Comparing Curried and Uncurried Rewriting

Currying is a transformation of term rewrite systems which may contain symbols of arbitrary arity into systems which contain only nullary symbols, together with a single binary symbol called application. We show that for all term rewrite systems (whether orthogonal or not) the following properties are preserved by this transformation: strong normalization, weak normalization, weak Church-Rosser, completeness, semi-completeness, and the non-convertibility of distinct normal forms. Under the condition of leftlinearity we show preservation of the properties NF (if a term is reducible to a normal form, then its reducts are all reducible to the same normal form) and UN (a term is reducible to at most one normal form). We exhibit counterexamples to the preservation of NF and UN for non-left linear systems. The results extend to partial currying (where some subset of the symbols are curried), and imply some modularity properties for unions of applicative systems.