Coq

For the coenzyme and dietary supplement, see Coenzyme Q10.

For the abbreviated placename, see Coquihalla.

Coq (software)

Developer(s)	The Coq development team
Initial release	May 1, 1989 (1989-05-01) (version 4.10)
Stable release	8.4pl6 / April 2015 (2015-04)
Development status	Active
Written in	OCaml
Operating system	Cross-platform
Available in	English
Type	Proof assistant
License	LGPL 2.1
Website	coq.inria.fr

Coq (programming language)

Paradigm	Functional
First appeared	1984
Typing discipline	static, strong
Filename extensions	.v
Website	{{#property:P856}}
Dialects
LEGO (proof assistant)
Influenced by
ML (programming), LCF (proof methods), Automath (hybrid programming/proving), System F and intuitionistic type theory (language)
Influenced
Agda, Matita, Albatross

An interactive proof session in CoqIDE, showing the proof script on the left and the proof state on the right.

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 presented Coquand, Huet, Paulin-Mohring, Barras, Filliâtre, Herbelin, Murthy, Bertot, Castéran with the 2013 ACM Software System Award for Coq.

Overview

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 90's, 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 a couple of 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). 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 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.

Applications

• CompCert: an optimizing compiler for a subset 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.

See also

• Calculus of constructions
• Curry–Howard correspondence
• Intuitionistic type theory

External links

• The Coq proof assistant – the official English website
• coq/coq – the project's source code repository on GitHub
• Cocorico!, the Coq Wiki
• MSR Inria math components – hosts the Ssreflect extension
• Constructive Coq Repository at Nijmegen
• Math Classes

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.