Proof assistant

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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[edit]

Name Latest version Developer(s) Implementation language Features
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[2] Already executable
Agda 2.6.3 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
Coq 8.19.0 INRIA OCaml Yes Yes Yes Yes Yes Yes
F* repository Microsoft Research and INRIA F* Yes Yes No Yes Yes[3] 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 Isabelle2021 (February 2021) Larry Paulson (Cambridge), Tobias Nipkow (München) and Makarius Wenzel Standard ML, Scala Yes No Yes Yes Yes Yes
Lean v4.1.0 (official release) v4.2.0-rc1 (pre-release)[4] Leonardo de Moura (Microsoft Research) C++, Lean Yes Yes Yes Yes Yes Yes
LEGO (not affiliated with Lego) 1.3.1 Randy Pollack (Edinburgh) Standard ML Yes Yes Yes No No No
Metamath v0.198[5] Norman Megill ANSI C
Mizar 8.1.05 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.
  • 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:
  • 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[edit]

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 Isabelle by Makarius Wenzel,[7] and for Lean 4 by the leanprover developers.[8]

Formalization extent[edit]

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.[9][10]

Notable formalized proofs[edit]

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

Theorem Proof assistant Year
Four color theorem[11] 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 [16] Lean 2023

See also[edit]

Notes[edit]

  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. ^ Search for "proofs by reflection": arXiv:1803.06547
  4. ^ "Lean 4 Releases Page". GitHub. Retrieved 15 October 2023.
  5. ^ "Release v0.198 · metamath/Metamath-exe". GitHub.
  6. ^ 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.
  7. ^ Wenzel, Makarius. "Isabelle". Retrieved 2 November 2019.
  8. ^ "VS Code Lean 4". GitHub. Retrieved 15 October 2023.
  9. ^ Wiedijk, Freek (15 September 2023). "Formalizing 100 Theorems".
  10. ^ 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.
  11. ^ 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
  12. ^ "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.
  13. ^ 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.
  14. ^ "Math Problem 3,500 Years In The Making Finally Gets A Solution". IFLScience. 2022-03-11. Retrieved 2024-02-09.
  15. ^ Avigad, Jeremy. "MATHEMATICS AND THE FORMAL TURN" (PDF). arXiv.
  16. ^ Sloman, Leila (2023-12-06). "'A-Team' of Math Proves a Critical Link Between Addition and Sets". Quanta Magazine. Retrieved 2023-12-07.

References[edit]

External links[edit]

Catalogues