User blog:Hyp cos/Fundamental Sequences in Taranovsky's Notation

Here come two sets of fundamental sequences (FS) on Taranovsky's ordinal notation. One is for ideal use, and the other is for reality use.

Ideal use - FS in Bignum Bakeoff
Here's a code to calculate fast-growing function based on Taranovsky's notation. This code is used for Bignum Bakeoff. Here're some explanation. As we all know, ordinals in Taranovsky's notation are made up from two constants: \(0,\Omega_n\), and a binary function C. In that code, ordinals are coded to integers. So l(x) give the left component of x, and r(x) give the right component of x. This trick is similar to Loader's one. Function g(x,y) compare two ordinals, and it returns true iff x > y. b is for built-from-below, and it's identical to Taranovsky's original one. t judges an ordinal in standard form or not. However, not all positive integers can be decoded to ordinals, so we need function s.
 * 0 is coded to 1.
 * \(\Omega_n\) is coded to 2.
 * \(C(\alpha,\beta)\) is coded to \(2^a\times(2b+1)\), where \(\alpha\) and \(\beta\) are coded to a and b respectively.

And function f is for FS. Here, the definition of FS on Taranovsky's notation is:
 * \(\alpha[k]\) is the largest ordinal \(\beta\) such that \(\beta<\alpha\) in and \(\beta\) is coded to a number smaller than \(k\).

This definition is irregular, and we surely can't use this to evaluate FS in reality. For example, we may need a program to evaluate FS, and use it for analysis of Taranovsky's notation. Due to the long calculating time of FS, we must think about a new way.