A Transfinite Knuth–Bendix Order for Lambda-Free Higher-Order Terms

  • Heiko Becker
  • Jasmin Christian BlanchetteEmail author
  • Uwe Waldmann
  • Daniel Wand
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10395)


We generalize the Knuth–Bendix order (KBO) to higher-order terms without \(\lambda \)-abstraction. The restriction of this new order to first-order terms coincides with the traditional KBO. The order has many useful properties, including transitivity, the subterm property, compatibility with contexts (monotonicity), stability under substitution, and well-foundedness. Transfinite weights and argument coefficients can also be supported. The order appears promising as the basis of a higher-order superposition calculus.



We are grateful to Stephan Merz, Tobias Nipkow, and Christoph Weidenbach for making this research possible; to Mathias Fleury and Dmitriy Traytel for helping us formalize the syntactic ordinals; to Andrei Popescu and Christian Sternagel for advice with extending a partial well-founded order to a total one in the mechanized proof of Lemma 3; to Andrei Voronkov for the enlightening discussion about KBO at the IJCAR 2016 banquet; and to Carsten Fuhs, Mark Summerfield, and the anonymous reviewers for suggesting many textual improvements.

Blanchette has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 713999, Matryoshka). Wand is supported by the Deutsche Forschungsgemeinschaft (DFG) grant Hardening the Hammer (NI 491/14-1).


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Heiko Becker
    • 1
  • Jasmin Christian Blanchette
    • 2
    • 3
    • 4
    Email author
  • Uwe Waldmann
    • 4
  • Daniel Wand
    • 4
    • 5
  1. 1.Max-Planck-Institut für SoftwaresystemeSaarbrückenGermany
  2. 2.Vrije Universiteit AmsterdamAmsterdamThe Netherlands
  3. 3.Inria Nancy – Grand EstVillers-lès-NancyFrance
  4. 4.Max-Planck-Institut für InformatikSaarbrückenGermany
  5. 5.Technische Universität MünchenMunichGermany

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