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## REPLACED

This blog has been replaced. Simpler and more rigorous calculations of the Strong D Function can be found at another blog that can be accessed by this link.

## Strong D Function Overview

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().

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

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

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

Rule: N1 $$D(1,0,n) >> f_{\omega}^{n+1}(3) = f_{\omega}^{n-2}(f_{\omega+1}(3))$$ when n>2

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

## D(1,m,n) Calculations

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

$$D(1,1,1) = D(0,D(1,1,0),D(1,1,0)) = D(0,D(1,0,2),D(1,0,2)) = D(1,0,3)$$

$$D(1,1,n) = D(1,0,n+2)$$

$$D(1,2,0) = D(0,D(1,1,2),D(1,1,2)) = D(1,1,3) = D(1,0,5)$$

$$D(1,2,n) = D(1,1,n+3) = D(1,0,n+5)$$

$$D(1,3,n) = D(1,2,n+4) = D(1,1,n+7) = D(1,0,n+9)$$

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

then

$$D(1,2,2) = D(1,0,2-1+4.3/2) = f_{\omega}^{2-3+6}(f_{\omega+1}(3)) = f_{\omega}^5(f_{\omega+1}(3))$$

and

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

## D(2,0,n) Calculations

$$D(2,0,0) = D(1,D(1,2,2),D(1,2,2)) >>$$ very low bound $$>> D(1,0,D(1,2,2))$$

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

and

$$D(2,0,0) = D(1,D(1,2,2),D(1,2,2)) >> f_{\omega}^{m-3+(m+2).(m+1)/2}(f_{\omega+1}(3))$$ where $$m >> f_{\omega}^5(f_{\omega+1}(3))$$

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

$$>> f_{\omega}^{m^2/2+2m}(a) >> f_{\omega+1}(a)$$ need to check this

then

$$>> f_{\omega}^{f_{\omega}^5(a)^2/2+f_{\omega}^5(a).2}(a) >> f_{\omega+1}(a)$$ need to check this

$$D(2,0,1) = D(1,D(2,0,0),D(2,0,0)) >> f_{\omega}^{m-3+(m+2).(m+1)/2}(f_{\omega+1}(3))$$ where $$m >> f_{\omega+1}^2(3)$$

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

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

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

$$D(2,0,2) = D(1,D(2,0,1),D(2,0,1)) >> f_{\omega}^{m^2/2+m.2}(f_{\omega+1}(3))$$ 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 $$D(l-1,0,1) = f_{\phi}(3)$$

## General Proof of Strong D Function Growth Rate

A general proof of the Strong D Function Growth Rate is available on another blog post.