Equations - a function definition plugin.

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Copyright 2009-2019 Matthieu Sozeau matthieu.sozeau@inria.fr
Copyright 2015-2018 Cyprien Mangin cyprien.mangin@m4x.org

Distributed under the terms of the GNU Lesser General Public License Version 2.1 (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 demonstration

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:

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) 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 https://coq.inria.fr/opam/released

and run:

opam install coq-equations

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

opam repo add coq-core-dev https://coq.inria.fr/opam/core-dev

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

opam repo add coq-extra-dev https://coq.inria.fr/opam/extra-dev

Install from source

Alternatively, to compile Equations, simply run:

coq_makefile -f _CoqProject -o Makefile

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

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.