User:180.190.91.169/Layered Hierarchy

Example
Layer $$n = x(a,b(a,b(a,b...[n]a,b...(a,b-1) (a,b)x$$

Layer 1
$$X+(x+(x+(x+(x+...))))) = x[y]x$$

Layer 2
$$x[...(x[(x[x]x)]x)...]x = x[(x,y)]x$$

Layer 3
$$x[(a,(a,(a,(a,(a,(a,...))))))]x = x[(x,y[x]y)]x$$

Layer 4
Starts to get weirder $$x[(a,[(a,[(a,[(a,...)])])])])]x = x[a[a(a,x)]b]x$$

Layer 5
$$x[([a[([a...[(a,b,...)])]...)]x = x[a(a,b)[a(a,b)]b]x$$

Layer 6
$$x[((a,b)^a)[a(a,b)^a[a(a,b[...]] = x[x(a,b,...)[x(a,b,...)]]x$$

Layer 7
$$x[(a^(a,b)a,b[a,b^a[a,b^[a(a,b)...]]]] = x^{[x(a,b,...)[x(a(a,b)x...[x(a,b-1)^{x^(a,b)}]} $$

Layer 8
$$x[a,b^{a,b}[a^{a,b}[a,b^{a,b}[...]]]] = x^{[a[a[a...[a,b]...]}x[(a,b)]x$$

Layer 9
$$x[a,b^{[a,b...^{[a,b^{...}]}]}x^{x^{x[(a,b)}}]x = x^^{[a,b,c...[a,b,...[a...]]]}x $$