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.

中文翻译：

---

单项式的可接受排序是有充分根据的：一个建设性的证明

摘要

在本文中，我们考虑多元多项式的范式（NF）算法终止的构造性证明，以及单项式上允许排序 < \(_{e}\)的相关概念。在经典数学中，关系 < \(_{e}\)的良好准序性质是从迪克森引理导出的，这足以证明 NF 算法的终止是合理的。在基于构造性类型理论（Coq和Agda）的可证明编程中，需要对排序（在其构造性版本中）的充分根据有一个更强的条件（在构造性数学中）。我们针对 < \(_{e}\)提出该定理 (T) 的建设性证明，它基于我们在此称为“模式方法”的已知方法。这个关于任意可接受排序的有充分根据的定理本身也很重要，与 NF 算法无关。我们不知道有任何其他关于该定理的构造性证明的著作。然而，事实证明，从其他研究人员在 2003 年取得的成果中得出结论并不困难。我们以 Agda 语言以我们的库AdmissiblePPO-wellFounded的形式对该证明进行编程可证明的计算代数程序。这一发展还使用该定理来证明多项式 NF 算法的终止。因此，该库还包含一组可证明的多项式代数程序，其数量明显大于证明定理 T 所需的程序。