View source for Proof assistant

{{Short description|Software tool to assist with the development of formal proofs by human–machine collaboration}}
{{for|verification in computer science|formal verification}}
{{for|the academic conference|Interactive Theorem Proving (conference)}}
{{distinguish|Interactive proof system}}
{{missing information|[[automated proof checking]]|date=February 2024}}
{{more footnotes|date=November 2018}}

[[Image:CoqProofOfDecidablityOfEqualityOnNaturalNumbers.png|thumb|upright=1.7|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 proof]]s by human–machine collaboration. This involves some sort of interactive proof editor, or other [[User interface|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.<ref>{{Cite web |last=Ornes |first=Stephen |date=August 27, 2020 |title=Quanta Magazine – How Close Are Computers to Automating Mathematical Reasoning? |url=https://www.quantamagazine.org/how-close-are-computers-to-automating-mathematical-reasoning-20200827/}}</ref>

== {{anchor|Comparison}}System comparison ==
{{see also|Dependent type#Comparison|Automated theorem proving#Comparison}}
<!-- Need to add at least [[Automath]], [[PhoX]] -->
{| class=wikitable
|-
! rowspan=2 | Name !! rowspan=2 | Latest version !! rowspan=2 | Developer(s) !! rowspan=2 | Implementation language !! colspan=6 | Features
|-
! [[Higher-order logic]] !! [[Dependent type]]s !! [[de Bruijn criterion|Small kernel]] !! [[Proof automation]] !! [[Proof by reflection]] !! [[Code generation (compiler)|Code generation]]
|-
| [[ACL2]] || 8.3 || [[Matt Kaufmann]] and [[J Strother Moore]] ||  [[Common Lisp]]     || {{no}}  || {{n/a|Untyped}}  || {{no}}  || {{yes}} || {{yes}}<ref>{{cite book|last=Hunt|first=Warren|author2=Matt Kaufmann |author3=Robert Bellarmine Krug |author4=J Moore |author5=Eric W. Smith |title=Theorem Proving in Higher Order Logics|chapter=Meta Reasoning in ACL2|series=Lecture Notes in Computer Science|year=2005|volume=3603|pages=163–178|doi=10.1007/11541868_11|isbn=978-3-540-28372-0|chapter-url=http://www.cs.utexas.edu/~moore/publications/meta-05.pdf}}</ref> || {{n/a|Already executable}}
|-
| [[Agda (programming language)|Agda]] || 2.6.4.3<ref name="github-agda" />
|| Ulf Norell, Nils Anders Danielsson, and Andreas Abel ([[Chalmers University of Technology|Chalmers]] and [[Gothenburg University|Gothenburg]])<ref name="github-agda" /> ||[[Haskell (programming language)|Haskell]]<ref name="github-agda" /> || {{yes}}{{Cn|date=July 2024}} || {{yes}}<ref name="agdaWiki" /> || {{yes}}{{Cn|date=July 2024}} || {{no}}{{Cn|date=July 2024}} || {{partial}}{{Cn|date=July 2024}} || {{n/a|Already executable}}{{Cn|date=July 2024}}
|-
| [[Albatross (programming language)|Albatross]] || 0.4
|| Helmut Brandl ||[[OCaml]]|| {{yes}} || {{no}} || {{yes}} || {{yes}} || {{unknown}} || {{not yet}} Implemented
|-
| [[Coq (software)|Coq]] || 8.20.0 || [[INRIA]] || [[OCaml]] || {{yes}} || {{yes}} || {{yes}} || {{yes}} || {{yes}} || {{yes}}
|-
| [[F* (programming language)|F*]] || repository || [[Microsoft Research]] and [[INRIA]] || [[F* (programming language)|F*]] || {{yes}} || {{yes}} || {{no}} || {{yes}} || {{yes}}<ref>Search for "proofs by reflection": {{ArXiv|1803.06547}}</ref> || {{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 (programming language)|Idris]] || 2 0.6.0. || Edwin Brady || [[Idris (programming language)|Idris]] || {{yes}} || {{yes}} || {{yes}} || {{unknown}} || {{partial}} || {{yes}}
|-
| [[Isabelle (proof assistant)|Isabelle]] || Isabelle2024 (May 2024) ||[[Larry Paulson]] ([[University of Cambridge|Cambridge]]), [[Tobias Nipkow]] ([[Technische Universität München|München]]) and [[Makarius Wenzel]] ||  [[Standard ML]], [[Scala (programming language)|Scala]]    || {{yes}} || {{no}}  || {{yes}} || {{yes}} || {{yes}} || {{yes}}
|-
|[[Lean (proof assistant)|Lean]]
|v4.7.0<ref>{{Cite web|url=https://github.com/leanprover/lean4/releases|title=Lean 4 Releases Page |website=GitHub |access-date=15 October 2023}}</ref>
|[[Leonardo de Moura]] ([[Microsoft Research]])
|[[C++]], Lean
|{{yes}}
|{{yes}}
|{{yes}}
|{{yes}}
|{{yes}}
|{{yes}}
|-
| [[LEGO (proof assistant)|LEGO]] || 1.3.1 || [[Randy Pollack]] ([[University of Edinburgh|Edinburgh]]) || [[Standard ML]] || {{yes}} || {{yes}} || {{yes}} || {{no}} || {{no}} || {{no}}
|-
| [[Metamath]] || v0.198<ref>{{cite web | url=https://github.com/metamath/metamath-exe/releases/tag/v0.198 | title=Release v0.198 · metamath/Metamath-exe | website=[[GitHub]] }}</ref> || Norman Megill || [[ANSI C]] ||  ||  ||  ||  ||  || 
|-
| [[Mizar system|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}}
|-
| [[Prototype Verification System|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]]&nbsp;– a programming language, a first-order logical theory, and a theorem prover (with both interactive and automatic modes) in the Boyer–Moore tradition.
* [[Coq (software)|Coq]]&nbsp;– 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 prover]]s&nbsp;– 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]]&nbsp;– The "primary descendant", still under active development. Support for both [[Moscow ML]] and [[Poly/ML]]. Has a [[BSD-style license]].
**[[HOL Light]]&nbsp;– A thriving "minimalist fork". [[OCaml]] based.
**ProofPower&nbsp;– Went proprietary, then returned to open source. Based on [[Standard ML]].
* IMPS, An Interactive Mathematical Proof System.<ref>{{cite journal |last1=Farmer |first1=William M. |last2=Guttman |first2=Joshua D. |last3=Thayer |first3=F. Javier |title=IMPS: An interactive mathematical proof system |journal=Journal of Automated Reasoning |date=1993 |volume=11 |issue=2 |pages=213–248 |doi=10.1007/BF00881906 |s2cid=3084322 |access-date=22 January 2020|url=https://core.ac.uk/display/23376340}}</ref>
* [[Isabelle theorem prover|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 (software)|Jape]]&nbsp;– Java based.
* [[Lean (proof assistant)|Lean]]
* [[LEGO (proof assistant)|LEGO]]
* [[Matita]]&nbsp;– A light system based on the Calculus of Inductive Constructions.
* [[MINLOG]]&nbsp;– A proof assistant based on first-order minimal logic.
* [[Mizar system|Mizar]]&nbsp;– A proof assistant based on first-order logic, in a [[natural deduction]] style, and [[Tarski–Grothendieck set theory]].
* [[PhoX]]&nbsp;– A proof assistant based on higher-order logic which is eXtensible.
* [[Prototype Verification System]] (PVS)&nbsp;– a proof language and system based on higher-order logic.
* [[Theorem Proving System|TPS]] and ETPS&nbsp;– Interactive theorem provers also based on simply-typed lambda calculus, but based on an independent [[Q0 Logic|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 (programming language)|Scala]] infrastructure for document-oriented proof processing. More recently, [[Visual Studio Code]] extensions have been developed for Coq,<ref>{{Cite web|url=https://github.com/coq-community/vscoq|title=coq-community/vscoq|date=July 29, 2024|via=GitHub}}</ref> Isabelle by Makarius Wenzel,<ref>{{cite web |last1=Wenzel |first1=Makarius |title=Isabelle |url=https://marketplace.visualstudio.com/items?itemName=makarius.isabelle |access-date=2 November 2019}}</ref> and for Lean 4 by the leanprover developers.<ref>{{cite web |title=VS Code Lean 4 |url=https://github.com/leanprover/vscode-lean4 |website=GitHub |access-date=15 October 2023}}</ref>

== 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.<ref>{{cite web |url=https://www.cs.ru.nl/~freek/100/ |title=Formalizing 100 Theorems |first=Freek |last=Wiedijk |date=15 September 2023 }}</ref><ref>{{cite journal |url=https://www.ias.ac.in/article/fulltext/sadh/034/01/0003-0025 |title=Proof assistants: History, ideas and future |first=Herman |last=Geuvers |journal=Sādhanā |volume=34 |issue=1 |date=February 2009 |pages=3–25 |doi= 10.1007/s12046-009-0001-5|s2cid=14827467 |doi-access=free |hdl=2066/75958 |hdl-access=free }}</ref>

== Notable formalized proofs ==

{{See also|Computer-assisted proof#Theorems proved with the help of computer programs}}

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

{| class=wikitable
! scope="col" | Theorem
! scope="col" | Proof assistant
! scope="col" | Year
|-
| [[Four color theorem]]<ref>{{Citation |last=Gonthier |first=Georges |author-link=Georges Gonthier |title=Formal Proof—The Four-Color Theorem |journal=[[Notices of the American Mathematical Society]] |volume=55 |year=2008 |url=https://www.ams.org/notices/200811/tx081101382p.pdf |archive-url=https://web.archive.org/web/20110805094909/http://www.ams.org/notices/200811/tx081101382p.pdf |archive-date=2011-08-05 |url-status=live |issue=11 |pages=1382–1393 |mr=2463991 }}</ref> || Coq || 2005
|-
| [[Feit–Thompson theorem]]<ref>{{Cite web |date=2016-11-19 |title=Feit thomson proved in coq - Microsoft Research Inria Joint Centre |url=http://www.msr-inria.fr/news/feit-thomson-proved-in-coq/ |access-date=2023-12-07 |archive-url=https://web.archive.org/web/20161119094854/http://www.msr-inria.fr/news/feit-thomson-proved-in-coq/ |archive-date=2016-11-19 }}</ref> || Coq || 2012
|-
| [[Fundamental group]] of the [[circle]]<ref>{{Cite book |title=2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science |url=https://ieeexplore.ieee.org/document/6571554 |access-date=2023-12-07 |doi=10.1109/lics.2013.28 |date=2013 |last1=Licata |first1=Daniel R. |last2=Shulman |first2=Michael |chapter=Calculating the Fundamental Group of the Circle in Homotopy Type Theory |pages=223–232 |arxiv=1301.3443 |isbn=978-1-4799-0413-6 |s2cid=5661377 }}</ref> || Coq || 2013
|-
|[[Erdős–Graham problem]]<ref>{{Cite web |date=2022-03-11 |title=Math Problem 3,500 Years In The Making Finally Gets A Solution |url=https://www.iflscience.com/math-problem-3500-years-in-the-making-finally-gets-a-solution-62925 |access-date=2024-02-09 |website=IFLScience |language=en}}</ref><ref>{{Cite arXiv |last=Avigad |first=Jeremy |date=2023 |class=math.HO |title=Mathematics and the formal turn |eprint=2311.00007 }}</ref>
|Lean
|2022
|- 
| Polynomial Freiman-Ruzsa conjecture over <math>\mathbb F_2</math><ref>{{Cite web |last=Sloman |first=Leila |date=2023-12-06 |title='A-Team' of Math Proves a Critical Link Between Addition and Sets |url=https://www.quantamagazine.org/a-team-of-math-proves-a-critical-link-between-addition-and-sets-20231206/ |access-date=2023-12-07 |website=Quanta Magazine |language=en}}</ref> || Lean || 2023
|-
|[[Busy Beaver|BB(5)]] = 47,176,870<ref>{{Cite web |date=2024-07-02 |title=We have proved “BB(5) = 47,176,870” |url=https://discuss.bbchallenge.org/t/july-2nd-2024-we-have-proved-bb-5-47-176-870/237 |access-date=2024-07-09 |website=The Busy Beaver Challenge |language=en}}</ref>
|Coq
|2024
|}

== See also ==
* {{annotated link|Automated theorem proving}}
* {{annotated link|Computer-assisted proof}}
* {{annotated link|Formal verification}}
* {{annotated link|QED manifesto}}
* {{annotated link|Satisfiability modulo theories}}
* [[Prover9]] – is an [[Automated theorem proving|automated theorem prover]] for [[First-order logic|first-order and equational logic]]

== Notes ==
{{Reflist|refs=
<ref name="agdaWiki">{{cite web |title=The Agda Wiki |url=https://wiki.portal.chalmers.se/agda/pmwiki.php |access-date=31 July 2024}}</ref>
<ref name="github-agda">{{cite web |title=agda/agda:  Agda is a dependently typed programming language / interactive theorem prover. |url=https://github.com/agda/agda |website=GitHub |access-date=31 July 2024}}</ref>
}}

== References ==
*{{cite book |author1-link=Henk Barendregt |first1=Henk |last1=Barendregt |first2=Herman |last2=Geuvers |chapter=18. Proof-assistants using Dependent Type Systems |chapter-url=http://www.ncc.up.pt/~nam/aulas/0506/t_coq/barendregt01proofassistants.pdf |editor1-first=Alan J. A. |editor1-last=Robinson |editor2-first=Andrei |editor2-last=Voronkov |title=Handbook of Automated Reasoning |publisher=Elsevier |volume=2 |date=2001 |isbn=978-0-444-50812-6 |pages=1149– |archive-url=https://web.archive.org/web/20070727062855/http://www.ncc.up.pt/~nam/aulas/0506/t_coq/barendregt01proofassistants.pdf |archive-date=2007-07-27 |ref={{harvid|Handbook vol 2|2001}}}}
*{{cite book |author1-link=Frank Pfenning |first1=Frank |last1=Pfenning |chapter-url=https://www.cs.cmu.edu/~fp/papers/handbook01.pdf |chapter=17. Logical frameworks |title={{harvnb|Handbook vol 2|2001}} |pages=1065–1148}}
*{{cite book |first=Frank |last=Pfenning |chapter=The practice of logical frameworks |chapter-url= |editor-first=H. |editor-last=Kirchner |title=Trees in Algebra and Programming – CAAP '96 |publisher=Springer |series=Lecture Notes in Computer Science |volume=1059 |date=1996 |isbn=3-540-61064-2 |pages=119–134 |doi=10.1007/3-540-61064-2_33}}
*{{cite book |author1-link=Robert L. Constable |first=Robert L. |last=Constable |chapter=X. Types in computer science, philosophy and logic |chapter-url={{GBurl|MfTMDeCq7ukC|p=683}} |editor-first=S. R. |editor-last=Buss |title=Handbook of Proof Theory |publisher=Elsevier |series=Studies in Logic |volume=137 |date=1998 |isbn=978-0-08-053318-6 |pages=683–786 |url=}}
*{{cite web |first=Freek |last=Wiedijk |title=The Seventeen Provers of the World |date=2005 |publisher=Radboud University Nijmegen |url=https://www.cs.ru.nl/~freek/comparison/comparison.pdf }}

== External links ==
{{external links|date=December 2022}}
* [https://theoremprover-museum.github.io/ Theorem Prover Museum]
* [http://adam.chlipala.net/cpdt/html/Intro.html "Introduction"] in ''Certified Programming with Dependent Types''.
* [http://video.ias.edu/univalent/appel Introduction to the Coq Proof Assistant] (with a general introduction to interactive theorem proving)
* [http://www.cs.swan.ac.uk/~csetzer/lectures/intertheo/07/interactiveTheoremProvingForAgdaUsers.html Interactive Theorem Proving for Agda Users]
* [https://github.com/johnyf/tool_lists/blob/master/verification_synthesis.md#theorem-provers A list of theorem proving tools]

; Catalogues
* [https://www.cs.ru.nl/~freek/digimath/bycategory.html#tacticprover Digital Math by Category: Tactic Provers]
* [http://www.mcs.anl.gov/research/projects/AR/others.html Automated Deduction Systems and Groups]
* [https://www.cs.cmu.edu/afs/cs/project/ai-repository/ai/areas/reasonng/atp/systems/0.html Theorem Proving and Automated Reasoning Systems]
* [http://www-formal.stanford.edu/clt/ARS/Pages/systems.html Database of Existing Mechanized Reasoning Systems]
* [http://www.nuprl.org/Intro/others.html NuPRL: Other Systems]
* {{cite web | title=Specific Logical Frameworks and Implementations | url=https://www.cs.cmu.edu/~fp/lfs-impl.html | access-date=15 February 2024 | archive-date=10 April 2022 | archive-url=https://web.archive.org/web/20220410151836/https://www.cs.cmu.edu/~fp/lfs-impl.html | url-status=dead }} (By Frank Pfenning).
* [[DMOZ]]: [http://www.dmoz.org/Science/Math/Logic_and_Foundations/Computational_Logic/Logical_Frameworks/ Science: Math: Logic and Foundations: Computational Logic: Logical Frameworks]

[[Category:Argument technology]]
[[Category:Automated theorem proving]]
[[Category:Proof assistants| ]]

[[de:Maschinengestütztes Beweisen]]