The Normalization Theorem

A weakly normalizing term is one such that there is a reduction sequence which terminates in a normal form. An example (from the slides) which is weakly but not strongly normalizing is (λxy.y)Ω. This is because the lazy reduction realizes there is no x into which I should substitute the Ω in y so this reduces to λy.y. But trying to evaluate the Ω will never yield a result. The lazy way to beta-reduce is to reduce the left-most, outermost redex. This always works if the term is weakly normalizing. The eager way is to do the outermost redex only when the right hand side is a value (call-by-value). This does not always find the reduction of a weakly normalizing term.