User:Wythagoras/Dollar function/Bracket notation

This is the first part of Dollar function. It uses only normal brackets. It works quite similiar to the Hardy Hierarchy and Kirby-Paris Hydra.

Formal definition
\(\bullet\) can be anything

\(\circ\) is a group of brackets.

1. If there is nothing after the $, the array is solved. The value of the array is the number before the $.

2. \(a\$b\bullet=(a+b)\$\bullet\)

3. \(a\$\circ[0]\bullet\circ=a\$\circ a\bullet\circ\)

4. \(a\$\circ[\bullet+1]_c\bullet\circ=a\$\circ[\bullet]_c[\bullet]_c...[\bullet]_c[\bullet]_c\bullet\circ\) with a \(\bullet\)'s (in case you want to know, the subscripts are for the next part)

5. If the bracket contains a zero and the bracket has other content, you can remove the zero.

Analysis
\(a\$[0] = f_1(a)\)

\(a\$[1] = f_2(a)\)

\(a\$[b] = f_{b+1}(a)\)

\(a\$ 0 > f_{\omega}(a)\)

\(a\$[[0]1] > f_{\omega+1}(a)\)

\(a\$0][0 > f_{\omega2}(a)\)

\(a\$ 1 > f_{\omega^2}(a)\)

\(a\$[[b]c] > f_{\omega^b+c}(a)\)

\(a\$[ 0 ] > f_{\omega^\omega}(a)\)

\(a\$[[[b]c]d] > f_{\omega^\omega^b+\omega^c+d}(a)\)

\(a\$ [[0]] > f_{\omega^{\omega^\omega}}(a)\)

\(a\$0]_2] = [[...[[0...]] = f_{\varepsilon_0}(a)\)