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## Bracket Notation

\(\circ\) will be anything

\(\bullet\) is group of brackets

0. \(a\$\) = \(a\)

1. \(a\$b\circ\) = \((a+b)\$\circ\)

2. \(a\$\bullet [0]\circ \bullet\) = \(a\$\bullet a\circ \bullet\)

3. \(a\$\bullet [b]\circ \bullet\) = \(a\$\bullet \underbrace{[b-1]...[b-1]}_{a}\circ \bullet\)

## Extended Bracket Notation

3a. \(a\$\bullet [b]_c\circ \bullet\) = \(a\$\bullet \underbrace{[b-1]_c ... [b-1]_c}_{a}\circ \bullet\), if c != 1

4. \(a\$\bullet [\circ]_1 \bullet\) = \(a\$\bullet [\circ] \bullet\)

5. \(a\$\bullet [0]_c\circ \bullet\) = \(a\$\bullet [... \bullet [0]_{c-1}\circ \bullet ...]_{c-1}\circ \bullet\) (a nested)

## Linear Array Notation

6. \(a\$\bullet [b(\circ)0...0,c+1 \circ]_1\circ \bullet\) = \(a\$\bullet [[0](\circ)0...[0(\circ)0...0,c \circ]_{...[0(\circ)0...0,c \circ]...},c \circ]_1\circ \bullet\) (b \([0(\circ)0...0,c \circ]\)'s)

7. \(a\$\bullet [\circ 1]_\circ \bullet\) = \(a\$\bullet [\circ]_\circ \bullet\)

## Dimensional Array Notation

8. \(a\$\bullet [b(\circ)0...0(d+1)c+1 \circ]_1\circ \bullet\) = \(a\$\bullet [[0](\circ)0...[0](d)[0]...[0](d)[0](d+1)c \circ]_1\circ \bullet\) (b [0] between (d)'s)

9. \(a\$\bullet [\circ x(0)y \circ]_\circ \bullet\) = \(a\$\bullet [\circ x,y \circ]_\circ \bullet\)