\[(0) = 1\] \[(0)(0) = 2\] 不难发现，(0)(1)展开为(0)(0)(0)…，是所有自然数的极限，因此这里是第一个超限序数omega。 \[(0)(1) = \w = \text{FTO}\]

\[(0)(1)(0) = \w+1\] 又一次 \[(0)(1)(0)(1) = \w2\] 同样的有 \[(0)(1)(0)(1)(0)(1) = \w3\] 接下来，(0)(1)(1)展开成(0)(1)(0)(1)(0)(1)…，相当于\(\sup\{\w\m n|n<\w\}\)。 \[(0)(1)(1) = \w^2\] \[(0)(1)(1)(0)(1)(1) = \w^2 2\] (0)(1)(1)(1)相当于\(\sup\{\w^2\m n|n<\w\}\)。 \[(0)(1)(1)(1) = \w^3\] \[(0)(1)(2) = \w^\w\] \[(0)(1)(2)(0)(1)(2) = \w^\w 2\] \[(0)(1)(2)(1) = \w^{\w+1}\] 这里我们可以发现一个规律。对于(0)X,且X不包含全0项的矩阵，在后面添加一个(1),可以使这个序数乘以omega。 \[(0)(1)(2)(1)(0)(1)(2)(1) = \w^{\w+1}2\] \[(0)(1)(2)(1)(1) = \w^{\w+2}\] \[(0)(1)(2)(1)(1)(1) = \w^{\w+3}\] \[(0)(1)(2)(1)(2) = \w^{\w2}\] 这里的(1)(2)(1)(2)相当于(0)(1)(0)(1)=omega2,所以(0)(1)(2)(1)(2)结果omega的指数为omega2 \[(0)(1)(2)(1)(2)(1) = \w^{\w2+1}\] \[(0)(1)(2)(1)(2)(1)(2) = \w^{\w3}\] \[(0)(1)(2)(2) = \w^{\w^2}\] \[(0)(1)(2)(2)(1) = \w^{\w^2+1}\] \[(0)(1)(2)(2)(1)(2)(2) = \w^{\w^2 2}\] \[(0)(1)(2)(2)(2) = \w^{\w^3}\] \[(0)(1)(2)(3) = \w^{\w^{\w}}\] \[(0)(1)(2)(3)(1)(2)(3) = \w^{\w^{\w}2}\] \[(0)(1)(2)(3)(2) = \w^{\w^{\w+1}}\] \[(0)(1)(2)(3)(2)(3) = \w^{\w^{\w2}}\] \[(0)(1)(2)(3)(3) = \w^{\w^{\w^2}}\] \[(0)(1)(2)(3)(4) = \w^{\w^{\w^\w}}\] \[(0)(1)(2)(3)(4)(5)(6)(5)(4)(3)(3)(3)(3)(2)(2)(2)(2)(2)(1)(0)(0)(0)(0) = \w^{\w^{\w^{\w^{\w^{\w+1}+1}+4}+5}+1}+4\]

这里(0)(1,1)展开成(0)(1)(2)(3)….，我们这里使用的序数坍缩函数是Buchholz-like的。对于大部分序数\(\alpha\)有\(\psi(\alpha+1)=\psi(\alpha)\m\w\)。 \(\psi(\W)\)是第一个\(\alpha\mapsto\w^{\alpha}\)不动点 \[(0)(1,1) = \psi(\W) = \text{SCO}\] \[(0)(1,1)(0)(1,1) = \psi(\W)2\] \[(0)(1,1)(1) = \psi(\W+1)\] \[(0)(1,1)(1)(1) = \psi(\W+2)\] \[(0)(1,1)(1)(2) = \psi(\W+\w)\] \[(0)(1,1)(1)(2)(3) = \psi(\W+\w^\w)\] 这里的(1)(2,1)展开成(1)(2)(3)(4)… \[(0)(1,1)(1)(2,1) = \psi(\W+\psi(\W))\] \[(0)(1,1)(1)(2,1)(1) = \psi(\W+\psi(\W)+1)\] \[(0)(1,1)(1)(2,1)(1)(2) = \psi(\W+\psi(\W)+\w)\] \[(0)(1,1)(1)(2,1)(1)(2,1) = \psi(\W+\psi(\W)2)\] \[(0)(1,1)(1)(2,1)(1)(2,1)(1)(2,1) = \psi(\W+\psi(\W)3)\] 这里的(1)(2,1)(2)相当于(0)(1,1)(1)。 \[(0)(1,1)(1)(2,1)(2) = \psi(\W+\psi(\W+1))\] \[(0)(1,1)(1)(2,1)(2)(2) = \psi(\W+\psi(\W+2))\] \[(0)(1,1)(1)(2,1)(2)(3) = \psi(\W+\psi(\W+\w))\] \[(0)(1,1)(1)(2,1)(2)(3,1) = \psi(\W+\psi(\W+\psi(\W)))\] \[(0)(1,1)(1)(2,1)(2)(3,1)(3) = \psi(\W+\psi(\W+\psi(\W+1)))\] 第二个\(\alpha\mapsto\w^{\alpha}\)不动点 \[(0)(1,1)(1,1) = \psi(\W2)\] \[(0)(1,1)(1,1)(1)(2,1) = \psi(\W2+\psi(\W))\] \[(0)(1,1)(1,1)(1)(2,1)(2) = \psi(\W2+\psi(\W+1))\] \[(0)(1,1)(1,1)(1)(2,1)(2)(3,1) = \psi(\W2+\psi(\W+\psi(\W)))\] \[(0)(1,1)(1,1)(1)(2,1)(2,1) = \psi(\W2+\psi(\W2))\] \[(0)(1,1)(1,1)(1)(2,1)(2,1)(1)(2,1)(2,1) = \psi(\W2+\psi(\W2)2)\] \[(0)(1,1)(1,1)(1)(2,1)(2,1)(2) = \psi(\W2+\psi(\W2+1))\] \[(0)(1,1)(1,1)(1)(2,1)(2,1)(2)(3) = \psi(\W2+\psi(\W2+\w))\] \[(0)(1,1)(1,1)(1)(2,1)(2,1)(2)(3,1)(3,1) = \psi(\W2+\psi(\W2+\psi(\W2)))\] 第三个\(\alpha\mapsto\w^{\alpha}\)不动点 \[(0)(1,1)(1,1)(1,1) = \psi(\W3)\] 第\(\w\)个\(\alpha\mapsto\w^{\alpha}\)不动点 \[(0)(1,1)(2) = \psi(\W\w)\] \[(0)(1,1)(2)(1)(2,1) = \psi(\W\w+\psi(\W))\] \[(0)(1,1)(2)(1)(2,1)(2,1) = \psi(\W\w+\psi(\W2))\] \[(0)(1,1)(2)(1)(2,1)(3) = \psi(\W\w+\psi(\W\w))\] \[(0)(1,1)(2)(1,1) = \psi(\W\m(\w+1))\] \[(0)(1,1)(2)(1,1)(1,1) = \psi(\W\m(\w+2))\] \[(0)(1,1)(2)(1,1)(2) = \psi(\W\w2)\] \[(0)(1,1)(2)(1,1)(2)(1,1)(2) = \psi(\W\w3)\] \[(0)(1,1)(2)(2) = \psi(\W\w^2)\] \[(0)(1,1)(2)(2)(2) = \psi(\W\w^3)\] \[(0)(1,1)(2)(3) = \psi(\W\w^\w)\] \[(0)(1,1)(2)(3)(4) = \psi(\W\w^{\w^{\w}})\] \[(0)(1,1)(2)(3,1) = \psi(\W\psi(\W))\] \[(0)(1,1)(2)(3,1)(2) = \psi(\W\psi(\W+1))\] \[(0)(1,1)(2)(3,1)(2)(3,1) = \psi(\W\psi(\W+\psi(\W)))\] \[(0)(1,1)(2)(3,1)(3) = \psi(\W\psi(\W+\psi(\W+1)))\] \[(0)(1,1)(2)(3,1)(3,1) = \psi(\W\psi(\W2))\] \[(0)(1,1)(2)(3,1)(4) = \psi(\W\psi(\W\w))\] \[(0)(1,1)(2)(3,1)(4)(4) = \psi(\W\psi(\W\w^2))\] \[(0)(1,1)(2)(3,1)(4)(5,1) = \psi(\W\psi(\W\psi(\W)))\] \[(0)(1,1)(2)(3,1)(4)(5,1)(6)(7,1) = \psi(\W\psi(\W\psi(\W\psi(\W))))\]

2-Row Iteration

第1个\(\alpha\mapsto\psi(\W\alpha)\)不动点 \[(0)(1,1)(2,1) = \psi(\W^2) = \text{CO}\] \[(0)(1,1)(2,1)(1)(2,1) = \psi(\W^2+\psi(\W))\] \[(0)(1,1)(2,1)(1)(2,1)(3) = \psi(\W^2+\psi(\W\w))\] \[(0)(1,1)(2,1)(1)(2,1)(3,1) = \psi(\W^2+\psi(\W^2))\] \[(0)(1,1)(2,1)(1,1) = \psi(\W^2+\W)\] \[(0)(1,1)(2,1)(1,1)(1,1) = \psi(\W^2+\W2)\] \[(0)(1,1)(2,1)(1,1)(2) = \psi(\W^2+\W\w)\] \[(0)(1,1)(2,1)(1,1)(2)(3,1) = \psi(\W^2+\W\psi(\W))\] \[(0)(1,1)(2,1)(1,1)(2)(3,1)(4) = \psi(\W^2+\W\psi(\W\w))\] \[(0)(1,1)(2,1)(1,1)(2)(3,1)(4)(5,1) = \psi(\W^2+\W\psi(\W\psi(\W)))\] \[(0)(1,1)(2,1)(1,1)(2)(3,1)(4,1) = \psi(\W^2+\W\psi(\W^2))\] \[(0)(1,1)(2,1)(1,1)(2)(3,1)(4,1)(3,1) = \psi(\W^2+\W\psi(\W^2+\W))\] \[(0)(1,1)(2,1)(1,1)(2)(3,1)(4,1)(3,1)(4) = \psi(\W^2+\W\psi(\W^2+\W\w))\] \[(0)(1,1)(2,1)(1,1)(2)(3,1)(4,1)(3,1)(4)(5,1)(6,1)(5,1) = \psi(\W^2+\W\psi(\W^2+\W\psi(\W^2+\W)))\] \[(0)(1,1)(2,1)(1,1)(2,1) = \psi(\W^2 2)\] 这一段对应着$\zeta_0$到$\zeta_1$无穷的$\varepsilon_0$嵌套过程

\[(0)(1,1)(2,1)(1,1)(2,1)(1,1) = \psi(\W^2 2+\W)\] \[(0)(1,1)(2,1)(1,1)(2,1)(1,1)(2)(3,1)(4,1) = \psi(\W^2 2+\W\psi(\W^2))\] \[(0)(1,1)(2,1)(1,1)(2,1)(1,1)(2)(3,1)(4,1)(3,1)(4,1) = \psi(\W^2 2+\W\psi(\W^2 2))\] \[(0)(1,1)(2,1)(1,1)(2,1)(1,1)(2)(3,1)(4,1)(3,1)(4,1)(3,1)(4) = \psi(\W^2 2+\W\psi(\W^2 2 + \W\w))\] \[(0)(1,1)(2,1)(1,1)(2,1)(1,1)(2,1) = \psi(\W^2 3)\] \[(0)(1,1)(2,1)(2) = \psi(\W^2 \w)\] \[(0)(1,1)(2,1)(2)(1,1) = \psi(\W^2 \w + \W)\] \[(0)(1,1)(2,1)(2)(1,1)(2)(3,1)(4,1)(4)(3,1) = \psi(\W^2 \w + \W\psi(\W^2 \w + \W))\] \[(0)(1,1)(2,1)(2)(1,1)(2,1) = \psi(\W^2\m(\w+1))\] \[(0)(1,1)(2,1)(2)(1,1)(2,1)(1,1)(2,1) = \psi(\W^2\m(\w+2))\] \[(0)(1,1)(2,1)(2)(1,1)(2,1)(2) = \psi(\W^2\w2)\] \[(0)(1,1)(2,1)(2)(2) = \psi(\W^2\w^2)\] \[(0)(1,1)(2,1)(2)(3) = \psi(\W^2\w^\w)\] \[(0)(1,1)(2,1)(2)(3,1) = \psi(\W^2\psi(\W))\] \[(0)(1,1)(2,1)(2)(3,1)(4) = \psi(\W^2\psi(\W\w))\] \[(0)(1,1)(2,1)(2)(3,1)(4,1) = \psi(\W^2\psi(\W^2))\] \[(0)(1,1)(2,1)(2,1) = \psi(\W^3) = \text{LCO}\] \[(0)(1,1)(2,1)(2,1)(1,1) = \psi(\W^3+\W)\] \[(0)(1,1)(2,1)(2,1)(1,1)(2)(3,1)(4,1)(4,1) = \psi(\W^3+\W\psi(\W^3))\] \[(0)(1,1)(2,1)(2,1)(1,1)(2)(3,1)(4,1)(4,1)(3,1)(4) = \psi(\W^3+\W\psi(\W^3+\W\w))\] \[(0)(1,1)(2,1)(2,1)(1,1)(2,1) = \psi(\W^3+\W^2)\] \[(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2) = \psi(\W^3+\W^2\w)\] \[(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2)(3,1)(4,1)(4,1)(3,1)(4,1) = \psi(\W^3+\W^2\psi(\W^3+\W^2))\] \[(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1) = \psi(\W^3 2)\] \[(0)(1,1)(2,1)(2,1)(2) = \psi(\W^3 \w)\] \[(0)(1,1)(2,1)(2,1)(2,1) = \psi(\W^4)\] \[(0)(1,1)(2,1)(3) = \psi(\W^\w) = \text{HCO}\] \[(0)(1,1)(2,1)(3)(2) = \psi(\W^\w \w)\] \[(0)(1,1)(2,1)(3)(2,1) = \psi(\W^{\w+1})\] \[(0)(1,1)(2,1)(3)(2,1)(3) = \psi(\W^{\w2})\] \[(0)(1,1)(2,1)(3)(3) = \psi(\W^{\w^2})\] \[(0)(1,1)(2,1)(3)(4,1) = \psi(\W^{\psi(\W)})\] \[(0)(1,1)(2,1)(3)(4,1)(4,1) = \psi(\W^{\psi(\W2)})\] \[(0)(1,1)(2,1)(3)(4,1)(5) = \psi(\W^{\psi(\W\w)})\] \[(0)(1,1)(2,1)(3)(4,1)(5,1) = \psi(\W^{\psi(\W^2)})\] \[(0)(1,1)(2,1)(3)(4,1)(5,1)(5,1) = \psi(\W^{\psi(\W^3)})\] \[(0)(1,1)(2,1)(3)(4,1)(5,1)(6) = \psi(\W^{\psi(\W^\w)})\] \[(0)(1,1)(2,1)(3)(4,1)(5,1)(6)(7,1) = \psi(\W^{\psi(\W^{\psi(\W)})})\] \[(0)(1,1)(2,1)(3,1) = \psi(\W^\W) = \text{FSO}\] \[(0)(1,1)(2,1)(3,1)(2,1) = \psi(\W^{\W+1})\] \[(0)(1,1)(2,1)(3,1)(2,1)(3) = \psi(\W^{\W+\w})\] \[(0)(1,1)(2,1)(3,1)(2,1)(3)(4,1)(5,1)(6,1)(5,1)(6) = \psi(\W^{\W+\psi(\W^{\W+\w})})\] \[(0)(1,1)(2,1)(3,1)(2,1)(3,1) = \psi(\W^{\W2})\] \[(0)(1,1)(2,1)(3,1)(2,1)(3,1)(2,1)(3,1) = \psi(\W^{\W3})\] \[(0)(1,1)(2,1)(3,1)(3) = \psi(\W^{\W\w})\] \[(0)(1,1)(2,1)(3,1)(3,1) = \psi(\W^{\W^2})\] \[(0)(1,1)(2,1)(3,1)(3,1)(2,1) = \psi(\W^{\W^2+1})\] \[(0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1) = \psi(\W^{\W^2+\W})\] \[(0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1)(3,1) = \psi(\W^{\W^2 2})\] \[(0)(1,1)(2,1)(3,1)(3,1)(3,1) = \psi(\W^{\W^3})\] \[(0)(1,1)(2,1)(3,1)(4) = \psi(\W^{\W^\w}) = \text{SVO}\] \[(0)(1,1)(2,1)(3,1)(4)(2,1)(3,1)(4) = \psi(\W^{\W^\w 2})\] \[(0)(1,1)(2,1)(3,1)(4)(3) = \psi(\W^{\W^\w \w})\] \[(0)(1,1)(2,1)(3,1)(4)(3,1) = \psi(\W^{\W^{\w+1}})\] \[(0)(1,1)(2,1)(3,1)(4)(3,1)(4) = \psi(\W^{\W^{\w2}})\] \[(0)(1,1)(2,1)(3,1)(4)(4) = \psi(\W^{\W^{\w^2}})\] \[(0)(1,1)(2,1)(3,1)(4)(5,1) = \psi(\W^{\W^{\psi(\W)}})\] \[(0)(1,1)(2,1)(3,1)(4,1) = \psi(\W^{\W^{\W}}) = \text{LVO}\] \[(0)(1,1)(2,1)(3,1)(4,1)(3,1) = \psi(\W^{\W^{\W+1}})\] \[(0)(1,1)(2,1)(3,1)(4,1)(3,1)(4,1) = \psi(\W^{\W^{\W2}})\] \[(0)(1,1)(2,1)(3,1)(4,1)(4) = \psi(\W^{\W^{\W\w}})\] \[(0)(1,1)(2,1)(3,1)(4,1)(4,1) = \psi(\W^{\W^{\W^2}})\] \[(0)(1,1)(2,1)(3,1)(4,1)(5) = \psi(\W^{\W^{\W^{\w}}})\] \[(0)(1,1)(2,1)(3,1)(4,1)(5,1) = \psi(\W^{\W^{\W^{\W}}})\]

Post MOCF limit

\[(0)(1,1)(2,2) = \psi(\W_2) = \text{BHO}\] \[(0)(1,1)(2,2)(1,1) = \psi(\W_2+\W)\] \[(0)(1,1)(2,2)(1,1)(2) = \psi(\W_2+\W\w)\] \[(0)(1,1)(2,2)(1,1)(2,1) = \psi(\W_2+\W^2)\] \[(0)(1,1)(2,2)(1,1)(2,1)(3,1) = \psi(\W_2+\W^\W)\] \[(0)(1,1)(2,2)(1,1)(2,2) = \psi(\W_2+\psi_1(\W_2))\] 注意这里不是\(\W_2 2\) \[(0)(1,1)(2,2)(1,1)(2,2)(1,1)(2,1)(3,1) = \psi(\W_2+\psi_1(\W_2)+\W^\W)\] \[(0)(1,1)(2,2)(1,1)(2,2)(1,1)(2,2) = \psi(\W_2+\psi_1(\W_2)2)\] 这里有藏层/bx \[(0)(1,1)(2,2)(2) = \psi(\W_2+\psi_1(\W_2+1))\] \[(0)(1,1)(2,2)(2,1) = \psi(\W_2+\psi_1(\W_2+\W))\] \[(0)(1,1)(2,2)(2,1)(1,1)(2,2) = \psi(\W_2+\psi_1(\W_2+\W)+\psi_1(\W_2))\] \[(0)(1,1)(2,2)(2,1)(1,1)(2,2)(2,1) = \psi(\W_2+\psi_1(\W_2+\W)2)\] \[(0)(1,1)(2,2)(2,1)(2) = \psi(\W_2+\psi_1(\W_2+\W+1))\] \[(0)(1,1)(2,2)(2,1)(2,1) = \psi(\W_2+\psi_1(\W_2+\W2))\] \[(0)(1,1)(2,2)(2,1)(3) = \psi(\W_2+\psi_1(\W_2+\W\w))\] \[(0)(1,1)(2,2)(2,1)(3,1) = \psi(\W_2+\psi_1(\W_2+\W^2))\] \[(0)(1,1)(2,2)(2,1)(3,1)(2,1) = \psi(\W_2+\psi_1(\W_2+\W^2+\W))\] \[(0)(1,1)(2,2)(2,1)(3,1)(2,1)(3,1) = \psi(\W_2+\psi_1(\W_2+\W^2 2))\] \[(0)(1,1)(2,2)(2,1)(3,1)(3) = \psi(\W_2+\psi_1(\W_2+\W^2 \w))\] \[(0)(1,1)(2,2)(2,1)(3,1)(3,1) = \psi(\W_2+\psi_1(\W_2+\W^3))\] \[(0)(1,1)(2,2)(2,1)(3,1)(4) = \psi(\W_2+\psi_1(\W_2+\W^\w))\] \[(0)(1,1)(2,2)(2,1)(3,1)(4,1) = \psi(\W_2+\psi_1(\W_2+\W^\W))\] \[(0)(1,1)(2,2)(2,1)(3,1)(4,1)(3,1) = \psi(\W_2+\psi_1(\W_2+\W^{\W+1}))\] \[(0)(1,1)(2,2)(2,1)(3,1)(4,1)(3,1)(4,1) = \psi(\W_2+\psi_1(\W_2+\W^{\W2}))\] \[(0)(1,1)(2,2)(2,1)(3,1)(4,1)(4) = \psi(\W_2+\psi_1(\W_2+\W^{\W\w}))\] \[(0)(1,1)(2,2)(2,1)(3,1)(4,1)(4,1) = \psi(\W_2+\psi_1(\W_2+\W^{\W^2}))\] \[(0)(1,1)(2,2)(2,1)(3,1)(4,1)(5) = \psi(\W_2+\psi_1(\W_2+\W^{\W^{\w}}))\] \[(0)(1,1)(2,2)(2,1)(3,1)(4,1)(5,1) = \psi(\W_2+\psi_1(\W_2+\W^{\W^{\W}}))\] \[(0)(1,1)(2,2)(2,1)(3,2) = \psi(\W_2+\psi_1(\W_2+\psi_1(\W_2)))\] \[(0)(1,1)(2,2)(2,1)(3,2)(2,1)(3,2) = \psi(\W_2+\psi_1(\W_2+\psi_1(\W_2)2))\] \[(0)(1,1)(2,2)(2,1)(3,2)(3,1) = \psi(\W_2+\psi_1(\W_2+\psi_1(\W_2+\W)))\] \[(0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,1) = \psi(\W_2+\psi_1(\W_2+\psi_1(\W_2+\W^2)))\] \[(0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2) = \psi(\W_2+\psi_1(\W_2+\psi_1(\W_2+\psi_1(\W_2))))\] \[(0)(1,1)(2,2)(2,2) = \psi(\W_2 2)\] \[(0)(1,1)(2,2)(2,2)(2,1)(3,2)(3,2) = \psi(\W_2 2+\psi_1(\W_2 2))\] \[(0)(1,1)(2,2)(2,2)(2,2) = \psi(\W_2 3)\] \[(0)(1,1)(2,2)(3) = \psi(\W_2 \w)\] \[(0)(1,1)(2,2)(3)(2,2) = \psi(\W_2\m( \w+1))\] \[(0)(1,1)(2,2)(3)(2,2)(3) = \psi(\W_2 \w2)\] \[(0)(1,1)(2,2)(3)(3) = \psi(\W_2 \w^2)\] \[(0)(1,1)(2,2)(3)(4,1) = \psi(\W_2\psi(\W))\] \[(0)(1,1)(2,2)(3)(4,1)(5,1) = \psi(\W_2\psi(\W^2))\] \[(0)(1,1)(2,2)(3)(4,1)(5,2) = \psi(\W_2\psi(\W_2))\] \[(0)(1,1)(2,2)(3,1) = \psi(\W_2\W)\] \[(0)(1,1)(2,2)(3,1)(2,2)(3,1) = \psi(\W_2\W2)\] \[(0)(1,1)(2,2)(3,1)(3,1) = \psi(\W_2\W^2)\] \[(0)(1,1)(2,2)(3,1)(4) = \psi(\W_2\W^\w)\] \[(0)(1,1)(2,2)(3,1)(4,1) = \psi(\W_2\W^\W)\] \[(0)(1,1)(2,2)(3,1)(4,2) = \psi(\W_2\psi_1(\W_2))\] \[(0)(1,1)(2,2)(3,1)(4,2)(4,2) = \psi(\W_2\psi_1(\W_2 2))\] \[(0)(1,1)(2,2)(3,1)(4,2)(5,1) = \psi(\W_2\psi_1(\W_2 \W))\] \[(0)(1,1)(2,2)(3,1)(4,2)(5,1)(6,2)(7,1) = \psi(\W_2\psi_1(\W_2 \psi_1(\W_2 \W)))\] \[(0)(1,1)(2,2)(3,2) = \psi(\W_2^2)\] \[(0)(1,1)(2,2)(3,2)(2,2) = \psi(\W_2^2+\W_2)\] \[(0)(1,1)(2,2)(3,2)(2,2)(3,1) = \psi(\W_2^2+\W_2\W)\] \[(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2) = \psi(\W_2^2+\W_2\psi_1(\W_2))\] \[(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5) = \psi(\W_2^2+\W_2\psi_1(\W_2 \w))\] \[(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2) = \psi(\W_2^2+\W_2\psi_1(\W_2^2))\] \[(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2)(4,2) = \psi(\W_2^2+\W_2\psi_1(\W_2^2 + \W_2))\] \[(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2)(4,2)(5,1)(6,2) = \psi(\W_2^2+\W_2\psi_1(\W_2^2 + \W_2\psi_1(\W_2)))\] \[(0)(1,1)(2,2)(3,2)(2,2)(3,1)(4,2)(5,2)(4,2)(5,1)(6,2)(7,2)(6,1)(7,2) = \psi(\W_2^2+\W_2\psi_1(\W_2^2 + \W_2\psi_1(\W_2^2 + \W_2\psi_1(\W_2))))\] \[(0)(1,1)(2,2)(3,2)(2,2)(3,2) = \psi(\W_2^2 2)\] \[(0)(1,1)(2,2)(3,2)(3) = \psi(\W_2^2 \w)\] \[(0)(1,1)(2,2)(3,2)(3,1) = \psi(\W_2^2 \W)\] \[(0)(1,1)(2,2)(3,2)(3,1)(2,2)(3,2)(3,1) = \psi(\W_2^2 \W2)\] \[(0)(1,1)(2,2)(3,2)(3,1)(3) = \psi(\W_2^2 \W\w)\] \[(0)(1,1)(2,2)(3,2)(3,1)(3,1) = \psi(\W_2^2 \W^2)\] \[(0)(1,1)(2,2)(3,2)(3,1)(4) = \psi(\W_2^2 \W^\w)\] \[(0)(1,1)(2,2)(3,2)(3,1)(4,1) = \psi(\W_2^2 \W^\W)\] \[(0)(1,1)(2,2)(3,2)(3,1)(4,2) = \psi(\W_2^2\psi_1(\W_2))\] \[(0)(1,1)(2,2)(3,2)(3,1)(4,2)(5,2) = \psi(\W_2^2\psi_1(\W_2^2))\] \[(0)(1,1)(2,2)(3,2)(3,1)(4,2)(5,2)(5,1)(6,2)(7,2) = \psi(\W_2^2\psi_1(\W_2^2))\] \[(0)(1,1)(2,2)(3,2)(3,2) = \psi(\W_2^3)\] \[(0)(1,1)(2,2)(3,2)(4) = \psi(\W_2^\w)\] \[(0)(1,1)(2,2)(3,2)(4)(3,2) = \psi(\W_2^{\w+1})\] \[(0)(1,1)(2,2)(3,2)(4)(4) = \psi(\W_2^{\w^2})\] \[(0)(1,1)(2,2)(3,2)(4)(5,1) = \psi(\W_2^{\psi(\W)})\] \[(0)(1,1)(2,2)(3,2)(4,1) = \psi(\W_2^{\W})\] \[(0)(1,1)(2,2)(3,2)(4,1)(4,1) = \psi(\W_2^{\W^2})\] \[(0)(1,1)(2,2)(3,2)(4,1)(5,2) = \psi(\W_2^{\psi_1(\W_2)})\] \[(0)(1,1)(2,2)(3,2)(4,2) = \psi(\W_2^{\W_2})\] \[(0)(1,1)(2,2)(3,2)(4,2)(3,2)(4,2) = \psi(\W_2^{\W_2 2})\] \[(0)(1,1)(2,2)(3,2)(4,2)(4) = \psi(\W_2^{\W_2 \w})\] \[(0)(1,1)(2,2)(3,2)(4,2)(4,2) = \psi(\W_2^{\W_2^2})\] \[(0)(1,1)(2,2)(3,2)(4,2)(5) = \psi(\W_2^{\W_2^{\w}})\] \[(0)(1,1)(2,2)(3,2)(4,2)(5,1) = \psi(\W_2^{\W_2^{\W}})\] \[(0)(1,1)(2,2)(3,2)(4,2)(5,2) = \psi(\W_2^{\W_2^{\W_2}})\] \[(0)(1,1)(2,2)(3,3) = \psi(\W_3)\] \[(0)(1,1)(2,2)(3,3)(2,2)(3,2) = \psi(\W_3+\W_2^{2})\] \[(0)(1,1)(2,2)(3,3)(2,2)(3,3) = \psi(\W_3+\psi_2(\W_3))\] \[(0)(1,1)(2,2)(3,3)(3) = \psi(\W_3+\psi_2(\W_3+1))\] \[(0)(1,1)(2,2)(3,3)(3,3) = \psi(\W_3 2)\] \[(0)(1,1)(2,2)(3,3)(4) = \psi(\W_3 \w)\] \[(0)(1,1)(2,2)(3,3)(4,1) = \psi(\W_3 \W)\] \[(0)(1,1)(2,2)(3,3)(4,2) = \psi(\W_3 \W_2)\] \[(0)(1,1)(2,2)(3,3)(4,3) = \psi(\W_3 ^2)\] \[(0)(1,1)(2,2)(3,3)(4,4) = \psi(\W_4)\] \[(0)(1,1)(2,2)(3,3)(4,4)(5,5) = \psi(\W_5)\] \[(0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6) = \psi(\W_6)\] \[(0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6)(7,7) = \psi(\W_7)\] \[(0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6)(7,7)(8,8) = \psi(\W_8)\]

\[(0)(1,1)(2,2)(3,3)(4,4)(5,5)(6,6)(7,7)(8,8)(9,9)=\psi(\W_9)\] \[(0)(1,1,1)=\psi(\W_\w)\]