Higher Order Hypercascading Hyper Notation (HHCHN) is the 14th part of HN.

Rules

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

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

Arrays

Definitions from HpCHN apply.

Y_1 = Y. Any Y_0 is removed. Also Y_A = Y<(A)>.

Y<A> (if A is sequence array) is Y<A>-cascader of X_Y<A-1><x>

Y is hypercascader of X<A>.

Y<B> (if B is sequence array) is hypercascader of X_Y<B-1>_<A>, that means it uses X_Y<B-1>_<A> and not X<A>.

Y<A> (if A is limit array) is hypercascader and Y<A>-cascader of Y<FS(A,x)>, that means it uses Y<FS(A,x)> and not X<x>.

X_Y<A><B> (if B and A are sequence arrays) is Y<A>-cascader of X_Y<A><B>, that means it uses X_Y<A-1><X_Y<A><B-1>,N> and not X_Y<A-1><N>.

X_Y<A><B> (if B is sequence array and A is limit array) is Y<A>-cascader of X_Y<A><B>, that means it uses X_Y<FS(A,x)><X_Y<A><B-1>> and not X_Y<FS(A,x)>.

X_Y<A><B> (if B is limit array and A is sequence array) is Y<A>-cascader of X_Y<A><FS(B,x)>, that means it uses X_Y<A><FS(B,x)> and not X_Y<A-1><x>.

X_Y<A><B> (if B and A are limit arrays) is Y<A>-cascader of X_Y<A><FS(B,x)>, that means it uses X_Y<A><FS(B,x)> and not Y<FS(A,x)>.

To compare arrays, use HpCHN rules and when comparing Y<A> and Y<B>, compare A and B, and that is the result.