An example of proof written as a functional program: the proof of commutativity of addition on natural numbers in the proof assistant Coq

In computer science, Coq is a proof assistant application. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their Interactive theorem proving is the field of Computer science and Mathematical logic concerned with tools to develop Formal proofs by man-machine collaboration 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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a constructive proof is a method of proof that demonstrates the existence of a Mathematical object with certain properties by creating Coq works within the theory of the calculus of inductive constructions, a derivative of the calculus of constructions. The calculus of inductive constructions is the underlying core language of the Coq Proof Assistant. The calculus of constructions (CoC is a higher-order Typed lambda calculus, initially developed by Thierry Coquand, where types are First-class values Coq is not an automated theorem prover but includes automatic theorem proving tactics and various decision procedures. Automated theorem proving ( ATP) or automated deduction, currently the most well-developed subfield of Automated reasoning (AR is the

It is developed in France, in the TypiCal (ex-LogiCal) project, jointly operated by INRIA, École Polytechnique, University Paris XI and CNRS. This article is about the country For a topic outline on this subject see List of basic France topics. The Institut national de recherche en informatique et en automatique ( INRIA) ( English National Institute for Research in Computer Science and Control) is For other Écoles Polytechniques see École Polytechnique de Montréal and École Polytechnique Fédérale de Lausanne. There was also formerly a group at École Normale Supérieure de Lyon. The École Normale Supérieure de Lyon (also known as ENSL, ENS-Lyon or Normale Sup' Lyon) is an Elite Grande école The team leader is Senior Scientist Benjamin Werner. Coq is implemented in Objective Caml. Objective Caml ( OCaml) is the main implementation of the Caml Programming language, created by Xavier Leroy, Jérôme Vouillon

The word coq means "cock" (rooster) in French, and stems from a tradition of naming French research development tools with animal names. A rooster (also called a cock or chanticleer) is a male Chicken ( Gallus gallus) the female being called a Hen. French ( français,) is a Romance language spoken around the world by 118 million people as a native language and by about 180 to 260 million people It is also a reference to Thierry Coquand, who developed the aforementioned calculus of constructions along with Gérard Huet.

Four color theorem and ssreflect extension

Benjamin Werner (of INRIA) and Georges Gonthier (of Microsoft Research, in Cambridge, England) used Coq to create a surveyable proof of the four color theorem, which was completed in September 2004. The Institut national de recherche en informatique et en automatique ( INRIA) ( English National Institute for Research in Computer Science and Control) is Microsoft Research (MSR is a division of Microsoft created in 1991 for researching various Computer science topics and issues The city of Cambridge (ˈkeɪmbrɪdʒ is a university town and the administrative centre of the county of Cambridgeshire, England England is a Country which is part of the United Kingdom. Its inhabitants account for more than 83% of the total UK population whilst its mainland The four color theorem (also known as the four color map theorem) states that given any plane separated into regions such as a political map of the states of a country September 2004: January - February - March - April - May - June - July - August ^[1]

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. 1 is freely available under the open source CeCill-B license, and is compatible with Coq 8. CeCILL (from " CE A C NRS I NRIA L ogiciel L ibre") is a Free software licence adapted to both international 1 (patch levels 2 and 3). ^[2]

See also

• Curry-Howard
• Intuitionistic type theory

References

External links

• The Coq proof assistant
• Cocorico!, the Coq Wiki
• MSR Inria math components - hosts the Ssreflect extension