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Coq logo.png
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
Filename extension(s) .v
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] Also, at first it was simply called Coc, the acronym of the calculus of construction, and a reference to Thierry Coquand, who developed the aforementioned calculus of constructions along with Gérard Huet.

Coq implements Gallina.[5] Programs written in Gallina have the strong 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. [6]

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.[7]

Based on this work, a significant extension to Coq was developed called Ssreflect (which stands for "small scale reflection").[8] 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.[9]


See also[edit]


  1. ^ What is Coq ? | The Coq Proof Assistant. Retrieved on 2013-07-21.
  2. ^ A short introduction to Coq.
  3. ^ PI.R2
  4. ^ Coq Version 8.0 for the Clueless (174 Hints). Retrieved on 2013-11-07.
  5. ^ Adam Chlipala. "Certified Programming with Dependent Types": "Library Universes".
  6. ^ Adam Chlipala. "Certified Programming with Dependent Types": "Library GeneralRec". "Library InductiveTypes".
  7. ^ Development of theories and tactics: Four Color Theorem
  8. ^ Georges Gonthier, Assia Mahboubi. "An introduction to small scale reflection in Coq": "Journal of Formalized Reasoning".
  9. ^ "Ssreflect 1.4 has been released - Microsoft Research Inria Joint Centre". Retrieved 2014-01-27. 
  10. ^ "Feit-Thompson theorem has been totally checked in Coq". 2012-09-20. Retrieved 2012-09-25. 

External links[edit]