HSS Hydra 分析

该分析来自梅天狸的知乎文章，使用 BMS 与之对照。

$p_0=1$

\begin{align}s\\&p_0(p_0)=(0)(1)\\&p_0(p_0(p_0))=(0)(1)(2)\\&p_0(p_1)=(0,0)(1,1)\\&p_0(p_1+p_0)=(0,0)(1,1)(1,0)\\&p_0(p_1+p_0(p_0))=(0,0)(1,1)(1,0)(2,0)\\&p_0(p_1+p_0(p_0(p_1)))=(0,0)(1,1)(1,0)(2,1)\\&p_0(p_1+p_0(p_0(p_1))+p_0(p_0(p_1)))=(0,0)(1,1)(1,0)(2,1)(1,0)(2,1)\\&p_0(p_1+p_0(p_0(p_1)+p_0))=(0,0)(1,1)(1,0)(2,1)(2,0)\\&p_0(p_1+p_0(p_0(p_1)+p_0(p_0(p_1))))=(0,0)(1,1)(1,0)(2,1)(2,0)(3,1)\\&p_0(p_1+p_0(p_1))=(0,0)(1,1)(1,1)\\&p_0(p_1+p_0(p_1)+p_0(p_0(p_1)+p_0(p_1)))=(0,0)(1,1)(1,1)(1,0)(2,1)(2,1)\\&p_0(p_1+p_0(p_1)+p_0(p_1))=(0,0)(1,1)(1,1)(1,1)\\&p_0(p_1+p_0(p_1+p_0))=(0,0)(1,1)(2,0)\\&p_0(p_1+p_0(p_1+p_0(p_0)))=(0,0)(1,1)(2,0)(3,0)\\&p_0(p_1+p_0(p_1+p_0(p_0(p_1))))=(0,0)(1,1)(2,0)(3,1)\\&p_0(p_1+p_0(p_1+p_0(p_0(p_1)+p_0(p_1))))=(0,0)(1,1)(2,0)(3,1)(3,1)\\&p_0(p_1+p_0(p_1+p_0(p_1)))=(0,0)(1,1)(2,1)\\&p_0(p_1+p_0(p_1+p_0(p_1))+p_0(p_0(p_1+p_0(p_1+p_0(p_1)))))=(0,0)(1,1)(2,1)(1,0)(2,1)(3,1)\\&p_0(p_1+p_0(p_1+p_0(p_1))+p_0(p_1))=(0,0)(1,1)(2,1)(1,1)\\&p_0(p_1+p_0(p_1+p_0(p_1))+p_0(p_1+p_0(p_0(p_1+p_0(p_1+p_0(p_1))))))=(0,0)(1,1)(2,1)(1,1)(2,0)(3,1)(4,1)\\&p_0(p_1+p_0(p_1+p_0(p_1))+p_0(p_1+p_0(p_1)))=(0,0)(1,1)(2,1)(1,1)(2,1)\\&p_0(p_1+p_0(p_1+p_0(p_1)+p_0))=(0,0)(1,1)(2,1)(2,0)\\&p_0(p_1+p_0(p_1+p_0(p_1)+p_0(p_0(p_1+p_0(p_1+p_0(p_1))))))=(0,0)(1,1)(2,1)(2,0)(3,1)(4,1)\\&p_0(p_1+p_0(p_1+p_0(p_1)+p_0(p_1)))=(0,0)(1,1)(2,1)(2,1)\\&p_0(p_1+p_0(p_1+p_0(p_1+p_0)))=(0,0)(1,1)(2,1)(3,0)\\&p_0(p_1+p_0(p_1+p_0(p_1+p_0(p_1))))=(0,0)(1,1)(2,1)(3,1)\\&p_0(p_1+p_0(p_1+p_0(p_1+p_0(p_1)))+p_0(p_1+p_0(p_1+p_0(p_1))))=(0,0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)\\&p_0(p_1+p_0(p_1+p_0(p_1+p_0(p_1))+p_0(p_1+p_0(p_1))))=(0,0)(1,1)(2,1)(3,1)(2,1)(3,1)\\&p_0(p_1+p_0(p_1+p_0(p_1+p_0(p_1)+p_0(p_1))))=(0,0)(1,1)(2,1)(3,1)(3,1)\\&p_0(p_1+p_0(p_1+p_0(p_1+p_0(p_1+p_0))))=(0,0)(1,1)(2,1)(3,1)(4,0)\\&p_0(p_1+p_0(p_1+p_0(p_1+p_0(p_1+p_0(p_1)))))=(0,0)(1,1)(2,1)(3,1)(4,1)\\&p_0(p_1+p_1)=(0,0)(1,1)(2,2)\end{align}

\begin{align}s\\&p_0(p_1+p_1+p_0(p_0(p_1+p_1)))=(0,0)(1,1)(2,2)(1,0)(2,1)(3,2)\\&p_0(p_1+p_1+p_0(p_1))=(0,0)(1,1)(2,2)(1,1)\\&p_0(p_1+p_1+p_0(p_1+p_0(p_1)))=(0,0)(1,1)(2,2)(1,1)(2,1)\\&p_0(p_1+p_1+p_0(p_1+p_0(p_1+p_1)))=(0,0)(1,1)(2,2)(1,1)(2,2)\\&p_0(p_1+p_1+p_0(p_1+p_0(p_1+p_1)+p_0))=(0,0)(1,1)(2,2)(2,0)\\&p_0(p_1+p_1+p_0(p_1+p_0(p_1+p_1)+p_0(p_1)))=(0,0)(1,1)(2,2)(2,1)\\&p_0(p_1+p_1+p_0(p_1+p_0(p_1+p_1)+p_0(p_1+p_0(p_1+p_1))))=(0,0)(1,1)(2,2)(2,1)(3,2)\\&p_0(p_1+p_1+p_0(p_1+p_1))=(0,0)(1,1)(2,2)(2,2)\\&p_0(p_1+p_1+p_0(p_1+p_1+p_0))=(0,0)(1,1)(2,2)(3,0)\\&p_0(p_1+p_1+p_0(p_1+p_1+p_0(p_1)))=(0,0)(1,1)(2,2)(3,1)\\&p_0(p_1+p_1+p_0(p_1+p_1+p_0(p_1+p_0(p_1+p_1+p_0(p_1+p_1)))))=(0,0)(1,1)(2,2)(3,1)(4,2)(4,2)\\&p_0(p_1+p_1+p_0(p_1+p_1+p_0(p_1+p_1)))=(0,0)(1,1)(2,2)(3,2)\\&p_0(p_1+p_1+p_0(p_1+p_1+p_0(p_1+p_1+p_0(p_1+p_1))))=(0,0)(1,1)(2,2)(3,2)(4,2)\\&p_0(p_1+p_1+p_1)=(0,0)(1,1)(2,2)(3,3)\\&p_0(p_1+p_1+p_1+p_0(p_1+p_0(p_1+p_1+p_1)))=(0,0)(1,1)(2,2)(3,3)(1,1)(2,2)(3,3)\\&p_0(p_1+p_1+p_1+p_0(p_1+p_1+p_0(p_1+p_1+p_1)))=(0,0)(1,1)(2,2)(3,3)(2,2)(3,3)\\&p_0(p_1+p_1+p_1+p_0(p_1+p_1+p_1))=(0,0)(1,1)(2,2)(3,3)(3,3)\\&p_0(p_1+p_1+p_1+p_1)=(0,0)(1,1)(2,2)(3,3)(4,4)\\&p_0(p_1(p_0))=(0,0,0)(1,1,1)\end{align}

\begin{align}s\\&p_0(p_1(p_0)+p_0(p_0(p_1(p_0))))=(0,0,0)(1,1,1)(1,0,0)(2,1,1)\\&p_0(p_1(p_0)+p_0(p_1))=(0,0,0)(1,1,1)(1,1,0)\\&p_0(p_1(p_0)+p_0(p_1+p_0(p_1(p_0))))=(0,0,0)(1,1,1)(1,1,0)(2,2,1)\\&p_0(p_1(p_0)+p_0(p_1+p_1))=(0,0,0)(1,1,1)(1,1,0)(2,2,1)(2,2,0)\\&p_0(p_1(p_0)+p_0(p_1+p_1+p_0(p_1(p_0))))=(0,0,0)(1,1,1)(1,1,0)(2,2,1)(2,2,0)(3,3,1)\\&p_0(p_1(p_0)+p_0(p_1+p_1+p_1))=(0,0,0)(1,1,1)(1,1,0)(2,2,1)(2,2,0)(3,3,1)(3,3,0)\\&p_0(p_1(p_0)+p_0(p_1(p_0))))=(0,0,0)(1,1,1)(1,1,1)\\&p_0(p_1(p_0)+p_0(p_1(p_0)+p_0(p_1)))=(0,0,0)(1,1,1)(2,1,0)\\&p_0(p_1(p_0)+p_0(p_1(p_0)+p_0(p_1))+p_0(p_1+p_0(p_1(p_0)+p_0(p_1(p_0)+p_0(p_1)))))=(0,0,0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)\\&p_0(p_1(p_0)+p_0(p_1(p_0)+p_0(p_1))+p_0(p_1+p_1+p_0(p_1(p_0)+p_0(p_1(p_0)+p_0(p_1)))))=(0,0,0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,0)(3,3,1)(4,1,0)\\&p_0(p_1(p_0)+p_0(p_1(p_0)+p_0(p_1))+p_0(p_1(p_0)))=(0,0,0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(2,2,1)\\&p_0(p_1(p_0)+p_0(p_1(p_0)+p_0(p_1)+p_0(p_1)))=(0,0,0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,1,0)(3,1,0)\\&p_0(p_1(p_0)+p_0(p_1(p_0)+p_0(p_1+p_1)))=(0,0,0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,2,0)\\&p_0(p_1(p_0)+p_0(p_1(p_0)+p_0(p_1+p_1+p_1)))=(0,0,0)(1,1,1)(2,1,0)(1,1,0)(2,2,1)(3,2,0)(2,2,0)(3,3,1)(3,3,0)\\&p_0(p_1(p_0)+p_0(p_1(p_0)+p_0(p_1(p_0))))=(0,0,0)(1,1,1)(2,1,0)(1,1,1)\\&p_0(p_1(p_0)+p_0(p_1(p_0)+p_0(p_1(p_0)))+p_0(p_1(p_0)+p_0(p_1(p_0))))=(0,0,0)(1,1,1)(2,1,0)(1,1,1)(2,1,0)(1,1,1)\\&p_0(p_1(p_0)+p_0(p_1(p_0)+p_0(p_1(p_0))+p_0(p_1)))=(0,0,0)(1,1,1)(2,1,0)(1,1,1)(2,1,0)(2,1,0)\\&p_0(p_1(p_0)+p_0(p_1(p_0)+p_0(p_1(p_0))+p_0(p_1(p_0))))=(0,0,0)(1,1,1)(2,1,0)(2,1,0)(1,1,1)\\&p_0(p_1(p_0)+p_0(p_1(p_0)+p_0(p_1(p_0)+p_0(p_1(p_0)))))=(0,0,0)(1,1,1)(2,1,0)(3,1,0)(1,1,1)\\&p_0(p_1(p_0)+p_1)=(0,0,0)(1,1,1)(2,1,0)(3,2,0)\\&p_0(p_1(p_0)+p_1+p_0(p_1(p_0)+p_1))=(0,0,0)(1,1,1)(2,1,0)(3,2,0)(3,2,0)\\&p_0(p_1(p_0)+p_1+p_1)=(0,0,0)(1,1,1)(2,1,0)(3,2,0)(4,3,0)\\&p_0(p_1(p_0)+p_1(p_0))=(0,0,0)(1,1,1)(2,1,0)(3,2,1)\\&p_0(p_1(p_0)+p_1(p_0)+p_1)=(0,0,0)(1,1,1)(2,1,0)(3,2,1)(4,2,0)(5,3,0)\\&p_0(p_1(p_0)+p_1(p_0)+p_1(p_0))=(0,0,0)(1,1,1)(2,1,0)(3,2,1)(4,2,0)(5,3,1)\\&p_0(p_1(p_0+p_0))=(0,0,0)(1,1,1)(2,1,1)\\&p_0(p_1(p_0+p_0)+p_0(p_1(p_0+p_0)))=(0,0,0)(1,1,1)(2,1,1)(1,1,1)(2,1,1)\\&p_0(p_1(p_0+p_0)+p_0(p_1(p_0+p_0)+p_0(p_1)))=(0,0,0)(1,1,1)(2,1,1)(2,1,0)\\&p_0(p_1(p_0+p_0)+p_0(p_1(p_0+p_0)+p_0(p_1(p_0))))=(0,0,0)(1,1,1)(2,1,1)(2,1,0)(1,1,1)\\&p_0(p_1(p_0+p_0)+p_0(p_1(p_0+p_0)+p_0(p_1(p_0+p_0))))=(0,0,0)(1,1,1)(2,1,1)(2,1,0)(1,1,1)(2,1,1)\\&p_0(p_1(p_0+p_0)+p_1)=(0,0,0)(1,1,1)(2,1,1)(2,1,0)(3,2,0)\\&p_0(p_1(p_0+p_0)+p_1(p_0+p_0))=(0,0,0)(1,1,1)(2,1,1)(2,1,0)(3,2,1)(4,2,1)\\&p_0(p_1(p_0+p_0+p_0))=(0,0,0)(1,1,1)(2,1,1)(2,1,1)\\&p_0(p_1(p_0(p_0)))=(0,0,0)(1,1,1)(2,1,1)(3,0,0)\\&p_0(p_1(p_0(p_0(p_1))))=(0,0,0)(1,1,1)(2,1,1)(3,0,0)(4,1,0)\\&p_0(p_1(p_0(p_0(p_1(p_0)))))=(0,0,0)(1,1,1)(2,1,1)(3,0,0)(4,1,1)\\&p_0(p_1(p_0(p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)\\&p_0(p_1(p_0(p_1))+p_0(p_1+p_0(p_1(p_0(p_1)))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,0)(2,2,1)(3,2,1)(4,1,0)\\&p_0(p_1(p_0(p_1))+p_0(p_1(p_0)))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,0)(2,2,1)(3,2,1)(4,1,0)(2,2,1)\\&p_0(p_1(p_0(p_1))+p_0(p_1(p_0(p_1))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,0)(2,2,1)(3,2,1)(4,1,0)(2,2,1)(3,2,1)(4,1,0)\\&p_0(p_1(p_0(p_1))+p_1)=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,0)(2,2,1)(3,2,1)(4,1,0)(3,2,0)(4,3,0)\\&p_0(p_1(p_0(p_1))+p_1(p_0(p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,0)(2,2,1)(3,2,1)(4,1,0)(3,2,0)(4,3,1)(5,3,1)(6,1,0)\\&p_0(p_1(p_0(p_1)+p_0))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,0)(2,2,1)(3,2,1)(4,1,0)(3,2,1)\\&p_0(p_1(p_0(p_1)+p_0(p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,0)(2,2,1)(3,2,1)(4,1,0)(3,2,1)(4,1,0)\\&p_0(p_1(p_0(p_1+p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,0)(2,2,1)(3,2,1)(4,2,0)\\&p_0(p_1(p_0(p_1(p_0))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,1)\\&p_0(p_1(p_0(p_1(p_0+p_0))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,1)(2,1,1)\\&p_0(p_1(p_0(p_1(p_0(p_1)))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,1)(2,1,1)(3,1,0)\\&p_0(p_1(p_1))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)\end{align}

\begin{align}s\\&p_0(p_1(p_1)+p_0(p_1))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)(1,1,0)\\&p_0(p_1(p_1)+p_0(p_1(p_0)))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)(1,1,1)\\&p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1)))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1)))+p_0))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)(2,0,0)\\&p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1)))+p_0(p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,0)\\&p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1)))+p_0(p_1(p_1))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1)))+p_1))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,0)(3,2,0)\\&p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1)))+p_1(p_0)))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,0)(3,2,1)\\&p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1)))+p_1(p_0(p_1(p_1)))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,0)(3,2,1)(4,2,1)(5,1,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1))+p_0)))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,0)(3,2,1)(4,2,1)(5,1,0)(4,2,1)\\&p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1))+p_0(p_1(p_1)))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,0)(3,2,1)(4,2,1)(5,1,0)(4,2,1)(5,1,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1)+p_0))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,0)(3,2,1)(4,2,1)(5,1,0)(5,0,0)\\&p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1))))))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,0)(3,2,1)(4,2,1)(5,1,0)(5,1,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1)))+p_0(p_1(p_0(p_1(p_1)))+p_1))))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,0)(3,2,1)(4,2,1)(5,1,0)(6,2,0)\\&p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1)))+p_1))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,0)(3,2,1)(4,2,1)(5,2,0)(3,2,1)(4,2,1)(5,2,0)\\&p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1)))+p_1(p_0(p_1(p_1)))))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,0)(3,2,1)(4,2,1)(5,2,0)(3,2,1)(4,2,1)(5,1,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_1)+p_0(p_1(p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,0)(3,2,1)(4,2,1)(5,2,0)(4,0,0)\\&p_0(p_1(p_1)+p_0(p_1(p_1))+p_0(p_1(p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,0)(3,2,1)(4,2,1)(5,2,0)(4,2,0)(5,3,1)(6,3,1)(7,3,0)(6,0,0)\\&p_0(p_1(p_1)+p_0(p_1(p_1)+p_0))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,1)\\&p_0(p_1(p_1)+p_0(p_1(p_1)+p_0+p_0))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,1)(2,1,1)\\&p_0(p_1(p_1)+p_0(p_1(p_1)+p_0(p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,1)(3,1,0)\\&p_0(p_1(p_1)+p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1))))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,1)(3,1,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_1)+p_0(p_1(p_1)+p_0(p_1(p_0(p_1(p_1)+p_0(p_1(p_1)+p_0))))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,1)(3,1,0)(1,1,1)(2,1,1)(3,1,0)(2,1,1)\\&p_0(p_1(p_1)+p_0(p_1(p_1)+p_0(p_1(p_1))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_1)+p_0(p_1(p_1)+p_0(p_1(p_1))+p_0))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,1)(3,1,0)(2,1,1)\\&p_0(p_1(p_1)+p_0(p_1(p_1)+p_0(p_1(p_1))+p_0(p_1(p_1))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,1,1)(3,1,0)(2,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_1)+p_0(p_1(p_1)+p_0(p_1(p_1)+p_0)))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(3,0,0)\\&p_0(p_1(p_1)+p_0(p_1(p_1)+p_0(p_1(p_1)+p_0(p_1(p_1)))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(3,1,0)(2,0,0)\\&p_0(p_1(p_1)+p_1)=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,0)\\&p_0(p_1(p_1)+p_1+p_1)=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,0)(5,3,0)\\&p_0(p_1(p_1)+p_1(p_0))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,1)\\&p_0(p_1(p_1)+p_1(p_0(p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,1,0)\\&p_0(p_1(p_1)+p_1(p_0(p_1(p_1))))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,1,0)(2,0,0)\\&p_0(p_1(p_1)+p_1(p_0(p_1(p_1))+p_1))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,0)\\&p_0(p_1(p_1)+p_1(p_1))=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,0)(5,0,0)\\&p_0(p_1(p_1+p_0))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)\\&p_0(p_1(p_1+p_0+p_0))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)\\&p_0(p_1(p_1+p_0(p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,0)\\&p_0(p_1(p_1+p_0(p_1(p_0(p_1(p_1))))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_1+p_0(p_1(p_0(p_1(p_1+p_0))))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,0)(1,1,1)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)\\&p_0(p_1(p_1+p_0(p_1(p_0(p_1(p_1+p_0(p_1)))))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,0)(1,1,1)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1,0)\\&p_0(p_1(p_1+p_0(p_1(p_1))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,0)(1,1,1)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1,0)(2,0,0)\\&p_0(p_1(p_1+p_0(p_1(p_1+p_0))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,0)(1,1,1)(2,1,1)(3,1,1)\\&p_0(p_1(p_1+p_1))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_1+p_1)+p_0(p_1(p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,0)(2,1,0)(3,2,1)(4,2,1)(5,2,0)(4,0,0)\\&p_0(p_1(p_1+p_1)+p_0(p_1(p_1+p_0)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,0)(2,1,0)(3,2,1)(4,2,1)(5,2,1)\\&p_0(p_1(p_1+p_1)+p_0(p_1(p_1+p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,0)(2,1,0)(3,2,1)(4,2,1)(5,2,1)(4,2,1)(5,2,0)(4,0,0)\\&p_0(p_1(p_1+p_1)+p_0(p_1(p_1+p_1)+p_0))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,0)(2,1,1)\\&p_0(p_1(p_1+p_1)+p_1)=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,0)(4,2,0)\\&p_0(p_1(p_1+p_1)+p_1(p_1))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,0)(5,0,0)\\&p_0(p_1(p_1+p_1)+p_1(p_1+p_0))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)\\&p_0(p_1(p_1+p_1)+p_1(p_1+p_1))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,2,0)(5,0,0)\\&p_0(p_1(p_1+p_1+p_0))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,1)\\&p_0(p_1(p_1+p_1+p_1))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_1(p_0)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,0,0)\\&p_0(p_1(p_1(p_0(p_1))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)\\&p_0(p_1(p_1(p_0(p_1(p_1(p_0))))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(1,1,1)(2,1,1)(3,1,1)(3,0,0)\\&p_0(p_1(p_1(p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,0,0)\end{align}

\begin{align}s\\&p_0(p_1(p_1(p_1))+p_0(p_1))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,0,0)(1,1,0)\\&p_0(p_1(p_1(p_1))+p_0(p_1(p_0(p_1(p_1(p_1))))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)(6,1,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_1(p_1))+p_0(p_1(p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,0,0)(1,1,1)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)(6,1,0)(2,0,0)\\&p_0(p_1(p_1(p_1))+p_0(p_1(p_1+p_0)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,0,0)(1,1,1)(2,1,1)(3,1,1)\\&p_0(p_1(p_1(p_1))+p_0(p_1(p_1(p_0))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,0,0)(1,1,1)(2,1,1)(3,1,1)(3,0,0)\\&p_0(p_1(p_1(p_1))+p_0(p_1(p_1(p_0(p_1)))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)\\&p_0(p_1(p_1(p_1))+p_0(p_1(p_1(p_0(p_1(p_1(p_1)))))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_1(p_1))+p_0(p_1(p_1(p_0(p_1(p_1(p_1)))))+p_1))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,1,0)(3,2,0)\\&p_0(p_1(p_1(p_1))+p_0(p_1(p_1(p_1))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,1,0)(3,2,1)(4,2,1)(5,2,1)(5,2,0)(4,0,0)\\&p_0(p_1(p_1(p_1))+p_0(p_1(p_1(p_1))+p_0))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,1,1)\\&p_0(p_1(p_1(p_1))+p_0(p_1(p_1(p_1))+p_0(p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,1,1)(3,1,0)\\&p_0(p_1(p_1(p_1))+p_0(p_1(p_1(p_1))+p_0(p_1(p_1(p_1)))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_1(p_1))+p_1)=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,1,1)(3,1,0)(4,2,0)\\&p_0(p_1(p_1(p_1))+p_1(p_1))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,0)(5,0,0)\\&p_0(p_1(p_1(p_1))+p_1(p_1(p_0)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)(6,0,0)\\&p_0(p_1(p_1(p_1))+p_1(p_1(p_0(p_1(p_1(p_1))))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)(6,1,0)(2,0,0)\\&p_0(p_1(p_1(p_1))+p_1(p_1(p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)(6,2,0)(5,0,0)\\&p_0(p_1(p_1(p_1)+p_0))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,1,1)(3,1,1)\\&p_0(p_1(p_1(p_1)+p_1))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,1,1)(3,1,1)(2,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_1(p_1)+p_1(p_0)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,1,1)(3,1,1)(3,0,0)\\&p_0(p_1(p_1(p_1)+p_1(p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(2,1,1)(3,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_1(p_1+p_0)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(3,0,0)\\&p_0(p_1(p_1(p_1+p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(3,1,0)(2,0,0)\\&p_0(p_1(p_1(p_1(p_1))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,1,0)(2,0,0)\\&p_0(p_1(p_2))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)\end{align}

\begin{align}s\\&p_0(p_1(p_2)+p_0(p_1(p_2)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)\\&p_0(p_1(p_2)+p_1)=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)(2,1,0)(3,2,0)\\&p_0(p_1(p_2)+p_1(p_1))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)(2,1,0)(3,2,1)(4,2,1)(5,2,0)(4,0,0)\\&p_0(p_1(p_2)+p_1(p_2))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)(2,1,0)(3,2,1)(4,2,1)(5,2,1)(5,2,0)(6,3,0)\\&p_0(p_1(p_2+p_0))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)(2,1,1)\\&p_0(p_1(p_2+p_0(p_1(p_2))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)(2,1,1)(3,1,0)(1,1,1)(2,1,1)(3,1,0)(4,2,0)\\&p_0(p_1(p_2+p_1))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)(2,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_2+p_1+p_0))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)(2,1,1)(3,1,1)\\&p_0(p_1(p_2+p_1(p_1))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)(2,1,1)(3,1,1)(3,1,0)(2,0,0)\\&p_0(p_1(p_2+p_1(p_1(p_2)))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)(2,1,1)(3,1,1)(3,1,0)(4,2,0)\\&p_0(p_1(p_2+p_1(p_1(p_2)+p_1))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)(3,1,0)(2,0,0)\\&p_0(p_1(p_2+p_1(p_1(p_2)+p_1(p_1(p_2))))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)(3,1,0)(4,2,0)\\&p_0(p_1(p_2+p_1(p_2)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)(4,2,0)\\&p_0(p_1(p_2+p_1(p_2+p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)(5,1,0)(2,0,0)\\&p_0(p_1(p_2+p_1(p_2+p_1(p_1(p_2)))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)(5,1,0)(6,2,0)\\&p_0(p_1(p_2+p_1(p_2+p_1(p_2))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)(5,2,0)\\&p_0(p_1(p_2+p_2))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)(5,3,0)\\&p_0(p_1(p_2(p_0)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,1)\\&p_0(p_1(p_2(p_0(p_1(p_2)))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,1)(5,2,1)(6,1,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,0)\\&p_0(p_1(p_2(p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,1)(5,2,1)(6,1,0)(2,0,0)\\&p_0(p_1(p_2(p_1(p_1(p_2)))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,1)(5,2,1)(6,1,0)(7,2,0)\\&p_0(p_1(p_2(p_1(p_2))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,0)\\&p_0(p_1(p_2(p_2)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,0)(5,0,0)\\&p_0(p_1(p_2(p_2+p_0)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)\\&p_0(p_1(p_2(p_2+p_1)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1,0)(2,0,0)\\&p_0(p_1(p_2(p_2+p_2)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,2,0)(5,0,0)\\&p_0(p_1(p_2(p_2(p_0))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)(6,0,0)\\&p_0(p_1(p_2(p_2(p_2))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)(6,2,0)(5,0,0)\\&p_0(p_1(p_2(p_3)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)(6,2,0)(7,3,0)\\&p_0(p_1(p_2(p_3+p_3)))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)(6,2,0)(7,3,0)(8,4,0)\\&p_0(p_1(p_2(p_3(p_3))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)(6,2,0)(7,3,1)(8,3,1)(9,3,0)(8,0,0)\\&p_0(p_1(p_2(p_3(p_4))))=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,0)(4,2,1)(5,2,1)(6,2,1)(6,2,0)(7,3,1)(8,3,1)(9,3,1)(9,3,0)(10,4,0)\\&limit=(0,0,0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)\end{align}