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Equations - a function definition plugin.

Build Status DOI Zulip Chat

Copyright 2009-2021 Matthieu Sozeau
Copyright 2015-2018 Cyprien Mangin

Distributed under the terms of the GNU Lesser General Public License Version 2.1 or later (see LICENSE for details).

Equations provides a notation for writing programs by dependent pattern-matching and (well-founded) recursion in Coq. It compiles everything down to eliminators for inductive types, equality and accessibility, providing a definitional extension to the Coq kernel.

Live demo

Try it now in your browser with JSCoq!


  • The reference manual provides an introduction and a summary of the commands and options. This introduction can also be followed interactively with Equations installed: equations_intro.v

  • A gallery of examples provides more consequent developments using Equations.


Two articles describing the system are available:

  • Equations Reloaded: High-Level Dependently-Typed Functional Programming and Proving in Coq. Matthieu Sozeau and Cyprien Mangin. 2019. Proc. ACM Program. Lang. 3, ICFP, Article 86 (August 2019), 29 pages. DOI. See Equations Reloaded for associated material, including a VM to run the examples.

  • Equations: A Dependent Pattern-Matching Compiler Matthieu Sozeau (2010) In: Kaufmann M., Paulson L.C. (eds) Interactive Theorem Proving. ITP 2010. Lecture Notes in Computer Science, vol 6172. Springer, Berlin, Heidelberg.

We did a case study on a proof of normalization for an hereditary substitution procedure:


The current development version works with Coq 8.8 (branch 8.8), Coq 8.9 (branch 8.9), Coq 8.10 (branch 8.10), and the current Coq master branch (branch master).

See releases for sources and official releases.

Install with OPAM

This package is available on OPAM. Activate the Coq repository if you didn't do it yet:

opam repo add coq-released

and run:

opam install coq-equations

To get the beta versions of Coq, activate the repository:

opam repo add coq-core-dev

To get the development version of Equations, activate the repository:

opam repo add coq-extra-dev

Install from source

Alternatively, to compile Equations, simply run:


in the toplevel directory, with coqc and ocamlc in your path.

Optionally, one can build the test-suite or examples:

make examples test-suite

Then add the paths to your .coqrc:

Add ML Path "/Users/mat/research/coq/equations/src".
Add Rec LoadPath "/Users/mat/research/coq/equations/theories" as Equations.

Or install it:

make install

As usual, you will need to run this command with the appropriate privileges if the version of Coq you are using is installed system-wide, rather than in your own directory. E.g. on Ubuntu, you would prefix the command with sudo and then enter your user account password when prompted.

HoTT Variant

The HoTT variant of Equations works with the coq-hott library for Coq 8.13 and up. When using opam, simply install first the coq-hott library and coq-equations will install its HoTT variant. From source, first install coq-hott and then use:

./ --enable-hott

This will compile the HoTT library variant in addition to the standard one. Then, after make install, one can import the plugin in Coq, using:

From Equations Require Import HoTT.All.