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LEVEL 1

Z=20

Z_{b}=2b

Za=20+3a

Z_{b}a=2b+3a

LEVEL 2

In the following, the uppercase A represents the rest of the expression

\text{A}\,a,b=\text{A}\,(b+(\text{A}\,a))

\text{A}\,a\#b=\text{A}(\text{A}\,a\#(b-1));\,\text{A}\,a\#1=\text{A}\,a

LEVEL 3

\text{A}\,a\text{X}b=\text{A}\,\underset{b}{\underbrace{a,...,a}}

\text{A}\,a\#^{c}b=\text{A}\,\underset{b}{\underbrace{a\#^{c-1}a...\#^{c-1}a}}; \text{A}\,a\#^{1}b=\text{A}\,a\#b

\text{A}\,a[c]b=\text{A}\,\underset{b}{\underbrace{a[c-1]a[c-1]a...a[c-1]a}};\text{A}\,a[0]b=\text{A}\,aXb

LEVEL 4

\text{A}\,a[\$]b=\text{A}\,a[b]a

\text{A}\,a\#\$b=\text{A}\,a\#^{b}a

More coming soon.

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