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A General Constructive Form of Higman’s Lemma

Authors Stefano Berardi , Gabriele Buriola, Peter Schuster

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Subject Classification

ACM Subject Classification
  • Theory of computation → Constructive mathematics
  • Theory of computation → Proof theory
  • Mathematics of computing → Discrete mathematics
Keywords
  • intuitionistic logic
  • constructive mathematics
  • formal proof
  • inductive predicate
  • bar induction
  • well quasi-order
  • Higman’s lemma

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Abstract

In logic and computer science one often needs to constructivize a theorem ∀ f ∃ g.P(f,g), stating that for every infinite sequence f there is an infinite sequence g such that P(f,g). Here P is a computable predicate but g is not necessarily computable from f. In this paper we propose the following constructive version of ∀ f ∃ g.P(f,g): for every f there is a "long enough" finite prefix g₀ of g such that P(f,g₀), where "long enough" is expressed by membership to a bar which is a free parameter of the constructive version.
Our approach with bars generalises the approaches to Higman’s lemma undertaken by Coquand-Fridlender, Murthy-Russell and Schwichtenberg-Seisenberger-Wiesnet. As a first test for our bar technique, we sketch a constructive theory of well-quasi orders. This includes yet another constructive version of Higman’s lemma: that every infinite sequence of words has an infinite ascending subsequence. As compared with the previous constructive versions of Higman’s lemma, our constructive proofs are closer to the original classical proofs.

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Stefano Berardi, Gabriele Buriola, and Peter Schuster. A General Constructive Form of Higman’s Lemma. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.CSL.2024.16

Author Details

Stefano Berardi
  • Dipartimento di Informatica, Università di Torino, Italy
Gabriele Buriola
  • Dipartimento di Informatica, Università di Verona, Italy
Peter Schuster
  • Dipartimento di Informatica, Università di Verona, Italy

Acknowledgements

The present study was started while the third author worked within the project "Reducing complexity in algebra, logic, combinatorics - REDCOM" belonging to the programme "Ricerca Scientifica di Eccellenza 2018" of the Fondazione Cariverona. The second and the third author are members of the "Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni" (GNSAGA) of the Istituto Nazionale di Alta Matematica (INdAM). The authors wish to thank Daniel Fridlender for his interest and suggestions, and to express their gratitude to Thierry Coquand, whose original idea set the basis for this work. Last but not least, the anonymous reviewers' meticulous reading and constructive critique have been extremely helpful.

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