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HigherOrder Rewriting
 12th Int. Conf. on Rewriting Techniques and Applications, LNCS 2051
, 1999
"... This paper will appear in the proceedings of the 10th international conference on rewriting techniques and applications (RTA'99). c flSpringer Verlag. ..."
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This paper will appear in the proceedings of the 10th international conference on rewriting techniques and applications (RTA'99). c flSpringer Verlag.
Trivial
"... i # 0, C i+1 = # # C i [1]. (See below for examples.) Remark that for any prefix D of s, # # D is a prefix of s again [3]. Hence to show l# # C # s, it su#ces by C 0 = l# to show monotonicity: C i # C i+1 , #i # 0, by induction on i. The base case l# # C 1 holds since the headsymbo ..."
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i # 0, C i+1 = # # C i [1]. (See below for examples.) Remark that for any prefix D of s, # # D is a prefix of s again [3]. Hence to show l# # C # s, it su#ces by C 0 = l# to show monotonicity: C i # C i+1 , #i # 0, by induction on i. The base case l# # C 1 holds since the headsymbol of r traces back to any position in l. In the induction step, suppose p # C i for some i > 0. By definition of C i , there exists some q # C i1<
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"... Reduce to the max A cofinal strategy for weakly orthogonal higherorder pattern rewrite systems (WOPRSs). Notions and results needed for WOPRSs can be found in [Oos95, Oos99]. Definition 1 A simultaneous set U of redex(occurrenc)es in a term s is maximal, if any redex v in s overlaps the head of so ..."
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Reduce to the max A cofinal strategy for weakly orthogonal higherorder pattern rewrite systems (WOPRSs). Notions and results needed for WOPRSs can be found in [Oos95, Oos99]. Definition 1 A simultaneous set U of redex(occurrenc)es in a term s is maximal, if any redex v in s overlaps the head of some redex in U. By the tree structure of terms maximal sets can be constructed insideout, but need not be unique. Lemma 2 If s −→V ◦ t then t − → ◦ s ∗ , where s ∗ is obtained from s by contracting a maximal set U. Proof By maximality of U and simultaneity of V, we can define an injection ι mapping every redex v ∈ V to a redex ι(v) ∈ U such that v overlaps the head of ι(v). By weak orthogonality s − → ◦ ι(V) t, so t − → ◦ U/ι(V) s ∗ [Oos95, Theorem 5, Prism]. ✷ Note that distinct maximal sets may exist, but these must be equipollent hence lead to the same result, justifying our notation s ∗. The lemma is a generalisation of [BBKV76, Lemma 3.2.2], [Tak95, Section 1, property (5)], [Nip96, Section 5.2], and [Raa96, Lemma 5.3.3]. Only slightly relaxing weakorthogonality invalidates the theorem, as witnessed by the term f(a) in the parallelclosed TRS a → a, f(a) → f(b). As a standard corollary, we have that the maximal strategy is (hyper)(head)normalising and cofinal for WOPRSs, where a strategy is maximal if it contracts maximal sets. This generalises (results for) the GrossKnuth strategy for λcalculus and the fullsubstitution strategy for orthogonal TRSs in the papers cited. Note that it also applies to λβηcalculus, i.e. fullextendedness is not required, so the result does not follow from [Oos99, Theorem 1]. The proof of the lemma avoids notions such as chain of λ’s [BBKV76], chain [Vri87] and cluster [BKV98] of redexes, which were introduced to set up a satisfactory residual theory for λβηcalculus, having the same ‘nice ’ properties as that of λβcalculus. Instead the proof is based on the more general notion of weakly orthogonal projection [Oos99], which applies to all WOPRSs.