User blog:Wythagoras/Dollar function: final version

Inspired by Hyp cos I decided to remake everything exepted for Bracket Notation.

Which rule should you use?

 * 1) If there is nothing after the $, use rule 1
 * 2) If there are any non-nested non-subscript numbers, use rule 2
 * 3) If there are any non-nested non-subscript [0]'s, use rule 3
 * 4) If there are any non-nested non-subscript [b]'s, use rule 4
 * 5) If the previous things doesn't apply but the lowest level bracket can be solved with normal bracket notation:
 * 6) Search in the bracket for the least nested lowest level bracket or number
 * 7) If it is a 0:
 * 8) If the zero is the only content, use rule 3
 * 9) Otherwise, use rule 5
 * 10) If it is another number, use rule 4
 * 11) If it is a bracket: Return to step 5
 * 12) If the lowest level bracket can be solved with extended bracket notation:
 * 13) Is the number in the typed bracket a 0, use rule 6 and the subrule if needed
 * 14) Otherwise, use rule 4

Extended Bracket Notation
This works now like the Buchholz hydra, and the limit is \(\psi(\psi_I(0))\) 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

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

6. If the active bracket has level k, search for the least nested bracket with level (k-1).

S1: The outermost bracket is always level 1

S2: If there is no bracket with level (k-1), add it.