User blog:Bubby3/Notation provably equivalent to BMS pair sequences

On this wiki, there are several functions that behave similarily to pair sequence system, which people don't doubt the termination of, such as the original R function, and dollar function extended bracket notation. However, people seriously doubt the termination of pair sequences of BM4, which act very similarily to them.

Here is my notation trying to reformalize pair sequences. The notation has a number (n) and a group of nested brackets (#) with an order or subscript. It is notated n:#. 0th order brackets don't have to have a subcript. Square brackets or [] are shorthand 1st order brackets and braces {} are shorthand for 2nd order brackets.

The rules are:
 * 1) Replace n with n^2
 * 2) The active bracket of  defined by the following process:
 * 3) If the last bracket of the expression is empty, that is the active bracket
 * 4) Othserside, repeat this process and jump into the last bracket
 * 5) Let A0 be the active bracket and A(d+1) be the bracket containing Ad, until we get to an orphan bracket (a bracket not contained within another bracket)
 * 6) The cases for solving the expression are:
 * 7) If the active bracket is an orphan bracket, delete
 * 8) Otherwise, if the active bracket has an order of 0, delete it and make n copies of the bracket contained within the former bracket.
 * 9) Otherwise, find the lowest m such that Am has an order of the lower than the active bracket.
 * 10) If no m exists, delete the active bracket
 * 11) Otherwise, replace the active bracket with i_n, where i_0 is empty and i_(d+1) is A(m) with the active bracket replaced with i_(d)