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Equations - a function definition plugin.

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Copyright 2009-2022 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. The plugin can be used with Coq's standard logic in Prop for a proof-irrelevant, erasable interpretation of pattern-matching, or with a polymorphic logic in Type or re-using the prelude of the HoTT/Coq library for a proof-relevant interpretation. In all cases, the resulting definitions are axiom-free.

Table of Contents

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.

Papers and presentations


The latest version works with Coq 8.13 (branch 8.13), Coq 8.14 (branch 8.14), Coq 8.15 (branch 8.15), and the current Coq main branch (branch main).

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.