SGH与FGH对照

本条目展示 SGH 与 FGH 的对照，使用 Veblen 函数和 MOCF。本条目分析来自梅天狸。

Part 1

• $g_{{\epsilon }_0\times 2}(n)=g_{{\epsilon }_0+{\epsilon }_0[n]}(n)=(n\uparrow \uparrow n)\times 2$
• $g_{{\epsilon }_0\times \omega }(n)=g_{{\epsilon }_0\times n}(n)=(n\uparrow \uparrow n)\times n$
• $g_{{\epsilon }_0\times {\omega }^2}(n)=g_{{\epsilon }_0\times \omega \times n}(n)=(n\uparrow \uparrow n)\times n^2$
• $g_{{\epsilon }_0\times {\omega }^{\omega }}(n)=g_{{\epsilon }_0\times {\omega }^n}(n)=(n\uparrow \uparrow n)\times n^n$
• $g_{{\epsilon }_0\times {\omega }^{{\omega }^{\omega }}}(n)=(n\uparrow \uparrow n)\times (n\uparrow \uparrow 3)$
• $g_{{\epsilon }_0^2}(n)=g_{{\epsilon }_0\times {\epsilon }_0[n]}(n)=(n\uparrow \uparrow n{)}^2$
• $g_{{\epsilon }_0^3}(n)=(n\uparrow \uparrow n{)}^3$
• $g_{{\epsilon }_0^{\omega }}(n)=g_{{\epsilon }_0^n}(n)=(n\uparrow \uparrow n{)}^n$
• $g_{{\epsilon }_0^{{\omega }^{\omega }}}(n)=(n\uparrow \uparrow n{)}^{n^n}$
• $g_{{\epsilon }_0^{{\epsilon }_0}}(n)=(n\uparrow \uparrow n{)}^{n\uparrow \uparrow n}=(n\uparrow \uparrow n)\uparrow \uparrow 2$
• $g_{{\epsilon }_0^{{\epsilon }_0^{{\epsilon }_0}}}(n)=(n\uparrow \uparrow n)\uparrow \uparrow 3$
• $g_{{\epsilon }_1}(n)=(n\uparrow \uparrow n)\uparrow \uparrow n\approx n\uparrow \uparrow (n\times 2)$
• $g_{{\epsilon }_2}(n)=((n\uparrow \uparrow n)\uparrow \uparrow n)\uparrow \uparrow n\approx n\uparrow \uparrow (n\times 3)$
• $n\uparrow \uparrow (n\times 3)\sim {\epsilon }_2$
• $n\uparrow \uparrow (n\times 3+1)\sim {\epsilon }_2^{{\epsilon }_2}$
• $n\uparrow \uparrow (n\times 3+2)\sim {\epsilon }_2^{{\epsilon }_2^{{\epsilon }_2}}$
• $n\uparrow \uparrow (n\times 4)\sim {\epsilon }_3$
• $n\uparrow \uparrow (n\times 5)\sim {\epsilon }_4$
• $n\uparrow \uparrow (n^2)\sim {\epsilon }_{\omega }$
• $n\uparrow \uparrow (n^2+1)\sim {\epsilon }_{\omega }^{{\epsilon }_{\omega }}$
• $n\uparrow \uparrow (n^2+n)\sim {\epsilon }_{\omega +1}$
• $n\uparrow \uparrow (n^2\times 2)\sim {\epsilon }_{\omega \times 2}$
• $n\uparrow \uparrow (n^3)\sim {\epsilon }_{{\omega }^2}$
• $n\uparrow \uparrow (n^n)=n\uparrow \uparrow n\uparrow \uparrow 2\sim {\epsilon }_{{\omega }^{\omega }}$
• $n\uparrow \uparrow (n^n+n)\sim {\epsilon }_{{\omega }^{\omega }+1}$
• $n\uparrow \uparrow (n^{n+1})\sim {\epsilon }_{{\omega }^{\omega +1}}$
• $n\uparrow \uparrow (n^{n^2})\sim {\epsilon }_{{\omega }^{{\omega }^2}}$
• $n\uparrow \uparrow n\uparrow \uparrow 3\sim {\epsilon }_{{\omega }^{{\omega }^{\omega }}}$
• $n\uparrow \uparrow n\uparrow \uparrow n=n\uparrow \uparrow \uparrow 3\sim {\epsilon }_{{\epsilon }_0}$
• $n\uparrow \uparrow (n\uparrow \uparrow n+n)\sim {\epsilon }_{{\epsilon }_0+1}$
• $n\uparrow \uparrow ((n\uparrow \uparrow n)\times 2)\sim {\epsilon }_{{\epsilon }_0\times 2}$
• $n\uparrow \uparrow ((n\uparrow \uparrow n)\uparrow \uparrow 2)\approx n\uparrow \uparrow n\uparrow \uparrow (n+1)\sim {\epsilon }_{{\epsilon }_0^{{\epsilon }_0}}$
• $n\uparrow \uparrow n\uparrow \uparrow (n\times 2)\sim {\epsilon }_{{\epsilon }_1}$
• $n\uparrow \uparrow n\uparrow \uparrow (n^2)\sim {\epsilon }_{{\epsilon }_{\omega }}$
• $n\uparrow \uparrow n\uparrow \uparrow n\uparrow \uparrow n=n\uparrow \uparrow \uparrow 4\sim {\epsilon }_{{\epsilon }_{{\epsilon }_0}}$
• $n\uparrow \uparrow \uparrow n\sim {\zeta }_0$
• $(n\uparrow \uparrow \uparrow n)\uparrow \uparrow n\sim {\epsilon }_{{\zeta }_0+1}$
• $(n\uparrow \uparrow \uparrow n)\uparrow \uparrow (n\times 2)\sim {\epsilon }_{{\zeta }_0+2}$
• $(n\uparrow \uparrow \uparrow n)\uparrow \uparrow (n\uparrow \uparrow n)\sim {\epsilon }_{{\zeta }_0+{\epsilon }_0}$
• $(n\uparrow \uparrow \uparrow n)\uparrow \uparrow (n\uparrow \uparrow \uparrow n)\approx n\uparrow \uparrow \uparrow (n+1)\sim {\epsilon }_{{\zeta }_0\times 2}$
• $(n\uparrow \uparrow \uparrow n)\uparrow \uparrow (n\uparrow \uparrow \uparrow n)\uparrow \uparrow n\sim {\epsilon }_{{\epsilon }_{{\zeta }_0+1}}$
• $(n\uparrow \uparrow \uparrow n)\uparrow \uparrow (n\uparrow \uparrow \uparrow n)\uparrow \uparrow (n\uparrow \uparrow \uparrow n)\sim {\epsilon }_{{\epsilon }_{{\zeta }_0\times 2}}$
• $(n\uparrow \uparrow \uparrow n)\uparrow \uparrow \uparrow n\approx n\uparrow \uparrow \uparrow (n\times 2)\sim {\zeta }_1$
• $(n\uparrow \uparrow \uparrow (n\times 2))\uparrow \uparrow n\sim {\epsilon }_{{\zeta }_1+1}$
• $(n\uparrow \uparrow \uparrow (n\times 2))\uparrow \uparrow (n\uparrow \uparrow \uparrow n)\sim {\epsilon }_{{\zeta }_1+{\zeta }_0}$
• $(n\uparrow \uparrow \uparrow (n\times 2))\uparrow \uparrow (n\uparrow \uparrow \uparrow (n\times 2))\sim {\epsilon }_{{\zeta }_1\times 2}$
• $(n\uparrow \uparrow \uparrow (n\times 2))\uparrow \uparrow \uparrow n\approx n\uparrow \uparrow \uparrow (n\times 3)\sim {\zeta }_2$
• $n\uparrow \uparrow \uparrow (n^2)\sim {\zeta }_{\omega }$
• $n\uparrow \uparrow \uparrow (n^n)\sim {\zeta }_{{\omega }^{\omega }}$
• $n\uparrow \uparrow \uparrow (n\uparrow \uparrow n)\sim {\zeta }_{{\epsilon }_0}$
• $n\uparrow \uparrow \uparrow n\uparrow \uparrow \uparrow n\sim {\zeta }_{{\zeta }_0}$
• $n\uparrow \uparrow \uparrow \uparrow n\sim {\eta }_0$
• $(n\uparrow \uparrow \uparrow \uparrow n)\uparrow \uparrow \uparrow n\sim {\zeta }_{{\eta }_0+1}$
• $n\uparrow \uparrow \uparrow \uparrow (n\times 2)\sim {\eta }_1$
• $n{\uparrow }^{5}3\sim {\eta }_{{\eta }_0}$
• $n{\uparrow }^{5}n\sim \phi (4,0)$
• $n{\uparrow }^5(n\times 2)\sim \phi (4,1)$
• $n{\uparrow }^{6}n\sim \phi (5,0)$
• $n{\uparrow }^{n}n\sim \phi (\omega ,0)$

Part 2

• $f_3(f_{\omega }(n))\approx (n{\uparrow }^{n}n)\uparrow \uparrow n\sim \psi ({\mathrm{\Omega }}^{\omega }+1)$
• $f_3^2(f_{\omega }(n))\approx (n{\uparrow }^{n}n)\uparrow \uparrow (n\times 2)\sim \psi ({\mathrm{\Omega }}^{\omega }+2)$
• $f_3^n(f_{\omega }(n))\approx (n{\uparrow }^{n}n)\uparrow \uparrow (n^2)\sim \psi ({\mathrm{\Omega }}^{\omega }+\omega )$
• $f_3^{f_3(n)}(f_{\omega }(n))\approx (n{\uparrow }^{n}n)\uparrow \uparrow (n\uparrow \uparrow n)\sim \psi ({\mathrm{\Omega }}^{\omega }+\psi (0))$
• $f_4(f_{\omega }(n))=f_3^{f_{\omega }(n)}(f_{\omega }(n))\approx (n{\uparrow }^{n}n)\uparrow \uparrow \uparrow 2\sim \psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega }))$
• $f_3(f_4(f_{\omega }(n)))\approx ((n{\uparrow }^{n}n)\uparrow \uparrow \uparrow 2)\uparrow \uparrow n\sim \psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega })+1)$
• $f_3^{f_{\omega }(n)}(f_4(f_{\omega }(n)))\approx ((n{\uparrow }^{n}n)\uparrow \uparrow \uparrow 2)\uparrow \uparrow (n{\uparrow }^{n}n)\sim \psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega })\times 2)$
• $f_3^{f_{\omega }(n)\times 2}(f_4(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega })\times 3)$
• $f_3^{f_3(f_{\omega }(n))}(f_4(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega }+1))$
• $f_3^{f_3^2(f_{\omega }(n))}(f_4(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega }+2))$
• $f_4^2(f_{\omega }(n))=f_3^{f_3^{f_{\omega }(n)}(f_{\omega }(n))}(f_4(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega })))$
• $f_3(f_4^2(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega }))+1)$
• $f_3^{f_4(f_{\omega }(n))}(f_4^2(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega }))\times 2)$
• $f_3^{f_3(f_4(f_{\omega }(n)))}(f_4^2(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega })+1))$
• $f_3^{f_3^{f_{\omega }(n)}(f_4(f_{\omega }(n)))}(f_4^2(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega })\times 2))$
• $f_4^3(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega }))))$
• $f_4^n(f_{\omega }(n))\approx (n{\uparrow }^{n}n)\uparrow \uparrow \uparrow n\sim \psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega })$
• $f_4^{n+1}(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }+\psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }))$
• $f_4^{n+2}(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }+\psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }+\psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega })))$
• $f_4^{n\times 2}(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }\times 2)$
• $f_4^{n^2}(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }\times \omega )$
• $f_4^{f_3(n)}(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }\times \psi (0))$
• $f_5(f_{\omega }(n))=f_4^{f_{\omega }(n)}(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^{\omega }))$
• $f_3(f_5(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^{\omega })+1)$
• $f_3^{f_{\omega }(n)}(f_5(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^{\omega })+\psi ({\mathrm{\Omega }}^{\omega }))$
• $f_4(f_5(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^{\omega })+\psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^{\omega })))$
• $f_4^n(f_5(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^{\omega })+\mathrm{\Omega })$
• $f_4^{f_{\omega }(n)}(f_5(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^{\omega })\times 2)$
• $f_4^{f_{\omega }(n)\times 2}(f_5(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^{\omega })\times 3)$
• $f_4^{f_3(f_{\omega }(n))}(f_5(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^{\omega }+1))$
• $f_4^{f_4(f_{\omega }(n))}(f_5(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }))$
• $f_5^2(f_{\omega }(n))=f_4^{f_5(f_{\omega }(n))}(f_5(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^{\omega })))$
• $f_5^n(f_{\omega }(n))\approx (n{\uparrow }^{n}n)\uparrow \uparrow \uparrow \uparrow n\sim \psi ({\mathrm{\Omega }}^{\omega }+{\mathrm{\Omega }}^2)$
• $f_5^{n\times 2}(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega }+{\mathrm{\Omega }}^2\times 2)$
• $f_6(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega }+{\mathrm{\Omega }}^2\times \psi ({\mathrm{\Omega }}^{\omega }))$
• $f_6^n(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega }+{\mathrm{\Omega }}^3)$
• $f_7^n(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega }+{\mathrm{\Omega }}^4)$
• $f_n(f_{\omega }(n))\approx (n{\uparrow }^{n}n){\uparrow }^{n}n\approx n{\uparrow }^n(2n)\sim \psi ({\mathrm{\Omega }}^{\omega }\times 2)$
• $f_n^2(f_{\omega }(n))\approx n{\uparrow }^n(3n)\sim \psi ({\mathrm{\Omega }}^{\omega }\times 3)$
• $f_n^n(f_{\omega }(n))\approx n{\uparrow }^n(n^2)\sim \psi ({\mathrm{\Omega }}^{\omega }\times \omega )$
• $f_n^{f_3(n)}(f_{\omega }(n))\approx n{\uparrow }^{n}n\uparrow \uparrow n\sim \psi ({\mathrm{\Omega }}^{\omega }\times \psi (0))$
• $f_{n+1}(f_{\omega }(n))=f_n^{f_{\omega }(n)}(f_{\omega }(n))\approx n{\uparrow }^{n}n{\uparrow }^{n}n\sim \psi ({\mathrm{\Omega }}^{\omega }\times \psi ({\mathrm{\Omega }}^{\omega }))$
• $f_n(f_{n+1}(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }\times \psi ({\mathrm{\Omega }}^{\omega })+{\mathrm{\Omega }}^{\omega })$
• $f_n^{f_{\omega }(n)}(f_{n+1}(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }\times \psi ({\mathrm{\Omega }}^{\omega })\times 2)$
• $f_n^{f_3(f_{\omega }(n))}(f_{n+1}(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }\times \psi ({\mathrm{\Omega }}^{\omega }+1))$
• $f_n^{f_n(f_{\omega }(n))}(f_{n+1}(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega }\times \psi ({\mathrm{\Omega }}^{\omega }\times 2))$
• $f_{n+1}^2(f_{\omega }(n))\approx n{\uparrow }^{n}n{\uparrow }^{n}n{\uparrow }^{n}n\sim \psi ({\mathrm{\Omega }}^{\omega }\times \psi ({\mathrm{\Omega }}^{\omega }\times \psi ({\mathrm{\Omega }}^{\omega })))$
• $f_{n+1}^n(f_{\omega }(n))\approx n{\uparrow }^{n+1}n\sim \psi ({\mathrm{\Omega }}^{\omega +1})$
• $f_n(f_{n+1}^n(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega +1}+{\mathrm{\Omega }}^{\omega })$
• $f_{n+1}^{n+1}(f_{\omega }(n))=f_n^{f_{n+1}^n(f_{\omega }(n))}(f_{n+1}^n(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega +1}+{\mathrm{\Omega }}^{\omega }\times \psi ({\mathrm{\Omega }}^{\omega +1}))$
• $f_{n+1}^{n\times 2}(f_{\omega }(n))\approx n{\uparrow }^{n+1}(2n)\sim \psi ({\mathrm{\Omega }}^{\omega +1}\times 2)$
• $f_{n+2}(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega +1}\times \psi ({\mathrm{\Omega }}^{\omega }))$
• $f_n(f_{n+2}(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega +1}\times \psi ({\mathrm{\Omega }}^{\omega })+{\mathrm{\Omega }}^{\omega })$
• $f_n^{f_{n+1}^n(f_{\omega }(n))}(f_{n+2}(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega +1}\times \psi ({\mathrm{\Omega }}^{\omega })+{\mathrm{\Omega }}^{\omega }\times \psi ({\mathrm{\Omega }}^{\omega +1}))$
• $f_{n+1}(f_{n+2}(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega +1}\times \psi ({\mathrm{\Omega }}^{\omega })+{\mathrm{\Omega }}^{\omega }\times \psi ({\mathrm{\Omega }}^{\omega +1}\times \psi ({\mathrm{\Omega }}^{\omega })))$
• $f_{n+1}^n(f_{n+2}(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega +1}\times \psi ({\mathrm{\Omega }}^{\omega })+{\mathrm{\Omega }}^{\omega +1})$
• $f_{n+1}^{f_{\omega }(n)}(f_{n+2}(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega +1}\times \psi ({\mathrm{\Omega }}^{\omega })\times 2)$
• $f_{n+1}^{f_{n+1}(f_{\omega }(n))}(f_{n+2}(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega +1}\times \psi ({\mathrm{\Omega }}^{\omega }\times \psi ({\mathrm{\Omega }}^{\omega })))$
• $f_{n+1}^{f_{n+1}^n(f_{\omega }(n))}(f_{n+2}(f_{\omega }(n)))\sim \psi ({\mathrm{\Omega }}^{\omega +1}\times \psi ({\mathrm{\Omega }}^{\omega +1}))$
• $f_{n+2}^2(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega +1}\times \psi ({\mathrm{\Omega }}^{\omega +1}\times \psi ({\mathrm{\Omega }}^{\omega })))$
• $f_{n+2}^n(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega +2})$
• $f_{n+2}^{n+1}(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega +2}+{\mathrm{\Omega }}^{\omega +1}\times \psi ({\mathrm{\Omega }}^{\omega }))$
• $f_{n+2}^{n\times 2}(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega +2}\times 2)$
• $f_{n+3}(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega +2}\times \psi ({\mathrm{\Omega }}^{\omega }))$
• $f_{n+3}^2(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega +2}\times \psi ({\mathrm{\Omega }}^{\omega +2}\times \psi ({\mathrm{\Omega }}^{\omega })))$
• $f_{n+3}^n(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega +3})$
• $f_{n\times 2}(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega \times 2})$
• $f_{n\times 2+1}(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega \times 2}\times \psi ({\mathrm{\Omega }}^{\omega }))$
• $f_{n\times 2+1}^n(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega \times 2+1})$
• $f_{n\times 3}(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{\omega \times 3})$
• $f_{n^2}(f_{\omega }(n))\sim \psi ({\mathrm{\Omega }}^{{\omega }^2})$
• $f_{f_3(n)}(f_{\omega }(n))\approx n{\uparrow }^{n\uparrow \uparrow n}n\sim \psi ({\mathrm{\Omega }}^{\psi (0)})$
• $f_{f_4(n)}(f_{\omega }(n))\approx n{\uparrow }^{n\uparrow \uparrow \uparrow n}n\sim \psi ({\mathrm{\Omega }}^{\psi (\mathrm{\Omega })})$
• $f_{\omega }^2(n)=f_{f_{\omega }(n)}(f_{\omega }(n))\approx n{\uparrow }^{n{\uparrow }^{n}n}n\sim \psi ({\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\omega })})$
• $f_n(f_{\omega }^2(n))\sim \psi ({\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\omega })}+{\mathrm{\Omega }}^{\omega })$
• $f_{n+1}^n(f_{\omega }^2(n))\sim \psi ({\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\omega })}+{\mathrm{\Omega }}^{\omega +1})$
• $f_{f_{\omega }(n)}(f_{\omega }^2(n))\sim \psi ({\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\omega })}\times 2)$
• $f_{f_{\omega }(n)}^n(f_{\omega }^2(n))\sim \psi ({\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\omega })}\times \omega )$
• $f_{f_{\omega }(n)+1}(f_{\omega }^2(n))\sim \psi ({\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\omega })}\times \psi ({\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\omega })}))$
• $f_{f_{\omega }(n)+1}^n(f_{\omega }^2(n))\sim \psi ({\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\omega })+1})$
• $f_{f_{\omega }(n)+n}(f_{\omega }^2(n))\sim \psi ({\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\omega })+\omega })$
• $f_{f_{\omega }(n)\times 2}(f_{\omega }^2(n))\sim \psi ({\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\omega })\times 2})$
• $f_{f_3(f_{\omega }(n))}(f_{\omega }^2(n))\sim \psi ({\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\omega }+1)})$
• $f_{f_{n+1}^n(f_{\omega }(n))}(f_{\omega }^2(n))\sim \psi ({\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\omega +1})})$
• $f_{\omega }^3(n)=f_{f_{f_{\omega }(n)}(f_{\omega }(n))}(f_{\omega }^2(n))\sim \psi ({\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\omega })})})$
• $f_{\omega +1}(n)=f_{\omega }^n(n)\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})=\phi (1,0,0)$

Part 3

• $f_3(f_{\omega +1}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+1)$
• $f_4(f_{\omega +1}(n))=f_3^{f_{\omega +1}(n)}(f_{\omega +1}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}))$
• $f_4^n(f_{\omega +1}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+\mathrm{\Omega })$
• $f_5^n(f_{\omega +1}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^2)$
• $f_n(f_{\omega +1}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\omega })$
• $f_{n+1}^n(f_{\omega +1}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\omega +1})$
• $f_{f_3(n)}(f_{\omega +1}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi (0)})$
• $f_{f_{\omega }(n)}(f_{\omega +1}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\omega })})$
• $f_{\omega }(f_{\omega +1}(n))=f_{f_{\omega +1}(n)}(f_{\omega +1}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})})=\phi (\phi (1,0,0),1)$
• $f_{f_{\omega +1}(n)}(f_{\omega }(f_{\omega +1}(n)))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})}\times 2)$
• $f_{f_{\omega +1}(n)+1}^n(f_{\omega }(f_{\omega +1}(n)))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})+1})$
• $f_{f_3(f_{\omega +1}(n))}(f_{\omega }(f_{\omega +1}(n)))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+1)})$
• $f_{f_n(f_{\omega +1}(n))}(f_{\omega }(f_{\omega +1}(n)))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\omega })})$
• $f_{\omega }^2(f_{\omega +1}(n))=f_{f_{\omega }(f_{\omega +1}(n))}(f_{\omega }(f_{\omega +1}(n)))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})})})$
• $f_{\omega }^n(f_{\omega +1}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}\times 2)=\phi (1,0,1)$
• $f_{\omega }^{n\times 2}(f_{\omega +1}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}\times 3)$
• $f_{\omega +1}^2(n)=f_{\omega }^{f_{\omega +1}(n)}(f_{\omega +1}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}))$
• $f_{\omega }^n(f_{\omega +1}^2(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})+{\mathrm{\Omega }}^{\mathrm{\Omega }})$
• $f_{\omega }^{f_{\omega +1}(n)}(f_{\omega +1}^2(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})\times 2)$
• $f_{\omega }^{f_n(f_{\omega +1}(n))}(f_{\omega +1}^2(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\omega }))$
• $f_{\omega }^{f_{\omega }(f_{\omega +1}(n))}(f_{\omega +1}^2(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})}))$
• $f_{\omega }^{f_{\omega }^n(f_{\omega +1}(n))}(f_{\omega +1}^2(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}\times 2))$
• $f_{\omega +1}^3(n)\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})))$
• $f_{\omega +2}(n)=f_{\omega +1}^n(n)\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1})=\phi (1,1,0)$
• $f_{\omega }(f_{\omega +2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1})})$
• $f_{\omega }^n(f_{\omega +2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1}+{\mathrm{\Omega }}^{\mathrm{\Omega }})$
• $f_{\omega }^{n\times 2}(f_{\omega +2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1}+{\mathrm{\Omega }}^{\mathrm{\Omega }}\times 2)$
• $f_{\omega +1}(f_{\omega +2}(n))=f_{\omega }^{f_{\omega +2}(n)}(f_{\omega +2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1}+{\mathrm{\Omega }}^{\mathrm{\Omega }}\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1}))$
• $f_{\omega }^{f_{\omega +2}(n)}(f_{\omega +1}(f_{\omega +2}(n)))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1}+{\mathrm{\Omega }}^{\mathrm{\Omega }}\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1})\times 2)$
• $f_{\omega }^{f_{\omega }^n(f_{\omega +2}(n))}(f_{\omega +1}(f_{\omega +2}(n)))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1}+{\mathrm{\Omega }}^{\mathrm{\Omega }}\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1}+{\mathrm{\Omega }}^{\mathrm{\Omega }}))$
• $f_{\omega +1}^2(f_{\omega +2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1}+{\mathrm{\Omega }}^{\mathrm{\Omega }}\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1}+{\mathrm{\Omega }}^{\mathrm{\Omega }}\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1})))$
• $f_{\omega +1}^n(f_{\omega +2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1}\times 2)$
• $f_{\omega +2}^2(n)\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1}\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1}))$
• $f_{\omega +3}(n)=f_{\omega +2}^n(n)\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+2})$
• $f_{\omega +4}(n)\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+3})$
• $f_{\omega \times 2}(n)=f_{\omega +n}(n)\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\omega })$
• $f_{\omega }(f_{\omega \times 2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\omega }+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\omega })})$
• $f_{\omega }^n(f_{\omega \times 2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\omega }+{\mathrm{\Omega }}^{\mathrm{\Omega }})$
• $f_{\omega +1}(f_{\omega \times 2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\omega }+{\mathrm{\Omega }}^{\mathrm{\Omega }}\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\omega }))$
• $f_{\omega +1}^n(f_{\omega \times 2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\omega }+{\mathrm{\Omega }}^{\mathrm{\Omega }+1})$
• $f_{\omega +2}^n(f_{\omega \times 2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\omega }+{\mathrm{\Omega }}^{\mathrm{\Omega }+2})$
• $f_{\omega +n}(f_{\omega \times 2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\omega }\times 2)$
• $f_{\omega +n+1}(f_{\omega \times 2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\omega }\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\omega }))$
• $f_{\omega +n+1}^n(f_{\omega \times 2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\omega +1})$
• $f_{\omega +n\times 2}(f_{\omega \times 2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\omega \times 2})$
• $f_{\omega +f_{\omega +1}(n)}(f_{\omega \times 2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})})$
• $f_{\omega +f_{\omega +2}(n)}(f_{\omega \times 2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1})})$
• $f_{\omega \times 2}^2(n)=f_{\omega +f_{\omega +n}(n)}(f_{\omega \times 2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\omega })})$
• $f_{\omega \times 2}^3(n)\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\omega })})})$
• $f_{\omega \times 2+1}(n)\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times 2})$
• $f_{\omega }^n(f_{\omega \times 2+1}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times 2}+{\mathrm{\Omega }}^{\mathrm{\Omega }})$
• $f_{\omega \times 2}^n(f_{\omega \times 2+1}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times 2}\times 2)$
• $f_{\omega \times 2+1}^2(n)\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times 2}\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times 2}))$
• $f_{\omega \times 2+2}(n)\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times 2+1})$
• $f_{\omega \times 3}(n)\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times 2+\omega })$
• $f_{\omega \times 3}^2(n)\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times 2+\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times 2+\omega })})$
• $f_{\omega \times 3+1}(n)\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times 3})$
• $f_{\omega \times 4}(n)\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times 3+\omega })$
• $f_{\omega \times 4+1}(n)\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times 4})$
• $f_{{\omega }^2}(n)\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega })=\phi (\omega ,0,0)$
• $f_{\omega }^n(f_{{\omega }^2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega }+{\mathrm{\Omega }}^{\mathrm{\Omega }})$
• $f_{\omega \times 2}^n(f_{{\omega }^2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega }+{\mathrm{\Omega }}^{\mathrm{\Omega }\times 2})$
• $f_{\omega \times n}(f_{{\omega }^2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega }\times 2)$
• $f_{\omega \times n+1}(f_{{\omega }^2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega }\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega }))$
• $f_{\omega \times n+1}^n(f_{{\omega }^2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega +1})$
• $f_{\omega \times n+2}^n(f_{{\omega }^2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega +2})$
• $f_{\omega \times n+n}(f_{{\omega }^2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega +\omega })$
• $f_{\omega \times n+f_{{\omega }^2}(n)}(f_{{\omega }^2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega +\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega })})$
• $f_{\omega \times n+\omega }(f_{{\omega }^2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega +\mathrm{\Omega }})$
• $f_{\omega \times n+\omega \times 2}(f_{{\omega }^2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega +\mathrm{\Omega }\times 2})$
• $f_{\omega \times n\times 2}(f_{{\omega }^2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega \times 2})$
• $f_{{\omega }^2}^2(n)=f_{\omega \times f_{{\omega }^2}(n)}(f_{{\omega }^2}(n))\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega })})$
• $f_{{\omega }^2}^3(n)\sim \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega })})})$
• $f_{{\omega }^2+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2})=\phi (1,0,0,0)$
• $f_{\omega }^n(f_{{\omega }^2+1}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2}+{\mathrm{\Omega }}^{\mathrm{\Omega }})$
• $f_{\omega \times 2}^n(f_{{\omega }^2+1}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2}+{\mathrm{\Omega }}^{\mathrm{\Omega }\times 2})$
• $f_{\omega \times n}(f_{{\omega }^2+1}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2}+{\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega })$
• $f_{{\omega }^2}(f_{{\omega }^2+1}(n))=f_{\omega \times f_{{\omega }^2+1}(n)}(f_{{\omega }^2+1}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2}+{\mathrm{\Omega }}^{\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2})})=\phi (\phi (1,0,0,0),0,1)$
• $f_{{\omega }^2}^2(f_{{\omega }^2+1}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2}+{\mathrm{\Omega }}^{\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2}+{\mathrm{\Omega }}^{\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2})})})$
• $f_{{\omega }^2}^n(f_{{\omega }^2+1}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2}\times 2)$
• $f_{{\omega }^2+1}^2(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2}\times \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2}))$
• $f_{{\omega }^2+2}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2+1})$
• $f_{{\omega }^2+\omega }(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2+\omega })$
• $f_{{\omega }^2+\omega +1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2+\mathrm{\Omega }})$
• $f_{{\omega }^2+\omega +2}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2+\mathrm{\Omega }+1})$
• $f_{{\omega }^2+\omega \times 2+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2+\mathrm{\Omega }\times 2})$
• $f_{{\omega }^2\times 2}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2+\mathrm{\Omega }\times \omega })$
• $f_{{\omega }^2\times 2+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2\times 2})$
• $f_{{\omega }^2\times 3+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2\times 3})$
• $f_{{\omega }^3}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2\times \omega })$
• $f_{{\omega }^2\times n}(f_{{\omega }^3}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2\times \omega }\times 2)$
• $f_{{\omega }^2\times n+1}^n(f_{{\omega }^3}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2\times \omega +1})$
• $f_{{\omega }^2\times n+n}(f_{{\omega }^3}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2\times \omega +\omega })$
• $f_{{\omega }^2\times n+\omega }(f_{{\omega }^3}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2\times \omega +\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2\times \omega })})$
• $f_{{\omega }^2\times n+\omega }^n(f_{{\omega }^3}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2\times \omega +\mathrm{\Omega }})$
• $f_{{\omega }^2\times n+\omega \times n}(f_{{\omega }^3}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2\times \omega +\mathrm{\Omega }\times \omega })$
• $f_{{\omega }^2\times n+{\omega }^2}^n(f_{{\omega }^3}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2\times \omega +{\mathrm{\Omega }}^2})$
• $f_{{\omega }^2\times n\times 2}(f_{{\omega }^3}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2\times \omega \times 2})$
• $f_{{\omega }^3}^2(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2\times \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2})})$
• $f_{{\omega }^3+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^3})$
• $f_{{\omega }^3+2}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^3+1})$
• $f_{{\omega }^3+\omega +1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^3+\mathrm{\Omega }})$
• $f_{{\omega }^3+{\omega }^2+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^3+{\mathrm{\Omega }}^2})$
• $f_{{\omega }^3\times 2+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^3\times 2})$
• $f_{{\omega }^4}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^3\times \omega })$
• $f_{{\omega }^4+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^4})$
• \(f_{\omega^\omega}(n)\sim\psi(\Omega^{\Omega^\omega})=\varphi(1@\omega)\)

Part 4

• $f_{{\omega }^{\omega }}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }})$
• $f_3(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}+1)$
• $f_4^n(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}+\mathrm{\Omega })$
• $f_5^n(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}+{\mathrm{\Omega }}^2)$
• $f_n(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}+{\mathrm{\Omega }}^{\omega })$
• $f_{\omega }(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }})})$
• $f_{\omega }^n(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}+{\mathrm{\Omega }}^{\mathrm{\Omega }})$
• $f_{\omega +1}^n(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}+{\mathrm{\Omega }}^{\mathrm{\Omega }+1})$
• $f_{\omega +n}(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}+{\mathrm{\Omega }}^{\mathrm{\Omega }+\omega })$
• $f_{\omega \times 2}^n(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}+{\mathrm{\Omega }}^{\mathrm{\Omega }\times 2})$
• $f_{\omega \times n}(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}+{\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega })$
• $f_{{\omega }^2}^n(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}+{\mathrm{\Omega }}^{{\mathrm{\Omega }}^2})$
• $f_{{\omega }^3}^n(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}+{\mathrm{\Omega }}^{{\mathrm{\Omega }}^3})$
• $f_{{\omega }^n}(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}\times 2)$
• $f_{{\omega }^n}^2(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}\times 3)$
• $f_{{\omega }^n}^n(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}\times \omega )$
• $f_{{\omega }^n+1}(f_{{\omega }^{\omega }}(n))=f_{{\omega }^n}^{f_{{\omega }^{\omega }}(n)}(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}\times \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}))$
• $f_{{\omega }^n+1}^n(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }+1})$
• $f_{{\omega }^n+2}^n(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }+2})$
• $f_{{\omega }^n+n}(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }+\omega })$
• $f_{{\omega }^n+\omega }^n(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }})$
• $f_{{\omega }^n+\omega \times 2}^n(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }+\mathrm{\Omega }\times 2})$
• $f_{{\omega }^n+{\omega }^2}^n(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }+{\mathrm{\Omega }}^2})$
• $f_{{\omega }^n\times 2}(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }\times 2})$
• $f_{{\omega }^n\times 3}(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }\times 3})$
• $f_{{\omega }^n\times n}(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }\times \omega })$
• $f_{{\omega }^{n+1}}(f_{{\omega }^{\omega }}(n))=f_{{\omega }^n\times f_{{\omega }^{\omega }}(n)}(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega +1}})$
• $f_{{\omega }^n}(f_{{\omega }^{n+1}}(f_{{\omega }^{\omega }}(n)))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega +1}}+{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }})$
• $f_{{\omega }^n+1}^n(f_{{\omega }^{n+1}}(f_{{\omega }^{\omega }}(n)))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega +1}}+{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }+1})$
• $f_{{\omega }^n\times n}(f_{{\omega }^{n+1}}(f_{{\omega }^{\omega }}(n)))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega +1}}+{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }\times \omega })$
• $f_{{\omega }^{n+1}}^2(f_{{\omega }^{\omega }}(n))=f_{{\omega }^n\times f_{{\omega }^{n+1}}(f_{{\omega }^{\omega }}(n))}(f_{{\omega }^{n+1}}(f_{{\omega }^{\omega }}(n)))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega +1}}\times 2)$
• $f_{{\omega }^{n+1}}^n(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega +1}}\times \omega )$
• $f_{{\omega }^{n+1}+1}^n(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega +1}+1})$
• $f_{{\omega }^{n+1}+{\omega }^n}(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega +1}+{\mathrm{\Omega }}^{\omega }})$
• $f_{{\omega }^{n+1}\times 2}^n(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega +1}\times 2})$
• $f_{{\omega }^{n+1}\times n}(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega +1}\times \omega })$
• $f_{{\omega }^{n+2}}^n(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega +2}})$
• $f_{{\omega }^{n\times 2}}(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega \times 2}})$
• $f_{{\omega }^{n^2}}(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{{\omega }^2}})$
• $f_{{\omega }^{f_3(n)}}(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\psi (0)}})$
• $f_{{\omega }^{\omega }}^2(n)=f_{{\omega }^{f_{{\omega }^{\omega }}(n)}}(f_{{\omega }^{\omega }}(n))\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }})}})$
• $f_{{\omega }^{\omega }}^3(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }})}})}})$
• $f_{{\omega }^{\omega }+1}(n)=f_{{\omega }^{\omega }}^n(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}})$
• $f_{{\omega }^{\omega }+2}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}+1})$
• $f_{{\omega }^{\omega }+3}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}+2})$
• $f_{{\omega }^{\omega }+\omega }(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}+\omega })$
• $f_{{\omega }^{\omega }+\omega +1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}+\mathrm{\Omega }})$
• $f_{{\omega }^{\omega }+\omega \times 2}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}+\mathrm{\Omega }+\omega })$
• $f_{{\omega }^{\omega }+\omega \times 2+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}+\mathrm{\Omega }\times 2})$
• $f_{{\omega }^{\omega }+{\omega }^2}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}+\mathrm{\Omega }\times \omega })$
• $f_{{\omega }^{\omega }+{\omega }^2+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^2})$
• $f_{{\omega }^{\omega }+{\omega }^3+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^3})$
• $f_{{\omega }^{\omega }\times 2}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\omega }})$
• $f_{{\omega }^{\omega }\times 2+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}\times 2})$
• $f_{{\omega }^{\omega }\times 3+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}\times 3})$
• $f_{{\omega }^{\omega +1}}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}\times \omega })$
• $f_{{\omega }^{\omega +1}+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }+1}})$
• $f_{{\omega }^{\omega +1}+\omega +1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }+1}+\mathrm{\Omega }})$
• $f_{{\omega }^{\omega +1}+{\omega }^{\omega }+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }+1}+{\mathrm{\Omega }}^{\mathrm{\Omega }}})$
• $f_{{\omega }^{\omega +1}\times 2+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }+1}\times 2})$
• $f_{{\omega }^{\omega +2}}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }+1}\times \omega })$
• $f_{{\omega }^{\omega +2}+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }+2}})$
• $f_{{\omega }^{\omega \times 2}}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }+\omega }})$
• $f_{{\omega }^{\omega \times 2}+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }\times 2}})$
• $f_{{\omega }^{\omega \times 3}+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }\times 3}})$
• $f_{{\omega }^{{\omega }^2}}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }\times \omega }})$
• $f_{{\omega }^{{\omega }^2}+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^2}})$
• $f_{{\omega }^{{\omega }^2}\times 2+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^2}\times 2})$
• $f_{{\omega }^{{\omega }^2+1}+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^2+1}})$
• $f_{{\omega }^{{\omega }^2+\omega }+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^2+\mathrm{\Omega }}})$
• $f_{{\omega }^{{\omega }^2\times 2}+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^2\times 2}})$
• $f_{{\omega }^{{\omega }^3}+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^3}})$
• $f_{{\omega }^{{\omega }^{\omega }}}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}})$
• $f_{{\omega }^{{\omega }^{\omega }}+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}}})$
• $f_{{\omega }^{{\omega }^{{\omega }^{\omega }}}+1}(n)\sim \psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}}}})$
• $f_{\psi (0)}(n)\sim \psi ({\psi }_1(0))$
• $f_3(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)+1)$
• $f_4^n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)+\mathrm{\Omega })$
• $f_n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)+{\mathrm{\Omega }}^{\omega })$
• $f_{\omega }(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)+{\mathrm{\Omega }}^{\psi ({\psi }_1(0))})$
• $f_{\omega }^n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)+{\mathrm{\Omega }}^{\mathrm{\Omega }})$
• $f_{\omega +1}^n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)+{\mathrm{\Omega }}^{\mathrm{\Omega }+1})$
• $f_{\omega \times 2}^n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)+{\mathrm{\Omega }}^{\mathrm{\Omega }\times 2})$
• $f_{{\omega }^{\omega }}^n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)+{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}})$
• $f_{{\omega }^{\omega }+1}^n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)+{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}+1})$
• $f_{{\omega }^{{\omega }^{\omega }}}^n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)+{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}}})$
• $f_{{\omega }^{{\omega }^{{\omega }^{\omega }}}}^n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)+{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}}}})$
• $f_{\psi (0)[n]}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)\times 2)$
• $f_{\psi (0)[n]}^2(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)\times 3)$
• $f_{\psi (0)[n]}^n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)\times \omega )$
• $f_{\psi (0)[n]+1}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)\times \psi ({\psi }_1(0)))$
• $f_{\psi (0)[n]+1}^n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)\times \mathrm{\Omega })$
• $f_{\psi (0)[n]}(f_{\psi (0)[n]+1}^n(f_{\psi (0)}(n)))\sim \psi ({\psi }_1(0)\times \mathrm{\Omega }+{\psi }_1(0))$
• $f_{\psi (0)[n]+1}^{n+1}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)\times \mathrm{\Omega }+{\psi }_1(0)\times \psi ({\psi }_1(0)\times \mathrm{\Omega }))$
• $f_{\psi (0)[n]+1}^{n\times 2}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)\times \mathrm{\Omega }\times 2)$
• $f_{\psi (0)[n]+2}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)\times \mathrm{\Omega }\times \psi ({\psi }_1(0)))$
• $f_{\psi (0)[n]+2}^n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)\times {\mathrm{\Omega }}^2)$
• $f_{\psi (0)[n]+n}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)\times {\mathrm{\Omega }}^{\omega })$
• $f_{\psi (0)[n]+\omega }^n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0)\times {\mathrm{\Omega }}^{\mathrm{\Omega }})$
• $f_{\psi (0)[n]\times 2}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^2)$
• $f_{\psi (0)[n]\times n}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{\omega })$
• $f_{\psi (0)[n]\times \omega }^n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{\mathrm{\Omega }})$
• $f_{\psi (0)[n]\times {\omega }^2}^n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\mathrm{\Omega }}^2})$
• $f_{\psi (0)[n]\times {\omega }^{\omega }}^n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}})$
• $f_{\psi (0)[n{]}^2}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0)})$
• $f_{\psi (0)[n{]}^2+\psi (0)[n]}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0)+1})$
• $f_{\psi (0)[n{]}^2\times 2}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0)\times 2})$
• $f_{\psi (0)[n{]}^2\times n}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0)\times \omega })$
• $f_{\psi (0)[n{]}^2\times \omega }^n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0)\times \mathrm{\Omega }})$
• $f_{\psi (0)[n{]}^3}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0{)}^2})$
• $f_{\psi (0)[n{]}^n}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0{)}^{\omega }})$
• $f_{\psi (0)[n{]}^{\omega }}^n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0{)}^{\mathrm{\Omega }}})$
• $f_{\psi (0)[n{]}^{\psi (0)[n]}}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0)}})$
• $f_{\psi (0)[n+1]}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0)}})$
• $f_{\psi (0)[n+1]+\psi (0)[n]}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0)}+1})$
• $f_{\psi (0)[n+1]+\psi (0)[n{]}^2}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0)}+{\psi }_1(0)})$
• $f_{\psi (0)[n+1]+\psi (0)[n{]}^{\omega }}^n(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0)}+{\psi }_1(0{)}^{\mathrm{\Omega }}})$
• $f_{\psi (0)[n+1]\times 2}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0)}\times 2})$
• $f_{\psi (0)[n+1]\times \psi (0)[n]}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0)+1}})$
• $f_{\psi (0)[n+1]\times \psi (0)[n{]}^2}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0)+2}})$
• $f_{\psi (0)[n+1{]}^2}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0)\times 2}})$
• $f_{\psi (0)[n+1{]}^{\psi (0)[n]}}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0{)}^2}})$
• $f_{\psi (0)[n+1{]}^{\psi (0)[n]+1}}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0{)}^2+{\psi }_1(0)}})$
• $f_{\psi (0)[n+1{]}^{\psi (0)[n]\times 2}}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0{)}^2\times 2}})$
• $f_{\psi (0)[n+1{]}^{\psi (0)[n{]}^2}}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0{)}^3}})$
• $f_{\psi (0)[n+2]}(f_{\psi (0)}(n))=f_{\psi (0)[n+1{]}^{\psi (0)[n+1]}}(f_{\psi (0)}(n))=f_{\psi (0)[n+1{]}^{\psi (0)[n{]}^{\psi (0)[n]}}}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0)}}})$
• $f_{\psi (0)[n+3]}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0)}}}})$
• $f_{\psi (0)[n\times 2]}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(1))$
• $f_{\psi (0)[n]}(f_{\psi (0)[n\times 2]}(f_{\psi (0)}(n)))\sim \psi ({\psi }_1(1)+{\psi }_1(0))$
• $f_{\psi (0)[n\times 2]}^2(f_{\psi (0)}(n))\sim \psi ({\psi }_1(1)\times 2)$
• $f_{\psi (0)[n\times 2]+\psi (0)[n]}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(1)\times {\psi }_1(0))$
• $f_{\psi (0)[n\times 2]+\psi (0)[n+1]}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(1)\times {\psi }_1(0{)}^{{\psi }_1(0{)}^{{\psi }_1(0)}})$
• $f_{\psi (0)[n\times 2]\times 2}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(1{)}^2)$
• $f_{\psi (0)[n\times 2]\times \psi (0)[n]}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(1{)}^{{\psi }_1(0)})$
• $f_{\psi (0)[n\times 2{]}^2}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(1{)}^{{\psi }_1(1)})$
• $f_{\psi (0)[n\times 2{]}^3}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(1{)}^{{\psi }_1(1{)}^2})$
• $f_{\psi (0)[n\times 2+1]}(f_{\psi (0)}(n))=f_{\psi (0)[n\times 2{]}^{\psi (0)[n\times 2]}}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(1{)}^{{\psi }_1(1{)}^{{\psi }_1(1)}})$
• $f_{\psi (0)[n\times 3]}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(2))$
• $f_{\psi (0)[n^2]}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(\omega ))$
• $f_{\psi (0)[n^2+n]}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(\omega +1))$
• $f_{\psi (0)[n^3]}(f_{\psi (0)}(n))\sim \psi ({\psi }_1({\omega }^2))$
• $f_{\psi (0)[n^n]}(f_{\psi (0)}(n))\sim \psi ({\psi }_1({\omega }^{\omega }))$
• $f_{\psi (0)[f_3(n)]}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(\psi (0)))$
• $f_{\psi (0)}^2(n)=f_{\psi (0)[f_{\psi (0)}(n)]}(f_{\psi (0)}(n))\sim \psi ({\psi }_1(\psi ({\psi }_1(0))))$
• $f_{\psi (0)[n]}(f_{\psi (0)}^2(n))\sim \psi ({\psi }_1(\psi ({\psi }_1(0)))\times {\psi }_1(0))$
• $f_{\psi (0)[n\times 2]}(f_{\psi (0)}^2(n))\sim \psi ({\psi }_1(\psi ({\psi }_1(0)))\times {\psi }_1(1))$
• $f_{\psi (0)[f_{\psi (0)}(n)]}(f_{\psi (0)}^2(n))\sim \psi ({\psi }_1(\psi ({\psi }_1(0)){)}^2)$
• $f_{\psi (0)[f_{\psi (0)}(n)+1]}(f_{\psi (0)}^2(n))\sim \psi ({\psi }_1(\psi ({\psi }_1(0)){)}^{{\psi }_1(\psi ({\psi }_1(0)){)}^{{\psi }_1(\psi ({\psi }_1(0)))}})$
• $f_{\psi (0)[f_{\psi (0)}(n)+n]}(f_{\psi (0)}^2(n))\sim \psi ({\psi }_1(\psi ({\psi }_1(0))+1))$
• $f_{\psi (0)[f_{\psi (0)}(n)\times 2]}(f_{\psi (0)}^2(n))\sim \psi ({\psi }_1(\psi ({\psi }_1(0))\times 2))$
• $f_{\psi (0)[f_3(f_{\psi (0)}(n))]}(f_{\psi (0)}^2(n))\sim \psi ({\psi }_1(\psi ({\psi }_1(0)+1)))$
• $f_{\psi (0)[f_{\psi (0)[n]}(f_{\psi (0)}(n))]}(f_{\psi (0)}^2(n))\sim \psi ({\psi }_1(\psi ({\psi }_1(0)\times 2)))$
• $f_{\psi (0)[f_{\psi (0)[n\times 2]}(f_{\psi (0)}(n))]}(f_{\psi (0)}^2(n))\sim \psi ({\psi }_1(\psi ({\psi }_1(1))))$
• $f_{\psi (0)[f_{\psi (0)[n^2]}(f_{\psi (0)}(n))]}(f_{\psi (0)}^2(n))\sim \psi ({\psi }_1(\psi ({\psi }_1(\omega ))))$
• $f_{\psi (0)}^3(n)\sim \psi ({\psi }_1(\psi ({\psi }_1(\psi ({\psi }_1(0))))))$
• $f_{\psi (0)+1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }))$
• $f_{\psi (0)[n]}(f_{\psi (0)+1}(n))\sim \psi ({\psi }_1(\mathrm{\Omega })+{\psi }_1(0))$
• $f_{\psi (0)[n\times 2]}(f_{\psi (0)+1}(n))\sim \psi ({\psi }_1(\mathrm{\Omega })+{\psi }_1(1))$
• $f_{\psi (0)[f_{\psi (0)}(n)]}(f_{\psi (0)+1}(n))\sim \psi ({\psi }_1(\mathrm{\Omega })+{\psi }_1(\psi ({\psi }_1(0))))$
• $f_{\psi (0)}(f_{\psi (0)+1}(n))=f_{\psi (0)[f_{\psi (0)+1}(n)]}(f_{\psi (0)+1}(n))\sim \psi ({\psi }_1(\mathrm{\Omega })+{\psi }_1(\psi ({\psi }_1(\mathrm{\Omega }))))$
• $f_{\psi (0)}^2(f_{\psi (0)+1}(n))\sim \psi ({\psi }_1(\mathrm{\Omega })+{\psi }_1(\psi ({\psi }_1(\mathrm{\Omega })+{\psi }_1(\psi ({\psi }_1(\mathrm{\Omega }))))))$
• $f_{\psi (0)}^n(f_{\psi (0)+1}(n))\sim \psi ({\psi }_1(\mathrm{\Omega })\times 2)$
• $f_{\psi (0)}^{n^2}(f_{\psi (0)+1}(n))\sim \psi ({\psi }_1(\mathrm{\Omega })\times \omega )$
• $f_{\psi (0)+1}^2(n)\sim \psi ({\psi }_1(\mathrm{\Omega })\times \psi ({\psi }_1(\mathrm{\Omega })))$
• $f_{\psi (0)+2}(n)\sim \psi ({\psi }_1(\mathrm{\Omega })\times \mathrm{\Omega })$
• $f_{\psi (0)+3}(n)\sim \psi ({\psi }_1(\mathrm{\Omega })\times {\mathrm{\Omega }}^2)$
• $f_{\psi (0)+\omega }(n)\sim \psi ({\psi }_1(\mathrm{\Omega })\times {\mathrm{\Omega }}^{\omega })$
• $f_{\psi (0)+\omega +1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega })\times {\mathrm{\Omega }}^{\mathrm{\Omega }})$
• $f_{\psi (0)+{\omega }^{\omega }+1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega })\times {\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}})$
• $f_{\psi (0)\times 2}(n)\sim \psi ({\psi }_1(\mathrm{\Omega })\times {\psi }_1(0))$
• $f_{\psi (0)\times 2+1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }{)}^2)$
• $f_{\psi (0)\times 3+1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }{)}^3)$
• $f_{\psi (0)\times \omega }(n)\sim \psi ({\psi }_1(\mathrm{\Omega }{)}^{\omega })$
• $f_{\psi (0)\times \omega +1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }{)}^{\mathrm{\Omega }})$
• $f_{\psi (0{)}^2}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }{)}^{{\psi }_1(0)})$
• $f_{\psi (0{)}^2+1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }{)}^{{\psi }_1(\mathrm{\Omega })})$
• $f_{\psi (0{)}^2+\psi (0)+1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }{)}^{{\psi }_1(\mathrm{\Omega })+1})$
• $f_{\psi (0{)}^2\times 2+1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }{)}^{{\psi }_1(\mathrm{\Omega })\times 2})$
• $f_{\psi (0{)}^3+1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }{)}^{{\psi }_1(\mathrm{\Omega }{)}^2})$
• $f_{\psi (0{)}^{\omega }+1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }{)}^{{\psi }_1(\mathrm{\Omega }{)}^{\mathrm{\Omega }}})$
• $f_{\psi (0{)}^{\psi (0)}}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }{)}^{{\psi }_1(\mathrm{\Omega }{)}^{{\psi }_1(0)}})$
• $f_{\psi (0{)}^{\psi (0)}+1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }{)}^{{\psi }_1(\mathrm{\Omega }{)}^{{\psi }_1(\mathrm{\Omega })}})$
• $f_{\psi (1)}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }+1))$
• $f_{\psi (0)[n]}(f_{\psi (1)}(n))\sim \psi ({\psi }_1(\mathrm{\Omega }+1)+{\psi }_1(0))$
• $f_{\psi (0)}^n(f_{\psi (1)}(n))\sim \psi ({\psi }_1(\mathrm{\Omega }+1)+{\psi }_1(\mathrm{\Omega }))$
• $f_{\psi (0{)}^{\psi (0)}}^n(f_{\psi (1)}(n))\sim \psi ({\psi }_1(\mathrm{\Omega }+1)+{\psi }_1(\mathrm{\Omega }{)}^{{\psi }_1(\mathrm{\Omega }{)}^{{\psi }_1(\mathrm{\Omega })}})$
• $f_{\psi (1)[n]}(f_{\psi (1)}(n))\sim \psi ({\psi }_1(\mathrm{\Omega }+1)\times 2)$
• $f_{\psi (1)[n]}^2(f_{\psi (1)}(n))\sim \psi ({\psi }_1(\mathrm{\Omega }+1)\times 3)$
• $f_{\psi (1)[n]+1}(f_{\psi (1)}(n))\sim \psi ({\psi }_1(\mathrm{\Omega }+1)\times \psi ({\psi }_1(\mathrm{\Omega }+1)))$
• $f_{\psi (1)[n]+1}^n(f_{\psi (1)}(n))\sim \psi ({\psi }_1(\mathrm{\Omega }+1)\times \mathrm{\Omega })$
• $f_{\psi (1)[n]+\psi (0)[n]}(f_{\psi (1)}(n))\sim \psi ({\psi }_1(\mathrm{\Omega }+1)\times {\psi }_1(0))$
• $f_{\psi (1)[n]+\psi (0)}^n(f_{\psi (1)}(n))\sim \psi ({\psi }_1(\mathrm{\Omega }+1)\times {\psi }_1(\mathrm{\Omega }))$
• $f_{\psi (1)[n]\times 2}(f_{\psi (1)}(n))\sim \psi ({\psi }_1(\mathrm{\Omega }+1{)}^2)$
• $f_{\psi (1)[n]\times \psi (0)[n]}(f_{\psi (1)}(n))\sim \psi ({\psi }_1(\mathrm{\Omega }+1{)}^{{\psi }_1(0)})$
• $f_{\psi (1)[n{]}^2}(f_{\psi (1)}(n))\sim \psi ({\psi }_1(\mathrm{\Omega }+1{)}^{{\psi }_1(\mathrm{\Omega }+1)})$
• $f_{\psi (1)[n+1]}(f_{\psi (1)}(n))\sim \psi ({\psi }_1(\mathrm{\Omega }+1{)}^{{\psi }_1(\mathrm{\Omega }+1{)}^{{\psi }_1(\mathrm{\Omega }+1)}})$
• $f_{\psi (1)[n\times 2]}(f_{\psi (1)}(n))\sim \psi ({\psi }_1(\mathrm{\Omega }+2))$
• $f_{\psi (1)}^2(n)\sim \psi ({\psi }_1(\mathrm{\Omega }+\psi ({\psi }_1(\mathrm{\Omega }))))$
• $f_{\psi (1)+1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }\times 2))$
• $f_{\psi (1)+2}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }\times 2)\times \mathrm{\Omega })$
• $f_{\psi (1)+\psi (0)+1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }\times 2)\times {\psi }_1(\mathrm{\Omega }))$
• $f_{\psi (1)\times 2+1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }\times 2{)}^2)$
• $f_{\psi (1)\times \psi (0)+1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }\times 2{)}^{{\psi }_1(\mathrm{\Omega })})$
• $f_{\psi (1{)}^2+1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }\times 2{)}^{{\psi }_1(\mathrm{\Omega }\times 2)})$
• $f_{\psi (1{)}^{\psi (1)}+1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }\times 2{)}^{{\psi }_1(\mathrm{\Omega }\times 2{)}^{{\psi }_1(\mathrm{\Omega }\times 2)}})$
• $f_{\psi (2)}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }\times 2+1))$
• $f_{\psi (2)+1}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }\times 3))$
• $f_{\psi (3)}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }\times 3+1))$
• $f_{\psi (4)}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }\times 4+1))$
• $f_{\psi (\omega )}(n)\sim \psi ({\psi }_1(\mathrm{\Omega }\times \omega ))$
• $f_{\psi (\omega )+1}(n)\sim \psi ({\psi }_1({\mathrm{\Omega }}^2))$
• $f_{\psi (\omega )+2}(n)\sim \psi ({\psi }_1({\mathrm{\Omega }}^2)\times \mathrm{\Omega })$
• $f_{\psi (\omega )\times 2+1}(n)\sim \psi ({\psi }_1({\mathrm{\Omega }}^2{)}^2)$
• $f_{\psi (\omega {)}^2+1}(n)\sim \psi ({\psi }_1({\mathrm{\Omega }}^2{)}^{{\psi }_1({\mathrm{\Omega }}^2)})$
• $f_{\psi (\omega +1)}(n)\sim \psi ({\psi }_1({\mathrm{\Omega }}^2+1))$
• $f_{\psi (\omega +1)+1}(n)\sim \psi ({\psi }_1({\mathrm{\Omega }}^2+\mathrm{\Omega }))$
• $f_{\psi (\omega +2)+1}(n)\sim \psi ({\psi }_1({\mathrm{\Omega }}^2+\mathrm{\Omega }\times 2))$
• $f_{\psi (\omega \times 2)+1}(n)\sim \psi ({\psi }_1({\mathrm{\Omega }}^2\times 2))$
• $f_{\psi ({\omega }^2)+1}(n)\sim \psi ({\psi }_1({\mathrm{\Omega }}^3))$
• $f_{\psi ({\omega }^{\omega })}(n)\sim \psi ({\psi }_1({\mathrm{\Omega }}^{\omega }))$
• $f_{\psi ({\omega }^{\omega })+1}(n)\sim \psi ({\psi }_1({\mathrm{\Omega }}^{\mathrm{\Omega }}))$
• $f_{\psi ({\omega }^{{\omega }^{\omega }})+1}(n)\sim \psi ({\psi }_1({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}}))$
• $f_{\psi (\psi (0))}(n)\sim \psi ({\psi }_1({\psi }_1(0)))$
• $f_{\psi (\psi (0))+1}(n)\sim \psi ({\psi }_1({\psi }_1(\mathrm{\Omega })))$
• $f_{\psi (\psi (1))+1}(n)\sim \psi ({\psi }_1({\psi }_1(\mathrm{\Omega }\times 2)))$
• $f_{\psi (\psi (\psi (0)))+1}(n)\sim \psi ({\psi }_1({\psi }_1({\psi }_1(\mathrm{\Omega }))))$
• $f_{\psi (\mathrm{\Omega })}(n)\sim \psi ({\mathrm{\Omega }}_2)$
• $f_3(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+1)$
• $f_{\omega }^n(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\mathrm{\Omega }}^{\mathrm{\Omega }})$
• $f_{\psi (0)[n]}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1(0))$
• $f_{\psi (0)}^n(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1(\mathrm{\Omega }))$
• $f_{\psi (1)[n]}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1(\mathrm{\Omega }+1))$
• $f_{\psi (1)}^n(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1(\mathrm{\Omega }\times 2))$
• $f_{\psi (\omega )}^n(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}^2))$
• $f_{\psi (\psi (0))}^n(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\psi }_1(\mathrm{\Omega })))$
• $f_{\psi (\psi (\psi (0)))}^n(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\psi }_1({\psi }_1(\mathrm{\Omega }))))$
• $f_{\psi (\mathrm{\Omega })[n]}^n(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2)\times \omega )$
• $f_{\psi (\mathrm{\Omega })[n]+1}^n(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2)\times \mathrm{\Omega })$
• $f_{\psi (\mathrm{\Omega })[n]\times 2}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2{)}^2)$
• $f_{\psi (\mathrm{\Omega })[n]\times \omega }^n(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2{)}^{\mathrm{\Omega }})$
• $f_{\psi (\mathrm{\Omega })[n]\times \psi (0)+1}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2{)}^{{\psi }_1(\mathrm{\Omega })})$
• $f_{\psi (\mathrm{\Omega })[n{]}^2}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2{)}^{{\psi }_1({\mathrm{\Omega }}_2)})$
• $f_{\psi (\mathrm{\Omega })[n{]}^{\psi (\mathrm{\Omega })[n]}}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2{)}^{{\psi }_1({\mathrm{\Omega }}_2{)}^{{\psi }_1({\mathrm{\Omega }}_2)}})$
• $f_{\psi (\psi (\mathrm{\Omega })[n-1]+1)[n]}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+1))$
• $f_{\psi (\psi (\mathrm{\Omega })[n-1]+1)}^n(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+\mathrm{\Omega }))$
• $f_{\psi (\psi (\mathrm{\Omega })[n-1]+n)}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+\mathrm{\Omega }\times \omega ))$
• $f_{\psi (\psi (\mathrm{\Omega })[n-1]+\omega )}^n(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+{\mathrm{\Omega }}^2))$
• $f_{\psi (\psi (\mathrm{\Omega })[n-1]+\psi (0))}^n(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+{\psi }_1(\mathrm{\Omega })))$
• $f_{\psi (\psi (\mathrm{\Omega })[n-1]+\psi (\psi (0)))}^n(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+{\psi }_1({\psi }_1(\mathrm{\Omega }))))$
• $f_{\psi (\mathrm{\Omega })[n+1]}(f_{\psi (\mathrm{\Omega })}(n))=f_{\psi (\psi (\mathrm{\Omega })[n])}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2)))$
• $f_{\psi (\psi (\mathrm{\Omega })[n]+1)}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2)+1))$
• $f_{\psi (\psi (\mathrm{\Omega })[n]\times 2)}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2)\times 2))$
• $f_{\psi (\psi (\mathrm{\Omega })[n{]}^2)}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2{)}^2))$
• $f_{\psi (\psi (\psi (\mathrm{\Omega })[n-1]+1))}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+1)))$
• $f_{\psi (\mathrm{\Omega })[n+2]}(f_{\psi (\mathrm{\Omega })}(n))=f_{\psi (\psi (\psi (\mathrm{\Omega })[n]))}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2))))$
• $f_{\psi (\mathrm{\Omega })[n\times 2]}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2\times 2)$
• $f_{\psi (\mathrm{\Omega })[n]}(f_{\psi (\mathrm{\Omega })[n\times 2]}(f_{\psi (\mathrm{\Omega })}(n)))\sim \psi ({\mathrm{\Omega }}_2\times 2+{\psi }_1({\mathrm{\Omega }}_2))$
• $f_{\psi (\mathrm{\Omega })[n\times 2]}^2(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2\times 2+{\psi }_1({\mathrm{\Omega }}_2\times 2))$
• $f_{\psi (\mathrm{\Omega })[n\times 2]+1}^n(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2\times 2+{\psi }_1({\mathrm{\Omega }}_2\times 2)\times \mathrm{\Omega })$
• $f_{\psi (\mathrm{\Omega })[n\times 2]\times 2}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2\times 2+{\psi }_1({\mathrm{\Omega }}_2\times 2{)}^2)$
• $f_{\psi (\mathrm{\Omega })[n\times 2{]}^2}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2\times 2+{\psi }_1({\mathrm{\Omega }}_2\times 2{)}^{{\psi }_{!}({\mathrm{\Omega }}_2\times 2)})$
• $f_{\psi (\psi (\mathrm{\Omega })[n\times 2-1]+1)}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2\times 2+{\psi }_1({\mathrm{\Omega }}_2\times 2+1))$
• $f_{\psi (\mathrm{\Omega })[n\times 2+1]}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2\times 2+{\psi }_1({\mathrm{\Omega }}_2\times 2+{\psi }_1({\mathrm{\Omega }}_2\times 2)))$
• $f_{\psi (\mathrm{\Omega })[n\times 3]}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2\times 3)$
• $f_{\psi (\mathrm{\Omega })[n^2]}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2\times \omega )$
• $f_{\psi (\mathrm{\Omega })[n^2+1]}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2\times \omega +{\psi }_1({\mathrm{\Omega }}_2\times \omega +{\psi }_1({\mathrm{\Omega }}_2\times \omega )))$
• $f_{\psi (\mathrm{\Omega })[n^2+n]}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2\times \omega +{\mathrm{\Omega }}_2)$
• $f_{\psi (\mathrm{\Omega })[n^3]}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2\times {\omega }^2)$
• $f_{\psi (\mathrm{\Omega })[f_3(n)]}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2\times \psi (0))$
• $f_{\psi (\mathrm{\Omega })}^2(n)=f_{\psi (\mathrm{\Omega })[f_{\psi (\mathrm{\Omega })}(n)]}(f_{\psi (\mathrm{\Omega })}(n))\sim \psi ({\mathrm{\Omega }}_2\times \psi ({\mathrm{\Omega }}_2))$
• $f_{\psi (\mathrm{\Omega })}^3(n)\sim \psi ({\mathrm{\Omega }}_2\times \psi ({\mathrm{\Omega }}_2\times \psi ({\mathrm{\Omega }}_2)))$
• $f_{\psi (\mathrm{\Omega })+1}(n)=f_{\psi (\mathrm{\Omega })}^n(n)\sim \psi ({\mathrm{\Omega }}_2\times \mathrm{\Omega })$
• $f_{\psi (\mathrm{\Omega })}(f_{\psi (\mathrm{\Omega })+1}(n))\sim \psi ({\mathrm{\Omega }}_2\times \mathrm{\Omega }+{\mathrm{\Omega }}_2\times \psi ({\mathrm{\Omega }}_2\times \mathrm{\Omega }))$
• $f_{\psi (\mathrm{\Omega })}^n(f_{\psi (\mathrm{\Omega })+1}(n))\sim \psi ({\mathrm{\Omega }}_2\times \mathrm{\Omega }\times 2)$
• $f_{\psi (\mathrm{\Omega })+1}^2(n)\sim \psi ({\mathrm{\Omega }}_2\times \mathrm{\Omega }\times \psi ({\mathrm{\Omega }}_2\times \mathrm{\Omega }))$
• $f_{\psi (\mathrm{\Omega })+2}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\mathrm{\Omega }}^2)$
• $f_{\psi (\mathrm{\Omega })+\omega }(n)\sim \psi ({\mathrm{\Omega }}_2\times {\mathrm{\Omega }}^{\omega })$
• $f_{\psi (\mathrm{\Omega })+\omega +1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\mathrm{\Omega }}^{\mathrm{\Omega }})$
• $f_{\psi (\mathrm{\Omega })+{\omega }^{\omega }+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}})$
• $f_{\psi (\mathrm{\Omega })+\psi (0)+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1(\mathrm{\Omega }))$
• $f_{\psi (\mathrm{\Omega })+\psi (1)+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1(\mathrm{\Omega }\times 2))$
• $f_{\psi (\mathrm{\Omega })+\psi (\psi (0))+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\psi }_1(\mathrm{\Omega })))$
• $f_{\psi (\mathrm{\Omega })\times 2}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2))$
• $f_{\psi (\mathrm{\Omega })}^n(f_{\psi (\mathrm{\Omega })\times 2}(n))\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2)+{\mathrm{\Omega }}_2\times \mathrm{\Omega })$
• $f_{\psi (\mathrm{\Omega })+\psi (0)}^n(f_{\psi (\mathrm{\Omega })\times 2}(n))\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2)+{\mathrm{\Omega }}_2\times {\psi }_1(\mathrm{\Omega }))$
• $f_{\psi (\mathrm{\Omega })+\psi (\mathrm{\Omega })[n]}(f_{\psi (\mathrm{\Omega })\times 2}(n))\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2)\times 2)$
• $f_{\psi (\mathrm{\Omega })+\psi (\mathrm{\Omega })[n]}^2(f_{\psi (\mathrm{\Omega })\times 2}(n))\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2)\times 3)$
• $f_{\psi (\mathrm{\Omega })+\psi (\mathrm{\Omega })[n]+1}(f_{\psi (\mathrm{\Omega })\times 2}(n))\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2)\times \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2)))$
• $f_{\psi (\mathrm{\Omega })+\psi (\mathrm{\Omega })[n]+1}^n(f_{\psi (\mathrm{\Omega })\times 2}(n))\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2)\times \mathrm{\Omega })$
• $f_{\psi (\mathrm{\Omega })+\psi (\mathrm{\Omega })[n]\times 2}(f_{\psi (\mathrm{\Omega })\times 2}(n))\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2{)}^2)$
• $f_{\psi (\mathrm{\Omega })+\psi (\psi (\mathrm{\Omega })[n-1]+1)[n]}(f_{\psi (\mathrm{\Omega })\times 2}(n))\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2+1))$
• $f_{\psi (\mathrm{\Omega })+\psi (\mathrm{\Omega })[n+1]}(f_{\psi (\mathrm{\Omega })\times 2}(n))\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2)))$
• $f_{\psi (\mathrm{\Omega })+\psi (\mathrm{\Omega })[n\times 2]}(f_{\psi (\mathrm{\Omega })\times 2}(n))\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times 2))$
• $f_{\psi (\mathrm{\Omega })\times 2}^2(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2))))$
• $f_{\psi (\mathrm{\Omega })\times 2+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }))$
• $f_{\psi (\mathrm{\Omega })\times 2}(f_{\psi (\mathrm{\Omega })\times 2+1}(n))\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega })+{\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }))))$
• $f_{\psi (\mathrm{\Omega })\times 2}^n(f_{\psi (\mathrm{\Omega })\times 2+1}(n))\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega })\times 2)$
• $f_{\psi (\mathrm{\Omega })\times 2+1}^2(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega })\times \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega })))$
• $f_{\psi (\mathrm{\Omega })\times 2+2}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega })\times \mathrm{\Omega })$
• $f_{\psi (\mathrm{\Omega })\times 3}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega })\times {\psi }_1({\mathrm{\Omega }}_2))$
• $f_{\psi (\mathrm{\Omega })\times 3+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }{)}^2)$
• $f_{\psi (\mathrm{\Omega })\times \omega +1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }{)}^{\mathrm{\Omega }})$
• $f_{\psi (\mathrm{\Omega })\times \omega +\psi (\mathrm{\Omega })+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }{)}^{\mathrm{\Omega }+1})$
• $f_{\psi (\mathrm{\Omega })\times \psi (0)}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }{)}^{{\psi }_1(0)})$
• $f_{\psi (\mathrm{\Omega }{)}^2}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }{)}^{{\psi }_1({\mathrm{\Omega }}_2)})$
• $f_{\psi (\mathrm{\Omega }{)}^2+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }{)}^{{\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega })})$
• $f_{\psi (\mathrm{\Omega }{)}^{\psi (\mathrm{\Omega })}+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }{)}^{{\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }{)}^{{\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega })}})$
• $f_{\psi (\mathrm{\Omega }+1)}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }+1))$
• $f_{\psi (\mathrm{\Omega }+1)+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }+\mathrm{\Omega }))$
• $f_{\psi (\mathrm{\Omega }+2)}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }+\mathrm{\Omega }+1))$
• $f_{\psi (\mathrm{\Omega }+2)+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }+\mathrm{\Omega }\times 2))$
• $f_{\psi (\mathrm{\Omega }+\omega )+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }+{\mathrm{\Omega }}^2))$
• $f_{\psi (\mathrm{\Omega }+\psi (0))}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }+{\psi }_1(0)))$
• $f_{\psi (\mathrm{\Omega }+\psi (0))+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }+{\psi }_1(\mathrm{\Omega })))$
• $f_{\psi (\mathrm{\Omega }+\psi (\mathrm{\Omega }))}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }+{\psi }_1({\mathrm{\Omega }}_2)))$
• $f_{\psi (\mathrm{\Omega }+\psi (\mathrm{\Omega }))[n\times 2]}(f_{\psi (\mathrm{\Omega }+\psi (\mathrm{\Omega }))}(n))\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }+{\psi }_1({\mathrm{\Omega }}_2\times 2)))$
• $f_{\psi (\mathrm{\Omega }+\psi (\mathrm{\Omega }))+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }+{\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega })))$
• $f_{\psi (\mathrm{\Omega }+\psi (\mathrm{\Omega })+1)}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }+{\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega })+1))$
• $f_{\psi (\mathrm{\Omega }+\psi (\mathrm{\Omega }+1))}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }+{\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }+1)))$
• $f_{\psi (\mathrm{\Omega }+\psi (\mathrm{\Omega }+\psi (\mathrm{\Omega })))+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }+{\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }+{\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }))))$
• $f_{\psi (\mathrm{\Omega }\times 2)}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }+{\mathrm{\Omega }}_2))$
• $f_{\psi (\mathrm{\Omega }\times 2)+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }\times 2))$
• $f_{\psi (\mathrm{\Omega }\times 2)\times 2+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }\times 2{)}^2)$
• $f_{\psi (\mathrm{\Omega }\times 2+1)}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }\times 2+1))$
• $f_{\psi (\mathrm{\Omega }\times 2+1)+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }\times 2+\mathrm{\Omega }))$
• $f_{\psi (\mathrm{\Omega }\times 2+\psi (\mathrm{\Omega }\times 2))+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }\times 2+{\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }\times 2)))$
• $f_{\psi (\mathrm{\Omega }\times 3)}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }\times 2+{\mathrm{\Omega }}_2))$
• $f_{\psi (\mathrm{\Omega }\times 3)+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }\times 3))$
• $f_{\psi (\mathrm{\Omega }\times \omega )}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }\times \omega ))$
• $f_{\psi (\mathrm{\Omega }\times \omega )+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times {\mathrm{\Omega }}^2))$
• $f_{\psi (\mathrm{\Omega }\times \omega +1)}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times {\mathrm{\Omega }}^2+1))$
• $f_{\psi (\mathrm{\Omega }\times \omega +\mathrm{\Omega })+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times {\mathrm{\Omega }}^2+{\mathrm{\Omega }}_2\times \mathrm{\Omega }))$
• $f_{\psi (\mathrm{\Omega }\times \omega \times 2)+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times {\mathrm{\Omega }}^2\times 2))$
• $f_{\psi (\mathrm{\Omega }\times {\omega }^2)+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times {\mathrm{\Omega }}^3))$
• $f_{\psi (\mathrm{\Omega }\times {\omega }^{\omega })+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times {\mathrm{\Omega }}^{\mathrm{\Omega }}))$
• $f_{\psi (\mathrm{\Omega }\times \psi (0))}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times {\psi }_1(0)))$
• $f_{\psi (\mathrm{\Omega }\times \psi (0))+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times {\psi }_1(\mathrm{\Omega })))$
• $f_{\psi (\mathrm{\Omega }\times \psi (1))}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times {\psi }_1(\mathrm{\Omega }+1)))$
• $f_{\psi (\mathrm{\Omega }\times \psi (\mathrm{\Omega }))}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2)))$
• $f_{\psi (\mathrm{\Omega }\times \psi (\mathrm{\Omega }))+1}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega })))$
• $f_{\psi (\mathrm{\Omega }\times \psi (\mathrm{\Omega }+1))}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }+1)))$
• $f_{\psi (\mathrm{\Omega }\times \psi (\mathrm{\Omega }\times 2))}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times \mathrm{\Omega }\times 2)))$
• $f_{\psi (\mathrm{\Omega }\times \psi (\mathrm{\Omega }\times \psi (0)))}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times {\psi }_1(0))))$
• $f_{\psi (\mathrm{\Omega }\times \psi (\mathrm{\Omega }\times \psi (\mathrm{\Omega })))}(n)\sim \psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2))))$
• $f_{\psi ({\mathrm{\Omega }}^2)}(n)\sim \psi ({\mathrm{\Omega }}_2^2)$
• $f_{\psi ({\mathrm{\Omega }}^2)[n]}(f_{\psi ({\mathrm{\Omega }}^2)}(n))\sim \psi ({\mathrm{\Omega }}_2^2+{\psi }_1({\mathrm{\Omega }}_2^2))$
• $f_{\psi (\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^2)[n-1]+1)}(f_{\psi ({\mathrm{\Omega }}^2)}(n))\sim \psi ({\mathrm{\Omega }}_2^2+{\psi }_1({\mathrm{\Omega }}_2^2+1))$
• $\psi ({\mathrm{\Omega }}^2)[n]=\psi (\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^2)[n-1])$
• $f_{\psi (\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^2)[n-1]+\mathrm{\Omega })[n]}(f_{\psi ({\mathrm{\Omega }}^2)}(n))\sim \psi ({\mathrm{\Omega }}_2^2+{\mathrm{\Omega }}_2)$
• $f_{\psi (\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^2)[n-1]+\mathrm{\Omega })}(f_{\psi ({\mathrm{\Omega }}^2)}(n))\sim \psi ({\mathrm{\Omega }}_2^2+{\mathrm{\Omega }}_2\times \mathrm{\Omega })$
• $f_{\psi (\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^2)[n-1]+\mathrm{\Omega }\times \psi (\mathrm{\Omega }))}(f_{\psi ({\mathrm{\Omega }}^2)}(n))\sim \psi ({\mathrm{\Omega }}_2^2+{\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2))$
• $f_{\psi ({\mathrm{\Omega }}^2)[n+1]}(f_{\psi ({\mathrm{\Omega }}^2)}(n))\sim \psi ({\mathrm{\Omega }}_2^2+{\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2^2))$
• $f_{\psi ({\mathrm{\Omega }}^2)[n\times 2]}(f_{\psi ({\mathrm{\Omega }}^2)}(n))\sim \psi ({\mathrm{\Omega }}_2^2\times 2)$
• $f_{\psi ({\mathrm{\Omega }}^2)+1}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times \mathrm{\Omega })$
• $f_{\psi ({\mathrm{\Omega }}^2)+2}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\mathrm{\Omega }}^2)$
• $f_{\psi ({\mathrm{\Omega }}^2)+\psi (0)}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1(0))$
• $f_{\psi ({\mathrm{\Omega }}^2)+\psi (\mathrm{\Omega })}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2))$
• $f_{\psi ({\mathrm{\Omega }}^2)+\psi (\mathrm{\Omega }\times \psi (\mathrm{\Omega }))}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2)))$
• $f_{\psi ({\mathrm{\Omega }}^2)\times 2}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2^2))$
• $f_{\psi ({\mathrm{\Omega }}^2)\times 2+1}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2^2\times \mathrm{\Omega }))$
• $f_{\psi ({\mathrm{\Omega }}^2{)}^2+1}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2^2\times \mathrm{\Omega }{)}^{{\psi }_1({\mathrm{\Omega }}_2^2\times \mathrm{\Omega })})$
• $f_{\psi ({\mathrm{\Omega }}^2+1)}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2^2\times \mathrm{\Omega }+1))$
• $f_{\psi ({\mathrm{\Omega }}^2+1)+1}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2^2\times \mathrm{\Omega }+\mathrm{\Omega }))$
• $f_{\psi ({\mathrm{\Omega }}^2+\psi ({\mathrm{\Omega }}^2))+1}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2^2\times \mathrm{\Omega }+{\psi }_1({\mathrm{\Omega }}_2^2\times \mathrm{\Omega })))$
• $f_{\psi ({\mathrm{\Omega }}^2+\mathrm{\Omega })}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2^2\times \mathrm{\Omega }+{\mathrm{\Omega }}_2))$
• $f_{\psi ({\mathrm{\Omega }}^2+\mathrm{\Omega })+1}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2^2\times \mathrm{\Omega }+{\mathrm{\Omega }}_2\times \mathrm{\Omega }))$
• $f_{\psi ({\mathrm{\Omega }}^2+\mathrm{\Omega }\times 2)+1}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2^2\times \mathrm{\Omega }+{\mathrm{\Omega }}_2\times \mathrm{\Omega }\times 2))$
• $f_{\psi ({\mathrm{\Omega }}^2+\mathrm{\Omega }\times \psi (0))+1}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2^2\times \mathrm{\Omega }+{\mathrm{\Omega }}_2\times {\psi }_1(\mathrm{\Omega })))$
• $f_{\psi ({\mathrm{\Omega }}^2+\mathrm{\Omega }\times \psi ({\mathrm{\Omega }}^2))+1}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2^2\times \mathrm{\Omega }+{\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2^2\times \mathrm{\Omega })))$
• $f_{\psi ({\mathrm{\Omega }}^2\times 2)}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2^2\times \mathrm{\Omega }\times 2))$
• $f_{\psi ({\mathrm{\Omega }}^2\times \omega )+1}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2^2\times {\mathrm{\Omega }}^2))$
• $f_{\psi ({\mathrm{\Omega }}^2\times \psi (0))+1}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2^2\times {\psi }_1(0)))$
• $f_{\psi ({\mathrm{\Omega }}^2\times \psi ({\mathrm{\Omega }}^2))}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2^2)))$
• $f_{\psi ({\mathrm{\Omega }}^2\times \psi ({\mathrm{\Omega }}^2))+1}(n)\sim \psi ({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2^2\times {\psi }_1({\mathrm{\Omega }}_2^2\times \mathrm{\Omega })))$
• $f_{\psi ({\mathrm{\Omega }}^3)}(n)\sim \psi ({\mathrm{\Omega }}_2^3)$
• $f_{\psi ({\mathrm{\Omega }}^4)}(n)\sim \psi ({\mathrm{\Omega }}_2^4)$
• $f_{\psi ({\mathrm{\Omega }}^{\omega })}(n)\sim \psi ({\mathrm{\Omega }}_2^{\omega })$
• $f_{\psi ({\mathrm{\Omega }}^{\omega })+1}(n)\sim \psi ({\mathrm{\Omega }}_2^{\mathrm{\Omega }})$
• $f_{\psi ({\mathrm{\Omega }}^{\omega })+2}(n)\sim \psi ({\mathrm{\Omega }}_2^{\mathrm{\Omega }}\times \mathrm{\Omega })$
• $f_{\psi ({\mathrm{\Omega }}^{\omega })\times 2}(n)\sim \psi ({\mathrm{\Omega }}_2^{\mathrm{\Omega }}\times {\psi }_1({\mathrm{\Omega }}_2^{\omega }))$
• $f_{\psi ({\mathrm{\Omega }}^{\omega })\times 2+1}(n)\sim \psi ({\mathrm{\Omega }}_2^{\mathrm{\Omega }}\times {\psi }_1({\mathrm{\Omega }}_2^{\mathrm{\Omega }}))$
• $f_{\psi ({\mathrm{\Omega }}^{\omega })\times 2+2}(n)\sim \psi ({\mathrm{\Omega }}_2^{\mathrm{\Omega }}\times {\psi }_1({\mathrm{\Omega }}_2^{\mathrm{\Omega }})\times \mathrm{\Omega })$
• $f_{\psi ({\mathrm{\Omega }}^{\omega }{)}^2+1}(n)\sim \psi ({\mathrm{\Omega }}_2^{\mathrm{\Omega }}\times {\psi }_1({\mathrm{\Omega }}_2^{\mathrm{\Omega }}{)}^{{\psi }_1({\mathrm{\Omega }}_2^{\mathrm{\Omega }})})$
• $f_{\psi ({\mathrm{\Omega }}^{\omega }+1)}(n)\sim \psi ({\mathrm{\Omega }}_2^{\mathrm{\Omega }}\times {\psi }_1({\mathrm{\Omega }}_2^{\mathrm{\Omega }}+1))$
• $f_{\psi ({\mathrm{\Omega }}^{\omega }+\psi ({\mathrm{\Omega }}^{\omega }))+1}(n)\sim \psi ({\mathrm{\Omega }}_2^{\mathrm{\Omega }}\times {\psi }_1({\mathrm{\Omega }}_2^{\mathrm{\Omega }}+{\psi }_1({\mathrm{\Omega }}_2^{\mathrm{\Omega }})))$
• $f_{\psi ({\mathrm{\Omega }}^{\omega }+\mathrm{\Omega })+1}(n)\sim \psi ({\mathrm{\Omega }}_2^{\mathrm{\Omega }}\times {\psi }_1({\mathrm{\Omega }}_2^{\mathrm{\Omega }}+{\mathrm{\Omega }}_2\times \mathrm{\Omega }))$
• $f_{\psi ({\mathrm{\Omega }}^{\omega }\times 2)}(n)\sim \psi ({\mathrm{\Omega }}_2^{\mathrm{\Omega }}\times {\psi }_1({\mathrm{\Omega }}_2^{\mathrm{\Omega }}+{\mathrm{\Omega }}_2^{\omega }))$
• $f_{\psi ({\mathrm{\Omega }}^{\omega }\times 2)+1}(n)\sim \psi ({\mathrm{\Omega }}_2^{\mathrm{\Omega }}\times {\psi }_1({\mathrm{\Omega }}_2^{\mathrm{\Omega }}\times 2))$
• $f_{\psi ({\mathrm{\Omega }}^{\omega }\times \psi (0))+1}(n)\sim \psi ({\mathrm{\Omega }}_2^{\mathrm{\Omega }}\times {\psi }_1({\mathrm{\Omega }}_2^{\mathrm{\Omega }}\times {\psi }_1(\mathrm{\Omega })))$
• $f_{\psi ({\mathrm{\Omega }}^{\omega }\times \psi ({\mathrm{\Omega }}^{\omega }))+1}(n)\sim \psi ({\mathrm{\Omega }}_2^{\mathrm{\Omega }}\times {\psi }_1({\mathrm{\Omega }}_2^{\mathrm{\Omega }}\times {\psi }_1({\mathrm{\Omega }}^{\mathrm{\Omega }})))$
• $f_{\psi ({\mathrm{\Omega }}^{\omega +1})}(n)\sim \psi ({\mathrm{\Omega }}_2^{\mathrm{\Omega }+1})$
• $f_{\psi ({\mathrm{\Omega }}^{\omega \times 2})+1}(n)\sim \psi ({\mathrm{\Omega }}_2^{\mathrm{\Omega }\times 2})$
• $f_{\psi ({\mathrm{\Omega }}^{{\omega }^2})+1}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}^2})$
• $f_{\psi ({\mathrm{\Omega }}^{{\omega }^{\omega }})+1}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}^{\mathrm{\Omega }}})$
• $f_{\psi ({\mathrm{\Omega }}^{\psi (0)})}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\psi }_1(0)})$
• $f_{\psi ({\mathrm{\Omega }}^{\psi (0)})+1}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\psi }_1(\mathrm{\Omega })})$
• $f_{\psi ({\mathrm{\Omega }}^{\psi (\mathrm{\Omega })})}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\psi }_1({\mathrm{\Omega }}_2)})$
• $f_{\psi ({\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^2)})}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\psi }_1({\mathrm{\Omega }}_2^2)})$
• $f_{\psi ({\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\psi (\mathrm{\Omega })})})}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\psi }_1({\mathrm{\Omega }}_2^{{\psi }_1({\mathrm{\Omega }}_2)})})$
• $f_{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2})$
• $f_{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})+1}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2}\times \mathrm{\Omega })$
• $f_{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})\times 2}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2}\times {\psi }_1({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2}))$
• $f_{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+1)+1}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2}\times {\psi }_1({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2}\times \mathrm{\Omega }+\mathrm{\Omega }))$
• $f_{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+\mathrm{\Omega })+1}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2}\times {\psi }_1({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2}\times \mathrm{\Omega }+{\mathrm{\Omega }}_2\times \mathrm{\Omega }))$
• $f_{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}+{\mathrm{\Omega }}^{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})})}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2}\times {\psi }_1({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2}\times \mathrm{\Omega }+{\mathrm{\Omega }}_2^{{\psi }_1({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2})}))$
• $f_{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}\times 2)}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2}\times {\psi }_1({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2}\times \mathrm{\Omega }\times 2))$
• $f_{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }}\times \psi (0))}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2}\times {\psi }_1({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2}\times {\psi }_1(0)))$
• $f_{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+1})}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2+1})$
• $f_{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+2})}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2+2})$
• $f_{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\omega })+1}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2+\mathrm{\Omega }})$
• $f_{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }+\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})})}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2})})$
• $f_{\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }\times 2})}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2\times 2})$
• $f_{\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^2})}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2^2})$
• $f_{\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }})}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2^{\omega }})$
• $f_{\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }})+1}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2^{\mathrm{\Omega }}})$
• $f_{\psi ({\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}})}(n)\sim \psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2}})$
• $f_{\psi ({\psi }_1(0))}(n)\sim \psi ({\psi }_2(0))$
• $f_{\psi ({\psi }_1(0)\times \mathrm{\Omega })}(n)\sim \psi ({\psi }_2(0)\times {\mathrm{\Omega }}_2)$
• $f_{\psi ({\psi }_1(0{)}^2)}(n)\sim \psi ({\psi }_2(0{)}^2)$
• $f_{\psi ({\psi }_1(0{)}^{\mathrm{\Omega }})}(n)\sim \psi ({\psi }_2(0{)}^{{\mathrm{\Omega }}_2})$
• $f_{\psi ({\psi }_1(0{)}^{{\psi }_1(0)})}(n)\sim \psi ({\psi }_2(0{)}^{{\psi }_2(0)})$
• $f_{\psi ({\psi }_1(1))}(n)\sim \psi ({\psi }_2(1))$
• $f_{\psi ({\psi }_1(\omega ))+1}(n)\sim \psi ({\psi }_2(\mathrm{\Omega }))$
• $f_{\psi ({\psi }_1(\psi (0)))}(n)\sim \psi ({\psi }_2({\psi }_1(0)))$
• $f_{\psi ({\psi }_1(\psi (\mathrm{\Omega })))}(n)\sim \psi ({\psi }_2({\psi }_1({\mathrm{\Omega }}_2)))$
• $f_{\psi ({\psi }_1(\psi ({\mathrm{\Omega }}^{\mathrm{\Omega }})))}(n)\sim \psi ({\psi }_2({\psi }_1({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2})))$
• $f_{\psi ({\psi }_1(\psi ({\psi }_1(0))))}(n)\sim \psi ({\psi }_2({\psi }_1({\psi }_2(0))))$
• $f_{\psi ({\psi }_1(\mathrm{\Omega }))}(n)\sim \psi ({\psi }_2({\mathrm{\Omega }}_2))$
• $f_{\psi ({\psi }_1({\psi }_1(0)))}(n)\sim \psi ({\psi }_2({\psi }_2(0)))$
• $f_{\psi ({\mathrm{\Omega }}_2)}(n)\sim \psi ({\mathrm{\Omega }}_3)$
• $f_{\psi ({\mathrm{\Omega }}_2)[n]}(f_{\psi ({\mathrm{\Omega }}_2)}(n))\sim \psi ({\mathrm{\Omega }}_3+{\psi }_2({\mathrm{\Omega }}_3))$
• $f_{\psi ({\mathrm{\Omega }}_2)[n+1]}(f_{\psi ({\mathrm{\Omega }}_2)}(n))\sim \psi ({\mathrm{\Omega }}_3+{\psi }_2({\mathrm{\Omega }}_3+{\psi }_2({\mathrm{\Omega }}_3)))$
• $f_{\psi ({\mathrm{\Omega }}_2)[n\times 2]}(f_{\psi ({\mathrm{\Omega }}_2)}(n))\sim \psi ({\mathrm{\Omega }}_3\times 2)$
• $f_{\psi ({\mathrm{\Omega }}_2)+1}(n)\sim \psi ({\mathrm{\Omega }}_3\times \mathrm{\Omega })$
• $f_{\psi ({\mathrm{\Omega }}_2)+\psi (0)}(n)\sim \psi ({\mathrm{\Omega }}_3\times {\psi }_1(0))$
• $f_{\psi ({\mathrm{\Omega }}_2)+\psi (\mathrm{\Omega })}(n)\sim \psi ({\mathrm{\Omega }}_3\times {\psi }_1({\mathrm{\Omega }}_2))$
• $f_{\psi ({\mathrm{\Omega }}_2)\times 2}(n)\sim \psi ({\mathrm{\Omega }}_3\times {\psi }_1({\mathrm{\Omega }}_3))$
• $f_{\psi ({\mathrm{\Omega }}_2)\times 2+1}(n)\sim \psi ({\mathrm{\Omega }}_3\times {\psi }_1({\mathrm{\Omega }}_3\times \mathrm{\Omega }))$
• $f_{\psi ({\mathrm{\Omega }}_2)\times 3+1}(n)\sim \psi ({\mathrm{\Omega }}_3\times {\psi }_1({\mathrm{\Omega }}_3\times \mathrm{\Omega }{)}^2)$
• $f_{\psi ({\mathrm{\Omega }}_2{)}^2+1}(n)\sim \psi ({\mathrm{\Omega }}_3\times {\psi }_1({\mathrm{\Omega }}_3\times \mathrm{\Omega }{)}^{{\psi }_1({\mathrm{\Omega }}_3\times \mathrm{\Omega })})$
• $f_{\psi ({\mathrm{\Omega }}_2+1)}(n)\sim \psi ({\mathrm{\Omega }}_3\times {\psi }_1({\mathrm{\Omega }}_3\times \mathrm{\Omega }+1))$
• $f_{\psi ({\mathrm{\Omega }}_2+\mathrm{\Omega })}(n)\sim \psi ({\mathrm{\Omega }}_3\times {\psi }_1({\mathrm{\Omega }}_3\times \mathrm{\Omega }+{\mathrm{\Omega }}_2))$
• $f_{\psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2))}(n)\sim \psi ({\mathrm{\Omega }}_3\times {\psi }_1({\mathrm{\Omega }}_3\times \mathrm{\Omega }+{\psi }_2({\mathrm{\Omega }}_3)))$
• $f_{\psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2))+1}(n)\sim \psi ({\mathrm{\Omega }}_3\times {\psi }_1({\mathrm{\Omega }}_3\times \mathrm{\Omega }+{\psi }_2({\mathrm{\Omega }}_3\times \mathrm{\Omega })))$
• $f_{\psi ({\mathrm{\Omega }}_2+{\psi }_1({\mathrm{\Omega }}_2+1))}(n)\sim \psi ({\mathrm{\Omega }}_3\times {\psi }_1({\mathrm{\Omega }}_3\times \mathrm{\Omega }+{\psi }_2({\mathrm{\Omega }}_3\times \mathrm{\Omega }+1)))$
• $f_{\psi ({\mathrm{\Omega }}_2\times 2)}(n)\sim \psi ({\mathrm{\Omega }}_3\times {\psi }_1({\mathrm{\Omega }}_3\times \mathrm{\Omega }+{\mathrm{\Omega }}_3))$
• $f_{\psi ({\mathrm{\Omega }}_2\times 2)+1}(n)\sim \psi ({\mathrm{\Omega }}_3\times {\psi }_1({\mathrm{\Omega }}_3\times \mathrm{\Omega }\times 2))$
• $f_{\psi ({\mathrm{\Omega }}_2\times \psi ({\mathrm{\Omega }}_2))}(n)\sim \psi ({\mathrm{\Omega }}_3\times {\psi }_1({\mathrm{\Omega }}_3\times {\psi }_1({\mathrm{\Omega }}_3)))$
• $f_{\psi ({\mathrm{\Omega }}_2\times \mathrm{\Omega })}(n)\sim \psi ({\mathrm{\Omega }}_3\times {\mathrm{\Omega }}_2)$
• $f_{\psi ({\mathrm{\Omega }}_2\times {\psi }_1(0))}(n)\sim \psi ({\mathrm{\Omega }}_3\times {\psi }_2(0))$
• $f_{\psi ({\mathrm{\Omega }}_2\times {\psi }_1({\mathrm{\Omega }}_2))}(n)\sim \psi ({\mathrm{\Omega }}_3\times {\psi }_2({\mathrm{\Omega }}_3))$
• $f_{\psi ({\mathrm{\Omega }}_2^2)}(n)\sim \psi ({\mathrm{\Omega }}_3^2)$
• $f_{\psi ({\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2})}(n)\sim \psi ({\mathrm{\Omega }}_3^{{\mathrm{\Omega }}_3})$
• $f_{\psi ({\psi }_2(0))}(n)\sim \psi ({\psi }_3(0))$
• $f_{\psi ({\mathrm{\Omega }}_3)}(n)\sim \psi ({\mathrm{\Omega }}_4)$
• $f_{\psi ({\mathrm{\Omega }}_4)}(n)\sim \psi ({\mathrm{\Omega }}_5)$
• $f_{\psi ({\mathrm{\Omega }}_{\omega })}\sim \psi ({\mathrm{\Omega }}_{\omega })=\text{BO}$