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: \(L(\&_{n-1}) = \#^n, n>1\).

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

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

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