Arrow Cascading Hyper Notation (ACHN) is the 9th part of HN.

Rules

An expression is of the form H<n,A>, where n is the base and A is an array.

1. If there is no array, H<n> = n^n
2. If the array's first entry is 1, delete it.
3. If the array is a sequence array, decrease it by one, then nest the whole expression into the base n times.
4. If the array is a limit array, change the array into the nth member of it's FS.

Arrays

This part uses the same definitions as CHN.

Define {x} is a string of x carats. Define N is an array.

N{x}1 = N

N{x}y = N{x-1}(N{x}y-1)

Else:

If A is sequence array:

(N{x}(A-1)){x-1}N = N{x}A

If A is limit array, the yth member of it's FS is N{x}FS(A,y)

To compare arrays A1 and A2:

First look at the array after the {x} for A1 (which will be A1_x) and A2 (which will be A2_x), and compare them, and that is your result, unless they're the same, then:

Look at the array after the {x-1} after A1_x for A1 (which will be A1_(x-1)) and after A2_x for A2 (which will be A2_(x-1)), and compare them, and that is your result, unless they're the same, then:

Look at the array after the {x-2} after A1_(x-1) for A1 (which will be A1_(x-2)) and after A2_(x-1) for A2 (which will be A2_(x-2)), and compare them, and that is your result, unless they're the same, then:

Look at the array after the {x-3} after A1_(x-2) for A1 (which will be A1_(x-3)) and after A2_(x-2) for A2 (which will be A2_(x-3)), and compare them, and that is your result, unless they're the same, then:

...

Continue until you get to {1}, then use CHN rules.