Developer(s) | The Coq development team |
---|---|
Initial release | 1 May 1989 (version 4.10) |
Stable release | |
Repository | github |
Written in | OCaml |
Operating system | Cross-platform |
Available in | English |
Type | Proof assistant |
License | LGPLv2.1 |
Website | coq |
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.
The name "Coq" is a wordplay on the name of Thierry Coquand, Calculus of Constructions or "CoC" and follows the French computer science tradition of naming software after animals (coq in French meaning rooster). [2] On October 11th, 2023, the development team announced that Coq will be renamed "The Rocq Prover" in the coming months, and has started updating the code base, website and associated tools. [3]
When viewed as a programming language, Coq implements a dependently typed functional programming language; [4] 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. [5]
The name coq means 'rooster' in French and stems from a French tradition of naming research development tools after animals. [6] 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. [7]
Coq provides a specification language called Gallina [8] ("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, [9] 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=funnm:nat=>nat_ind(funn0:nat=>n0+m=m+n0)(plus_n_0m)(fun(y:nat)(H:y+m=m+y)=>eq_ind(S(m+y))(funn0:nat=>S(y+m)=n0)(f_equalSH)(m+Sy)(plus_n_Smmy))n:forallnm: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 and .
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. [10] Their work led to the development of the SSReflect ("Small Scale Reflection") package, which was a significant extension to Coq. [11] 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:
set
tactic with more powerful matchingSSReflect 1.11 is freely available, dual-licensed under the open source CeCILL-B or CeCILL-2.0 license, and compatible with Coq 8.11. [12]
In addition to constructing Gallina terms explicitly, Coq supports the use of tactics written in the built-in language Ltac or in OCaml. These tactics automate the construction of proofs, carrying out trivial or obvious steps in proofs. [15] Several tactics implement decision procedures for various theories. For example, the "ring" tactic decides the theory of equality modulo ring or semiring axioms via associative-commutative rewriting. [16] For example, the following proof establishes a complex equality in the ring of integers in just one line of proof: [17]
RequireImportZArith.OpenScopeZ_scope.Goalforallabc:Z,(a+b+c)^2=a*a+b^2+c*c+2*a*b+2*a*c+2*b*c.intros;ring.Qed.
Built-in decision procedures are also available for the empty theory ("congruence"), propositional logic ("tauto"), quantifier-free linear integer arithmetic ("lia"), and linear rational/real arithmetic ("lra"). [18] [19] Further decision procedures have been developed as libraries, including one for Kleene algebras [20] and another for certain geometric goals. [21]
In algorithmic information theory, the Kolmogorov complexity of an object, such as a piece of text, is the length of a shortest computer program that produces the object as output. It is a measure of the computational resources needed to specify the object, and is also known as algorithmic complexity, Solomonoff–Kolmogorov–Chaitin complexity, program-size complexity, descriptive complexity, or algorithmic entropy. It is named after Andrey Kolmogorov, who first published on the subject in 1963 and is a generalization of classical information theory.
OCaml is a general-purpose, high-level, multi-paradigm programming language which extends the Caml dialect of ML with object-oriented features. OCaml was created in 1996 by Xavier Leroy, Jérôme Vouillon, Damien Doligez, Didier Rémy, Ascánder Suárez, and others.
In mathematics and theoretical computer science, a type theory is the formal presentation of a specific type system. Type theory is the academic study of type systems.
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In programming language theory and proof theory, the Curry–Howard correspondence is the direct relationship between computer programs and mathematical proofs.
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In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human-machine collaboration. This involves some sort of interactive proof editor, or other interface, with which a human can guide the search for proofs, the details of which are stored in, and some steps provided by, a computer.
In computer science and logic, a dependent type is a type whose definition depends on a value. It is an overlapping feature of type theory and type systems. In intuitionistic type theory, dependent types are used to encode logic's quantifiers like "for all" and "there exists". In functional programming languages like Agda, ATS, Coq, F*, Epigram, Idris, and Lean, dependent types help reduce bugs by enabling the programmer to assign types that further restrain the set of possible implementations.
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Agda is a dependently typed functional programming language originally developed by Ulf Norell at Chalmers University of Technology with implementation described in his PhD thesis. The original Agda system was developed at Chalmers by Catarina Coquand in 1999. The current version, originally known as Agda 2, is a full rewrite, which should be considered a new language that shares a name and tradition.
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Thierry Coquand is a French computer scientist and mathematician who is currently a professor of computer science at the University of Gothenburg, having previously worked at INRIA. He is known for his work in constructive mathematics, especially the calculus of constructions.
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