Coq
This article or section appears to contradict itself on the date when the surveyable proof of the 4 Colour Theorem was completed.
(January 2017)

Developer(s)  The Coq development team 

Initial release  May 1, 1989  (version 4.10)
Stable release 
8.7.2^{[1]} / February 17, 2018

Repository  github.com/coq/coq 
Development status  Active 
Written in  OCaml 
Operating system  Crossplatform 
Available in  English 
Type  Proof assistant 
License  LGPL 2.1 
Website  coq.inria.fr 
Paradigm  Functional 

First appeared  1984^{[2]} 
Typing discipline  static, strong 
Filename extensions  .v 
Website  coq.inria.fr 
Dialects  
LEGO (proof assistant)  
Influenced by  
ML (programming), LCF (proof methods), Automath (hybrid programming/proving), System F and intuitionistic type theory (language)  
Influenced  
Agda, Idris, Matita, Albatross 
In computer science, Coq is an interactive theorem prover. It 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. 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 and various decision procedures.
The Association for Computing Machinery rewarded Thierry Coquand, Gérard Pierre Huet, Christine PaulinMohring, Bruno Barras, JeanChristophe Filliâtre, Hugo Herbelin, Chetan Murthy, Yves Bertot and Pierre Castéran with the 2013 ACM Software System Award for Coq.
Contents
Overview
Seen as a programming language, Coq implements a dependently typed functional programming language,^{[3]} while seen as a logical system, it implements a higherorder type theory. The development of Coq has been supported since 1984 by INRIA, now in collaboration with École Polytechnique, University of ParisSud, Paris Diderot University and CNRS. In the 1990s, École Normale Supérieure de Lyon was also part of the project. The development of Coq was initiated by Gérard Pierre Huet and Thierry Coquand, after which more than 40 people, mainly researchers, contributed features of the core system. The implementation team was successively coordinated by Gérard Pierre Huet, Christine PaulinMohring and Hugo Herbelin. Coq is for the most part implemented in OCaml with a bit of C. The core system can be extended thanks to a mechanism of plugins.
The word coq means "rooster" in French, and stems from a local tradition of naming French research development tools with animal names.^{[4]} Up to 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 changed from CoC to Coq, also an indirect reference to Thierry Coquand who developed the Calculus of Constructions along with Gérard Pierre Huet and the Calculus of Inductive Constructions along with Christine PaulinMohring.
Coq provides a specification language called Gallina^{[5]} (meaning hen in Spanish and Italian). Programs written in Gallina have the weak normalization property – they always terminate. This is one way to avoid the halting problem. This may be surprising, since infinite loops (nontermination) are common in other programming languages.^{[6]}
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 September 2004.^{[7]}
Based on this work, a significant extension to Coq was developed called Ssreflect (which stands for "small scale reflection").^{[8]} Despite the name, most of the new features added to Coq by Ssreflect are generalpurpose features, useful not merely for the computational reflection style of proof. These 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 higherorder functions
 Concise anonymous arguments
 An improved
set
tactic with more powerful matching  Support for reflection
Ssreflect 1.4 is freely available duallicensed under the open source CeCILLB or CeCILL2.0 license, and is compatible with Coq 8.4.^{[9]}
Applications
 CompCert: an optimizing compiler for almost all of the C programming language which is fully programmed and proved in Coq.
 Disjointset data structure: correctness proof in Coq was published in 2007.^{[10]}
 Feit–Thompson theorem: formal proof using Coq was completed in September 2012.^{[11]}
 Four color theorem: formal proof using Coq was completed in 2005.^{[12]}
See also
 Calculus of constructions
 Curry–Howard correspondence
 Isabelle (proof assistant)  similar/competing software
 Intuitionistic type theory
References
 ^ "Coq 8.7.2 is out". 20180217.
 ^ What is Coq?. Coq.inria.fr. Retrieved on 20130721.
 ^ A short introduction to Coq,
 ^ "Frequently Asked Questions". Retrieved 20170609.
 ^ Adam Chlipala. "Certified Programming with Dependent Types": "Library Universes".
 ^ Adam Chlipala. "Certified Programming with Dependent Types": "Library GeneralRec". "Library InductiveTypes".
 ^ "Development of theories and tactics: Four Color Theorem". Archived from the original on 20070222.
 ^ Georges Gonthier, Assia Mahboubi. "An introduction to small scale reflection in Coq": "Journal of Formalized Reasoning".
 ^ "Ssreflect 1.4 has been released – Microsoft Research Inria Joint Centre". Msrinria.fr. Retrieved 20140127.
 ^ Conchon, Sylvain; Filliâtre, JeanChristophe (October 2007), "A Persistent UnionFind Data Structure", ACM SIGPLAN Workshop on ML, Freiburg, Germany
 ^ "FeitThompson theorem has been totally checked in Coq". Msrinria.inria.fr. 20120920. Archived from the original on 20161119. Retrieved 20120925.
 ^ Gonthier, Georges (2008), "Formal Proof—The FourColor Theorem" (PDF), Notices of the American Mathematical Society, 55 (11), pp. 1382–1393, MR 2463991
External links
Wikimedia Commons has media related to Coq. 
 The Coq proof assistant – the official English website
 coq/coq – the project's source code repository on GitHub
 JsCoq Interactive Online System  allows Coq to be run in a web browser, without the need for any software installation
 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
 Textbooks
 The Coq'Art – a book on Coq by Yves Bertot and Pierre Castéran
 Certified Programming with Dependent Types – online and printed textbook by Adam Chlipala
 Software Foundations – online textbook by Benjamin C. Pierce et al.
 An introduction to small scale reflection in Coq – a tutorial on SSreflect by Georges Gonthier and Assia Mahboubi
 Tutorials
 Introduction to the Coq Proof Assistant – video lecture by Andrew Appel at Institute for Advanced Study
 Video tutorials for the Coq proof assistant by Andrej Bauer.