Start by showing how proxying is copyright infringement. Or even search results...

Republishing copyrighted content on another domain without permission is copyright infringement. I don't know if Google's terms allow their republishing of copyrighted content via AMP, but most people in charge of those decisions at their companies probably aren't paying attention to the domain name or the implications of what they are allowing. At the very least, companies are being coerced to allow the republication of the content.

Because the authors opt-in?

But how? Where is the opt-in? Is including the javascript from ampproject.org considered an opt-in to give gogole a license to republish your copyrighted content?

There is no notice or terms and conditions/license or any kind of other legal agreement that I found on the amp website. [EDIT: actually the code on github is licensed as Apache 2: https://github.com/ampproject/amphtml]

So I don't understand how this works legally. You surely don't automatically sign over your copyright to somebody else by including their javascript library in your page?