Higman’s Lemma and Its Computational Content

  • Helmut SchwichtenbergEmail author
  • Monika Seisenberger
  • Franziskus Wiesnet
Part of the Progress in Computer Science and Applied Logic book series (PCS, volume 28)


Higman’s Lemma is a fascinating result in infinite combinatorics, with manyfold applications in logic and computer science. It has been proven several times using different formulations and methods. The aim of this paper is to look at Higman’s Lemma from a computational and comparative point of view. We give a proof of Higman’s Lemma that uses the same combinatorial idea as Nash-Williams’ indirect proof using the so-called minimal bad sequence argument, but which is constructive. For the case of a two letter alphabet such a proof was given by Coquand. Using more flexible structures, we present a proof that works for an arbitrary well-quasiordered alphabet. We report on a formalization of this proof in the proof assistant Minlog, and discuss machine extracted terms (in an extension of Gödel’s system T) expressing its computational content.


Higman’s Lemma Inductive definitions Minimal bad sequence Program extraction 



We would like to thank Daniel Fridlender and Iosif Petrakis for helpful contributions and comments.


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Helmut Schwichtenberg
    • 1
    Email author
  • Monika Seisenberger
    • 2
  • Franziskus Wiesnet
    • 1
  1. 1.Mathematisches InstitutLMUMunichGermany
  2. 2.Department of Computer ScienceSwansea UniversitySwansea SA28PPUK

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