Proof assistant

Software tool to assist with the development of formal proofs by human–machine collaboration

An interactive proof session in CoqIDE, showing the proof script on the left and the proof state on the right

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

• 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.^[8]
• 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,^[9] Isabelle by Makarius Wenzel,^[10] and for Lean 4 by the leanprover developers.^[11]

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.^[12][13]

Notable formalized proofs

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

See also

• Automated theorem proving – Subfield of automated reasoning and mathematical logic
• Computer-assisted proof – Mathematical proof at least partially generated by computer
• Formal verification – Proving or disproving the correctness of certain intended algorithms
• QED manifesto – Proposal for a computer-based database of all mathematical knowledge
• Satisfiability modulo theories – Logical problem studied in computer science
• Prover9 – is an automated theorem prover for first-order and equational logic

Notes

1. Ornes, Stephen (August 27, 2020). "Quanta Magazine – How Close Are Computers to Automating Mathematical Reasoning?".
2. Hunt, Warren; Matt Kaufmann; Robert Bellarmine Krug; J Moore; Eric W. Smith (2005). "Meta Reasoning in ACL2" (PDF). Theorem Proving in Higher Order Logics. Lecture Notes in Computer Science. Vol. 3603. pp. 163–178. doi:10.1007/11541868_11. ISBN 978-3-540-28372-0.
3. "agda/agda: Agda is a dependently typed programming language / interactive theorem prover". GitHub. Retrieved 31 July 2024.
4. "The Agda Wiki". Retrieved 31 July 2024.
5. Search for "proofs by reflection": arXiv:1803.06547
6. "Lean 4 Releases Page". GitHub. Retrieved 15 October 2023.
7. "Release v0.198 · metamath/Metamath-exe". GitHub.
8. Farmer, William M.; Guttman, Joshua D.; Thayer, F. Javier (1993). "IMPS: An interactive mathematical proof system". Journal of Automated Reasoning. 11 (2): 213–248. doi:10.1007/BF00881906. S2CID 3084322. Retrieved 22 January 2020.
9. "coq-community/vscoq". July 29, 2024 – via GitHub.
10. Wenzel, Makarius. "Isabelle". Retrieved 2 November 2019.
11. "VS Code Lean 4". GitHub. Retrieved 15 October 2023.
12. Wiedijk, Freek (15 September 2023). "Formalizing 100 Theorems".
13. Geuvers, Herman (February 2009). "Proof assistants: History, ideas and future". Sādhanā. 34 (1): 3–25. doi:10.1007/s12046-009-0001-5. hdl:2066/75958. S2CID 14827467.
14. Gonthier, Georges (2008), "Formal Proof—The Four-Color Theorem" (PDF), Notices of the American Mathematical Society, 55 (11): 1382–1393, MR 2463991, archived (PDF) from the original on 2011-08-05
15. "Feit thomson proved in coq - Microsoft Research Inria Joint Centre". 2016-11-19. Archived from the original on 2016-11-19. Retrieved 2023-12-07.
16. Licata, Daniel R.; Shulman, Michael (2013). "Calculating the Fundamental Group of the Circle in Homotopy Type Theory". 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science. pp. 223–232. arXiv:1301.3443. doi:10.1109/lics.2013.28. ISBN 978-1-4799-0413-6. S2CID 5661377. Retrieved 2023-12-07.
17. "Math Problem 3,500 Years In The Making Finally Gets A Solution". IFLScience. 2022-03-11. Retrieved 2024-02-09.
18. Avigad, Jeremy (2023). "Mathematics and the formal turn". arXiv:2311.00007 [math.HO].
19. Sloman, Leila (2023-12-06). "'A-Team' of Math Proves a Critical Link Between Addition and Sets". Quanta Magazine. Retrieved 2023-12-07.
20. "We have proved "BB(5) = 47,176,870"". The Busy Beaver Challenge. 2024-07-02. Retrieved 2024-07-09.

References

• Barendregt, Henk; Geuvers, Herman (2001). "18. Proof-assistants using Dependent Type Systems" (PDF). In Robinson, Alan J. A.; Voronkov, Andrei (eds.). Handbook of Automated Reasoning. Vol. 2. Elsevier. pp. 1149–. ISBN 978-0-444-50812-6. Archived from the original (PDF) on 2007-07-27.
• Pfenning, Frank. "17. Logical frameworks" (PDF). Handbook vol 2 2001. pp. 1065–1148.
• Pfenning, Frank (1996). "The practice of logical frameworks". In Kirchner, H. (ed.). Trees in Algebra and Programming – CAAP '96. Lecture Notes in Computer Science. Vol. 1059. Springer. pp. 119–134. doi:10.1007/3-540-61064-2_33. ISBN 3-540-61064-2.
• Constable, Robert L. (1998). "X. Types in computer science, philosophy and logic". In Buss, S. R. (ed.). Handbook of Proof Theory. Studies in Logic. Vol. 137. Elsevier. pp. 683–786. ISBN 978-0-08-053318-6.
• Wiedijk, Freek (2005). "The Seventeen Provers of the World" (PDF). 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
• "Specific Logical Frameworks and Implementations". Archived from the original on 10 April 2022. Retrieved 15 February 2024. (By Frank Pfenning).
• DMOZ: Science: Math: Logic and Foundations: Computational Logic: Logical Frameworks