Orthogonal higher-order rewrite systems are confluent
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  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  case(U,λx.F(inl(x)),λy.G(inr(y))) → F(U) is outside our framework, whichever way it is oriented.
KeywordsNormal Form Induction Hypothesis Logic Program Free Variable Critical Pair
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