Equations - a function definition plugin.

Copyright 2009-2022 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 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

• Documentation
• Papers
• Gallery
• Installation
• HoTT Variant

Learning Equations

• You can discover and learn Equations through interactive tutorials directly in your browser, or locally with Equations installed.

Documentation

• The reference manual provides a summary of the commands and options, and a brief introduction. This introduction can also be followed interactively with Equations installed, equations_intro.v or interactively in your browser.

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

Papers and presentations

• Equations Reloaded: High-Level Dependently-Typed Functional Programming and Proving in Coq. Matthieu Sozeau and Cyprien Mangin. In: Proc. ACM Program. Lang. 3, ICFP, Article 86 (August 2019), 29 pages. DOI, slides.

  This presents version 1.2 and above of the package. See Equations Reloaded for associated material, including a VM to run the examples.

• Equations for HoTT. Matthieu Sozeau, Talk given at the Homotopy Type Theory 2019 Conference in Pittsburgh, PA, August 2019.

  This explains the no-confusion principle and strong equivalences used by Equations and Jesper Cockx's version of dependent pattern-matching in Agda in terms of HoTT.

• Equations for Hereditary Substitution in Leivant's Predicative System F: A Case Study. Cyprien Mangin and Matthieu Sozeau. In: Proceedings Tenth International Workshop on Logical Frameworks and Meta Languages: Theory and Practice. Volume 185 of EPTCS. May 2015 - LFMTP'15.

  This is a case study on a proof of normalization for an hereditary substitution procedure on a variant of System F.

• 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.

  This presents an earlier version of the package.

Installation

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 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:

./configure.sh
make

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:

./configure.sh --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.