constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
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 Coq), and the calculus of inductive constructions (current Coq). More recently a new wave of such work is being done in homotopy type theory as another step in this direction. Coq’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 type theory:
based on modal type theory:
For monoidal category theory:
projects for formalization of mathematics with proof assistants:
Archive of Formal Proofs (using Isabelle)
ForMath project (using Coq)
UniMath project (using Coq and Agda)
Xena project (using Lean)
Other proof assistants
Historical projects that died out:
Wikipedia, Proof assistant
Oscar Lanford III., Computer assisted proofs in analysis, Proceedings of the International Congress of Mathematicians, 1986 (pdf)
Freek Wiedijk, Digital math by alphabet (web)
Carlos Simpson, Verification and creation of proofs by computer
Jeremy Avigad, Interactive Theorem Proving,
Automated Reasoning, and Mathematical Computation_, 2012 pdf slides
Conference Series on Intelligent Computer Mathematics
(2014)
Conference series on Interactive theorem proving
(2014)
Parts of the above text are taken from this MO comment by Jacques Carette.
Proof assistants specifically for homotopy type theory:
See also
on formal proof and proof assistants in undergraduate mathematics.
Last revised on June 3, 2021 at 12:13:53. See the history of this page for a list of all contributions to it.