# Coq

Developer(s) The Coq development team 1 May 1989; 33 years ago (version 4.10) 8.16.0[1]  / 5 September 2022; 23 days ago github.com/coq/coq OCaml Cross-platform English Proof assistant LGPLv2.1 coq.inria.fr
An interactive proof session in CoqIDE, showing the proof script on the left and the proof state on the right.

Coq is an interactive theorem prover first released in 1989. It allows for expressing mathematical assertions, mechanically checks proofs of these assertions, helps 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 (procedures) and various decision procedures.

The Association for Computing Machinery awarded Thierry Coquand, Gérard 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.

Coq is a wordplay on the name of Thierry Coquand, Calculus of Constructions or "CoC" and is following the French tradition to name tools after animals (coq in French meaning rooster).[2]

## Overview

When viewed as a programming language, Coq implements a dependently typed functional programming language;[3] when viewed 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, ENS Lyon was also part of the project. The development of Coq was initiated by Gérard Huet and Thierry Coquand, and more than 40 people, mainly researchers, have contributed features to the core system since its inception. The implementation team has successively been coordinated by Gérard Huet, Christine Paulin-Mohring, Hugo Herbelin, and Matthieu Sozeau. Coq is mainly implemented in OCaml with a bit of C. The core system can be extended by way of a plug-in mechanism.[4]

The name coq means "rooster" in French and stems from a French tradition of naming research development tools after animals.[5] Up until 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 was changed from CoC to Coq in an indirect reference to Coquand, who developed the Calculus of Constructions along with Gérard Huet and contributed to the Calculus of Inductive Constructions with Christine Paulin-Mohring.[6]

Coq provides a specification language called Gallina[7] ("hen" in Latin, Spanish, Italian and Catalan). Programs written in Gallina have the weak normalization property, implying that they always terminate. This is a distinctive property of the language, since infinite loops (non-terminating programs) are common in other programming languages,[8] and is one way to avoid the halting problem.

As an example, a proof of commutativity of addition on natural numbers in Coq:

```plus_comm =
fun n m : nat =>
nat_ind (fun n0 : nat => n0 + m = m + n0)
(plus_n_0 m)
(fun (y : nat) (H : y + m = m + y) =>
eq_ind (S (m + y))
(fun n0 : nat => S (y + m) = n0)
(f_equal S H)
(m + S y)
(plus_n_Sm m y)) n
: forall n m : nat, n + m = m + n
```

`nat_ind` stands for mathematical induction, `eq_ind` for substitution of equals, and `f_equal` for taking the same function on both sides of the equality. Earlier theorems are referenced showing ${\displaystyle m=m+0}$ and ${\displaystyle S(m+y)=m+Sy}$.

## Notable uses

### Four color theorem and SSReflect extension

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 2002.[9] Their work led to the development of the SSReflect ("Small Scale Reflection") package, which was a significant extension to Coq.[10] Despite its name, most of the features added to Coq by SSReflect are general-purpose features and are not limited to the computational reflection style of proof. These features 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 1.11 is freely available, dual-licensed under the open source CeCILL-B or CeCILL-2.0 license, and compatible with Coq 8.11.[11]

## References

1. ^ "Release Coq 8.16.0". 5 September 2022. Retrieved 5 September 2022.
2. ^ "Alternative names · coq/coq Wiki". GitHub. Retrieved 2021-11-11.
3. ^ A short introduction to Coq
4. ^ Avigad, Jeremy; Mahboubi, Assia (3 July 2018). Interactive Theorem Proving: 9th International Conference, ITP 2018, Held as ... ISBN 9783319948218. Retrieved 21 October 2018.
5. ^ "Frequently Asked Questions". Retrieved 2019-05-08.
6. ^ "Introduction to the Calculus of Inductive Constructions". Retrieved 21 May 2019.
7. ^ Adam Chlipala. "Certified Programming with Dependent Types": "Library Universes".
8. ^ Adam Chlipala. "Certified Programming with Dependent Types": "Library GeneralRec". "Library InductiveTypes".
9. ^ Gonthier, Georges (2008), "Formal Proof—The Four-Color Theorem" (PDF), Notices of the American Mathematical Society, vol. 55, no. 11, pp. 1382–1393, MR 2463991
10. ^ Georges Gonthier, Assia Mahboubi. "An introduction to small scale reflection in Coq": "Journal of Formalized Reasoning".
11. ^
12. ^ Conchon, Sylvain; Filliâtre, Jean-Christophe (October 2007), "A Persistent Union-Find Data Structure", ACM SIGPLAN Workshop on ML, Freiburg, Germany
13. ^ "Feit-Thompson theorem has been totally checked in Coq". Msr-inria.inria.fr. 2012-09-20. Archived from the original on 2016-11-19. Retrieved 2012-09-25.