An Application of Klop's Counterexample to a Higher-order Rewrite System

In 1978, Klop demonstrated that a rewrite system constructed by adding the untyped lambda calculus, which has the Church-Rosser property, to a Church-Rosser rst-order algebraic rewrite system may not be Church-Rosser. In contrast, Breazu-Tannen recently showed that augmenting any Church-Rosser rst-order algebraic rewrite system with the simply-typed lambda calculus results in a Church-Rosser rewrite system. In addition, Breazu-Tannen and Gallier have shown that the second-order polymorphic lambda calculus can be added to such rewrite systems without compromising the Church-Rosser property (for terms which can be provably typed). There are other systems for which a Church-Rosser result would be desirable, among them being SP F IX, the simply-typed lambda calculus extended with surjective pairing and xed points. This paper will show that Klop's untyped counterexample can be lifted to a typed system to demonstrate that SP F IX is not Church-Rosser.