# Proof assistant

In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human–machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer.
A recent effort within this field is making these tools use artificial intelligence to automate the formalization of ordinary mathematics.

## System comparison

| Name | Latest version | Developer(s) | Implementation language | Features | Features | Features | Features | Features | Features |
| Name | Latest version | Developer(s) | Implementation language | Higher-order logic | Dependent types | Small kernel | Proof automation | Proof by reflection | Code generation |
| ACL2 | 8.3 | Matt Kaufmann and J Strother Moore | Common Lisp | No | Untyped | No | Yes | Yes | Already executable |
| Agda | 2.6.4.3 | Ulf Norell, Nils Anders Danielsson, and Andreas Abel (Chalmers and Gothenburg) | Haskell | Yes | Yes | Yes | No | Partial | Already executable |
| Albatross | 0.4 | Helmut Brandl | OCaml | Yes | No | Yes | Yes | Un­known | Not yet Implemented |
| Rocq (formerly known as Coq) | 9.0 | INRIA | OCaml | Yes | Yes | Yes | Yes | Yes | Yes |
| F* | repository | Microsoft Research and INRIA | F* | Yes | Yes | No | Yes | Yes | Yes |
| HOL Light | repository | John Harrison | OCaml | Yes | No | Yes | Yes | No | No |
| HOL4 | Kananaskis-13 (or repo) | Michael Norrish, Konrad Slind, and others | Standard ML | Yes | No | Yes | Yes | No | Yes |
| Idris | 2 0.6.0. | Edwin Brady | Idris | Yes | Yes | Yes | Un­known | Partial | Yes |
| Isabelle | Isabelle2024 (May 2024) | Larry Paulson (Cambridge), Tobias Nipkow (München) and Makarius Wenzel | Standard ML, Scala | Yes | No | Yes | Yes | Yes | Yes |
| Lean | v4.7.0 | Leonardo de Moura (Microsoft Research) | C++, Lean | Yes | Yes | Yes | Yes | Yes | Yes |
| LEGO | 1.3.1 | Randy Pollack (Edinburgh) | Standard ML | Yes | Yes | Yes | No | No | No |
| Metamath | v0.198 | Norman Megill | ANSI C | | | | | | |
| Mizar | 8.1.11 | Białystok University | Free Pascal | Partial | Yes | No | No | No | No |
| Nqthm | | | | | | | | | |
| NuPRL | 5 | Cornell University | Common Lisp | Yes | Yes | Yes | Yes | Un­known | Yes |
| PVS | 6.0 | SRI International | Common Lisp | Yes | Yes | No | Yes | No | Un­known |
| Twelf | 1.7.1 | Frank Pfenning and Carsten Schürmann | Standard ML | Yes | Yes | Un­known | No | No | Un­known |
* ACL2 – a programming language, a first-order logical theory, and a theorem prover (with both interactive and automatic modes) in the Boyer–Moore tradition.
* Rocq (formerly known as Coq) – Allows the expression of mathematical assertions, mechanically checks proofs of these assertions, helps to find formal proofs, and extracts a certified program from the constructive proof of its formal specification.
* HOL theorem provers – A family of tools ultimately derived from the LCF theorem prover. In these systems the logical core is a library of their programming language. Theorems represent new elements of the language and can only be introduced via "strategies" which guarantee logical correctness. Strategy composition gives users the ability to produce significant proofs with relatively few interactions with the system. Members of the family include:
** HOL4 – The "primary descendant", still under active development. Support for both Moscow ML and Poly/ML. Has a BSD-style license.
** HOL Light – A thriving "minimalist fork". OCaml based.
** ProofPower – Went proprietary, then returned to open source. Based on Standard ML.
* IMPS, An Interactive Mathematical Proof System.
* Isabelle is an interactive theorem prover, successor of HOL. The main code-base is BSD-licensed, but the Isabelle distribution bundles many add-on tools with different licenses.
* Jape – Java based.
* Lean
* LEGO
* Matita – A light system based on the Calculus of Inductive Constructions.
* MINLOG – A proof assistant based on first-order minimal logic.
* Mizar – A proof assistant based on first-order logic, in a natural deduction style, and Tarski–Grothendieck set theory.
* PhoX – A proof assistant based on higher-order logic which is eXtensible.
* Prototype Verification System (PVS) – a proof language and system based on higher-order logic.
* TPS and ETPS – Interactive theorem provers also based on simply-typed lambda calculus, but based on an independent formulation of the logical theory and independent implementation.

## User interfaces

A popular front-end for proof assistants is the Emacs-based Proof General, developed at the University of Edinburgh.
Coq includes CoqIDE, which is based on OCaml/Gtk. Isabelle includes Isabelle/jEdit, which is based on jEdit and the Isabelle/Scala infrastructure for document-oriented proof processing. More recently, Visual Studio Code extensions have been developed for Coq, Isabelle by Makarius Wenzel, and for Lean 4 by the leanprover developers.

## Formalization extent

Freek Wiedijk has been keeping a ranking of proof assistants by the amount of formalized theorems out of a list of 100 well-known theorems. As of September 2023, only five systems have formalized proofs of more than 70% of the theorems, namely Isabelle, HOL Light, Rocq, Lean, and Metamath.

## Notable formalized proofs

The following is a list of notable proofs that have been formalized within proof assistants.
| Theorem | Proof assistant | Year |
| Four color theorem | Coq | 2005 |
| Feit–Thompson theorem | Coq | 2012 |
| Fundamental group of the circle | Coq | 2013 |
| Erdős–Graham problem | Lean | 2022 |
| Polynomial Freiman-Ruzsa conjecture over $\mathbb {F} _{2}$ | Lean | 2023 |
| BB(5) = 47,176,870 | Coq | 2024 |