User:12AbBa/"Normal" OCFs vs a different version of R function

In this article, we will be rigorously defining OCFs, and comparing them to R Function. The real thing is too cumbersome for me to understand, so I rewrite the expressions in arrows. It turns out that Extended Up-Arrow Notation is curiously similar to R function, so I can do that.

Part I: Below \(\varepsilon_0\)
In this part we have no OCFs. All notations are evaluated starting from the right, but arrows start from the left.

Basic Notation
\(\uparrow_0\) is short for \(\uparrow\).

Rules:

1. Base Case: \(a\uparrow b=a^b\)

2. Recursive case: \(a\uparrow_nb=a\uparrow_{n-1}a\uparrow_n(b-1)\)

Nested Notation
We change \(\uparrow_n\) into \(\uparrow_{\underbrace{\uparrow\uparrow\ddots\uparrow}_n}\).

Rules: scan from left to right. Stop at the first "valley", i.e. the first level that is lower than all levels before it.

1.