Aarex Hydra

Aarex Hydra going to have round brackets only. Every game have a tree, contains empty or round brackets with other tree inside, starting in step 1.

Let n be the step number. Each time the process completed, increase the step number by 1.

If there is () in the end, remove the last ().
If there is (#()) in the end, follow the rule:
- Define A0 as (#()).
- If there is a round bracket before Ax (where x is the maximum value which Ax is defined), define that as An+1.
- Else if there is round bracket containing all A0, A1, ... Ax-1, Ax inside, define that as Ax.
- Find m as the minimum value which A0 > Am, and call Am as B. If not, define B as whole tree.
- Replace innermost A0 with Am, n times. If Am is not equal to (#), do not use it and replace innermost A0 with (#)Am instead, n times.
- Finally, remove the last () in innermost A0.
Else, follow the process in the tree in the last ().

In comparison, () containing nothing always be greater than the empty string and we compare trees inside of () first then we compare the next ().

Let AH(n) be n, where n is the final steps when the tree is empty after the starting tree (((...()...))) with n ()s inside out.

Analysis

In analysis, a tree in the left will be empty after amount of steps in the right.

Empty - 0
() - 1
()() - 2
()()() - 3
()()()... with n ()s - n
(()) - w
- We will analyse the whole tree only.
(())() - w+1
(())()() - w+2
(())()() - w+3
(())()()()... with n ()s - w+n
(())()(()) - w2
(())()(())() - w2+1
(())()(())()(()) - w3
(())()(())()(())()(()) - w4
(())()(())()(())()(())()(()) - w5
(())()(())()(())()... with n (())()s - wn
(())(()) - w^2
(())(())() - w^2+1
(())(())()(()) - w^2+w
(())(())()(())()(()) - w^2+w2
(())(())()(())()(())()(()) - w^2+w3
(())(())()(())()(())()(())... with n ()(())s - w^2+wn
(())(())()(())(()) - w^2*2
(())(())()(())(())() - w^2*2+1
(())(())()(())(())()(()) - w^2*2+w
(())(())()(())(())()(())(()) - w^2*3
(())(())()(())(())()(())(())()(())(()) - w^2*4
(())(())()(())(())()(())(())()(())(())()(())(()) - w^2*5
(())(())()(())(())()(())(())()... with n (())(())()s - w^2*n
(())(())(()) - w^3
(())(())(())() - w^3+1
(())(())(())()(()) - w^3+w
(())(())(())()(())(()) - w^3+w^2
(())(())(())()(())(())(()) - w^3*2
(())(())(())(()) - w^4
(())(())(())(())(()) - w^5
(())(())(())(())(())(()) - w^6
(())(())(())... with (())s - w^n
(()()) - w^w
(()())() - w^w+1
(()())()(()) - w^w+w
(()())()(())(()) - w^w+w^2
(()())()(())(())(()) - w^w+w^3
(()())()(())(())(())... with n (())s - w^w+w^n
(()())()(()()) - w^w*2
(()())()(()())()(()) - w^w*2+w
(()())()(()())()(()()) - w^w*3
(()())()(()())()(()())()(()()) - w^w*4
(()())()(()())()(()())()(()())()(()()) - w^w*5
(()())()(()())()(()())()... with n (()())()s - w^w*n
(()())(()) - w^(w+1)
(()())(())()(()())(()) - w^(w+1)2
(()())(())(()) - w^(w+2)
(()())(())(())(()) - w^(w+3)
(()())(())(())(())(()) - w^(w+4)
(()())(())(())(())... with n (())s - w^(w+n)
(()())(())(()()) - w^(w2)
(()())(())(()())(()) - w^(w2+1)
(()())(())(()())(())(()()) - w^(w3)
(()())(()()) - w^w^2
(()())(()())(()()) - w^w^3
(()())(()())(()())(()()) - w^w^4
(()())(()())(()())... with n (()())s - w^w^n
(()()()) - w^w^w
(()()())()(()()()) - w^w^w*2
(()()())(())(()()()) - w^(w^w*2)
(()()())(()())(()()()) - w^w^(w*2)
(()()())(()()()) - w^w^w^2
(()()())(()()())(()()()) - w^w^w^3
(()()())(()()())(()()())(()()()) - w^w^w^4
(()()())(()()())(()()())... with n (()()())s - w^w^w^n
(()()()()) - w^w^w^w
(()()()())(()()()()) - w^w^w^w^2
(()()()()()) - w^w^w^w^w
(()()()()()()) - w^w^w^w^w^w
(()()()()()()()) - w^w^w^w^w^w^w
(()()()...) with n ()s - w^^n
(()(())) - e0
(()(()))()(()(())) - e0*2
(()(()))(())(()(())) - w^(e0*2)
(()(()))(()())(()(())) - w^w^(e0*2)
(()(()))(()()())(()(())) - w^w^w^(e0*2)
(()(()))(()()()...)(()(())) with n ()s - w^w^...^w^w^(e0*2) with n 'w^'s
(()(()))(()(())) - e1
(()(()))(()(()))()(()(()))(()(())) - e1*2
(()(()))(()(()))()(()(()))(()(())) - w^(e1*2)
(()(()))(()(()))(()(())) - e2
(()(()))(()(()))(()(()))(()(())) - e3
(()(()))(()(()))(()(()))(()(()))(()(())) - e4
(()(()))(()(()))(()(()))... with n (()(()))s - e(n-1)
(()(())()) - ew
(()(())())(()(())) - e(w+1)
(()(())())(()(()))(()(())()) - e(w*2)
(()(())())(()(())()) - e(w^2)
(()(())()()) - e(w^w)
(()(())()()()) - e(w^w^w)
(()(())()()()()) - e(w^w^w^w)
(()(())()()()...) with n ()s - e(w^^n)
(()(())()(())) - ee0

Buchholz-Aarex Hydra

This is the mix of Buchholz hydra and my hydra. In this hydra, we use Buchholz rules, but all Kirby-Paris hydra rules were replaced to all my hydra rules.

If there is (0) in the end, remove the last (0).
If there is (n) in the end, follow the rule:
- Define A0 as (n).
- If there is a round bracket before Ax (where x is the maximum value which Ax is defined), define that as An+1.
- Else if there is round bracket containing all A0, A1, ... Ax-1, Ax inside, define that as Ax.
- Define Bx as a number before tree in Ax.
- Find m as the minimum value which B0 > Bm, and call Bm as C. If not, define C as whole tree.
- Replace innermost A0 with (n-1 Am), n times.
- Finally, replace the last (n) with (n-1) in innermost A0.
If there is (#()) in the end, follow the rule:
- Define A0 as (#()).
- If there is a round bracket before Ax (where x is the maximum value which Ax is defined), define that as An+1.
- Else if there is round bracket containing all A0, A1, ... Ax-1, Ax inside, define that as Ax.
- Find m as the minimum value which A0 > Am, and call Am as B. If not, define B as whole tree.
- Replace innermost A0 with Am, n times. If Am is not equal to (#), do not use it and replace innermost A0 with (#)Am instead, n times.
- Finally, remove the last () in innermost A0.
Else, follow the process in the tree in the last (0).

There is also nested Buchholz-Aarex Hydra, and I will replace numbers with trees and use Nested Buchholz Hydra rules. We simply replace (nA) with (A)_n in this extension and replace numbers with amount of ()s.

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