User blog:Wythagoras/Buchholz hydra

Extended definition:
 * 1) If \(a\) has label 0, we proceed as in Kirby-Paris' game. Call the node's parent \(b\), and its grandparent \(c\). First we delete \(a\). If \(c\) exists (i.e. \(b\) is not the root), we make \(n\) copies of \(b\) and all its children and attach them to \(c\).
 * 2) If \(a\) has label \(\# + 1\), (where # is the remainder of the label) we go down the tree looking for a node \(b\) with label \(v \leq u\) (which is guaranteed to exist, as nodes directly above root are all 0's).

Consider \(b\) and subtree with root at \(b\), and call that subtree \(S\).

Create a copy of \(S\), call it \(S'\). Within \(S'\), we relabel \(b\) with \(u\) and relabel \(a\) with \(0\). Back in the original tree, replace \(a\) with \(S'\).
 * 1) If \(a\) has label \(\\), we simply relabel it with \(n + 1\).


 * 1) If \(a\) has a hydra as label, we solve the hydra.