Very interesting. I'm actually the most struck by the potential that this system might offer for math education, with its human-readable and human-like proofs. Math games have so far been focused on teaching how to do relatively mechanical things like arithmetic and algebraic manipulation because it's easy to automatically verify the correctness of those, but maybe something like this could be used to teach the creative side of mathematics as well? Let the student dabble around with various kinds of proofs, and let the system tell them which ones are correct, and provide guidance if the student is stuck.
Ganesalingam and Gowers recently released an ArXiV preprint of their article "A fully automatic problem solver with human-style output" about a theorem-proving program they have written that generates human-readable output. Gowers had blogged about the program in April 2013, after soliciting responses from readers regarding answers generated by the program and by humans (without explicitly revealing that one of the answerers was a machine) in March 2013.
There do exist theorem-proving systems such as Coq. In fact, it was announced about a year ago that the proof of the Feit-Thompson odd-order theorem, which originally appeared in a 255-page journal article in 1963, had been completely verified using Coq. But the Gowers-Ganesalingam program differs in that the output of this program is a human-style proof (for better or worse) whereas Coq uses a formal language syntax (see here for a small part of the proof of the Feit-Thompson theorem using Coq).
The ArXiV preprint contains details of the program and how the authors believe it improves on existing programs. One of the ways they believe they are better than past provers is their ability to more explicitly code for picking promising lines of reasoning.
I'd appreciate comments from LessWrong readers about the specifics of the Ganesalingam-Gowers program, and any broader ramifications of their ability to create this program for predictions related to artificial intelligence.