Recursive Factory Function

This is the function I made so far, but at the moment it only has two of the intended four arguments that I want to eventually add.

*The order of priority for rules is put in order of the numbers below.*

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First I need to clarify Conway chained arrows(I will try to define them as an array notation):

-For any amount of up-arrows n, 0^ⁿ0 = 1, and for any positive-integers a and b, 0^ⁿa = 0, a^ⁿ0 = 1, and a^⁰b = ab.

-I define a recursive function C(x) for any possibly-empty string x, of non-negative integers.

1. If x is empty or x = 1 or x = 0, then C(x) = 1.
2. *Apply this rule to the leftmost example* If x = #,1,#a or x = #,0,#a for (possibly empty)strings # and #a of non-negative integers, then C(x) = C(#).
3. If x = a, for a non-negative integer a, then C(x) = a+1.
4. If x = a,b for non-negative integers a and b, then C(x) = a^b.
5. If x =a,b,c for non-negative integers a, b, and c, then C(x) = $a\uparrow^c b$
6. If x = #,a,b ,for non-negative integers a, b, and a string # of two or more non-negative numbers, then C(#,a,b) = C(#,C(#,a-1,b),b-1)

-I define a notation (x)&(y) for any two non-negative integers x and y.

1. If x=0 and x=0, then (x)&(y) = 1, and if x=a and y=0, then (x)&(y) = 1
2. (x)&(y) = C(st(x,y))

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-I define "a function $st:\mathbb {N} ^{2}\rightarrow S,~(x,y)\mapsto st(x,y)$, where$~S$denotes the set of strings of non-negative integers"

1. If y=0, then st(x,y) is empty
2. If y>0, then st(x,y) = st(x,y-1),x

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Now for the operator function:

-I define g^i(x) for any non-negative integer i and any possibly-empty string x of non-negative integers in the following recursive way:

1. If i=0 and x is empty or x = 1 or x = 0, then gⁱ(x) = 3.
2. *Apply this rule to the leftmost example* If i=0 and x = #,1,#a or #,0,#a for strings # and #a of non-negative integers, then gⁱ(#)
3. If i=0 and x = a, for a non-negative integer a, then gⁱ(x) = (a+1)&(a+1).
4. If i=0 and x = a,b for non-negative integers a and b, then gⁱ(x) = g^a+1(a-1,g^a+1(a-1,b+1)).
5. If i=0 and x =a,b,c for non-negative integers a, b, and c, then gⁱ(x) = g^a+1(a-1,g^a+1(a-1,b+1,c+1),g^a+1(a-1,b+1,c+1))
6. If i=0 and x = #,a,b,c ,for non-negative integers a, b, c, and a non-empty string # of non-negative numbers, then gⁱ(x) = g^a+1(#,a,g^a+1(#,a,b-1,c),c-1)
7. If i>0 and x is empty, then gⁱ(x) = g^i-1(g^i-1())
8. If i>0 and and x is empty or x = 1 or x = 0, then gⁱ(x) = g^i-1(g^i-1(x)).
9. If i>0 and x=a, for a non-negative integer a, then gⁱ(x) = g^i-1(g^i-1(a+1))
10. If i>0 and x=#,a for a non-negative integer a, and a non-empty string # of non-negative integers, then gⁱ(x) = g^i-1(#,g^i-1(#,a+1))

-I define a notation (x)&&(y), for any two non-negative integers x and y:

1. If x=0 and x=0, then (x)&&(y) = 1, and if x=a and y=0, then (x)&&(y) = 1
2. (x)&&(y) = g(st(x,y))

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Now for the actual function:

-I define a function f(c,z) for any two non-negative integers z and c:

1. f(0,z) = (z)&&(z+1)
2. $f(c,z)=f(c-1,h(c-1,z+1,z+1))$

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-I define a function h for any three non-negative integers a, b, and x.

1. $h(0,0,x)=$(x+1)&( f(x+1) )
2. $h(a,0,x)=f(a,h(a-1,x+1,f(1,x+1)))$
3. $h(a,b,x)=h(a,b-1,h(a,b-1,x+1)))$

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