## An Inductive Version of Nash-Williams’ Minimal-Bad-Sequence Argument for Higman’s Lemma (2001)

Venue: | IN P. CALLAGHAN, E.AL., TYPES FOR PROOFS AND PROGRAMS, LECTURE NOTES IN COMPUTER SCIENCE 2277 |

Citations: | 3 - 1 self |

### BibTeX

@INPROCEEDINGS{Seisenberger01aninductive,

author = {Monika Seisenberger},

title = {An Inductive Version of Nash-Williams’ Minimal-Bad-Sequence Argument for Higman’s Lemma},

booktitle = {IN P. CALLAGHAN, E.AL., TYPES FOR PROOFS AND PROGRAMS, LECTURE NOTES IN COMPUTER SCIENCE 2277},

year = {2001},

pages = {233--242},

publisher = {Springer}

}

### OpenURL

### Abstract

Higman’s lemma has a very elegant, non-constructive proof due to Nash-Williams [NW63] using the so-called minimal-bad-sequence argument. The objective of the present paper is to give a proof that uses the same combinatorial idea, but is constructive. For a two letter alphabet this was done by Coquand and Fridlender [CF94]. Here we present a proof in a theory of inductive definitions that works for arbitrary decidable well quasiorders.

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Citation Context ...and and Fridlender [CF94]. Here we present a proof in a theory of inductive definitions that works for arbitrary decidable well quasiorders. 1 Introduction This paper is concerned with Higman’s lemma =-=[Hig52]-=-, usually formulated in terms of well quasi orders. If (A, ≤A) is a well quasiorder, then so is the set A ∗ of finite sequences in A, together with the embeddability relation ≤A ∗, where a sequence [a... |

64 |
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Citation Context ...tisches Institut der Universität München ⋆ ⋆ ⋆ 2 Department of Computer Science, University of Wales Swansea † Abstract. Higman’s lemma has a very elegant, non-constructive proof due to Nash-Williams =-=[NW63]-=- using the so-called minimal-bad-sequence argument. The objective of the present paper is to give a proof that uses the same combinatorial idea, but is constructive. For a two letter alphabet this was... |

52 |
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Citation Context ... of Nash-Williams using the so-called minimal-bad-sequence argument is considered most elegant. A variant of this proof was translated by Murthy via Friedman’s A-translation into a constructive proof =-=[Mur91]-=-, however resulting in a huge proof whose computational content couldn’t yet be discovered. More direct constructive proofs were given by Schütte/Simpson [SS85], Murthy/Russell [MR90], and Richman/Sto... |

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Citation Context ...nstance Kruskal’s tree theorem and the so-called extended Kruskal theorem, also known as Kruskal’s theorem with gap condition. Both have proofs using a minimal-bad-sequence argument(see [NW63] resp. =-=[Sim85]-=-), however no constructive proof at all is known for the latter. Kruskal’s theorem was proved constructively (see [RW93] for a proof using ordinal notations or [Sei01] for an inductive reformulation o... |

23 | Proof-theoretic investigations on Kruskal’s theorem
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Citation Context ...tion. Both have proofs using a minimal-bad-sequence argument(see [NW63] resp. [Sim85]), however no constructive proof at all is known for the latter. Kruskal’s theorem was proved constructively (see =-=[RW93]-=- for a proof using ordinal notations or [Sei01] for an inductive reformulation of this proof, and [Vel00] for a proof not requiring decidability). These proofs, however, are quite involved in comparis... |

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Citation Context ...structive proof [Mur91], however resulting in a huge proof whose computational content couldn’t yet be discovered. More direct constructive proofs were given by Schütte/Simpson [SS85], Murthy/Russell =-=[MR90]-=-, and Richman/Stolzenberg [RS93]. The Schütte/Simpson proof uses ordinal notations up to ɛ0 and is related to an earlier proof by Schmidt [Sch79], the other proofs are carried out in a (proof theoreti... |

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Citation Context ...-translation into a constructive proof [Mur91], however resulting in a huge proof whose computational content couldn’t yet be discovered. More direct constructive proofs were given by Schütte/Simpson =-=[SS85]-=-, Murthy/Russell [MR90], and Richman/Stolzenberg [RS93]. The Schütte/Simpson proof uses ordinal notations up to ɛ0 and is related to an earlier proof by Schmidt [Sch79], the other proofs are carried o... |

10 | On the logical strength of Nash-Williams’ theorem on transfinite sequences
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Citation Context ...G Graduiertenkolleg “Logik in der Informatik” † Research supported by the British EPSRCwas shown by Marcone, however it is open whether the special form used for Higman’s lemma has the same strength =-=[Mar96]-=-.) The objective of this paper is to present a constructive proof that captures the combinatorial idea behind Nash-Williams’ proof. For an alphabet A consisting of two letters this was done by Coquand... |

9 | A proof of Higman's lemma by structural induction
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(Show Context)
Citation Context ...-sequence argument. The objective of the present paper is to give a proof that uses the same combinatorial idea, but is constructive. For a two letter alphabet this was done by Coquand and Fridlender =-=[CF94]-=-. Here we present a proof in a theory of inductive definitions that works for arbitrary decidable well quasiorders. 1 Introduction This paper is concerned with Higman’s lemma [Hig52], usually formulat... |

8 | An intuitionistic proof of Kruskal’s theorem
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Citation Context ...a which in contrast to all proofs mentioned above does not require decidability of the given relation ≤A was given by Fridlender [Fri97]. His proof is based on a proof by Veldman that can be found in =-=[Vel00]-=-. In our formulation of Higman’s lemma we will also use an accessibility notion, as it was done in Fridlender’s proof. 2 Basic Definitions and an Inductive Characterization of Well Quasiorders In the ... |

5 | Higman’s lemma in type theory
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(Show Context)
Citation Context ...s necessary, as we will describe in section 3. A proof of Higman’s lemma which in contrast to all proofs mentioned above does not require decidability of the given relation ≤A was given by Fridlender =-=[Fri97]-=-. His proof is based on a proof by Veldman that can be found in [Vel00]. In our formulation of Higman’s lemma we will also use an accessibility notion, as it was done in Fridlender’s proof. 2 Basic De... |

3 |
Kruskal’s tree theorem in a constructive theory of inductive definitions
- Seisenberger
- 2001
(Show Context)
Citation Context ...uence argument(see [NW63] resp. [Sim85]), however no constructive proof at all is known for the latter. Kruskal’s theorem was proved constructively (see [RW93] for a proof using ordinal notations or =-=[Sei01]-=- for an inductive reformulation of this proof, and [Vel00] for a proof not requiring decidability). These proofs, however, are quite involved in comparison with the minimal-bad-sequence proof. We do n... |

2 |
Well–orderings and their maximal order types
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(Show Context)
Citation Context ...fs were given by Schütte/Simpson [SS85], Murthy/Russell [MR90], and Richman/Stolzenberg [RS93]. The Schütte/Simpson proof uses ordinal notations up to ɛ0 and is related to an earlier proof by Schmidt =-=[Sch79]-=-, the other proofs are carried out in a (proof theoretically stronger) theory of inductive definitions. However, their computational content is essentially the same, but does not correspond to that on... |