User blog comment:Chronolegends/improved Egg Notation - Analysis/@comment-28606698-20170427135113

I propose only two rules to reach epsilon. it is Dollar-Egg Notation, since it is mix of Egg-notation and definition of Dollar-function

1)	$$\bullet_1[]\bullet_2^a=\bullet_1^a \bullet_2^a$$ where $$\bullet_2$$ is empty of right brackets ]]..]] and $$X^a$$ denotes XX…X i.e. a X’s

2)	$$\bullet_1[\bullet_2^{b+1}]\bullet_3^{a}=\bullet_1[\bullet_2^b]\bullet_3^a$$ where $$\bullet_3$$ is empty of right brackets ]]..]]

Examples

$$[]^a=^a^a=^{2a}=f_1(a)$$

$$[][]^a=[]^{2a}=^{4a} =f_1(f_1(a))$$

$$[]=[]^a ^a= f_2(a)$$

$$[]^a=[]^a ^a= f_3(a)$$

$$^a =[^a] ^a \approx f_a(a)= f_\omega(a)$$

$$]]^a=[[^a ^a \approx f_{\omega+1}(a)$$

$$]^a]^a=[[][^a \approx f_{\omega+\omega}(a)$$

$$]^a](^a)=[[^a \approx f_{\omega^2}(a)$$

$$^a^a=[]^a \approx f_{\omega^\omega}(a)$$