Brace Operator Notation

This is a notation from GFE with several parts. There were parts before NPON, but they are contained within NPON.

Operators are applied to non-zero natural numbers, however some operators are numbers as well, for example 3 1 5 (using the "1" operator)=243. There are a few ways to distinguish the operator from the numbers, including putting spaces between them (for example 3 1 5) or underlining the operator (for example 3 1 5)

Nested Parentheses Operator Notation
Here, operators are made using natural numbers, parentheses (nested to any level), and concatenation. For example, "((5)(4)2)(3)4" and "((((2)3)4)5)6" are operators in this notation.

Rules
Operators are of the form "n", "PQ", or "(P)" where n is a natural number, and P and Q are operators, and $$Q\neq 0$$. R and S are strings such as "((5)(4)" or "(2)3" or "6))4)(3)2" (they don't have to have balanced parentheses), and T is also a string, but it's either empty or only right parentheses. R, S, and T are allowed to be empty.


 * a 0 b = a*b
 * a P 1 = a
 * a Pn+1 (b+1) = a Pn (a Pn+1 b)
 * a R(Sn+1)T b=a R$$\underbrace{(Sn)(Sn)\cdots(Sn)(Sn)}_{b}$$T a

Shorter rules (with more ellipses)
Using more ellipses, it's possible to define this in 3 rules.


 * a 0 b = a*b
 * a Pn+1 b = $$\underbrace{a\; Pn\;(a\; Pn\;(\cdots a\; Pn\;(a\; Pn\; a}_{b\; a's})\cdots))$$
 * a R(Sn+1)T b=a R$$\underbrace{(Sn)(Sn)\cdots(Sn)(Sn)}_{b}$$T a

Examples
Here, the separators are underlined for clarity.


 * 3 (3)5 4 = 3 (3)4 (3 (3)4 (3 (3)4 3))
 * 3 (((5)4)) 4 = 3 (((5)3)((5)3)((5)3)((5)3)) 3
 * 2 (((7)6)3) 3 = 2 (((7)6)2)(((7)6)2)(((7)6)2) 2

Collapsing Operator Notation
Soon