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}$