Please let \(f(x)\) equal \((9\def\t{\uparrow}\t\t\t x)\def\e{\Lambda}\e^{\t\t\t}\), where \(x\e^{\def\{{\underbrace{\t\t\cdots\t\t}}\{_y}\) is defined for \(y\in\Bbb N\) as the Bouncing Factorial of x, but instead of starting at 1, you start at 2, and you replace the multiplication in the definition with (y+2)-ation.
Please let \(g(x)\) equal \(f^{x\e^{\{_{f^{f^x(x)}x}}}(x)\).
Please let \(\def\\{c_\def\({\alpha}\(}\\\) be the Catullus hierarchy with these rules:
\(c_0(t)=\) \(\phi\)\((g^{g^t(t)}(t),g^{g^{g^{g^{t\e^{\{_{g^{g^{g^t(t)}(t)}(t)}}}(t)}(t)}(t)}(t),\def\ {\omega}\ ^{\ ^{\ ^9}})\)
\(c_{\(+1}(t)=\phi(\\^{\\^{\\^t(t)}(t)}(t),\\^{\\^{\\(t)\t^{\\^{\\^{\\^{t\e^{\{_{\\^{\\^{\\^{\\^{t\e^\t}(t)}(t)}(t)}(t)}}(t)}(t)}(t)}}\\^t(t)}(t)}(t),\ ^{\ ^{\ ^{\ ^\(}}})\)
\(\\(t)=c_{\([t]}(t)\iff\(\in\text{Lim}\)
I define hyper bouncing guppy as \(c_{\def\}{\varepsilon_0}\}}(\)\(\text{guppy}\)\()+1\) with respect to the Wainer Hierachy system of fundamental sequences and \(\}[n]={}^n\ \).
Please let \(g(x)\) equal \(f^{x\e^{\{_{f^{f^x(x)}x}}}(x)\).
Please let \(\def\\{c_\def\({\alpha}\(}\\\) be the Catullus hierarchy with these rules:
\(c_0(t)=\) \(\phi\)\((g^{g^t(t)}(t),g^{g^{g^{g^{t\e^{\{_{g^{g^{g^t(t)}(t)}(t)}}}(t)}(t)}(t)}(t),\def\ {\omega}\ ^{\ ^{\ ^9}})\)
\(c_{\(+1}(t)=\phi(\\^{\\^{\\^t(t)}(t)}(t),\\^{\\^{\\(t)\t^{\\^{\\^{\\^{t\e^{\{_{\\^{\\^{\\^{\\^{t\e^\t}(t)}(t)}(t)}(t)}}(t)}(t)}(t)}}\\^t(t)}(t)}(t),\ ^{\ ^{\ ^{\ ^\(}}})\)
\(\\(t)=c_{\([t]}(t)\iff\(\in\text{Lim}\)
I define hyper bouncing guppy as \(c_{\def\}{\varepsilon_0}\}}(\)\(\text{guppy}\)\()+1\) with respect to the Wainer Hierachy system of fundamental sequences and \(\}[n]={}^n\ \).
Please remember to stay hydrated.
ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\
ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\
