In order to define Alfeu, we'll start with defining an extension for BAN:
{2:3} = {2,2,2}
{3:3} = {3,3,3}
{5:4} = {5,5,5,5}
{a:b} = {a,a,a,a...a,a,a,a} (w/ b a's) = {a,b[2]2}
Therefore, {a:b} = f_ω^ω(n)
We'll continue such:
{3::2} = {3:3}
{3::3} = {3:{3:3}}
{3::4} = {3:{3:{3:3}}}
{a::b} = {a:{a:{a:{a...{a:{a:a}}}...}}} (w/ b a's) = f_ω^ω+ω(n)
To simplify, we'll continue as:
{3:#:2} = {3::3}
{3:#:3} = {3:::3}
{3:#:4} = {3::::3}
{a:#:b} = {a::::...::::a} (w/ b :'s)
This is approximated to be smaller than f_ω^ω+ω^2(n)
Alfeu is equal to {100:#:100} < f_ω^ω+ω^2(100).