User blog:B1mb0w/Summary of Strong D Function Growth Rates

This blog presents a summary of the growth rates for the Strong D Function growth rate. Refer to the blog with the general proof for all the detail.

Growth Rates of the Strong D Functions
The growth rate of the 2 parameter Strong D Function is:

\(D(n+1,n+1) >> f_n(n+3) >> f_n(n) = f_{\omega}(n)\)

The growth rate of the 2 parameter Strong D Function is:

\(D(1,0,n) >> f_{\omega}^4(n)\) Refer to First Proof

\(D(1,m,0) >> f_{\omega}^m(f_{\omega+1}(m))\) Refer to Second Proof

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

\(D(2,0,n) >> f_{\omega+1}^2(n)\) Refer to Third Proof

\(D(2,0,5) >> D(2,0,0) >> f_{\omega+1}^2(5)\)

assume

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

then

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

\(>> f_{\omega+1}(f_{\omega+1}^2(n-1)) = f_{\omega+1}^2(f_{\omega+1}(n-1))\)

\(>> f_{\omega+1}^2(n)\)

Next calculation - getting to \(\omega+2\)

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

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

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

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

Fourth proof: \(D(2,m,0) >> f_{\omega+1}^m(f_{\omega+2}(m))\)

\(D(2,3,0) = D(1,D(2,2,3),D(2,2,3)) >> D(1,D(2,0,8),D(2,0,8))\)

\(>> f_{\omega}^{D(2,0,8).2}(f_{\omega+1}(D(2,0,8))) >> f_{\omega+1}(D(2,0,8))\)

\(= f_{\omega+1}(D(2,0,5+5-2)) >> f_{\omega+1}(f_{\omega+2}(5))\)

\(= f_{\omega+1}(f_{\omega+1}^5(5)) = f_{\omega+1}^3(f_{\omega+1}^3(5)) >> f_{\omega+1}(f_{\omega+1}^3(3))\)

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

assume

\(D(2,m-1,0) >> f_{\omega+1}^{m-1}(f_{\omega+2}(m-1))\)

then

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

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

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

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

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

\(= f_{\omega+1}^{m.4-3}(f_{\omega+1}(m-1)) >> f_{\omega+1}^{m.4-3}(m) = f_{\omega+1}^{m.3-3}(f_{\omega+1}^m(m))\)

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

Next calculation - general formula for \(D(2,m,n)\)

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

\(>> f_{\omega+1}(f_{\omega+1}^m(f_{\omega+2}(m))) = f_{\omega+1}^{m+1}(f_{\omega+2}(m))\)

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

Next calculation - general formula for \(D(2,n,n)\)

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

Fifth proof: \(D(l,0,n) >> f_{\phi}^2(n)\)

assume

\(D(l-1,0,n) >> f_{\phi-1}^2(n))\)

\(D(l-1,m,n) >> f_{\phi-1}^{m+n}(f_{\phi}(m))\)

then

\(D(l,0,0) = D(l-1,D(l,l-1,l-1),D(l,l-1,l-1)) >> f_{\phi-1}^{D(l,l-1,l-1))+D(l,l-1,l-1))}(f_{\phi}(D(l,l-1,l-1))))\)

\(>> f_{\phi}(D(l,l-1,l-1)) >> f_{\phi}(D(l-1,D(l,l-1,l-2),D(l,l-1,l-2)))\)

\(>> f_{\phi}(f_{\phi-1}^{D(l,l-1,l-2)+D(l,l-1,l-2)}(f_{\phi}(D(l,l-1,l-2)))))\)

\(>> f_{\phi}(f_{\phi-1}(f_{\phi}(D(l,l-1,l-2)))) >> f_{\phi}(f_{\phi}(D(l,l-1,l-2))) >> f_{\phi}^2(D(l,l-1,l-2))\)

\(>> f_{\phi}^2(3)\)

\(D(l,0,3) >> D(l,0,0) >> f_{\phi}^2(3))\)

assume

\(D(l,0,n-1) >> f_{\phi}^2(n-1)\)

then

\(D(l,0,n) = D(l-1,D(l,0,n-1),D(l,0,n-1)) >> f_{\phi-1}^{D(l,0,n-1))+D(l,0,n-1))}(f_{\phi}(D(l,0,n-1))))\)

\(>> f_{\phi}(D(l,0,n-1)) >> f_{\phi}(f_{\phi}^2(n-1))\)

\(= f_{\phi}^2(f_{\phi}(n-1)) >> f_{\phi}^2(n)\)

Sixth proof: \(D(l,m,n) >> f_{\phi}^{m+n}(f_{\phi+1}(m))\)

assume

\(D(l-1,m,n) >> f_{\phi-1}^{m+n}(f_{\phi}(m))\)

\(D(l,0,n) >> f_{\phi}^2(n)\)

then

\(D(l,3,0) = D(l-1,D(l,2,3),D(l,2,3)) >> f_{\phi-1}^{D(l,2,3).2}(f_{\phi}(D(l,2,3)))\)

\(>> f_{\phi}(D((l,2,3)) >> f_{\phi}(D(l-1,D(l,2,2),D(l,2,2)))\)

\(>> f_{\phi}(f_{\phi-1}^{D(l,2,2)+D(l,2,2)}(f_{\phi}(D(l,2,2))))\)

\(>> f_{\phi}(f_{\phi}(D(l,2,2))) >> f_{\phi}^2(D(l-1,D(l,2,1),D(l,2,1))\)

\(>> f_{\phi}^2(D(l,2,1)) >> f_{\phi}^2(f_{\phi}(D(l-1,D(l,2,0),D(l,2,0)))\)

\(>> f_{\phi}^3(D(l,2,0)) >> f_{\phi}^3(f_{\phi}(D(l-1,D(l,1,2),D(l,1,2)))\)

\(>> f_{\phi}^4(D(l,1,2)) >> f_{\phi}^4(D(l,0,3)) >> f_{\phi}^4(f_{\phi}^2(3)) = f_{\phi}^3(f_{\phi}^3(3))\)

\(= f_{\phi}^3(f_{\phi+1}(3))\)

assume

\(D(l,m-1,0) >> f_{\phi}^m(f_{\phi+1}(m-1))\)

then

\(D(l,m,n) = D(l-1,D(l,m,n-1),D(l,m,n-1)) >> f_{\phi-1}^{D(l,m,n-1)+D(l,m,n-1)}(f_{\phi}(D(l,m,n-1)))\)

\(>> f_{\phi}(D((l,m,n-1)) >> f_{\phi}(D(l-1,D(l,m,n-2),D(l,m,n-2)))\)

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

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

\(>> f_{\phi}^2(D(l,m,n-3)) >> f_{\phi}^2(f_{\phi}(D(l-1,D(l,m,n-4),D(l,m,n-4)))\)

...

\(>> f_{\phi}^{n-1}(D(l,m,n-n)) >> f_{\phi}^{n-1}(D(l,m,0) >> f_{\phi}^{n-1}(f_{\phi}(D(l-1,D(l,m-1,m),D(l,m-1,m)))\)

\(>> f_{\phi}^n(D((l,m-1,m)) >> f_{\phi}^n(D(l-1,D(l,m-1,m-1),D(l,m-1,m-1)))\)

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

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

...

\(>> f_{\phi}^{n+m}(D((l,m-1,m-m)) >> f_{\phi}^{n+m}(D(l-1,m-1,0)) >> f_{\phi}^{n+m}(f_{\phi}^m(f_{\phi+1}(m-1)))\)

\(= f_{\phi}^{n+m.2}(f_{\phi+1}(m-1)) = f_{\phi}^{n+m.2}(f_{\phi}^{m-1}(m-1))\)

\(= f_{\phi}^{n+m.3-2}(f_{\phi}(m-1)) >> f_{\phi}^{n+m.3-2}(m) = f_{\phi}^{n+m.2-2}(f_{\phi}^{m}(m))\)

\(= f_{\phi}^{n+m.2-2}(f_{\phi+1}(m))\)