Formalisation of Constructive Univalent Type Theory

Introduction

There has been in the past 15 years remarkable achievements in the field of interactive theorem proving, both for checking complex software and checking non-trivial mathematical proofs.

Achievements in Software Correctness

For software correctness, X. Leroy (INRIA and College de France) has been leading since 2006 the CompCert project, which features a fully verified C compiler.

Achievements in Mathematical Proofs

For mathematical proofs, these systems could handle complex arguments, such as:

• The proof of the 4 color theorem
• The formal proof of the Feit-Thompson Theorem

More recently, the Xena project, led by K. Buzzard, is developing a large library of mathematical facts and has been able to help the mathematician P. Scholze (field medalist 2018) to check a highly non-trivial proof.

Theoretical Foundations

All these examples have been carried out in systems based on the formalism of dependent type theory and on early work of the PI. In parallel to these works, also around 15 years ago, a remarkable and unexpected correspondence was discovered between this formalism and the abstract study of homotopy theory and higher categorical structures.

Special Year and Research Directions

A special year 2012-2013 at the Institute of Advanced Study (Princeton) was organized by the late V. Voevodsky (field medalist 2002, Princeton), S. Awodey (CMU), and the PI. Preliminary results indicate that this research direction is productive, both for the understanding of dependent type systems and higher category theory, and suggest several crucial open questions.

Objectives of the Proposal

The objective of this proposal is to analyze these questions, with the ultimate goal of formulating a new way to look at mathematical objects and potentially a new foundation of mathematics. This could, in turn, be crucial for the design of future proof systems able to handle complex highly modular software systems and mathematical proofs.