User blog:Alemagno12/Making an OCF, attempt 2

I got inspired by this blog post to make another OCF, or at least something similar to it. I will call my previous OCF the JKL OCF and this one the lambda OCF.

Adding 2 more rules makes this notation extremely strong:
 * @ is the rest of the ordinal inside of G, excluding ends of parentheses...
 * ...and $ is a copy of that @
 * A function F eventually overgrows another function G if there's some value y < α -> F(α) such that F(x) ≥ G(x) for all x ≥ y and x does not contain a diagonalizer
 * A function F is eventually overgrowed by another function G if G eventually overgrows F
 * X is a diagonalizer of a function F if, for any function G that is eventually overgrowed by function H(x) = λx and eventually overgrows F, G(@X) = sup(G($),G(@G($)),G(@G($G($))),...)
 * Ψ(0) = ε0
 * Ψ(x,0) = Ψ(x)
 * Ψ0(x) = Ψ(x)
 * Ψx(0) = x
 * Ψx(y+1) = sup(Ψx(y),Ψx(y)Ψx(y),Ψx(y)Ψx(y) Ψx(y) ,...)
 * For limit ordinal y, Ψx(y,z) = sup(Ψx(y[1],z),Ψx(y[2],z),Ψx(y[3],z),...)
 * Ψ(Ψ(a,b),c) can be shortened to Ψ(a,c)
 * λx is the diagonalizer of the function F(y) = Ψ(y,x)
 * Ψ(0,x) = λx
 * Ψ(Ψ(...Ψ(@λx,y)...,2),1) is the diagonalizer of function F(z) = Ψ(Ψ(...Ψ($Ψ(z,x),y)...,2),1)

[WIP]