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Coq (software)
Developer(s)	The Coq development team
Initial release	1 May 1989; 35 years ago (version 4.10)
Stable release	8.20.0 / 3 September 2024; 41 days ago
Repository	github.com/coq/coq
Written in	OCaml
Operating system	Cross-platform
Available in	English
Type	Proof assistant
License	LGPLv2.1
Website	coq.inria.fr

Coq is an interactive theorem prover first released in 1989. It allows for expressing mathematical assertions, mechanically checks proofs of these assertions, helps find formal proofs, and extracts a certified program from the constructive proof of its formal specification. Coq works within the theory of the calculus of inductive constructions, a derivative of the calculus of constructions. Coq is not an automated theorem prover but includes automatic theorem proving tactics (procedures) and various decision procedures.

The Association for Computing Machinery awarded Thierry Coquand, Gérard Huet, Christine Paulin-Mohring, Bruno Barras, Jean-Christophe Filliâtre, Hugo Herbelin, Chetan Murthy, Yves Bertot, and Pierre Castéran with the 2013 ACM Software System Award for Coq.

The name "Coq" is a wordplay on the name of Thierry Coquand, Calculus of Constructions or "CoC" and follows the French computer science tradition of naming software after animals (coq in French meaning rooster).^[2] On October 11th, 2023, the development team announced that Coq will be renamed "The Rocq Prover" in the coming months, and has started updating the code base, website and associated tools.^[3]

Overview

When viewed as a programming language, Coq implements a dependently typed functional programming language;^[4] when viewed as a logical system, it implements a higher-order type theory. The development of Coq has been supported since 1984 by INRIA, now in collaboration with École Polytechnique, University of Paris-Sud, Paris Diderot University, and CNRS. In the 1990s, ENS Lyon was also part of the project. The development of Coq was initiated by Gérard Huet and Thierry Coquand, and more than 40 people, mainly researchers, have contributed features to the core system since its inception. The implementation team has successively been coordinated by Gérard Huet, Christine Paulin-Mohring, Hugo Herbelin, and Matthieu Sozeau. Coq is mainly implemented in OCaml with a bit of C. The core system can be extended by way of a plug-in mechanism.^[5]

The name coq means 'rooster' in French and stems from a French tradition of naming research development tools after animals.^[6] Up until 1991, Coquand was implementing a language called the Calculus of Constructions and it was simply called CoC at this time. In 1991, a new implementation based on the extended Calculus of Inductive Constructions was started and the name was changed from CoC to Coq in an indirect reference to Coquand, who developed the Calculus of Constructions along with Gérard Huet and contributed to the Calculus of Inductive Constructions with Christine Paulin-Mohring.^[7]

Coq provides a specification language called Gallina^[8] ("hen" in Latin, Spanish, Italian and Catalan). Programs written in Gallina have the weak normalization property, implying that they always terminate. This is a distinctive property of the language, since infinite loops (non-terminating programs) are common in other programming languages,^[9] and is one way to avoid the halting problem.

As an example, consider a proof of a lemma that taking the successor of a natural number flips its parity. The fold-unfold tactic introduced by Danvy^[10] is used to help keep the proof simple.

Ltac fold_unfold_tactic name := intros; unfold name; fold name; reflexivity.

Require Import Arith Nat Bool.

Fixpoint is_even (n : nat) : bool :=
  match n with
  | 0 =>
    true
  | S n' =>
    eqb (is_even n') false
  end.

Lemma fold_unfold_is_even_0:
  is_even 0 = true.

Proof.
  fold_unfold_tactic is_even.
Qed.

Lemma fold_unfold_is_even_S:
  forall n' : nat,
    is_even (S n') = eqb (is_even n') false.

Proof.
  fold_unfold_tactic is_even.
Qed.

Lemma successor_flips_evenness:
  forall n : nat,
    is_even n = negb (is_even (S n)).

Proof.
  intro n.
  rewrite -> (fold_unfold_is_even_S n).
  destruct (is_even n).

  * simpl.
    reflexivity.

  * simpl.
    reflexivity.
Qed.

Notable uses

Four color theorem and SSReflect extension

Georges Gonthier of Microsoft Research in Cambridge, England and Benjamin Werner of INRIA used Coq to create a surveyable proof of the four color theorem, which was completed in 2002.^[11] Their work led to the development of the SSReflect ("Small Scale Reflection") package, which was a significant extension to Coq.^[12] Despite its name, most of the features added to Coq by SSReflect are general-purpose features and are not limited to the computational reflection style of proof. These features include:

Additional convenient notations for irrefutable and refutable pattern matching, on inductive types with one or two constructors
Implicit arguments for functions applied to zero arguments, which is useful when programming with higher-order functions
Concise anonymous arguments
An improved set tactic with more powerful matching
Support for reflection

SSReflect 1.11 is freely available, dual-licensed under the open source CeCILL-B or CeCILL-2.0 license, and compatible with Coq 8.11.^[13]

Other applications

CompCert: an optimizing compiler for almost all of the C programming language which is largely programmed and proven correct in Coq.
Disjoint-set data structure: correctness proof in Coq was published in 2007.^[14]
Feit–Thompson theorem: formal proof using Coq was completed in September 2012.^[15]
Busy beaver: The value of the 5-state winning busy beaver was discovered by Heiner Marxen and Jürgen Buntrock in 1989, but only proved to be the winning fifth busy beaver — stylized as BB(5) — in 2024 using a proof in Coq.^[16]^[17]

Tactic language

In addition to constructing Gallina terms explicitly, Coq supports the use of tactics written in the built-in language Ltac or in OCaml. These tactics automate the construction of proofs, carrying out trivial or obvious steps in proofs.^[18] Several tactics implement decision procedures for various theories. For example, the "ring" tactic decides the theory of equality modulo ring or semiring axioms via associative-commutative rewriting.^[19] For example, the following proof establishes a complex equality in the ring of integers in just one line of proof:^[20]

Require Import ZArith.
Open Scope Z_scope.
Goal forall a b c:Z,
    (a + b + c) ^ 2 =
     a * a + b ^ 2 + c * c + 2 * a * b + 2 * a * c + 2 * b * c.
  intros; ring.
Qed.

Built-in decision procedures are also available for the empty theory ("congruence"), propositional logic ("tauto"), quantifier-free linear integer arithmetic ("lia"), and linear rational/real arithmetic ("lra").^[21]^[22] Further decision procedures have been developed as libraries, including one for Kleene algebras^[23] and another for certain geometric goals.^[24]

References

^ "Release Coq 8.20.0". 3 September 2024.
^ "Alternative names · coq/coq Wiki". GitHub. Retrieved 3 March 2023.
^ "Coq roadmap 069". GitHub.
^ A short introduction to Coq
^ Avigad, Jeremy; Mahboubi, Assia (3 July 2018). Interactive Theorem Proving: 9th International Conference, ITP 2018, Held as ... Springer. ISBN 9783319948218. Retrieved 21 October 2018.
^ "Frequently Asked Questions". GitHub. Retrieved 2019-05-08.
^ "Introduction to the Calculus of Inductive Constructions". Retrieved 21 May 2019.
^ Adam Chlipala. "Certified Programming with Dependent Types": "Library Universes".
^ Adam Chlipala. "Certified Programming with Dependent Types": "Library GeneralRec". "Library InductiveTypes".
^ Danvy, Olivier (2022). "Fold–unfold lemmas for reasoning about recursive programs using the Coq proof assistant". Journal of Functional Programming. 32. doi:10.1017/S0956796822000107. ISSN 0956-7968.
^ Gonthier, Georges (2008). "Formal Proof—The Four-Color Theorem" (PDF). Notices of the American Mathematical Society. 55 (11): 1382–1393. MR 2463991.
^ Gonthier, Georges; Mahboubi, Assia (2010). "An introduction to small scale reflection in Coq". Journal of Formalized Reasoning. 3 (2): 95–152. doi:10.6092/ISSN.1972-5787/1979.
^ "The Mathematical Components Library 1.11.0". GitHub.
^ Conchon, Sylvain; Filliâtre, Jean-Christophe (2007). "A persistent union-find data structure". In Russo, Claudio V.; Dreyer, Derek (eds.). Proceedings of the ACM Workshop on ML, 2007, Freiburg, Germany, October 5, 2007. Association for Computing Machinery. pp. 37–46. doi:10.1145/1292535.1292541.
^ "Feit-Thompson theorem has been totally checked in Coq". Msr-inria.inria.fr. 2012-09-20. Archived from the original on 2016-11-19. Retrieved 2012-09-25.
^ "[July 2nd 2024] We have proved "BB(5) = 47,176,870"". The Busy Beaver Challenge. 2024-07-02. Retrieved 2024-07-02.
^ "The Busy Beaver Challenge". bbchallenge.org. Retrieved 2024-07-02.
^ Kaiser, Jan-Oliver; Ziliani, Beta; Krebbers, Robbert; Régis-Gianas, Yann; Dreyer, Derek (2018-07-30). "Mtac2: typed tactics for backward reasoning in Coq". Proceedings of the ACM on Programming Languages. 2 (ICFP): 78:1–78:31. doi:10.1145/3236773. hdl:21.11116/0000-0003-2E8E-B.
^ Grégoire, Benjamin; Mahboubi, Assia (2005). "Proving Equalities in a Commutative Ring Done Right in Coq". In Hurd, Joe; Melham, Tom (eds.). Theorem Proving in Higher Order Logics: 18th International Conference, TPHOLs 2005, Oxford, UK, August 22–25, 2005, Proceedings. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer. pp. 98–113. doi:10.1007/11541868_7. ISBN 978-3-540-31820-0.
^ "The ring and field tactic families — Coq 8.11.1 documentation". coq.inria.fr. Retrieved 2023-12-04.
^ Besson, Frédéric (2007). "Fast Reflexive Arithmetic Tactics the Linear Case and Beyond". In Altenkirch, Thorsten; McBride, Conor (eds.). Types for Proofs and Programs: International Workshop, TYPES 2006, Nottingham, UK, April 18–21, 2006, Revised Selected Papers. Lecture Notes in Computer Science. Vol. 4502. Berlin, Heidelberg: Springer. pp. 48–62. doi:10.1007/978-3-540-74464-1_4. ISBN 978-3-540-74464-1.
^ "Micromega: solvers for arithmetic goals over ordered rings — Coq 8.18.0 documentation". coq.inria.fr. Retrieved 2023-12-04.
^ Braibant, Thomas; Pous, Damien (2010). Kaufmann, Matt; Paulson, Lawrence C. (eds.). An Efficient Coq Tactic for Deciding Kleene Algebras. Interactive Theorem Proving: First International Conference, ITP 2010 Edinburgh, UK, July 11-14, 2010, Proceedings. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer. pp. 163–178. doi:10.1007/978-3-642-14052-5_13. ISBN 978-3-642-14052-5. S2CID 3566183.
^ Narboux, Julien (2004). "A Decision Procedure for Geometry in Coq". In Slind, Konrad; Bunker, Annette; Gopalakrishnan, Ganesh (eds.). Theorem Proving in Higher Order Logics: 17th International Conference, TPHOLS 2004, Park City, Utah, USA, September 14–17, 2004, Proceedings. Lecture Notes in Computer Science. Vol. 3223. Berlin, Heidelberg: Springer. pp. 225–240. doi:10.1007/978-3-540-30142-4_17. ISBN 978-3-540-30142-4. S2CID 11238876.

External links

Official website, in English
Coq on GitHub, source code repository
JsCoq Interactive Online System – allows Coq to run in a web browser, with no need to install extra software
Alectryon – a library to process Coq snippets embedded in documents, showing goals and messages for each Coq sentence
Coq Wiki
Mathematical Components library – widely used library of mathematical structures, part of which is the SSReflect proof language
Constructive Coq Repository at Nijmegen
Math Classes
Coq at Open Hub