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