Coq is a proof assistant which handles 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 a theorem prover but includes automatic theorem proving tactics. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object with certain properties by creating or providing a method for creating such an object. ... The calculus of constructions (CoC) is a higher-order typed lambda calculus where types are first-class values. ...

It is developed in France, in the LogiCal project, jointly operated by INRIA, École Polytechnique, University Paris XI and CNRS (there was also formerly a group at École Normale Supérieure de Lyon). The team leaders are Prof. Gilles Dowek and Prof. Christine Paulin-Mohring. Coq is written in the Ocaml programming language. The Institut national de recherche en informatique et en automatique (INRIA) is a French national research institution focusing on computer science, control theory and applied mathematics. ... For other Écoles Polytechniques, see École Polytechnique de Montréal and École Polytechnique Fédérale de Lausanne. ... The Centre National de la Recherche Scientifique (CNRS) is one of the most prominent scientific research institutions in France. ... The École Normale Supérieure de Lyon (also known as ENSL, ENS-lyon or Normale Sup Lyon) is an elite grande école located in Lyon, France, in the district of Gerland. ... Objective Caml, also known as OCaml or OCaml for short, is an advanced programming language that is part of the ML family. ...

Coq means "rooster" in French - and Thierry Coquand (along with Gérard Huet) developed the aforementioned calculus of constructions. This article concerns biological roosters. ... The calculus of constructions (CoC) is a higher-order typed lambda calculus where types are first-class values. ...

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) is a French national research institution focusing on computer science, control theory and applied mathematics. ... Microsoft Corporation (NASDAQ: MSFT, HKEx: 4338) is the worlds largest software company, with 2005 global annual sales of almost $40 billion USD and nearly 60,000 employees in 85 countries and regions. ... Map of the Cambridgeshire area (1904) The city of Cambridge is an old English university town and the administrative centre of the county of Cambridgeshire. ... Royal motto (French): Dieu et mon droit (Translated: God and my right) Englands location within the British Isles Official language English de facto Capital London de facto Largest city London Area – Total Ranked 1st UK 130,395 km² Population – Total (mid-2004) – Total (2001 Census) – Density Ranked 1st UK... Example of a four color map Example of a map with non-contiguous regions The four color theorem states that given any plane separated into regions, such as a political map of the counties of a state, the regions may be colored using no more than four colors in such...

See also

• Intuitionistic Type Theory

Intuitionistic Type Theory, or Constructive Type Theory, or Martin-Löf Type Theory or just Type Theory (with capital letters) is at the same time a functional programming language, a logic and a set theory based on the principles of mathematical constructivism. ...

External links

• The Coq proof assistant
• Development of theories and tactics: Four Color Theorem
• A Wiki for Coq