Fast Iteration Hierarchy

The Fast iteration hierarchy is a system similar to the Fast-growing hierarchy.

Definition
\(f(a){[1]b} = f(f(...)) \mathrm{with} b f\)

\(f(a){[b+1]c} = f(a){[b]a[b]a...} \mathrm{with} c [b]\)

\(f(a){[b]c} = f(a){[a\mathrm{th member of the fundamental sequence of b}]c} \mathrm{iff b is a limit ordinal.}\)

\(f(a){[b]c[d]e...} = g(a){[d]e...} \mathrm{where} g(a) = f(a){[b]c}\)