coq-community / buchberger

Verified implementation in Coq of Buchberger's algorithm for computing Gröbner bases [maintainer=@palmskog]

Buchberger

A verified implementation of Buchberger's algorithm in Coq, which computes the Gröbner basis associated with a polynomial ideal. Also includes a constructive proof of Dickson's lemma.

Meta

• Author(s):
  • Laurent Théry (initial)
  • Henrik Persson (initial)
• Coq-community maintainer(s):
  • Karl Palmskog (@palmskog)
• License: GNU Lesser General Public License v2.1 or later
• Compatible Coq versions: 8.17 or later
• Additional dependencies: none
• Coq namespace: Buchberger
• Related publication(s):
  • A machine checked implementation of Buchberger's algorithm doi:10.1023/A:1026518331905
  • An Integrated Development of Buchberger's Algorithm in Coq
  • Gröbner Bases in Type Theory doi:10.1007/3-540-48167-2_3

Building and installation instructions

The easiest way to install the latest released version of Buchberger is via OPAM:

opam repo add coq-released https://coq.inria.fr/opam/released
opam install coq-buchberger

To instead build and install manually, do:

git clone https://github.com/coq-community/buchberger.git
cd buchberger
make   # or make -j <number-of-cores-on-your-machine> 
make install

Documentation

This project contains a Coq formalization of Buchberger's algorithm. It is composed of:

• A proof of correctness of the algorithm as described in A machine checked implementation of Buchberger's algorithm, Journal of Automated Reasoning, January 2001.
• An implementation of the algorithm. With respect to the paper, terms are not abstracted but built directly from coef and monomials.
• A constructive proof of Dickson's lemma due to Henrik Persson.

In the file Extract.v, it is explained how the extracted code found in sin_num.ml can be used to compute Gröbner bases.

Related work

An alternative formalization of Gröbner bases in Coq using the SSReflect proof language and the Mathematical Components library is available elsewhere.

buchberger's Issues

Use well-ordering facts from Stdlib once they are axiom free

In coq/coq#15254, Gaëtan Gilbert makes the proofs of well-ordering in Stdlib axiom free. If this PR is merged, the reimplementation here of some Stdlib facts should be removed. This also makes the whole project LGPL-2.1-or-later again.

Purge Relation_Operators_compat

The module Relation_Operators_compat was introduced in this commit to work around changes to Desc. It would be better for the workaround to be restricted to a few lemmas instead.