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])])