\(\bullet\) can be anything
\(\circ\) is a group of brackets.
\(\diamond\) contains only zeroes and seperators
the s function indicates solving a limit directly by the rules, so that it has a lower level than the limit structure.
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 and a zero in it, search for the least nested bracket with level (k-1) with the same array in it, nest that bracket a times in the place of the level k bracket.
7. \([[b\bullet,c\bullet,\text{◆}]] = [[0,c-1\bullet,\bullet]_{[b-1\bullet,c\bullet,\bullet]1}]\)
8. \([[\diamond,b\bullet,c\bullet,\bullet]] = [[\diamond,[\diamond,b\bullet,c-1\bullet,\bullet]_{[\diamond,b-1\bullet,c\bullet,\bullet]},c-1,\bullet]\)
10. \([0,c\bullet,\bullet]_1 = [0]_1\)
11. \([0]_1e^{\bullet}0 = [0]\)
12. \([0]e^{\bullet}(b\bullet) = [0(\bullet)0...0(\bullet)1]e^c(b-1\bullet)\) a entries
13. \([0(c,\bullet)b] = [0(c,\bullet)b-1]e^{c-1,\bullet}([0(c,\bullet)b-1]e^{c-1,\bullet}([0(c,\bullet)b-1]e^{c-1,\bullet}(...)))\) where the \(e\) operator works on the first dimension before \((c,\bullet)\)
14. \([b\bullet(\diamond,0,c)1] = [0(\diamond,[b-1\bullet(\diamond,0,c)1][b-1\bullet(\diamond,0,c)1],c-1)1]\)
15. \([0(\diamond,0,c)1]_1 = [0]\)
16. \([0(0,\bullet)b] = [[0](s(0,\bullet))1(0,\bullet)b-1]\)
S1. The outermost bracket is always level 1
S2. If there is no bracket with level (k-1), add it directly after the level k bracket.
S3. If before a seperator are only lower level seperators and zeroes, you can replace the all the lower level seperators with a zero.
An comma is can be written shorthand for (0).
What is active bracket?[]
The active bracket is the bracket with the lowest level. The brackets can be ordered by level in FGH, and then removing outermost bracket. Or you look to: the smallest seperator, shortest array, smallest number in array, smallest bracket type, least number of nestings, smallest number inside the bracket.