A direct proof of the finite developments theorem

Extract

Let M be a term of the type free λ-calculus and let be a set of occurrences of redexes in M. A reduction sequence from M which first contracts a member of and afterwards only residuals of is called a development (of M with respect to ). The finite developments theorem says that developments are always finite.

There are several proofs of this theorem in the literature. A plausible strategy is to define some kind of measure for pairs (M, ), which—if M′ results from M by contracting a redex occurrence in and ′ is the set of residuals of in M′— decreases in passing from (M, ) to (M′, ′). This procedure is followed as a matter of fact in the proofs in Hyland [4] and in Barendregt [1] (both are covered in Klop [5]). If, as in the latter proof, the natural numbers are used as measures, then the measure of (M, ) will actually denote an upper bound of the number of reduction steps in a development of M with respect to .

In the present proof we straightforwardly define for each pair (M, ) a natural number, which can easily be seen to indicate the exact number of reduction steps in a development of maximal length of M with respect to .