Residuals have been studied for various forms of rewriting\ud and residual systems have been defined to capture residuals in an\ud abstract setting. In this article we study residuals in orthogonal Pattern\ud Rewriting Systems (PRSs). First, the rewrite relation is defined\ud by means of a higher-order rewriting logic, and proof terms are defined\ud that witness reductions. Then, we have the formal machinery to define\ud a residual operator for PRSs, and we will prove that an orthogonal\ud PRS together with the residual operator mentioned above, is a residual\ud system. As a side-effect, all results of (abstract) residual theory are\ud inherited by orthogonal PRSs, such as confluence, and the notion of\ud permutation equivalence of reductions
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