\(\bullet\) 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.

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