\( \newcommand\a{\alpha} \newcommand\z{\zeta} \newcommand\x{\xi} \newcommand\n{\nu} \newcommand\m{\mu} \newcommand\O{\Omega} \newcommand\v{\varphi} \newcommand\p[2]{\psi_{#1}(#2)} \newcommand\b[2]{b_{#1}(#2)} \)
概要[]
Rpakrさんが作成した拡張ブーフホルツのψ関数にφ関数を加えただけの関数を解析してみました。(削除依頼でたら削除します)
定義(ユーザーブログ:Rpakr/拡張ブーフホルツのψ関数にφ関数を加えただけの関数より引用)[]
(この定義は、拡張ブーフホルツのψ関数の定義をコピペして \(\{\varphi(\x,\z)\mid\x\in C_\n^n(\a)\land \z\in C_\n^n(\a)\}\) を付け加えただけである。)
集合 \(C_\m\) と関数 \(\psi_\m\) は超限再帰によって以下のように定義される。
\(\begin{align*} C_\n^0(\a)=&\O_\n\\ C_\n^{n+1}(\a)=&\{\x+\z\mid\x\in C_\n^n(\a)\land \z\in C_\n^n(\a)\}\cup\\ &\{\varphi(\x,\z)\mid\x\in C_\n^n(\a)\land \z\in C_\n^n(\a)\}\cup\\ &\{\psi_\m(\x)\mid \m\in C_\n^n(\a)\land\x\in C_\n^n(\a)\cap \a\}\\ C_\n(\a)=&\bigcup_{n < \omega} C_\n^n (\a)\\ \psi_\n(\a)=&\min\{\x \mid \x \notin C_\n(\a)\} \end{align*}\)
ここで \(\O_0:=1,\O_\x:=\aleph_\x(\x>0)\) とする。
解析[]
\(\v(0,\x)\)を\(\v(\x)\)と略記する。
有限多変数ヴェブレン関数との比較[]
| \( \p{0}{0} \) | \( \v(1,0,0) \) |
| \( \p{0}{1} \) | \( \v(1,0,1) \) |
| \( \p{0}{\p{0}{0}} \) | \( \v(1,0,\v(1,0,0)) \) |
| \( \p{0}{\p{1}{0}} \) | \( \v(1,1,0) \) |
| \( \p{0}{\p{1}{0}+1} \) | \( \v(1,0,\v(1,1,0)+1) \) |
| \( \p{0}{\p{1}{0}+2} \) | \( \v(1,0,\v(1,1,0)+2) \) |
| \( \p{0}{\p{1}{0}+\p{0}{\p{1}{0}}} \) | \( \v(1,0,\v(1,1,0)+\v(1,1,0)) \) |
| \( \p{0}{\p{1}{0}+\v(\p{0}{\p{1}{0}})} \) | \( \v(1,0,\v(\v(1,1,0))) \) |
| \( \p{0}{\p{1}{0}+\v(\p{0}{\p{1}{0}}+1)} \) | \( \v(1,0,\v(\v(1,1,0)+1)) \) |
| \( \p{0}{\p{1}{0}+\v(\v(\p{0}{\p{1}{0}}+1))} \) | \( \v(1,0,\v(\v(\v(1,1,0)+1))) \) |
| \( \p{0}{\p{1}{0}+\v(1,\p{0}{\p{1}{0}}+1)} \) | \( \v(1,0,\v(1,\v(1,1,0)+1)) \) |
| \( \p{0}{\p{1}{0}+\v(\v(1),\p{0}{\p{1}{0}}+1)} \) | \( \v(1,0,\v(\v(1),\v(1,1,0)+1)) \) |
| \( \p{0}{\p{1}{0}+\p{0}{\p{1}{0}+1}} \) | \( \v(1,0,\v(1,0,\v(1,1,0)+1)) \) |
| \( \p{0}{\p{1}{0}+\p{1}{0}} \) | \( \v(1,1,1) \) |
| \( \p{0}{\p{1}{0}+\p{1}{0}+1} \) | \( \v(1,0,\v(1,1,1)+1) \) |
| \( \p{0}{\p{1}{0}+\p{1}{0}+\p{1}{0}} \) | \( \v(1,1,2) \) |
| \( \p{0}{\v(\p{1}{0}+1)} \) | \( \v(1,1,\v(1)) \) |
| \( \p{0}{\v(\p{1}{0}+\p{0}{\p{1}{0}})} \) | \( \v(1,1,\v(1,1,0)) \) |
| \( \p{0}{\v(\p{1}{0}+\p{1}{0})} \) | \( \v(1,2,0) \) |
| \( \p{0}{\v(\p{1}{0}+\p{1}{0})+\p{1}{0}} \) | \( \v(1,1,\v(1,2,0)+1) \) |
| \( \p{0}{\v(\p{1}{0}+\p{1}{0})+\v(\p{1}{0}+\p{0}{\v(\p{1}{0}+\p{1}{0})})} \) | \( \v(1,1,\v(1,2,0)+\v(1,2,0)) \) |
| \( \p{0}{\v(\p{1}{0}+\p{1}{0})+\v(\p{1}{0}+\p{1}{0})} \) | \( \v(1,2,1) \) |
| \( \p{0}{\v(\p{1}{0}+\p{1}{0}+1)} \) | \( \v(1,2,\v(1)) \) |
| \( \p{0}{\v(\p{1}{0}+\p{1}{0}+\p{0}{\v(\p{1}{0}+\p{1}{0})})} \) | \( \v(1,2,\v(1,2,0)) \) |
| \( \p{0}{\v(\p{1}{0}+\p{1}{0}+\p{1}{0})} \) | \( \v(1,3,0) \) |
| \( \p{0}{\v(\v(\p{1}{0}+1))} \) | \( \v(1,\v(1),0) \) |
| \( \p{0}{\v(\v(\p{1}{0}+\p{0}{0}))} \) | \( \v(1,\v(1,0,0),0) \) |
| \( \p{0}{\v(\v(\p{1}{0}+\p{1}{0}))} \) | \( \v(2,0,0) \) |
| \( \p{0}{\v(\v(\p{1}{0}+\p{1}{0}))+\v(\v(\p{1}{0}+\p{1}{0}))} \) | \( \v(2,0,1) \) |
| \( \p{0}{\v(\v(\p{1}{0}+\p{1}{0})+1)} \) | \( \v(2,0,\v(1)) \) |
| \( \p{0}{\v(\v(\p{1}{0}+\p{1}{0})+\p{1}{0})} \) | \( \v(2,1,0) \) |
| \( \p{0}{\v(\v(\p{1}{0}+\p{1}{0})+\v(\p{1}{0}+\p{1}{0}))} \) | \( \v(3,0,0) \) |
| \( \p{0}{\v(\v(\p{1}{0}+\p{1}{0}+1))} \) | \( \v(\v(1),0,0) \) |
| \( \p{0}{\v(\v(\p{1}{0}+\p{1}{0}+\p{1}{0}))} \) | \( \v(1,0,0,0) \) |
| \( \p{0}{\v(\v(\v(\p{1}{0}+1)))} \) | \( \v(1,0,0,\dots,0,0) \) |
拡張ブーフホルツの\(\psi\)関数との比較[]
以下拡張ブーフホルツの\(\psi\)関数を\(\b{\n}{\a}\)と表記する。
| \( \p{0}{\v(\v(\v(\p{1}{0}+1)))} \) | \( \b{0}{\b{1}{\b{1}{\b{1}{1}}}} \) |
| \( \p{0}{\v(\v(\v(\p{1}{0}+\p{1}{0})))} \) | \( \b{0}{\b{1}{\b{1}{\b{1}{\b{1}{0}}}}} \) |
| \( \p{0}{\v(1,\p{1}{0}+1)} \) | \( \b{0}{\b{1}{\b{2}{0}}}=\b{0}{\b{2}{0}} \) |
| \( \p{0}{\v(1,\p{1}{0}+1)+\v(1,\p{1}{0}+1)} \) | \( \b{0}{\b{2}{0}+\b{1}{\b{2}{0}}} \) |
| \( \p{0}{\v(\v(1,\p{1}{0}+1)+1)} \) | \( \b{0}{\b{2}{0}+\b{1}{\b{2}{0}+1}} \) |
| \( \p{0}{\v(\v(\v(1,\p{1}{0}+1)+1))} \) | \( \b{0}{\b{2}{0}+\b{1}{\b{2}{0}+\b{1}{\b{2}{0}+1}}} \) |
| \( \p{0}{\v(1,\p{1}{0}+2)} \) | \( \b{0}{\b{2}{0}+\b{2}{0}} \) |
| \( \p{0}{\v(1,\p{1}{0}+3)} \) | \( \b{0}{\b{2}{0}+\b{2}{0}+\b{2}{0}} \) |
| \( \p{0}{\v(1,\p{1}{0}+\v(1))} \) | \( \b{0}{\b{2}{1}} \) |
| \( \p{0}{\v(1,\v(1,\p{1}{0}+1))} \) | \( \b{0}{\b{2}{\b{1}{\b{2}{0}}}} \) |
| \( \p{0}{\v(2,\p{1}{0}+1)} \) | \( \b{0}{\b{2}{\b{1}{\b{2}{\b{2}{0}}}}}=\b{0}{\b{2}{\b{2}{0}}} \) |
| \( \p{0}{\v(1,\v(2,\p{1}{0}+1)+1)} \) | \( \b{0}{\b{2}{\b{2}{0}}+\b{2}{0}} \) |
| \( \p{0}{\v(1,\v(2,\p{1}{0}+1)+\v(2,\p{1}{0}+1))} \) | \( \b{0}{\b{2}{\b{2}{0}}+\b{2}{\b{1}{\b{2}{\b{2}{0}}}}} \) |
| \( \p{0}{\v(2,\p{1}{0}+2)} \) | \( \b{0}{\b{2}{\b{2}{0}}+\b{2}{\b{2}{0}}} \) |
| \( \p{0}{\v(2,\p{1}{0}+\v(1))} \) | \( \b{0}{\b{2}{\b{2}{0}+1}} \) |
| \( \p{0}{\v(3,\p{1}{0}+1)} \) | \( \b{0}{\b{2}{\b{2}{0}+\b{2}{0}}} \) |
| \( \p{0}{\v(\v(1),\p{1}{0}+1)} \) | \( \b{0}{\b{2}{\b{2}{1}}} \) |
| \( \p{0}{\v(\p{0}{\v(\v(1),\p{1}{0}+1)},\p{1}{0}+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{0}{\b{2}{\b{2}{1}}}}}} \) |
| \( \p{0}{\v(\p{1}{0},1)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}}} \) |
| \( \p{0}{\v(1,\v(\p{1}{0},1)+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}}+\b{2}{0}} \) |
| \( \p{0}{\v(2,\v(\p{1}{0},1)+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}}+\b{2}{\b{2}{0}}} \) |
| \( \p{0}{\v(3,\v(\p{1}{0},1)+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}}+\b{2}{\b{2}{0}+\b{2}{0}}} \) |
| \( \p{0}{\v(\v(1),\v(\p{1}{0},1)+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}}+\b{2}{\b{2}{1}}} \) |
| \( \p{0}{\v(\p{0}{\v(2,\v(\p{1}{0},1)+1)},\v(\p{1}{0},1)+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}}+\b{2}{\b{2}{\b{0}{\b{2}{\b{2}{\b{1}{0}}}+\b{2}{\b{2}{0}}}}}} \) |
| \( \p{0}{\v(\p{1}{0},2)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}}+\b{2}{\b{2}{\b{1}{0}}}} \) |
| \( \p{0}{\v(\p{1}{0},3)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}}+\b{2}{\b{2}{\b{1}{0}}}+\b{2}{\b{2}{\b{1}{0}}}} \) |
| \( \p{0}{\v(\p{1}{0},\v(1))} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}+1}} \) |
| \( \p{0}{\v(\p{1}{0},\p{1}{0})} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}+\b{1}{0}}} \) |
| \( \p{0}{\v(\p{1}{0},\v(\p{1}{0},1))} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}+\b{1}{\b{2}{\b{2}{\b{1}{0}}}}}} \) |
| \( \p{0}{\v(\p{1}{0}+1,0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}+\b{2}{0}}} \) |
| \( \p{0}{\v(\p{1}{0}+1,1)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}+\b{2}{0}}+\b{2}{\b{2}{\b{1}{0}}+\b{2}{0}}} \) |
| \( \p{0}{\v(\p{1}{0}+1,\v(1))} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}+\b{2}{0}+1}} \) |
| \( \p{0}{\v(\p{1}{0}+1,\v(\p{1}{0}+1,0))} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}+\b{2}{0}+\b{1}{\b{2}{\b{2}{\b{1}{0}}+\b{2}{0}}}}} \) |
| \( \p{0}{\v(\p{1}{0}+2,0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}+\b{2}{0}+\b{2}{0}}} \) |
| \( \p{0}{\v(\p{1}{0}+\v(1),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}+\b{2}{1}}} \) |
| \( \p{0}{\v(\p{1}{0}+\p{1}{0},0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}+\b{2}{\b{1}{0}}}} \) |
| \( \p{0}{\v(\p{1}{0}+\p{1}{0}+1,0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}+\b{2}{\b{1}{0}}+1}} \) |
| \( \p{0}{\v(\p{1}{0}+\p{1}{0}+\p{1}{0},0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}}+\b{2}{\b{1}{0}}+\b{2}{\b{1}{0}}}} \) |
| \( \p{0}{\v(\v(\p{1}{0}+1),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}+1}}} \) |
| \( \p{0}{\v(\v(\p{1}{0}+2),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}+2}}} \) |
| \( \p{0}{\v(\v(\p{1}{0}+\p{1}{0}),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{0}+\p{1}{0}}}} \) |
| \( \p{0}{\v(\v(\v(\p{1}{0}+1)),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{1}}}} \) |
| \( \p{0}{\v(\v(\v(\p{1}{0}+2)),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{2}}}} \) |
| \( \p{0}{\v(\v(\v(\p{1}{0}+\p{1}{0})),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{1}{0}}}}} \) |
| \( \p{0}{\v(\v(\v(\v(\p{1}{0}+1))),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{1}{1}}}}} \) |
| \( \p{0}{\v(\v(1,\p{1}{0}+1),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{0}}}}} \) |
| \( \p{0}{\v(\v(1,\p{1}{0}+1)+1,0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{0}}}+\b{2}{0}}} \) |
| \( \p{0}{\v(\v(1,\p{1}{0}+1)+\v(1,\p{1}{0}+1),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{0}}}+\b{2}{\b{1}{\b{2}{0}}}}} \) |
| \( \p{0}{\v(\v(\v(1,\p{1}{0}+1)+1),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{0}}+1}}} \) |
| \( \p{0}{\v(\v(\v(1,\p{1}{0}+1)+\v(1,\p{1}{0}+1)),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{0}}+\b{1}{\b{2}{0}}}}} \) |
| \( \p{0}{\v(\v(\v(\v(1,\p{1}{0}+1)+1)),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{0}+1}}}} \) |
| \( \p{0}{\v(\v(\v(\v(1,\p{1}{0}+1)+\v(1,\p{1}{0}+1)),0)} \) | \(\b{0}{\b{2}{\b{2}{\b{1}{\b{2}{0}+\b{1}{\b{2}{0}}}}}}\) |
| \( \p{0}{\v(\v(\v(\v(\v(1,\p{1}{0}+1)+1))),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{0}+\b{1}{\b{2}{0}+1}}}}} \) |
| \( \p{0}{\v(\v(1,\p{1}{0}+2),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{0}+\b{2}{0}}}}} \) |
| \( \p{0}{\v(\v(1,\p{1}{0}+3),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{0}+\b{2}{0}+\b{2}{0}}}}} \) |
| \( \p{0}{\v(\v(1,\p{1}{0}+\v(1)),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{1}}}}} \) |
| \( \p{0}{\v(\v(1,\p{1}{0}+\p{1}{0}),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{1}{0}}}}}} \) |
| \( \p{0}{\v(\v(1,\v(\p{1}{0}+1)),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{1}{1}}}}}} \) |
| \( \p{0}{\v(\v(1,\v(\p{1}{0}+\p{1}{0})),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{1}{\b{1}{0}}}}}}} \) |
| \( \p{0}{\v(\v(1,\v(\v(\p{1}{0}+1))),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{1}{\b{1}{1}}}}}}} \) |
| \( \p{0}{\v(\v(1,\v(1,\p{1}{0}+1)),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{1}{\b{2}{0}}}}}}} \) |
| \( \p{0}{\v(\v(1,\v(1,\p{1}{0}+\v(1,\p{1}{0}+1))),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{1}{\b{2}{\b{1}{\b{2}{0}}}}}}}}} \) |
| \( \p{0}{\v(\v(2,\p{1}{0}+1),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{0}}}}}} \) |
| \( \p{0}{\v(\v(1,\v(2,\p{1}{0}+1)+1),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{0}}+\b{2}{0}}}}} \) |
| \( \p{0}{\v(\v(2,\p{1}{0}+2),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{0}}+\b{2}{\b{2}{0}}}}}} \) |
| \( \p{0}{\v(\v(2,\p{1}{0}+\v(1)),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{0}+1}}}}} \) |
| \( \p{0}{\v(\v(3,\p{1}{0}+1),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{0}+\b{2}{0}}}}}} \) |
| \( \p{0}{\v(\v(4,\p{1}{0}+1),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{0}+\b{2}{0}+\b{2}{0}}}}}} \) |
| \( \p{0}{\v(\v(\v(1),\p{1}{0}+1),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{1}}}}}} \) |
| \( \p{0}{\v(\v(\p{1}{0},1),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{\b{1}{0}}}}}}} \) |
| \( \p{0}{\v(\v(\p{1}{0}+1,0),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{\b{1}{0}}+\b{2}{0}}}}}} \) |
| \( \p{0}{\v(\v(\p{1}{0}+\p{1}{0},0),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{\b{1}{0}}+\b{2}{\b{1}{0}}}}}}} \) |
| \( \p{0}{\v(\v(\v(\p{1}{0}+1),0),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{\b{1}{0}+1}}}}}} \) |
| \( \p{0}{\v(\v(\v(1,\p{1}{0}+1),0),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{\b{1}{\b{2}{0}}}}}}}} \) |
| \( \p{0}{\v(\v(\v(\v(2,\p{1}{0}+1),0),0),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{0}}}}}}}}}}}} \) |
| \( \p{0}{\p{1}{1}} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}} \) |
| \( \v(\p{0}{\p{1}{1}}+1) \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+1} \) |
| \( \v(\p{0}{\p{1}{1}}+\p{0}{\p{1}{1}}) \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{0}{\b{2}{\b{2}{\b{2}{0}}}}} \) |
| \( \v(\v(\p{0}{\p{1}{1}}+1)) \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{0}{\b{2}{\b{2}{\b{2}{0}}}+1}} \) |
| \( \v(1,\p{0}{\p{1}{1}}+1) \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{0}} \) |
| \( \v(\v(1,\p{0}{\p{1}{1}}+1)+1) \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{0}+1} \) |
| \( \v(1,\p{0}{\p{1}{1}}+2) \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{0}+\b{1}{0}} \) |
| \( \v(1,\p{0}{\p{1}{1}}+\v(1)) \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{1}} \) |
| \( \v(1,\v(1,\p{0}{\p{1}{1}}+1)) \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{0}}}} \) |
| \( \v(2,\p{0}{\p{1}{1}}+1) \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{0}}} \) |
| \( \v(1,\v(2,\p{0}{\p{1}{1}}+1)+1) \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{0}}+\b{1}{0}} \) |
| \( \v(2,\p{0}{\p{1}{1}}+2) \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{0}}+\b{1}{\b{1}{0}}} \) |
| \( \v(2,\p{0}{\p{1}{1}}+\v(1)) \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{0}+1}} \) |
| \( \v(3,\p{0}{\p{1}{1}}+1) \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{0}+\b{1}{0}}} \) |
| \( \v(\v(1),\p{0}{\p{1}{1}}+1) \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{1}}} \) |
| \( \v(\v(\v(1),\p{0}{\p{1}{1}}+1),\p{0}{\p{1}{1}}+1) \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{\b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{1}}}}}} \) |
| \( \p{0}{\p{1}{1}+1} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{\b{1}{0}}}} \) |
| \( \p{0}{\p{1}{1}+2} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{\b{1}{0}}}+\b{1}{\b{1}{\b{1}{0}}}} \) |
| \( \p{0}{\p{1}{1}+\v(1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{\b{1}{0}}+1}} \) |
| \( \p{0}{\p{1}{1}+\p{1}{0}} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{\b{1}{0}}+\b{1}{0}}} \) |
| \( \p{0}{\p{1}{1}+\p{1}{0}+\p{1}{0}} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{\b{1}{0}}+\b{1}{0}+\b{1}{0}}} \) |
| \( \p{0}{\p{1}{1}+\v(\p{1}{0}+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{\b{1}{0}}+\b{1}{1}}} \) |
| \( \p{0}{\p{1}{1}+\v(\p{1}{0}+\p{1}{0})} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{\b{1}{0}}+\b{1}{\b{1}{0}}}} \) |
| \( \p{0}{\p{1}{1}+\v(\p{1}{0}+\p{1}{0}+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{\b{1}{0}}+\b{1}{\b{1}{0}}+\b{1}{1}}} \) |
| \( \p{0}{\p{1}{1}+\v(\p{1}{0}+\p{1}{0}+\p{1}{0})} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{\b{1}{0}}+\b{1}{\b{1}{0}}+\b{1}{\b{1}{0}}}} \) |
| \( \p{0}{\p{1}{1}+\v(\v(\p{1}{0}+1))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{\b{1}{0}+1}}} \) |
| \( \p{0}{\p{1}{1}+\v(\v(\p{1}{0}+\p{1}{0}))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{\b{1}{0}+\b{1}{0}}}} \) |
| \( \p{0}{\p{1}{1}+\v(\v(\v(\p{1}{0}+1)))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{\b{1}{1}}}} \) |
| \( \p{0}{\p{1}{1}+\v(\v(\v(\p{1}{0}+\p{1}{0})))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{1}{\b{1}{\b{1}{0}}}}} \) |
| \( \p{0}{\p{1}{1}+\v(1,\p{1}{0}+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{0}}} \) |
| \( \p{0}{\p{1}{1}+\v(1,\p{1}{0}+1)+1} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{0}}+\b{1}{\b{1}{\b{1}{0}}}} \) |
| \( \p{0}{\p{1}{1}+\v(1,\p{1}{0}+1)+\v(1,\p{1}{0}+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{0}}+\b{1}{\b{2}{0}}} \) |
| \( \p{0}{\p{1}{1}+\v(\v(1,\p{1}{0}+1)+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{0}+1}} \) |
| \( \p{0}{\p{1}{1}+\v(\v(1,\p{1}{0}+1)+\v(1,\p{1}{0}+1))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{0}+\b{1}{\b{2}{0}}}} \) |
| \( \p{0}{\p{1}{1}+\v(\v(\v(1,\p{1}{0}+1)+1))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{0}+\b{1}{\b{2}{0}+1}}} \) |
| \( \p{0}{\p{1}{1}+\v(\v(\v(\v(1,\p{1}{0}+1)+1)))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{0}+\b{1}{\b{2}{0}+\b{1}{\b{2}{0}+1}}}} \) |
| \( \p{0}{\p{1}{1}+\v(1,\p{1}{0}+2)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{0}+\b{2}{0}}} \) |
| \( \p{0}{\p{1}{1}+\v(1,\p{1}{0}+3)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{0}+\b{2}{0}+\b{2}{0}}} \) |
| \( \p{0}{\p{1}{1}+\v(1,\p{1}{0}+\v(1))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{1}}} \) |
| \( \p{0}{\p{1}{1}+\v(1,\v(1,\p{1}{0}+1))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{1}{\b{2}{0}}}}} \) |
| \( \p{0}{\p{1}{1}+\v(2,\p{1}{0}+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{0}}}} \) |
| \( \p{0}{\p{1}{1}+\v(1,\v(2,\p{1}{0}+1)+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{0}}+\b{2}{0}}} \) |
| \( \p{0}{\p{1}{1}+\v(2,\p{1}{0}+2)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{0}}+\b{2}{\b{2}{0}}}} \) |
| \( \p{0}{\p{1}{1}+\v(2,\p{1}{0}+\v(1))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{0}+1}}} \) |
| \( \p{0}{\p{1}{1}+\v(2,\p{1}{0}+\v(2,\p{1}{0}+1))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{0}+\b{1}{\b{2}{\b{2}{0}}}}}} \) |
| \( \p{0}{\p{1}{1}+\v(3,\p{1}{0}+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{0}+\b{2}{0}}}} \) |
| \( \p{0}{\p{1}{1}+\v(\v(1),\p{1}{0}+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{1}}}} \) |
| \( \p{0}{\p{1}{1}+\v(\p{1}{0},1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{\b{1}{0}}}}} \) |
| \( \p{0}{\p{1}{1}+\v(\v(1),\v(\p{1}{0},1)+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{\b{1}{0}}}+\b{2}{\b{2}{1}}}} \) |
| \( \p{0}{\p{1}{1}+\v(\p{1}{0},2)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{\b{1}{0}}}+\b{2}{\b{2}{\b{1}{0}}}}} \) |
| \( \p{0}{\p{1}{1}+\v(\p{1}{0},\v(1))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{\b{1}{0}}+1}}} \) |
| \( \p{0}{\p{1}{1}+\v(\p{1}{0},\v(\p{1}{0},\v(1)))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{\b{1}{0}}+\b{1}{\b{2}{\b{2}{\b{1}{0}}+1}}}}} \) |
| \( \p{0}{\p{1}{1}+\v(\p{1}{0}+1,0)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{\b{1}{0}}+\b{2}{0}}}} \) |
| \( \p{0}{\p{1}{1}+\v(\p{1}{0}+2,0)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{\b{1}{0}}+\b{2}{0}+\b{2}{0}}}} \) |
| \( \p{0}{\p{1}{1}+\v(\p{1}{0}+\p{1}{0},0)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{\b{1}{0}}+\b{2}{\b{1}{0}}}}} \) |
| \( \p{0}{\p{1}{1}+\v(\v(\p{1}{0}+1),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{\b{1}{0}+1}}}} \) |
| \( \p{0}{\p{1}{1}+\v(\v(\p{1}{0},1),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{\b{1}{0}}}}}}}} \) |
| \( \p{0}{\p{1}{1}+\v(\v(\v(\p{1}{0}+1),0),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{\b{1}{0}}}}}}}}}}} \) |
| \( \p{0}{\p{1}{1}+\p{1}{1}} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{\b{2}{0}}}}} \) |
| \( \p{0}{\p{1}{1}+\p{1}{1}+1} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{\b{2}{0}}}}+\b{1}{\b{1}{\b{1}{0}}}} \) |
| \( \p{0}{\p{1}{1}+\p{1}{1}+\p{1}{1}} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{\b{2}{0}}}}+\b{1}{\b{2}{\b{2}{\b{2}{0}}}}} \) |
| \( \p{0}{\v(\p{1}{1}+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{\b{2}{0}}}+1}} \) |
| \( \p{0}{\v(\p{1}{1}+\p{1}{1})} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{\b{2}{0}}}+\b{1}{\b{2}{\b{2}{\b{2}{0}}}}}} \) |
| \( \p{0}{\v(1,\p{1}{1}+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{2}{0}} \) |
| \( \p{0}{\v(2,\p{1}{1}+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{2}{\b{2}{0}}} \) |
| \( \p{0}{\v(\v(1),\p{1}{1}+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{2}{\b{2}{1}}} \) |
| \( \p{0}{\v(\p{1}{1},1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{2}{\b{2}{\b{1}{0}}}} \) |
| \( \p{0}{\v(\v(\p{1}{1},1),0)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{2}{\b{2}{\b{1}{\b{2}{\b{2}{\b{2}{0}}}+\b{2}{\b{2}{\b{1}{0}}}}}}} \) |
| \( \p{0}{\p{1}{2}} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}}+\b{2}{\b{2}{\b{2}{0}}}} \) |
| \( \p{0}{\p{1}{\v(1)}} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+1}} \) |
| \( \p{0}{\p{1}{\p{1}{0}}} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+\b{1}{0}}} \) |
| \( \p{0}{\p{1}{\p{1}{\p{1}{0}}}} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+\b{1}{\b{2}{\b{2}{\b{2}{0}}+\b{1}{0}}}}} \) |
| \( \p{0}{\p{2}{0}} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+\b{2}{0}}} \) |
| \( \p{0}{\p{2}{0}+\p{1}{0}} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+\b{2}{0}}+\b{1}{0}} \) |
| \( \p{0}{\p{2}{0}+\p{1}{1}} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+\b{2}{0}}+\b{1}{\b{2}{\b{2}{\b{2}{0}}}}} \) |
| \( \p{0}{\p{2}{0}+\p{1}{\p{1}{0}}} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+\b{2}{0}}+\b{1}{\b{2}{\b{2}{\b{2}{0}}+\b{1}{0}}}} \) |
| \( \p{0}{\p{2}{0}+\p{1}{\p{2}{0}}} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+\b{2}{0}}+\b{1}{\b{2}{\b{2}{\b{2}{0}}+\b{2}{0}}}} \) |
| \( \p{0}{\p{2}{0}+\p{2}{0}} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+\b{2}{0}}+\b{2}{0}} \) |
| \( \p{0}{\p{2}{0}+\p{2}{0}+\p{1}{0}} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+\b{2}{0}}+\b{2}{0}+\b{1}{0}} \) |
| \( \p{0}{\p{2}{0}+\p{2}{0}+\p{2}{0}} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+\b{2}{0}}+\b{2}{0}+\b{2}{0}} \) |
| \( \p{0}{\v(\p{2}{0}+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+\b{2}{0}}+\b{2}{1}} \) |
| \( \p{0}{\v(\p{2}{0}+\p{1}{0})} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+\b{2}{0}}+\b{2}{\b{1}{0}}} \) |
| \( \p{0}{\v(\p{2}{0}+\p{2}{0})} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+\b{2}{0}}+\b{2}{\b{2}{0}}} \) |
| \( \p{0}{\v(\p{2}{0}+\p{2}{0}+\p{2}{0})} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+\b{2}{0}}+\b{2}{\b{2}{0}+\b{2}{0}}} \) |
| \( \p{0}{\v(\v(\p{2}{0}+\p{2}{0}))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+\b{2}{0}}+\b{2}{\b{2}{\b{2}{0}}}} \) |
| \( \p{0}{\v(\v(\p{2}{0}+\p{2}{0})+\p{2}{0})} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+\b{2}{0}}+\b{2}{\b{2}{\b{2}{0}}+\b{2}{0}}} \) |
| \( \p{0}{\v(\v(\p{2}{0}+\p{2}{0})+\p{2}{0}+1)} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+\b{2}{0}+1}} \) |
| \( \p{0}{\v(\v(\p{2}{0}+\p{2}{0})+\v(\p{2}{0}+\p{2}{0}))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}}+\b{2}{\b{2}{0}}}} \) |
| \( \p{0}{\v(\v(\p{2}{0}+\p{2}{0}+1))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}+1}}} \) |
| \( \p{0}{\v(\v(\p{2}{0}+\p{2}{0}+\p{2}{0}))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{0}+\b{2}{0}}}} \) |
| \( \p{0}{\v(\v(\v(\p{2}{0}+1)))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{1}}}} \) |
| \( \p{0}{\v(\v(\v(\p{2}{0}+\p{2}{0})))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{\b{2}{0}}}}} \) |
| \( \p{0}{\v(\v(\v(\v(\p{2}{0}+\p{2}{0}))))} \) | \( \b{0}{\b{2}{\b{2}{\b{2}{\b{2}{\b{2}{0}}}}}} \) |
| \( \p{0}{\v(1,\p{2}{0}+1)} \) | \( \b{0}{\b{3}{0}} \) |
| \( \p{0}{\v(2,\p{2}{0}+1)} \) | \( \b{0}{\b{3}{\b{3}{0}}} \) |
| \( \p{0}{\v(3,\p{2}{0}+1)} \) | \( \b{0}{\b{3}{\b{3}{0}+\b{3}{0}}} \) |
| \( \p{0}{\v(\v(1),\p{2}{0}+1)} \) | \( \b{0}{\b{3}{\b{3}{1}}} \) |
| \( \p{0}{\v(\p{2}{0},1)} \) | \( \b{0}{\b{3}{\b{3}{\b{2}{0}}}} \) |
| \( \p{0}{\v(\v(\p{2}{0},1),0)} \) | \( \b{0}{\b{3}{\b{3}{\b{2}{\b{3}{\b{3}{\b{2}{0}}}}}}} \) |
| \( \p{0}{\v(\v(\v(\p{2}{0},1),0),0)} \) | \( \b{0}{\b{3}{\b{3}{\b{2}{\b{3}{\b{3}{\b{2}{\b{3}{\b{3}{\b{2}{0}}}}}}}}}} \) |
| \( \p{0}{\p{2}{1}} \) | \( \b{0}{\b{3}{\b{3}{\b{3}{0}}}} \) |
| \( \p{0}{\p{2}{2}} \) | \( \b{0}{\b{3}{\b{3}{\b{3}{0}}}+\b{3}{\b{3}{\b{3}{0}}}} \) |
| \( \p{0}{\p{2}{\v(1)}} \) | \( \b{0}{\b{3}{\b{3}{\b{3}{0}}+1}} \) |
| \( \p{0}{\p{2}{\p{2}{0}}} \) | \( \b{0}{\b{3}{\b{3}{\b{3}{0}}+\b{2}{0}}} \) |
| \( \p{0}{\p{2}{\p{2}{\p{2}{0}}}} \) | \( \b{0}{\b{3}{\b{3}{\b{3}{0}}+\b{2}{\b{3}{\b{3}{\b{3}{0}}+\b{2}{0}}}}} \) |
| \( \p{0}{\p{3}{0}} \) | \( \b{0}{\b{3}{\b{3}{\b{3}{0}}+\b{3}{0}}} \) |
| \( \p{0}{\v(1,\p{3}{0}+1)} \) | \( \b{0}{\b{4}{0}} \) |
| \( \p{0}{\v(1,\p{\v(1)}{0}+1)} \) | \( \b{0}{\b{\b{0}{1}+1}{0}} \) |
| \( \p{0}{\v(1,\p{\v(1)+1}{0}+1)} \) | \( \b{0}{\b{\b{0}{1}+2}{0}} \) |
| \( \p{0}{\v(1,\p{\v(1)+\v(1)}{0}+1)} \) | \( \b{0}{\b{\b{0}{1}+\b{0}{1}}{0}} \) |
| \( \p{0}{\v(1,\p{\p{1}{0}}{0}+1)} \) | \( \b{0}{\b{\b{1}{0}}{0}} \) |
| \( \p{0}{\v(1,\p{\p{\p{1}{0}}{0}}{0}+1)} \) | \( \b{0}{\b{\b{\b{1}{0}}{0}}{0}} \) |
最小のオメガ不動点を\(\Delta\)とする。
| \( \p{0}{\Delta} \) | \( \b{0}{\Delta} \) |