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LPrSSψ分析:修订间差异

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第286行: 第286行:
|(0)(1,1,1)(2)(1,1,1)
|(0)(1,1,1)(2)(1,1,1)
|-
|-
|<math>\psi(\Omega_2\times\psi(\Omega))</math>
|<math>\psi(\Omega_2\times\varepsilon_0)</math>
|<math>\psi(\Omega_\omega\times\psi(\Omega))</math>
|<math>\psi(\Omega_\omega\times\varepsilon_0)</math>
|(0)(1,1,1)(2)(3,1)
|(0)(1,1,1)(2)(3,1)
|-
|-
第300行: 第300行:


== 分析3:(0)(1,1,1)(2,1)~(0)(1,1,1)(2,1)(1,1,1) ==
== 分析3:(0)(1,1,1)(2,1)~(0)(1,1,1)(2,1)(1,1,1) ==
分析进入提升阶段。可见LPrSSψ中的一个Ω_2对应的不是BOCF中的一个Ω_ω,而是BMS中的一个(1,1,1)。这里注意LPrSSψ一般不单独定义Ω_0,展开方式,Ω_2展开出ψ_1时,把外面的Ω变成了里面的Ω_2,以此类推,但ω不提升。因为阶差为d时,Ω_(n+1)变出的ψ_n只把外面的x>=n的Ω_x提升为Ω_(x+d)。不然,若定义Ω_0=ω,并在这种位置将其提升为Ω,会与没有【特别地,如果坏根中元素不是坏部中某项的祖先项,则该项在复制过程中将保持不变。】规则的BMS一样,发生无穷降链。此外,若无此种提升,可能可以得到类似IBMS的ILPrSSψ。
分析进入提升阶段。可见LPrSSψ中,ψ中的一个Ω_2对应的不是BOCF中的一个Ω_ω,而是BMS中的一个(1,1,1)。类似的,BGO前许多阶差2的表达式对应BMS第三行的1。这里注意LPrSSψ一般不单独定义Ω_0,展开方式,Ω_2展开出ψ_1时,把外面的Ω变成了里面的Ω_2,以此类推,但ω不提升。因为阶差为d时,Ω_(n+1)变出的ψ_n只把外面的x>=n的Ω_x提升为Ω_(x+d)。不然,若定义Ω_0=ω,并在这种位置将其提升为Ω,会与没有【特别地,如果坏根中元素不是坏部中某项的祖先项,则该项在复制过程中将保持不变。】规则的BMS一样,发生无穷降链。此外,若无此种提升,可能可以得到类似IBMS的ILPrSSψ。
{| class="wikitable"
{| class="wikitable"
|+分析3:(0)(1,1,1)(2,1)~(0)(1,1,1)(2,1)(1,1,1)
|+分析3:(0)(1,1,1)(2,1)~(0)(1,1,1)(2,1)(1,1,1)
第443行: 第443行:
|(0)(1,1,1)(2,1)(1,1,1)
|(0)(1,1,1)(2,1)(1,1,1)
|}
|}
未完待续
 
== 分析4:(0)(1,1,1)(2,1)(1,1,1)~TFBO ==
此阶段难度相对较低,LPrSSψ与BMS对应较为简单,BOCF有随时出现的Ω变Ω_ω提升。
{| class="wikitable"
|+分析4:(0)(1,1,1)(2,1)(1,1,1)~TFBO
!LPrSSψ
!BOCF
!BMS
|-
|<math>\psi(\Omega_2\times(\Omega+1))</math>
|<math>\psi(\Omega_\omega^2)</math>
|(0)(1,1,1)(2,1)(1,1,1)
|-
|<math>\psi(\Omega_2\times(\Omega+1)+\psi_1(\Omega_3))</math>
|<math>\psi(\Omega_\omega^2+\psi_1(\Omega_\omega))</math>
|(0)(1,1,1)(2,1)(1,1,1)(1,1)(2,2,1)
|-
|<math>\psi(\Omega_2\times(\Omega+1)+\psi_1(\Omega_3\Omega))</math>
|<math>\psi(\Omega_\omega^2+\psi_1(\Omega_\omega\Omega))</math>
|(0)(1,1,1)(2,1)(1,1,1)(1,1)(2,2,1)(3,1)
|-
|<math>\psi(\Omega_2\times(\Omega+1)+\psi_1(\Omega_3\Omega_2))</math>
|<math>\psi(\Omega_\omega^2+\psi_1(\Omega_\omega\Omega_2))</math>
|(0)(1,1,1)(2,1)(1,1,1)(1,1)(2,2,1)(3,2)
|-
|<math>\psi(\Omega_2\times(\Omega+1)+\psi_1(\Omega_3\times(\Omega_2+1)))</math>
|<math>\psi(\Omega_\omega^2+\psi_1(\Omega_\omega^2))</math>
|(0)(1,1,1)(2,1)(1,1,1)(1,1)(2,2,1)(3,2)(2,2,1)
|-
|<math>\psi(\Omega_2\times(\Omega+1)+\psi_1(\Omega_3\times(\Omega_2+1)+\Omega_2))</math>
|<math>\psi(\Omega_\omega^2+\Omega_2)</math>
|(0)(1,1,1)(2,1)(1,1,1)(1,1)(2,2,1)(3,2)(2,2,1)(2,2)
|-
|<math>\psi(\Omega_2\times(\Omega+2))</math>
|<math>\psi(\Omega_\omega^2+\Omega_\omega)</math>
|(0)(1,1,1)(2,1)(1,1,1)(1,1,1)
|-
|<math>\psi(\Omega_2\times(\Omega+\omega))</math>
|<math>\psi(\Omega_\omega^2+\Omega_\omega\omega)</math>
|(0)(1,1,1)(2,1)(1,1,1)(2)
|-
|<math>\psi(\Omega_2\times\Omega2)</math>
|<math>\psi(\Omega_\omega^2+\Omega_\omega\Omega)</math>
|(0)(1,1,1)(2,1)(1,1,1)(2,1)
|-
|<math>\psi(\Omega_2\times\Omega2+\psi_1(\Omega_3\times(\Omega_2+\Omega)))</math>
|<math>\psi(\Omega_\omega^2+\Omega_\omega\Omega+\psi_1(\Omega_\omega^2+\Omega_\omega\Omega))</math>
|(0)(1,1,1)(2,1)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)(2,2,1)(3,1)
|-
|<math>\psi(\Omega_2\times\Omega2+\psi_1(\Omega_3\times(\Omega_2+\Omega)+\Omega_2))</math>
|<math>\psi(\Omega_\omega^2+\Omega_\omega\times(\Omega+1))</math>
|(0)(1,1,1)(2,1)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)(2,2,1)(3,1)(2,2,1)
|-
|<math>\psi(\Omega_2\times\Omega2+\psi_1(\Omega_3\times\Omega_22))</math>
|<math>\psi(\Omega_\omega^2+\Omega_\omega\Omega_2)</math>
|(0)(1,1,1)(2,1)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)(2,2,1)(3,2)
|-
|<math>\psi(\Omega_2\times(\Omega2+1))</math>
|<math>\psi(\Omega_\omega^22)</math>
|(0)(1,1,1)(2,1)(1,1,1)(2,1)(1,1,1)
|-
|<math>\psi(\Omega_2\times\Omega\omega)</math>
|<math>\psi(\Omega_\omega^2\omega)</math>
|(0)(1,1,1)(2,1)(2)
|-
|<math>\psi(\Omega_2\times\Omega^2)</math>
|<math>\psi(\Omega_\omega^2\Omega)</math>
|(0)(1,1,1)(2,1)(2,1)
|-
|<math>\psi(\Omega_2\times\Omega^2+\psi_1(\Omega_3\Omega_2\Omega))</math>
|<math>\psi(\Omega_\omega^2\Omega+\psi_1(\Omega_\omega^2\Omega))</math>
|(0)(1,1,1)(2,1)(2,1)(1,1)(2,2,1)(3,2)(3,1)
|-
|<math>\psi(\Omega_2\times\Omega^2+\psi_1(\Omega_3\Omega_2\Omega+\Omega_2))</math>
|<math>\psi(\Omega_\omega^2\Omega+\Omega_\omega)</math>
|(0)(1,1,1)(2,1)(2,1)(1,1)(2,2,1)(3,2)(3,1)(2,2,1)
|-
|<math>\psi(\Omega_2\times\Omega^2+\psi_1(\Omega_3\times\Omega_2^2))</math>
|<math>\psi(\Omega_\omega^2\Omega_2)</math>
|(0)(1,1,1)(2,1)(2,1)(1,1)(2,2,1)(3,2)(3,2)
|-
|<math>\psi(\Omega_2\times(\Omega^2+1))</math>
|<math>\psi(\Omega_\omega^3)</math>
|(0)(1,1,1)(2,1)(2,1)(1,1,1)
|-
|<math>\psi(\Omega_2\times\Omega^\omega)</math>
|<math>\psi(\Omega_\omega^\omega)</math>
|(0)(1,1,1)(2,1)(3)
|-
|<math>\psi(\Omega_2\times\Omega^\Omega)</math>
|<math>\psi(\Omega_\omega^\Omega)</math>
|(0)(1,1,1)(2,1)(3,1)
|-
|<math>\psi(\Omega_2\times\Omega^\Omega+\psi_1(\Omega_3\times\Omega_2^\Omega))</math>
|<math>\psi(\Omega_\omega^\Omega+\psi_1(\Omega_\omega^\Omega))</math>
|(0)(1,1,1)(2,1)(3,1)(1,1)(2,2,1)(3,2)(4,1)
|-
|<math>\psi(\Omega_2\times\Omega^\Omega+\psi_1(\Omega_3\times\Omega_2^\Omega+\Omega_2))</math>
|<math>\psi(\Omega_\omega^\Omega+\Omega_\omega)</math>
|(0)(1,1,1)(2,1)(3,1)(1,1)(2,2,1)(3,2)(4,1)(2,2,1)
|-
|<math>\psi(\Omega_2\times\Omega^\Omega+\psi_1(\Omega_3\times\Omega_2^{\Omega_2}))</math>
|<math>\psi(\Omega_\omega^{\Omega_2})</math>
|(0)(1,1,1)(2,1)(3,1)(1,1)(2,2,1)(3,2)(4,2)
|-
|<math>\psi(\Omega_2\times(\Omega^\Omega+1))</math>
|<math>\psi(\Omega_\omega^{\Omega_\omega})</math>
|(0)(1,1,1)(2,1)(3,1)(1,1,1)
|-
|<math>\psi(\Omega_2\times\Omega^\Omega2)</math>
|<math>\psi(\Omega_\omega^{\Omega_\omega}+\Omega_\omega^\Omega)</math>
|(0)(1,1,1)(2,1)(3,1)(1,1,1)(2,1)(3,1)
|-
|<math>\psi(\Omega_2\times\Omega^{\Omega+1})</math>
|<math>\psi(\Omega_\omega^{\Omega_\omega}\Omega)</math>
|(0)(1,1,1)(2,1)(3,1)(2,1)
|-
|<math>\psi(\Omega_2\times\Omega^{\Omega2})</math>
|<math>\psi(\Omega_\omega^{\Omega_\omega+\Omega})</math>
|(0)(1,1,1)(2,1)(3,1)(2,1)(3,1)
|-
|<math>\psi(\Omega_2\times\Omega^{\Omega^2})</math>
|<math>\psi(\Omega_\omega^{\Omega_\omega\Omega})</math>
|(0)(1,1,1)(2,1)(3,1)(3,1)
|-
|<math>\psi(\Omega_2\times\Omega^{\Omega^\Omega})</math>
|<math>\psi(\Omega_\omega^{\Omega_\omega^\Omega})</math>
|(0)(1,1,1)(2,1)(3,1)(4,1)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_2))=\psi(\Omega_2\times\varepsilon_{\Omega+1})</math>
|<math>\psi(\Omega_{\omega+1})</math>
|(0)(1,1,1)(2,1)(3,2)
|}
 
== 分析5:TFBO~BiO ==
此阶段若对照BMS分析,难度相对较低,而BOCF有随时出现的各级提升。
{| class="wikitable"
|+分析5:TFBO~BiO
!LPrSSψ
!BOCF
!BMS
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_2))</math>
|<math>\psi(\Omega_{\omega+1})</math>
|(0)(1,1,1)(2,1)(3,2)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_2)+\psi_1(\Omega_3\times\psi_2(\Omega_3)))</math>
|<math>\psi(\Omega_{\omega+1}+\psi_1(\Omega_{\omega+1}))</math>
|(0)(1,1,1)(2,1)(3,2)(1,1)(2,2,1)(3,2)(4,3)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_2)+\psi_1(\Omega_3\times\psi_2(\Omega_3)+\Omega_2))</math>
|<math>\psi(\Omega_{\omega+1}+\Omega_2)</math>
|(0)(1,1,1)(2,1)(3,2)(1,1)(2,2,1)(3,2)(4,3)(2,2)
|-
|<math>\psi(\Omega_2\times(\psi_1(\Omega_2)+1))</math>
|<math>\psi(\Omega_{\omega+1}+\Omega_\omega)</math>
|(0)(1,1,1)(2,1)(3,2)(1,1,1)
|-
|<math>\psi(\Omega_2\times(\psi_1(\Omega_2)+\Omega))</math>
|<math>\psi(\Omega_{\omega+1}+\Omega_\omega\Omega)</math>
|(0)(1,1,1)(2,1)(3,2)(1,1,1)(2,1)
|-
|<math>\psi(\Omega_2\times(\psi_1(\Omega_2)+\Omega+1))</math>
|<math>\psi(\Omega_{\omega+1}+\Omega_\omega^2)</math>
|(0)(1,1,1)(2,1)(3,2)(1,1,1)(2,1)(1,1,1)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_2)\times2)</math>
|<math>\psi(\Omega_{\omega+1}+\psi_\omega(\Omega_{\omega+1}))</math>
|(0)(1,1,1)(2,1)(3,2)(1,1,1)(2,1)(3,2)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_2+1))</math>
|<math>\psi(\Omega_{\omega+1}+\psi_\omega(\Omega_{\omega+1}+1))</math>
|(0)(1,1,1)(2,1)(3,2)(2)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_2+\Omega))</math>
|<math>\psi(\Omega_{\omega+1}+\psi_\omega(\Omega_{\omega+1}+\Omega))</math>
|(0)(1,1,1)(2,1)(3,2)(2,1)
|-
|<math>\psi(\Omega_2\times(\psi_1(\Omega_2+\Omega)+1))</math>
|<math>\psi(\Omega_{\omega+1}+\psi_\omega(\Omega_{\omega+1}+\Omega_\omega))</math>
|(0)(1,1,1)(2,1)(3,2)(2,1)(1,1,1)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_2+\psi_1(\Omega_2)))</math>
|<math>\psi(\Omega_{\omega+1}+\psi_\omega(\Omega_{\omega+1}+\psi_\omega(\Omega_{\omega+1})))</math>
|(0)(1,1,1)(2,1)(3,2)(2,1)(3,2)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_2+\psi_1(\Omega_2+\Omega)))</math>
|<math>\psi(\Omega_{\omega+1}+\psi_\omega(\Omega_{\omega+1}+\psi_\omega(\Omega_{\omega+1}+\Omega)))</math>
|(0)(1,1,1)(2,1)(3,2)(3,1)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2))))</math>
|<math>\psi(\Omega_{\omega+1}+\psi_\omega(\Omega_{\omega+1}+\psi_\omega(\Omega_{\omega+1}+\psi_\omega(\Omega_{\omega+1}))))</math>
|(0)(1,1,1)(2,1)(3,2)(3,1)(4,2)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_22))</math>
|<math>\psi(\Omega_{\omega+1}2)</math>
|(0)(1,1,1)(2,1)(3,2)(3,2)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_2\omega))</math>
|<math>\psi(\Omega_{\omega+1}\omega)</math>
|(0)(1,1,1)(2,1)(3,2)(4)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_2\Omega))</math>
|<math>\psi(\Omega_{\omega+1}\Omega)</math>
|(0)(1,1,1)(2,1)(3,2)(4,1)
|-
|<math>\psi(\Omega_2\times(\psi_1(\Omega_2\Omega)+1))</math>
|<math>\psi(\Omega_{\omega+1}\Omega_\omega)</math>
|(0)(1,1,1)(2,1)(3,2)(4,1)(1,1,1)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_2\times\psi_1(\Omega_2)))</math>
|<math>\psi(\Omega_{\omega+1}\times\psi_\omega(\Omega_{\omega+1}))</math>
|(0)(1,1,1)(2,1)(3,2)(4,1)(5,2)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_2^2))</math>
|<math>\psi(\Omega_{\omega+1}^2)</math>
|(0)(1,1,1)(2,1)(3,2)(4,2)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_2^\omega))</math>
|<math>\psi(\Omega_{\omega+1}^\omega)</math>
|(0)(1,1,1)(2,1)(3,2)(4,2)(5)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_2^\Omega))</math>
|<math>\psi(\Omega_{\omega+1}^\Omega)</math>
|(0)(1,1,1)(2,1)(3,2)(4,2)(5,1)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_2^{\Omega_2}))</math>
|<math>\psi(\Omega_{\omega+1}^{\Omega_{\omega+1}})</math>
|(0)(1,1,1)(2,1)(3,2)(4,2)(5,2)
|-
|<math>\psi(\Omega_2\times\psi_1(\psi_2(\Omega_3)))</math>
|<math>\psi(\Omega_{\omega+2})</math>
|(0)(1,1,1)(2,1)(3,2)(4,3)
|-
|<math>\psi(\Omega_2\times\psi_1(\psi_2(\Omega_3)\times2))</math>
|<math>\psi(\Omega_{\omega+2}+\psi_{\omega+1}(\Omega_{\omega+2}))</math>
|(0)(1,1,1)(2,1)(3,2)(4,3)(3,2)(4,3)
|-
|<math>\psi(\Omega_2\times\psi_1(\psi_2(\Omega_32)))</math>
|<math>\psi(\Omega_{\omega+2}2)</math>
|(0)(1,1,1)(2,1)(3,2)(4,3)(4,3)
|-
|<math>\psi(\Omega_2\times\psi_1(\psi_2(\Omega_3^2)))</math>
|<math>\psi(\Omega_{\omega+2}^2)</math>
|(0)(1,1,1)(2,1)(3,2)(4,3)(5,3)
|-
|<math>\psi(\Omega_2\times\psi_1(\psi_2(\psi_3(\Omega_4))))</math>
|<math>\psi(\Omega_{\omega+3})</math>
|(0)(1,1,1)(2,1)(3,2)(4,3)(5,4)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_3))</math>
|<math>\psi(\Omega_{\omega2})</math>
|(0)(1,1,1)(2,1)(3,2,1)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_3)\times2)</math>
|<math>\psi(\Omega_{\omega2}+\psi_\omega(\Omega_{\omega2}))</math>
|(0)(1,1,1)(2,1)(3,2,1)(1,1,1)(2,1)(3,2,1)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_3+1))</math>
|<math>\psi(\Omega_{\omega2}+\psi_\omega(\Omega_{\omega2}+1))</math>
|(0)(1,1,1)(2,1)(3,2,1)(2)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_3+\Omega))</math>
|<math>\psi(\Omega_{\omega2}+\psi_\omega(\Omega_{\omega2}+\Omega))</math>
|(0)(1,1,1)(2,1)(3,2,1)(2,1)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_3+\psi_1(\Omega_3)))</math>
|<math>\psi(\Omega_{\omega2}+\psi_\omega(\Omega_{\omega2}+\psi_\omega(\Omega_{\omega2})))</math>
|(0)(1,1,1)(2,1)(3,2,1)(2,1)(3,2,1)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_3+\Omega_2))</math>
|<math>\psi(\Omega_{\omega2}+\Omega_{\omega+1})</math>
|(0)(1,1,1)(2,1)(3,2,1)(3,2)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_32))</math>
|<math>\psi(\Omega_{\omega2}2)</math>
|(0)(1,1,1)(2,1)(3,2,1)(3,2,1)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_3\omega))</math>
|<math>\psi(\Omega_{\omega2}\omega)</math>
|(0)(1,1,1)(2,1)(3,2,1)(4)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_3\Omega))</math>
|<math>\psi(\Omega_{\omega2}\Omega)</math>
|(0)(1,1,1)(2,1)(3,2,1)(4,1)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_3\Omega_2))</math>
|<math>\psi(\Omega_{\omega2}\Omega_{\omega+1})</math>
|(0)(1,1,1)(2,1)(3,2,1)(4,2)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_3\times(\Omega_2+1)))</math>
|<math>\psi(\Omega_{\omega2}^2)</math>
|(0)(1,1,1)(2,1)(3,2,1)(4,2)(3,2,1)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_3\times\Omega_2^2))</math>
|<math>\psi(\Omega_{\omega2}^2\times\Omega_{\omega+1})</math>
|(0)(1,1,1)(2,1)(3,2,1)(4,2)(4,2)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_3\times\Omega_2^{\Omega_2}))</math>
|<math>\psi(\Omega_{\omega2}^{\Omega_{\omega+1}})</math>
|(0)(1,1,1)(2,1)(3,2,1)(4,2)(5,2)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_3\times\psi_2(\Omega_3)))</math>
|<math>\psi(\Omega_{\omega2+1})</math>
|(0)(1,1,1)(2,1)(3,2,1)(4,2)(5,3)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_3\times\psi_2(\psi_3(\Omega_4))))</math>
|<math>\psi(\Omega_{\omega2+2})</math>
|(0)(1,1,1)(2,1)(3,2,1)(4,2)(5,3)(6,4)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_3\times\psi_2(\Omega_4)))</math>
|<math>\psi(\Omega_{\omega3})</math>
|(0)(1,1,1)(2,1)(3,2,1)(4,2)(5,3,1)
|-
|<math>\psi(\Omega_2\times\psi_1(\Omega_3\times\psi_2(\Omega_4\times\psi_3(\Omega_5))))</math>
|<math>\psi(\Omega_{\omega4})</math>
|(0)(1,1,1)(2,1)(3,2,1)(4,2)(5,3,1)(6,3)(7,4,1)
|-
|<math>\psi(\Omega_2^2)</math>
|<math>\psi(\Omega_{\omega^2})</math>
|(0)(1,1,1)(2,1,1)
|-
|<math>\psi(\Omega_2^2+\Omega_2)</math>
|<math>\psi(\Omega_{\omega^2}+\Omega_\omega)</math>
|(0)(1,1,1)(2,1,1)(1,1,1)
|-
|<math>\psi(\Omega_2^2+\psi_1(\Omega_3))</math>
|<math>\psi(\Omega_{\omega^2}+\psi_\omega(\Omega_{\omega2}))</math>
|(0)(1,1,1)(2,1,1)(1,1,1)(2,1)(3,2,1)
|-
|<math>\psi(\Omega_2^2+\psi_1(\Omega_3^2))</math>
|<math>\psi(\Omega_{\omega^2}+\psi_\omega(\Omega_{\omega^2}))</math>
|(0)(1,1,1)(2,1,1)(1,1,1)(2,1)(3,2,1)(4,2,1)
|-
|<math>\psi(\Omega_2^2+\psi_1(\Omega_3^2+\Omega_3))</math>
|<math>\psi(\Omega_{\omega^2}+\Omega_{\omega2})</math>
|(0)(1,1,1)(2,1,1)(1,1,1)(2,1)(3,2,1)(4,2,1)(3,2,1)
|-
|<math>\psi(\Omega_2^22)</math>
|<math>\psi(\Omega_{\omega^2}2)</math>
|(0)(1,1,1)(2,1,1)(1,1,1)(2,1,1)
|-
|<math>\psi(\Omega_2^2\Omega)</math>
|<math>\psi(\Omega_{\omega^2}\Omega)</math>
|(0)(1,1,1)(2,1,1)(2,1)
|-
|<math>\psi(\Omega_2^2\Omega+\Omega_2)</math>
|<math>\psi(\Omega_{\omega^2}\Omega_\omega)</math>
|(0)(1,1,1)(2,1,1)(2,1)(1,1,1)
|-
|<math>\psi(\Omega_2^2\times(\Omega+1))</math>
|<math>\psi(\Omega_{\omega^2}^2)</math>
|(0)(1,1,1)(2,1,1)(2,1)(1,1,1)(2,1,1)
|-
|<math>\psi(\Omega_2^2\times\psi_1(\Omega_2))</math>
|<math>\psi(\Omega_{\omega^2+1})</math>
|(0)(1,1,1)(2,1,1)(2,1)(3,2)
|-
|<math>\psi(\Omega_2^2\times\psi_1(\Omega_3))</math>
|<math>\psi(\Omega_{\omega^2+\omega})</math>
|(0)(1,1,1)(2,1,1)(2,1)(3,2,1)
|-
|<math>\psi(\Omega_2^2\times\psi_1(\Omega_3^2))</math>
|<math>\psi(\Omega_{\omega^22})</math>
|(0)(1,1,1)(2,1,1)(2,1)(3,2,1)(4,2,1)
|-
|<math>\psi(\Omega_2^3)</math>
|<math>\psi(\Omega_{\omega^3})</math>
|(0)(1,1,1)(2,1,1)(2,1,1)
|-
|<math>\psi(\Omega_2^\omega)</math>
|<math>\psi(\Omega_{\omega^\omega})</math>
|(0)(1,1,1)(2,1,1)(3)
|-
|<math>\psi(\Omega_2^{\varepsilon_0})</math>
|<math>\psi(\Omega_{\varepsilon_0})</math>
|(0)(1,1,1)(2,1,1)(3)(4,1)
|-
|<math>\psi(\Omega_2^{\psi(\Omega_2)})</math>
|<math>\psi(\Omega_{\psi(\Omega_\omega)})</math>
|(0)(1,1,1)(2,1,1)(3)(4,1,1)
|-
|<math>\psi(\Omega_2^{\psi(\Omega_2^{\psi(\Omega_2)})})</math>
|<math>\psi(\Omega_{\psi(\Omega_{\psi(\Omega_\omega)})})</math>
|(0)(1,1,1)(2,1,1)(3)(4,1,1)(5,1,1)(6)(7,1,1)
|-
|<math>\psi(\Omega_2^\Omega)</math>
|<math>\psi(\Omega_\Omega)</math>
|(0)(1,1,1)(2,1,1)(3,1)
|}
 
== 分析6:BiO~EBO ==
此阶段分析对照BMS难度较低。至此,我们可以总结出一些规律:BMS与LPrSSψ之间,第一行决定层数,即从最外面到这个位置套了几层ψ函数。第二行决定基数,即这里应该有个Ω_n的n,当没有第二行,说明这个位置有后继,0则整体为后继序数,>0则后继在ψ中,提到外面后变成×ω。第三行决定阶差,即这里的基数比所在的ψ输出的基数大多少,当没有第三行,说明这个位置阶差<=1,当第三行为1,在BGO前,对应一个阶差2的展开。
{| class="wikitable"
|+分析6:BiO~EBO
!LPrSSψ
!BOCF
!BMS
|-
|<math>\psi(\Omega_2^\Omega)</math>
|<math>\psi(\Omega_\Omega)</math>
|(0)(1,1,1)(2,1,1)(3,1)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3))</math>
|<math>\psi(\Omega_\Omega+\psi_1(\Omega_\omega))</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^\Omega))</math>
|<math>\psi(\Omega_\Omega+\psi_1(\Omega_\Omega))</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^\Omega+\Omega_2))</math>
|<math>\psi(\Omega_\Omega+\Omega_2)</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(2,2)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^\Omega+\Omega_3))</math>
|<math>\psi(\Omega_\Omega+\Omega_\omega)</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(2,2,1)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^\Omega2))</math>
|<math>\psi(\Omega_\Omega2)</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(2,2,1)(3,2,1)(4,1)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^\Omega\omega))</math>
|<math>\psi(\Omega_\Omega\omega)</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^\Omega\Omega))</math>
|<math>\psi(\Omega_\Omega\Omega)</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,1)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^\Omega\Omega_2))</math>
|<math>\psi(\Omega_\Omega\Omega_2)</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,2)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^\Omega\times(\Omega_2+1)))</math>
|<math>\psi(\Omega_\Omega^2)</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,2)(2,2,1)(3,2,1)(4,1)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^\Omega\times\psi_2(\Omega_3)))</math>
|<math>\psi(\Omega_{\Omega+1})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,2)(4,3)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^\Omega\times\psi_2(\Omega_4)))</math>
|<math>\psi(\Omega_{\Omega+\omega})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,2)(4,3,1)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^\Omega\times\psi_2(\Omega_4^2)))</math>
|<math>\psi(\Omega_{\Omega+\omega^\omega})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,2)(4,3,1)(5,4,1)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^\Omega\times\psi_2(\Omega_4^\Omega)))</math>
|<math>\psi(\Omega_{\Omega2})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,2)(4,3,1)(5,4,1)(6,1)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^{\Omega+1}))</math>
|<math>\psi(\Omega_{\Omega\omega})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,2,1)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^{\Omega+2}))</math>
|<math>\psi(\Omega_{\Omega\times\omega^\omega})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,2,1)(3,2,1)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^{\Omega2}))</math>
|<math>\psi(\Omega_{\Omega^2})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,2,1)(4,1)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^{\Omega^2}))</math>
|<math>\psi(\Omega_{\Omega^\Omega})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(4,1)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^{\psi_1(\Omega_2)}))</math>
|<math>\psi(\Omega_{\psi_1(\Omega_2)})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(5,2)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^{\psi_1(\Omega_3)}))</math>
|<math>\psi(\Omega_{\psi_1(\Omega_\omega)})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(5,2,1)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^{\psi_1(\Omega_3^\Omega)}))</math>
|<math>\psi(\Omega_{\psi_1(\Omega_\Omega)})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(5,2,1)(6,2,1)(7,1)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^{\Omega_2}))</math>
|<math>\psi(\Omega_{\Omega_2})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,2)
|-
|<math>\psi(\Omega_2^\Omega+\psi_1(\Omega_3^{\Omega_2}+\psi_2(\Omega_4^{\Omega_3})))</math>
|<math>\psi(\Omega_{\Omega_3})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,2)(2,2)(3,3,1)(4,3,1)(5,3)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_2)</math>
|<math>\psi(\Omega_{\Omega_\omega})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_22)</math>
|<math>\psi(\Omega_{\Omega_\omega}+\Omega_\omega)</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(1,1,1)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_2\times\psi_1(\Omega_2))</math>
|<math>\psi(\Omega_{\Omega_\omega}+\psi_\omega(\Omega_{\omega+1}))</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_2\times\psi_1(\Omega_3))</math>
|<math>\psi(\Omega_{\Omega_\omega}+\psi_\omega(\Omega_{\omega2}))</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_2\times\psi_1(\Omega_3^\Omega))</math>
|<math>\psi(\Omega_{\Omega_\omega}+\psi_\omega(\Omega_{\Omega}))</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_2\times\psi_1(\Omega_3^\Omega+\Omega_2))</math>
|<math>\psi(\Omega_{\Omega_\omega}+\Omega_{\omega+1})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)(3,2)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_2\times\psi_1(\Omega_3^\Omega+\Omega_3))</math>
|<math>\psi(\Omega_{\Omega_\omega}+\Omega_{\omega2})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)(3,2,1)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_2\times\psi_1(\Omega_3^\Omega2))</math>
|<math>\psi(\Omega_{\Omega_\omega}+\Omega_{\Omega})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)(3,2,1)(4,3,1)(5,1)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_2\times\psi_1(\Omega_3^\Omega2+\Omega_3))</math>
|<math>\psi(\Omega_{\Omega_\omega}2)</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)(3,2,1)(4,3,1)(5,1)(1,1,1)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_2\times\psi_1(\Omega_3^\Omega\Omega))</math>
|<math>\psi(\Omega_{\Omega_\omega}\Omega)</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)(4,1)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_2\times\psi_1(\Omega_3^\Omega\Omega_2))</math>
|<math>\psi(\Omega_{\Omega_\omega}\Omega_{\omega+1})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)(4,2)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_2\times\psi_1(\Omega_3^\Omega\times\psi_2(\Omega_3)))</math>
|<math>\psi(\Omega_{\Omega_\omega+1})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)(4,2)(5,3)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_2\times\psi_1(\Omega_3^\Omega\times\psi_2(\Omega_4)))</math>
|<math>\psi(\Omega_{\Omega_\omega+\omega})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)(4,2)(5,3,1)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_2\times\psi_1(\Omega_3^{\Omega+1}))</math>
|<math>\psi(\Omega_{\Omega_\omega\omega})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)(4,2,1)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_2\times\psi_1(\Omega_3^{\Omega_2}))</math>
|<math>\psi(\Omega_{\Omega_{\omega+1}})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,2)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_2\times\psi_1(\Omega_3^{\Omega_2}+\psi_2(\Omega_4^{\Omega_3})))</math>
|<math>\psi(\Omega_{\Omega_{\omega+2}})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,2)(3,2)(4,3,1)(5,3,1)(6,3)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_2\times\psi_1(\Omega_3^{\Omega_2}+\Omega_3))</math>
|<math>\psi(\Omega_{\Omega_{\omega2}})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,2)(3,2,1)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_2^2)</math>
|<math>\psi(\Omega_{\Omega_{\omega^2}})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)
|-
|<math>\psi(\Omega_2^\Omega+\Omega_2^\omega)</math>
|<math>\psi(\Omega_{\Omega_{\omega^\omega}})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3)
|-
|<math>\psi(\Omega_2^\Omega2)</math>
|<math>\psi(\Omega_{\Omega_\Omega})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)
|-
|<math>\psi(\Omega_2^\Omega2+\Omega_2)</math>
|<math>\psi(\Omega_{\Omega_{\Omega_\omega}})</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(1,1,1)
|-
|<math>\psi(\Omega_2^\Omega3)</math>
|<math>\psi(\Omega\downarrow\downarrow4)</math>
|(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)
|-
|<math>\psi(\Omega_2^\Omega\omega)</math>
|<math>\psi(I)</math>
|(0)(1,1,1)(2,1,1)(3,1)(2)
|}
 
 
下篇:[[LPrSSψ分析Part2]]
[[分类:分析]]
[[分类:分析]]

2026年5月6日 (三) 11:13的最新版本

以下为长初等序列序数折叠函数(LPrSSψ)的分析。LPrSSψ与BOCF在BHO前完全一致,但强度大得多,可达(0)(1,1,1)(2,2,1)(3)以上。

分析1:BHO~BO

此阶段分析很简单,仅需将每一层从对应的ψ拆出来即可。

分析1:BHO~BO
LPrSSψ BOCF BMS
ψ(ψ1(Ω2))=ψ(εΩ+1) ψ(Ω2) (0)(1,1)(2,2)
ψ(ψ1(Ω2)+1) ψ(Ω2+1) (0)(1,1)(2,2)(1)
ψ(ψ1(Ω2)+Ω) ψ(Ω2+Ω) (0)(1,1)(2,2)(1,1)
ψ(ψ1(Ω2)×2) ψ(Ω2+ψ1(Ω2)) (0)(1,1)(2,2)(1,1)(2,2)
ψ(ψ1(Ω2)×3) ψ(Ω2+ψ1(Ω2)×2) (0)(1,1)(2,2)(1,1)(2,2)(1,1)(2,2)
ψ(ψ1(Ω2+1)) ψ(Ω2+ψ1(Ω2+1)) (0)(1,1)(2,2)(2)
ψ(ψ1(Ω2+ω)) ψ(Ω2+ψ1(Ω2+ω)) (0)(1,1)(2,2)(2)(3)
ψ(ψ1(Ω2+Ω)) ψ(Ω2+ψ1(Ω2+Ω)) (0)(1,1)(2,2)(2,1)
ψ(ψ1(Ω2+ψ1(Ω2))) ψ(Ω2+ψ1(Ω2+ψ1(Ω2))) (0)(1,1)(2,2)(2,1)(3,2)
ψ(ψ1(Ω2+ψ1(Ω2+ψ1(Ω2)))) ψ(Ω2+ψ1(Ω2+ψ1(Ω2+ψ1(Ω2)))) (0)(1,1)(2,2)(2,1)(3,2)(3,1)(4,2)
ψ(ψ1(Ω22))=ψ(εΩ+2) ψ(Ω22) (0)(1,1)(2,2)(2,2)
ψ(ψ1(Ω22)×2) ψ(Ω22+ψ1(Ω22)) (0)(1,1)(2,2)(2,2)(1,1)(2,2)(2,2)
ψ(ψ1(Ω22+1)) ψ(Ω22+ψ1(Ω22+1)) (0)(1,1)(2,2)(2,2)(2)
ψ(ψ1(Ω23)) ψ(Ω23) (0)(1,1)(2,2)(2,2)(2,2)
ψ(ψ1(Ω2ω))=ψ(εΩ+ω) ψ(Ω2ω) (0)(1,1)(2,2)(3)
ψ(ψ1(Ω2Ω))=ψ(εΩ2) ψ(Ω2Ω) (0)(1,1)(2,2)(3,1)
ψ(ψ1(Ω2×ψ1(Ω2)))=ψ(εεΩ+1) ψ(Ω2×ψ1(Ω2)) (0)(1,1)(2,2)(3,1)(4,2)
ψ(ψ1(Ω22))=ψ(ζΩ+1) ψ(Ω22) (0)(1,1)(2,2)(3,2)
ψ(ψ1(Ω22)×2) ψ(Ω22+ψ1(Ω22)) (0)(1,1)(2,2)(3,2)(1,1)(2,2)(3,2)
ψ(ψ1(Ω22+Ω2)) ψ(Ω22+Ω2) (0)(1,1)(2,2)(3,2)(2,2)
ψ(ψ1(Ω222)) ψ(Ω222) (0)(1,1)(2,2)(3,2)(2,2)(3,2)
ψ(ψ1(Ω23)) ψ(Ω23) (0)(1,1)(2,2)(3,2)(3,2)
ψ(ψ1(Ω2ω))=ψ(φ(ω,Ω+1)) ψ(Ω2ω) (0)(1,1)(2,2)(3,2)(4)
ψ(ψ1(Ω2Ω))=ψ(φ(Ω,1)) ψ(Ω2Ω) (0)(1,1)(2,2)(3,2)(4,1)
ψ(ψ1(Ω2ψ1(Ω2))) ψ(Ω2ψ1(Ω2)) (0)(1,1)(2,2)(3,2)(4,1)(5,2)
ψ(ψ1(Ω2Ω2))=ψ(ΓΩ+1) ψ(Ω2Ω2) (0)(1,1)(2,2)(3,2)(4,2)
ψ(ψ1(Ω23)) ψ(Ω23) (0)(1,1)(2,2)(3,2)(4,2)(5,2)
ψ(ψ1(ψ2(Ω3)))=ψ(ψ1(εΩ2+1)) ψ(Ω3) (0)(1,1)(2,2)(3,3)
ψ(ψ1(ψ2(Ω3))×2) ψ(Ω3+ψ1(Ω3)) (0)(1,1)(2,2)(3,3)(1,1)(2,2)(3,3)
ψ(ψ1(ψ2(Ω3)+1)) ψ(Ω3+ψ1(Ω3+1)) (0)(1,1)(2,2)(3,3)(2)
ψ(ψ1(ψ2(Ω3)+Ω2)) ψ(Ω3+Ω2) (0)(1,1)(2,2)(3,3)(2,2)
ψ(ψ1(ψ2(Ω3)×2)) ψ(Ω3+ψ2(Ω3)) (0)(1,1)(2,2)(3,3)(2,2)(3,3)
ψ(ψ1(ψ2(Ω3+1))) ψ(Ω3+ψ2(Ω3+1)) (0)(1,1)(2,2)(3,3)(3)
ψ(ψ1(ψ2(Ω3+Ω2))) ψ(Ω3+ψ2(Ω3+Ω2)) (0)(1,1)(2,2)(3,3)(3,2)
ψ(ψ1(ψ2(Ω3+ψ2(Ω3)))) ψ(Ω3+ψ2(Ω3+ψ2(Ω3))) (0)(1,1)(2,2)(3,3)(3,2)(4,3)
ψ(ψ1(ψ2(Ω32))) ψ(Ω32) (0)(1,1)(2,2)(3,3)(3,3)
ψ(ψ1(ψ2(Ω3ω))) ψ(Ω3ω) (0)(1,1)(2,2)(3,3)(4)
ψ(ψ1(ψ2(Ω32))) ψ(Ω32) (0)(1,1)(2,2)(3,3)(4,3)
ψ(ψ1(ψ2(Ω3Ω3))) ψ(Ω3Ω3) (0)(1,1)(2,2)(3,3)(4,3)(5,3)
ψ(ψ1(ψ2(ψ3(Ω4)))) ψ(Ω4) (0)(1,1)(2,2)(3,3)(4,4)
ψ(ψ1(ψ2(ψ3(ψ4(Ω5))))) ψ(Ω5) (0)(1,1)(2,2)(3,3)(4,4)(5,5)
ψ(Ω2)=ψ(ψ1(ψ2(ψ3(ψ4(...))))) ψ(Ωω) (0)(1,1,1)

分析2:BO~(0)(1,1,1)(2,1)

到(0)(1,1,1)(2,1)的分析仍较简单,仅需找层即可,但此后,LPrSSψ的行为更类似BMS,而不是BOCF,因为具有(0)(1,1,1)(2,1)(1,1,1)提升。

分析2:BO~FBLO
LPrSSψ BOCF BMS
ψ(Ω2) ψ(Ωω) (0)(1,1,1)
ψ(Ω2+1) ψ(Ωω+1) (0)(1,1,1)(1)
ψ(Ω2+Ω) ψ(Ωω+Ω) (0)(1,1,1)(1,1)
ψ(Ω2+ψ1(Ω2)) ψ(Ωω+ψ1(Ω2)) (0)(1,1,1)(1,1)(2,2)
ψ(Ω2+ψ1(ψ2(Ω3))) ψ(Ωω+ψ1(Ω3)) (0)(1,1,1)(1,1)(2,2)(3,3)
ψ(Ω2+ψ1(Ω3))=ψ(Ω2+ψ1(ψ2(ψ3(...)))) ψ(Ωω+ψ1(Ωω)) (0)(1,1,1)(1,1)(2,2,1)
ψ(Ω2+ψ1(Ω3)×2) ψ(Ωω+ψ1(Ωω)×2) (0)(1,1,1)(1,1)(2,2,1)(1,1)(2,2,1)
ψ(Ω2+ψ1(Ω3+1)) ψ(Ωω+ψ1(Ωω+1)) (0)(1,1,1)(1,1)(2,2,1)(2)
ψ(Ω2+ψ1(Ω3+Ω)) ψ(Ωω+ψ1(Ωω+Ω)) (0)(1,1,1)(1,1)(2,2,1)(2,1)
ψ(Ω2+ψ1(Ω3+ψ1(Ω2))) ψ(Ωω+ψ1(Ωω+ψ1(Ω2))) (0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2)
ψ(Ω2+ψ1(Ω3+ψ1(Ω3))) ψ(Ωω+ψ1(Ωω+ψ1(Ωω))) (0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2,1)
ψ(Ω2+ψ1(Ω3+Ω2)) ψ(Ωω+Ω2) (0)(1,1,1)(1,1)(2,2,1)(2,2)
ψ(Ω2+ψ1(Ω3+Ω22)) ψ(Ωω+Ω22) (0)(1,1,1)(1,1)(2,2,1)(2,2)(2,2)
ψ(Ω2+ψ1(Ω3+Ω22)) ψ(Ωω+Ω22) (0)(1,1,1)(1,1)(2,2,1)(2,2)(3,2)
ψ(Ω2+ψ1(Ω3+ψ2(Ω3))) ψ(Ωω+ψ2(Ω3)) (0)(1,1,1)(1,1)(2,2,1)(2,2)(3,3)
ψ(Ω2+ψ1(Ω3+ψ2(Ω4))) ψ(Ωω+ψ2(Ωω)) (0)(1,1,1)(1,1)(2,2,1)(2,2)(3,3,1)
ψ(Ω2+ψ1(Ω3+ψ2(Ω4+Ω3))) ψ(Ωω+Ω3) (0)(1,1,1)(1,1)(2,2,1)(2,2)(3,3,1)(3,3)
ψ(Ω2+ψ1(Ω3+ψ2(Ω4+ψ3(Ω5)))) ψ(Ωω+ψ3(Ωω)) (0)(1,1,1)(1,1)(2,2,1)(2,2)(3,3,1)(3,3)(4,4,1)
ψ(Ω22)=ψ(Ω2+ψ1(Ω3+ψ2(Ω4+ψ3(Ω5+...)))) ψ(Ωω2) (0)(1,1,1)(1,1,1)
ψ(Ω22+ψ1(Ω3)) ψ(Ωω2+ψ1(Ωω)) (0)(1,1,1)(1,1,1)(1,1)(2,2,1)
ψ(Ω22+ψ1(Ω32)) ψ(Ωω2+ψ1(Ωω2)) (0)(1,1,1)(1,1,1)(1,1)(2,2,1)(2,2,1)
ψ(Ω22+ψ1(Ω32+Ω2)) ψ(Ωω2+Ω2) (0)(1,1,1)(1,1,1)(1,1)(2,2,1)(2,2,1)(2,2)
ψ(Ω23) ψ(Ωω3) (0)(1,1,1)(1,1,1)(1,1,1)
ψ(Ω2ω) ψ(Ωωω) (0)(1,1,1)(2)
ψ(Ω2×(ω+1)) ψ(Ωω×(ω+1)) (0)(1,1,1)(2)(1,1,1)
ψ(Ω2×ε0) ψ(Ωω×ε0) (0)(1,1,1)(2)(3,1)
ψ(Ω2×ψ(Ω2)) ψ(Ωω×ψ(Ωω)) (0)(1,1,1)(2)(3,1,1)
ψ(Ω2Ω) ψ(ΩωΩ) (0)(1,1,1)(2,1)

分析3:(0)(1,1,1)(2,1)~(0)(1,1,1)(2,1)(1,1,1)

分析进入提升阶段。可见LPrSSψ中,ψ中的一个Ω_2对应的不是BOCF中的一个Ω_ω,而是BMS中的一个(1,1,1)。类似的,BGO前许多阶差2的表达式对应BMS第三行的1。这里注意LPrSSψ一般不单独定义Ω_0,展开方式,Ω_2展开出ψ_1时,把外面的Ω变成了里面的Ω_2,以此类推,但ω不提升。因为阶差为d时,Ω_(n+1)变出的ψ_n只把外面的x>=n的Ω_x提升为Ω_(x+d)。不然,若定义Ω_0=ω,并在这种位置将其提升为Ω,会与没有【特别地,如果坏根中元素不是坏部中某项的祖先项,则该项在复制过程中将保持不变。】规则的BMS一样,发生无穷降链。此外,若无此种提升,可能可以得到类似IBMS的ILPrSSψ。

分析3:(0)(1,1,1)(2,1)~(0)(1,1,1)(2,1)(1,1,1)
LPrSSψ BOCF BMS
ψ(Ω2Ω) ψ(ΩωΩ) (0)(1,1,1)(2,1)
ψ(Ω2Ω+ψ1(Ω3)) ψ(ΩωΩ+ψ1(Ωω)) (0)(1,1,1)(2,1)(1,1)(2,2,1)
ψ(Ω2Ω+ψ1(Ω3Ω)) ψ(ΩωΩ+ψ1(ΩωΩ)) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)
ψ(Ω2Ω+ψ1(Ω3Ω+1)) ψ(ΩωΩ+ψ1(ΩωΩ+1)) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2)
ψ(Ω2Ω+ψ1(Ω3Ω+Ω)) ψ(ΩωΩ+ψ1(ΩωΩ+Ω)) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,1)
ψ(Ω2Ω+ψ1(Ω3Ω+ψ1(Ω3Ω))) ψ(ΩωΩ+ψ1(ΩωΩ+ψ1(ΩωΩ))) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,1)(3,2,1)(4,1)
ψ(Ω2Ω+ψ1(Ω3Ω+Ω2)) ψ(ΩωΩ+Ω2) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,2)
ψ(Ω2Ω+ψ1(Ω3Ω+ψ2(Ω4))) ψ(ΩωΩ+ψ2(Ωω)) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,2)(3,3,1)
ψ(Ω2Ω+ψ1(Ω3Ω+ψ2(Ω4Ω))) ψ(ΩωΩ+ψ2(ΩωΩ)) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,2)(3,3,1)(4,1)
ψ(Ω2Ω+ψ1(Ω3Ω+ψ2(Ω4Ω+Ω3))) ψ(ΩωΩ+Ω3) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,2)(3,3,1)(4,1)(3,3)
ψ(Ω2Ω+ψ1(Ω3×(Ω+1)))=ψ(Ω2Ω+ψ1(Ω3Ω+ψ2(Ω4Ω+...))) ψ(Ωω×(Ω+1)) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,2,1)
ψ(Ω2Ω+ψ1(Ω3×(Ω+2))) ψ(Ωω×(Ω+2)) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,2,1)(2,2,1)
ψ(Ω2Ω+ψ1(Ω3×(Ω+ω))) ψ(Ωω×(Ω+ω)) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,2,1)(3)
ψ(Ω2Ω+ψ1(Ω3×Ω2)) ψ(Ωω×Ω2) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,2,1)(3,1)
ψ(Ω2Ω+ψ1(Ω3×Ωω)) ψ(Ωω×Ωω) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(3)
ψ(Ω2Ω+ψ1(Ω3×Ω2)) ψ(Ωω×Ω2) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(3,1)
ψ(Ω2Ω+ψ1(Ω3×Ωω)) ψ(Ωω×Ωω) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(4)
ψ(Ω2Ω+ψ1(Ω3×ΩΩ)) ψ(Ωω×ΩΩ) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(4,1)
ψ(Ω2Ω+ψ1(Ω3×ψ1(Ω2))) ψ(Ωω×ψ1(Ω2)) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(4,2)
ψ(Ω2Ω+ψ1(Ω3×ψ1(Ω3))) ψ(Ωω×ψ1(Ωω)) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(4,2,1)
ψ(Ω2Ω+ψ1(Ω3×ψ1(Ω3×ψ1(Ω3)))) ψ(Ωω×ψ1(Ωω×ψ1(Ωω))) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(4,2,1)(5,1)(6,2,1)
ψ(Ω2Ω+ψ1(Ω3Ω2)) ψ(ΩωΩ2) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)
ψ(Ω2Ω+ψ1(Ω3Ω2+Ω2)) ψ(ΩωΩ2+Ω2) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)(2,2)
ψ(Ω2Ω+ψ1(Ω3Ω2+ψ2(Ω3))) ψ(ΩωΩ2+ψ2(Ω3)) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)(2,2)(3,3)
ψ(Ω2Ω+ψ1(Ω3Ω2+ψ2(Ω4))) ψ(ΩωΩ2+ψ2(Ωω)) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)(2,2)(3,3,1)
ψ(Ω2Ω+ψ1(Ω3Ω2+ψ2(Ω4Ω))) ψ(ΩωΩ2+ψ2(ΩωΩ)) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)(2,2)(3,3,1)(4,1)
ψ(Ω2Ω+ψ1(Ω3Ω2+ψ2(Ω4Ω2))) ψ(ΩωΩ2+ψ2(ΩωΩ2)) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)(2,2)(3,3,1)(4,2)
ψ(Ω2Ω+ψ1(Ω3Ω2+ψ2(Ω4Ω2+Ω3))) ψ(ΩωΩ2+Ω3) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)(2,2)(3,3,1)(4,2)(3,3)
ψ(Ω2Ω+ψ1(Ω3Ω2+ψ2(Ω4×(Ω2+1)))) ψ(Ωω×(Ω2+1)) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)(2,2)(3,3,1)(4,2)(3,3,1)
ψ(Ω2Ω+ψ1(Ω3Ω2+ψ2(Ω4×Ω22))) ψ(Ωω×Ω22) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)(2,2)(3,3,1)(4,2)(3,3,1)(4,2)
ψ(Ω2Ω+ψ1(Ω3Ω2+ψ2(Ω4×ψ2(Ω4)))) ψ(Ωω×ψ2(Ωω)) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)(2,2)(3,3,1)(4,2)(5,3,1)
ψ(Ω2Ω+ψ1(Ω3Ω2+ψ2(Ω4Ω3))) ψ(ΩωΩ3) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)(2,2)(3,3,1)(4,3)
ψ(Ω2Ω+ψ1(Ω3Ω2+ψ2(Ω4Ω3+ψ3(Ω5Ω4)))) ψ(ΩωΩ4) (0)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)(2,2)(3,3,1)(4,3)(3,3)(4,4,1)(5,4)
ψ(Ω2×(Ω+1))=ψ(Ω2Ω+ψ1(Ω3Ω2+ψ2(Ω4Ω3+...))) ψ(Ωω2) (0)(1,1,1)(2,1)(1,1,1)

分析4:(0)(1,1,1)(2,1)(1,1,1)~TFBO

此阶段难度相对较低,LPrSSψ与BMS对应较为简单,BOCF有随时出现的Ω变Ω_ω提升。

分析4:(0)(1,1,1)(2,1)(1,1,1)~TFBO
LPrSSψ BOCF BMS
ψ(Ω2×(Ω+1)) ψ(Ωω2) (0)(1,1,1)(2,1)(1,1,1)
ψ(Ω2×(Ω+1)+ψ1(Ω3)) ψ(Ωω2+ψ1(Ωω)) (0)(1,1,1)(2,1)(1,1,1)(1,1)(2,2,1)
ψ(Ω2×(Ω+1)+ψ1(Ω3Ω)) ψ(Ωω2+ψ1(ΩωΩ)) (0)(1,1,1)(2,1)(1,1,1)(1,1)(2,2,1)(3,1)
ψ(Ω2×(Ω+1)+ψ1(Ω3Ω2)) ψ(Ωω2+ψ1(ΩωΩ2)) (0)(1,1,1)(2,1)(1,1,1)(1,1)(2,2,1)(3,2)
ψ(Ω2×(Ω+1)+ψ1(Ω3×(Ω2+1))) ψ(Ωω2+ψ1(Ωω2)) (0)(1,1,1)(2,1)(1,1,1)(1,1)(2,2,1)(3,2)(2,2,1)
ψ(Ω2×(Ω+1)+ψ1(Ω3×(Ω2+1)+Ω2)) ψ(Ωω2+Ω2) (0)(1,1,1)(2,1)(1,1,1)(1,1)(2,2,1)(3,2)(2,2,1)(2,2)
ψ(Ω2×(Ω+2)) ψ(Ωω2+Ωω) (0)(1,1,1)(2,1)(1,1,1)(1,1,1)
ψ(Ω2×(Ω+ω)) ψ(Ωω2+Ωωω) (0)(1,1,1)(2,1)(1,1,1)(2)
ψ(Ω2×Ω2) ψ(Ωω2+ΩωΩ) (0)(1,1,1)(2,1)(1,1,1)(2,1)
ψ(Ω2×Ω2+ψ1(Ω3×(Ω2+Ω))) ψ(Ωω2+ΩωΩ+ψ1(Ωω2+ΩωΩ)) (0)(1,1,1)(2,1)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)(2,2,1)(3,1)
ψ(Ω2×Ω2+ψ1(Ω3×(Ω2+Ω)+Ω2)) ψ(Ωω2+Ωω×(Ω+1)) (0)(1,1,1)(2,1)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)(2,2,1)(3,1)(2,2,1)
ψ(Ω2×Ω2+ψ1(Ω3×Ω22)) ψ(Ωω2+ΩωΩ2) (0)(1,1,1)(2,1)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)(2,2,1)(3,2)
ψ(Ω2×(Ω2+1)) ψ(Ωω22) (0)(1,1,1)(2,1)(1,1,1)(2,1)(1,1,1)
ψ(Ω2×Ωω) ψ(Ωω2ω) (0)(1,1,1)(2,1)(2)
ψ(Ω2×Ω2) ψ(Ωω2Ω) (0)(1,1,1)(2,1)(2,1)
ψ(Ω2×Ω2+ψ1(Ω3Ω2Ω)) ψ(Ωω2Ω+ψ1(Ωω2Ω)) (0)(1,1,1)(2,1)(2,1)(1,1)(2,2,1)(3,2)(3,1)
ψ(Ω2×Ω2+ψ1(Ω3Ω2Ω+Ω2)) ψ(Ωω2Ω+Ωω) (0)(1,1,1)(2,1)(2,1)(1,1)(2,2,1)(3,2)(3,1)(2,2,1)
ψ(Ω2×Ω2+ψ1(Ω3×Ω22)) ψ(Ωω2Ω2) (0)(1,1,1)(2,1)(2,1)(1,1)(2,2,1)(3,2)(3,2)
ψ(Ω2×(Ω2+1)) ψ(Ωω3) (0)(1,1,1)(2,1)(2,1)(1,1,1)
ψ(Ω2×Ωω) ψ(Ωωω) (0)(1,1,1)(2,1)(3)
ψ(Ω2×ΩΩ) ψ(ΩωΩ) (0)(1,1,1)(2,1)(3,1)
ψ(Ω2×ΩΩ+ψ1(Ω3×Ω2Ω)) ψ(ΩωΩ+ψ1(ΩωΩ)) (0)(1,1,1)(2,1)(3,1)(1,1)(2,2,1)(3,2)(4,1)
ψ(Ω2×ΩΩ+ψ1(Ω3×Ω2Ω+Ω2)) ψ(ΩωΩ+Ωω) (0)(1,1,1)(2,1)(3,1)(1,1)(2,2,1)(3,2)(4,1)(2,2,1)
ψ(Ω2×ΩΩ+ψ1(Ω3×Ω2Ω2)) ψ(ΩωΩ2) (0)(1,1,1)(2,1)(3,1)(1,1)(2,2,1)(3,2)(4,2)
ψ(Ω2×(ΩΩ+1)) ψ(ΩωΩω) (0)(1,1,1)(2,1)(3,1)(1,1,1)
ψ(Ω2×ΩΩ2) ψ(ΩωΩω+ΩωΩ) (0)(1,1,1)(2,1)(3,1)(1,1,1)(2,1)(3,1)
ψ(Ω2×ΩΩ+1) ψ(ΩωΩωΩ) (0)(1,1,1)(2,1)(3,1)(2,1)
ψ(Ω2×ΩΩ2) ψ(ΩωΩω+Ω) (0)(1,1,1)(2,1)(3,1)(2,1)(3,1)
ψ(Ω2×ΩΩ2) ψ(ΩωΩωΩ) (0)(1,1,1)(2,1)(3,1)(3,1)
ψ(Ω2×ΩΩΩ) ψ(ΩωΩωΩ) (0)(1,1,1)(2,1)(3,1)(4,1)
ψ(Ω2×ψ1(Ω2))=ψ(Ω2×εΩ+1) ψ(Ωω+1) (0)(1,1,1)(2,1)(3,2)

分析5:TFBO~BiO

此阶段若对照BMS分析,难度相对较低,而BOCF有随时出现的各级提升。

分析5:TFBO~BiO
LPrSSψ BOCF BMS
ψ(Ω2×ψ1(Ω2)) ψ(Ωω+1) (0)(1,1,1)(2,1)(3,2)
ψ(Ω2×ψ1(Ω2)+ψ1(Ω3×ψ2(Ω3))) ψ(Ωω+1+ψ1(Ωω+1)) (0)(1,1,1)(2,1)(3,2)(1,1)(2,2,1)(3,2)(4,3)
ψ(Ω2×ψ1(Ω2)+ψ1(Ω3×ψ2(Ω3)+Ω2)) ψ(Ωω+1+Ω2) (0)(1,1,1)(2,1)(3,2)(1,1)(2,2,1)(3,2)(4,3)(2,2)
ψ(Ω2×(ψ1(Ω2)+1)) ψ(Ωω+1+Ωω) (0)(1,1,1)(2,1)(3,2)(1,1,1)
ψ(Ω2×(ψ1(Ω2)+Ω)) ψ(Ωω+1+ΩωΩ) (0)(1,1,1)(2,1)(3,2)(1,1,1)(2,1)
ψ(Ω2×(ψ1(Ω2)+Ω+1)) ψ(Ωω+1+Ωω2) (0)(1,1,1)(2,1)(3,2)(1,1,1)(2,1)(1,1,1)
ψ(Ω2×ψ1(Ω2)×2) ψ(Ωω+1+ψω(Ωω+1)) (0)(1,1,1)(2,1)(3,2)(1,1,1)(2,1)(3,2)
ψ(Ω2×ψ1(Ω2+1)) ψ(Ωω+1+ψω(Ωω+1+1)) (0)(1,1,1)(2,1)(3,2)(2)
ψ(Ω2×ψ1(Ω2+Ω)) ψ(Ωω+1+ψω(Ωω+1+Ω)) (0)(1,1,1)(2,1)(3,2)(2,1)
ψ(Ω2×(ψ1(Ω2+Ω)+1)) ψ(Ωω+1+ψω(Ωω+1+Ωω)) (0)(1,1,1)(2,1)(3,2)(2,1)(1,1,1)
ψ(Ω2×ψ1(Ω2+ψ1(Ω2))) ψ(Ωω+1+ψω(Ωω+1+ψω(Ωω+1))) (0)(1,1,1)(2,1)(3,2)(2,1)(3,2)
ψ(Ω2×ψ1(Ω2+ψ1(Ω2+Ω))) ψ(Ωω+1+ψω(Ωω+1+ψω(Ωω+1+Ω))) (0)(1,1,1)(2,1)(3,2)(3,1)
ψ(Ω2×ψ1(Ω2+ψ1(Ω2+ψ1(Ω2)))) ψ(Ωω+1+ψω(Ωω+1+ψω(Ωω+1+ψω(Ωω+1)))) (0)(1,1,1)(2,1)(3,2)(3,1)(4,2)
ψ(Ω2×ψ1(Ω22)) ψ(Ωω+12) (0)(1,1,1)(2,1)(3,2)(3,2)
ψ(Ω2×ψ1(Ω2ω)) ψ(Ωω+1ω) (0)(1,1,1)(2,1)(3,2)(4)
ψ(Ω2×ψ1(Ω2Ω)) ψ(Ωω+1Ω) (0)(1,1,1)(2,1)(3,2)(4,1)
ψ(Ω2×(ψ1(Ω2Ω)+1)) ψ(Ωω+1Ωω) (0)(1,1,1)(2,1)(3,2)(4,1)(1,1,1)
ψ(Ω2×ψ1(Ω2×ψ1(Ω2))) ψ(Ωω+1×ψω(Ωω+1)) (0)(1,1,1)(2,1)(3,2)(4,1)(5,2)
ψ(Ω2×ψ1(Ω22)) ψ(Ωω+12) (0)(1,1,1)(2,1)(3,2)(4,2)
ψ(Ω2×ψ1(Ω2ω)) ψ(Ωω+1ω) (0)(1,1,1)(2,1)(3,2)(4,2)(5)
ψ(Ω2×ψ1(Ω2Ω)) ψ(Ωω+1Ω) (0)(1,1,1)(2,1)(3,2)(4,2)(5,1)
ψ(Ω2×ψ1(Ω2Ω2)) ψ(Ωω+1Ωω+1) (0)(1,1,1)(2,1)(3,2)(4,2)(5,2)
ψ(Ω2×ψ1(ψ2(Ω3))) ψ(Ωω+2) (0)(1,1,1)(2,1)(3,2)(4,3)
ψ(Ω2×ψ1(ψ2(Ω3)×2)) ψ(Ωω+2+ψω+1(Ωω+2)) (0)(1,1,1)(2,1)(3,2)(4,3)(3,2)(4,3)
ψ(Ω2×ψ1(ψ2(Ω32))) ψ(Ωω+22) (0)(1,1,1)(2,1)(3,2)(4,3)(4,3)
ψ(Ω2×ψ1(ψ2(Ω32))) ψ(Ωω+22) (0)(1,1,1)(2,1)(3,2)(4,3)(5,3)
ψ(Ω2×ψ1(ψ2(ψ3(Ω4)))) ψ(Ωω+3) (0)(1,1,1)(2,1)(3,2)(4,3)(5,4)
ψ(Ω2×ψ1(Ω3)) ψ(Ωω2) (0)(1,1,1)(2,1)(3,2,1)
ψ(Ω2×ψ1(Ω3)×2) ψ(Ωω2+ψω(Ωω2)) (0)(1,1,1)(2,1)(3,2,1)(1,1,1)(2,1)(3,2,1)
ψ(Ω2×ψ1(Ω3+1)) ψ(Ωω2+ψω(Ωω2+1)) (0)(1,1,1)(2,1)(3,2,1)(2)
ψ(Ω2×ψ1(Ω3+Ω)) ψ(Ωω2+ψω(Ωω2+Ω)) (0)(1,1,1)(2,1)(3,2,1)(2,1)
ψ(Ω2×ψ1(Ω3+ψ1(Ω3))) ψ(Ωω2+ψω(Ωω2+ψω(Ωω2))) (0)(1,1,1)(2,1)(3,2,1)(2,1)(3,2,1)
ψ(Ω2×ψ1(Ω3+Ω2)) ψ(Ωω2+Ωω+1) (0)(1,1,1)(2,1)(3,2,1)(3,2)
ψ(Ω2×ψ1(Ω32)) ψ(Ωω22) (0)(1,1,1)(2,1)(3,2,1)(3,2,1)
ψ(Ω2×ψ1(Ω3ω)) ψ(Ωω2ω) (0)(1,1,1)(2,1)(3,2,1)(4)
ψ(Ω2×ψ1(Ω3Ω)) ψ(Ωω2Ω) (0)(1,1,1)(2,1)(3,2,1)(4,1)
ψ(Ω2×ψ1(Ω3Ω2)) ψ(Ωω2Ωω+1) (0)(1,1,1)(2,1)(3,2,1)(4,2)
ψ(Ω2×ψ1(Ω3×(Ω2+1))) ψ(Ωω22) (0)(1,1,1)(2,1)(3,2,1)(4,2)(3,2,1)
ψ(Ω2×ψ1(Ω3×Ω22)) ψ(Ωω22×Ωω+1) (0)(1,1,1)(2,1)(3,2,1)(4,2)(4,2)
ψ(Ω2×ψ1(Ω3×Ω2Ω2)) ψ(Ωω2Ωω+1) (0)(1,1,1)(2,1)(3,2,1)(4,2)(5,2)
ψ(Ω2×ψ1(Ω3×ψ2(Ω3))) ψ(Ωω2+1) (0)(1,1,1)(2,1)(3,2,1)(4,2)(5,3)
ψ(Ω2×ψ1(Ω3×ψ2(ψ3(Ω4)))) ψ(Ωω2+2) (0)(1,1,1)(2,1)(3,2,1)(4,2)(5,3)(6,4)
ψ(Ω2×ψ1(Ω3×ψ2(Ω4))) ψ(Ωω3) (0)(1,1,1)(2,1)(3,2,1)(4,2)(5,3,1)
ψ(Ω2×ψ1(Ω3×ψ2(Ω4×ψ3(Ω5)))) ψ(Ωω4) (0)(1,1,1)(2,1)(3,2,1)(4,2)(5,3,1)(6,3)(7,4,1)
ψ(Ω22) ψ(Ωω2) (0)(1,1,1)(2,1,1)
ψ(Ω22+Ω2) ψ(Ωω2+Ωω) (0)(1,1,1)(2,1,1)(1,1,1)
ψ(Ω22+ψ1(Ω3)) ψ(Ωω2+ψω(Ωω2)) (0)(1,1,1)(2,1,1)(1,1,1)(2,1)(3,2,1)
ψ(Ω22+ψ1(Ω32)) ψ(Ωω2+ψω(Ωω2)) (0)(1,1,1)(2,1,1)(1,1,1)(2,1)(3,2,1)(4,2,1)
ψ(Ω22+ψ1(Ω32+Ω3)) ψ(Ωω2+Ωω2) (0)(1,1,1)(2,1,1)(1,1,1)(2,1)(3,2,1)(4,2,1)(3,2,1)
ψ(Ω222) ψ(Ωω22) (0)(1,1,1)(2,1,1)(1,1,1)(2,1,1)
ψ(Ω22Ω) ψ(Ωω2Ω) (0)(1,1,1)(2,1,1)(2,1)
ψ(Ω22Ω+Ω2) ψ(Ωω2Ωω) (0)(1,1,1)(2,1,1)(2,1)(1,1,1)
ψ(Ω22×(Ω+1)) ψ(Ωω22) (0)(1,1,1)(2,1,1)(2,1)(1,1,1)(2,1,1)
ψ(Ω22×ψ1(Ω2)) ψ(Ωω2+1) (0)(1,1,1)(2,1,1)(2,1)(3,2)
ψ(Ω22×ψ1(Ω3)) ψ(Ωω2+ω) (0)(1,1,1)(2,1,1)(2,1)(3,2,1)
ψ(Ω22×ψ1(Ω32)) ψ(Ωω22) (0)(1,1,1)(2,1,1)(2,1)(3,2,1)(4,2,1)
ψ(Ω23) ψ(Ωω3) (0)(1,1,1)(2,1,1)(2,1,1)
ψ(Ω2ω) ψ(Ωωω) (0)(1,1,1)(2,1,1)(3)
ψ(Ω2ε0) ψ(Ωε0) (0)(1,1,1)(2,1,1)(3)(4,1)
ψ(Ω2ψ(Ω2)) ψ(Ωψ(Ωω)) (0)(1,1,1)(2,1,1)(3)(4,1,1)
ψ(Ω2ψ(Ω2ψ(Ω2))) ψ(Ωψ(Ωψ(Ωω))) (0)(1,1,1)(2,1,1)(3)(4,1,1)(5,1,1)(6)(7,1,1)
ψ(Ω2Ω) ψ(ΩΩ) (0)(1,1,1)(2,1,1)(3,1)

分析6:BiO~EBO

此阶段分析对照BMS难度较低。至此,我们可以总结出一些规律:BMS与LPrSSψ之间,第一行决定层数,即从最外面到这个位置套了几层ψ函数。第二行决定基数,即这里应该有个Ω_n的n,当没有第二行,说明这个位置有后继,0则整体为后继序数,>0则后继在ψ中,提到外面后变成×ω。第三行决定阶差,即这里的基数比所在的ψ输出的基数大多少,当没有第三行,说明这个位置阶差<=1,当第三行为1,在BGO前,对应一个阶差2的展开。

分析6:BiO~EBO
LPrSSψ BOCF BMS
ψ(Ω2Ω) ψ(ΩΩ) (0)(1,1,1)(2,1,1)(3,1)
ψ(Ω2Ω+ψ1(Ω3)) ψ(ΩΩ+ψ1(Ωω)) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)
ψ(Ω2Ω+ψ1(Ω3Ω)) ψ(ΩΩ+ψ1(ΩΩ)) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)
ψ(Ω2Ω+ψ1(Ω3Ω+Ω2)) ψ(ΩΩ+Ω2) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(2,2)
ψ(Ω2Ω+ψ1(Ω3Ω+Ω3)) ψ(ΩΩ+Ωω) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(2,2,1)
ψ(Ω2Ω+ψ1(Ω3Ω2)) ψ(ΩΩ2) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(2,2,1)(3,2,1)(4,1)
ψ(Ω2Ω+ψ1(Ω3Ωω)) ψ(ΩΩω) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3)
ψ(Ω2Ω+ψ1(Ω3ΩΩ)) ψ(ΩΩΩ) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,1)
ψ(Ω2Ω+ψ1(Ω3ΩΩ2)) ψ(ΩΩΩ2) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,2)
ψ(Ω2Ω+ψ1(Ω3Ω×(Ω2+1))) ψ(ΩΩ2) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,2)(2,2,1)(3,2,1)(4,1)
ψ(Ω2Ω+ψ1(Ω3Ω×ψ2(Ω3))) ψ(ΩΩ+1) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,2)(4,3)
ψ(Ω2Ω+ψ1(Ω3Ω×ψ2(Ω4))) ψ(ΩΩ+ω) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,2)(4,3,1)
ψ(Ω2Ω+ψ1(Ω3Ω×ψ2(Ω42))) ψ(ΩΩ+ωω) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,2)(4,3,1)(5,4,1)
ψ(Ω2Ω+ψ1(Ω3Ω×ψ2(Ω4Ω))) ψ(ΩΩ2) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,2)(4,3,1)(5,4,1)(6,1)
ψ(Ω2Ω+ψ1(Ω3Ω+1)) ψ(ΩΩω) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,2,1)
ψ(Ω2Ω+ψ1(Ω3Ω+2)) ψ(ΩΩ×ωω) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,2,1)(3,2,1)
ψ(Ω2Ω+ψ1(Ω3Ω2)) ψ(ΩΩ2) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(3,2,1)(4,1)
ψ(Ω2Ω+ψ1(Ω3Ω2)) ψ(ΩΩΩ) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(4,1)
ψ(Ω2Ω+ψ1(Ω3ψ1(Ω2))) ψ(Ωψ1(Ω2)) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(5,2)
ψ(Ω2Ω+ψ1(Ω3ψ1(Ω3))) ψ(Ωψ1(Ωω)) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(5,2,1)
ψ(Ω2Ω+ψ1(Ω3ψ1(Ω3Ω))) ψ(Ωψ1(ΩΩ)) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,1)(5,2,1)(6,2,1)(7,1)
ψ(Ω2Ω+ψ1(Ω3Ω2)) ψ(ΩΩ2) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,2)
ψ(Ω2Ω+ψ1(Ω3Ω2+ψ2(Ω4Ω3))) ψ(ΩΩ3) (0)(1,1,1)(2,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,2)(2,2)(3,3,1)(4,3,1)(5,3)
ψ(Ω2Ω+Ω2) ψ(ΩΩω) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)
ψ(Ω2Ω+Ω22) ψ(ΩΩω+Ωω) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(1,1,1)
ψ(Ω2Ω+Ω2×ψ1(Ω2)) ψ(ΩΩω+ψω(Ωω+1)) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2)
ψ(Ω2Ω+Ω2×ψ1(Ω3)) ψ(ΩΩω+ψω(Ωω2)) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)
ψ(Ω2Ω+Ω2×ψ1(Ω3Ω)) ψ(ΩΩω+ψω(ΩΩ)) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)
ψ(Ω2Ω+Ω2×ψ1(Ω3Ω+Ω2)) ψ(ΩΩω+Ωω+1) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)(3,2)
ψ(Ω2Ω+Ω2×ψ1(Ω3Ω+Ω3)) ψ(ΩΩω+Ωω2) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)(3,2,1)
ψ(Ω2Ω+Ω2×ψ1(Ω3Ω2)) ψ(ΩΩω+ΩΩ) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)(3,2,1)(4,3,1)(5,1)
ψ(Ω2Ω+Ω2×ψ1(Ω3Ω2+Ω3)) ψ(ΩΩω2) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)(3,2,1)(4,3,1)(5,1)(1,1,1)
ψ(Ω2Ω+Ω2×ψ1(Ω3ΩΩ)) ψ(ΩΩωΩ) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)(4,1)
ψ(Ω2Ω+Ω2×ψ1(Ω3ΩΩ2)) ψ(ΩΩωΩω+1) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)(4,2)
ψ(Ω2Ω+Ω2×ψ1(Ω3Ω×ψ2(Ω3))) ψ(ΩΩω+1) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)(4,2)(5,3)
ψ(Ω2Ω+Ω2×ψ1(Ω3Ω×ψ2(Ω4))) ψ(ΩΩω+ω) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)(4,2)(5,3,1)
ψ(Ω2Ω+Ω2×ψ1(Ω3Ω+1)) ψ(ΩΩωω) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,1)(4,2,1)
ψ(Ω2Ω+Ω2×ψ1(Ω3Ω2)) ψ(ΩΩω+1) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,2)
ψ(Ω2Ω+Ω2×ψ1(Ω3Ω2+ψ2(Ω4Ω3))) ψ(ΩΩω+2) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,2)(3,2)(4,3,1)(5,3,1)(6,3)
ψ(Ω2Ω+Ω2×ψ1(Ω3Ω2+Ω3)) ψ(ΩΩω2) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,3,1)(5,2)(3,2,1)
ψ(Ω2Ω+Ω22) ψ(ΩΩω2) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)
ψ(Ω2Ω+Ω2ω) ψ(ΩΩωω) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3)
ψ(Ω2Ω2) ψ(ΩΩΩ) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)
ψ(Ω2Ω2+Ω2) ψ(ΩΩΩω) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(1,1,1)
ψ(Ω2Ω3) ψ(Ω4) (0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)
ψ(Ω2Ωω) ψ(I) (0)(1,1,1)(2,1,1)(3,1)(2)


下篇:LPrSSψ分析Part2