambda-Prolog. In: ICLP/SLP, pp. 810–827 (1988)
Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: A Proof Assistant for Higher-Order Logic. LNCS, vol. 2283. Springer, Heidelberg (2002)
Norrish, M.: Recursive function definition for types with binders. In: Slind, K., Bunker, A., Gopalakrishnan, G.C. (eds.) TPHOLs 2004. LNCS, vol. 3223, pp. 241–256. Springer, Heidelberg (2004)
Popescu, A., Gunter, E.L.: Recursion principles for syntax with bindings and substitution. In: ICFP, pp. 346–358 (2011)
Schropp, A.: Instantiating deeply embedded many-sorted theories into HOL types in Isabelle. Master’s thesis, Technische Universität München (2012), http://home.in.tum.de/~schropp/master-thesis.pdf
Schropp, A., Popescu, A.: Nonfree datatypes: metatheory, implementation and examples, http://bitbucket.org/isaspecops/nonfree-data/downloads/cpp2013_bundle.zip
Shulman, M., Licata, D., Lumsdaine, P.L., et al.: Higher inductive types on the homotopy type theory blog, http://homotopytypetheory.org/category/higher-inductive-types/
Breazu-Tannen, V., Subrahmanyam, R.: Logical and computational aspects of programming with sets/bags/lists. In: Leach Albert, J., Monien, B., Rodríguez-Artalejo, M. (eds.) ICALP 1991. LNCS, vol. 510, pp. 60–75. Springer, Heidelberg (1991)
Traytel, D., Popescu, A., Blanchette, J.C.: Foundational, compositional (co)datatypes for higher-order logic—Category theory applied to theorem proving. In: LICS 2012, pp. 596–605 (2012)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this paper
Cite this paper
Schropp, A., Popescu, A. (2013). Nonfree Datatypes in Isabelle/HOL. In: Gonthier, G., Norrish, M. (eds) Certified Programs and Proofs. CPP 2013. Lecture Notes in Computer Science, vol 8307. Springer, Cham. https://doi.org/10.1007/978-3-319-03545-1_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-03545-1_8
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-03544-4
Online ISBN: 978-3-319-03545-1
eBook Packages: Computer ScienceComputer Science (R0)