TITLE: Has anyone thought about creating a formal proof wiki with verifier?
QUESTION [59 upvotes]: Mathematics has undergone some rather nice developments recently with the adoption of new techologies, things like on-line journals, the arXiv, this website, etc.  I imagine there must be many further developments that could be quite useful.
What I'm thinking of is a website where anyone can contribute formal proofs of theorems.  In particular there would be many proofs of the same theorem provided the proof is different -- like a constructive proof of Brouwer's fixed point theorem, and non-constructive proof, etc.  
The idea would be to build up a large web of formal proofs, one building on another so that one could eventually do searches through this space of formal proofs to find out what the most efficient proofs are, in the sense of how many ASCII characters it would take to write-up the proof using Zermelo-Frankel set theory.  One hope would be to have a big, active database of verified formal proofs.  Another would be to have a webpage where you could hope to discover whether or not there are simpler proofs of theorems you know, that you may have not been be aware of.
Being a web-page there would be certain useful efficiencies -- the webpage could "compile" your proof and check to see it's valid.  Being a wiki would make it relatively easy for people to contribute and build on an existing infrastructure.  And you'd be free to use pre-existing proofs (provided they've been verified as valid) in any subsequent proofs.  One could readily check what axioms a proof needs -- for example to what extent a proof needs the axiom of choice, and so on. 
Is there any efforts towards such a development?   Such a tool would hopefully function like the publishing arm of some sort of modern internet-era Bourbaki.

REPLY [2 votes]: What I'm thinking of is a website where anyone can contribute formal proofs of theorems. 

I'm currently working on openmathematics.org although nothing is live yet. It will allow projects developed with the Occam proof assistant to be packaged and published in a similar vein to http://npmjs.com. The command line tool, similar to the npm tool, is called open and should be at least partially functional later this year.

In particular there would be many proofs of the same theorem provided the proof is different -- like a constructive proof of Brouwer's fixed point theorem, and non-constructive proof, etc.

There would be nothing to stop people contributing different proofs of the same theorem and nothing to stop users choosing between them. However, all would be developed with Occam, rather than divers proof assistants.

The idea would be to build up a large web of formal proofs, one building on another so that one could eventually do searches through this space of formal proofs...

One thing I've been working on with Occam is to turn all labels and references into hyperlinks. If you click on a reference, you go straight to that theorem or axiom. If you click on a label, you get a list of all the places where it's referenced. This functionality is in place in the Occam proof assistant but could surface on the openmathematics.org site at some point, too.

Being a web-page there would be certain useful efficiencies -- the webpage could "compile" your proof and check to see it's valid. 

Occam will verify proofs on the fly but there could also be a step in the publishing process that verifies the project with a tool written in a language that can be trusted, rather than JavaScript. This is far in the future, however.

Being a wiki would make it relatively easy for people to contribute and build on an existing infrastructure. And you'd be free to use pre-existing proofs (provided they've been verified as valid) in any subsequent proofs. One could readily check what axioms a proof needs -- for example to what extent a proof needs the axiom of choice, and so on.

Actually what I decided to do is allow issues in much the vein as http://github.com. They seem to be a great way to foster communication and collaboration. The openmathematics.org site will leverage the GitHub API to allow users to create issues directly, and wash the text through KaTeX to support mathematical markup.

Is there any efforts towards such a development? Such a tool would hopefully function like the publishing arm of some sort of modern internet-era Bourbaki.

Well, yes, myself. At the moment I think I have another two years work ahead of me, it's three years and counting so far. Occam is worth a look now, but isn't verifying anything yet.