Cardinalities of Finite Relations in Coq

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9807)

Abstract

We present an extension of a Coq library for relation algebras, where we provide support for cardinals in a point-free way. This makes it possible to reason purely algebraically, which is well-suited for mechanisation. We discuss several applications in the area of graph theory and program verification.

References

  1. 1.
    Berghammer, R., Höfner, P., Stucke, I.: Tool-based verification of a relational vertex coloring program. In: Kahl, W., Winter, M., Oliveira, J. (eds.) RAMiCS 2015. LNCS, vol. 9348, pp. 275–292. Springer, Heidelberg (2015). doi:10.1007/978-3-319-24704-5_17 Google Scholar
  2. 2.
    Berghammer, R., Stucke, I., Winter, M.: Investigating and computing bipartitions with algebraic means. In: Kahl, W., Winter, M., Oliveira, J. (eds.) RAMiCS 2015. LNCS, vol. 9348, pp. 257–274. Springer, Heidelberg (2015). doi:10.1007/978-3-319-24704-5_16 CrossRefGoogle Scholar
  3. 3.
    Brunet, P., Pous, D., Stucke, I.: Cardinalities of relations in Coq. Coq Development and full version of this extended abstract (2016). http://media.informatik.uni-kiel.de/cardinal/
  4. 4.
    Furusawa, H.: Algebraic formalisations of fuzzy relations and their representation theorems. Ph.D. thesis, Department of Informatics, Kyushu University (1998)Google Scholar
  5. 5.
    Galatos, N., Jipsen, P., Kowalski, T., Ono, H., Lattices, R.: An Algebraic Glimpse at Substructural Logics. Elsevier, Oxford (2007)MATHGoogle Scholar
  6. 6.
    Kahl, W.: Calculational relation-algebraic proofs in Isabelle/Isar. In: Berghammer, R., Möller, B., Struth, G. (eds.) RelMiCS 2003. LNCS, vol. 3051, pp. 178–190. Springer, Heidelberg (2004)Google Scholar
  7. 7.
    Kahl, W.: Dependently-typed formalisation of relation-algebraic abstractions. In: de Swart, H. (ed.) RAMICS 2011. LNCS, vol. 6663, pp. 230–247. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Kawahara, Y.: On the cardinality of relations. In: Schmidt, R.A. (ed.) RelMiCS/AKA 2006. LNCS, vol. 4136, pp. 251–265. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Pous, D.: Relation Algebra and KAT in Coq. http://perso.ens-lyon.fr/damien.pous/ra/
  10. 10.
    Pous, D.: Kleene algebra with tests and coq tools for while programs. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 180–196. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  11. 11.
    Schmidt, G., Ströhlein, T.: Relation algebras: concept of points and representability. Discrete Math. 54(1), 83–92 (1985)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Schmidt, G., Ströhlein, T.: Relations and Graphs - Discrete Mathematics for Computer Scientists. EATCS Monographs on Theoretical Computer Science. Springer, Berlin (1993)MATHGoogle Scholar
  13. 13.
    Sozeau, M.: A new look at generalized rewriting in type theory. J. Formalized Reason. 2(1), 41–62 (2009)MathSciNetMATHGoogle Scholar
  14. 14.
    Tarski, A.: On the calculus of relations. J. Symbolic Log. 6(3), 73–89 (1941)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Tarski, A., Givant, S.: A Formalization of Set Theory without Variables, vol. 41. Colloquium Publications, AMS, Providence, Rhode Island (1987)MATHGoogle Scholar
  16. 16.
    Wei, V.: A lower bound for the stability number of a simple graph. Bell Laboratories Technical Memorandum 81–11217-9 (1981)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Univ Lyon, CNRS, ENS de Lyon, UCB Lyon 1, LIPLyonFrance
  2. 2.Institut für InformatikChristian-Albrechts-Universität zu KielKielGermany

Personalised recommendations