User blog:B1mb0w/General Proof of Strong D Function Growth Rate

This is the general proof of the Strong D Function growth rate. It will focus on 3 parameter functions of the form D(l,m,n) and the proof will be extended later for larger numbers of parameters.

Strong D Functions D(m,n) with 2 Parameters
For 2 parameters the Strong D Function is the same as the Deeply Nested Ackermann Function which uses the small d notation. Refer to the Comparison Rule C1 on that blog for the proof of the following:

\(d(m,n) >> f_{m-1}(n+2)\)

then

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

\(D(n+1,n+1) >> f_n(n+3) >> f_n(n) = f_{\omega}(n)\)  This is the growth rate of the 2 parameter function

Strong D Functions D(l,m,n) with 3 Parameters
\(D(1,0,0) = D(0,D(0,1,1),D(0,1,1)) = D(D(1,1),D(1,1)) = D(4,4) >> f_{\omega}(3)\)

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

\(>> f_{f_{\omega}(3)}(f_{\omega}(3)) = f_{\omega}(f_{\omega}(3)) >> f_{\omega}(a)\) for any \(a < f_{\omega}(3)\)

\(D(1,0,2) = D(0,D(1,0,1),D(1,0,1)) >> D(f_{\omega}(a),f_{\omega}(a)) >> f_{f_{\omega}(a)}(f_{\omega}(a))\)

\(= f_{\omega}(f_{\omega}(a)) >> f_{\omega}(b)\) for any \(b < f_{\omega}(a)\)

First calculation - getting to \(\omega\)

\(D(1,0,n) >> f_{\omega}(n+\delta)\) where \(\delta >> n\)

and

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

Next calculation - getting to \(\omega+1\)

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

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

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

Next calculation - general formula for \(D(1,m,0)\)

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

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

and

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

Using Rule: NL \(D(l,m,n) = D(l,0,n-1+(m+2).(m+1)/2)\) or

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

then

\(D(1,m,0) = D(1,0,0-1+(m+2).(m+1)/2) = D(1,0,(m^2+m.3+2-2)/2)\)

\(>> D(1,0,m+m+m)\) when m > 1 and

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

Next calculation - general formula for \(D(1,D(1,0,n),0)\)

By definition when \(m = D(1,0,n) > n\) then

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

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

Next calculation - general formula for \(D(2,0,n)\)

\(D(2,0,0) = D(1,D(1,2,2),D(1,2,2)) = D(1,D(1,0,7),D(1,0,7))\)

\(>> D(1,D(1,0,7),0) >> f_{\omega+1}^2(7.2) >> f_{\omega+1}^2(14)\)

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

\(>> D(1,D(1,D(1,2,2),D(1,2,2)),0) >> D(1,D(1,D(1,2,2),0),0)\)

\(>> D(1,D(1,D(1,0,7),0),0) >> f_{\omega+1}^2(D(1,D(1,0,7),0)) >> f_{\omega+1}^2(f_{\omega+1}^2(14))\)

\(>> f_{\omega+1}^2(a)\) where \(a < f_{\omega+1}^2(14)\)

then

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

and

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

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

\(D(2,0,n+1) = D(1,D(2,0,n),D(2,0,n)) >> D(1,D(2,0,n),D(1,D(2,0,n-1),D(2,0,n-1))\)

\(>> D(1,D(2,0,n),D(2,0,n)) >> D(1,D(2,0,n),D(1,D(2,0,n-1),D(2,0,n-1))\)

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

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

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

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

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

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

The rest of this blog is a Work In Progress
\(D(2,0,2) = D(1,D(2,0,1),D(2,0,1)) >> f_{\omega}^{m^2/2+m.2}(f_{\omega+1}(3))\) where \(m >> f_{\omega+1}^3(3)\)

\(>> f_{\omega}^{m.2}(a)\) where \(m >> f_{\omega+1}^2(a)\) and \(a >> f_{\omega+1}(3))\)

\(>> f_{\omega}^{f_{\omega+1}^2(a).2}(a) >> f_{\omega+1}^3(a) = f_{\omega+1}^4(3) = f_{\omega+1}(f_{\omega+2}(3))\)

and

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

D(l,0,n) Calculations
Let \(D(l-1,0,n) = f_{\phi}^{n-1}(f_{\phi+1}(3))\) where \(\phi\) is any ordinal up to \(\epsilon_0\)

then

\(D(l,0,0) = D(l-1,D(l-1,l,l),D(l-1,1,1)) >> D(l-1,D(l-1,l,l),0)\)

Using Rule: NL from above then

\(= D(l,0,(D(l-1,l,l)+2).(D(l-1,l,l)+1)/2-1) >> D(l,0,D(l-1,l,l)+1)\)

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

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

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

and

\(D(l,0,1) = D(l-1,D(l,0,0),D(l,0,0)) >> D(l-1,D(l,0,0),0)\)

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

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

and

\(D(l,0,2) >> D(l-1,D(l,0,1),D(l,0,1)) >> D(l-1,D(l,0,1),0)\)

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

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

\(= f_{\phi+1}(f_{\phi+2}(3))\)

and when

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

then

\(D(l,0,n) >> D(l-1,D(l,0,n-1),D(l,0,n-1)) >> D(l-1,D(l,0,n-1),0)\)

\(>> D(l,0,D(l,0,n-1)+D(l,0,n-2)+D(l,0,n-3)+ ... +D(l,0,1)+D(l-1,0,1)+1)\)

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

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

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

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

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

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

and

Rule: LN \(D(l,0,n) >> f_{\phi+1}((D(l,0,n-1)))\) where  \(D(l,0,n-1) = f_{\phi+1}^{n-2}(f_{\phi+2}(3))\)

and

Rule: LG \(D(l,0,n) >> f_{\mu}^{n-1}(f_{\mu+1}(3))\) where  \(\mu=\phi+1\) and

\(D(l-1,0,n) = f_{\phi}^{n-1}(f_{\phi+1}(3))\)

and

Rule: L1 \(D(l,0,1) >> f_{\phi+1}(3)\) where  \(D(l-1,0,1) = f_{\phi}(3)\)

D(l,0,n) Calculations for \(f_{\phi}^{n-1}(f_{\phi+1}(p))\)
The purpose of this section is to generalize the rule LN above further and to consider values of p other than 3.

Let \(D(l-1,0,n) >> f_{\phi}^{n-q}(f_{\phi+1}(p))\) where \(\phi\) is any ordinal up to \(\epsilon_0\) and q is a small number (say 100)

then

\(D(l,0,0) = D(l-1,D(l-1,l,l),D(l-1,1,1)) >> D(l-1,D(l-1,l,l),0)\)

\(= D(l,0,(D(l-1,l,l)+2).(D(l-1,l,l)+1)/2-1) >> D(l,0,D(l-1,l,l)+K)\) where K is a big number >> q (say 2,000,000)

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

\(>> f_{\phi}^{f_{\phi+1}(p)}(f_{\phi+1}(p)) = f_{\phi+1}(f_{\phi+1}(p))\)

\(= f_{\phi+1}^2(p)\)

and

\(D(l,0,1) = D(l-1,D(l,0,0),D(l,0,0)) >> D(l-1,D(l,0,0),0)\)

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

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

and

\(D(l,0,2) >> D(l-1,D(l,0,1),D(l,0,1)) >> D(l-1,D(l,0,1),0)\)

\(>> D(l,0,D(1,0,1)+D(l,0,0)+D(l-1,0,1)+K) >> f_{\phi+1}^4(p)\)

and

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

\(D(l,0,p-1) = f_{\phi+1}(f_{\phi+2}(p))\)

\(D(l,0,p) = f_{\phi+1}^2(f_{\phi+2}(p))\)

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

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

then  ... work in progress

\(D(l,0,n) >> D(l-1,D(l,0,n-1),D(l,0,n-1)) >> D(l-1,D(l,0,n-1),0)\)

\(>> D(l,0,D(l,0,n-1)+D(l,0,n-2)+D(l,0,n-3)+ ... +D(l,0,1)+D(l-1,0,1)+1)\)

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

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

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

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

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

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

and  ... work in progress

Rule: LP \(D(l,0,n) >> f_{\phi+1}((D(l,0,n-1)))\) where  \(D(l,0,n-1) = f_{\phi+1}^{n-2}(f_{\phi+2}(p))\)