Simple Type Theory is not too Simple: Grothendieck's Schemes without Dependent Types
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.
Cite
@article{arxiv.2104.09366,
title = {Simple Type Theory is not too Simple: Grothendieck's Schemes without Dependent Types},
author = {Anthony Bordg and Lawrence Paulson and Wenda Li},
journal= {arXiv preprint arXiv:2104.09366},
year = {2022}
}
Comments
Final version accepted for publication in the journal Experimental Mathematics. Our code can be found in the Archive of Formal Proofs, see https://www.isa-afp.org/entries/Grothendieck_Schemes.html