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

Abstract

We report on a formalization of schemes in the proof assistant Isabelle/HOL, and we discuss the design choices made in the process. Schemes are sophisticated mathematical objects in algebraic geometry introduced by Alexander Grothendieck in 1960. This experiment shows that the simple type theory implemented in Isabelle can handle such elaborate constructions despite doubts raised about Isabelle's capability in that direction. 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.

A Formalised Theorem in the Partition Calculus

“…Thus, types can take other types as parameters, but they can't take other values as parameters: there are no "dependent types". My colleagues and I are pursuing the thesis that simple type theory is not merely sufficient to formalise mathematics [32] but superior to strong type theories that make automation difficult and introduce complications such as intensional equality.…”