Simple Type Theory is not too Simple: Grothendieck’s Schemes Without Dependent Types

IF 0.7 4区数学 Q2 MATHEMATICS

Experimental Mathematics Pub Date : 2021-04-19 DOI:10.1080/10586458.2022.2062073

Anthony Bordg, Lawrence Charles Paulson, Wenda Li

引用次数: 11

Abstract

Abstract Church’s simple type theory is often deemed too simple for elaborate mathematical constructions. In particular, doubts were raised whether schemes could be formalized in this setting and a challenge was issued. Schemes are sophisticated mathematical objects in algebraic geometry introduced by Alexander Grothendieck in 1960. In this article we report on a successful formalization of schemes in the simple type theory of the proof assistant Isabelle/HOL, and we discuss the design choices which make this work possible. We show in the particular case of schemes how the powerful dependent types of Coq or Lean can be traded for a minimalist apparatus called locales.

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简单类型理论不是太简单:没有依赖类型的Grothendieck方案

丘奇的简单类型理论通常被认为过于简单，无法进行复杂的数学构造。特别是，有人怀疑在这种情况下能否正式确定各项计划，并提出了一项挑战。方案是代数几何中复杂的数学对象，由Alexander Grothendieck于1960年引入。本文报道了证明助手Isabelle/HOL的简单类型理论中方案的成功形式化，并讨论了使这项工作成为可能的设计选择。在方案的特殊情况下，我们展示了如何将Coq或Lean的强大依赖类型交换为称为locale的极简设备。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Experimental Mathematics 数学-数学

CiteScore

1.70

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Experimental Mathematics publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses. Experiment has always been, and increasingly is, an important method of mathematical discovery. (Gauss declared that his way of arriving at mathematical truths was "through systematic experimentation.") Yet this tends to be concealed by the tradition of presenting only elegant, fully developed, and rigorous results. Experimental Mathematics was founded in the belief that theory and experiment feed on each other, and that the mathematical community stands to benefit from a more complete exposure to the experimental process. The early sharing of insights increases the possibility that they will lead to theorems: An interesting conjecture is often formulated by a researcher who lacks the techniques to formalize a proof, while those who have the techniques at their fingertips have been looking elsewhere. Even when the person who had the initial insight goes on to find a proof, a discussion of the heuristic process can be of help, or at least of interest, to other researchers. There is value not only in the discovery itself, but also in the road that leads to it.