User blog:B1mb0w/Strong D Function Calculations

The Strong D Function will be calculated in this blog hopefully up to a growth rate of \(f_{\epsilon_0}\). This blog will also refer to the omega to epsilon nought ordinal hierarchy index of f.

Strong D Function Overview
From the Strong D Function blog, we can start with these formulas:

\(D(4,1) = D(3,D(4,0)) = D(3,D(3,D(3,4))) = d(4,1)\)

Using the Comparison Rule C1 \(d(m,n) >> f_{m-1}(n+2)\) we get

\(D(4,1) >> f_{3}(3) = f_{\omega}(3)\)

D(4,n) Calculations
\(D(4,2) = f_{3}(4)\)

Using formula 15 \(f_n(a+1) >> f_{n-1}(f_{n}(a))\) we get

\(D(4,2) = f_{3}(4) >> f_{2}(f_{3}(3)) = f_{2}(f_{\omega}(3))\)

similarly

\(D(4,2) = D(3,D(4,1)) >> D(3,f_{\omega}(3)) >> f_{3-1}(f_{\omega}(3)+2) = f_2(f_{\omega}(3)+2) >> f_2(f_{\omega}(3))\)

\(D(4,3) = D(3,D(4,2)) >> D(3,f_2(f_{\omega}(3))) >> f_2^2(f_{\omega}(3))\)

\(D(4,n) >> f_2^{n-1}(f_{\omega}(3))\)

\(D(4,D(4,1)) >> D(4,f_{\omega}(3)+1)) >> f_2^{f_{\omega}(3))+1-1}(f_{\omega}(3)) = f_3(f_{\omega}(3))\)

D(5,n) Calculations
\(D(5,0) = D(4,D(4,5)) >> f_3(D(4,5)) >> f_3(f_2^{5-1}(f_{\omega}(3))) >>\) very low bound \(>> f_3(f_{\omega}(3))\)

\(D(5,1) = D(4,D(5,0)) >> f_3(f_3(f_{\omega}(3))) = f_3^2(f_{\omega}(3))\)

\(D(5,2) = D(4,D(5,1)) >> f_3(f_3^2(f_{\omega}(3))) = f_3^3(f_{\omega}(3))\)

\(D(5,n) = D(4,D(5,n-1)) >> f_3(f_3^n(f_{\omega}(3))) = f_3^{n+1}(f_{\omega}(3))\)

\(D(5,D(4,1)) = D(4,f_{\omega}(3)) >> f_3^{f_{\omega}(3)}(f_{\omega}(3)) = f_4(f_{\omega}(3))\)

D(m,n) Calculations
\(D(m,0) >> f_{m-2}(f_{\omega}(3))\)

\(D(m,n) >> f_{m-2}^{n+1}(f_{\omega}(3))\)

\(D(D(4,1)+1,D(4,1)-1) >> f_{f_{\omega}(3)+1-2}^{f_{\omega}(3)-1+1}(f_{\omega}(3)) = f_{f_{\omega}(3)-1}^{f_{\omega}(3)}(f_{\omega}(3)) = f_{f_{\omega}(3)}(f_{\omega}(3)) = f_{\omega}(f_{\omega}(3)) = f_{\omega}^2(3)\)

D(1,0,n) Calculations
\(D(1,0,0) = D(0,D(0,1,1),D(0,1,1)) = D(D(1,1),D(1,1)) = D(4,4) >> f_2^3(f_{\omega}(3))\)

\(D(1,0,1) = D(0,D(1,0,0),D(1,0,0)) = D(D(4,4),D(4,4)) >> D(f_2^3(f_{\omega}(3)),f_2^3(f_{\omega}(3)))\)

\(>> f_{f_2^3(f_{\omega}(3))}(f_2^3(f_{\omega}(3))) = f_{\omega}(f_2^3(f_{\omega}(3))) >>\) very low bound \(>> f_{\omega}^2(3)\)

\(D(1,0,1) = D(D(4,4),D(4,4)) >>\) very low bound \(>> D(D(4,1)+1,D(4,1)-1) >> f_{\omega}^2(3)\)

\(D(1,0,2) = D(0,D(1,0,1),D(1,0,1)) >> D(f_{\omega}^2(3),f_{\omega}^2(3)) >> f_{f_{\omega}^2(3)}(f_{\omega}^2(3)) = f_{\omega}(f_{\omega}^2(3)) >> f_{\omega}^3(3) = f_{\omega+1}(3)\)

\(D(1,0,n) >> f_{\omega}^{n+1}(3) = f_{\omega}^{n-2}(f_{\omega+1}(3))\) when n>2

D(1,m,n) Calculations
From the Strong D Function blog, we can start with these formulas:

\(D(1,m,n) >> f_{\omega}^{n-3+(m+2).(m+1)/2)}(f_{\omega+1}(3))\)

e.g. \(D(1,2,2) >> f_{\omega}^{2-3+(2+2).(2+1)/2)}(f_{\omega+1}(3)) = f_{\omega}^5(f_{\omega+1}(3))\)

D(2,0,n) Calculations
\(D(2,0,0) = D(1,D(1,2,2),D(1,2,2))\)

\(>> f_{\omega}^{m-3+(m+2).(m+1)/2)}(f_{\omega+1}(3))\) where \(m >> f_{\omega}^5(f_{\omega+1}(3))\)

\(= f_{\omega}^{m-3+m^2/2+3m/2+2/2}(f_{\omega+1}(3))\)

\(= f_{\omega}^{m^2/2+5m/2-2}(f_{\omega+1}(3))\)

work in progress

D(2,m,n) Calculations
\(D(2,m,n) = D(2,0,n-1+(m+2).(m+1)/2)\)

The reference below will be edited or deleted
Refer to User blog:B1mb0w/Mapping D(2,0,n) to epsilon nought

\(D(4,D(4,2)) >> D(4,f_2(f_{\omega}(3))+1) >> f_2^{f_2(f_{\omega}(3))}(f_{\omega}(3))\)

Using formula 14 \(f_{b}^{f_{b+1}^n(a).2}(a) >> f_{b+1}^{n+1}(a)\) we get \(f_2^{a.2}(a) >> f_{b+1}^{n+1}(a)\)

\(D(4,D(4,2)) >> f_2^{f_{\omega}(3).2}(f_{\omega}(3)) = f_3^2(f_{\omega}(3))\)

\(>> f_2^{f_{\omega}(3).2}(f_{\omega}(3)) = f_2^{f_{\omega}(3)}(f_2^{f_{\omega}(3)}(f_{\omega}(3))) = f_2^{f_{\omega}(3)}(f_3(f_{\omega}(3)))\)

Using formula 24 \(f_{\omega}^{f_{\omega+1}^n(3).2}(3) >> f_{\omega+1}^{n+1}(3)\)