Chained arrow notation

Chained Arrow Notation is a generalization of Arrow Notation.

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