By Ariya Isihara.

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Then, by the above lemma, we can find t ∈ T having no infinitary normal form. By non-emptiness of the algorithmic system, we can choose vS ∈ V of the sort S, for each S ∈ S. Let σ(x) = vout(x) for every x ∈ X. Then, tσ can have no infinitary normal form, by functionality of the algorithmic system. Hence, the system is not WN. 34 Chapter 1. Algorithmic term rewriting systems 35 Chapter 2 Technical preliminaries In this chapter, we present examples of sorts and algorithmic systems that will be used throughout the thesis.

Tn ≡ s Thus, we can use the rule which is applied in t → t on all the copied redexes in ti , the positions of which are all parallel, to obtain the diagram shown in Figure 12. We refer to [46, Chapter 4] for a more detailed explanation. Lemma 49 In an orthogonal term rewriting system, if t → → s and t → → s , then there exists u ∈ T such that s → → u and s → → u, viz. every orthogonal system has the finitary confluence property. Proof: By mathematical induction on the length of the reduction t → → s.

Inductive construction. We write t I 3. Coinductive construction. We write t Note that whenever t s, one of D, I, or s if t( ) ∈ D. s if t( ) ∈ C and srt(s) ∈ SI . C s if t( ) ∈ C and srt(s) ∈ SC . C holds between t and s. Then, a path in a term t can be specified as an infinite sequence of immediate subterming: t ≡ t0 t1 t2 ... If there occur infinitely many D - or I -steps in the above sequence, the path is vicious. Hence, for proper terms, there exists some ordinal assignment θ : T → On such that t s implies θ(t) θ(s), and that t D s or I t s implies θ(t) > θ(s).

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