Skip to main content

A Verified Decision Procedure for Orders in Isabelle/HOL

  • 242 Accesses

Part of the Lecture Notes in Computer Science book series (LNPSE,volume 12971)

Abstract

We present the first verified implementation of a decision procedure for the quantifier-free theory of partial and linear orders. We formalise the procedure in Isabelle/HOL and provide a specification that is made executable using Isabelle’s code generator. The procedure is already part of the development version of Isabelle as a sub-procedure of the simplifier.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-88885-5_9
  • Chapter length: 17 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   59.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-88885-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   79.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.

Notes

  1. 1.

    File path of the procedure in the Isabelle2021 distribution: src/Provers/order.ML.

  2. 2.

    Introduced in commit https://isabelle-dev.sketis.net/rISABELLEa3cc9fa129.

References

  1. Berghofer, S., Nipkow, T.: Proof terms for simply typed higher order logic. In: Aagaard, M., Harrison, J. (eds.) TPHOLs 2000. LNCS, vol. 1869, pp. 38–52. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-44659-1_3

    CrossRef  MATH  Google Scholar 

  2. Bossert, W., Suzumura, K.: Consistency, Choice, and Rationality. Harvard University Press, Cambridge (2010)

    CrossRef  Google Scholar 

  3. Ehrenfeucht, A.: Decidability of the theory of linear order. Not. Am. Math. Soc. 6, 268–269 (1959)

    Google Scholar 

  4. Haftmann, F., Krauss, A., Kunčar, O., Nipkow, T.: Data refinement in Isabelle/HOL. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 100–115. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39634-2_10

    CrossRef  Google Scholar 

  5. Janiczak, A.: Undecidability of some simple formalized theories. Fundam. Math. 40, 131–139 (1953)

    MathSciNet  CrossRef  Google Scholar 

  6. Kreisel, G.: Review of “Undecidability of some simple formalized theories’’. Math. Rev. 15, 669–670 (1954)

    Google Scholar 

  7. Läuchli, H., Leonard, J.: On the elementary theory of linear order. Fundam. Math. 59, 109–116 (1966)

    MathSciNet  CrossRef  Google Scholar 

  8. Negri, S., Von Plato, J., Coquand, T.: Proof-theoretical analysis of order relations. Arch. Math. Logic 43(3), 297–309 (2004)

    MathSciNet  CrossRef  Google Scholar 

  9. Nipkow, T.: Linear quantifier elimination. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 18–33. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-71070-7_3

    CrossRef  Google Scholar 

  10. Nipkow, T., Wenzel, M., Paulson, L.C.: Isabelle/HOL–A Proof Assistant for Higher-Order Logic. LNCS, vol. 2283. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45949-9

    CrossRef  MATH  Google Scholar 

  11. Nipkow, T., Roßkopf, S.: Isabelle’s metalogic: Formalization and proof checker (2021). https://arxiv.org/abs/2104.12224

  12. Stevens, L., Nipkow, T.: A verified decision procedure for orders. https://www21.in.tum.de/team/stevensl/assets/atva-2021-artifact.zip. Formal proof development

  13. Szpilrajn, E.: Sur l’extension de l’ordre partiel. Fundam. Math. 1(16), 386–389 (1930)

    CrossRef  Google Scholar 

  14. Zeller, P., Stevens, L.: Order extension and Szpilrajn’s theorem. Archive of Formal Proofs (2021). https://devel.isa-afp.org/entries/Szpilrajn.html. Formal proof development

Download references

Acknowledgements

We thank Kevin Kappelmann and the anonymous reviewers for their comments.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Lukas Stevens .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Stevens, L., Nipkow, T. (2021). A Verified Decision Procedure for Orders in Isabelle/HOL. In: Hou, Z., Ganesh, V. (eds) Automated Technology for Verification and Analysis. ATVA 2021. Lecture Notes in Computer Science(), vol 12971. Springer, Cham. https://doi.org/10.1007/978-3-030-88885-5_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-88885-5_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-88884-8

  • Online ISBN: 978-3-030-88885-5

  • eBook Packages: Computer ScienceComputer Science (R0)