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) with the same array in it.

S1: The outermost bracket is always level 1

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

Linear Array Notation
Here are no $, but this are just rules what you should do if that kind of array is the lowest level array.

7. \( b\bullet,c = [[0,c-1]_{[b-1\bullet,c]1}]\)

8. To diagonalize in the nth position with bracket types, you must use \([\underbrace{0,0...0,1}_n]_k\) They diagonalize in the last entry.

9. \( \diamond,b\bullet,c,\bullet = [[\diamond,[\diamond,b\bullet,c-1,\bullet]_{[\diamond,b-1\bullet,c,\bullet]},c-1,\bullet]\)

10. \([0,c,\bullet] = [0]\)

S3. Zeroes at the and of the array must be removed

Analysis
\([0],1\) has level \(\psi(\psi_I(0))\)

\([0][0],1\) has level \(\psi(\psi_I(1))\)

\([1],1\) has level \(\psi(\psi_I(\omega))\)

\([[0]_2],1\) has level \(\psi(\psi_I(\varepsilon_0))\)

\([[[0]_2]_2],1\) has level \(\psi(\psi_I(\zeta_0))\)

\([[[0]_3]_2],1\) has level \(\psi(\psi_I(\varphi(\omega,0)))\)

\([[[0],1,1]]\) has level \(\psi(\psi_I(\psi(\psi_I(0))))\)

\([[[[[0],1,1]],1]]\) has level \(\psi(\psi_I(\psi(\psi_I(\psi(\psi_I(0))))))\)

\([0]_2,1\) has level \(\psi(\psi_I(\Omega))\)

\([0]_{[0]},1\) has level \(\psi(\psi_I(\Omega_\omega))\)

\([[0],1],1\) has level \(\psi(\psi_I(\psi_I(0)))\)

\([0,1]_2,1\) has level \(\psi(\psi_I(I))\)

\([1,1]_2,1\) has level \(\psi(\psi_I(I\omega))\)

\([[0]_2,1]_2,1\) has level \(\psi(\psi_I(I\Omega))\)

\([[0,1],1]_2,1\) has level \(\psi(\psi_I(I\psi_I(0)))\)

\([[0,1]_2,1]_2,1\) has level \(\psi(\psi_I(I^2))\)

\([[1,1]_2,1]_2,1\) has level \(\psi(\psi_I(I^\omega))\)

\([[[0,1]_2,1]_2,1]_2,1\) has level \(\psi(\psi_I(I^I))\)

\([0,1]_3,1\) has level \(\psi(\psi_I(\varepsilon_{I+1}))\)

\([0,1]_4,1\) has level \(\psi(\psi_I(\varphi(\omega,I+1)))\)

\([[0,1]_4[0,1]_4]_3,1\) has level \(\psi(\psi_I(\Omega_{I+1}))\)

\([[0,1]_5[0,1]_5]_4,1\) has level \(\psi(\psi_I(\Omega_{I+2}))\)

\([0],2\) has level \(\psi(\psi_{I_2}(0))\)

\([0],[0]\) has level \(\psi(\psi_{I_\omega}(0))\)

\([0],[0]_2\) has level \(\psi(\psi_{I_\Omega}(0))\)

\([0],[0,1]\) has level \(\psi(\psi_{I_{\psi_I(0)}(0))\)

\([0],[0,2]\) has level \(\psi(\psi_{I_{\psi_{I_2}(0)}(0))\)

\([0],[0,1]_2\) has level \(\psi(\psi_{I_{I}(0))\)

\(0,[0],1\) has level \(\psi(\psi_{\chi(1)}(0))\)

\(0,[0][0],1\) has level \(\psi(\psi_{\chi(1)}(1))\)

\(0,[0]_2,1\) has level \(\psi(\psi_{\chi(1)}(\Omega))\)

\(0,[0,0,1]_2,1\) has level \(\psi(\psi_{\chi(1)}(\chi(1)))\)

\( 0,0,2 \) has level \(\psi(\psi_{\chi(2)}(0))\)

\( 0,0,3 \) has level \(\psi(\psi_{\chi(3)}(0))\)

\( 0,0,[0] \) has level \(\psi(\psi_{\chi(\omega)}(0))\)

\( 0,0,[0]_2 \) has level \(\psi(\psi_{\chi(\Omega)}(0))\)

\( 0,0,[0,1] \) has level \(\psi(\psi_{\chi(\psi_I(0))}(0))\)

\( 0,0,[0,0,1]_2 \) has level \(\psi(\psi_{\chi(M)}(0))\)

\(0,0,[0],1\) has level \(\psi(\Psi_{\Xi(3,0)}(0))\)

\(0,0,0,[0],1\) has level \(\psi(\Psi_{\Xi(4,0)}(0))\)

limit of linear arrays has level \(\psi(\Psi_{\Xi(\omega,0)}(0))\)