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Formalizing Monotone Algebras for Certification of Termination and Complexity Proofs

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8560)


Monotone algebras are frequently used to generate reduction orders in automated termination and complexity proofs. To be able to certify these proofs, we formalized several kinds of interpretations in the proof assistant Isabelle/HOL. We report on our integration of matrix interpretations, arctic interpretations, and nonlinear polynomial interpretations over various domains, including the reals.


  • Formal Proof
  • Proof Assistant
  • Termination Proof
  • Dependency Pair
  • Derivational Complexity

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Supported by the Austrian Science Fund (FWF) projects P22767 and J3202.

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  • DOI: 10.1007/978-3-319-08918-8_30
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  1. Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press (1998)

    Google Scholar 

  2. Ballarin, C.: Locales: A module system for mathematical theories. J. Autom. Reasoning 52(2), 123–153 (2014), doi:10.1007/s10817-013-9284-7

  3. Blanqui, F., Koprowski, A.: CoLoR: a Coq library on well-founded rewrite relations and its application to the automated verification of termination certificates. Math. Struct. Comp. Sci. 21(4), 827–859 (2011), , doi:10.1017/S0960129511000120

  4. Cichon, A., Lescanne, P.: Polynomial interpretations and the complexity of algorithms. In: Kapur, D. (ed.) CADE 1992. LNCS, vol. 607, pp. 139–147. Springer, Heidelberg (1992), , doi:10.1007/3-540-55602-8_161

  5. Cohen, C.: Construction of real algebraic numbers in coq. In: Beringer, L., Felty, A. (eds.) ITP 2012. LNCS, vol. 7406, pp. 67–82. Springer, Heidelberg (2012), , doi:10.1007/978-3-642-32347-8_6

  6. Contejean, E., Courtieu, P., Forest, J., Pons, O., Urbain, X.: Automated certified proofs with CIME3. In: Proc. 22nd RTA. LIPIcs, vol. 10, pp. 21–30. Schloss Dagstuhl (2011), , doi:10.4230/LIPIcs.RTA.2011.21

  7. Courtieu, P., Gbedo, G., Pons, O.: Improved matrix interpretation. In: van Leeuwen, J., Muscholl, A., Peleg, D., Pokorný, J., Rumpe, B. (eds.) SOFSEM 2010. LNCS, vol. 5901, pp. 283–295. Springer, Heidelberg (2010), , doi:10.1007/978-3-642-11266-9_24

  8. Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of term rewriting. J. Autom. Reasoning 40(2-3), 195–220 (2008), , doi:10.1007/s10817-007-9087-9

  9. Giesl, J., Thiemann, R., Schneider-Kamp, P.: The dependency pair framework: Combining techniques for automated termination proofs. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 301–331. Springer, Heidelberg (2005), , doi:10.1007/978-3-540-32275-7_21

  10. 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), , doi:10.1007/978-3-642-39634-2_10

  11. Haftmann, F., Nipkow, T.: Code generation via higher-order rewrite systems. In: Blume, M., Kobayashi, N., Vidal, G. (eds.) FLOPS 2010. LNCS, vol. 6009, pp. 103–117. Springer, Heidelberg (2010), , doi:10.1007/978-3-642-12251-4_9

  12. Hirokawa, N., Moser, G.: Automated complexity analysis based on the dependency pair method. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 364–379. Springer, Heidelberg (2008), , doi:10.1007/978-3-540-71070-7_32

  13. Hofbauer, D.: Termination Proofs and Derivation Lengths in Term Rewriting Systems. Dissertation, Technische Universität Berlin, Germany (1991), Available as Technical Report 92-46, TU Berlin (1992)

    Google Scholar 

  14. Hofbauer, D., Lautemann, C.: Termination proofs and the length of derivations. In: Dershowitz, N. (ed.) RTA 1989. LNCS, vol. 355, pp. 167–177. Springer, Heidelberg (1989), , doi:10.1007/3-540-51081-8_107

  15. Huffman, B., Kunčar, O.: Lifting and transfer: A modular design for quotients in isabelle/HOL. In: Gonthier, G., Norrish, M. (eds.) CPP 2013. LNCS, vol. 8307, pp. 131–146. Springer, Heidelberg (2013), , doi:10.1007/978-3-319-03545-1_9

  16. Koprowski, A., Waldmann, J.: Arctic termination … below zero. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 202–216. Springer, Heidelberg (2008), , doi:10.1007/978-3-540-70590-1_14

  17. Korp, M., Sternagel, C., Zankl, H., Middeldorp, A.: Tyrolean Termination Tool 2. In: Treinen, R. (ed.) RTA 2009. LNCS, vol. 5595, pp. 295–304. Springer, Heidelberg (2009), , doi:10.1007/978-3-642-02348-4_21

  18. Lankford, D.: On proving term rewriting systems are Noetherian. Technical Report MTP-3, Louisiana Technical University, Ruston, LA, USA (1979)

    Google Scholar 

  19. Lochbihler, A.: Light-weight containers for Isabelle: Efficient, extensible, nestable. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 116–132. Springer, Heidelberg (2013), , doi:10.1007/978-3-642-39634-2_11

  20. Lucas, S.: On the relative power of polynomials with real, rational, and integer coefficients in proofs of termination of rewriting. Appl. Algebr. Eng. Comm. 17(1), 49–73 (2006), , doi:10.1007/s00200-005-0189-5

  21. Lucas, S.: Practical use of polynomials over the reals in proofs of termination. In: Proc. 9th PPDP, pp. 39–50. ACM (2007), , doi:10.1145/1273920.1273927

  22. Moser, G., Schnabl, A., Waldmann, J.: Complexity analysis of term rewriting based on matrix and context dependent interpretations. In: Proc. 28th FSTTCS. LIPIcs, vol. 2, pp. 304–315. Schloss Dagstuhl (2008), , doi:10.4230/LIPIcs.FSTTCS.2008.1762

  23. Neurauter, F., Middeldorp, A.: On the domain and dimension hierarchy of matrix interpretations. In: Bjørner, N., Voronkov, A. (eds.) LPAR-18 2012. LNCS, vol. 7180, pp. 320–334. Springer, Heidelberg (2012), , doi:10.1007/978-3-642-28717-6_25

  24. Neurauter, F., Middeldorp, A., Zankl, H.: Monotonicity criteria for polynomial interpretations over the naturals. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS, vol. 6173, pp. 502–517. Springer, Heidelberg (2010), , doi:10.1007/978-3-642-14203-1_42

  25. Nipkow, T., Paulson, L.C., Wenzel, M.T.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002), , doi:10.1007/3-540-45949-9

  26. Porter, B.: Cauchy’s mean theorem and the Cauchy-Schwarz inequality. Archive of Formal Proofs (March 2006),

  27. Schnabl, A.: Derivational Complexity Analysis Revisited. PhD thesis, University of Innsbruck, Austria (2011)

    Google Scholar 

  28. Sternagel, C., Thiemann, R.: Certification extends termination techniques. In: Proc. 11th WST (2010), arXiv:1208.1594

    Google Scholar 

  29. Sternagel, C., Thiemann, R.: Executable matrix operations on matrices of arbitrary dimensions. Archive of Formal Proofs (June 2010),

  30. Sternagel, C., Thiemann, R.: Executable multivariate polynomials. Archive of Formal Proofs (August 2010),

  31. Sternagel, C., Thiemann, R.: Signature extensions preserve termination. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 514–528. Springer, Heidelberg (2010), , doi:10.1007/978-3-642-15205-4_39

  32. Thiemann, R.: Formalizing bounded increase. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 245–260. Springer, Heidelberg (2013), , doi:10.1007/978-3-642-39634-2_19

  33. Thiemann, R.: Implementing field extensions of the form \(\mathbb{Q}\sqrt{b}\). Archive of Formal Proofs (February 2014),

  34. Thiemann, R., Sternagel, C.: Certification of termination proofs using CeTA. In: Proc. 22nd TPHOLs. LNCS, vol. 5674, pp. 452–468. Springer, Heidelberg (2009),

  35. Zankl, H., Korp, M.: Modular complexity analysis via relative complexity. In: Proc. 21st RTA. LIPIcs, vol. 6, pp. 385–400. Schloss Dagstuhl (2010), , doi:10.4230/LIPIcs.RTA.2010.385

  36. Zankl, H., Middeldorp, A.: Satisfiability of non-linear (Ir)rational arithmetic. In: Clarke, E.M., Voronkov, A. (eds.) LPAR-16 2010. LNCS, vol. 6355, pp. 481–500. Springer, Heidelberg (2010), , doi:10.1007/978-3-642-17511-4_27

  37. Zankl, H., Thiemann, R., Middeldorp, A.: Satisfiability of non-linear arithmetic over algebraic numbers. In: Proc. SCSS. RISC-Linz Technical Report, vol. 10-10, pp. 19–24 (2010)

    Google Scholar 

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Sternagel, C., Thiemann, R. (2014). Formalizing Monotone Algebras for Certification of Termination and Complexity Proofs. In: Dowek, G. (eds) Rewriting and Typed Lambda Calculi. RTA TLCA 2014 2014. Lecture Notes in Computer Science, vol 8560. Springer, Cham.

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