User blog:Cefe origol/My yet unnamed arithmetic

My yet unnamed arithmetic has 6 main functions for f(a,b): zero: returns 0 a: returns a b: returns b Sa: returns a+1 Sb: returns b+1 sum: returns a+b

New functions are defined as $$f_g(a,b)\ \begin{cases}{f_h(f_i(f_j(a,b),f_k(a,b)),f_l(f_g(a-1,b),f_g(a,b-1)))}\ & a,b > 0\\b & a= 0\\a & b=0\end{cases}$$ where g is greater than than h,i,j,k and l. Which gets abreviated to $$\text\$$.

Define the function $$\text{MYUA(a,b)}\$$ as the maximum number that can be expressed in b formulas with the set of original functions in $$\text{x}_a\$$. Where $$\text{x}_a=\begin{cases}Original functions & x = 0\\\text{x}_a+{MYUA(a,b)} & x > 0\end{cases}\$$.