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weak Veblen 函数：修订间差异

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U3Vzc3liYWth​（留言 | 贡献）
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小 修复公式错误
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小 撤销U3Vzc3liYWth（讨论）的修订版本2511
标签：撤销
 
第14行： 第14行：
<big>'''<math>+1\ \phi</math>'''</big>
<big>'''<math>+1\ \phi</math>'''</big>


# <math>\varphi(\#,(\alpha+1)</math>@<math>(\beta+1),0</math>@<math>0)[0]=0</math>。
# \(\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)[0]=0\)。
# <math>\varphi(\#,(\alpha+1)</math>@<math>(\beta+1),0</math>@<math>0)[n+1]=\varphi(\#,\alpha</math>@<math>(\beta+1),\varphi(\#,(\alpha+1)</math>@<math>(\beta+1),0</math>@<math>0)[n]</math>@<math>\beta)</math>。
# \(\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)。[n+1]=\varphi(\#,\alpha\text{@}(\beta+1),\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)[n]\text{@}\beta)\)。
# 对于极限序数<math>\alpha</math>，<math>\varphi(\#,\alpha</math>@<math>\beta)[n]=\varphi(\#,\alpha[n]</math>@<math>\beta)</math>。
# 对于极限序数\(\alpha\)，\(\varphi(\#,\alpha\text{@}\beta)[n]=\varphi(\#,\alpha[n]\text{@}\beta)\)。
# 对于极限序数<math>\beta</math>，<math>\varphi(\#,(\alpha+1)</math>@<math>\beta,0</math>@<math>0)[n]=\varphi(\#,\alpha</math>@<math>\beta,1</math>@<math>\beta[n])</math>。
# 对于极限序数\(\beta\)，\(\varphi(\#,(\alpha+1)\text{@}\beta,0\text{@}0)。[n]=\varphi(\#,\alpha\text{@}\beta,1\text{@}\beta[n])\)。
# <math>\varphi(\#,(\gamma+1)</math>@<math>0)=\varphi(\#,\gamma</math>@<math>0)+1</math>。
# \(\varphi(\#,(\gamma+1)\text{@}0)=\varphi(\#,\gamma\text{@}0)+1\)。


<big>'''<math>\times\omega\ \phi</math>'''</big>
<big>'''<math>\times\omega\ \phi</math>'''</big>
# <math>\varphi(\#,(\alpha+1)</math>@<math>(\beta+1),0</math>@<math>0)[0]=0</math>。
# \(\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)[0]=0\)。
# <math>\varphi(\#,(\alpha+1)</math>@<math>(\beta+1),0</math>@<math>0)[n+1]=\varphi(\#,\alpha</math>@<math>(\beta+1),\varphi(\#,(\alpha+1)</math>@<math>(\beta+1),0</math>@<math>0)[n]</math>@<math>\beta)</math>。
# \(\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)。[n+1]=\varphi(\#,\alpha\text{@}(\beta+1),\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)[n]\text{@}\beta)\)。
# 对于极限序数<math>\alpha</math>，<math>\varphi(\#,\alpha</math>@<math>\beta)[n]=\varphi(\#,\alpha[n]</math>@<math>\beta)</math>。
# 对于极限序数\(\alpha\)，\(\varphi(\#,\alpha\text{@}\beta)[n]=\varphi(\#,\alpha[n]\text{@}\beta)\)。
# 对于极限序数<math>\beta</math>，<math>\varphi(\#,(\alpha+1)</math>@<math>\beta,0</math>@<math>0)[n]=\varphi(\#,\alpha</math>@<math>\beta,1</math>@<math>\beta[n])</math>。
# 对于极限序数\(\beta\)，\(\varphi(\#,(\alpha+1)\text{@}\beta,0\text{@}0)[n]=\varphi(\#,\alpha\text{@}\beta,1\text{@}\beta[n])\)。
# <math>\varphi(\#,(\gamma+1)</math>@<math>0)=\varphi(\#,\gamma</math>@<math>0)\cdot\varphi(1)</math>。
# \(\varphi(\#,(\gamma+1)\text{@}0)=\varphi(\#,\gamma\text{@}0)\cdot\varphi(1)\)。


=== 分析 ===
=== 分析 ===
以下分析中，左为 <math>\mathrm{FP}\ \phi</math>，中为 <math>\times\omega\ \phi</math>，右为 <math>+1\ \phi</math>。
以下分析中，左为 <math>\mathrm{FP}\ \phi</math>，中为 <math>\times\omega\ \phi</math>，右为 <math>+1\ \phi</math>。


<math>\varphi(0)=\varphi(0)=\varphi(0)</math>
\(\varphi(0)=\varphi(0)=\varphi(0)\)


<math>\varphi(0)\cdot2=\varphi(0)\cdot2=\varphi(\varphi(0))</math>
\(\varphi(0)\cdot2=\varphi(0)\cdot2=\varphi(\varphi(0))\)


<math>\varphi(0)\cdot3=\varphi(0)\cdot3=\varphi(\varphi(\varphi(0)))</math>
\(\varphi(0)\cdot3=\varphi(0)\cdot3=\varphi(\varphi(\varphi(0)))\)


<math>\varphi(1)=\varphi(1)=\varphi(1,0)=</math> [[FTO|ω]]
\(\varphi(1)=\varphi(1)=\varphi(1,0)=\) [[FTO|ω]]


<math>\varphi(1)+1=\varphi(1)+1=\varphi(1,1)</math>
\(\varphi(1)+1=\varphi(1)+1=\varphi(1,1)\)


<math>\varphi(1)+2=\varphi(1)+2=\varphi(1,2)</math>
\(\varphi(1)+2=\varphi(1)+2=\varphi(1,2)\)


<math>\varphi(1)\cdot2=\varphi(1)\cdot2=\varphi(1,\varphi(1,0))</math>
\(\varphi(1)\cdot2=\varphi(1)\cdot2=\varphi(1,\varphi(1,0))\)


<math>\varphi(1)\cdot2+1=\varphi(1)\cdot2+1=\varphi(1,\varphi(1,1))</math>
\(\varphi(1)\cdot2+1=\varphi(1)\cdot2+1=\varphi(1,\varphi(1,1))\)


<math>\varphi(1)\cdot2+2=\varphi(1)\cdot2+2=\varphi(1,\varphi(1,2))</math>
\(\varphi(1)\cdot2+2=\varphi(1)\cdot2+2=\varphi(1,\varphi(1,2))\)


<math>\varphi(1)\cdot3=\varphi(1)\cdot3=\varphi(1,\varphi(1,\varphi(1,0)))</math>
\(\varphi(1)\cdot3=\varphi(1)\cdot3=\varphi(1,\varphi(1,\varphi(1,0)))\)


<math>\varphi(2)=\varphi(2)=\varphi(2,0)</math>
\(\varphi(2)=\varphi(2)=\varphi(2,0)\)


<math>\varphi(2)+1=\varphi(2)+1=\varphi(2,1)</math>
\(\varphi(2)+1=\varphi(2)+1=\varphi(2,1)\)


<math>\varphi(2)+2=\varphi(2)+2=\varphi(2,2)</math>
\(\varphi(2)+2=\varphi(2)+2=\varphi(2,2)\)


<math>\varphi(2)+\varphi(1)=\varphi(2)+\varphi(1)=\varphi(2,\varphi(1,0))</math>
\(\varphi(2)+\varphi(1)=\varphi(2)+\varphi(1)=\varphi(2,\varphi(1,0))\)


<math>\varphi(2)\cdot2=\varphi(2)\cdot2=\varphi(2,\varphi(2,0))</math>
\(\varphi(2)\cdot2=\varphi(2)\cdot2=\varphi(2,\varphi(2,0))\)


<math>\varphi(2)\cdot3=\varphi(2)\cdot3=\varphi(2,\varphi(2,\varphi(2,0)))</math>
\(\varphi(2)\cdot3=\varphi(2)\cdot3=\varphi(2,\varphi(2,\varphi(2,0)))\)


<math>\varphi(3)=\varphi(3)=\varphi(3,0)</math>
\(\varphi(3)=\varphi(3)=\varphi(3,0)\)


<math>\varphi(3)+\varphi(2)=\varphi(3)+\varphi(2)=\varphi(3,\varphi(2,0))</math>
\(\varphi(3)+\varphi(2)=\varphi(3)+\varphi(2)=\varphi(3,\varphi(2,0))\)


<math>\varphi(4)=\varphi(4)=\varphi(4,0)</math>
\(\varphi(4)=\varphi(4)=\varphi(4,0)\)


<math>\varphi(\varphi(1))=\varphi(\varphi(1))=\varphi(\varphi(1,0),0)</math>
\(\varphi(\varphi(1))=\varphi(\varphi(1))=\varphi(\varphi(1,0),0)\)


<math>\varphi(\varphi(1))+1=\varphi(\varphi(1))+1=\varphi(\varphi(1,0),1)</math>
\(\varphi(\varphi(1))+1=\varphi(\varphi(1))+1=\varphi(\varphi(1,0),1)\)


<math>\varphi(\varphi(1))+\varphi(1)=\varphi(\varphi(1))+\varphi(1)=\varphi(\varphi(1,0),\varphi(1,0))</math>
\(\varphi(\varphi(1))+\varphi(1)=\varphi(\varphi(1))+\varphi(1)=\varphi(\varphi(1,0),\varphi(1,0))\)


<math>\varphi(\varphi(1))+\varphi(2)=\varphi(\varphi(1))+\varphi(2)=\varphi(\varphi(1,0),\varphi(2,0))</math>
\(\varphi(\varphi(1))+\varphi(2)=\varphi(\varphi(1))+\varphi(2)=\varphi(\varphi(1,0),\varphi(2,0))\)


<math>\varphi(\varphi(1))\cdot2=\varphi(\varphi(1))\cdot2=\varphi(\varphi(1,0),\varphi(\varphi(1,0),0))</math>
\(\varphi(\varphi(1))\cdot2=\varphi(\varphi(1))\cdot2=\varphi(\varphi(1,0),\varphi(\varphi(1,0),0))\)


<math>\varphi(\varphi(1)+1)=\varphi(\varphi(1)+1)=\varphi(\varphi(1,1),0)</math>
\(\varphi(\varphi(1)+1)=\varphi(\varphi(1)+1)=\varphi(\varphi(1,1),0)\)


<math>\varphi(\varphi(1)+2)=\varphi(\varphi(1)+2)=\varphi(\varphi(1,2),0)</math>
\(\varphi(\varphi(1)+2)=\varphi(\varphi(1)+2)=\varphi(\varphi(1,2),0)\)


<math>\varphi(\varphi(1)+3)=\varphi(\varphi(1)+3)=\varphi(\varphi(1,3),0)</math>
\(\varphi(\varphi(1)+3)=\varphi(\varphi(1)+3)=\varphi(\varphi(1,3),0)\)


<math>\varphi(\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2)=\varphi(\varphi(1,\varphi(1,0)),0)</math>
\(\varphi(\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2)=\varphi(\varphi(1,\varphi(1,0)),0)\)


<math>\varphi(\varphi(1)\cdot2+1)=\varphi(\varphi(1)\cdot2+1)=\varphi(\varphi(1,\varphi(1,1)),0)</math>
\(\varphi(\varphi(1)\cdot2+1)=\varphi(\varphi(1)\cdot2+1)=\varphi(\varphi(1,\varphi(1,1)),0)\)


<math>\varphi(\varphi(1)\cdot3)=\varphi(\varphi(1)\cdot3)=\varphi(\varphi(1,\varphi(1,\varphi(1,0))),0)</math>
\(\varphi(\varphi(1)\cdot3)=\varphi(\varphi(1)\cdot3)=\varphi(\varphi(1,\varphi(1,\varphi(1,0))),0)\)


<math>\varphi(\varphi(2))=\varphi(\varphi(2))=\varphi(\varphi(2,0),0)</math>
\(\varphi(\varphi(2))=\varphi(\varphi(2))=\varphi(\varphi(2,0),0)\)


<math>\varphi(\varphi(2)+1)=\varphi(\varphi(2)+1)=\varphi(\varphi(2,1),0)</math>
\(\varphi(\varphi(2)+1)=\varphi(\varphi(2)+1)=\varphi(\varphi(2,1),0)\)


<math>\varphi(\varphi(2)+\varphi(1))=\varphi(\varphi(2)+\varphi(1))=\varphi(\varphi(2,\varphi(1,0)),0)</math>
\(\varphi(\varphi(2)+\varphi(1))=\varphi(\varphi(2)+\varphi(1))=\varphi(\varphi(2,\varphi(1,0)),0)\)


<math>\varphi(\varphi(3))=\varphi(\varphi(3))=\varphi(\varphi(3,0),0)</math>
\(\varphi(\varphi(3))=\varphi(\varphi(3))=\varphi(\varphi(3,0),0)\)


<math>\varphi(\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1,0),0),0)</math>
\(\varphi(\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1,0),0),0)\)


<math>\varphi(\varphi(\varphi(1))+1)=\varphi(\varphi(\varphi(1))+1)=\varphi(\varphi(\varphi(1,0),1),0)</math>
\(\varphi(\varphi(\varphi(1))+1)=\varphi(\varphi(\varphi(1))+1)=\varphi(\varphi(\varphi(1,0),1),0)\)


<math>\varphi(\varphi(\varphi(1))+\varphi(1))=\varphi(\varphi(\varphi(1))+\varphi(1))=\varphi(\varphi(\varphi(1,0),\varphi(1,0)),0)</math>
\(\varphi(\varphi(\varphi(1))+\varphi(1))=\varphi(\varphi(\varphi(1))+\varphi(1))=\varphi(\varphi(\varphi(1,0),\varphi(1,0)),0)\)


<math>\varphi(\varphi(\varphi(1))\cdot2)=\varphi(\varphi(\varphi(1))\cdot2)=\varphi(\varphi(\varphi(1,0),\varphi(\varphi(1,0),0)),0)</math>
\(\varphi(\varphi(\varphi(1))\cdot2)=\varphi(\varphi(\varphi(1))\cdot2)=\varphi(\varphi(\varphi(1,0),\varphi(\varphi(1,0),0)),0)\)


<math>\varphi(\varphi(\varphi(1)+1))=\varphi(\varphi(\varphi(1)+1))=\varphi(\varphi(\varphi(1,1),0),0)</math>
\(\varphi(\varphi(\varphi(1)+1))=\varphi(\varphi(\varphi(1)+1))=\varphi(\varphi(\varphi(1,1),0),0)\)


<math>\varphi(\varphi(\varphi(2)))=\varphi(\varphi(\varphi(2)))=\varphi(\varphi(\varphi(2,0),0),0)</math>
\(\varphi(\varphi(\varphi(2)))=\varphi(\varphi(\varphi(2)))=\varphi(\varphi(\varphi(2,0),0),0)\)


<math>\varphi(\varphi(\varphi(3)))=\varphi(\varphi(\varphi(3)))=\varphi(\varphi(\varphi(3,0),0),0)</math>
\(\varphi(\varphi(\varphi(3)))=\varphi(\varphi(\varphi(3)))=\varphi(\varphi(\varphi(3,0),0),0)\)


<math>\varphi(\varphi(\varphi(\varphi(1))))=\varphi(\varphi(\varphi(\varphi(1))))=\varphi(\varphi(\varphi(\varphi(1,0),0),0),0)</math>
\(\varphi(\varphi(\varphi(\varphi(1))))=\varphi(\varphi(\varphi(\varphi(1))))=\varphi(\varphi(\varphi(\varphi(1,0),0),0),0)\)


<math>\varphi(\varphi(\varphi(\varphi(\varphi(1)))))=\varphi(\varphi(\varphi(\varphi(\varphi(1)))))=\varphi(\varphi(\varphi(\varphi(\varphi(1,0),0),0),0),0)</math>
\(\varphi(\varphi(\varphi(\varphi(\varphi(1)))))=\varphi(\varphi(\varphi(\varphi(\varphi(1)))))=\varphi(\varphi(\varphi(\varphi(\varphi(1,0),0),0),0),0)\)


<math>\varphi(1,0)=\varphi(1,0)=\varphi(1,0,0)=</math> [[SCO|ε0]]
\(\varphi(1,0)=\varphi(1,0)=\varphi(1,0,0)=\) [[SCO|ε0]]


<math>\varphi(1,0)+1=\varphi(1,0)+1=\varphi(1,0,1)</math>
\(\varphi(1,0)+1=\varphi(1,0)+1=\varphi(1,0,1)\)


<math>\varphi(1,0)+\varphi(1)=\varphi(1,0)+\varphi(1)=\varphi(1,0,\varphi(1,0))</math>
\(\varphi(1,0)+\varphi(1)=\varphi(1,0)+\varphi(1)=\varphi(1,0,\varphi(1,0))\)


<math>\varphi(1,0)\cdot2=\varphi(1,0)\cdot2=\varphi(1,0,\varphi(1,0,0))</math>
\(\varphi(1,0)\cdot2=\varphi(1,0)\cdot2=\varphi(1,0,\varphi(1,0,0))\)


<math>\varphi(1,0)\cdot3=\varphi(1,0)\cdot3=\varphi(1,0,\varphi(1,0,\varphi(1,0,0)))</math>
\(\varphi(1,0)\cdot3=\varphi(1,0)\cdot3=\varphi(1,0,\varphi(1,0,\varphi(1,0,0)))\)


<math>\varphi(1,0)\cdot\varphi(1)=\varphi(1,1)=\varphi(1,1,0)</math>
\(\varphi(1,0)\cdot\varphi(1)=\varphi(1,1)=\varphi(1,1,0)\)


<math>\varphi(1,0)\cdot\varphi(1)\cdot2=\varphi(1,1)\cdot2=\varphi(1,1,\varphi(1,1,0))</math>
\(\varphi(1,0)\cdot\varphi(1)\cdot2=\varphi(1,1)\cdot2=\varphi(1,1,\varphi(1,1,0))\)


<math>\varphi(1,0)\cdot\varphi(2)=\varphi(1,2)=\varphi(1,2,0)</math>
\(\varphi(1,0)\cdot\varphi(2)=\varphi(1,2)=\varphi(1,2,0)\)


<math>\varphi(1,0)\cdot\varphi(\varphi(1))=\varphi(1,\varphi(1))=\varphi(1,\varphi(1,0),0)</math>
\(\varphi(1,0)\cdot\varphi(\varphi(1))=\varphi(1,\varphi(1))=\varphi(1,\varphi(1,0),0)\)


<math>\varphi(1,0)\cdot\varphi(\varphi(1)+1)=\varphi(1,\varphi(1)+1)=\varphi(1,\varphi(1,1),0)</math>
\(\varphi(1,0)\cdot\varphi(\varphi(1)+1)=\varphi(1,\varphi(1)+1)=\varphi(1,\varphi(1,1),0)\)


<math>\varphi(1,0)\cdot\varphi(\varphi(2))=\varphi(1,\varphi(2))=\varphi(1,\varphi(2,0),0)</math>
\(\varphi(1,0)\cdot\varphi(\varphi(2))=\varphi(1,\varphi(2))=\varphi(1,\varphi(2,0),0)\)


<math>\varphi(1,0)\cdot\varphi(\varphi(\varphi(1)))=\varphi(1,\varphi(\varphi(1)))=\varphi(1,\varphi(\varphi(1,0),0),0)</math>
\(\varphi(1,0)\cdot\varphi(\varphi(\varphi(1)))=\varphi(1,\varphi(\varphi(1)))=\varphi(1,\varphi(\varphi(1,0),0),0)\)


<math>\varphi(1,0)^{2}=\varphi(1,\varphi(1,0))=\varphi(1,\varphi(1,0,0),0)</math>
\(\varphi(1,0)^{2}=\varphi(1,\varphi(1,0))=\varphi(1,\varphi(1,0,0),0)\)


<math>\varphi(1,0)^{2}\cdot\varphi(1)=\varphi(1,\varphi(1,0)+1)=\varphi(1,\varphi(\varphi(1,0,0)),0)</math>
\(\varphi(1,0)^{2}\cdot\varphi(1)=\varphi(1,\varphi(1,0)+1)=\varphi(1,\varphi(\varphi(1,0,0)),0)\)


<math>\varphi(1,0)^{2}\cdot\varphi(\varphi(\varphi(1)))=\varphi(1,\varphi(1,0)+\varphi(\varphi(1)))=\varphi(1,\varphi(1,0,\varphi(\varphi(1,0),0)),0)</math>
\(\varphi(1,0)^{2}\cdot\varphi(\varphi(\varphi(1)))=\varphi(1,\varphi(1,0)+\varphi(\varphi(1)))=\varphi(1,\varphi(1,0,\varphi(\varphi(1,0),0)),0)\)


<math>\varphi(1,0)^{3}=\varphi(1,\varphi(1,0)\cdot2)=\varphi(1,\varphi(1,0,\varphi(1,0,0),0),0)</math>
\(\varphi(1,0)^{3}=\varphi(1,\varphi(1,0)\cdot2)=\varphi(1,\varphi(1,0,\varphi(1,0,0),0),0)\)


<math>\varphi(1,0)^{\varphi(1)}=\varphi(1,\varphi(1,1))=\varphi(1,\varphi(1,1,0),0)</math>
\(\varphi(1,0)^{\varphi(1)}=\varphi(1,\varphi(1,1))=\varphi(1,\varphi(1,1,0),0)\)


<math>\varphi(1,0)^{\varphi(1)}\cdot\varphi(1)=\varphi(1,\varphi(1,1)+1)=\varphi(1,\varphi(\varphi(1,1,0)),0)</math>
\(\varphi(1,0)^{\varphi(1)}\cdot\varphi(1)=\varphi(1,\varphi(1,1)+1)=\varphi(1,\varphi(\varphi(1,1,0)),0)\)


<math>\varphi(1,0)^{\varphi(1)}\cdot\varphi(\varphi(1))=\varphi(1,\varphi(1,1)+2)=\varphi(1,\varphi(\varphi(\varphi(1,1,0))),0)</math>
\(\varphi(1,0)^{\varphi(1)}\cdot\varphi(\varphi(1))=\varphi(1,\varphi(1,1)+2)=\varphi(1,\varphi(\varphi(\varphi(1,1,0))),0)\)


<math>\varphi(1,0)^{(\varphi(1)+1)}=\varphi(1,\varphi(1,1)+\varphi(1,0))=\varphi(1,\varphi(1,1,\varphi(1,0,0)),0)</math>
\(\varphi(1,0)^{(\varphi(1)+1)}=\varphi(1,\varphi(1,1)+\varphi(1,0))=\varphi(1,\varphi(1,1,\varphi(1,0,0)),0)\)


<math>\varphi(1,0)^{(\varphi(1)\cdot2)}=\varphi(1,\varphi(1,1)\cdot2)=\varphi(1,\varphi(1,1,\varphi(1,1,0)),0)</math>
\(\varphi(1,0)^{(\varphi(1)\cdot2)}=\varphi(1,\varphi(1,1)\cdot2)=\varphi(1,\varphi(1,1,\varphi(1,1,0)),0)\)


<nowiki><math>\varphi(1,0)^{\varphi(1)^{2}}=\varphi(1,\varphi(1,2))=\varphi(1,\varphi(1,2,0),0)</math></nowiki>
<nowiki>\(\varphi(1,0)^{\varphi(1)^{2}}=\varphi(1,\varphi(1,2))=\varphi(1,\varphi(1,2,0),0)\)</nowiki>


<nowiki><math>\varphi(1,0)^{\varphi(1)^{\varphi(1)}}=\varphi(1,\varphi(1,\varphi(1)))=\varphi(1,\varphi(1,\varphi(1,0),0),0)</math></nowiki>
<nowiki>\(\varphi(1,0)^{\varphi(1)^{\varphi(1)}}=\varphi(1,\varphi(1,\varphi(1)))=\varphi(1,\varphi(1,\varphi(1,0),0),0)\)</nowiki>


<math>\varphi(1,0)^{\varphi(1,0)}=\varphi(1,\varphi(1,\varphi(1,0)))=\varphi(1,\varphi(1,\varphi(1,0,0),0),0)</math>
\(\varphi(1,0)^{\varphi(1,0)}=\varphi(1,\varphi(1,\varphi(1,0)))=\varphi(1,\varphi(1,\varphi(1,0,0),0),0)\)


<math>\varphi(1,0)^{(\varphi(1,0)+1)}=\varphi(1,\varphi(1,\varphi(1,0))+\varphi(1,0))=\varphi(1,\varphi(1,\varphi(1,0,0),\varphi(1,0,0)),0)</math>
\(\varphi(1,0)^{(\varphi(1,0)+1)}=\varphi(1,\varphi(1,\varphi(1,0))+\varphi(1,0))=\varphi(1,\varphi(1,\varphi(1,0,0),\varphi(1,0,0)),0)\)


<math>\varphi(1,0)^{(\varphi(1,0)\cdot\varphi(1))}=\varphi(1,\varphi(1,\varphi(1,0))+\varphi(1,1))=\varphi(1,\varphi(1,\varphi(1,0,0),\varphi(1,1,0)),0)</math>
\(\varphi(1,0)^{(\varphi(1,0)\cdot\varphi(1))}=\varphi(1,\varphi(1,\varphi(1,0))+\varphi(1,1))=\varphi(1,\varphi(1,\varphi(1,0,0),\varphi(1,1,0)),0)\)


<nowiki><math>\varphi(1,0)^{\varphi(1,0)^{2}}=\varphi(1,\varphi(1,\varphi(1,2)))=\varphi(1,\varphi(1,\varphi(1,2,0),0),0)</math></nowiki>
<nowiki>\(\varphi(1,0)^{\varphi(1,0)^{2}}=\varphi(1,\varphi(1,\varphi(1,2)))=\varphi(1,\varphi(1,\varphi(1,2,0),0),0)\)</nowiki>


<nowiki><math>\varphi(1,0)^{\varphi(1,0)^{\varphi(1)}}=\varphi(1,\varphi(1,\varphi(1,\varphi(1))))=\varphi(1,\varphi(1,\varphi(1,\varphi(1,0),0),0),0)</math></nowiki>
<nowiki>\(\varphi(1,0)^{\varphi(1,0)^{\varphi(1)}}=\varphi(1,\varphi(1,\varphi(1,\varphi(1))))=\varphi(1,\varphi(1,\varphi(1,\varphi(1,0),0),0),0)\)</nowiki>


<nowiki><math>\varphi(1,0)^{\varphi(1,0)^{\varphi(1,0)}}=\varphi(1,\varphi(1,\varphi(1,\varphi(1,0))))=\varphi(1,\varphi(1,\varphi(1,\varphi(1,0,0),0),0),0)</math></nowiki>
<nowiki>\(\varphi(1,0)^{\varphi(1,0)^{\varphi(1,0)}}=\varphi(1,\varphi(1,\varphi(1,\varphi(1,0))))=\varphi(1,\varphi(1,\varphi(1,\varphi(1,0,0),0),0),0)\)</nowiki>


<math>\varphi(1,1)=\varphi(2,0)=\varphi(2,0,0)</math>
\(\varphi(1,1)=\varphi(2,0)=\varphi(2,0,0)\)


<math>\varphi(1,1)\cdot\varphi(1,0)=\varphi(2,\varphi(1,0))=\varphi(2,\varphi(1,0,0),0)</math>
\(\varphi(1,1)\cdot\varphi(1,0)=\varphi(2,\varphi(1,0))=\varphi(2,\varphi(1,0,0),0)\)


<math>\varphi(1,1)\cdot\varphi(1,0)^{\varphi(1)}=\varphi(2,\varphi(1,\varphi(1,1)))=\varphi(2,\varphi(1,\varphi(1,1,0),0),0)</math>
\(\varphi(1,1)\cdot\varphi(1,0)^{\varphi(1)}=\varphi(2,\varphi(1,\varphi(1,1)))=\varphi(2,\varphi(1,\varphi(1,1,0),0),0)\)


<math>\varphi(1,1)\cdot\varphi(1,0)^{\varphi(1,0)}=\varphi(2,\varphi(1,\varphi(1,\varphi(1,0))))=\varphi(2,\varphi(1,\varphi(1,\varphi(1,0,0),0),0),0)</math>
\(\varphi(1,1)\cdot\varphi(1,0)^{\varphi(1,0)}=\varphi(2,\varphi(1,\varphi(1,\varphi(1,0))))=\varphi(2,\varphi(1,\varphi(1,\varphi(1,0,0),0),0),0)\)


<math>\varphi(1,1)^{2}=\varphi(2,\varphi(2,0))=\varphi(2,\varphi(2,0,0),0)</math>
\(\varphi(1,1)^{2}=\varphi(2,\varphi(2,0))=\varphi(2,\varphi(2,0,0),0)\)


<math>\varphi(1,1)^{\varphi(1)}=\varphi(2,\varphi(2,1))=\varphi(2,\varphi(2,\varphi(1,0),0),0)</math>
\(\varphi(1,1)^{\varphi(1)}=\varphi(2,\varphi(2,1))=\varphi(2,\varphi(2,\varphi(1,0),0),0)\)


<math>\varphi(1,1)^{\varphi(1,0)}=\varphi(2,\varphi(2,\varphi(1,0)))=\varphi(2,\varphi(2,\varphi(1,0,0),0),0)</math>
\(\varphi(1,1)^{\varphi(1,0)}=\varphi(2,\varphi(2,\varphi(1,0)))=\varphi(2,\varphi(2,\varphi(1,0,0),0),0)\)


<math>\varphi(1,1)^{\varphi(1,1)}=\varphi(2,\varphi(2,\varphi(2,0)))=\varphi(2,\varphi(2,\varphi(2,0,0),0),0)</math>
\(\varphi(1,1)^{\varphi(1,1)}=\varphi(2,\varphi(2,\varphi(2,0)))=\varphi(2,\varphi(2,\varphi(2,0,0),0),0)\)


<math>\varphi(1,2)=\varphi(3,0)=\varphi(3,0,0)</math>
\(\varphi(1,2)=\varphi(3,0)=\varphi(3,0,0)\)


<math>\varphi(1,\varphi(1))=\varphi(\varphi(1),0)=\varphi(\varphi(1,0),0,0)</math>
\(\varphi(1,\varphi(1))=\varphi(\varphi(1),0)=\varphi(\varphi(1,0),0,0)\)


<math>\varphi(1,\varphi(1))\cdot\varphi(1,0)=\varphi(\varphi(1),\varphi(1,0))=\varphi(\varphi(1,0),\varphi(1,0,0),0)</math>
\(\varphi(1,\varphi(1))\cdot\varphi(1,0)=\varphi(\varphi(1),\varphi(1,0))=\varphi(\varphi(1,0),\varphi(1,0,0),0)\)


<math>\varphi(1,\varphi(1))\cdot\varphi(1,1)=\varphi(\varphi(1),\varphi(1,1))=\varphi(\varphi(1,0),\varphi(1,1,0),0)</math>
\(\varphi(1,\varphi(1))\cdot\varphi(1,1)=\varphi(\varphi(1),\varphi(1,1))=\varphi(\varphi(1,0),\varphi(1,1,0),0)\)


<math>\varphi(1,\varphi(1))^{2}=\varphi(\varphi(1),\varphi(1,\varphi(1)))=\varphi(\varphi(1,0),\varphi(1,\varphi(1,0),0))</math>
\(\varphi(1,\varphi(1))^{2}=\varphi(\varphi(1),\varphi(1,\varphi(1)))=\varphi(\varphi(1,0),\varphi(1,\varphi(1,0),0))\)


<math>\varphi(1,\varphi(1)+1)=\varphi(\varphi(1)+1,0)=\varphi(\varphi(1,1),0,0)</math>
\(\varphi(1,\varphi(1)+1)=\varphi(\varphi(1)+1,0)=\varphi(\varphi(1,1),0,0)\)


<math>\varphi(1,\varphi(1)+2)=\varphi(\varphi(1)+2,0)=\varphi(\varphi(1,2),0,0)</math>
\(\varphi(1,\varphi(1)+2)=\varphi(\varphi(1)+2,0)=\varphi(\varphi(1,2),0,0)\)


<math>\varphi(1,\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2,0)=\varphi(\varphi(1,\varphi(1,0)),0,0)</math>
\(\varphi(1,\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2,0)=\varphi(\varphi(1,\varphi(1,0)),0,0)\)


<math>\varphi(1,\varphi(1)^{2})=\varphi(\varphi(2),0)=\varphi(\varphi(2,0),0,0)</math>
\(\varphi(1,\varphi(1)^{2})=\varphi(\varphi(2),0)=\varphi(\varphi(2,0),0,0)\)


<math>\varphi(1,\varphi(1)^{3})=\varphi(\varphi(3),0)=\varphi(\varphi(3,0),0,0)</math>
\(\varphi(1,\varphi(1)^{3})=\varphi(\varphi(3),0)=\varphi(\varphi(3,0),0,0)\)


<math>\varphi(1,\varphi(1)^{\varphi(1)})=\varphi(\varphi(\varphi(1)),0)=\varphi(\varphi(\varphi(1,0),0),0,0)</math>
\(\varphi(1,\varphi(1)^{\varphi(1)})=\varphi(\varphi(\varphi(1)),0)=\varphi(\varphi(\varphi(1,0),0),0,0)\)


<math>\varphi(1,\varphi(1,0))=\varphi(\varphi(1,0),0)=\varphi(\varphi(1,0,0),0,0)</math>
\(\varphi(1,\varphi(1,0))=\varphi(\varphi(1,0),0)=\varphi(\varphi(1,0,0),0,0)\)


<math>\varphi(1,\varphi(1,0)+1)=\varphi(\varphi(1,0)+1,0)=\varphi(\varphi(1,0,1),0,0)</math>
\(\varphi(1,\varphi(1,0)+1)=\varphi(\varphi(1,0)+1,0)=\varphi(\varphi(1,0,1),0,0)\)


<math>\varphi(1,\varphi(1,0)\cdot\varphi(1))=\varphi(\varphi(1,1),0)=\varphi(\varphi(1,1,0),0,0)</math>
\(\varphi(1,\varphi(1,0)\cdot\varphi(1))=\varphi(\varphi(1,1),0)=\varphi(\varphi(1,1,0),0,0)\)


<math>\varphi(1,\varphi(1,0)^{\varphi(1)})=\varphi(\varphi(1,\varphi(1,1)),0)=\varphi(\varphi(1,\varphi(1,1,0),0),0,0)</math>
\(\varphi(1,\varphi(1,0)^{\varphi(1)})=\varphi(\varphi(1,\varphi(1,1)),0)=\varphi(\varphi(1,\varphi(1,1,0),0),0,0)\)


<math>\varphi(1,\varphi(1,1))=\varphi(\varphi(2,0),0)=\varphi(\varphi(2,0,0),0,0)</math>
\(\varphi(1,\varphi(1,1))=\varphi(\varphi(2,0),0)=\varphi(\varphi(2,0,0),0,0)\)


<math>\varphi(1,\varphi(1,\varphi(1)))=\varphi(\varphi(\varphi(1),0),0)=\varphi(\varphi(\varphi(1,0),0,0),0,0)</math>
\(\varphi(1,\varphi(1,\varphi(1)))=\varphi(\varphi(\varphi(1),0),0)=\varphi(\varphi(\varphi(1,0),0,0),0,0)\)


<math>\varphi(1,\varphi(1,\varphi(1,0)))=\varphi(\varphi(\varphi(1,0),0),0)=\varphi(\varphi(\varphi(1,0,0),0,0),0,0)</math>
\(\varphi(1,\varphi(1,\varphi(1,0)))=\varphi(\varphi(\varphi(1,0),0),0)=\varphi(\varphi(\varphi(1,0,0),0,0),0,0)\)


<math>\varphi(2,0)=\varphi(1,0,0)=\varphi(1,0,0,0)=</math> [[CO|ζ0]]
\(\varphi(2,0)=\varphi(1,0,0)=\varphi(1,0,0,0)=\) [[CO|ζ0]]


<math>\varphi(1,\varphi(2,0)+1)=\varphi(\varphi(1,0,0),0)=\varphi(\varphi(1,0,0,0),0,0)</math>
\(\varphi(1,\varphi(2,0)+1)=\varphi(\varphi(1,0,0),0)=\varphi(\varphi(1,0,0,0),0,0)\)


<math>\varphi(1,\varphi(2,0)+\varphi(1))=\varphi(\varphi(1,0,0)+\varphi(1),0)=\varphi(\varphi(1,0,0,\varphi(1,0)),0,0)</math>
\(\varphi(1,\varphi(2,0)+\varphi(1))=\varphi(\varphi(1,0,0)+\varphi(1),0)=\varphi(\varphi(1,0,0,\varphi(1,0)),0,0)\)


<math>\varphi(1,\varphi(2,0)+\varphi(1,0))=\varphi(\varphi(1,0,0)+\varphi(1,0),0)=\varphi(\varphi(1,0,0,\varphi(1,0,0)),0,0)</math>
\(\varphi(1,\varphi(2,0)+\varphi(1,0))=\varphi(\varphi(1,0,0)+\varphi(1,0),0)=\varphi(\varphi(1,0,0,\varphi(1,0,0)),0,0)\)


<math>\varphi(1,\varphi(2,0)+\varphi(1,0)\cdot\varphi(1))=\varphi(\varphi(1,0,0)+\varphi(1,1),0)=\varphi(\varphi(1,0,0,\varphi(1,1,0)),0,0)</math>
\(\varphi(1,\varphi(2,0)+\varphi(1,0)\cdot\varphi(1))=\varphi(\varphi(1,0,0)+\varphi(1,1),0)=\varphi(\varphi(1,0,0,\varphi(1,1,0)),0,0)\)


<math>\varphi(1,\varphi(2,0)+\varphi(1,1))=\varphi(\varphi(1,0,0)+\varphi(2,0),0)=\varphi(\varphi(1,0,0,\varphi(2,0,0)),0,0)</math>
\(\varphi(1,\varphi(2,0)+\varphi(1,1))=\varphi(\varphi(1,0,0)+\varphi(2,0),0)=\varphi(\varphi(1,0,0,\varphi(2,0,0)),0,0)\)


<math>\varphi(1,\varphi(2,0)+\varphi(1,\varphi(1,0)))=\varphi(\varphi(1,0,0)+\varphi(\varphi(1,0),0),0)=\varphi(\varphi(1,0,0,\varphi(\varphi(1,0,0),0,0)),0,0))</math>
\(\varphi(1,\varphi(2,0)+\varphi(1,\varphi(1,0)))=\varphi(\varphi(1,0,0)+\varphi(\varphi(1,0),0),0)=\varphi(\varphi(1,0,0,\varphi(\varphi(1,0,0),0,0)),0,0))\)


<math>\varphi(1,\varphi(2,0)\cdot2)=\varphi(\varphi(1,0,0)\cdot2,0)=\varphi(\varphi(1,0,1,0),0,0)</math>
\(\varphi(1,\varphi(2,0)\cdot2)=\varphi(\varphi(1,0,0)\cdot2,0)=\varphi(\varphi(1,0,1,0),0,0)\)


<math>\varphi(1,\varphi(2,0)\cdot\varphi(1))=\varphi(\varphi(1,1,0),0)=\varphi(\varphi(1,0,\varphi(1,0),0),0,0)</math>
\(\varphi(1,\varphi(2,0)\cdot\varphi(1))=\varphi(\varphi(1,1,0),0)=\varphi(\varphi(1,0,\varphi(1,0),0),0,0)\)


<math>\varphi(1,\varphi(2,0)^{2})=\varphi(\varphi(1,0,\varphi(1,0,0)),0)=\varphi(\varphi(1,0,\varphi(1,0,0,0),0),0,0)</math>
\(\varphi(1,\varphi(2,0)^{2})=\varphi(\varphi(1,0,\varphi(1,0,0)),0)=\varphi(\varphi(1,0,\varphi(1,0,0,0),0),0,0)\)


<math>\varphi(1,\varphi(1,\varphi(2,0)+1))=\varphi(\varphi(\varphi(1,0,0),0),0)=\varphi(\varphi(\varphi(1,0,0,0),0,0),0,0)</math>
\(\varphi(1,\varphi(1,\varphi(2,0)+1))=\varphi(\varphi(\varphi(1,0,0),0),0)=\varphi(\varphi(\varphi(1,0,0,0),0,0),0,0)\)


<math>\varphi(1,\varphi(1,\varphi(1,\varphi(2,0)+1)))=\varphi(\varphi(\varphi(\varphi(1,0,0),0),0),0)=\varphi(\varphi(\varphi(\varphi(1,0,0,0),0,0),0,0),0,0)</math>
\(\varphi(1,\varphi(1,\varphi(1,\varphi(2,0)+1)))=\varphi(\varphi(\varphi(\varphi(1,0,0),0),0),0)=\varphi(\varphi(\varphi(\varphi(1,0,0,0),0,0),0,0),0,0)\)


<math>\varphi(2,1)=\varphi(2,0,0)=\varphi(2,0,0,0)</math>
\(\varphi(2,1)=\varphi(2,0,0)=\varphi(2,0,0,0)\)


<math>\varphi(1,\varphi(2,1)+1)=\varphi(\varphi(2,0,0),0)=\varphi(\varphi(2,0,0,0),0,0)</math>
\(\varphi(1,\varphi(2,1)+1)=\varphi(\varphi(2,0,0),0)=\varphi(\varphi(2,0,0,0),0,0)\)


<math>\varphi(1,\varphi(1,\varphi(2,1)+1))=\varphi(\varphi(\varphi(2,0,0),0),0)=\varphi(\varphi(\varphi(2,0,0,0),0,0),0,0)</math>
\(\varphi(1,\varphi(1,\varphi(2,1)+1))=\varphi(\varphi(\varphi(2,0,0),0),0)=\varphi(\varphi(\varphi(2,0,0,0),0,0),0,0)\)


<math>\varphi(2,2)=\varphi(3,0,0)=\varphi(3,0,0,0)</math>
\(\varphi(2,2)=\varphi(3,0,0)=\varphi(3,0,0,0)\)


<math>\varphi(2,\varphi(1))=\varphi(\varphi(1),0,0)=\varphi(\varphi(1,0),0,0,0)</math>
\(\varphi(2,\varphi(1))=\varphi(\varphi(1),0,0)=\varphi(\varphi(1,0),0,0,0)\)


<math>\varphi(2,\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2,0,0)=\varphi(\varphi(1,\varphi(1,0)),0,0,0)</math>
\(\varphi(2,\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2,0,0)=\varphi(\varphi(1,\varphi(1,0)),0,0,0)\)


<math>\varphi(2,\varphi(2))=\varphi(\varphi(2),0,0)=\varphi(\varphi(2,0),0,0,0)</math>
\(\varphi(2,\varphi(2))=\varphi(\varphi(2),0,0)=\varphi(\varphi(2,0),0,0,0)\)


<math>\varphi(2,\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1)),0,0)=\varphi(\varphi(\varphi(1,0),0),0,0,0)</math>
\(\varphi(2,\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1)),0,0)=\varphi(\varphi(\varphi(1,0),0),0,0,0)\)


<math>\varphi(2,\varphi(1,0))=\varphi(\varphi(1,0),0,0)=\varphi(\varphi(1,0,0),0,0,0)</math>
\(\varphi(2,\varphi(1,0))=\varphi(\varphi(1,0),0,0)=\varphi(\varphi(1,0,0),0,0,0)\)


<math>\varphi(2,\varphi(1,1))=\varphi(\varphi(2,0),0,0)=\varphi(\varphi(2,0,0),0,0,0)</math>
\(\varphi(2,\varphi(1,1))=\varphi(\varphi(2,0),0,0)=\varphi(\varphi(2,0,0),0,0,0)\)


<math>\varphi(2,\varphi(1,\varphi(1)))=\varphi(\varphi(\varphi(1),0),0,0)=\varphi(\varphi(\varphi(1,0),0,0),0,0,0)</math>
\(\varphi(2,\varphi(1,\varphi(1)))=\varphi(\varphi(\varphi(1),0),0,0)=\varphi(\varphi(\varphi(1,0),0,0),0,0,0)\)


<math>\varphi(2,\varphi(2,0))=\varphi(\varphi(1,0,0),0,0)=\varphi(\varphi(1,0,0,0),0,0,0)</math>
\(\varphi(2,\varphi(2,0))=\varphi(\varphi(1,0,0),0,0)=\varphi(\varphi(1,0,0,0),0,0,0)\)


<math>\varphi(2,\varphi(2,\varphi(2,0)))=\varphi(\varphi(\varphi(1,0,0),0,0),0,0)=\varphi(\varphi(\varphi(1,0,0,0),0,0,0),0,0,0)</math>
\(\varphi(2,\varphi(2,\varphi(2,0)))=\varphi(\varphi(\varphi(1,0,0),0,0),0,0)=\varphi(\varphi(\varphi(1,0,0,0),0,0,0),0,0,0)\)


<math>\varphi(3,0)=\varphi(1,0,0,0)=\varphi(1,0,0,0,0)=</math> [[LCO|η0]]
\(\varphi(3,0)=\varphi(1,0,0,0)=\varphi(1,0,0,0,0)=\) [[LCO|η0]]


<math>\varphi(1,\varphi(3,0)+1)=\varphi(\varphi(1,0,0,0),0)=\varphi(\varphi(1,0,0,0,0),0,0)</math>
\(\varphi(1,\varphi(3,0)+1)=\varphi(\varphi(1,0,0,0),0)=\varphi(\varphi(1,0,0,0,0),0,0)\)


<math>\varphi(2,\varphi(3,0)+1)=\varphi(\varphi(1,0,0,0),0,0)=\varphi(\varphi(1,0,0,0,0),0,0,0)</math>
\(\varphi(2,\varphi(3,0)+1)=\varphi(\varphi(1,0,0,0),0,0)=\varphi(\varphi(1,0,0,0,0),0,0,0)\)


<math>\varphi(3,1)=\varphi(2,0,0,0)=\varphi(2,0,0,0,0)</math>
\(\varphi(3,1)=\varphi(2,0,0,0)=\varphi(2,0,0,0,0)\)


<math>\varphi(3,\varphi(1))=\varphi(\varphi(1),0,0,0)=\varphi(\varphi(1,0),0,0,0,0)</math>
\(\varphi(3,\varphi(1))=\varphi(\varphi(1),0,0,0)=\varphi(\varphi(1,0),0,0,0,0)\)


<math>\varphi(3,\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1)),0,0,0)=\varphi(\varphi(\varphi(1,0),0),0,0,0,0)</math>
\(\varphi(3,\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1)),0,0,0)=\varphi(\varphi(\varphi(1,0),0),0,0,0,0)\)


<math>\varphi(3,\varphi(1,0))=\varphi(\varphi(1,0),0,0,0)=\varphi(\varphi(1,0,0),0,0,0,0)</math>
\(\varphi(3,\varphi(1,0))=\varphi(\varphi(1,0),0,0,0)=\varphi(\varphi(1,0,0),0,0,0,0)\)


<math>\varphi(3,\varphi(2,0))=\varphi(\varphi(1,0,0),0,0,0)=\varphi(\varphi(1,0,0,0),0,0,0,0)</math>
\(\varphi(3,\varphi(2,0))=\varphi(\varphi(1,0,0),0,0,0)=\varphi(\varphi(1,0,0,0),0,0,0,0)\)


<math>\varphi(3,\varphi(3,0))=\varphi(\varphi(1,0,0,0),0,0,0)=\varphi(\varphi(1,0,0,0,0),0,0,0,0)</math>
\(\varphi(3,\varphi(3,0))=\varphi(\varphi(1,0,0,0),0,0,0)=\varphi(\varphi(1,0,0,0,0),0,0,0,0)\)


<math>\varphi(4,0)=\varphi(1,0,0,0,0)=\varphi(1,0,0,0,0,0)</math>
\(\varphi(4,0)=\varphi(1,0,0,0,0)=\varphi(1,0,0,0,0,0)\)


<math>\varphi(\varphi(1),0)=\varphi(1</math>@<math>\varphi(1))=\varphi(1</math>@<math>\varphi(1,0))=</math> [[HCO]]
\(\varphi(\varphi(1),0)=\varphi(1@\varphi(1))=\varphi(1@\varphi(1,0))=\) [[HCO]]


<math>\varphi(1,\varphi(\varphi(1),0)+1)=\varphi(\varphi(1</math>@<math>\varphi(1)),0)=\varphi(\varphi(1</math>@<math>\varphi(1,0)),0,0)</math>
\(\varphi(1,\varphi(\varphi(1),0)+1)=\varphi(\varphi(1@\varphi(1)),0)=\varphi(\varphi(1@\varphi(1,0)),0,0)\)


<math>\varphi(2,\varphi(\varphi(1),0)+1)=\varphi(\varphi(1</math>@<math>\varphi(1)),0,0)=\varphi(\varphi(1</math>@<math>\varphi(1,0)),0,0,0)</math>
\(\varphi(2,\varphi(\varphi(1),0)+1)=\varphi(\varphi(1@\varphi(1)),0,0)=\varphi(\varphi(1@\varphi(1,0)),0,0,0)\)


<math>\varphi(3,\varphi(\varphi(1),0)+1)=\varphi(\varphi(1</math>@<math>\varphi(1)),0,0,0)=\varphi(\varphi(1</math>@<math>\varphi(1,0)),0,0,0,0)</math>
\(\varphi(3,\varphi(\varphi(1),0)+1)=\varphi(\varphi(1@\varphi(1)),0,0,0)=\varphi(\varphi(1@\varphi(1,0)),0,0,0,0)\)


<math>\varphi(\varphi(1),1)=\varphi(2</math>@<math>\varphi(1))=\varphi(2</math>@<math>\varphi(1,0))</math>
\(\varphi(\varphi(1),1)=\varphi(2@\varphi(1))=\varphi(2@\varphi(1,0))\)


<math>\varphi(1,\varphi(\varphi(1),1)+1)=\varphi(\varphi(2</math>@<math>\varphi(1)),0)=\varphi(\varphi(2</math>@<math>\varphi(1,0)),0,0)</math>
\(\varphi(1,\varphi(\varphi(1),1)+1)=\varphi(\varphi(2@\varphi(1)),0)=\varphi(\varphi(2@\varphi(1,0)),0,0)\)


<math>\varphi(\varphi(1),2)=\varphi(3</math>@<math>\varphi(1))=\varphi(3</math>@<math>\varphi(1,0))</math>
\(\varphi(\varphi(1),2)=\varphi(3@\varphi(1))=\varphi(3@\varphi(1,0))\)


<math>\varphi(\varphi(1),\varphi(1))=\varphi(\varphi(1)</math>@<math>\varphi(1))=\varphi(\varphi(1,0)</math>@<math>\varphi(1,0))</math>
\(\varphi(\varphi(1),\varphi(1))=\varphi(\varphi(1)@\varphi(1))=\varphi(\varphi(1,0)@\varphi(1,0))\)


<math>\varphi(\varphi(1),\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2</math>@<math>\varphi(1))=\varphi(\varphi(1,0)\cdot2</math>@<math>\varphi(1,0))</math>
\(\varphi(\varphi(1),\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2@\varphi(1))=\varphi(\varphi(1,0)\cdot2@\varphi(1,0))\)


<math>\varphi(\varphi(1),\varphi(2))=\varphi(\varphi(2)</math>@<math>\varphi(1))=\varphi(\varphi(2,0)</math>@<math>\varphi(1,0))</math>
\(\varphi(\varphi(1),\varphi(2))=\varphi(\varphi(2)@\varphi(1))=\varphi(\varphi(2,0)@\varphi(1,0))\)


<math>\varphi(\varphi(1),\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1))</math>@<math>\varphi(1))=\varphi(\varphi(\varphi(1,0),0)</math>@<math>\varphi(1,0))</math>
\(\varphi(\varphi(1),\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1))@\varphi(1))=\varphi(\varphi(\varphi(1,0),0)@\varphi(1,0))\)


<math>\varphi(\varphi(1),\varphi(1,0))=\varphi(\varphi(1,0)</math>@<math>\varphi(1))=\varphi(\varphi(1,0,0)</math>@<math>\varphi(1,0))</math>
\(\varphi(\varphi(1),\varphi(1,0))=\varphi(\varphi(1,0)@\varphi(1))=\varphi(\varphi(1,0,0)@\varphi(1,0))\)


<math>\varphi(\varphi(1),\varphi(2,0))=\varphi(\varphi(1,0,0)</math>@<math>\varphi(1))=\varphi(\varphi(1,0,0,0)</math>@<math>\varphi(1,0))</math>
\(\varphi(\varphi(1),\varphi(2,0))=\varphi(\varphi(1,0,0)@\varphi(1))=\varphi(\varphi(1,0,0,0)@\varphi(1,0))\)


<math>\varphi(\varphi(1),\varphi(3,0))=\varphi(\varphi(1,0,0,0)</math>@<math>\varphi(1))=\varphi(\varphi(1,0,0,0,0)</math>@<math>\varphi(1,0))</math>
\(\varphi(\varphi(1),\varphi(3,0))=\varphi(\varphi(1,0,0,0)@\varphi(1))=\varphi(\varphi(1,0,0,0,0)@\varphi(1,0))\)


<math>\varphi(\varphi(1),\varphi(\varphi(1),0))=\varphi(\varphi(1</math>@<math>\varphi(1))</math>@<math>\varphi(1))=\varphi(\varphi(1</math>@<math>\varphi(1,0))</math>@<math>\varphi(1,0))</math>
\(\varphi(\varphi(1),\varphi(\varphi(1),0))=\varphi(\varphi(1@\varphi(1))@\varphi(1))=\varphi(\varphi(1@\varphi(1,0))@\varphi(1,0))\)


<math>\varphi(\varphi(1)+1,0)=\varphi(1</math>@<math>\varphi(1)+1)=\varphi(1</math>@<math>\varphi(1,1))</math>
\(\varphi(\varphi(1)+1,0)=\varphi(1@\varphi(1)+1)=\varphi(1@\varphi(1,1))\)


<math>\varphi(\varphi(1)+1,\varphi(\varphi(1)+1,0))=\varphi(\varphi(1</math>@<math>\varphi(1)+1)</math>@<math>\varphi(1)+1)=\varphi(\varphi(1</math>@<math>\varphi(1,1))</math>@<math>\varphi(1,1))</math>
\(\varphi(\varphi(1)+1,\varphi(\varphi(1)+1,0))=\varphi(\varphi(1@\varphi(1)+1)@\varphi(1)+1)=\varphi(\varphi(1@\varphi(1,1))@\varphi(1,1))\)


<math>\varphi(\varphi(1)+2,0)=\varphi(1</math>@<math>\varphi(1)+2)=\varphi(1</math>@<math>\varphi(1,2))</math>
\(\varphi(\varphi(1)+2,0)=\varphi(1@\varphi(1)+2)=\varphi(1@\varphi(1,2))\)


<math>\varphi(\varphi(1)\cdot2,0)=\varphi(1</math>@<math>\varphi(1)\cdot2)=\varphi(1</math>@<math>\varphi(1,\varphi(1)))</math>
\(\varphi(\varphi(1)\cdot2,0)=\varphi(1@\varphi(1)\cdot2)=\varphi(1@\varphi(1,\varphi(1)))\)


<math>\varphi(\varphi(1)\cdot3,0)=\varphi(1</math>@<math>\varphi(1)\cdot3)=\varphi(1</math>@<math>\varphi(1,\varphi(1,\varphi(1,0))))</math>
\(\varphi(\varphi(1)\cdot3,0)=\varphi(1@\varphi(1)\cdot3)=\varphi(1@\varphi(1,\varphi(1,\varphi(1,0))))\)


<math>\varphi(\varphi(2),0)=\varphi(1</math>@<math>\varphi(2))=\varphi(1</math>@<math>\varphi(2,0))</math>
\(\varphi(\varphi(2),0)=\varphi(1@\varphi(2))=\varphi(1@\varphi(2,0))\)


<math>\varphi(\varphi(\varphi(1)),0)=\varphi(1</math>@<math>\varphi(\varphi(1)))=\varphi(1</math>@<math>\varphi(\varphi(1,0),0))</math>
\(\varphi(\varphi(\varphi(1)),0)=\varphi(1@\varphi(\varphi(1)))=\varphi(1@\varphi(\varphi(1,0),0))\)


<math>\varphi(\varphi(1,0),0)=\varphi(1</math>@<math>\varphi(1,0))=\varphi(1</math>@<math>\varphi(1,0,0))</math>
\(\varphi(\varphi(1,0),0)=\varphi(1@\varphi(1,0))=\varphi(1@\varphi(1,0,0))\)


<math>\varphi(\varphi(2,0),0)=\varphi(1</math>@<math>\varphi(1,0,0))=\varphi(1</math>@<math>\varphi(1,0,0,0))</math>
\(\varphi(\varphi(2,0),0)=\varphi(1@\varphi(1,0,0))=\varphi(1@\varphi(1,0,0,0))\)


<math>\varphi(\varphi(3,0),0)=\varphi(1</math>@<math>\varphi(1,0,0,0))=\varphi(1</math>@<math>\varphi(1,0,0,0,0))</math>
\(\varphi(\varphi(3,0),0)=\varphi(1@\varphi(1,0,0,0))=\varphi(1@\varphi(1,0,0,0,0))\)


<math>\varphi(\varphi(\varphi(1),0),0)=\varphi(1</math>@<math>\varphi(1</math>@<math>\varphi(1)))=\varphi(1</math>@<math>\varphi(1</math>@<math>\varphi(1,0)))</math>
\(\varphi(\varphi(\varphi(1),0),0)=\varphi(1@\varphi(1@\varphi(1)))=\varphi(1@\varphi(1@\varphi(1,0)))\)


<math>\varphi(\varphi(\varphi(1,0),0),0)=\varphi(1</math>@<math>\varphi(1</math>@<math>\varphi(1,0)))=\varphi(1</math>@<math>\varphi(1</math>@<math>\varphi(1,0,0)))</math>
\(\varphi(\varphi(\varphi(1,0),0),0)=\varphi(1@\varphi(1@\varphi(1,0)))=\varphi(1@\varphi(1@\varphi(1,0,0)))\)


<math>\varphi(\varphi(\varphi(2,0),0),0)=\varphi(1</math>@<math>\varphi(1</math>@<math>\varphi(1,0,0)))=\varphi(1</math>@<math>\varphi(1</math>@<math>\varphi(1,0,0,0)))</math>
\(\varphi(\varphi(\varphi(2,0),0),0)=\varphi(1@\varphi(1@\varphi(1,0,0)))=\varphi(1@\varphi(1@\varphi(1,0,0,0)))\)


<math>\varphi(1,0,0)=\varphi(1</math>@<math>(1,0))=\varphi(1</math>@<math>(1,0))=</math> [[FSO]]
\(\varphi(1,0,0)=\varphi(1@(1,0))=\varphi(1@(1,0))=\) [[FSO]]


<math>\varphi(1,\varphi(1,0,0)+1)=\varphi(\varphi(1</math>@<math>(1,0)),0)=\varphi(\varphi(1</math>@<math>(1,0)),0,0)</math>
\(\varphi(1,\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0)),0)=\varphi(\varphi(1@(1,0)),0,0)\)


<math>\varphi(2,\varphi(1,0,0)+1)=\varphi(\varphi(1</math>@<math>(1,0)),0,0)=\varphi(\varphi(1</math>@<math>(1,0)),0,0,0)</math>
\(\varphi(2,\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0)),0,0)=\varphi(\varphi(1@(1,0)),0,0,0)\)


<math>\varphi(\varphi(1),\varphi(1,0,0)+1)=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1))=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1,0))</math>
\(\varphi(\varphi(1),\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0))@\varphi(1))=\varphi(\varphi(1@(1,0))@\varphi(1,0))\)


<math>\varphi(\varphi(1,0),\varphi(1,0,0)+1)=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1,0))=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1,0,0))</math>
\(\varphi(\varphi(1,0),\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0))@\varphi(1,0))=\varphi(\varphi(1@(1,0))@\varphi(1,0,0))\)


<math>\varphi(\varphi(2,0),\varphi(1,0,0)+1)=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1,0,0))=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1,0,0,0))</math>
\(\varphi(\varphi(2,0),\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0))@\varphi(1,0,0))=\varphi(\varphi(1@(1,0))@\varphi(1,0,0,0))\)


<math>\varphi(\varphi(1,0,0),1)=\varphi(2</math>@<math>\varphi(1</math>@<math>(1,0)))=\varphi(2</math>@<math>\varphi(1</math>@<math>(1,0)))</math>
\(\varphi(\varphi(1,0,0),1)=\varphi(2@\varphi(1@(1,0)))=\varphi(2@\varphi(1@(1,0)))\)


<math>\varphi(\varphi(1,0,0),2)=\varphi(3</math>@<math>\varphi(1</math>@<math>(1,0)))=\varphi(3</math>@<math>\varphi(1</math>@<math>(1,0)))</math>
\(\varphi(\varphi(1,0,0),2)=\varphi(3@\varphi(1@(1,0)))=\varphi(3@\varphi(1@(1,0)))\)


<math>\varphi(\varphi(1,0,0),\varphi(1))=\varphi(\varphi(1)</math>@<math>\varphi(1</math>@<math>(1,0)))=\varphi(\varphi(1,0)</math>@<math>\varphi(1</math>@<math>(1,0)))</math>
\(\varphi(\varphi(1,0,0),\varphi(1))=\varphi(\varphi(1)@\varphi(1@(1,0)))=\varphi(\varphi(1,0)@\varphi(1@(1,0)))\)


<math>\varphi(\varphi(1,0,0),\varphi(1,0))=\varphi(\varphi(1,0)</math>@<math>\varphi(1</math>@<math>(1,0)))=\varphi(\varphi(1,0,0)</math>@<math>\varphi(1</math>@<math>(1,0)))</math>
\(\varphi(\varphi(1,0,0),\varphi(1,0))=\varphi(\varphi(1,0)@\varphi(1@(1,0)))=\varphi(\varphi(1,0,0)@\varphi(1@(1,0)))\)


<math>\varphi(\varphi(1,0,0),\varphi(1,0,0))=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1</math>@<math>(1,0)))=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1</math>@<math>(1,0)))</math>
\(\varphi(\varphi(1,0,0),\varphi(1,0,0))=\varphi(\varphi(1@(1,0))@\varphi(1@(1,0)))=\varphi(\varphi(1@(1,0))@\varphi(1@(1,0)))\)


<math>\varphi(\varphi(1,0,0)+1,0)=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0))+1)=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0),1</math>@<math>0))</math>
\(\varphi(\varphi(1,0,0)+1,0)=\varphi(1@\varphi(1@(1,0))+1)=\varphi(1@\varphi(1@(1,0),1@0))\)


<math>\varphi(\varphi(1,0,0)+\varphi(1),0)=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0))+\varphi(1))=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0),\varphi(1,0)</math>@<math>0))</math>
\(\varphi(\varphi(1,0,0)+\varphi(1),0)=\varphi(1@\varphi(1@(1,0))+\varphi(1))=\varphi(1@\varphi(1@(1,0),\varphi(1,0)@0))\)


<math>\varphi(\varphi(1,0,0)+\varphi(1,0),0)=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0))+\varphi(1,0))=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0),\varphi(1,0,0)</math>@<math>0))</math>
\(\varphi(\varphi(1,0,0)+\varphi(1,0),0)=\varphi(1@\varphi(1@(1,0))+\varphi(1,0))=\varphi(1@\varphi(1@(1,0),\varphi(1,0,0)@0))\)


<math>\varphi(\varphi(1,0,0)\cdot2,0)=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0))\cdot2)=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0),\varphi(1</math>@<math>(1,0))</math>@<math>0))</math>
\(\varphi(\varphi(1,0,0)\cdot2,0)=\varphi(1@\varphi(1@(1,0))\cdot2)=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0))@0))\)


<math>\varphi(\varphi(1,0,0)\cdot\varphi(1),0)=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0),1</math>@<math>0))=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0),1</math>@<math>1))</math>
\(\varphi(\varphi(1,0,0)\cdot\varphi(1),0)=\varphi(1@\varphi(1@(1,0),1@0))=\varphi(1@\varphi(1@(1,0),1@1))\)


<math>\varphi(\varphi(1,0,0)^{2},0)=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0),\varphi(1</math>@<math>(1,0))</math>@<math>0))=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0),\varphi(1</math>@<math>(1,0))</math>@<math>0))</math>
\(\varphi(\varphi(1,0,0)^{2},0)=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0))@0))=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0))@0))\)


<math>\varphi(\varphi(1,0,0)^{\varphi(1)},0)=\varphi(1</math>@<math>\varphi(1,\varphi(1</math>@<math>(1,0),1</math>@<math>0),0))=\varphi(1</math>@<math>\varphi(1,\varphi(1</math>@<math>(1,0),1</math>@<math>1),0))</math>
\(\varphi(\varphi(1,0,0)^{\varphi(1)},0)=\varphi(1@\varphi(1,\varphi(1@(1,0),1@0),0))=\varphi(1@\varphi(1,\varphi(1@(1,0),1@1),0))\)


<math>\varphi(\varphi(1,0,0)^{\varphi(1,0)},0)=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0),\varphi(1</math>@<math>(1,0),\varphi(1,0)</math>@<math>0)</math>@<math>0))=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0),\varphi(1</math>@<math>(1,0),\varphi(1,0)</math>@<math>0)</math>@<math>0))</math>
\(\varphi(\varphi(1,0,0)^{\varphi(1,0)},0)=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0),\varphi(1,0)@0)@0))=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0),\varphi(1,0)@0)@0))\)


<math>\varphi(\varphi(1,0,0)^{\varphi(1,0,0)},0)=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0),\varphi(1</math>@<math>(1,0),\varphi(1</math>@<math>(1,0))</math>@<math>0)</math>@<math>0))=\varphi(1</math>@<math>\varphi(1</math>@<math>(1,0),\varphi(1</math>@<math>(1,0),\varphi(1</math>@<math>(1,0))</math>@<math>0)</math>@<math>0))</math>
\(\varphi(\varphi(1,0,0)^{\varphi(1,0,0)},0)=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0),\varphi(1@(1,0))@0)@0))=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0),\varphi(1@(1,0))@0)@0))\)


<math>\varphi(\varphi(1,\varphi(1,0,0)+1),0)=\varphi(1</math>@<math>\varphi(\varphi(1</math>@<math>(1,0)),0))=\varphi(1</math>@<math>\varphi(\varphi(1</math>@<math>(1,0)),0,0))</math>
\(\varphi(\varphi(1,\varphi(1,0,0)+1),0)=\varphi(1@\varphi(\varphi(1@(1,0)),0))=\varphi(1@\varphi(\varphi(1@(1,0)),0,0))\)


<math>\varphi(\varphi(1,\varphi(1,0,0)+\varphi(1)),0)=\varphi(1</math>@<math>\varphi(\varphi(1</math>@<math>(1,0))+\varphi(1),0))=\varphi(1</math>@<math>\varphi(\varphi(1</math>@<math>(1,0))+\varphi(1,0),0))</math>
\(\varphi(\varphi(1,\varphi(1,0,0)+\varphi(1)),0)=\varphi(1@\varphi(\varphi(1@(1,0))+\varphi(1),0))=\varphi(1@\varphi(\varphi(1@(1,0))+\varphi(1,0),0))\)


<math>\varphi(\varphi(\varphi(1),\varphi(1,0,0)+1),0)=\varphi(1</math>@<math>\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1)))=\varphi(1</math>@<math>\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>\varphi(1,0)))</math>
\(\varphi(\varphi(\varphi(1),\varphi(1,0,0)+1),0)=\varphi(1@\varphi(\varphi(1@(1,0))@\varphi(1)))=\varphi(1@\varphi(\varphi(1@(1,0))@\varphi(1,0)))\)


<math>\varphi(1,0,1)=\varphi(1</math>@<math>(1,1))=\varphi(1</math>@<math>(1,1))</math>
\(\varphi(1,0,1)=\varphi(1@(1,1))=\varphi(1@(1,1))\)


<math>\varphi(1,0,2)=\varphi(2</math>@<math>(1,1))=\varphi(2</math>@<math>(1,1))</math>
\(\varphi(1,0,2)=\varphi(2@(1,1))=\varphi(2@(1,1))\)


<math>\varphi(1,0,\varphi(1))=\varphi(\varphi(1)</math>@<math>(1,1))=\varphi(\varphi(1,0)</math>@<math>(1,1))</math>
\(\varphi(1,0,\varphi(1))=\varphi(\varphi(1)@(1,1))=\varphi(\varphi(1,0)@(1,1))\)


<math>\varphi(1,0,\varphi(1,0))=\varphi(\varphi(1,0)</math>@<math>(1,1))=\varphi(\varphi(1,0,0)</math>@<math>(1,1))</math>
\(\varphi(1,0,\varphi(1,0))=\varphi(\varphi(1,0)@(1,1))=\varphi(\varphi(1,0,0)@(1,1))\)


<math>\varphi(1,0,\varphi(1,0,0))=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>(1,1))=\varphi(\varphi(1</math>@<math>(1,0))</math>@<math>(1,1))</math>
\(\varphi(1,0,\varphi(1,0,0))=\varphi(\varphi(1@(1,0))@(1,1))=\varphi(\varphi(1@(1,0))@(1,1))\)


<math>\varphi(1,1,0)=\varphi(1</math>@<math>(1,1))=\varphi(1</math>@<math>(1,1))</math>
\(\varphi(1,1,0)=\varphi(1@(1,1))=\varphi(1@(1,1))\)


<math>\varphi(1,1,\varphi(1,1,0))=\varphi(\varphi(1</math>@<math>(1,1))</math>@<math>(1,1))=\varphi(\varphi(1</math>@<math>(1,1))</math>@<math>(1,1))</math>
\(\varphi(1,1,\varphi(1,1,0))=\varphi(\varphi(1@(1,1))@(1,1))=\varphi(\varphi(1@(1,1))@(1,1))\)


<math>\varphi(1,2,0)=\varphi(1</math>@<math>(1,2))=\varphi(1</math>@<math>(1,2))</math>
\(\varphi(1,2,0)=\varphi(1@(1,2))=\varphi(1@(1,2))\)


<math>\varphi(1,\varphi(1),0)=\varphi(1</math>@<math>(1,\varphi(1)))=\varphi(1</math>@<math>(1,\varphi(1,0)))</math>
\(\varphi(1,\varphi(1),0)=\varphi(1@(1,\varphi(1)))=\varphi(1@(1,\varphi(1,0)))\)


<math>\varphi(1,\varphi(1,0),0)=\varphi(1</math>@<math>(1,\varphi(1,0)))=\varphi(1</math>@<math>(1,\varphi(1,0,0)))</math>
\(\varphi(1,\varphi(1,0),0)=\varphi(1@(1,\varphi(1,0)))=\varphi(1@(1,\varphi(1,0,0)))\)


<math>\varphi(1,\varphi(1,0,0),0)=\varphi(1</math>@<math>(1,\varphi(1</math>@<math>(1,0))))=\varphi(1</math>@<math>(1,\varphi(1</math>@<math>(1,0))))</math>
\(\varphi(1,\varphi(1,0,0),0)=\varphi(1@(1,\varphi(1@(1,0))))=\varphi(1@(1,\varphi(1@(1,0))))\)


<math>\varphi(2,0,0)=\varphi(1</math>@<math>(2,0))=\varphi(1</math>@<math>(2,0))</math>
\(\varphi(2,0,0)=\varphi(1@(2,0))=\varphi(1@(2,0))\)


<math>\varphi(2,0,1)=\varphi(2</math>@<math>(2,0))=\varphi(2</math>@<math>(2,0))</math>
\(\varphi(2,0,1)=\varphi(2@(2,0))=\varphi(2@(2,0))\)


<math>\varphi(2,1,0)=\varphi(1</math>@<math>(2,1))=\varphi(1</math>@<math>(2,1))</math>
\(\varphi(2,1,0)=\varphi(1@(2,1))=\varphi(1@(2,1))\)


<math>\varphi(2,\varphi(2,0,0),0)=\varphi(1</math>@<math>(2,\varphi(1</math>@<math>(2,0))))=\varphi(1</math>@<math>(2,\varphi(1</math>@<math>(2,0))))</math>
\(\varphi(2,\varphi(2,0,0),0)=\varphi(1@(2,\varphi(1@(2,0))))=\varphi(1@(2,\varphi(1@(2,0))))\)


<math>\varphi(3,0,0)=\varphi(1</math>@<math>(3,0))=\varphi(1</math>@<math>(3,0))</math>
\(\varphi(3,0,0)=\varphi(1@(3,0))=\varphi(1@(3,0))\)


<math>\varphi(\varphi(1),0,0)=\varphi(1</math>@<math>(\varphi(1),0))=\varphi(1</math>@<math>(\varphi(1,0),0))</math>
\(\varphi(\varphi(1),0,0)=\varphi(1@(\varphi(1),0))=\varphi(1@(\varphi(1,0),0))\)


<math>\varphi(\varphi(1,0),0,0)=\varphi(1</math>@<math>(\varphi(1,0),0))=\varphi(1</math>@<math>(\varphi(1,0,0),0))</math>
\(\varphi(\varphi(1,0),0,0)=\varphi(1@(\varphi(1,0),0))=\varphi(1@(\varphi(1,0,0),0))\)


<math>\varphi(\varphi(1,0,0),0,0)=\varphi(1</math>@<math>(\varphi(1</math>@<math>(1,0)),0))=\varphi(1</math>@<math>(\varphi(1</math>@<math>(1,0)),0))</math>
\(\varphi(\varphi(1,0,0),0,0)=\varphi(1@(\varphi(1@(1,0)),0))=\varphi(1@(\varphi(1@(1,0)),0))\)


<math>\varphi(1,0,0,0)=\varphi(1</math>@<math>(1,0,0))=\varphi(1</math>@<math>(1,0,0))=</math> [[ACO]]
\(\varphi(1,0,0,0)=\varphi(1@(1,0,0))=\varphi(1@(1,0,0))=\) [[ACO]]


<math>\varphi(1,0,0,1)=\varphi(2</math>@<math>(1,0,0))=\varphi(2</math>@<math>(1,0,0))</math>
\(\varphi(1,0,0,1)=\varphi(2@(1,0,0))=\varphi(2@(1,0,0))\)


<math>\varphi(1,0,1,0)=\varphi(1</math>@<math>(1,0,1))=\varphi(1</math>@<math>(1,0,1))</math>
\(\varphi(1,0,1,0)=\varphi(1@(1,0,1))=\varphi(1@(1,0,1))\)


<math>\varphi(1,1,0,0)=\varphi(1</math>@<math>(1,1,0))=\varphi(1</math>@<math>(1,1,0))</math>
\(\varphi(1,1,0,0)=\varphi(1@(1,1,0))=\varphi(1@(1,1,0))\)


<math>\varphi(2,0,0,0)=\varphi(1</math>@<math>(2,0,0))=\varphi(1</math>@<math>(2,0,0))</math>
\(\varphi(2,0,0,0)=\varphi(1@(2,0,0))=\varphi(1@(2,0,0))\)


<math>\varphi(\varphi(1),0,0,0)=\varphi(1</math>@<math>(\varphi(1),0,0))=\varphi(1</math>@<math>(\varphi(1,0),0,0))</math>
\(\varphi(\varphi(1),0,0,0)=\varphi(1@(\varphi(1),0,0))=\varphi(1@(\varphi(1,0),0,0))\)


<math>\varphi(\varphi(1,0,0,0),0,0,0)=\varphi(1</math>@<math>(\varphi(1</math>@<math>(1,0,0)),0,0))=\varphi(1</math>@<math>(\varphi(1</math>@<math>(1,0,0)),0,0))</math>
\(\varphi(\varphi(1,0,0,0),0,0,0)=\varphi(1@(\varphi(1@(1,0,0)),0,0))=\varphi(1@(\varphi(1@(1,0,0)),0,0))\)


<math>\varphi(1,0,0,0,0)=\varphi(1</math>@<math>(1,0,0,0))=\varphi(1</math>@<math>(1,0,0,0))</math>
\(\varphi(1,0,0,0,0)=\varphi(1@(1,0,0,0))=\varphi(1@(1,0,0,0))\)


<math>\varphi(1,0,0,0,0,0)=\varphi(1</math>@<math>(1,0,0,0,0))=\varphi(1</math>@<math>(1,0,0,0,0))</math>
\(\varphi(1,0,0,0,0,0)=\varphi(1@(1,0,0,0,0))=\varphi(1@(1,0,0,0,0))\)


<math>\varphi(1,1,4,5,1,4)=\varphi(5</math>@<math>(1,1,4,5,1))=\varphi(5</math>@<math>(1,1,4,5,1))</math>
\(\varphi(1,1,4,5,1,4)=\varphi(5@(1,1,4,5,1))=\varphi(5@(1,1,4,5,1))\)


<math>\varphi(1</math>@<math>\varphi(1))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1)))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,0)))=</math> [[SVO]]
\(\varphi(1@\varphi(1))=\varphi(1@(1@\varphi(1)))=\varphi(1@(1@\varphi(1,0)))=\) [[SVO]]


<math>\varphi(1,\varphi(1</math>@<math>\varphi(1))+1)=\varphi(1</math>@<math>(1</math>@<math>\varphi(1)),1</math>@<math>1)=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,0)),1</math>@<math>2)</math>
\(\varphi(1,\varphi(1@\varphi(1))+1)=\varphi(1@(1@\varphi(1)),1@1)=\varphi(1@(1@\varphi(1,0)),1@2)\)


<math>\varphi(1</math>@<math>\varphi(1),1</math>@<math>0)=\varphi(2</math>@<math>(1</math>@<math>\varphi(1)))=\varphi(2</math>@<math>(1</math>@<math>\varphi(1,0)))</math>
\(\varphi(1@\varphi(1),1@0)=\varphi(2@(1@\varphi(1)))=\varphi(2@(1@\varphi(1,0)))\)


<math>\varphi(1</math>@<math>\varphi(1),1</math>@<math>1)=\varphi(1</math>@<math>(1</math>@<math>\varphi(1),1</math>@<math>0))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,0),1</math>@<math>0))</math>
\(\varphi(1@\varphi(1),1@1)=\varphi(1@(1@\varphi(1),1@0))=\varphi(1@(1@\varphi(1,0),1@0))\)


<math>\varphi(1</math>@<math>\varphi(1),1</math>@<math>2)=\varphi(1</math>@<math>(1</math>@<math>\varphi(1),1</math>@<math>1))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,0),1</math>@<math>1))</math>
\(\varphi(1@\varphi(1),1@2)=\varphi(1@(1@\varphi(1),1@1))=\varphi(1@(1@\varphi(1,0),1@1))\)


<math>\varphi(2</math>@<math>\varphi(1))=\varphi(1</math>@<math>(2</math>@<math>\varphi(1)))=\varphi(1</math>@<math>(2</math>@<math>\varphi(1,0)))</math>
\(\varphi(2@\varphi(1))=\varphi(1@(2@\varphi(1)))=\varphi(1@(2@\varphi(1,0)))\)


<math>\varphi(\varphi(1)</math>@<math>\varphi(1))=\varphi(1</math>@<math>(\varphi(1)</math>@<math>\varphi(1)))=\varphi(1</math>@<math>(\varphi(1,0)</math>@<math>\varphi(1,0)))</math>
\(\varphi(\varphi(1)@\varphi(1))=\varphi(1@(\varphi(1)@\varphi(1)))=\varphi(1@(\varphi(1,0)@\varphi(1,0)))\)


<math>\varphi(1</math>@<math>\varphi(1)+1)=\varphi(1</math>@<math>(1</math>@<math>\varphi(1)+1))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,1)))</math>
\(\varphi(1@\varphi(1)+1)=\varphi(1@(1@\varphi(1)+1))=\varphi(1@(1@\varphi(1,1)))\)


<math>\varphi(1</math>@<math>\varphi(1)\cdot2)=\varphi(1</math>@<math>(1</math>@<math>\varphi(1)\cdot2))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,\varphi(1,0))))</math>
\(\varphi(1@\varphi(1)\cdot2)=\varphi(1@(1@\varphi(1)\cdot2))=\varphi(1@(1@\varphi(1,\varphi(1,0))))\)


<math>\varphi(1</math>@<math>\varphi(2))=\varphi(1</math>@<math>(1</math>@<math>\varphi(2)))=\varphi(1</math>@<math>(1</math>@<math>\varphi(2,0)))</math>
\(\varphi(1@\varphi(2))=\varphi(1@(1@\varphi(2)))=\varphi(1@(1@\varphi(2,0)))\)


<math>\varphi(1</math>@<math>\varphi(1,0))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,0)))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,0,0)))</math>
\(\varphi(1@\varphi(1,0))=\varphi(1@(1@\varphi(1,0)))=\varphi(1@(1@\varphi(1,0,0)))\)


<math>\varphi(1</math>@<math>\varphi(2,0))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,0,0)))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1,0,0,0)))</math>
\(\varphi(1@\varphi(2,0))=\varphi(1@(1@\varphi(1,0,0)))=\varphi(1@(1@\varphi(1,0,0,0)))\)


<math>\varphi(1</math>@<math>\varphi(1,0,0))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1</math>@<math>(1,0))))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1</math>@<math>(1,0))))</math>
\(\varphi(1@\varphi(1,0,0))=\varphi(1@(1@\varphi(1@(1,0))))=\varphi(1@(1@\varphi(1@(1,0))))\)


<math>\varphi(1</math>@<math>\varphi(1</math>@<math>\varphi(1)))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1</math>@<math>(1</math>@<math>\varphi(1)))))=\varphi(1</math>@<math>(1</math>@<math>\varphi(1</math>@<math>(1</math>@<math>\varphi(1,0)))))=</math> [[LVO]]
\(\varphi(1@\varphi(1@\varphi(1)))=\varphi(1@(1@\varphi(1@(1@\varphi(1)))))=\varphi(1@(1@\varphi(1@(1@\varphi(1,0)))))=\) [[LVO]]


=== 历史 ===
=== 历史 ===

2025年8月26日 (二) 18:30的最新版本

weak Veblen 函数（又称“弱 φ”、“弱 Veblen”等），是 Veblen 函数的变体。

定义

现在有两种 Veblen 函数的变体可被称为 weak Veblen。

如果称 Veblen 函数的末位 + 1 等价于跳到下一个 ε 点（不动点进制），定义 Veblen 函数为 FP φ，那么可以直观地定义下面两种weak Veblen：

  • 若 weak Veblen 函数的末位 + 1 对应于其序数值 ×ω，则称其为 ×ω ϕ。
  • 若 weak Veblen 函数的末位 + 1 对应于其序数值 +1，则称其为 +1 φ。

以下是两种 weak Veblen 直到序元级别的定义：

+1 ϕ

  1. \(\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)[0]=0\)。
  2. \(\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)。[n+1]=\varphi(\#,\alpha\text{@}(\beta+1),\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)[n]\text{@}\beta)\)。
  3. 对于极限序数\(\alpha\)，\(\varphi(\#,\alpha\text{@}\beta)[n]=\varphi(\#,\alpha[n]\text{@}\beta)\)。
  4. 对于极限序数\(\beta\)，\(\varphi(\#,(\alpha+1)\text{@}\beta,0\text{@}0)。[n]=\varphi(\#,\alpha\text{@}\beta,1\text{@}\beta[n])\)。
  5. \(\varphi(\#,(\gamma+1)\text{@}0)=\varphi(\#,\gamma\text{@}0)+1\)。

×ω ϕ

  1. \(\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)[0]=0\)。
  2. \(\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)。[n+1]=\varphi(\#,\alpha\text{@}(\beta+1),\varphi(\#,(\alpha+1)\text{@}(\beta+1),0\text{@}0)[n]\text{@}\beta)\)。
  3. 对于极限序数\(\alpha\)，\(\varphi(\#,\alpha\text{@}\beta)[n]=\varphi(\#,\alpha[n]\text{@}\beta)\)。
  4. 对于极限序数\(\beta\)，\(\varphi(\#,(\alpha+1)\text{@}\beta,0\text{@}0)[n]=\varphi(\#,\alpha\text{@}\beta,1\text{@}\beta[n])\)。
  5. \(\varphi(\#,(\gamma+1)\text{@}0)=\varphi(\#,\gamma\text{@}0)\cdot\varphi(1)\)。

分析

以下分析中，左为 FP ϕ，中为 ×ω ϕ，右为 +1 ϕ。

\(\varphi(0)=\varphi(0)=\varphi(0)\)

\(\varphi(0)\cdot2=\varphi(0)\cdot2=\varphi(\varphi(0))\)

\(\varphi(0)\cdot3=\varphi(0)\cdot3=\varphi(\varphi(\varphi(0)))\)

\(\varphi(1)=\varphi(1)=\varphi(1,0)=\) ω

\(\varphi(1)+1=\varphi(1)+1=\varphi(1,1)\)

\(\varphi(1)+2=\varphi(1)+2=\varphi(1,2)\)

\(\varphi(1)\cdot2=\varphi(1)\cdot2=\varphi(1,\varphi(1,0))\)

\(\varphi(1)\cdot2+1=\varphi(1)\cdot2+1=\varphi(1,\varphi(1,1))\)

\(\varphi(1)\cdot2+2=\varphi(1)\cdot2+2=\varphi(1,\varphi(1,2))\)

\(\varphi(1)\cdot3=\varphi(1)\cdot3=\varphi(1,\varphi(1,\varphi(1,0)))\)

\(\varphi(2)=\varphi(2)=\varphi(2,0)\)

\(\varphi(2)+1=\varphi(2)+1=\varphi(2,1)\)

\(\varphi(2)+2=\varphi(2)+2=\varphi(2,2)\)

\(\varphi(2)+\varphi(1)=\varphi(2)+\varphi(1)=\varphi(2,\varphi(1,0))\)

\(\varphi(2)\cdot2=\varphi(2)\cdot2=\varphi(2,\varphi(2,0))\)

\(\varphi(2)\cdot3=\varphi(2)\cdot3=\varphi(2,\varphi(2,\varphi(2,0)))\)

\(\varphi(3)=\varphi(3)=\varphi(3,0)\)

\(\varphi(3)+\varphi(2)=\varphi(3)+\varphi(2)=\varphi(3,\varphi(2,0))\)

\(\varphi(4)=\varphi(4)=\varphi(4,0)\)

\(\varphi(\varphi(1))=\varphi(\varphi(1))=\varphi(\varphi(1,0),0)\)

\(\varphi(\varphi(1))+1=\varphi(\varphi(1))+1=\varphi(\varphi(1,0),1)\)

\(\varphi(\varphi(1))+\varphi(1)=\varphi(\varphi(1))+\varphi(1)=\varphi(\varphi(1,0),\varphi(1,0))\)

\(\varphi(\varphi(1))+\varphi(2)=\varphi(\varphi(1))+\varphi(2)=\varphi(\varphi(1,0),\varphi(2,0))\)

\(\varphi(\varphi(1))\cdot2=\varphi(\varphi(1))\cdot2=\varphi(\varphi(1,0),\varphi(\varphi(1,0),0))\)

\(\varphi(\varphi(1)+1)=\varphi(\varphi(1)+1)=\varphi(\varphi(1,1),0)\)

\(\varphi(\varphi(1)+2)=\varphi(\varphi(1)+2)=\varphi(\varphi(1,2),0)\)

\(\varphi(\varphi(1)+3)=\varphi(\varphi(1)+3)=\varphi(\varphi(1,3),0)\)

\(\varphi(\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2)=\varphi(\varphi(1,\varphi(1,0)),0)\)

\(\varphi(\varphi(1)\cdot2+1)=\varphi(\varphi(1)\cdot2+1)=\varphi(\varphi(1,\varphi(1,1)),0)\)

\(\varphi(\varphi(1)\cdot3)=\varphi(\varphi(1)\cdot3)=\varphi(\varphi(1,\varphi(1,\varphi(1,0))),0)\)

\(\varphi(\varphi(2))=\varphi(\varphi(2))=\varphi(\varphi(2,0),0)\)

\(\varphi(\varphi(2)+1)=\varphi(\varphi(2)+1)=\varphi(\varphi(2,1),0)\)

\(\varphi(\varphi(2)+\varphi(1))=\varphi(\varphi(2)+\varphi(1))=\varphi(\varphi(2,\varphi(1,0)),0)\)

\(\varphi(\varphi(3))=\varphi(\varphi(3))=\varphi(\varphi(3,0),0)\)

\(\varphi(\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1,0),0),0)\)

\(\varphi(\varphi(\varphi(1))+1)=\varphi(\varphi(\varphi(1))+1)=\varphi(\varphi(\varphi(1,0),1),0)\)

\(\varphi(\varphi(\varphi(1))+\varphi(1))=\varphi(\varphi(\varphi(1))+\varphi(1))=\varphi(\varphi(\varphi(1,0),\varphi(1,0)),0)\)

\(\varphi(\varphi(\varphi(1))\cdot2)=\varphi(\varphi(\varphi(1))\cdot2)=\varphi(\varphi(\varphi(1,0),\varphi(\varphi(1,0),0)),0)\)

\(\varphi(\varphi(\varphi(1)+1))=\varphi(\varphi(\varphi(1)+1))=\varphi(\varphi(\varphi(1,1),0),0)\)

\(\varphi(\varphi(\varphi(2)))=\varphi(\varphi(\varphi(2)))=\varphi(\varphi(\varphi(2,0),0),0)\)

\(\varphi(\varphi(\varphi(3)))=\varphi(\varphi(\varphi(3)))=\varphi(\varphi(\varphi(3,0),0),0)\)

\(\varphi(\varphi(\varphi(\varphi(1))))=\varphi(\varphi(\varphi(\varphi(1))))=\varphi(\varphi(\varphi(\varphi(1,0),0),0),0)\)

\(\varphi(\varphi(\varphi(\varphi(\varphi(1)))))=\varphi(\varphi(\varphi(\varphi(\varphi(1)))))=\varphi(\varphi(\varphi(\varphi(\varphi(1,0),0),0),0),0)\)

\(\varphi(1,0)=\varphi(1,0)=\varphi(1,0,0)=\) ε0

\(\varphi(1,0)+1=\varphi(1,0)+1=\varphi(1,0,1)\)

\(\varphi(1,0)+\varphi(1)=\varphi(1,0)+\varphi(1)=\varphi(1,0,\varphi(1,0))\)

\(\varphi(1,0)\cdot2=\varphi(1,0)\cdot2=\varphi(1,0,\varphi(1,0,0))\)

\(\varphi(1,0)\cdot3=\varphi(1,0)\cdot3=\varphi(1,0,\varphi(1,0,\varphi(1,0,0)))\)

\(\varphi(1,0)\cdot\varphi(1)=\varphi(1,1)=\varphi(1,1,0)\)

\(\varphi(1,0)\cdot\varphi(1)\cdot2=\varphi(1,1)\cdot2=\varphi(1,1,\varphi(1,1,0))\)

\(\varphi(1,0)\cdot\varphi(2)=\varphi(1,2)=\varphi(1,2,0)\)

\(\varphi(1,0)\cdot\varphi(\varphi(1))=\varphi(1,\varphi(1))=\varphi(1,\varphi(1,0),0)\)

\(\varphi(1,0)\cdot\varphi(\varphi(1)+1)=\varphi(1,\varphi(1)+1)=\varphi(1,\varphi(1,1),0)\)

\(\varphi(1,0)\cdot\varphi(\varphi(2))=\varphi(1,\varphi(2))=\varphi(1,\varphi(2,0),0)\)

\(\varphi(1,0)\cdot\varphi(\varphi(\varphi(1)))=\varphi(1,\varphi(\varphi(1)))=\varphi(1,\varphi(\varphi(1,0),0),0)\)

\(\varphi(1,0)^{2}=\varphi(1,\varphi(1,0))=\varphi(1,\varphi(1,0,0),0)\)

\(\varphi(1,0)^{2}\cdot\varphi(1)=\varphi(1,\varphi(1,0)+1)=\varphi(1,\varphi(\varphi(1,0,0)),0)\)

\(\varphi(1,0)^{2}\cdot\varphi(\varphi(\varphi(1)))=\varphi(1,\varphi(1,0)+\varphi(\varphi(1)))=\varphi(1,\varphi(1,0,\varphi(\varphi(1,0),0)),0)\)

\(\varphi(1,0)^{3}=\varphi(1,\varphi(1,0)\cdot2)=\varphi(1,\varphi(1,0,\varphi(1,0,0),0),0)\)

\(\varphi(1,0)^{\varphi(1)}=\varphi(1,\varphi(1,1))=\varphi(1,\varphi(1,1,0),0)\)

\(\varphi(1,0)^{\varphi(1)}\cdot\varphi(1)=\varphi(1,\varphi(1,1)+1)=\varphi(1,\varphi(\varphi(1,1,0)),0)\)

\(\varphi(1,0)^{\varphi(1)}\cdot\varphi(\varphi(1))=\varphi(1,\varphi(1,1)+2)=\varphi(1,\varphi(\varphi(\varphi(1,1,0))),0)\)

\(\varphi(1,0)^{(\varphi(1)+1)}=\varphi(1,\varphi(1,1)+\varphi(1,0))=\varphi(1,\varphi(1,1,\varphi(1,0,0)),0)\)

\(\varphi(1,0)^{(\varphi(1)\cdot2)}=\varphi(1,\varphi(1,1)\cdot2)=\varphi(1,\varphi(1,1,\varphi(1,1,0)),0)\)

\(\varphi(1,0)^{\varphi(1)^{2}}=\varphi(1,\varphi(1,2))=\varphi(1,\varphi(1,2,0),0)\)

\(\varphi(1,0)^{\varphi(1)^{\varphi(1)}}=\varphi(1,\varphi(1,\varphi(1)))=\varphi(1,\varphi(1,\varphi(1,0),0),0)\)

\(\varphi(1,0)^{\varphi(1,0)}=\varphi(1,\varphi(1,\varphi(1,0)))=\varphi(1,\varphi(1,\varphi(1,0,0),0),0)\)

\(\varphi(1,0)^{(\varphi(1,0)+1)}=\varphi(1,\varphi(1,\varphi(1,0))+\varphi(1,0))=\varphi(1,\varphi(1,\varphi(1,0,0),\varphi(1,0,0)),0)\)

\(\varphi(1,0)^{(\varphi(1,0)\cdot\varphi(1))}=\varphi(1,\varphi(1,\varphi(1,0))+\varphi(1,1))=\varphi(1,\varphi(1,\varphi(1,0,0),\varphi(1,1,0)),0)\)

\(\varphi(1,0)^{\varphi(1,0)^{2}}=\varphi(1,\varphi(1,\varphi(1,2)))=\varphi(1,\varphi(1,\varphi(1,2,0),0),0)\)

\(\varphi(1,0)^{\varphi(1,0)^{\varphi(1)}}=\varphi(1,\varphi(1,\varphi(1,\varphi(1))))=\varphi(1,\varphi(1,\varphi(1,\varphi(1,0),0),0),0)\)

\(\varphi(1,0)^{\varphi(1,0)^{\varphi(1,0)}}=\varphi(1,\varphi(1,\varphi(1,\varphi(1,0))))=\varphi(1,\varphi(1,\varphi(1,\varphi(1,0,0),0),0),0)\)

\(\varphi(1,1)=\varphi(2,0)=\varphi(2,0,0)\)

\(\varphi(1,1)\cdot\varphi(1,0)=\varphi(2,\varphi(1,0))=\varphi(2,\varphi(1,0,0),0)\)

\(\varphi(1,1)\cdot\varphi(1,0)^{\varphi(1)}=\varphi(2,\varphi(1,\varphi(1,1)))=\varphi(2,\varphi(1,\varphi(1,1,0),0),0)\)

\(\varphi(1,1)\cdot\varphi(1,0)^{\varphi(1,0)}=\varphi(2,\varphi(1,\varphi(1,\varphi(1,0))))=\varphi(2,\varphi(1,\varphi(1,\varphi(1,0,0),0),0),0)\)

\(\varphi(1,1)^{2}=\varphi(2,\varphi(2,0))=\varphi(2,\varphi(2,0,0),0)\)

\(\varphi(1,1)^{\varphi(1)}=\varphi(2,\varphi(2,1))=\varphi(2,\varphi(2,\varphi(1,0),0),0)\)

\(\varphi(1,1)^{\varphi(1,0)}=\varphi(2,\varphi(2,\varphi(1,0)))=\varphi(2,\varphi(2,\varphi(1,0,0),0),0)\)

\(\varphi(1,1)^{\varphi(1,1)}=\varphi(2,\varphi(2,\varphi(2,0)))=\varphi(2,\varphi(2,\varphi(2,0,0),0),0)\)

\(\varphi(1,2)=\varphi(3,0)=\varphi(3,0,0)\)

\(\varphi(1,\varphi(1))=\varphi(\varphi(1),0)=\varphi(\varphi(1,0),0,0)\)

\(\varphi(1,\varphi(1))\cdot\varphi(1,0)=\varphi(\varphi(1),\varphi(1,0))=\varphi(\varphi(1,0),\varphi(1,0,0),0)\)

\(\varphi(1,\varphi(1))\cdot\varphi(1,1)=\varphi(\varphi(1),\varphi(1,1))=\varphi(\varphi(1,0),\varphi(1,1,0),0)\)

\(\varphi(1,\varphi(1))^{2}=\varphi(\varphi(1),\varphi(1,\varphi(1)))=\varphi(\varphi(1,0),\varphi(1,\varphi(1,0),0))\)

\(\varphi(1,\varphi(1)+1)=\varphi(\varphi(1)+1,0)=\varphi(\varphi(1,1),0,0)\)

\(\varphi(1,\varphi(1)+2)=\varphi(\varphi(1)+2,0)=\varphi(\varphi(1,2),0,0)\)

\(\varphi(1,\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2,0)=\varphi(\varphi(1,\varphi(1,0)),0,0)\)

\(\varphi(1,\varphi(1)^{2})=\varphi(\varphi(2),0)=\varphi(\varphi(2,0),0,0)\)

\(\varphi(1,\varphi(1)^{3})=\varphi(\varphi(3),0)=\varphi(\varphi(3,0),0,0)\)

\(\varphi(1,\varphi(1)^{\varphi(1)})=\varphi(\varphi(\varphi(1)),0)=\varphi(\varphi(\varphi(1,0),0),0,0)\)

\(\varphi(1,\varphi(1,0))=\varphi(\varphi(1,0),0)=\varphi(\varphi(1,0,0),0,0)\)

\(\varphi(1,\varphi(1,0)+1)=\varphi(\varphi(1,0)+1,0)=\varphi(\varphi(1,0,1),0,0)\)

\(\varphi(1,\varphi(1,0)\cdot\varphi(1))=\varphi(\varphi(1,1),0)=\varphi(\varphi(1,1,0),0,0)\)

\(\varphi(1,\varphi(1,0)^{\varphi(1)})=\varphi(\varphi(1,\varphi(1,1)),0)=\varphi(\varphi(1,\varphi(1,1,0),0),0,0)\)

\(\varphi(1,\varphi(1,1))=\varphi(\varphi(2,0),0)=\varphi(\varphi(2,0,0),0,0)\)

\(\varphi(1,\varphi(1,\varphi(1)))=\varphi(\varphi(\varphi(1),0),0)=\varphi(\varphi(\varphi(1,0),0,0),0,0)\)

\(\varphi(1,\varphi(1,\varphi(1,0)))=\varphi(\varphi(\varphi(1,0),0),0)=\varphi(\varphi(\varphi(1,0,0),0,0),0,0)\)

\(\varphi(2,0)=\varphi(1,0,0)=\varphi(1,0,0,0)=\) ζ0

\(\varphi(1,\varphi(2,0)+1)=\varphi(\varphi(1,0,0),0)=\varphi(\varphi(1,0,0,0),0,0)\)

\(\varphi(1,\varphi(2,0)+\varphi(1))=\varphi(\varphi(1,0,0)+\varphi(1),0)=\varphi(\varphi(1,0,0,\varphi(1,0)),0,0)\)

\(\varphi(1,\varphi(2,0)+\varphi(1,0))=\varphi(\varphi(1,0,0)+\varphi(1,0),0)=\varphi(\varphi(1,0,0,\varphi(1,0,0)),0,0)\)

\(\varphi(1,\varphi(2,0)+\varphi(1,0)\cdot\varphi(1))=\varphi(\varphi(1,0,0)+\varphi(1,1),0)=\varphi(\varphi(1,0,0,\varphi(1,1,0)),0,0)\)

\(\varphi(1,\varphi(2,0)+\varphi(1,1))=\varphi(\varphi(1,0,0)+\varphi(2,0),0)=\varphi(\varphi(1,0,0,\varphi(2,0,0)),0,0)\)

\(\varphi(1,\varphi(2,0)+\varphi(1,\varphi(1,0)))=\varphi(\varphi(1,0,0)+\varphi(\varphi(1,0),0),0)=\varphi(\varphi(1,0,0,\varphi(\varphi(1,0,0),0,0)),0,0))\)

\(\varphi(1,\varphi(2,0)\cdot2)=\varphi(\varphi(1,0,0)\cdot2,0)=\varphi(\varphi(1,0,1,0),0,0)\)

\(\varphi(1,\varphi(2,0)\cdot\varphi(1))=\varphi(\varphi(1,1,0),0)=\varphi(\varphi(1,0,\varphi(1,0),0),0,0)\)

\(\varphi(1,\varphi(2,0)^{2})=\varphi(\varphi(1,0,\varphi(1,0,0)),0)=\varphi(\varphi(1,0,\varphi(1,0,0,0),0),0,0)\)

\(\varphi(1,\varphi(1,\varphi(2,0)+1))=\varphi(\varphi(\varphi(1,0,0),0),0)=\varphi(\varphi(\varphi(1,0,0,0),0,0),0,0)\)

\(\varphi(1,\varphi(1,\varphi(1,\varphi(2,0)+1)))=\varphi(\varphi(\varphi(\varphi(1,0,0),0),0),0)=\varphi(\varphi(\varphi(\varphi(1,0,0,0),0,0),0,0),0,0)\)

\(\varphi(2,1)=\varphi(2,0,0)=\varphi(2,0,0,0)\)

\(\varphi(1,\varphi(2,1)+1)=\varphi(\varphi(2,0,0),0)=\varphi(\varphi(2,0,0,0),0,0)\)

\(\varphi(1,\varphi(1,\varphi(2,1)+1))=\varphi(\varphi(\varphi(2,0,0),0),0)=\varphi(\varphi(\varphi(2,0,0,0),0,0),0,0)\)

\(\varphi(2,2)=\varphi(3,0,0)=\varphi(3,0,0,0)\)

\(\varphi(2,\varphi(1))=\varphi(\varphi(1),0,0)=\varphi(\varphi(1,0),0,0,0)\)

\(\varphi(2,\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2,0,0)=\varphi(\varphi(1,\varphi(1,0)),0,0,0)\)

\(\varphi(2,\varphi(2))=\varphi(\varphi(2),0,0)=\varphi(\varphi(2,0),0,0,0)\)

\(\varphi(2,\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1)),0,0)=\varphi(\varphi(\varphi(1,0),0),0,0,0)\)

\(\varphi(2,\varphi(1,0))=\varphi(\varphi(1,0),0,0)=\varphi(\varphi(1,0,0),0,0,0)\)

\(\varphi(2,\varphi(1,1))=\varphi(\varphi(2,0),0,0)=\varphi(\varphi(2,0,0),0,0,0)\)

\(\varphi(2,\varphi(1,\varphi(1)))=\varphi(\varphi(\varphi(1),0),0,0)=\varphi(\varphi(\varphi(1,0),0,0),0,0,0)\)

\(\varphi(2,\varphi(2,0))=\varphi(\varphi(1,0,0),0,0)=\varphi(\varphi(1,0,0,0),0,0,0)\)

\(\varphi(2,\varphi(2,\varphi(2,0)))=\varphi(\varphi(\varphi(1,0,0),0,0),0,0)=\varphi(\varphi(\varphi(1,0,0,0),0,0,0),0,0,0)\)

\(\varphi(3,0)=\varphi(1,0,0,0)=\varphi(1,0,0,0,0)=\) η0

\(\varphi(1,\varphi(3,0)+1)=\varphi(\varphi(1,0,0,0),0)=\varphi(\varphi(1,0,0,0,0),0,0)\)

\(\varphi(2,\varphi(3,0)+1)=\varphi(\varphi(1,0,0,0),0,0)=\varphi(\varphi(1,0,0,0,0),0,0,0)\)

\(\varphi(3,1)=\varphi(2,0,0,0)=\varphi(2,0,0,0,0)\)

\(\varphi(3,\varphi(1))=\varphi(\varphi(1),0,0,0)=\varphi(\varphi(1,0),0,0,0,0)\)

\(\varphi(3,\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1)),0,0,0)=\varphi(\varphi(\varphi(1,0),0),0,0,0,0)\)

\(\varphi(3,\varphi(1,0))=\varphi(\varphi(1,0),0,0,0)=\varphi(\varphi(1,0,0),0,0,0,0)\)

\(\varphi(3,\varphi(2,0))=\varphi(\varphi(1,0,0),0,0,0)=\varphi(\varphi(1,0,0,0),0,0,0,0)\)

\(\varphi(3,\varphi(3,0))=\varphi(\varphi(1,0,0,0),0,0,0)=\varphi(\varphi(1,0,0,0,0),0,0,0,0)\)

\(\varphi(4,0)=\varphi(1,0,0,0,0)=\varphi(1,0,0,0,0,0)\)

\(\varphi(\varphi(1),0)=\varphi(1@\varphi(1))=\varphi(1@\varphi(1,0))=\) HCO

\(\varphi(1,\varphi(\varphi(1),0)+1)=\varphi(\varphi(1@\varphi(1)),0)=\varphi(\varphi(1@\varphi(1,0)),0,0)\)

\(\varphi(2,\varphi(\varphi(1),0)+1)=\varphi(\varphi(1@\varphi(1)),0,0)=\varphi(\varphi(1@\varphi(1,0)),0,0,0)\)

\(\varphi(3,\varphi(\varphi(1),0)+1)=\varphi(\varphi(1@\varphi(1)),0,0,0)=\varphi(\varphi(1@\varphi(1,0)),0,0,0,0)\)

\(\varphi(\varphi(1),1)=\varphi(2@\varphi(1))=\varphi(2@\varphi(1,0))\)

\(\varphi(1,\varphi(\varphi(1),1)+1)=\varphi(\varphi(2@\varphi(1)),0)=\varphi(\varphi(2@\varphi(1,0)),0,0)\)

\(\varphi(\varphi(1),2)=\varphi(3@\varphi(1))=\varphi(3@\varphi(1,0))\)

\(\varphi(\varphi(1),\varphi(1))=\varphi(\varphi(1)@\varphi(1))=\varphi(\varphi(1,0)@\varphi(1,0))\)

\(\varphi(\varphi(1),\varphi(1)\cdot2)=\varphi(\varphi(1)\cdot2@\varphi(1))=\varphi(\varphi(1,0)\cdot2@\varphi(1,0))\)

\(\varphi(\varphi(1),\varphi(2))=\varphi(\varphi(2)@\varphi(1))=\varphi(\varphi(2,0)@\varphi(1,0))\)

\(\varphi(\varphi(1),\varphi(\varphi(1)))=\varphi(\varphi(\varphi(1))@\varphi(1))=\varphi(\varphi(\varphi(1,0),0)@\varphi(1,0))\)

\(\varphi(\varphi(1),\varphi(1,0))=\varphi(\varphi(1,0)@\varphi(1))=\varphi(\varphi(1,0,0)@\varphi(1,0))\)

\(\varphi(\varphi(1),\varphi(2,0))=\varphi(\varphi(1,0,0)@\varphi(1))=\varphi(\varphi(1,0,0,0)@\varphi(1,0))\)

\(\varphi(\varphi(1),\varphi(3,0))=\varphi(\varphi(1,0,0,0)@\varphi(1))=\varphi(\varphi(1,0,0,0,0)@\varphi(1,0))\)

\(\varphi(\varphi(1),\varphi(\varphi(1),0))=\varphi(\varphi(1@\varphi(1))@\varphi(1))=\varphi(\varphi(1@\varphi(1,0))@\varphi(1,0))\)

\(\varphi(\varphi(1)+1,0)=\varphi(1@\varphi(1)+1)=\varphi(1@\varphi(1,1))\)

\(\varphi(\varphi(1)+1,\varphi(\varphi(1)+1,0))=\varphi(\varphi(1@\varphi(1)+1)@\varphi(1)+1)=\varphi(\varphi(1@\varphi(1,1))@\varphi(1,1))\)

\(\varphi(\varphi(1)+2,0)=\varphi(1@\varphi(1)+2)=\varphi(1@\varphi(1,2))\)

\(\varphi(\varphi(1)\cdot2,0)=\varphi(1@\varphi(1)\cdot2)=\varphi(1@\varphi(1,\varphi(1)))\)

\(\varphi(\varphi(1)\cdot3,0)=\varphi(1@\varphi(1)\cdot3)=\varphi(1@\varphi(1,\varphi(1,\varphi(1,0))))\)

\(\varphi(\varphi(2),0)=\varphi(1@\varphi(2))=\varphi(1@\varphi(2,0))\)

\(\varphi(\varphi(\varphi(1)),0)=\varphi(1@\varphi(\varphi(1)))=\varphi(1@\varphi(\varphi(1,0),0))\)

\(\varphi(\varphi(1,0),0)=\varphi(1@\varphi(1,0))=\varphi(1@\varphi(1,0,0))\)

\(\varphi(\varphi(2,0),0)=\varphi(1@\varphi(1,0,0))=\varphi(1@\varphi(1,0,0,0))\)

\(\varphi(\varphi(3,0),0)=\varphi(1@\varphi(1,0,0,0))=\varphi(1@\varphi(1,0,0,0,0))\)

\(\varphi(\varphi(\varphi(1),0),0)=\varphi(1@\varphi(1@\varphi(1)))=\varphi(1@\varphi(1@\varphi(1,0)))\)

\(\varphi(\varphi(\varphi(1,0),0),0)=\varphi(1@\varphi(1@\varphi(1,0)))=\varphi(1@\varphi(1@\varphi(1,0,0)))\)

\(\varphi(\varphi(\varphi(2,0),0),0)=\varphi(1@\varphi(1@\varphi(1,0,0)))=\varphi(1@\varphi(1@\varphi(1,0,0,0)))\)

\(\varphi(1,0,0)=\varphi(1@(1,0))=\varphi(1@(1,0))=\) FSO

\(\varphi(1,\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0)),0)=\varphi(\varphi(1@(1,0)),0,0)\)

\(\varphi(2,\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0)),0,0)=\varphi(\varphi(1@(1,0)),0,0,0)\)

\(\varphi(\varphi(1),\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0))@\varphi(1))=\varphi(\varphi(1@(1,0))@\varphi(1,0))\)

\(\varphi(\varphi(1,0),\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0))@\varphi(1,0))=\varphi(\varphi(1@(1,0))@\varphi(1,0,0))\)

\(\varphi(\varphi(2,0),\varphi(1,0,0)+1)=\varphi(\varphi(1@(1,0))@\varphi(1,0,0))=\varphi(\varphi(1@(1,0))@\varphi(1,0,0,0))\)

\(\varphi(\varphi(1,0,0),1)=\varphi(2@\varphi(1@(1,0)))=\varphi(2@\varphi(1@(1,0)))\)

\(\varphi(\varphi(1,0,0),2)=\varphi(3@\varphi(1@(1,0)))=\varphi(3@\varphi(1@(1,0)))\)

\(\varphi(\varphi(1,0,0),\varphi(1))=\varphi(\varphi(1)@\varphi(1@(1,0)))=\varphi(\varphi(1,0)@\varphi(1@(1,0)))\)

\(\varphi(\varphi(1,0,0),\varphi(1,0))=\varphi(\varphi(1,0)@\varphi(1@(1,0)))=\varphi(\varphi(1,0,0)@\varphi(1@(1,0)))\)

\(\varphi(\varphi(1,0,0),\varphi(1,0,0))=\varphi(\varphi(1@(1,0))@\varphi(1@(1,0)))=\varphi(\varphi(1@(1,0))@\varphi(1@(1,0)))\)

\(\varphi(\varphi(1,0,0)+1,0)=\varphi(1@\varphi(1@(1,0))+1)=\varphi(1@\varphi(1@(1,0),1@0))\)

\(\varphi(\varphi(1,0,0)+\varphi(1),0)=\varphi(1@\varphi(1@(1,0))+\varphi(1))=\varphi(1@\varphi(1@(1,0),\varphi(1,0)@0))\)

\(\varphi(\varphi(1,0,0)+\varphi(1,0),0)=\varphi(1@\varphi(1@(1,0))+\varphi(1,0))=\varphi(1@\varphi(1@(1,0),\varphi(1,0,0)@0))\)

\(\varphi(\varphi(1,0,0)\cdot2,0)=\varphi(1@\varphi(1@(1,0))\cdot2)=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0))@0))\)

\(\varphi(\varphi(1,0,0)\cdot\varphi(1),0)=\varphi(1@\varphi(1@(1,0),1@0))=\varphi(1@\varphi(1@(1,0),1@1))\)

\(\varphi(\varphi(1,0,0)^{2},0)=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0))@0))=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0))@0))\)

\(\varphi(\varphi(1,0,0)^{\varphi(1)},0)=\varphi(1@\varphi(1,\varphi(1@(1,0),1@0),0))=\varphi(1@\varphi(1,\varphi(1@(1,0),1@1),0))\)

\(\varphi(\varphi(1,0,0)^{\varphi(1,0)},0)=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0),\varphi(1,0)@0)@0))=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0),\varphi(1,0)@0)@0))\)

\(\varphi(\varphi(1,0,0)^{\varphi(1,0,0)},0)=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0),\varphi(1@(1,0))@0)@0))=\varphi(1@\varphi(1@(1,0),\varphi(1@(1,0),\varphi(1@(1,0))@0)@0))\)

\(\varphi(\varphi(1,\varphi(1,0,0)+1),0)=\varphi(1@\varphi(\varphi(1@(1,0)),0))=\varphi(1@\varphi(\varphi(1@(1,0)),0,0))\)

\(\varphi(\varphi(1,\varphi(1,0,0)+\varphi(1)),0)=\varphi(1@\varphi(\varphi(1@(1,0))+\varphi(1),0))=\varphi(1@\varphi(\varphi(1@(1,0))+\varphi(1,0),0))\)

\(\varphi(\varphi(\varphi(1),\varphi(1,0,0)+1),0)=\varphi(1@\varphi(\varphi(1@(1,0))@\varphi(1)))=\varphi(1@\varphi(\varphi(1@(1,0))@\varphi(1,0)))\)

\(\varphi(1,0,1)=\varphi(1@(1,1))=\varphi(1@(1,1))\)

\(\varphi(1,0,2)=\varphi(2@(1,1))=\varphi(2@(1,1))\)

\(\varphi(1,0,\varphi(1))=\varphi(\varphi(1)@(1,1))=\varphi(\varphi(1,0)@(1,1))\)

\(\varphi(1,0,\varphi(1,0))=\varphi(\varphi(1,0)@(1,1))=\varphi(\varphi(1,0,0)@(1,1))\)

\(\varphi(1,0,\varphi(1,0,0))=\varphi(\varphi(1@(1,0))@(1,1))=\varphi(\varphi(1@(1,0))@(1,1))\)

\(\varphi(1,1,0)=\varphi(1@(1,1))=\varphi(1@(1,1))\)

\(\varphi(1,1,\varphi(1,1,0))=\varphi(\varphi(1@(1,1))@(1,1))=\varphi(\varphi(1@(1,1))@(1,1))\)

\(\varphi(1,2,0)=\varphi(1@(1,2))=\varphi(1@(1,2))\)

\(\varphi(1,\varphi(1),0)=\varphi(1@(1,\varphi(1)))=\varphi(1@(1,\varphi(1,0)))\)

\(\varphi(1,\varphi(1,0),0)=\varphi(1@(1,\varphi(1,0)))=\varphi(1@(1,\varphi(1,0,0)))\)

\(\varphi(1,\varphi(1,0,0),0)=\varphi(1@(1,\varphi(1@(1,0))))=\varphi(1@(1,\varphi(1@(1,0))))\)

\(\varphi(2,0,0)=\varphi(1@(2,0))=\varphi(1@(2,0))\)

\(\varphi(2,0,1)=\varphi(2@(2,0))=\varphi(2@(2,0))\)

\(\varphi(2,1,0)=\varphi(1@(2,1))=\varphi(1@(2,1))\)

\(\varphi(2,\varphi(2,0,0),0)=\varphi(1@(2,\varphi(1@(2,0))))=\varphi(1@(2,\varphi(1@(2,0))))\)

\(\varphi(3,0,0)=\varphi(1@(3,0))=\varphi(1@(3,0))\)

\(\varphi(\varphi(1),0,0)=\varphi(1@(\varphi(1),0))=\varphi(1@(\varphi(1,0),0))\)

\(\varphi(\varphi(1,0),0,0)=\varphi(1@(\varphi(1,0),0))=\varphi(1@(\varphi(1,0,0),0))\)

\(\varphi(\varphi(1,0,0),0,0)=\varphi(1@(\varphi(1@(1,0)),0))=\varphi(1@(\varphi(1@(1,0)),0))\)

\(\varphi(1,0,0,0)=\varphi(1@(1,0,0))=\varphi(1@(1,0,0))=\) ACO

\(\varphi(1,0,0,1)=\varphi(2@(1,0,0))=\varphi(2@(1,0,0))\)

\(\varphi(1,0,1,0)=\varphi(1@(1,0,1))=\varphi(1@(1,0,1))\)

\(\varphi(1,1,0,0)=\varphi(1@(1,1,0))=\varphi(1@(1,1,0))\)

\(\varphi(2,0,0,0)=\varphi(1@(2,0,0))=\varphi(1@(2,0,0))\)

\(\varphi(\varphi(1),0,0,0)=\varphi(1@(\varphi(1),0,0))=\varphi(1@(\varphi(1,0),0,0))\)

\(\varphi(\varphi(1,0,0,0),0,0,0)=\varphi(1@(\varphi(1@(1,0,0)),0,0))=\varphi(1@(\varphi(1@(1,0,0)),0,0))\)

\(\varphi(1,0,0,0,0)=\varphi(1@(1,0,0,0))=\varphi(1@(1,0,0,0))\)

\(\varphi(1,0,0,0,0,0)=\varphi(1@(1,0,0,0,0))=\varphi(1@(1,0,0,0,0))\)

\(\varphi(1,1,4,5,1,4)=\varphi(5@(1,1,4,5,1))=\varphi(5@(1,1,4,5,1))\)

\(\varphi(1@\varphi(1))=\varphi(1@(1@\varphi(1)))=\varphi(1@(1@\varphi(1,0)))=\) SVO

\(\varphi(1,\varphi(1@\varphi(1))+1)=\varphi(1@(1@\varphi(1)),1@1)=\varphi(1@(1@\varphi(1,0)),1@2)\)

\(\varphi(1@\varphi(1),1@0)=\varphi(2@(1@\varphi(1)))=\varphi(2@(1@\varphi(1,0)))\)

\(\varphi(1@\varphi(1),1@1)=\varphi(1@(1@\varphi(1),1@0))=\varphi(1@(1@\varphi(1,0),1@0))\)

\(\varphi(1@\varphi(1),1@2)=\varphi(1@(1@\varphi(1),1@1))=\varphi(1@(1@\varphi(1,0),1@1))\)

\(\varphi(2@\varphi(1))=\varphi(1@(2@\varphi(1)))=\varphi(1@(2@\varphi(1,0)))\)

\(\varphi(\varphi(1)@\varphi(1))=\varphi(1@(\varphi(1)@\varphi(1)))=\varphi(1@(\varphi(1,0)@\varphi(1,0)))\)

\(\varphi(1@\varphi(1)+1)=\varphi(1@(1@\varphi(1)+1))=\varphi(1@(1@\varphi(1,1)))\)

\(\varphi(1@\varphi(1)\cdot2)=\varphi(1@(1@\varphi(1)\cdot2))=\varphi(1@(1@\varphi(1,\varphi(1,0))))\)

\(\varphi(1@\varphi(2))=\varphi(1@(1@\varphi(2)))=\varphi(1@(1@\varphi(2,0)))\)

\(\varphi(1@\varphi(1,0))=\varphi(1@(1@\varphi(1,0)))=\varphi(1@(1@\varphi(1,0,0)))\)

\(\varphi(1@\varphi(2,0))=\varphi(1@(1@\varphi(1,0,0)))=\varphi(1@(1@\varphi(1,0,0,0)))\)

\(\varphi(1@\varphi(1,0,0))=\varphi(1@(1@\varphi(1@(1,0))))=\varphi(1@(1@\varphi(1@(1,0))))\)

\(\varphi(1@\varphi(1@\varphi(1)))=\varphi(1@(1@\varphi(1@(1@\varphi(1)))))=\varphi(1@(1@\varphi(1@(1@\varphi(1,0)))))=\) LVO

历史

据信在 2024 年下旬及以前的 Weak Veblen 都是指的 ×ω φ，但在这之后 ×ω φ 的生态位快速被 +1 φ 所代替了，所以出现了一些较为混乱的局面。

检索自“http://wiki.googology.top/index.php?title=weak_Veblen_函数&oldid=2512”