Reductions of residuals are finite

Abstract

An important theorem of the λ β K \lambda \beta K -calculus which has not been fully appreciated up to now is D. E. Schroer’s finiteness theorem (1963), which states that all reductions of residuals are finite. The present paper gives a new proof of this theorem and extends it from λ β \lambda \beta -reduction to λ β η \lambda \beta \eta -reduction and reductions with certain extra operators added, for example the pairing, iteration and recursion operators. Combinatory weak reduction, with or without extra operators, is also included.