Proof assistant explained

Proof assistant should not be confused with Interactive proof system.

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.^[1]

System comparison

Name	Latest version	Developer(s)	Implementation language	Features
					Dependent types				Code generation
	8.3							^[2]
	2.6.4.3	Ulf Norell, Nils Anders Danielsson, and Andreas Abel (Chalmers and Gothenburg)
	0.4	Helmut Brandl	OCaml						Implemented
	8.19.0
	repository							^[3]
	repository	John Harrison
	Kananaskis-13 (or repo)	Michael Norrish, Konrad Slind, and others
	2 0.6.0.	Edwin Brady
	Isabelle2024 (May 2024)
Lean	v4.7.0^[4]	Leonardo de Moura (Microsoft Research)	C++, Lean
LEGO (not affiliated with Lego)	1.3.1	Randy Pollack (Edinburgh)
	v0.198^[5]	Norman Megill
	8.1.11
	5
	6.0
	1.7.1

• ACL2 – a programming language, a first-order logical theory, and a theorem prover (with both interactive and automatic modes) in the Boyer–Moore tradition.
• 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.^[6]
• 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,^[7] Isabelle by Makarius Wenzel,^[8] and for Lean 4 by the leanprover developers.^[9]

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, Coq, Lean, and Metamath.^{[10] [11]}

Notable formalized proofs

The following is a list of notable proofs that have been formalized within proof assistants.

Theorem	Proof assistant	Year
	Coq	2005
Feit–Thompson theorem^[12]	Coq	2012
Fundamental group of the circle^[13]	Coq	2013
Erdős–Graham problem^[14] ^[15]	Lean	2022
Polynomial Freiman-Ruzsa conjecture over F _{2} ^[16]	Lean	2023
BB(5) = 47,176,870^[17]	Coq	2024

See also

• • • • • • Prover9 – is an automated theorem prover for first-order and equational logic

References

• Book: Henk Barendregt . Henk . Barendregt . Herman . Geuvers . 18. Proof-assistants using Dependent Type Systems . http://www.ncc.up.pt/~nam/aulas/0506/t_coq/barendregt01proofassistants.pdf . Alan J. A. . Robinson . Andrei . Voronkov . Handbook of Automated Reasoning . Elsevier . 2 . 2001 . 978-0-444-50812-6 . 1149– . https://web.archive.org/web/20070727062855/http://www.ncc.up.pt/~nam/aulas/0506/t_coq/barendregt01proofassistants.pdf . 2007-07-27 . .
• Book: Frank Pfenning . Frank . Pfenning . https://www.cs.cmu.edu/~fp/papers/handbook01.pdf . 17. Logical frameworks . . 1065–1148.
• Book: Pfenning, Frank . The practice of logical frameworks . H. . Kirchner . Trees in Algebra and Programming – CAAP '96 . Springer . Lecture Notes in Computer Science . 1059 . 1996 . 3-540-61064-2 . 119–134 . 10.1007/3-540-61064-2_33.
• Book: Constable, Robert L. . Robert L. Constable . X. Types in computer science, philosophy and logic . . S. R. . Buss . Handbook of Proof Theory . Elsevier . Studies in Logic . 137 . 1998 . 978-0-08-053318-6 . 683–786 .
• Web site: Freek . Wiedijk . The Seventeen Provers of the World . 2005 . Radboud University Nijmegen .

External links

• Theorem Prover Museum
• "Introduction" in Certified Programming with Dependent Types.
• Introduction to the Coq Proof Assistant (with a general introduction to interactive theorem proving)
• Interactive Theorem Proving for Agda Users
• A list of theorem proving tools

Catalogues

• Digital Math by Category: Tactic Provers
• Automated Deduction Systems and Groups
• Theorem Proving and Automated Reasoning Systems
• Database of Existing Mechanized Reasoning Systems
• NuPRL: Other Systems
• Web site: Specific Logical Frameworks and Implementations . 15 February 2024 . 10 April 2022 . https://web.archive.org/web/20220410151836/https://www.cs.cmu.edu/~fp/lfs-impl.html . dead . (By Frank Pfenning).
• DMOZ: Science: Math: Logic and Foundations: Computational Logic: Logical Frameworks

Notes and References

1. Web site: Ornes . Stephen . August 27, 2020 . Quanta Magazine – How Close Are Computers to Automating Mathematical Reasoning? .
2. Book: Hunt, Warren. Matt Kaufmann . Robert Bellarmine Krug . J Moore . Eric W. Smith . Theorem Proving in Higher Order Logics. Meta Reasoning in ACL2. Lecture Notes in Computer Science. 2005. 3603. 163–178. 10.1007/11541868_11. 978-3-540-28372-0. http://www.cs.utexas.edu/~moore/publications/meta-05.pdf.
3. Search for "proofs by reflection":
4. Web site: Lean 4 Releases Page . GitHub . 15 October 2023.
5. Web site: Release v0.198 · metamath/Metamath-exe . .
6. Farmer . William M. . Guttman . Joshua D. . Thayer . F. Javier . IMPS: An interactive mathematical proof system . Journal of Automated Reasoning . 1993 . 11 . 2 . 213–248 . 10.1007/BF00881906 . 3084322 . 22 January 2020.
7. Web site: coq-community/vscoq. July 29, 2024. GitHub.
8. Web site: Wenzel . Makarius . Isabelle . 2 November 2019.
9. Web site: VS Code Lean 4 . GitHub . 15 October 2023.
10. Web site: Formalizing 100 Theorems . Freek . Wiedijk . 15 September 2023 .
11. Proof assistants: History, ideas and future . Herman . Geuvers . Sādhanā . 34 . 1 . February 2009 . 3–25 . 10.1007/s12046-009-0001-5. 14827467 . free . 2066/75958 . free .
12. Web site: 2016-11-19 . Feit thomson proved in coq - Microsoft Research Inria Joint Centre . 2023-12-07 . https://web.archive.org/web/20161119094854/http://www.msr-inria.fr/news/feit-thomson-proved-in-coq/ . 2016-11-19 .
13. Book: 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science . 2023-12-07 . 10.1109/lics.2013.28 . 2013 . Licata . Daniel R. . Shulman . Michael . Calculating the Fundamental Group of the Circle in Homotopy Type Theory . 223–232 . 1301.3443 . 978-1-4799-0413-6 . 5661377 .
14. Web site: 2022-03-11 . Math Problem 3,500 Years In The Making Finally Gets A Solution . 2024-02-09 . IFLScience . en.
15. Avigad . Jeremy . 2023 . math.HO . Mathematics and the formal turn . 2311.00007 .
16. Web site: Sloman . Leila . 2023-12-06 . 'A-Team' of Math Proves a Critical Link Between Addition and Sets . 2023-12-07 . Quanta Magazine . en.
17. Web site: 2024-07-02 . We have proved “BB(5) = 47,176,870” . 2024-07-09 . The Busy Beaver Challenge . en.