constructive mathematics, realizability, computability propositions as types, proofs as programs, computational trinitarianism topos, homotopy topos type theory, homotopy type theory canonical form, univalence Bishop set, h-set decidable equality, decidable subset, inhabited set, subsingleton A proof assistant or proof management system is a kind of software designed to help with proofs in formalized mathematics. Many proof assistants resemble and/or include a programming language. There are arguably two threads of current development in proof systems, which may be called “foundational” and “coverage”. The “foundational” work tries to find the best foundational theory to formalize mathematics (see also at foundations of mathematics). Out of that work first came dependent types (Automath, in the late 60s), then the calculus of constructions (early Rocq), and the calculus of inductive constructions (current Rocq). More recently a new wave of such work is being done in homotopy type theory as another step in this direction. Rocq’s library is not that large, except in the area of group theory where the results of the work on Feit-Thompson theorem has produced something larger. The “coverage” work tries to formalize as much as possible of mathematics in existing theories. For instance, for decades people have been building Mizar‘s library (Mizar is based on Tarski–Grothendieck set theory rather than type theory). Its library is a couple of orders of magnitude larger than anyone else’s. On the other hand, despite this quantity, it remains an issue to attack problems of contemporary research interest in these systems, see also at Mizar – problem of pertinence. Similar to Mizar is NuPRL, HOL light and Isabelle, which all have decently sized libraries. (Isabelle can be used with either material set theory, like Mizar, or higher-order type theory, like the others.) based on plain type theory/set theory: based on dependent type theory/homotopy type theory: based on cubical...