https://github.com/rocq-community/coqeal
The Coq Effective Algebra Library [maintainers=@CohenCyril,@proux01]
https://github.com/rocq-community/coqeal
coq coq-ci coq-platform mathcomp mathcomp-ci refinement
Last synced: 20 days ago
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The Coq Effective Algebra Library [maintainers=@CohenCyril,@proux01]
- Host: GitHub
- URL: https://github.com/rocq-community/coqeal
- Owner: rocq-community
- License: other
- Created: 2014-02-10T12:35:29.000Z (about 11 years ago)
- Default Branch: master
- Last Pushed: 2025-02-28T15:17:06.000Z (about 2 months ago)
- Last Synced: 2025-03-24T15:06:59.055Z (about 1 month ago)
- Topics: coq, coq-ci, coq-platform, mathcomp, mathcomp-ci, refinement
- Language: Coq
- Homepage:
- Size: 2.01 MB
- Stars: 69
- Watchers: 13
- Forks: 17
- Open Issues: 9
-
Metadata Files:
- Readme: README.md
- License: LICENSE
Awesome Lists containing this project
- awesome-coq - CoqEAL - Framework to ease change of data representations in proofs. (Projects / Frameworks)
README
# CoqEAL
[![Docker CI][docker-action-shield]][docker-action-link]
[![Contributing][contributing-shield]][contributing-link]
[![Code of Conduct][conduct-shield]][conduct-link]
[![Zulip][zulip-shield]][zulip-link]
[docker-action-shield]: https://github.com/coq-community/coqeal/actions/workflows/docker-action.yml/badge.svg?branch=master
[docker-action-link]: https://github.com/coq-community/coqeal/actions/workflows/docker-action.yml
[contributing-shield]: https://img.shields.io/badge/contributions-welcome-%23f7931e.svg
[contributing-link]: https://github.com/coq-community/manifesto/blob/master/CONTRIBUTING.md
[conduct-shield]: https://img.shields.io/badge/%E2%9D%A4-code%20of%20conduct-%23f15a24.svg
[conduct-link]: https://github.com/coq-community/manifesto/blob/master/CODE_OF_CONDUCT.md
[zulip-shield]: https://img.shields.io/badge/chat-on%20zulip-%23c1272d.svg
[zulip-link]: https://coq.zulipchat.com/#narrow/stream/237663-coq-community-devs.20.26.20users
This Coq library contains a subset of the work that was developed in the context
of the ForMath EU FP7 project (2009-2013). It has two parts:
- theory, which contains developments in algebra including normal forms of matrices,
and optimized algorithms on MathComp data structures.
- refinements, which is a framework to ease change of data representations during a proof.
## Meta
- Author(s):
- Guillaume Cano (initial)
- Cyril Cohen (initial)
- Maxime Dénès (initial)
- Érik Martin-Dorel
- Anders Mörtberg (initial)
- Damien Rouhling
- Pierre Roux
- Vincent Siles (initial)
- Coq-community maintainer(s):
- Cyril Cohen ([**@CohenCyril**](https://github.com/CohenCyril))
- Pierre Roux ([**@proux01**](https://github.com/proux01))
- License: [MIT License](LICENSE)
- Compatible Coq versions: 8.20 or later (use releases for other Coq versions)
- Additional dependencies:
- [Bignums](https://github.com/coq/bignums) same version as Coq
- [Coq-Elpi](https://github.com/LPCIC/coq-elpi) 2.4.1 or later
- [Hierarchy Builder](https://github.com/math-comp/hierarchy-builder) 1.4.0 or later
- [MathComp ssreflect](https://math-comp.github.io) 2.3 or later
- [MathComp algebra](https://math-comp.github.io) 2.1 or later
- [MathComp Multinomials](https://github.com/math-comp/multinomials) 2.0 or later
- [MathComp real-closed](https://math-comp.github.io) 2.0 or later
- Coq namespace: `CoqEAL`
- Related publication(s):
- [A refinement-based approach to computational algebra in Coq](https://hal.inria.fr/hal-00734505/document) doi:[10.1007/978-3-642-32347-8_7](https://doi.org/10.1007/978-3-642-32347-8_7)
- [Refinements for free!](https://hal.inria.fr/hal-01113453/document) doi:[10.1007/978-3-319-03545-1_10](https://doi.org/10.1007/978-3-319-03545-1_10)
- [A Coq Formalization of Finitely Presented Modules](https://hal.inria.fr/hal-01378905/document) doi:[10.1007/978-3-319-08970-6_13](https://doi.org/10.1007/978-3-319-08970-6_13)
- [Formalized Linear Algebra over Elementary Divisor Rings in Coq](https://hal.inria.fr/hal-01081908/document) doi:[10.2168/LMCS-12(2:7)2016](https://doi.org/10.2168/LMCS-12(2:7)2016)
- [A refinement-based approach to large scale reflection for algebra](https://hal.inria.fr/hal-01414881/document)
- [Interaction entre algèbre linéaire et analyse en formalisation des mathématiques](https://tel.archives-ouvertes.fr/tel-00986283/)
- [A formal proof of Sasaki-Murao algorithm](https://jfr.unibo.it/article/view/2615) doi:[10.6092/issn.1972-5787/2615](https://doi.org/10.6092/issn.1972-5787/2615)
- [Formalizing Refinements and Constructive Algebra in Type Theory](http://hdl.handle.net/2077/37325)
- [Coherent and Strongly Discrete Rings in Type Theory](https://staff.math.su.se/anders.mortberg/papers/coherent.pdf) doi:[10.1007/978-3-642-35308-6_21](https://doi.org/10.1007/978-3-642-35308-6_21)
## Building and installation instructions
The easiest way to install the latest released version of CoqEAL
is via [OPAM](https://opam.ocaml.org/doc/Install.html):
```shell
opam repo add coq-released https://coq.inria.fr/opam/released
opam install coq-coqeal
```
To instead build and install manually, do:
``` shell
git clone https://github.com/coq-community/coqeal.git
cd coqeal
make # or make -j
make install
```
## Theory
The theory directory has the following content:
- `ssrcomplements`, `minor` `mxstructure`, `polydvd`, `similar`,
`binetcauchy`, `ssralg_ring_tac`: Various extensions of the
Mathematical Components library.
- `dvdring`, `coherent`, `stronglydiscrete`, `edr`: Hierarchy of
structures with divisibility (from rings with divisibility, PIDs,
elementary divisor rings, etc.).
- `fpmod`: Formalization of finitely presented modules.
- `kaplansky`: For providing elementary divisor rings from the
Kaplansky condition.
- `closed_poly`: Polynomials with coefficients in a closed field.
- `companion`, `frobenius_form`, `jordan`, `perm_eq_image`,
`smith_complements`: Results on normal forms of matrices.
- `bareiss_dvdring`, `bareiss`, `gauss`, `karatsuba`, `rank`,
`strassen`, `toomcook`, `smithpid`, `smith`: Various efficient
algorithms for computing operations on polynomials or matrices.
## Refinements
The refinements directory has the following content:
- `refinements`: Classes for refinements and refines together with
operational typeclasses for common operations.
- `binnat`: Proof that the binary naturals of Coq (`N`) are a refinement
of the MathComp unary naturals (`nat`) together with basic operations.
- `binord`: Proof that the binary natural numbers of Coq (`N`) are a refinement
of the MathComp ordinals.
- `binint`: MathComp integers (`ssrint`) are refined to a new type
parameterized by positive numbers (represented by a sigma type) and
natural numbers. This means that proofs can be done using only
lemmas from the MathComp library which leads to simpler proofs than
previous versions of `binint` (e.g., `N`).
- `binrat`: Arbitrary precision rational numbers (`bigQ`) from the
[Bignums](https://github.com/coq/bignums) library are refined to
MathComp's rationals (`rat`).
- `rational`: The rational numbers of MathComp (`rat`) are refined to
pairs of elements refining integers using parametricity of
refinements.
- `seqmatrix` and `seqmx_complements`: Refinement of MathComp
matrices (`M[R]_(m,n)`) to lists of lists (`seq (seq R)`).
- `seqpoly`: Refinement of MathComp polynomials (`{poly R}`) to lists (`seq R`).
- `multipoly`: Refinement of
[MathComp multinomials](https://github.com/math-comp/multinomials)
and multivariate polynomials to Coq
[finite maps](https://github.com/coq/coq/blob/master/theories/FSets/FMapAVL.v).
Files should use the following conventions (w.r.t. `Local` and `Global` instances):
```coq
(** Part 1: Generic operations *)
Section generic_operations.
Global Instance generic_operation := ...
(** Part 2: Correctness proof for proof-oriented types and programs *)
Section theory.
Local Instance param_correctness : param ...
(** Part 3: Parametricity *)
Section parametricity.
Global Instance param_parametricity : param ...
Proof. exact: param_trans. Qed.
End parametricity.
End theory.
```