NOCF

NOCF（None Ordinal Collapse Function）是 OCF 的一种。

理念

由于 NOCF 没有完整的定义，这里给出它的理念：

${\psi }_{\alpha }(0)={\mathrm{\Omega }}_{\alpha }({\mathrm{\Omega }}_0=1)$，${\psi }_{\alpha }(\mathrm{♯}+1)={\psi }_{\alpha }(\mathrm{♯})+1$；在 OCF 内遇到 ${\mathrm{\Omega }}_{\alpha }$ 的处理方式与 MOCF 一致。

分析

下列分析的等号左侧为 NOCF，右侧为 MOCF。

• ${\psi }_0(0)=1$
• ${\psi }_0({\psi }_0(0))=2$
• ${\psi }_0({\psi }_0({\psi }_0(0)))=3$
• ${\psi }_0({\psi }_1(0))=\omega$
• ${\psi }_0({\psi }_1({\psi }_0(0)))=\omega +1$
• ${\psi }_0({\psi }_1({\psi }_0({\psi }_1(0))))=\omega \times 2$
• ${\psi }_0({\psi }_1({\psi }_1(0)))={\omega }^2$
• ${\psi }_0({\psi }_1({\psi }_1({\psi }_0({\psi }_1({\psi }_1(0))))))={\omega }^2\times 2$
• ${\psi }_0({\psi }_1({\psi }_1({\psi }_1(0))))={\omega }^3$
• ${\psi }_0({\psi }_1({\psi }_1({\psi }_1({\psi }_1(0)))))={\omega }^4$
• ${\psi }_0({\psi }_2(0))={\omega }^{\omega }$
• ${\psi }_0({\psi }_2({\psi }_0({\psi }_2(0))))={\omega }^{\omega }\times 2$
• ${\psi }_0({\psi }_2({\psi }_1(0)))={\omega }^{\omega +1}$
• ${\psi }_0({\psi }_2({\psi }_1({\psi }_1(0))))={\omega }^{\omega +2}$
• ${\psi }_0({\psi }_2({\psi }_1({\psi }_2(0))))={\omega }^{\omega \times 2}$
• ${\psi }_0({\psi }_2({\psi }_1({\psi }_2({\psi }_1({\psi }_2(0))))))={\omega }^{\omega \times 3}$
• ${\psi }_0({\psi }_2({\psi }_2(0))={\omega }^{{\omega }^2}$
• ${\psi }_0({\psi }_2({\psi }_2({\psi }_1(0)))={\omega }^{{\omega }^2+1}$
• ${\psi }_0({\psi }_2({\psi }_2({\psi }_1({\psi }_2({\psi }_2(0)))))={\omega }^{{\omega }^2\times 2}$
• ${\psi }_0({\psi }_2({\psi }_2({\psi }_2(0))))={\omega }^{{\omega }^3}$
• ${\psi }_0({\psi }_3(0))={\omega }^{{\omega }^{\omega }}$
• ${\psi }_0({\psi }_3({\psi }_1(0)))={\omega }^{{\omega }^{\omega }+1}$
• ${\psi }_0({\psi }_3({\psi }_2(0)))={\omega }^{{\omega }^{\omega +1}}$
• ${\psi }_0({\psi }_3({\psi }_3(0)))={\omega }^{{\omega }^{{\omega }^2}}$
• ${\psi }_0({\psi }_4(0))={\omega }^{{\omega }^{{\omega }^{\omega }}}$
• ${\psi }_0({\psi }_{\omega }(0))=\psi (0)$
• ${\psi }_0({\psi }_{\omega }({\psi }_0({\psi }_{\omega }(0))))=\psi (0)\times 2$
• ${\psi }_0({\psi }_{\omega }({\psi }_1(0)))={\omega }^{\psi (0)+1}$
• ${\psi }_0({\psi }_{\omega }({\psi }_1({\psi }_2(0))))={\omega }^{\psi (0)+\omega }$
• ${\psi }_0({\psi }_{\omega }({\psi }_1({\psi }_{\omega }(0))))={\omega }^{\psi (0)\times 2}$
• ${\psi }_0({\psi }_{\omega }({\psi }_2(0)))={\omega }^{{\omega }^{\psi (0)+1}}$
• ${\psi }_0({\psi }_{\omega }({\psi }_2({\psi }_{\omega }(0))))={\omega }^{{\omega }^{\psi (0)\times 2}}$
• ${\psi }_0({\psi }_{\omega }({\psi }_{\omega }(0)))=\psi (1)$
• ${\psi }_0({\psi }_{\omega }({\psi }_{\omega }({\psi }_{\omega }(0))))=\psi (2)$
• ${\psi }_0({\psi }_{\omega +1}(0))=\psi (\omega )$
• ${\psi }_0({\psi }_{\omega +1}({\psi }_{\omega }(0)))=\psi (\omega +1)$
• ${\psi }_0({\psi }_{\omega +1}({\psi }_{\omega }({\psi }_{\omega +1}(0))))=\psi (\omega \times 2)$
• ${\psi }_0({\psi }_{\omega +1}({\psi }_{\omega +1}(0)))=\psi ({\omega }^2)$
• ${\psi }_0({\psi }_{\omega +2}(0))=\psi ({\omega }^{\omega })$
• ${\psi }_0({\psi }_{\omega +3}(0))=\psi ({\omega }^{{\omega }^{\omega }})$
• ${\psi }_0({\psi }_{\omega \times 2}(0))=\psi (\psi (0))$
• ${\psi }_0({\psi }_{\omega \times 2}({\psi }_{\omega }(0)))=\psi (\psi (0)+1)$
• ${\psi }_0({\psi }_{\omega \times 2}({\psi }_{\omega }({\psi }_{\omega \times 2}(0))))=\psi (\psi (0)\times 2)$
• ${\psi }_0({\psi }_{\omega \times 2}({\psi }_{\omega +1}(0)))=\psi ({\omega }^{\psi (0)+1})$
• ${\psi }_0({\psi }_{\omega \times 2}({\psi }_{\omega \times 2}(0)))=\psi (\psi (1))$
• ${\psi }_0({\psi }_{\omega \times 2+1}(0))=\psi (\psi (\omega ))$
• ${\psi }_0({\psi }_{\omega \times 3}(0))=\psi (\psi (\psi (0)))$
• ${\psi }_0({\psi }_{{\omega }^2}(0))=\psi (\mathrm{\Omega })$
• ${\psi }_0({\psi }_{{\omega }^2}({\psi }_{\omega }(0)))=\psi (\mathrm{\Omega }+1)$
• ${\psi }_0({\psi }_{{\omega }^2}({\psi }_{\omega }({\psi }_{\omega \times 2}(0))))=\psi (\mathrm{\Omega }+\psi (0))$
• ${\psi }_0({\psi }_{{\omega }^2}({\psi }_{\omega }({\psi }_{{\omega }^2}(0))))=\psi (\mathrm{\Omega }+\psi (\mathrm{\Omega }))$
• ${\psi }_0({\psi }_{{\omega }^2}({\psi }_{\omega +1}(0)))=\psi (\mathrm{\Omega }+{\omega }^{\psi (\mathrm{\Omega })+1})$
• ${\psi }_0({\psi }_{{\omega }^2}({\psi }_{\omega \times 2}(0)))=\psi (\mathrm{\Omega }+\psi (\mathrm{\Omega }+1))$
• ${\psi }_0({\psi }_{{\omega }^2}({\psi }_{\omega \times 3}(0)))=\psi (\mathrm{\Omega }+\psi (\mathrm{\Omega }+\psi (\mathrm{\Omega }+1)))$
• ${\psi }_0({\psi }_{{\omega }^2}({\psi }_{{\omega }^2}(0)))=\psi (\mathrm{\Omega }\times 2)$
• ${\psi }_0({\psi }_{{\omega }^2}({\psi }_{{\omega }^2}({\psi }_{{\omega }^2}(0))))=\psi (\mathrm{\Omega }\times 3)$
• ${\psi }_0({\psi }_{{\omega }^2+1}(0))=\psi (\mathrm{\Omega }\times \omega )$
• ${\psi }_0({\psi }_{{\omega }^2+1}({\psi }_{{\omega }^2}(0)))=\psi (\mathrm{\Omega }\times \omega +\mathrm{\Omega })$
• ${\psi }_0({\psi }_{{\omega }^2+1}({\psi }_{{\omega }^2}({\psi }_{{\omega }^2+1}(0))))=\psi (\mathrm{\Omega }\times \omega \times 2)$
• ${\psi }_0({\psi }_{{\omega }^2+1}({\psi }_{{\omega }^2+1}(0)))=\psi (\mathrm{\Omega }\times {\omega }^2)$
• ${\psi }_0({\psi }_{{\omega }^2+2}(0))=\psi (\mathrm{\Omega }\times {\omega }^{\omega })$
• ${\psi }_0({\psi }_{{\omega }^2+\omega }(0))=\psi (\mathrm{\Omega }\times \psi (0))$
• ${\psi }_0({\psi }_{{\omega }^2\times 2}(0))=\psi (\mathrm{\Omega }\times \psi (\mathrm{\Omega }))$
• ${\psi }_0({\psi }_{{\omega }^3}(0))=\psi ({\mathrm{\Omega }}^2)$
• ${\psi }_0({\psi }_{{\omega }^3}({\psi }_{\omega }(0)))=\psi ({\mathrm{\Omega }}^2+1)$
• ${\psi }_0({\psi }_{{\omega }^3}({\psi }_{{\omega }^2}(0)))=\psi ({\mathrm{\Omega }}^2+\mathrm{\Omega })$
• ${\psi }_0({\psi }_{{\omega }^3}({\psi }_{{\omega }^2}({\psi }_{{\omega }^3}(0))))=\psi ({\mathrm{\Omega }}^2+\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^2))$
• ${\psi }_0({\psi }_{{\omega }^3}({\psi }_{{\omega }^3}(0)))=\psi ({\mathrm{\Omega }}^2\times 2)$
• ${\psi }_0({\psi }_{{\omega }^3+1}(0))=\psi ({\mathrm{\Omega }}^2\times \omega )$
• ${\psi }_0({\psi }_{{\omega }^3\times 2}(0))=\psi ({\mathrm{\Omega }}^2\times \psi ({\mathrm{\Omega }}^2))$
• ${\psi }_0({\psi }_{{\omega }^4}(0))=\psi ({\mathrm{\Omega }}^3)$
• ${\psi }_0({\psi }_{{\omega }^{\omega }}(0))=\psi ({\mathrm{\Omega }}^{\omega })$
• ${\psi }_0({\psi }_{{\psi }_0({\psi }_{\omega }(0))}(0))=\psi ({\mathrm{\Omega }}^{\psi (0)})$
• ${\psi }_0({\psi }_{{\psi }_0({\psi }_{{\omega }^{\omega }}(0))}(0))=\psi ({\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\omega })})$
• ${\psi }_0({\psi }_{{\psi }_0({\psi }_{{\psi }_0({\psi }_{\omega }(0))}(0))}(0))=\psi ({\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\psi (0)})})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}(0))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_0({\psi }_{{\psi }_1(0)}(0))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})\times 2$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1(0)))={\omega }^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})+1}$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{\omega }(0))))={\omega }^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})+\psi (0)}$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0))}(0))))={\omega }^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})\times 2}$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0))}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0))}(0))))))={\omega }^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})\times 3}$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0))}({\psi }_2(0)))))={\omega }^{{\omega }^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})+1}}$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0))}({\psi }_{\omega }(0)))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+1)$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0))}({\psi }_{\omega }({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0))}(0))))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0))}({\psi }_{\omega \times 2}({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0))}(0))))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0))}({\psi }_{{\omega }^2}(0)))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+\mathrm{\Omega })$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0))}({\psi }_{{\omega }^3}(0)))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^2)$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0))}({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0))}(0)))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0))}({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0))}({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0)}(0))))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})}\times 2)$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_0(0)))}(0))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})}\times \omega )$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_0({\psi }_1(0))))}(0))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})}\times \psi (0))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_0({\psi }_{{\psi }_1(0)}(0))))}(0))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})}\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1(0)))}(0))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})+1})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0))}(0)))}(0))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})\times 2})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0))}({\psi }_{\omega }(0))))}(0))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+1)})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_1(0)}(0))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}\times 2)$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_1(0)}(0))))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}\times 3)$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_2(0)))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}\times \omega )$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_2({\psi }_{{\psi }_0({\psi }_{{\psi }_1(0)}(0))}(0))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_2({\psi }_{{\psi }_1(0)}(0))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_2({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_1(0)}(0))))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1}+{\mathrm{\Omega }}^{\mathrm{\Omega }})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_2({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_1(0)}({\psi }_2(0)))))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1}+{\mathrm{\Omega }}^{\mathrm{\Omega }}\times \omega )$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_2({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_1(0)}({\psi }_2({\psi }_{{\psi }_1(0)}(0))))))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1}\times 2)$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_2({\psi }_{{\psi }_1(0)}({\psi }_2({\psi }_{{\psi }_1(0)}(0))))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+2})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_3(0)))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\omega })$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_3({\psi }_{{\psi }_1(0)}(0))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times 2})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_3({\psi }_{{\psi }_1(0)}({\psi }_2({\psi }_{{\psi }_1(0)}(0))))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times 2+1})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_3({\psi }_{{\psi }_1(0)}({\psi }_2({\psi }_{{\psi }_1(0)}({\psi }_3({\psi }_{{\psi }_1(0)}(0))))))))=\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times 3})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_3({\psi }_{{\psi }_1(0)}({\psi }_3({\psi }_{{\psi }_1(0)}(0))))))=\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_4(0)))=\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_4({\psi }_{{\psi }_1(0)}(0))))=\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{\omega }(0)))=\psi ({\psi }_1(0))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{\omega }({\psi }_1({\psi }_{{\psi }_1(0)}(0)))))=\psi ({\psi }_1(0)+{\mathrm{\Omega }}^{\mathrm{\Omega }})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{\omega }({\psi }_1({\psi }_{{\psi }_1(0)}({\psi }_{\omega }(0))))))=\psi ({\psi }_1(0)\times 2)$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{\omega }({\psi }_2(0))))=\psi ({\psi }_1(0)\times \omega )$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{\omega }({\psi }_2({\psi }_{{\psi }_1(0)}(0)))))=\psi ({\mathrm{\Omega }}^{{\psi }_1(0)+1})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{\omega }({\psi }_2({\psi }_{{\psi }_1(0)}({\psi }_{\omega }(0))))))=\psi ({\mathrm{\Omega }}^{{\psi }_1(0)\times 2})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{\omega }({\psi }_{\omega }(0))))=\psi ({\psi }_1(1))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{\omega }({\psi }_{\omega +1}(0))))=\psi ({\psi }_1(\omega ))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{\omega }({\psi }_{{\psi }_1(0)}(0))))=\psi ({\psi }_1(\mathrm{\Omega }))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{\omega +1}(0)))=\psi ({\psi }_1(\mathrm{\Omega }\times \omega ))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{\omega +1}({\psi }_{{\psi }_1(0)}(0))))=\psi ({\psi }_1({\mathrm{\Omega }}^2))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{\omega \times 2}(0)))=\psi ({\psi }_1({\psi }_1(0)))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{{\omega }^2}(0)))=\psi ({\mathrm{\Omega }}_2)$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{{\omega }^2}({\psi }_1({\psi }_{{\psi }_1(0)}({\psi }_{{\omega }^2}(0))))))=\psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{{\omega }^2}({\psi }_{\omega }(0))))=\psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+1))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{{\omega }^2}({\psi }_{\omega \times 2}(0))))=\psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+1)))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{{\omega }^2}({\psi }_{{\omega }^2}(0))))=\psi ({\mathrm{\Omega }}_2\times 2)$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{{\omega }^2}({\psi }_{{\omega }^2+1}(0))))=\psi ({\mathrm{\Omega }}_2\times \omega )$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{{\omega }^2}({\psi }_{{\psi }_1(0)}(0))))=\psi ({\mathrm{\Omega }}_2\times \mathrm{\Omega })$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{{\omega }^2+\omega }(0)))=\psi ({\mathrm{\Omega }}_2\times {\psi }_1(0))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{{\omega }^2\times 2}(0)))=\psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{{\omega }^3}(0)))=\psi ({{\mathrm{\Omega }}_2}^2)$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{{\omega }^{\omega }}(0)))=\psi ({{\mathrm{\Omega }}_2}^{\omega })$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{{\psi }_1(0)}(0)))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }})$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_1(0)}({\psi }_{{\psi }_1(0)}(0))))))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}+{\psi }_1({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{{\psi }_1(0)}({\psi }_{\omega }(0))))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}+{\psi }_1({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}+1))$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{{\psi }_1(0)}({\psi }_{\omega \times 2}(0))))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}+{\mathrm{\Omega }}_2)$
• ${\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{{\psi }_1(0)}({\psi }_{{\psi }_1(0)}(0))))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times 2)$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_0(0))}(0))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times \omega )$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_0({\psi }_1(0)))}(0))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times \psi (0))$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{\omega }(0))))}(0))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times \psi ({\psi }_1(0)))$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_0({\psi }_{{\psi }_1(0)}({\psi }_{{\psi }_1(0)}(0))))}(0))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times \psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}))$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1(0))}(0))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times \mathrm{\Omega })$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1(0))}({\psi }_1({\psi }_{{\psi }_1({\psi }_1(0))}(0))))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times \mathrm{\Omega }+{\psi }_1({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times \mathrm{\Omega }))$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1(0))}({\psi }_{\omega }(0)))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times \mathrm{\Omega }+{\psi }_1({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times \mathrm{\Omega }+1))$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1(0))}({\psi }_{{\omega }^2}(0)))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times \mathrm{\Omega }+{\mathrm{\Omega }}_2)$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1(0))}({\psi }_{{\psi }_1(0)}(0)))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times \mathrm{\Omega }+{{\mathrm{\Omega }}_2}^{\mathrm{\Omega }})$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1(0))}({\psi }_{{\psi }_1(0)}({\psi }_{{\psi }_1({\psi }_1(0))}(0))))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times \mathrm{\Omega }\times 2)$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1(0))}({\psi }_{{\psi }_1({\psi }_0(0))}(0)))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times \mathrm{\Omega }\times \omega )$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1(0))}({\psi }_{{\psi }_1({\psi }_0(0))}({\psi }_{{\psi }_1({\psi }_1(0))}(0))))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times {\mathrm{\Omega }}^2)$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1(0))}({\psi }_{{\psi }_1({\psi }_0({\psi }_1(0)))}(0)))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times {\psi }_1(0))$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1(0))}({\psi }_{{\psi }_1({\psi }_0({\psi }_1({\psi }_1(0))))}(0)))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times {\psi }_1({\mathrm{\Omega }}_2))$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1(0))}({\psi }_{{\psi }_1({\psi }_1(0))}(0)))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times {\psi }_1({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}))$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1(0))}({\psi }_{{\psi }_1({\psi }_1(0))}({\psi }_{{\psi }_1(0)}({\psi }_{{\psi }_1({\psi }_1(0))}({\psi }_{{\psi }_1({\psi }_1(0))}(0))))))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times {\psi }_1({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }})\times 2)$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1(0))}({\psi }_{{\psi }_1({\psi }_1(0))}({\psi }_{{\psi }_1({\psi }_0({\psi }_1(0)))}(0))))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times {\psi }_1({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}+1))$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1(0))}({\psi }_{{\psi }_1({\psi }_1(0))}({\psi }_{{\psi }_1({\psi }_1(0))}(0))))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times {\psi }_1({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times 2))$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1({\psi }_0(0)))}(0))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times {\psi }_1({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times \omega ))$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1({\psi }_1(0)))}(0))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times {\psi }_1({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times \mathrm{\Omega }))$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1({\psi }_1(0)))}({\psi }_{{\psi }_1(0)}({\psi }_{{\psi }_1({\psi }_1({\psi }_1(0)))}(0))))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times {\psi }_1({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times \mathrm{\Omega })\times 2)$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1({\psi }_1(0)))}({\psi }_{{\psi }_1({\psi }_1(0))}({\psi }_{{\psi }_1({\psi }_1({\psi }_1(0)))}(0))))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times {\psi }_1({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times \mathrm{\Omega }\times 2))$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1({\psi }_1(0)))}({\psi }_{{\psi }_1({\psi }_1({\psi }_1(0)))}(0)))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times {\psi }_1({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times {\psi }_1({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }})))$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_1({\psi }_1({\psi }_1(0))))}(0))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times {\psi }_1({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times {\psi }_1({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }}\times \mathrm{\Omega })))$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_2(0))}(0))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }+1})$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_2({\psi }_1({\psi }_2(0))))}(0))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }+2})$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_2({\psi }_2(0)))}(0))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }+\omega })$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_{{\psi }_1(0)}(0))}(0))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }\times 2})$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_{{\psi }_1(0)}({\psi }_1({\psi }_{{\psi }_1(0)}(0))))}(0))=\psi ({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }\times 3})$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_{{\psi }_1(0)}({\psi }_{\omega }(0)))}(0))=\psi ({{\mathrm{\Omega }}_2}^{{\psi }_1(0)})$
• ${\psi }_0({\psi }_{{\psi }_1({\psi }_{{\psi }_1(0)}({\psi }_{{\psi }_1(0)}(0)))}(0))=\psi ({{\mathrm{\Omega }}_2}^{{\psi }_1({{\mathrm{\Omega }}_2}^{\mathrm{\Omega }})})$
• ${\psi }_0({\psi }_{{\psi }_2(0)}(0))=\psi ({{\mathrm{\Omega }}_2}^{{\mathrm{\Omega }}_2})$
• ${\psi }_0({\psi }_{{\psi }_2(0)}({\psi }_1({\psi }_{{\psi }_2(0)}(0))))=\psi ({{\mathrm{\Omega }}_2}^{{\mathrm{\Omega }}_2}+{\psi }_1({{\mathrm{\Omega }}_2}^{{\mathrm{\Omega }}_2}))$
• ${\psi }_0({\psi }_{{\psi }_2(0)}({\psi }_2(0)))=\psi ({{\mathrm{\Omega }}_2}^{{\mathrm{\Omega }}_2}+{\psi }_1({{\mathrm{\Omega }}_2}^{{\mathrm{\Omega }}_2})\times \omega )$
• ${\psi }_0({\psi }_{{\psi }_2(0)}({\psi }_2({\psi }_{{\psi }_1({\psi }_{{\psi }_2(0)}(0))}(0))))=\psi ({{\mathrm{\Omega }}_2}^{{\mathrm{\Omega }}_2}+{\psi }_1({{\mathrm{\Omega }}_2}^{{\mathrm{\Omega }}_2}{)}^2)$
• ${\psi }_0({\psi }_{{\psi }_2(0)}({\psi }_2({\psi }_{{\psi }_1({\psi }_{{\psi }_2(0)}(0))}({\psi }_{\omega }(0)))))=\psi ({{\mathrm{\Omega }}_2}^{{\mathrm{\Omega }}_2}+{\psi }_1({{\mathrm{\Omega }}_2}^{{\mathrm{\Omega }}_2}+1))$
• ${\psi }_0({\psi }_{{\psi }_2(0)}({\psi }_2({\psi }_{{\psi }_1({\psi }_{{\psi }_2(0)}(0))}({\psi }_{{\omega }^2}(0)))))=\psi ({{\mathrm{\Omega }}_2}^{{\mathrm{\Omega }}_2}+{\mathrm{\Omega }}_2)$
• ${\psi }_0({\psi }_{{\psi }_2(0)}({\psi }_2({\psi }_{{\psi }_1({\psi }_{{\psi }_2(0)}(0))}({\psi }_{{\psi }_1({\psi }_{{\psi }_2(0)}(0))}(0)))))=\psi ({{\mathrm{\Omega }}_2}^{{\mathrm{\Omega }}_2}+{\mathrm{\Omega }}_2^{{\psi }_1({{\mathrm{\Omega }}_2}^{{\mathrm{\Omega }}_2})})$
• ${\psi }_0({\psi }_{{\psi }_2(0)}({\psi }_2({\psi }_{{\psi }_2(0)}(0))))=\psi ({{\mathrm{\Omega }}_2}^{{\mathrm{\Omega }}_2}\times 2)$
• ${\psi }_0({\psi }_{{\psi }_2(0)}({\psi }_{\omega }(0)))=\psi ({\psi }_2(0))$
• ${\psi }_0({\psi }_{{\psi }_3(0)}(0))=\psi ({{\mathrm{\Omega }}_3}^{{\mathrm{\Omega }}_3})$
• ${\psi }_0({\psi }_{{\psi }_3(0)}({\psi }_{\omega }(0)))=\psi ({\psi }_3(0))$
• ${\psi }_0({\psi }_{{\psi }_{\omega }(0)}(0))=\psi ({\mathrm{\Omega }}_{\omega })$
• ${\psi }_0({\psi }_{{\psi }_{\omega }(0)}({\psi }_1({\psi }_{{\psi }_{\omega }(0)}(0))))=\psi ({\mathrm{\Omega }}_{\omega }+{\psi }_1({\mathrm{\Omega }}_{\omega }))$
• ${\psi }_0({\psi }_{{\psi }_{\omega }(0)}({\psi }_2({\psi }_{{\psi }_{\omega }(0)}(0))))=\psi ({\mathrm{\Omega }}_{\omega }+{\psi }_2({\mathrm{\Omega }}_{\omega }))$
• ${\psi }_0({\psi }_{{\psi }_{\omega }(0)}({\psi }_{\omega }(0)))=\psi ({\mathrm{\Omega }}_{\omega }\times 2)$
• ${\psi }_0({\psi }_{{\psi }_{\omega }(0)}({\psi }_{\omega }({\psi }_{\omega }(0))))=\psi ({\mathrm{\Omega }}_{\omega }\times 3)$
• ${\psi }_0({\psi }_{{\psi }_{\omega }(0)}({\psi }_{\omega }({\psi }_{{\psi }_1(0)}(0))))=\psi ({\mathrm{\Omega }}_{\omega }\times \mathrm{\Omega })$
• ${\psi }_0({\psi }_{{\psi }_{\omega }(0)}({\psi }_{\omega }({\psi }_{{\psi }_{\omega }(0)}(0))))=\psi ({{\mathrm{\Omega }}_{\omega }}^2)$
• ${\psi }_0({\psi }_{{\psi }_{\omega }(0)}({\psi }_{\omega }({\psi }_{{\psi }_{\omega }(0)}({\psi }_{\omega }(0)))))=\psi ({{\mathrm{\Omega }}_{\omega }}^2+{\mathrm{\Omega }}_{\omega })$
• ${\psi }_0({\psi }_{{\psi }_{\omega }(0)}({\psi }_{\omega +1}(0)))=\psi ({{\mathrm{\Omega }}_{\omega }}^2\times \omega )$
• ${\psi }_0({\psi }_{{\psi }_{\omega }(0)}({\psi }_{\omega +1}({\psi }_{{\psi }_{\omega }(0)}(0))))=\psi ({{\mathrm{\Omega }}_{\omega }}^3)$
• ${\psi }_0({\psi }_{{\psi }_{\omega }(0)}({\psi }_{\omega +2}(0)))=\psi ({{\mathrm{\Omega }}_{\omega }}^{\omega })$
• ${\psi }_0({\psi }_{{\psi }_{\omega }(0)}({\psi }_{\omega +2}({\psi }_{{\psi }_{\omega }(0)}(0))))=\psi ({{\mathrm{\Omega }}_{\omega }}^{{\mathrm{\Omega }}_{\omega }})$
• ${\psi }_0({\psi }_{{\psi }_{\omega }(0)}({\psi }_{\omega \times 2}(0)))=\psi ({\psi }_{\omega }(0))$
• ${\psi }_0({\psi }_{{\psi }_{\omega +1}(0)}(0))=\psi ({{\mathrm{\Omega }}_{\omega +1}}^{{\mathrm{\Omega }}_{\omega +1}})$
• ${\psi }_0({\psi }_{{\psi }_{\omega \times 2}(0)}(0))=\psi ({\mathrm{\Omega }}_{\omega \times 2})$
• ${\psi }_0({\psi }_{{\psi }_{\omega \times 2}(0)}({\psi }_{\omega }(0)))=\psi ({\mathrm{\Omega }}_{\omega \times 2}+{\mathrm{\Omega }}_{\omega })$
• ${\psi }_0({\psi }_{{\psi }_{\omega \times 2}(0)}({\psi }_{\omega \times 2}(0)))=\psi ({\mathrm{\Omega }}_{\omega \times 2}\times 2)$
• ${\psi }_0({\psi }_{{\psi }_{\omega \times 2}(0)}({\psi }_{\omega \times 3}(0)))=\psi ({\psi }_{\omega \times 2}(0))$
• ${\psi }_0({\psi }_{{\psi }_{{\omega }^2}(0)}(0))=\psi ({\mathrm{\Omega }}_{{\omega }^2})$
• ${\psi }_0({\psi }_{{\psi }_{{\psi }_1(0)}(0)}(0))=\psi ({\mathrm{\Omega }}_{\mathrm{\Omega }})$
• ${\psi }_0({\psi }_{{\psi }_{{\psi }_2(0)}(0)}(0))=\psi ({\mathrm{\Omega }}_{{\mathrm{\Omega }}_2})$
• ${\psi }_0({\psi }_{{\psi }_{{\psi }_{\omega }(0)}(0)}(0))=\psi ({\mathrm{\Omega }}_{{\mathrm{\Omega }}_{\omega }})$
• ${\psi }_0({\psi }_{{\psi }_{{\psi }_{{\psi }_1(0)}(0)}(0)}(0))=\psi ({\mathrm{\Omega }}_{{\mathrm{\Omega }}_{\mathrm{\Omega }}})$
• $limit=\psi ({\psi }_I(0))$