My Psi Function

I will always use this psi function in my analysis

Rules below omega fixed point

• (a+b)[c] = a+b[c]
• a(b+1)[c] = ab+a[c]
• (ab)[c] = a(b[c])
• a^(b+1)[c] = a^b*a[c]
• (a^b)[c] = a^(b[c])
• Psi_W(0)[0] = 1
• Psi_W_(a+1)(0)[0] = W_a+1
• Psi_a(b+1)[0] = Psi_a(b)+1
• Psi_a(b+1)[c] = w^(Psi_a(b)[c-1])
• Psi_a(b)[c] = Psi_a(b[c])
• Psi_a(b[W_(c+1)])[0] = 0
• Psi_a(b[W_(c+1)])[d] = Psi_a(b[Psi_W_(c+1)(b[W_(c+1)])[d-1]])
• Psi_a(b[W_c])[d] = Psi_a(b[W_(c[d])])