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

arXiv: https://arxiv.org/abs/2104.09366 · HTML: https://mlsanity.com/paper/2104.09366/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Apr 2021.

## Built on by (most cited first, 2 of 2)

- [Data for Mathematical Copilots: Better Ways of Presenting Proofs for Machine Learning](https://mlsanity.com/paper/2412.15184/) (2024)
- [Munkres' General Topology Autoformalized in Isabelle/HOL](https://mlsanity.com/paper/2604.07455/) (2026)

## Abstract (arXiv)

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.