Simple Type Theory is not too Simple: Grothendieck's Schemes without Dependent Types

TL;DR

This paper demonstrates that complex algebraic geometry objects like schemes can be formalized within simple type theory, challenging assumptions about the theory's limitations.

Contribution

It presents a successful formalization of schemes in Isabelle/HOL's simple type theory, replacing dependent types with locales.

Findings

Schemes can be formalized in simple type theory.

Locales can replace dependent types for complex structures.

Formalization was achieved in Isabelle/HOL.

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.