Lambda calculus help

Question

So i'm totally stuck on this one part of a problem. It would be awesome if someone could help.........

Show that the term ZZ where Z is λz.λx. x(z z x) satisfies the requirement for fixed point combinators that ZZM =β M(ZZM).

Answer 1

This is completely trivial. You just apply the definition of β-reduction two times:

 Z Z M = (λz.λx. x(z z x)) Z M > (λx. x(Z Z x)) M > M (Z Z M) 

where > is the β-reduction.

Therefore Z Z M β-reduces to M (Z Z M) in two steps, hence Z Z M =β M (Z Z M).