Proof assistant
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). On October 11, 2023, the development team announced that Coq will be renamed "The Rocq Prover" in coming months, and began updating the code base, website, and associated tools.
Overview.
When viewed as a programming language, Coq implements a dependently typed functional programming model; when viewed as a logical system, it implements a higher-order type theory. The development of Coq has been supported since 1984 by French Institute for Research in Computer Science and Automation (INRIA), now in collaboration with École Polytechnique, University of Paris-Sud, Paris Diderot University, and French National Centre for Scientific Research (CNRS). In the 1990s, École normale supérieure de Lyon (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.
The name means 'rooster' in French and stems from a French tradition of naming research development tools after animals. Up until 1991, Coquand was implementing a language called the "calculus of constructions" and it was simply called "CoC" then. In 1991, a new implementation based on the extended "calculus of inductive constructions" was begun and the name 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.
Coq provides a specification language called Gallina ("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,
and is one way to avoid the halting problem.
As an example, consider a proof of a lemma that taking the successor of a natural number flips its parity. The fold-unfold tactic introduced by Danvy is used to help keep the proof simple.
Ltac fold_unfold_tactic name := intros; unfold name; fold name; reflexivity.
Require Import Arith Nat Bool.
Fixpoint is_even (n : nat) : bool :=
 match n with
 | 0 = ;
 true
 | S n' = ;
 eqb (is_even n') false
 end.
Lemma fold_unfold_is_even_0:
 is_even 0 = true.
Proof.
 fold_unfold_tactic is_even.
Qed.
Lemma fold_unfold_is_even_S:
 forall n' : nat,
 is_even (S n') = eqb (is_even n') false.
Proof.
 fold_unfold_tactic is_even.
Qed.
Lemma successor_flips_evenness:
 forall n : nat,
 is_even n = negb (is_even (S n)).
Proof.
 intro n.
 rewrite - ; (fold_unfold_is_even_S n).
 destruct (is_even n).
 * simpl.
 reflexivity.
 * simpl.
 reflexivity.
Qed.
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. Their work led to the development of the SSReflect ("Small Scale Reflection") package, which was a significant extension to Coq. Despite its name, most of the features added to Coq by SSReflect are general-purpose features and are not limited to the computational reflective programming style of proof. These features include:
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.
Tactic language.
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. 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. For example, the following proof establishes a complex equality in the ring of integers in just one line of proof:
Require Import ZArith.
Open Scope Z_scope.
Goal forall a b c: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"). Further decision procedures have been developed as libraries, including one for Kleene algebras and another for certain geometric goals.
References.
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