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Hyperstar X-Notation

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Complexed and very powerful notation with an unimaginable growth speed.

Current Extensions

• X-Notation
• Extended X-Notation
• Superextended X-Notation
• Hyperextended X-Notation

TAD

• Iterator - a chain of elements or other iterators (denoted as […])
• Element - positive natural number
• Chain - a set of iterators and elements
• Base - first element
• Recursion - repeating of an array ‘base’ times. ‘<…>’
• Iteration - repeating of elements of array ‘base’ times. ‘…’
• …(c) - repeating of elements of array ‘c’ times.
• # - a part of a chain that does not transformed.
• #_min - a part of a chain that does not transformed (minimal length).

All rules from previous array types is applicable to arrays of follow types, which contain iterators of previous type. All # are equal to each other correspondingly unless specified separately.

Superscript for braces - count of braces.

X-Notation

1. $ X{n} = n + 1 $
2. $ X{# ‘ 1} = X{#} $
3. $ X{n ‘ c + 1} = X{<…> ‘ c} $
4. $ X{#_{min} ‘ 1 ‘ c + 1 ‘ #} = X{# ‘ <…> ‘ c ‘ #} $
5. $ X{n[1]} = X{n ‘ n ‘ …} $
6. $ X{n[1] ‘ …(d) ‘ 1} = X{n ‘ n ‘ …[1] ‘ …(d)} $
7. $ X{n[1] ‘ …(d) ‘ c + 1} = X{n ‘ n ‘ …[1] ‘ …(d) ‘ c} $
8. $ X{n[1] ‘ …(d) ‘ #{min} ‘ 1 ‘ c + 1 ‘ #} = X{n[1] ‘ …(d) ‘ #{min} ‘ <…> ‘ c ‘ #} $
9. $ X{n[1] ‘ …(d) ‘ 1 ‘ 1 ‘ …(c) ‘ 2} = X{n[1] ‘ …(d) ‘ n ‘ n ‘ …(c)} $
10. $ X{n[1] ‘ …(d) ‘ [1]} = X{n[1] ‘ …(d) ‘ n ‘ n ‘ …} $
11. $ X{n[c + 1 ‘ #]} = X{n[c ‘ #] ‘ [c ‘ #] ‘ …} $
12. $ X{n[c + 1 ‘ #] ‘ …(d) ‘ 1} = X{n[c + 1 ‘ #] ‘ …(d)} $
13. $ X{n[c + 1 ‘ #] ‘ …(d) ‘ 2} = X{n[c ‘ #] ‘ [c ‘ #] ‘ …[c + 1 ‘ #] ‘ …(d)} $
14. $ X{n[c + 1 ‘ #] ‘ …(d) ‘ e + 1} = X{n[c ‘ #] ‘ [c ‘ #] ‘ …[c + 1 ‘ #] ‘ …(d) ‘ e} $
15. $ X{n[c + 1 ‘ #] ‘ …(d) ‘ #{min} ‘ 1 ‘ e + 1 ‘ #} = X{n[c + 1 ‘ #] ‘ …(d) ‘ #{min} ‘ <…> ‘ e ‘ #} $
16. $ X{n[c + 1 ‘ #] ‘ …(d) ‘ 1 ‘ 1 ‘ …(e) ‘ 2} = X{n[c + 1 ‘ #] ‘ …(d) ‘ n ‘ n ‘ …(e)} $
17. $ X{n[c + 1 ‘ #] ‘ …(d) ‘ [e + 1 ‘ #]} = X{n[c + 1 ‘ #] ‘ …(d) ‘ [e ‘ #] ‘ [e ‘ #] ‘ …} $
18. $ X{n[# ‘ 1]} = X{n[#]} $
19. $ X{n[1 ‘ 2]} = X{n[n]} $
20. $ X{n[#{min} ‘ 1 ‘ c + 1 ‘ #]} = X{n[#{min} ‘ <…> ‘ c ‘ #]} $
21. $ X{n[1 ‘ 1 ‘ …(c) ‘ 2]} = X{n[n ‘ n ‘ …(c)]} $

Extended X-Notation

1. $ X{n[[1]]} = X{n[n ‘ n ‘ …]} $
2. $ X{n[[[1]]]} = X{n[[n ‘ n ‘ …]]} $
3. $ X{n[[[[1]]]]} = X{n[[[n ‘ n ‘ …]]]} $

And etc. Note: all rules from X-Notation applicable inside iterator. Iterators are independent objects, and expansion X-Notation rules 6-19 working inside other iterators. like: X{n[[[1’1]]]} = X{n[[[n]]]}

Calculation order: from external to internal chains.

…[[1]]… = …[1[1]]… .

Common recursive rule:

1. $ X{n[^c[1]]^c} = X{n[^c n ‘ n ‘ …]^c} $

Superextended X-Notation

Note: in rule 3 of SEHSXN, … used for a chain with repeating iterators.

1. $ X{n[1 \backslash 1]} = X{n[1]^n} $
2. $ X{n[#{min} \backslash 1 \backslash c + 1 \backslash #]} = X{n[#{min} \backslash [<…>] \backslash c \backslash #]} $
3. $ X{n[1 \backslash 1 \backslash …(c + 1)]} = X{n[[<…>] \backslash [<…>] \backslash …(c)]} $

Hyperextended X-Notation

Note: in all rules of HEHSXN, … used for a chain with repeating iterators.

1. $ X{n[1 \backslash_2 1]} = X{n[[<…>] \backslash [<…>] \backslash …]} $
2. $ X{n[1 \backslash_{s + 1} 1]} = X{n[[<…>] \backslash_s [<…>] \backslash_s …]} $
3. $ X{n[1 \backslash_{1 ‘ 2} 1]} = X{n[1 \backslash_n 1]} $
4. $ X{n[1 \backslash_{1 → 1} 1]} = X{n[1 \backslash_{<…>} 1]} $
5. $ X{n[1 \backslash_{1 → c + 1} 1]} = X{n[1 \backslash_{<…> → c} 1]} $
6. $ X{n[1 \backslash_{#{min} → 1 → c + 1 → #} 1]} = X{n[1 \backslash{#_{min} → <…> → c → #} 1]} $
7. $ X{n[1 \backslash_{1 →2 1} 1]} = X{n[1 \backslash{n → n → n → …} 1]} $

8. $ X{n[1 \backslash_{1 →{c + 1} 1} 1]} = X{n[1 \backslash{n →_c n →_c n →_c …} 1]} $

9. $ X{n[1 \backslash_{1 →{1 → 1} 1} 1]} = X{n[1 \backslash{1 →_{<…>} 1} 1]} $