OK, I've spent literally two minutes as I'm in the middle of something urgent, and I'll come back to it later, but my impression is that while the result is interesting, and the authors may have used computer experiments to gain intuition about what was happening, it's not a computer proof. It's a tradtional proof of something that's relevant to computers.
If I'm wrong please provide a better summary so I can read it with your insights to guide me.
Maybe this will help: http://arxiv.org/abs/0808.1549