Coq

Paradigm(s)	Functional
Appeared in	1984[1]
Stable release	8.4 (August 2012 (2012-08))
Typing discipline	static, strong
Influenced by	ML and Standard ML
Influenced	Agda
OS	Cross-platform
License	LGPL 2.1
Usual filename extensions	.v
coq.inria.fr

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.

Coq implements a dependently typed functional programming language.^[2]

It is developed in France, in the PI.R2 team of the PPS laboratory,^[3] jointly operated by INRIA, École Polytechnique, Paris-Sud 11 University, Paris Diderot University and CNRS. There was also formerly a group at École Normale Supérieure de Lyon. The project manager of Coq is Hugo Herbelin. Coq is implemented in OCaml.

The word coq means "rooster" in French, and stems from a tradition of naming French research development tools with animal names.^[4] It is also a reference to Thierry Coquand, who developed the aforementioned calculus of constructions along with Gérard Huet. Also, at first it was simply called Coc, the acronym of calculus of construction.

Four color theorem and ssreflect extension [edit]

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.^[5]

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.^[6]

Applications [edit]

• Four color theorem: formal proof using Coq was completed in September 2004.
• Feit–Thompson theorem: formal proof using Coq was completed in September 2012.^[7]
• CompCert an optimizing compiler for C (programming language) which is fully programmed and proved in Coq.

See also [edit]

	Mathematics portal
	Free software portal

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

References [edit]

External links [edit]

Wikimedia Commons has media related to: Coq

• The Coq proof assistant – the official English website
• 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 draft textbook by Adam Chlipala
• Software Foundations – Online textbook by Benjamin C. Pierce et al.

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.