Cascading-E notation

The Cascading-E notation (E^ for short) is a further extension and generalization of Hyper-E notation introduced by Sbiis Saibian in January 22, 2013. It covers all functions up to \(f_{\varepsilon_0}(n)\) and comparable to tetrational arrays in BEAF.

Hyper-product and cascaders
Separators in the form #^X*#^X...#^X*#^X are called hyper-product of cascaders and each #^X is a single cascader.

Key Band
Key Band is a cascader in the form \(#^n\) (n is a positive integer).

Follow this algorithm for finding Key Band:


 * 1) Begin at ground level, and go to step 2.
 * 2) At current level find the last cascader of hyper-product.
 * 3) If this cascader in the form \(#^n\), then it is Key Band by the definition.
 * 4) Go up to the next exponent level and back to the step 2.

Definition
Below are 5 formal rules need to define E^. Let \(L(\&\_n\) is defined to be hyper-product at the level n.

Rule 1. Condition: No hyperions.

\(E(a)b = a^b\)

Rule 2. Condition: \(L(\&\_{n-1} \neq #^n\).

\(E(a)b@X#^{(X#^{n})}@c = E(a)b@X#^{(X#^{n-1)}^b\)@b

Rule 3. Condition: b=1.

\(E@a#^{n}b = E@a\)

Rule 4. Condition: \(L(\&\_{n-1} = #^n\).

\(E@aX#^{n}b = E@aX#^{n-1}aX#^{n}(b-1)\).

Rule 5. Condition: Rules 1-4 doesn't apply.

\(E@a#b = E@(E@a#(b-1))\)