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. The limit of its growth rate is comparable to \(f_{\varepsilon_0}(n)\) in FGH, making it about as powerful as nested arrays in Bird's Array Notation.

E^ is part of the general Extensible-E System, which includes all future extensions of the notation, such as Extended Cascading-E Notation.

Hyper-product and cascaders
Separators of the form \(\#^{X}*\#^{X}\cdots\#^{X}*\#^{X}\) are known as a hyper-product of cascaders and each \(\#^{X}\) is a single cascader. X can be any positive integer or itself can be a hyper-product of cascaders. This recursive definition defines all possible separators in Cascading-E. Consequently, even separators in the form \(\#^{n}\) where n is a positive integer are considered a hyper-product of cascaders. Sometimes in basic numbers, like Godgahlah, we can say that En#^#n = En####...####n with n #'s..

Key Band
The Key Band is a cascader of the form \(\#^{n}\) (n is a positive integer), situated at the right most position of the hyper-product.

The following search-algorithm finds and defines the Key Band:


 * 1) Begin at ground level (level 0), and proceed to step 2.
 * 2) For the current level, find the last cascader of the current hyper-product, and proceed to step 3.
 * 3) If this cascader is in the form \(\#^{n}\), then it is the Key Band by definition, otherwise proceed to step 4.
 * 4) Go up to the next exponent level of the last cascader and go back to step 2.

Definition
Let Ea1&a2& ... &an be any expression in E^, where a1 through an are n positive integers, and all & are hyper-products (which may or may not be distinct.) Each individual & may be chosen from the set of legal separators.

Below are the 5 formal rules of E^. Let \(\&_k\) be the kth hyper-product and \(L(\&_k)\) be the last cascader of the kth hyper-product.


 * Rule 1. With no hyperions, we have \(E(a)b = a^b\).
 * Rule 2. If \(L(\&_{n-1}) \neq \#^n\) (the last cascader is not of the form \(\#^{n}\)):
 * \(E(a)b@X\#^{(X\#^{n})}@c = E(a)b@X\#^{(X\#^{n-1})^{^b}}@b\) (@ indicates the unchanged remainder of the expression.)
 * Rule 3. If the last argument is 1, it can be removed: \(E@a\#^{n}1 = E@a\)
 * Rule 4. \(L(\&_{n-1}) = \#^n\) and \(\&_k \neq \#\):
 * \(E@aX\#^{n}b = E@aX\#^{n-1}aX\#^{n}(b-1)\).
 * Rule 5. Otherwise:
 * \(E@a\#b = E@(E@a\#(b-1))\)

Examples

 * E10#^#3 = E10###10
 * E10#^#10#^#3 = E10#^#10###10
 * E10#^#*#3 = E10#^#10#^#10
 * E10#^##3 = E10#^#*#^#*#^#10
 * E10#^#^#3 = E10#^(###)10
 * E100#^#1 = E100#100