This article or section appears to contradict itself on the date when the surveyable proof of the four color theorem was completed.(January 2017)
|Developer(s)||The Coq development team|
|Initial release||1 May 1989(version 4.10)|
8.8.1 / 9 July 2018
|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.
The Association for Computing Machinery rewarded Thierry Coquand, Gérard Pierre 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.
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 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, 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 Paulin-Mohring 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 Pierre Huet and the Calculus of Inductive Constructions along with Christine Paulin-Mohring.
Coq provides a specification language called Gallina (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 (non-termination) are common in other programming languages.
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:
settactic with more powerful matching