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https://github.com/coq-community/hydra-battles

Variations on Kirby & Paris' hydra battles and other entertaining math in Coq (collaborative, documented, includes exercises) [maintainer=@Casteran]
https://github.com/coq-community/hydra-battles

coq coq-nix-toolbox discrete-mathematics docker-coq-action formal-proofs hydra-battles ordinal-notations primitive-recursive-functions

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Variations on Kirby & Paris' hydra battles and other entertaining math in Coq (collaborative, documented, includes exercises) [maintainer=@Casteran]

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README 

        
# Hydras & Co.



[![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/hydra-battles/workflows/Docker%20CI/badge.svg?branch=master

[docker-action-link]: https://github.com/coq-community/hydra-battles/actions?query=workflow:"Docker%20CI"



[contributing-shield]: https://img.shields.io/badge/contributions-welcome-%23f7931e.svg

[contributing-link]: https://github.com/coq-community/hydra-battles#contributions-are-welcome



[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-based project has four parts:



- An exploration of some properties of Kirby and Paris' hydra

  battles, including the study of several representations of

  ordinal numbers and a part of the so-called _Ketonen and Solovay

  machinery_ (combinatorial properties of epsilon0).



  This part also hosts a formalization by Russell O'Connor of

  primitive recursive functions and Peano Arithmetic (PA).



- Some algorithms for computing _x^n_ with as few multiplications as

  possible (using _addition chains_).



- A bridge to definitions and results in the

  [Gaia project](https://github.com/coq-community/gaia), in particular

  on ordinals.



- A proof originally by Russell O'Connor of the Gödel-Rosser 1st

  incompleteness theorem, which says that any first order theory

  extending NN (which is PA without induction) that is complete is

  inconsistent.



Both the [documentation](https://coq-community.org/hydra-battles/doc/hydras.pdf)

and the Coq sources are _work continuously in progress_. For more information on

how the project is organized, maintained, and documented, see

[this paper](https://hal.archives-ouvertes.fr/hal-03404668) from

the proceedings of [JFLA 2022](http://jfla.inria.fr/jfla2022.html).



## Meta



- Author(s):

  - Yves Bertot

  - Pierre Castéran

  - Évelyne Contejean

  - Jeremy Damour

  - Stéphane Desarzens

  - Russell O'Connor

  - Karl Palmskog

  - Clément Pit-Claudel

  - Théo Zimmermann

- Coq-community maintainer(s):

  - Pierre Castéran ([**@Casteran**](https://github.com/Casteran))

- License: [MIT License](LICENSE)

- Compatible Coq versions: 8.14 or later

- Additional dependencies:

  - [Equations](https://github.com/mattam82/Coq-Equations) 1.2 or later

  - [Paramcoq](https://github.com/coq-community/paramcoq) 1.1.3 or later

  - [MathComp SSReflect](https://github.com/math-comp/math-comp) 1.13 or later

  - [MathComp Algebra](https://github.com/math-comp/math-comp)

  - [Gaia's Schütte ordinals](https://github.com/coq-community/gaia) 1.14 or later

  - [Mczify](https://github.com/math-comp/mczify)

  - [LibHyps](https://github.com/Matafou/LibHyps)

  - [CoqPrime](https://github.com/thery/coqprime)

  - [ZornsLemma](https://github.com/coq-community/zorns-lemma) 10.2.0 or later

- Coq namespace: `hydras`, `additions`, `gaia_hydras`, `Goedel`

- Related publication(s):

  - [Hydras & Co.: Formalized mathematics in Coq for inspiration and entertainment](https://hal.archives-ouvertes.fr/hal-03404668) 

  - [Essential Incompleteness of Arithmetic Verified by Coq](https://arxiv.org/abs/cs/0505034) doi:[10.1007/11541868_16](https://doi.org/10.1007/11541868_16)

  - [Accessible Independence Results for Peano Arithmetic](https://faculty.baruch.cuny.edu/lkirby/accessible_independence_results.pdf) doi:[10.1112/blms/14.4.285](https://doi.org/10.1112/blms/14.4.285)

  - [Rapidly Growing Ramsey Functions](https://www.jstor.org/stable/2006985) doi:[10.2307/2006985](https://doi.org/10.2307/2006985)

  - [Proof Theory](https://link.springer.com/book/10.1007/978-3-642-66473-1) doi:[10.1007/978-3-642-66473-1](https://doi.org/10.1007/978-3-642-66473-1)



## Building and installation



- To get the required dependencies, you can use [opam](https://opam.ocaml.org)

  or [Nix](https://nixos.org). With opam:

  - `opam install ./coq-hydra-battles.opam --deps-only` to get the _hydra battles_ dependencies;

  - `opam install ./coq-addition-chains.opam --deps-only` to get the _addition chains_ dependencies.

  - `opam install ./coq-gaia-hydras.opam --deps-only` to get the _gaia hydras_ dependencies.

  - `opam install ./coq-goedel.opam --deps-only` to get the _Goedel_ dependencies.



  With Nix, just run `nix-shell` to get all the dependencies

  (including for building the documentation). If you only want the

  dependencies to build a sub-package, you can run one of:

  - `nix-shell --argstr job hydra-battles`

  - `nix-shell --argstr job addition-chains`

  - `nix-shell --argstr job gaia-hydras`

  - `nix-shell --argstr job goedel`



- Building the PDF documentation also requires

  [Alectryon](https://github.com/cpitclaudel/alectryon) 1.4

  and [SerAPI](https://github.com/ejgallego/coq-serapi).

  See [`doc/movies/Readme.md`](doc/movies/Readme.md) for details.



- The general Makefile is in the top directory:

  - `make` : compilation of the Coq scripts

  - `make pdf` : generation of PDF documentation as `doc/hydras.pdf`

  - `make html` : generation of HTML documentation in `theories/html`



- You may also rely on `dune` to install just one part. Run:

  - `dune build coq-hydra-battles.install` to build only the _hydra battles_ part

  - `dune build coq-addition-chains.install` to build only the _addition chains_ part

  - `dune build coq-gaia-hydras.install` to build only the _gaia hydras_ part

  - `dune build coq-goedel.install` to build only the _goedel_ part



- Documentation for the `master` branch is continuously deployed at:

  - https://coq-community.org/hydra-battles/doc/hydras.pdf

  - https://coq-community.org/hydra-battles/theories/html/toc.html



## Contents



### Coq sources (directory theories)



- theories/ordinals



  - Hydra/*.v 

    - Representation in _Coq_ of hydras and hydra battles

    - A proof of termination of all hydra battles (using ordinal numbers below epsilon0)

    - A proof that no variant bounded by some ordinal less than epsilon0 can prove this termination

    - Comparison of the length of some kind of Hydra battles with the Hardy hierarchy of fast growing functions

    

  - OrdinalNotations/*.v

    - Generic formalization on ordinal notations (well-founded ordered datatypes with a comparison function)



  - Epsilon0/*.v

    - Data types for representing ordinals less than epsilon0 in Cantor normal form

    - The _Ketonen-Solovay machinery_: canonical sequences, accessibility, paths inside epsilon0

    - Representation of some hierarchies of fast growing functions



  - Schutte/*.v

    - An axiomatization of countable ordinals, after Kurt Schütte. It is intended to be a reference for the data types considered in theories/Epsilon0.



  - Gamma0/*.v

    - A data type for ordinals below Gamma0 in Veblen normal form (**draft**).



  - rpo/*.v

    - A contribution on _recursive path orderings_ by Evelyne Contejean.



  - Ackermann/*.v

    - Primitive recursive functions, first-order logic, NN, and PA



  - MoreAck/*.v

     - Complements to the legacy **Ackermann** library



  - Prelude/*.v

    - Various auxiliary definitions and lemmas



- theories/additions/*.v  

  - Addition chains



- theories/gaia/*.v

  - Bridge to the ordinals in Gaia that are encoded following Schütte and Ackermann



- theories/goedel/*.v

  - Gödel's 1st incompleteness theorem and its variations



### Exercises



- exercises/ordinals/*.v

  - Exercises on ordinals



- exercises/primrec/*.v

  - Exercises on primitive recursive functions



## Contributions are welcome



Any suggestion for improving the Coq scripts and/or the documentation will be taken into account.



- In particular, we would be delighted to replace proofs with simpler ones, and/or to propose various proofs or definitions of the same concept, in order to illustrate different techniques and patterns. New tactics for automatizing the proofs are welcome too.



- Along the text, we propose several _projects_, the solution of which is planned to be integrated in the development.



- Please do not hesitate to send your remarks as GitHub issues and your suggestions of improvements (including solutions of "projects") as pull requests.

- Of course, new topics are welcome !



- If you wish to contribute without having to clone the project /

  install the dependencies on your machine, you may use

  [Gitpod](https://gitpod.io/#https://github.com/coq-community/hydra-battles/)

  to get an editor and all the dependencies in your browser, with

  support to open pull requests as well.



- __Contact__ : pierre dot casteran at gmail dot com



A bibliography is at the end of the documentation. Please feel free to suggest more references to us.