https://github.com/math-comp/real-closed
Theorems for Real Closed Fields
https://github.com/math-comp/real-closed
coq mathcomp real-closed-fields ssreflect
Last synced: 5 months ago
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Theorems for Real Closed Fields
- Host: GitHub
- URL: https://github.com/math-comp/real-closed
- Owner: math-comp
- Created: 2018-04-18T08:51:18.000Z (almost 8 years ago)
- Default Branch: master
- Last Pushed: 2025-09-05T08:46:15.000Z (5 months ago)
- Last Synced: 2025-09-05T10:27:50.177Z (5 months ago)
- Topics: coq, mathcomp, real-closed-fields, ssreflect
- Language: Rocq Prover
- Homepage:
- Size: 455 KB
- Stars: 14
- Watchers: 12
- Forks: 11
- Open Issues: 9
-
Metadata Files:
- Readme: README.md
- Authors: AUTHORS
Awesome Lists containing this project
README
# Real closed fields
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[docker-action-link]: https://github.com/math-comp/real-closed/actions/workflows/docker-action.yml
This library contains definitions and theorems about real closed
fields, with a construction of the real closure and the algebraic
closure (including a proof of the fundamental theorem of
algebra). It also contains a proof of decidability of the first
order theory of real closed field, through quantifier elimination.
## Meta
- Author(s):
- Cyril Cohen (initial)
- Assia Mahboubi (initial)
- License: [CeCILL-B](CECILL-B)
- Compatible Coq versions: Coq 9.0 or later
- Additional dependencies:
- [MathComp ssreflect 2.3 or later](https://math-comp.github.io)
- [MathComp algebra](https://math-comp.github.io)
- [MathComp field](https://math-comp.github.io)
- [MathComp bigenough 1.0.0 or later](https://github.com/math-comp/bigenough)
- Coq namespace: `mathcomp.real_closed`
- Related publication(s):
- [Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination](https://hal.inria.fr/inria-00593738v4) doi:[10.2168/LMCS-8(1:2)2012](https://doi.org/10.2168/LMCS-8(1:2)2012)
- [Construction of real algebraic numbers in Coq](https://hal.inria.fr/hal-00671809v2) doi:[10.1007/978-3-642-32347-8_6](https://doi.org/10.1007/978-3-642-32347-8_6)
## Building and installation instructions
The easiest way to install the latest released version of Real closed fields
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-mathcomp-real-closed
```
To instead build and install manually, do:
``` shell
git clone https://github.com/math-comp/real-closed.git
cd real-closed
make # or make -j
make install
```
## Documentation
The repository contains
- the decision procedure `rcf_sat` and its corectness lemma [`rcf_satP`](https://github.com/math-comp/real-closed/blob/3721886fffb13ea9c80824043f119ffed0c780f2/theories/qe_rcf.v#L991) for the first order theory of real closed fields through
[certified quantifier elimination](https://hal.inria.fr/inria-00593738v4)
- the definition `{realclosure F}` , a [construction of the real closure of an archimedean field](https://hal.inria.fr/hal-00671809v2), which is canonically a [`rcfType`](https://github.com/math-comp/math-comp/blob/c1ec9cd8e7e50f73159613c492aad4c6c40bc3aa/mathcomp/algebra/ssrnum.v#L63) when `F` is an archimedean field, and the characteristic theorems of section [`RealClosureTheory`](https://github.com/math-comp/real-closed/blob/3721886fffb13ea9c80824043f119ffed0c780f2/theories/realalg.v#L1477).
- the definition `complex R`, a construction of the algebraic closure of a real closed field, which is canonically a [`numClosedFieldType`](https://github.com/math-comp/math-comp/blob/c1ec9cd8e7e50f73159613c492aad4c6c40bc3aa/mathcomp/algebra/ssrnum.v#L73) that additionally satisfies [`complexalg_algebraic`](https://github.com/math-comp/real-closed/blob/3721886fffb13ea9c80824043f119ffed0c780f2/theories/complex.v#L1324).
Except for the end-results listed above, one should not rely on anything.
The formalization is based on the [Mathematical Components](https://github.com/math-comp/math-comp)
library for the [Coq](https://coq.inria.fr) proof assistant.
## Development instructions
### With nix.
1. Install nix:
- To install it on a single-user unix system where you have admin
rights, just type:
> sh <(curl https://nixos.org/nix/install)
You should run this under your usual user account, not as
root. The script will invoke `sudo` as needed.
For other configurations (in particular if multiple users share
the machine) or for nix uninstallation, go to the [appropriate
section of the nix
manual](https://nixos.org/nix/manual/#ch-installing-binary).
- You need to **log out of your desktop session and log in again** before you proceed to step 2.
- Step 1. only need to be done once on a same machine.
2. Open a new terminal. Navigate to the root of the Abel repository. Then type:
> nix-shell
- This will download and build the required packages, wait until
you get a shell.
- You need to type this command every time you open a new terminal.
- You can call `nixEnv` after you start the nix shell to see your
work environemnet (or call `nix-shell` with option `--arg
print-env true`).
3. You are now in the correct work environment. You can do
> make
and do whatever you are accustomed to do with Coq.
4. In particular, you can edit files using `emacs` or `coqide`.
- If you were already using emacs with proof general, make sure you
empty your `coq-prog-name` variables and any other proof general
options that used to be tied to a previous local installation of
Coq.
- If you do not have emacs installed, but want to use it, you can
go back to step 2. and call `nix-shell` with the following option
> nix-shell --arg withEmacs true
in order to get a temporary installation of emacs and
proof-general. Make sure you add `(require 'proof-site)` to your
`$HOME/.emacs`.