This is an attempt at rewriting Wythagoras' dollar function as a hydra.

The dollar function \(\$(H)\) takes one argument, a hydra (tree) where the root node and all the leaf nodes are labeled with nonnegative integers.

Basic rules

At each step we mutate the hydra until it is reduced to only a root node. The output of the hydra is the label of the root node at this step.

Let \(n\) be the label of the root node, and let \(X\) be the leftmost node of \(T\), or the first leaf. Let \(P\) represent its parent.

  1. If \(P\) is the root node and \(X\) has label \(a\), chop off \(X\) and relabel the root node to \(n + a\).
  2. If \(P\) is not the root node:
    1. If \(X\) has label 0:
      1. Chop off \(X\).
      2. If \(P\) is a leaf node now, label \(P\) with \(n\).
    2. If \(X\) has label \(a + 1\) for nonnegative \(a\):
      1. Relabel \(X\) with \(a\).
      2. Add \(n - 1\) copies of \(P\) to the right of \(P\).

We note that the dollar function up to this point is known to be at the \(\varepsilon_0\) level, as is the suspiciously similar Kirby-Paris hydra.

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