バシク行列の解析 by Kuroha

見積もりです。証明はありません。Rathjen の ψ を参照。強配列表記との対応は気持ち、厳密にはFGHみたいな対応を考えるべきなんだろうが。

\(Ω=χ_0(0)\)

\(I=χ_1(0)\)

\(Λ_0=ψ_{χ_M(0)}(0)\)

2行

(0)(1,1)	φ(1,0)
(0)(1,1)(2,1)	φ(2,0)
(0)(1,1)(2,1)(2,1)	φ(3,0)
(0)(1,1)(2,1)(3,1)	φ(1,0,0) \(=Ψ^0_Ω(0)\)
(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)	φ(1,0,1) \(=Ψ^0_Ω(1)\)
(0)(1,1)(2,1)(3,1)(2,1)	φ(1,1,0) Ψ^0_Ω(Ω)
(0)(1,1)(2,1)(3,1)(2,1)(1,1)(2,1)(3,1)	φ(1,0,φ(1,1,0)+1) \(=Ψ^0_Ω(Ψ^0_Ω(Ω))\)
(0)(1,1)(2,1)(3,1)(2,1)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)	φ(1,0,φ(1,1,0)+2) \(Ψ^0_Ω(Ψ^0_Ω(Ω)+1)\)
(0)(1,1)(2,1)(3,1)(2,1)(1,1)(2,1)(3,1)(2,1)	φ(1,1,1) \(=Ψ^0_Ω(Ω+1)\)
(0)(1,1)(2,1)(3,1)(2,1)(1,1)(2,1)(3,1)(2,1)(1,1)(2,1)(3,1)	φ(1,0,φ(1,1,1)+1) \(Ψ^0_Ω(Ψ^0_Ω(Ω+1))\)
(0)(1,1)(2,1)(3,1)(2,1)(1,1)(2,1)(3,1)(2,1)(1,1)(2,1)(3,1)(2,1)	φ(1,1,2) \(=Ψ^0_Ω(Ω+2)\)
(0)(1,1)(2,1)(3,1)(2,1)(2,1)	φ(1,2,0) \(=Ψ^0_Ω(Ω\cdot2)\)
(0)(1,1)(2,1)(3,1)(2,1)(2,1)(1,1)(2,1)(3,1)	φ(1,0,φ(1,2,0)+1) \(=Ψ^0_Ω(Ψ^0_Ω(Ω\cdot2))\)
(0)(1,1)(2,1)(3,1)(2,1)(2,1)(1,1)(2,1)(3,1)(2,1)	φ(1,1,φ(1,2,0)+1) \(=Ψ^0_Ω(Ω+Ψ^0_Ω(Ω\cdot2))\)
(0)(1,1)(2,1)(3,1)(2,1)(2,1)(1,1)(2,1)(3,1)(2,1)(1,1)(2,1)(3,1)	φ(1,0,φ(1,1,φ(1,2,0)+1)+1) \(=Ψ_Ω(Ψ^0_Ω(Ω+Ψ^0_Ω(Ω\cdot2)))\)
(0)(1,1)(2,1)(3,1)(2,1)(2,1)(1,1)(2,1)(3,1)(2,1)(1,1)(2,1)(3,1)(2,1)	φ(1,1,φ(1,2,0)+2) \(=Ψ^0_Ω(Ω+Ψ^0_Ω(Ω\cdot2)+1)\)
(0)(1,1)(2,1)(3,1)(2,1)(2,1)(1,1)(2,1)(3,1)(2,1)(2,1)	φ(1,2,1) \(=Ψ^0_Ω(Ω\cdot2+1)\)
(0)(1,1)(2,1)(3,1)(2,1)(2,1)(2,1)	φ(1,3,0) \(=Ψ^0_Ω(Ω\cdot3)\)
(0)(1,1)(2,1)(3,1)(2,1)(3,1)	φ(2,0,0) \(=Ψ^0_Ω(Ω^2)\)

3行 ψ_Ω(Λ_0)まで

(0)(1,1,1)	\(ψ_Ω(χ_0(ω))\\Ψ^0_Ω(Ω_ω)\\=\text{PTO}(Π^1_1-\text{CA}_0)\)
(0)(1,1,1)(1,1,1)	\(ψ_Ω(χ_0(ω)\cdot2)\)
(0)(1,1,1)(2,1)	\(ψ_Ω(χ_0(ω)\cdotχ_0(0))\)
(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)	\(ψ_Ω(χ_0(ω)\cdotχ_0(0)+ψ_{χ_0(1)}(χ_0(ω)\cdotχ_0(0)))\)
(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,2)(3,1)(4,2,1)(5,1)	\(ψ_Ω(χ_0(ω)\cdotχ_0(0)+ψ_{χ_0(1)}\\(χ_0(ω)\cdotχ_0(0)+ψ_{χ_0(1)}(χ_0(ω)\cdotχ_0(0))))\)
(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,2)(3,2)	\(ψ_Ω(χ_0(ω)\cdotχ_0(0)+χ_0(1))\)
(0)(1,1,1)(2,1,1)(3,1)(2)	\(ψ_Ω(Φ_1(0))\\=Ψ^0_Ω(Ψ^0_{Ξ(1)}(2))\)
(0)(1,1,1)(2,1,1)(3,1)(2,1,1)	\(ψ_Ω(Φ_1(ω))\\=Ψ^0_Ω(Ψ^0_{Ξ(1)}(ω))\)
(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(2)	\(ψ_Ω(Φ_2(0))\\=Ψ^0_Ω(Ξ(1))\)
(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(2)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(2)	\(ψ_Ω(Φ_2(0)\cdot2)\\=Ψ^0_Ω(Ξ(1)+Ψ^0_{Ξ(1)}(Ξ(1)))\)
(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(2,1)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(2)	\(ψ_Ω(Φ_2(0)^2)\\=Ψ^0_Ω(Ξ(1)+Ψ^0_{Ξ(1)}(Ξ(1))^2)\)
(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2)(4)	\(ψ_Ω(Φ_1(Φ_2(0)+1))\\=Ψ^0_Ω(Ξ(1)+Ψ^0_{Ξ(1)}(Ψ^0_{Ξ(1)}(Ξ(1))))\)
(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2)(4,2,1)	\(ψ_Ω(Φ_1(Φ_2(0)+ω))\\=Ψ^0_Ω(Ξ(1)+Ψ^0_{Ξ(1)}(Ψ^0_{Ξ(1)}(Ξ(1))+ω))\)
(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2)(4,2,1)(5,2)(4)	\(ψ_Ω(Φ_2(1))\\=Ψ^0_Ω(Ξ(1)+Ψ^0_{Ξ(1)}(Ξ(1)+1))\)
(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(2,1,1)	\(ψ_Ω(Φ_2(ω))\\=Ψ^0_Ω(Ξ(1)+Ψ^0_{Ξ(1)}(Ξ(1)+ω))\)
(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(2,1,1)(3,1)(2)	\(ψ_Ω(Φ_3(0))\\=Ψ^0_Ω(Ξ(1)\cdot2)\)
(0)(1,1,1)(2,1,1)(3,1)(3,1)(1,1,1)(2,1,1)(3,1)(2)	\(ψ_Ω(Φ_{Φ_1(0)}(0))\\=Ψ^0_Ω(Ξ(1)\cdotΨ^0_{Ξ(1)}(0))\)
(0)(1,1,1)(2,1,1)(3,1)(3,1)(1,1,1)(2,1,1)(3,1)(3,1)(1,1,1)(2,1,1)(3,1)(2)	\(ψ_Ω(Φ_{Φ_{Φ_1(0)}(0)}(0))\\=Ψ^0_Ω(Ξ(1)\cdotΨ^0_{Ξ(1)}(Ξ(1)\cdotΨ^0_{Ξ(1)}(0)))\)
(0)(1,1,1)(2,1,1)(3,1)(3,1)(2)	\(ψ_Ω(ψ_I(0))\\=Ψ^0(Ξ(1)^2)\)
(0)(1,1,1)(2,1,1)(3,1)(3,1)(2,1)(1,1,1)(2,1,1)(3,1)(3,1)(2)	\(ψ_Ω(ψ_I(0)^2)\\=Ψ^0(Ξ(1)^2+Ψ^0_{Ξ(1)}(Ξ(1)^2)^2)\)
(0)(1,1,1)(2,1,1)(3,1)(3,1)(2,1,1)	\(ψ_Ω(ψ_I(ω))\\=Ψ^0(Ξ(1)^2+Ψ^0_{Ξ(1)}(Ξ(1)^2+ω))\)
(0)(1,1,1)(2,1,1)(3,1)(3,1)(2,1,1)(3,1)(2)	\(ψ_Ω(I)\\=Ψ^0(Ξ(1)^2+Ξ(1))\)
(0)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2)(5)	\(ψ_Ω(Φ_1(I+1))\\=Ψ^0_Ω(Ψ^0_{Ξ(2)}(3))\)
(0)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2)(5,2)(6,3,1)(7,3,1)(8,3)(7)	\(ψ_Ω(Φ_1(I+2))\\=Ψ^0_Ω(Ψ^0_{Ψ^1_{Ξ(2)}(3)}(4))\)
(0)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2)(5,2,1)	\(ψ_Ω(Φ_1(I+ω))\\=Ψ^0_Ω(Ψ^0_{Ψ^1_{Ξ(2)}(3)}(ω))\)
(0)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2)(5,2,1)(6,2)(5)	\(ψ_Ω(Φ_2(I+1))\\=Ψ^0_Ω(Ψ^1_{Ξ(2)}(3))\)
(0)(1,1,1)(2,1,1)(3,1,1)	\(ψ_Ω(χ_1(ω))\\=Ψ^0_Ω(Ψ^0_{Ξ(2)}(ω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)	\(ψ_Ω(χ_1(ω)+ψ_I(χ_1(ω)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)	\(ψ_Ω(χ_1(ω)+ψ_I(χ_1(ω))^2)\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2)(4,2)(5,2)(4,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)	\(ψ_Ω(χ_1(ω)+ψ_{χ_0(ψ_I(χ_1(ω))+1)}(χ_1(ω)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2)(4,2)(5,2)(4,1)(3,2)(4,2)(5,2)(4,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)	\(ψ_Ω(χ_1(ω)+ψ_{χ_0(ψ_I(χ_1(ω))+1)}(χ_1(ω)\\+ψ_I(χ_1(ω))))\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2)(4,2)(5,2)(4,2)	\(ψ_Ω(χ_1(ω)+χ_0(ψ_I(χ_1(ω))+1))\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)	\(ψ_Ω(χ_1(ω)+χ_0(ψ_I(χ_1(ω))+ω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2)(5,2)(4)	\(ψ_Ω(χ_1(ω)+ψ_I(χ_1(ω)+1))\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2)(5,2)(4,2,1)(5,2)(4)	\(ψ_Ω(χ_1(ω)+I)\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2,1)	\(ψ_Ω(χ_1(ω)+ψ_{χ_0(I+1)}(χ_1(ω)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2)(4,2)(5,2)(4,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)	\(ψ_Ω(χ_1(ω)+ψ_{χ_0(I+1)}(χ_1(ω))\\+ψ_{χ_0(ψ_I(χ_1(ω))+1)}(χ_1(ω)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2)(4,2)(5,2)(4,2)	\(ψ_Ω(χ_1(ω)+ψ_{χ_0(I+1)}(χ_1(ω))\\+ψ_{χ_0(ψ_I(χ_1(ω))+1)}(χ_1(ω)+χ_0(ψ_I(χ_1(ω))+1)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)	\(ψ_Ω(χ_1(ω)+ψ_{χ_0(I+1)}(χ_1(ω))\\+ψ_{χ_0(ψ_I(χ_1(ω))+1)}(χ_1(ω)+χ_0(ψ_I(χ_1(ω))+ω)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2,1)	\(ψ_Ω(χ_1(ω)+ψ_{χ_0(I+1)}(χ_1(ω))\\+ψ_{χ_0(ψ_I(χ_1(ω))+1)}(χ_1(ω)+ψ_{χ_0(I+1)}(χ_1(ω))))\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(2,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)	\(ψ_Ω(χ_1(ω)+ψ_{χ_0(I+1)}(χ_1(ω))\\+ψ_{χ_0(ψ_I(χ_1(ω))+1)}(χ_1(ω)\\+ψ_{χ_0(I+1)}(χ_1(ω)))\cdotψ_I(χ_1(ω)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(2,1)(3,2,1)(4,2,1)(5,2,1)	\(ψ_Ω(χ_1(ω)+ψ_{χ_0(I+1)}(χ_1(ω))\\+ψ_{χ_0(ψ_I(χ_1(ω))+1)}(χ_1(ω)+ψ_{χ_0(I+1)}(χ_1(ω)))^2)\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(3,2)(4,2)(5,2)(4,2)	\(ψ_Ω(χ_1(ω)+ψ_{χ_0(I+1)}(χ_1(ω))\\+χ_0(ψ_I(χ_1(ω))+1))\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(3,2,1)	\(ψ_Ω(χ_1(ω)+ψ_{χ_0(I+1)}(χ_1(ω))\\+χ_0(ψ_I(χ_1(ω))+ω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(3,2,1)(4,2,1)(5,2)(6,3,1)(7,3,1)(8,3,1)	\(ψ_Ω(χ_1(ω)+ψ_{χ_0(I+1)}(χ_1(ω))\\+ψ_I(χ_1(ω)+ψ_{χ_0(I+1)}(χ_1(ω))))\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(2,1,1)	\(ψ_Ω(χ_1(ω)+ψ_{χ_0(I+1)}(χ_1(ω))\cdotω)\)
(0)(1,1,1)(2,1,1)(3,1,1)(1,1,1)(2,1,1)(3,1,1)	\(ψ_Ω(χ_1(ω)\cdot2)\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1)(3,2,1)(4,2,1)(5,2)(4)	\(ψ_Ω(Φ_1(χ_1(ω)+1))\\=Ψ^0_Ω(Ψ^0_{Ψ^1_{Ξ(2)}(ω)}(ω+1))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)	\(ψ_Ω(χ_1(ψ_{χ_1(0)}(χ_1(ω))))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(2)	\(ψ_Ω(χ_1(χ_1(0)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(2)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(2)	\(ψ_Ω(χ_1(χ_1(0))+ψ_I(χ_1(χ_1(0))))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(2,1)(3,2)(4,2)(5,2)(4,1)(3,2)(4,2)(5,2)(4,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(2)	\(ψ_Ω(χ_1(χ_1(0))+ψ_{χ_0(ψ_I(χ_1(χ_1(0)))+1)}(χ_1(χ_1(0))\\+ψ_{χ_1(0)}(χ_1(χ_1(0)))))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(2,1)(3,2)(4,2)(5,2)(4,2)	\(ψ_Ω(χ_1(χ_1(0))+χ_0(ψ_I(χ_1(χ_1(0)))+1))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(2,1)(3,2,1)(4,2,1)(5,2)(5,2)(4)	\(ψ_Ω(χ_1(χ_1(0))+ψ_I(χ_1(χ_1(0))+1))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(2,1)(3,2,1)(4,2,1)(5,2,1)	\(ψ_Ω(χ_1(χ_1(0))+ψ_{χ_0(I+1)}(χ_1(ω)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(3,2,1)(4,2,1)(5,2)(6,3,1)(7,3,1)(8,3,1)	\(ψ_Ω(χ_1(χ_1(0))+ψ_{χ_0(I+1)}(χ_1(ω))\\+ψ_I(χ_1(χ_1(0))+ψ_{χ_0(I+1)}(χ_1(ω))))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(3,2,1)(4,2,1)(5,2,1)	\(ψ_Ω(χ_1(χ_1(0))+ψ_{χ_0(I+1)}(χ_1(ω)\cdot2))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(4,2,1)(5,2)(3,2,1)(4,2,1)(5,2)(6,3,1)(7,3,1)(8,3,1)(7,3,1)(8,2)(4)	\(ψ_Ω(χ_1(χ_1(0))+ψ_{χ_0(I+1)}(χ_1(χ_1(0))))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(2,1,1)	\(ψ_Ω(χ_1(χ_1(0))+ψ_{χ_0(I+1)}(χ_1(χ_1(0)))\cdotω)\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(4,2,1)	\(ψ_Ω(χ_1(χ_1(0))+ψ_{χ_0(I+1)}(χ_1(χ_1(0))\\+χ_0(χ_1(χ_1(0))+ω)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,1)(4,2,1)(5,2,1)(6,2,1)	\(ψ_Ω(χ_1(χ_1(0))+χ_1(ω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,2)	\(ψ_Ω(χ_1(χ_0(χ_1(0)+1)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2,1)(5,2,1)(6,2)(4,2,1)(5,2,1)(6,2)(7,3,1)(8,3,1)(9,3,1)(8,3,1)(9,2)(5)	\(ψ_Ω(χ_1(χ_1(1)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1,1)	\(ψ_Ω(χ_1(χ_1(ω)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(2)	\(ψ_Ω(ψ_{χ_2(0)}(0))\\=Ψ^0_Ω(Ξ(2))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(2)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(2)	\(ψ_Ω(ψ_{χ_2(0)}(0)\cdot2)\\=Ψ^0_Ω(Ξ(2)+Ψ^0_{Ξ(2)}(Ξ(2)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2)(4)	\(ψ_Ω(Φ_1(ψ_{χ_2(0)}(0)+1))\\=Ψ^0_Ω(Ξ(2)+Ψ^0_{Ω_{Ψ^0_{Ξ(2)}(Ξ(2))+1}}(Ψ^0_{Ψ^1_{Ξ(2)}(Ψ^0_{Ξ(2)}(Ξ(2)))}\\(Ψ^0_{Ξ(2)}(Ξ(2))+1)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)	\(ψ_Ω(χ_1(ψ_{χ_2(0)}(0)+ω))\\=Ψ^0_Ω(Ξ(2)\\+Ψ^0_{Ω_{Ψ^0_{Ξ(2)}(Ξ(2))+1}}(Ψ^0_{Ξ(2)}(Ψ^0_{Ξ(2)}(Ξ(2))+ω)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(4,2,1)(5,2)(4)	\(ψ_Ω(ψ_{χ_2(0)}(1))\\=Ψ^0_Ω(Ξ(2)+Ψ^0_{Ω_{Ψ^0_{Ξ(2)}(Ξ(2))+1}}(Ξ(2)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(4,2,1)(5,2)(4)(3,2,1)(4,2,1)(5,2,1)(4,2,1)(5,2)(4)	\(ψ_Ω(ψ_{χ_2(0)}(1)\cdot2)\\=Ψ^0_Ω(Ξ(2)+Ψ^0_{Ψ^1_{Ξ(2)}(Ξ(2))}(Ξ(2)+1))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(2,1,1)	\(ψ_Ω(ψ_{χ_2(0)}(ω))\\=Ψ^0_Ω(Ξ(2)+Ψ^0_{Ψ^1_{Ξ(2)}(Ξ(2))}(Ξ(2)+ω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1)(2,1,1)(3,1)(2)	\(ψ_Ω(χ_2(0))\\=Ψ^0_Ω(Ξ(2)+Ψ^1_{Ξ(2)}(Ξ(2)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,1)	\(ψ_Ω(χ_2(ω))\\=Ψ^0_Ω(Ξ(2)+Ψ^0_{Ξ(2)}(Ξ(2)+ω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,1)	\(ψ_Ω(χ_3(ω))\\=Ψ^0_Ω(Ξ(2)\cdot2+Ψ^0_{Ξ(2)}(Ξ(2)\cdot2+ω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)	\(ψ_Ω(ψ_{χ_ω(0)}(0))\\=Ψ^0_Ω(Ξ(2)\cdotω)\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)(1,1,1)(2,1,1)(3,1,1)(3)	\(ψ_Ω(ψ_{χ_ω(0)}(0)\cdot2)\\=Ψ^0_Ω(Ξ(2)\cdotω+Ψ^0_{Ξ(2)}(Ξ(2)\cdotω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)(2,1)(1,1,1)(2,1,1)(3,1,1)(3)	\(ψ_Ω(ψ_{χ_ω(0)}(0)^2)\\=Ψ^0_Ω(Ξ(2)\cdotω+Ψ^0_{Ξ(2)}(Ξ(2)\cdotω)^2)\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)(2,1)(3,2,1)(4,2,1)(5,2)(4)	\(ψ_Ω(Φ_1(ψ_{χ_ω(0)}(0)+1))\\=Ψ^0_Ω(Ξ(2)\cdotω+Ψ^0_{Ω_{Ψ^0_{Ξ(2)}(Ξ(2)\cdotω)+1}}\\(Ψ^0_{Ψ^1_{Ξ(2)}(Ψ^0_{Ξ(2)}(Ξ(2)\cdotω))}(Ψ^0_{Ξ(2)}(Ξ(2)\cdotω)+1)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)(2,1)(3,2,1)(4,2,1)(5,2,1)	\(ψ_Ω(χ_1(ψ_{χ_ω(0)}(0)+ω))\\=Ψ^0_Ω(Ξ(2)\cdotω+Ψ^0_{Ω_{Ψ^0_{Ξ(2)}(Ξ(2)\cdotω)+1}}\\(Ψ^0_{Ξ(2)}(Ψ^0_{Ξ(2)}(Ξ(2)\cdotω)+ω)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)(2,1)(3,2,1)(4,2,1)(5,2,1)(4,2,1)(5,2)(4)	\(ψ_Ω(ψ_{χ_2(ψ_{χ_ω(0)}(0)+1)}(0))\\=Ψ^0_Ω(Ξ(2)\cdotω+Ψ^0_{Ω_{Ψ^0_{Ξ(2)}(Ξ(2)\cdotω)+1}}(Ξ(2)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)(2,1)(3,2,1)(4,2,1)(5,2,1)(4,2,1)(5,2,1)	\(ψ_Ω(χ_2(ψ_{χ_ω(0)}(0)+ω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)(2,1)(3,2,1)(4,2,1)(5,2,1)(5)	\(ψ_Ω(ψ_{χ_ω(0)}(1))\\=Ψ^0_Ω(Ξ(2)\cdotω+Ψ^0_{Ω_{Ψ^0_{Ξ(2)}(Ξ(2)\cdotω)+1}}(Ξ(2)\cdotω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)(2,1)(3,2,1)(4,2,1)(5,2,1)(5)(3,2,1)	\(Ψ^0_Ω(Ξ(2)\cdotω\\+Ω_{Ψ^0_{Ξ(2)}(Ξ(2)\cdotω)+ω})\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)(2,1)(3,2,1)(4,2,1)(5,2,1)(5)(3,2,1)(4,2,1)(5,2,1)(5)	\(Ψ^0_Ω(Ξ(2)\cdotω\\+Ψ^0_{Ψ^1_{Ξ(2)}(Ξ(2)\cdotω)}(Ξ(2)\cdotω+1))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)(2,1,1)	\(ψ_Ω(ψ_{χ_ω(0)}(ω))\\=Ψ^0_Ω(Ξ(2)\cdotω\\+Ψ^0_{Ψ^1_{Ξ(2)}(Ξ(2)\cdotω)}(Ξ(2)\cdotω+ω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)(2,1,1)(3,1)(2)	\(ψ_Ω(χ_ω(0))\\=Ψ^0_Ω(Ξ(2)\cdotω+Ψ^1_{Ξ(2)}(Ξ(2)\cdotω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)(2,1,1)(3,1)(2)(1,1,1)(2,1,1)(3,1,1)(3)(2,1,1)(3,1)(2)	\(ψ_Ω(χ_ω(0)+ψ_{χ_ω(0)}(χ_ω(0)))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5)	\(ψ_Ω(χ_ω(0)+ψ_{χ_ω(0)}(χ_ω(0)+1))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5)(4,2,1)(5,2)(4)	\(ψ_Ω(χ_ω(0)\cdot2)\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)(2,1,1)(3,1)(2,1,1)	\(ψ_Ω(χ_ω(0)\cdotω)\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)(2,1,1)(3,1,1)	\(ψ_Ω(χ_ω(ω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)(2,1,1)(3,1,1)(2,1,1)(3,1,1)	\(ψ_Ω(χ_{ω+1}(ω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3)(2,1,1)(3,1,1)(3)	\(ψ_Ω(ψ_{χ_{ω\cdot2}(0)}(0))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)	\(ψ_Ω(ψ_{χ_{χ_0(0)}(0)}(0))\\=Ψ^0_Ω(Ξ(2)\cdotΩ)\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,2,1)(4,1)(3,2,1)	\(ψ_Ω(ψ_{χ_{χ_0(0)}(0)}(ω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,2,1)(4,1)(3,2,1)(4,2)(3)	\(ψ_Ω(χ_{χ_0(0)}(0))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,2,1)(4,1)(3,2,1)(4,2,1)	\(ψ_Ω(χ_{χ_0(0)}(ω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,2,1)(4,1)(3,2,1)(4,2,1)(3,2,1)(4,2,1)	\(ψ_Ω(χ_{χ_0(0)+1}(ω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,2,1)(4,1)(3,2,1)(4,2,1)(4,1)	\(ψ_Ω(ψ_{χ_{χ_0(0)\cdot2}(0)}(0))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(1,1)(2,2,1)(3,2,1)(4,2,1)(4,2)	\(ψ_Ω(ψ_{χ_{χ_0(1)}(0)}(0))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(1,1,1)	\(ψ_Ω(ψ_{χ_{χ_0(ω)}(0)}(0))\\=Ψ^0_Ω(Ξ(2)\cdotΩ_ω)\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(1,1,1)	\(ψ_Ω(ψ_{χ_{χ_0(ω)}(0)}(0)\cdot2)\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(4,2,1)	\(ψ_Ω(ψ_{χ_{χ_0(ω)}(0)}(ω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(4,2,1)(5,2)(4)	\(ψ_Ω(χ_{χ_0(ω)}(0))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(4,2,1)(5,2,1)	\(ψ_Ω(χ_{χ_0(ω)}(ω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(4,2,1)(5,2,1)(4,2,1)(5,2,1)	\(ψ_Ω(χ_{χ_0(ω)+1}(ω))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,1)(4,2,1)(5,2,1)(5,1)(1,1,1)	\(ψ_Ω(ψ_{χ_{χ_0(ω)\cdot2}(0)}(0))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(1,1,1)(2,1)(3,2,1)(4,2,1)(5,2,1)(5,2)(3,2,1)	\(ψ_Ω(ψ_{χ_{χ_0(ω\cdot2)}(0)}(0))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(1,1,1)(2,1,1)	\(ψ_Ω(ψ_{χ_{χ_0(ω^2)}(0))}(0)\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(1,1,1)(2,1,1)(3,1,1)	\(ψ_Ω(ψ_{χ_{χ_1(ω)}(0)}(0))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(1,1,1)(2,1,1)(3,1,1)(3,1)	\(ψ_Ω(ψ_{χ_{χ_{χ_0(0)}(0)}(0)}(0))\)
(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2)	\(ψ_Ω(Λ_0)\\=Ψ^0_Ω(Ξ(2)^2)\)

\begin{array}&=&\\(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)\\(2,1,1)(3,1)(2)&=&Ψ^0_Ω(Ξ(3))\\(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)\\(2,1,1)(3,1)(2)(1,1,1)(2,1,1)\\(3,1,1)(3,1,1)(2,1,1)(3,1)(2)&=&Ψ^0_Ω(Ξ(3)+Ψ^0_{Ξ(3)}(Ξ(3)))\\(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)\\(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)\\(5,2)(4)&=&Ψ^0_Ω(Ξ(3)+Ψ^0_{Ψ^1_{Ξ(3)}(Ψ^0_{Ξ(3)}(Ξ(3)))}(Ξ(3)+1))\\(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)\\(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)\\(5,2)(6,3,1)(7,3,1)(8,3)(7)&=&Ψ^0_Ω(Ξ(3)+Ψ^0_{Ψ^2_{Ξ(3)}(Ψ^0_{Ξ(3)}(Ξ(3)))}(Ξ(3)+1))\\(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)\\(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)\\(5,2)(6,3,1)(7,3,1)(8,3)(7,3)\\(8,4,1)(9,4,1)(10,4)(9)&=&Ψ^0_Ω(Ξ(3)+Ψ^0_{Ψ^1_{Ψ^2_{Ξ(3)}(Ψ^0_{Ξ(3)}(Ξ(3)))}(Ξ(3)+1)}(Ξ(3)+2))\\(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)\\(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)\\(5,2,1)&=&Ψ^0_Ω(Ξ(3)+Ψ^0_{Ψ^2_{Ξ(3)}(Ψ^0_{Ξ(3)}(Ξ(3)))}(Ξ(3)+ω))\\(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)\\(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)\\(5,2,1)(4,2,1)(5,2)(4)&=&Ψ^0_Ω(Ξ(3)+Ψ^2_{Ξ(3)}(Ψ^0_{Ξ(3)}(Ξ(3))))\\(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)\\(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)\\(5,2,1)(5,2,1)&=&Ψ^0_Ω(Ξ(3)+Ψ^0_{Ξ(3)}(Ψ^0_{Ξ(3)}(Ξ(3))+ω))\\(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)\\(2,1,1)(3,1)(2,1)(3,2,1)(4,2,1)\\(5,2,1)(5,2,1)(4,2,1)(5,2)(4)&=&Ψ^0_Ω(Ξ(3)+Ψ^0_{Ψ^1_{Ξ(3)}(Ξ(3))}(Ξ(3)+1))\\(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)\\(2,1,1)(3,1)(2,1,1)&=&Ψ^0_Ω(Ξ(3)+Ψ^0_{Ψ^1_{Ξ(3)}(Ξ(3))}(Ξ(3)+ω))\\(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)\\(2,1,1)(3,1)(2,1,1)(3,1)(2)&=&Ψ^0_Ω(Ξ(3)+Ψ^1_{Ξ(3)}(Ξ(3)))\\(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)\\(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2)\\(5)&=&Ψ^0_Ω(Ξ(3)+Ψ^0_{Ψ^2_{Ξ(3)}(Ψ^1_{Ξ(3)}(Ξ(3)))}(Ξ(3)+1))\\(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)\\(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2)\\(5,2)(6,3,1)(7,3,1)(8,3)(7)&=&Ψ^0_Ω(Ξ(3)+Ψ^0_{Ψ^1_{Ψ^2_{Ξ(3)}(Ψ^1_{Ξ(3)}(Ξ(3)))}(Ξ(3)+1)}(Ξ(3)+2))\\ \\(0)(1,1,1)(2,1,1)(3,1,1)(4,1)\\(2)&=&Ψ^0_Ω(K)\\(0)(1,1,1)(2,1,1)(3,1,1)(4,1)\\(5,2)&=&Ψ^0_Ω(φ_1(K+1))\\(0)(1,1,1)(2,2)&=&\text{pDAN}\\(0)(1,1,1)(2,2,1)(2,2)&=&\text{sDAN}\\(0)(1,1,1)(2,2,1)(3)&=&\text{DAN}\end{array}