# Admissible Ordering on Monomials is Well-Founded: A Constructive Proof

### Abstract

In this paper, we consider a constructive proof of the termination of the normal form (NF) algorithm for multivariate polynomials, as well as the related concept of admissible ordering <$$_{e}$$ on monomials. In classical mathematics, the well-quasiorder property of relation <$$_{e}$$ is derived from Dickson’s lemma, and this is sufficient to justify the termination of the NF algorithm. In provable programming based on constructive type theory (Coq and Agda), a somewhat stronger condition (in constructive mathematics) of the well-foundedness of the ordering (in its constructive version) is required. We propose a constructive proof of this theorem (T) for <$$_{e}$$, which is based on a known method that we refer to here as the “pattern method.” This theorem on the well-foundedness of an arbitrary admissible ordering is also important in itself, independently of the NF algorithm. We are not aware of any other works on constructive proof of this theorem. However, it turns out that it follows, not very difficultly, from the results achieved by other researchers in 2003. We program this proof in the Agda language in the form of our library AdmissiblePPO-wellFounded of provable computational algebra programs. This development also uses the theorem to prove termination of the NF algorithm for polynomials. Thus, the library also contains a set of provable programs for polynomial algebra, which is significantly larger than that needed to prove Theorem T.

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## ACKNOWLEDGMENTS

• The author is grateful to Antonina N. Nepejvoda and Andrei P. Nemytykh for their useful remarks.

• The author thanks the participants of the Computer Algebra seminar lead by S. A. Abramov (Faculty of Computational Mathematics and Cybernetics of the Moscow State University) for the discussion on the subject of this paper.

## Funding

This work was supported by the Ministry of Science and Higher Education of the Russian Federation, project no. 122012700089-0.

## Author information

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Correspondence to S. D. Meshveliani.

## Ethics declarations

The author declares that he has no conflicts of interest.

Translated by Yu. Kornienko

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Meshveliani, S.D. Admissible Ordering on Monomials is Well-Founded: A Constructive Proof. Program Comput Soft 49, 199–214 (2023). https://doi.org/10.1134/S0361768823040102

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• DOI: https://doi.org/10.1134/S0361768823040102