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.

摘要

Church 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，提出了在这种情况下是否可以正式确定计划的疑问，并提出了质疑。方案是 Alexander Grothendieck 在 1960 年引入的代数几何中复杂的数学对象。在本文中，我们报告了证明助手 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. 我们在特定的方案案例中展示了如何将 Coq 或 Lean 的强大依赖类型转换为称为 locales 的极简设备。