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Hyperstar Hard X-Notation

Complexed and very powerful notation with an unimaginable growth speed. New version of X-Notation.

Current Extensions

• Hard X-Notation
• Extended Hard X-Notation (HCCF)
• Superextended Hard X-Notation (EHCCF + NHCCF)
• Hyperextended Hard X-Notation (HEHCCF)

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.

All rules from previous array types is applicable to arrays of follow types, which contain iterators of previous type.

Superscript for braces - count of braces.

Hard X-Notation

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

Hash Chain Collapse Function

Note: γ - base of HSHXN expression. At the end of recursion: 0. Recursion of #_ξ = <…>_ξ. #_0 = #

1. $ #(0) = γ $
2. $ #(c + 1 ‘ ::) = #(c ‘ ::) ‘ #(c ‘ ::) ‘ …(γ) $
3. $ 2 ‘ #(0) = #^γ(0) $
4. $ c + 1 ‘ :: ‘ #(0) = c ‘ :: ‘ #^γ(0) $
5. $ # ‘ 2(0) = <…> ‘ #(0) $
6. $ # ‘ c + 1 ‘ ::(0) = <…> ‘ # ‘ c ‘ ::(0) $
7. $ # ‘ …(d) ‘ #(0) = # ‘ …(d) ‘ <…>(0) $
8. $ # ‘ …(d) ‘ # ‘ 2(0) = # ‘ …(d) ‘ <…> ‘ #(0) $
9. $ # ‘ …(d) ‘ # ‘ c + 1 ‘ ::(0) = # ‘ …(d) ‘ <…> ‘ # ‘ c ‘ ::(0) $

Extended Hash Chain Collapse Function

Note: γ - base of HSHXN expression. At the end of recursion: 0. Recursion of #_ξ = <…>_ξ. #_0 = #

1. $ #_{c + 1}(0) = #_c $
2. $ #_c(e + 1 ‘ ::) = #_c(e ‘ ::) ‘ #_c(e ‘ ::) ‘ …(γ) $
3. $ #_c(#) = #_c(<…>) $
4. $ #_c(#_d) = #_c(<…>_d) $, d < c
5. $ 2 ‘ #_c(#) = #_c^γ(0) $
6. $ e + 1 ‘ :: ‘ #_c(#) = e ‘ :: ‘ #_c^γ(0) $
7. $ # ‘ #_c(#) = <…> ‘ #_c^γ(0) $
8. $ #_d ‘ #_c(#) = <…>_d ‘ #_c^γ(0) $, d < c
9. $ #_c ‘ 2(#) = <…>_c ‘ #_c(0) $
10. $ #_c ‘ e + 1 ‘ ::(#) = <…>_c ‘ #_c ‘ e ‘ ::(0) $
11. $ #_c ‘ #_d(#) = #_c ‘ <…>_d(0) $, d < c
12. $ #_c ‘ #_c(#) = #_c ‘ <…>_c(0) $

Nested Hash Chain Collapse Function

Note: γ - base of HSHXN expression. At the end of recursion: 0. Recursion of #_ξ = <…>_ξ. #_0 = #

1. $ ##(0) = #{<…>} $
2. $ #{# ‘ 2}(0) = ## $
3. $ #{# ‘ #}(0) = #{# ‘ <…>} $
4. $ #{#_1}(0) = #{<…>_0} $
5. $ #{#_2}(0) = #{<…>_1} $
6. $ #{#_3}(0) = #{<…>_2} $
7. $ #{##}(0) = #{#{<…>}} $

And etc.