Catching函数分析2：EBO~JO

本条目展示Catching函数在EBO到JO之间的过程。以下左为FGH，右为SGH。均为MOCF

$\begin{array}{r}.\\ & f_3(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(0)+1)\\ & f_{\omega }^n(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(0)+{\mathrm{\Omega }}^{\omega })\\ & f_{\psi (0)}^n(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(0)+{\psi }_1(\mathrm{\Omega }))\\ & f_{\psi ({\mathrm{\Omega }}_{\omega })}^n(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(0)+{\psi }_1({\mathrm{\Omega }}_{\mathrm{\Omega }}))\\ & f_{\psi ({\mathrm{\Omega }}_{{\mathrm{\Omega }}_{\omega }})}^n(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(0)+{\psi }_1({\mathrm{\Omega }}_{{\mathrm{\Omega }}_{\mathrm{\Omega }}}))\\ & f_{\psi ({\psi }_I(0))[n]}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(0)+{\psi }_1({\psi }_I(0)))\\ & f_{\psi ({\psi }_I(0)[n]+1)}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(0)+{\psi }_1({\psi }_I(0)+1))\\ & f_{\psi ({\psi }_I(0)[n]+\mathrm{\Omega })[n]}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(0)+{\mathrm{\Omega }}_2)\\ & f_{\psi ({\psi }_I(0)[n]+\mathrm{\Omega })}^n(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(0)+{\mathrm{\Omega }}_2\times \mathrm{\Omega })\\ & f_{\psi ({\psi }_I(0)[n]+{\mathrm{\Omega }}_2)}^n(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(0)+{\mathrm{\Omega }}_3\times \mathrm{\Omega })\\ & f_{\psi ({\psi }_I(0)[n]+{\mathrm{\Omega }}_{\omega })[n]}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(0)+{\mathrm{\Omega }}_{\omega })\\ & f_{\psi ({\psi }_I(0)[n]+{\mathrm{\Omega }}_{\omega })}^n(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(0)+{\mathrm{\Omega }}_{\mathrm{\Omega }})\\ & f_{\psi ({\psi }_I(0)[n]+{\mathrm{\Omega }}_{{\mathrm{\Omega }}_{\omega }})}^n(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(0)+{\mathrm{\Omega }}_{{\mathrm{\Omega }}_{\mathrm{\Omega }}})\\ & f_{\psi ({\psi }_I(0)[n]\times 2)}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(0)\times 2)\\ & f_{\psi ({\psi }_I(0)[n{]}^2)}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(0{)}^2)\\ & f_{\psi ({\psi }_I(0)[n{]}^{{\psi }_I(0)[n]})}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(0{)}^{{\psi }_I(0)})\\ & f_{\psi ({\psi }_{{\mathrm{\Omega }}_{{\psi }_I(0)[n-1]+1}}(0))}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_{{\mathrm{\Omega }}_{{\psi }_I(0)+1}}(0))\\ & f_{\psi ({\mathrm{\Omega }}_{{\psi }_I(0)[n-1]+1})}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\mathrm{\Omega }}_{{\psi }_I(0)+1})\\ & f_{\psi ({\mathrm{\Omega }}_{{\psi }_I(0)[n-1]+\omega })}^n(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\mathrm{\Omega }}_{{\psi }_I(0)+\mathrm{\Omega }})\\ & f_{\psi ({\mathrm{\Omega }}_{{\psi }_I(0)[n-1]+{\mathrm{\Omega }}_{\omega }})}^n(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\mathrm{\Omega }}_{{\psi }_I(0)+{\mathrm{\Omega }}_{\mathrm{\Omega }}})\\ & f_{\psi ({\psi }_I(0)[n+1])}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\mathrm{\Omega }}_{{\psi }_I(0)\times 2})\\ & f_{\psi ({\mathrm{\Omega }}_{{\psi }_I(0)[n]+1})}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\mathrm{\Omega }}_{{\psi }_I(0)\times 2+1})\\ & f_{\psi ({\mathrm{\Omega }}_{{\psi }_I(0)[n]\times 2})}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\mathrm{\Omega }}_{{\psi }_I(0)\times 3})\\ & f_{\psi ({\mathrm{\Omega }}_{{\psi }_I(0)[n]\times \omega })}^n(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\mathrm{\Omega }}_{{\psi }_I(0)\times \mathrm{\Omega }})\\ & f_{\psi ({\mathrm{\Omega }}_{{\psi }_I(0)[n{]}^2})}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\mathrm{\Omega }}_{{\psi }_I(0{)}^2})\\ & f_{\psi ({\mathrm{\Omega }}_{{\mathrm{\Omega }}_{{\psi }_I(0)[n]+1}})}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\mathrm{\Omega }}_{{\mathrm{\Omega }}_{{\psi }_I(0)+1}})\\ & f_{\psi ({\psi }_I(0)[n+2])}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\mathrm{\Omega }}_{{\mathrm{\Omega }}_{{\psi }_I(0)\times 2}})\\ & f_{\psi ({\psi }_I(0)[n\times 2])}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(1))\\ & f_{\psi ({\psi }_I(0)[n\times 2+1])}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\mathrm{\Omega }}_{{\psi }_I(1)\times 2})\\ & f_{\psi ({\psi }_I(0)[n\times 3])}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(2))\\ & f_{\psi ({\psi }_I(0)[n^2])}(f_{\psi ({\psi }_I(0))}(n))\sim \psi ({\psi }_I(\omega ))\\ & f_{\psi ({\psi }_I(0))}^2(n)\sim \psi ({\psi }_I(\psi ({\psi }_I(0))))\\ & f_{\psi ({\psi }_I(0))}^3(n)\sim \psi ({\psi }_I(\psi ({\psi }_I(\psi ({\psi }_I(0))))))\\ & f_{\psi ({\psi }_I(0))+1}(n)\sim \psi ({\psi }_I(\mathrm{\Omega }))\\ & f_{\psi ({\psi }_I(0))+2}(n)\sim \psi ({\psi }_I(\mathrm{\Omega })+{\psi }_{{\mathrm{\Omega }}_2}({\psi }_I(\mathrm{\Omega }))\times \mathrm{\Omega })\\ & f_{\psi ({\psi }_I(0)+\mathrm{\Omega })}(n)\sim \psi ({\psi }_I(\mathrm{\Omega })+{\mathrm{\Omega }}_2)\\ & f_{\psi ({\psi }_I(0)\times \mathrm{\Omega })}(n)\sim \psi ({\psi }_I(\mathrm{\Omega })\times {\mathrm{\Omega }}_2)\\ & f_{\psi ({\psi }_I(0{)}^2)+1}(n)\sim \psi ({\psi }_I(\mathrm{\Omega }{)}^2)\\ & f_{\psi ({\psi }_{{\mathrm{\Omega }}_{{\psi }_I(0)+1}}(0))}(n)\sim \psi ({\psi }_{{\mathrm{\Omega }}_{{\psi }_I(\mathrm{\Omega })+1}}(0))\\ & f_{\psi ({\mathrm{\Omega }}_{{\psi }_I(0)+1})}(n)\sim \psi ({\mathrm{\Omega }}_{{\psi }_I(\mathrm{\Omega })+1})\\ & f_{\psi ({\mathrm{\Omega }}_{{\mathrm{\Omega }}_{{\psi }_I(0)+1}})}(n)\sim \psi ({\mathrm{\Omega }}_{{\mathrm{\Omega }}_{{\psi }_I(\mathrm{\Omega })+1}})\\ & f_{\psi ({\psi }_I(1))}(n)\sim \psi ({\psi }_I(\mathrm{\Omega }+1))\\ & f_{\psi ({\psi }_I(1)[n\times 2])}(f_{\psi ({\psi }_I(1))}(n))\sim \psi ({\psi }_I(\mathrm{\Omega }+2))\\ & f_{\psi ({\psi }_I(1))}^2(n)\sim \psi ({\psi }_I(\mathrm{\Omega }+\psi ({\psi }_I(\mathrm{\Omega }))))\\ & f_{\psi ({\psi }_I(1))+1}(n)\sim \psi ({\psi }_I(\mathrm{\Omega }\times 2))\\ & f_{\psi ({\psi }_I(2))}(n)\sim \psi ({\psi }_I(\mathrm{\Omega }\times 2+1))\\ & f_{\psi ({\psi }_I(\omega ))}(n)\sim \psi ({\psi }_I(\mathrm{\Omega }\times \omega ))\\ & f_{\psi ({\psi }_I(\omega ))+1}(n)\sim \psi ({\psi }_I({\mathrm{\Omega }}^2))\\ & f_{\psi ({\psi }_I(\omega +1))+1}(n)\sim \psi ({\psi }_I({\mathrm{\Omega }}^2+\mathrm{\Omega }))\\ & f_{\psi ({\psi }_I({\omega }^{\omega }))+1}(n)\sim \psi ({\psi }_I({\mathrm{\Omega }}^{\mathrm{\Omega }}))\\ & f_{\psi ({\psi }_I(\psi (0)))+1}(n)\sim \psi ({\psi }_I({\psi }_{{\mathrm{\Omega }}_2}(\mathrm{\Omega })))\\ & f_{\psi ({\psi }_I(\psi ({\psi }_I(0))))+1}(n)\sim \psi ({\psi }_I({\psi }_{{\mathrm{\Omega }}_2}({\psi }_I(\mathrm{\Omega }))))\\ & f_{\psi ({\psi }_I(\mathrm{\Omega }))}(n)\sim \psi ({\psi }_I({\mathrm{\Omega }}_2))\\ & f_{\psi ({\psi }_I({\mathrm{\Omega }}_2))}(n)\sim \psi ({\psi }_I({\mathrm{\Omega }}_3))\\ & f_{\psi ({\psi }_I({\mathrm{\Omega }}_3))}(n)\sim \psi ({\psi }_I({\mathrm{\Omega }}_4))\\ & f_{\psi ({\psi }_I({\mathrm{\Omega }}_{\omega }))}(n)\sim \psi ({\psi }_I({\mathrm{\Omega }}_{\omega }))\\ & C(\omega +1)=\psi ({\psi }_I({\mathrm{\Omega }}_{\omega }))\end{array}$

\begin{align}.\\\&f_{\psi(\psi_I(\Omega_\omega))+1}(n)\sim\psi(\psi_I(\Omega_\Omega))\\\&f_{\psi(\psi_I(\Omega_\omega))+2}(n)\sim\psi(\psi_I(\Omega_\Omega)\times\Omega)\\\&f_{\psi(\psi_I(\Omega_\omega)+1)}(n)\sim\psi(\psi_I(\Omega_\Omega)\times\psi_{\Omega_2}(\psi_I(\Omega_\Omega)+1))\\\&f_{\psi(\psi_I(\Omega_\omega)\times\Omega)}(n)\sim\psi(\psi_I(\Omega_\Omega)\times\Omega_2)\\\&f_{\psi(\psi_I(\Omega_\omega)^2)+1}(n)\sim\psi(\psi_I(\Omega_\Omega)^2)\\\&f_{\psi(\Omega_{\psi_I(\Omega_\omega)+1})}(n)\sim\psi(\Omega_{\psi_I(\Omega_\Omega)+1})\\\&f_{\psi(\psi_I(\Omega_\omega+1))}(n)\sim\psi(\psi_I(\Omega_\Omega+1))\\\&f_{\psi(\psi_I(\Omega_\omega+1))+1}(n)\sim\psi(\psi_I(\Omega_\Omega+\Omega))\\\&f_{\psi(\psi_I(\Omega_\omega+\psi(0)))}(n)\sim\psi(\psi_I(\Omega_\Omega+\psi_1(0)))\\\&f_{\psi(\psi_I(\Omega_\omega+\Omega))}(n)\sim\psi(\psi_I(\Omega_\Omega+\Omega_2))\\\&f_{\psi(\psi_I(\Omega_\omega\times2))}(n)\sim\psi(\psi_I(\Omega_\Omega+\Omega_\omega))\\\&f_{\psi(\psi_I(\psi_{\Omega_{\omega+1}}(0)))}(n)\sim\psi(\psi_I(\psi_{\Omega_{\Omega+1}}(0)))\\\&f_{\psi(\psi_I(\Omega_{\omega+1}))}(n)\sim\psi(\psi_I(\Omega_{\Omega+1}))\\\&f_{\psi(\psi_I(\Omega_{\omega\times2}))+1}(n)\sim\psi(\psi_I(\Omega_{\Omega\times2}))\\\&f_{\psi(\psi_I(\Omega_{\psi(0)}))}(n)\sim\psi(\psi_I(\Omega_{\psi_{1}(0)}))\\\&f_{\psi(\psi_I(\Omega_{\Omega}))}(n)\sim\psi(\psi_I(\Omega_{\Omega_2}))\\\&f_{\psi(\psi_I(\Omega_{\Omega_2}))}(n)\sim\psi(\psi_I(\Omega_{\Omega_3}))\\\&f_{\psi(\psi_I(\Omega_{\Omega_\omega}))}(n)\sim\psi(\psi_I(\Omega_{\Omega_\omega}))\\\&\color{red}{C(\omega+2)=\psi(\psi_I(\Omega_{\Omega_\omega}))}\\\&\color{red}{C(\omega\times2)=\psi(\psi_I(\psi_I(0)))}\\\&f_{\psi(\psi_I(\psi_I(0)))+1}(n)\sim\psi(\psi_I(\psi_I(\Omega)))\\\&f_{\psi(\psi_I(\psi_I(1)))+1}(n)\sim\psi(\psi_I(\psi_I(\Omega\times2)))\\\&f_{\psi(\psi_I(\psi_I(\Omega)))}(n)\sim\psi(\psi_I(\psi_I(\Omega_2)))\\\&f_{\psi(\psi_I(\psi_I(\Omega_\omega)))}(n)\sim\psi(\psi_I(\psi_I(\Omega_\omega)))\\\&\color{red}{C(\omega\times2+1)=\psi(\psi_I(\psi_I(\Omega_\omega)))}\\\&\color{red}{C(\omega\times2+2)=\psi(\psi_I(\psi_I(\Omega_{\Omega_\omega})))}\\\&\color{red}{C(\omega\times3)=\psi(\psi_I(\psi_I(\psi_I(0))))}\\\&\color{red}{C(\omega\times4)=\psi(\psi_I(\psi_I(\psi_I(\psi_I(0)))))}\\\&\Large\color{red}{C(\omega^2)=\psi(I)}\\\&f_3(f_{\psi(I)}(n))\sim\psi(I+1)\\\&f_{\psi(0)}^n(f_{\psi(I)}(n))\sim\psi(I+\psi_{\Omega_2}(\Omega))\\\&f_{\psi(\Omega_\omega)[n]}(f_{\psi(I)}(n))\sim\psi(I+\psi_{\Omega_2}(\Omega_\omega))\\\&f_{\psi(\Omega_\omega)}^n(f_{\psi(I)}(n))\sim\psi(I+\psi_{\Omega_2}(\Omega_\Omega))\\\&f_{\psi(\psi_I(0))}^n(f_{\psi(I)}(n))\sim\psi(I+\psi_{\Omega_2}(\psi_I(\Omega)))\\\&f_{\psi(I)[n]}(f_{\psi(I)}(n))\sim\psi(I+\psi_{\Omega_2}(I))\\\&f_{\psi(\psi_I(I)[n]+1)}(f_{\psi(I)}(n))\sim\psi(I+\psi_{\Omega_2}(I+1))\\\&f_{\psi(\psi_I(I)[n]+\Omega)[n]}(f_{\psi(I)}(n))\sim\psi(I+\Omega_2)\\\&f_{\psi(\psi_I(I)[n]+\psi_I(0))[n]}(f_{\psi(I)}(n))\sim\psi(I+\psi_I(0))\\\&f_{\psi(\psi_I(I)[n]\times2)}(f_{\psi(I)}(n))\sim\psi(I+\psi_I(I))\\\&f_{\psi(\psi_{\Omega_{\psi_I(I)[n]+1}}(0))}(f_{\psi(I)}(n))\sim\psi(I+\psi_{\Omega_{\psi_I(I)+1}}(0))\\\&f_{\psi(\psi_{\Omega_{\psi_I(I)[n]+1}}(\psi_I(0)))}(f_{\psi(I)}(n))\sim\psi(I+\psi_{\Omega_{\psi_I(I)+1}}(\psi_I(0)))\\\&f_{\psi(\psi_{\Omega_{\psi_I(I)[n]+1}}(\psi_I(I)[n]))}(f_{\psi(I)}(n))\sim\psi(I+\psi_{\Omega_{\psi_I(I)+1}}(I))\\\&f_{\psi(\Omega_{\psi_I(I)[n]+1})}(f_{\psi(I)}(n))\sim\psi(I+\Omega_{\psi_I(I)+1})\\\&f_{\psi(\psi_I(\psi_I(I)[n-1]+1))}(f_{\psi(I)}(n))\sim\psi(I+\psi_I(I+1))\\\&f_{\psi(\psi_I(\psi_I(I)[n-1]+\Omega_\omega))}(f_{\psi(I)}(n))\sim\psi(I+\psi_I(I+\Omega_\omega))\\\&f_{\psi(\psi_I(\psi_I(I)[n-1]+\psi_I(0)))}(f_{\psi(I)}(n))\sim\psi(I+\psi_I(I+\psi_I(0)))\\\&f_{\psi(I)[n+1]}(f_{\psi(I)}(n))\sim\psi(I+\psi_I(I+\psi_I(I)))\\\&f_{\psi(I)[n+2]}(f_{\psi(I)}(n))\sim\psi(I+\psi_I(I+\psi_I(I+\psi_I(I))))\\\&f_{\psi(I)[n\times2]}(f_{\psi(I)}(n))\sim\psi(I\times2)\\\&f_{\psi(I)}^2(n)\sim\psi(I\times\psi(I))\\\&f_{\psi(I)+1}(n)\sim\psi(I\times\Omega)\\\&f_{\psi(I+1)}(n)\sim\psi(I\times\psi_{\Omega_2}(I+1))\\\&f_{\psi(I\times2)}(n)\sim\psi(I\times\psi_{\Omega_2}(I\times2))\\\&f_{\psi(I\times\Omega)}(n)\sim\psi(I\times\Omega_2)\\\&f_{\psi(I\times\Omega_2)}(n)\sim\psi(I\times\Omega_3)\\\&f_{\psi(I\times\Omega_\omega)}(n)\sim\psi(I\times\Omega_\omega)\\\&C(\omega^2+1)=\psi(I\times\Omega_\omega)\\\&C(\omega^2+2)=\psi(I\times\Omega_{\Omega_\omega})\\\&C(\omega^2+\omega)=\psi(I\times\psi_I(0))\\\&C(\omega^2+\omega+1)=\psi(I\times\psi_I(\Omega_\omega))\\\&C(\omega^2+\omega\times2)=\psi(I\times\psi_I(\psi_I(0)))\\\&{C(\omega^2\times2)=\psi(I\times\psi_I(I))\\\&f_{\psi(I\times\psi_I(I))+1}(n)\sim\psi(I\times\psi_I(I\times\Omega))\\\&f_{\psi(I\times\psi_I(I\times\Omega_\omega))}(n)\sim\psi(I\times\psi_I(I\times\Omega_\omega))\\\&{C(\omega^2\times2+1)=\psi(I\times\psi_I(I\times\Omega_\omega))\\\&{C(\omega^2\times2+\omega)=\psi(I\times\psi_I(I\times\psi_I(0)))\\\&{C(\omega^2\times3)=\psi(I\times\psi_I(I\times\psi_I(I)))\\\&{C(\omega^3)=\psi(I^2)\\\&f_{\psi(I^2)+1}(n)\sim\psi(I^2\times\Omega)\\\&{C(\omega^3+1)=\psi(I^2\times\Omega_\omega)\\\&{C(\omega^3+\omega)=\psi(I^2\times\psi_I(0))\\\&{C(\omega^3+\omega^2)=\psi(I^2\times\psi_I(I))\\\&{C(\omega^3\times2)=\psi(I^2\times\psi_I(I^2))\\\&{C(\omega^4)=\psi(I^3)\\\&{C(\omega^\omega)=\psi(I^\omega)\end{align}