Chained arrow notation

Chained Arrow Notation is a generalization of Arrow Notation devised by J. H. Conway and R. K. Guy. By Bird's Proof, Jonathan Bowers' Array Notation will generally produce larger values than Chained Arrow Notation.

Using this notation, Conway and Guy created a new function similar to the Ackermann function that defines $$cg(n) = \underbrace{n \rightarrow n \rightarrow \ldots \rightarrow n \rightarrow n}_n$$. Although it is equal to the Ackermann function for n from 1 to 3, $$cg(4) = 4\uparrow\uparrow\uparrow\uparrow 4\uparrow\uparrow\uparrow\uparrow 4\uparrow\uparrow\uparrow\uparrow 4 >> 4\uparrow\uparrow\uparrow\uparrow 4 = A(4)$$.

Formula
$$a \rightarrow b \rightarrow c = a\underbrace{\uparrow\ldots\uparrow}_cb$$ (in which Arrow Notation is used)

$$a \rightarrow\ldots\rightarrow b \rightarrow 1 = a \rightarrow\ldots\rightarrow b$$

$$a \rightarrow\ldots\rightarrow b \rightarrow 1 \rightarrow c = a \rightarrow\ldots\rightarrow b$$

$$a \rightarrow\ldots\rightarrow b \rightarrow (c + 1) \rightarrow (d + 1) = a \rightarrow\ldots\rightarrow b \rightarrow (a \rightarrow\ldots\rightarrow b \rightarrow c \rightarrow (d + 1) ) \rightarrow d$$