||This article or section appears to contradict itself about 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)|
8.6 / December 14, 2016
|Typing discipline||static, strong|
|LEGO (proof assistant)|
|ML (programming), LCF (proof methods), Automath (hybrid programming/proving), System F and intuitionistic type theory (language)|
|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.
Seen as a programming language, Coq implements a dependently typed functional programming language, while seen as a logical system, it implements a higher-order type theory. The development of Coq is supported since 1984 by INRIA, now in collaboration with École Polytechnique, University of Paris-Sud, Paris Diderot University and CNRS. In the 1990s, École Normale Supérieure de Lyon was also part of the project. The development of Coq has been initiated by Gérard Huet and Thierry Coquand, after which more than 40 people, mainly researchers, contributed features of the core system. The implementation team has been successively coordinated by Gérard Huet, Christine Paulin 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 plug-ins.
The word coq means "rooster" in French, and stems from a local tradition of naming French research development tools with animal names. 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 Huet and the Calculus of Inductive Constructions along with Christine Paulin.
Coq provides a specification language called Gallina (that means 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 (non-termination) are common in other programming languages. 
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.
Based on this work, a significant extension to Coq was developed called Ssreflect (which stands for "small scale reflection"). Despite the name, most of the new features added to Coq by Ssreflect are general purpose 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 higher-order functions
- Concise anonymous arguments
- An improved
settactic with more powerful matching
- Support for reflection
- CompCert: an optimizing compiler for almost all of the C programming language which is fully programmed and proved in Coq.
- Disjoint-set data structure: correctness proof in Coq was published in 2007.
- Feit–Thompson theorem: formal proof using Coq was completed in September 2012.
- Four color theorem: formal proof using Coq was completed in 2005.
- "Coq 8.6 is out". 2016-12-14.
- What is Coq ? | The Coq Proof Assistant. Coq.inria.fr. Retrieved on 2013-07-21.
- A short introduction to Coq,
- Coq Version 8.0 for the Clueless (174 Hints). Flint.cs.yale.edu. Retrieved on 2013-11-07.
- 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
- 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". Msr-inria.fr. Retrieved 2014-01-27.
- Conchon, Sylvain; Filliâtre, Jean-Christophe (October 2007), "A Persistent Union-Find Data Structure", ACM SIGPLAN Workshop on ML, Freiburg, Germany
- "Feit-Thompson theorem has been totally checked in Coq". Msr-inria.inria.fr. 2012-09-20. Retrieved 2012-09-25.
- Gonthier, Georges (2008), "Formal Proof—The Four-Color Theorem" (PDF), Notices of the American Mathematical Society, 55 (11), pp. 1382–1393, MR 2463991
|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
- Welcome to the JsCoq Interactive Online System!
- Cocorico!, the 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
- 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