User:Wythagoras/Dollar function/Linear Array Notation

Linear Array Notation is the third part of Dollar Function.

Formal definition
\(\bullet\) can be anything

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

\(\diamond\) is a group of zeroes.

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

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

9. \([0,\diamond,c,\bullet] = [[[0],\diamond,c-1,\bullet]_{[[0],\diamond,c-1,\bullet]_{[[0],\diamond,c-1,\bullet]_{...}}},\diamond,c-1,\bullet]\)

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

How to find the active bracket?
 * Start scanning form the left to the right, starting at the dollar sign.
 * If you find an number, stop. The opening bracket to the left is the active bracket.

Examples
\([0],1 = a,1 = [[0]_{[a-1,1][a-1,1]}]\)

\([0]1,1 = [[0]_{[[0],1][[0],1]}]\) and is therefore much more, it takes a long expanding before reaching the second \(\omega\) bracket.

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 \(\psi(\psi_I(\varepsilon_{\Omega+1}))\)

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

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

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

\([0],2\) 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 \(\psi(\psi_{\chi(1)}(1))\)

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

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

\( 0,1,1 \) has level \(\psi(\psi_{\chi(1)}(I_{\chi(1)+1}))\)

\( 0,0,2 \) 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 \(\psi(\psi_{\chi(M)}(0))\)

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

\( 0,0,[0,0,1]_3 \) has level \(\psi(\psi_{\chi(M_2)}(0))\)

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

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

The limit of linear arrays is \(\psi(\Psi_{\Xi(\omega,0)}(0))\)