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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
File path of the procedure in the Isabelle2021 distribution: src/Provers/order.ML.
- 2.
Introduced in commit https://isabelle-dev.sketis.net/rISABELLEa3cc9fa129.
References
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
Bossert, W., Suzumura, K.: Consistency, Choice, and Rationality. Harvard University Press, Cambridge (2010)
Ehrenfeucht, A.: Decidability of the theory of linear order. Not. Am. Math. Soc. 6, 268–269 (1959)
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
Janiczak, A.: Undecidability of some simple formalized theories. Fundam. Math. 40, 131–139 (1953)
Kreisel, G.: Review of “Undecidability of some simple formalized theories’’. Math. Rev. 15, 669–670 (1954)
Läuchli, H., Leonard, J.: On the elementary theory of linear order. Fundam. Math. 59, 109–116 (1966)
Negri, S., Von Plato, J., Coquand, T.: Proof-theoretical analysis of order relations. Arch. Math. Logic 43(3), 297–309 (2004)
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
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
Nipkow, T., Roßkopf, S.: Isabelle’s metalogic: Formalization and proof checker (2021). https://arxiv.org/abs/2104.12224
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
Szpilrajn, E.: Sur l’extension de l’ordre partiel. Fundam. Math. 1(16), 386–389 (1930)
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
Acknowledgements
We thank Kevin Kappelmann and the anonymous reviewers for their comments.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
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)