Two Decreasing Measures for Simply Typed Lambda-Terms (Extended Version)

This paper defines two decreasing measures for terms of the simply typed lambda-calculus, called the W-measure and the Tm-measure. A decreasing measure is a function that maps each typable lambda-term to an element of a well-founded ordering, in such a way that contracting any beta-redex decreases the value of the function, entailing strong normalization. Both measures are defined constructively, relying on an auxiliary calculus, a non-erasing variant of the lambda-calculus. In this system, dubbed the m-calculus, each beta-step creates a "wrapper" containing a copy of the argument that cannot be erased and cannot interact with the context in any other way. Both measures rely crucially on the observation, known to Turing and Prawitz, that contracting a redex cannot create redexes of higher degree, where the degree of a redex is defined as the height of the type of its lambda-abstraction. The W-measure maps each lambda-term to a natural number, and it is obtained by evaluating the term in the m-calculus and counting the number of remaining wrappers. The Tm-measure maps each lambda-term to a structure of nested multisets, where the nesting depth is proportional to the maximum redex degree.