Exercise 1.15
In Examples 1.9.3 we have seen that neither Ω nor ΔΔ reduces to a β-nf.
Prove that both λ-terms do not have a β-nf, as well.
The term Ω has only one possibility for β-reduction which leads to Ω again. This means there is only one β-reduction path, which is infinite in length. To be clear, no part of this chain has a term which is in β-normal-form. Therefore Ω does not have a β-nf.
The argument for ΔΔ is the same. The only difference is the detail that each β-reduction step actually increases the number of redexes in the resulting term.
(note: I think the textbook has a typo as example 1.9.3 shows ΔΔ does not reduce to a β-nf, not Δ, and it doesn't make sense to show a term Δ does not have a β-nf when it is in β-nf already)
