Best Free and Open Source Proof Assistants

Coq – formal proof management system

Coq is a formal proof management system. It provides a formal language to write mathematical definitions, executable algorithms and theorems together with an environment for semi-interactive development of machine-checked proofs.

Typical applications include the certification of properties of programming languages (e.g. the CompCert compiler certification project, the Verified Software Toolchain for verification of C programs, or the Iris framework for concurrent separation logic), the formalization of mathematics (e.g. the full formalization of the Feit-Thompson theorem, or homotopy type theory), and teaching.

Users generate proofs by entering a series of tactics that constitute steps in the proof. There are many built-in tactics, some of which are elementary, while others implement complex decision procedures (such as lia, a decision procedure for linear integer arithmetic). Ltac and its planned replacement, Ltac2, provide languages to define new tactics by combining existing tactics with looping and conditional constructs. These permit automation of large parts of proofs and sometimes entire proofs. Furthermore, users can add novel tactics or functionality by creating Coq plugins using OCaml.

The Coq kernel, a small part of Coq, does the final verification that the tactic-generated proof is valid. Usually the tactic-generated proof is indeed correct, but delegating proof verification to the kernel means that even if a tactic is buggy, it won’t be able to introduce an incorrect proof into the system.

Finally, Coq also supports extraction of verified programs to programming languages such as OCaml and Haskell. This provides a way of executing Coq code efficiently and can be used to create verified software libraries.

Support: Documentation, GitHub Code Repository
Developer: INRIA Formel, Coq, LogiCal, ProVal, TypiCal, Marelle, pi.r2, Ascola, Galinette projects (starting 1985),- Laboratoire de l’Informatique du Parallelisme (LIP)
associated to CNRS and ENS Lyon (Sep. 1989 to Aug. 1997), Laboratoire de Recherche en Informatique (LRI) associated to CNRS and university Paris Sud (since Sep. 1997), Laboratoire d’Informatique de l’Ecole Polytechnique (LIX) associated to CNRS and Ecole Polytechnique (since Jan. 2003). Laboratoire PPS associated to CNRS and University Paris Diderot
License: GNU Lesser General Public License v2.1

Coq is written in OCaml and Coq. Learn OCaml with our recommended free books and free tutorials.

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