User blog:King2218/BAL Notation Part 3

This next extension is pretty obvious. I turned the LEVEL to an array.

More Rules
Let $$\#$$ and $$\#^*$$ represent rest of array. (There MUST be a rest of array)

Let a, b, and c_n be NATURAL NUMBERS.

As usual, here are the new rules:


 * $$a(1)\{0\} = a$$ (I know. The LEVEL has a different numbering system unlike the ARRAY)
 * $$a(c_1,c_2,\cdots)\{\#, 0\} = a(c_1,c_2,\cdots)\{\#\}$$
 * $$a(c_1,c_2,\cdots)\{\underbrace{0, 0, \cdots, 0}_{n}, b, \#\} = a(c_1,c_2,\cdots)\{\underbrace{a, a, \cdots, a}_{n}, b-1, \#\}$$
 * $$a(c_1 + 1, c_2,\cdots)\{0\} = a(c_1, c_2,\cdots)\{\underbrace{a, a, \cdots, a}_{a}\}$$
 * $$a(\underbrace{1, 1, \cdots, 1}_{n}, c+1, \#)\{0\} = a(\underbrace{a, a, \cdots, a}_{n}, c, \#)\{0\}$$
 * $$a(c)\{b, \#\} = a\uparrow^ca(c)\{b-1, \#\}$$
 * $$a(\#, 1)\{\#^*\} = a(\#)\{\#^*\}$$

Simple enough.

Stuff
Let's try to evaluate $$2(1, 2)\{1\}$$.

$$2(1, 2)\{1\}$$
 * $$=4(1,2)\{0\}$$
 * $$=4(4,1)\{0\}$$
 * $$=4(4)\{0\}$$
 * $$=\text{Superquadribal}$$

We can rewrite our numbers too!


 * $$\text{Biblex} = \text{Bibal}(1,2)\{0\}$$
 * $$\text{Bidublex} = \text{Biblex}(1,2)\{0\}$$

Extending further
As you can see right now, this notation can be further extended to include a multidimensional ARRAY and a hyperdimensional LEVEL! We can even probably add multiple BASEs! Great!