Beginning the overhaul. Sample of first 250 lines.

This is merely a sampler. Some remarks:

* The new file compiles.

* According to Benedikt's suggestion, I have begun structruring proofs with bullets, so that tactics are only applied to the subgoals they were meant to. A *very* quick introduction is available in [1].

[1] http://poleiro.info/posts/2013-06-27-structuring-proofs-with-bullets.html
Why do you choose to put redundant parentheses in around the function?  I think it just adds visual clutter.