# Coq

**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.

Developer(s) | The Coq development team |
---|---|

Initial release | 1 May 1989 (version 4.10) |

Stable release | |

Repository | github |

Written in | OCaml |

Operating system | Cross-platform |

Available in | English |

Type | Proof assistant |

License | LGPLv2.1 |

Website | coq |

The Association for Computing Machinery rewarded 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.

## Overview

Seen as a programming language, Coq implements a dependently typed functional programming language,[2] while seen 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, after which more than 40 people, mainly researchers, have contributed features of the core system. The implementation team has successively been coordinated by Gérard Huet, Christine Paulin-Mohring, Hugo Herbelin, and Matthieu Sozeau. Coq is for the most part implemented in OCaml with a bit of C. The core system can be extended due to a mechanism of plug-ins.[3]

The name *coq* means "rooster" in French, and stems from a French tradition of naming research development tools after animals.[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 Huet and the Calculus of Inductive Constructions along with Christine Paulin-Mohring.[5]

Coq provides a specification language called Gallina[6] (meaning hen in Latin, Spanish, Italian and Catalan). 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.[7]

## 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 2005.[8]

Based on this work, a significant extension to Coq was developed called Ssreflect (which stands for "small scale reflection").[9] 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
`set`

tactic with more powerful matching - Support for reflection

Ssreflect 1.4 is freely available dual-licensed under the open source CeCILL-B or CeCILL-2.0 license, and is compatible with Coq 8.4.[10]

## Applications

- 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.[11]
- Feit–Thompson theorem: formal proof using Coq was completed in September 2012.[12]
- Four color theorem: formal proof using Coq was completed in 2005.[8]

## See also

- Nuprl
- Agda
- Calculus of constructions
- Curry–Howard correspondence
- Isabelle (proof assistant) – similar/competing software
- Intuitionistic type theory
- HOL (proof assistant)

## References

- "Coq 8.10.2 is out". 2019-11-29.
- A short introduction to Coq,
- Avigad, Jeremy; Mahboubi, Assia. "Interactive Theorem Proving: 9th International Conference, ITP 2018, Held as ..."
*Google Books*. Google Books. Retrieved 21 October 2018. - "Frequently Asked Questions". 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".
- Gonthier, Georges (2008), "Formal Proof—The Four-Color Theorem" (PDF),
*Notices of the American Mathematical Society*,**55**(11), pp. 1382–1393, MR 2463991 - 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. Archived from the original on 2014-02-02. 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. Archived from the original on 2016-11-19. Retrieved 2012-09-25.

## 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.