Orthogonal higher-order rewrite systems are confluent

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The results about higher-order critical pairs and the confluence of OHRSs provide a firm foundation for the further study of higher-order rewrite systems. It should now be interesting to lift more results and techniques both from term-rewriting and λ-calculus to the level of HRSs. For example termination proof techniques are much studied for TRSs and are urgently needed for HRSs; similarly the extension of our result to weakly orthogonal HRSs or even to Huet's “parallel closed” systems is highly desirable. Conversely, a large body of λ-calculus reduction theory has been lifted to CRSs [10] already and should be easy to carry over to HRSs.

Finally there is the need to extend the notion of an HRS to more general left-hand sides. For example the eta-rule for the case-construct on disjoint unions [15] case(U,λx.F(inl(x)),λy.G(inr(y))) F(U) is outside our framework, whichever way it is oriented.