On Mon, 23 Mar 2015, Guy Haworth wrote:

Let us say that, if Theorem 1 (T1) can be used to prove Theorem 2 (T2), then T1 --> T2.

Are there simple examples of 'equivalence', T1 --> T2 and T2 --> T1 (setting up loop T1 --> T2 --> T1) that could be explained to school children?

Are there cases of longer loops T1 --> ... --> Tn --> T1 which cannot, or cannot naturally, be shortened ? Schoolchildren-compatable preferred again !

Not quite an answer to your question, but a large part of Marvin Minsky's "Computation: Finite and Infinite Machines" is devoted to proving that a bunch of models of computation (M[1], M[2], M[3], ... M[n]) have equivalent power by using M[i] to simulate M[i+1 mod n]. Until you get to the last link, where M[n] simulates M[1], all you know is that none of the models gives you more power than the previous ones. In the end you know that they're all equally powerful. I remember reading this when I was 17 and thinking that it was the best story I'd ever seen. -- Tom Duff. Lo! the raptured arithmetician!