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If t → s, then t → -compatible’. s. This property is temporarily called ‘strongly 2. If t → s, then t →-compatible’. s. 4. Inductive height 43 3. The quasiorder is well-founded. In fact, we will see that an algorithmic term rewriting system is WN if and only if there exists a strongly → -compatible and weakly →-compatible well-founded quasiorder on initially proper ground terms (Theorems 96 and 97). For convenience, we introduce a ‘dual-compatibility notation’ as follows. Definition 59 (dual-compatibility) Let R and R be binary relations on a set A.

Proof: Let tι ι α be a convergent reduction sequence, and let X = {ι ∈ α | tι →pι tι+1 , |pι | = n} For a proof by contradiction, assume X is infinite. Then, there exists a limit ordinal λ < α such that X ∩ λ is also infinite. Choose the minimal λ with these properties. From the definition of convergent reduction, we can find β < λ such that β ι < λ ⇒ |pι | > n. Then, from the definition we have X ∩ λ ⊆ β. Since X ∩ λ is infinite, β should be also infinite. Thus, β is of the form β + k, where β is a limit ordinal and k ∈ N.

Tn ∈ V such that ti = ai . From adequacy of the semantics, we have fA (a1 , . . , an ) = f(t1 , . . , tn ) = cnf(f(t1 , . . , tn )) Thus, fA is unique if it exists and induces an adequate semantics. 2. Follows from the definition of weak soundness. 1. Natural numbers For the sort NAT and the constructors 0 : () → NAT and s : NAT → NAT, let ANAT = N 0A () = 0 sA (n) = n + 1 This semantics is full and strongly sound. Lists Given a sort S with AS , the sort LISTS and the constructors [ ]S : () → LISTS and consS fin : S × LISTS → LISTS , let ALISTS = A∗S A [ ]S () = A consS fin (a, l) = a; l where we overload the symbol ‘;’ to denote the construction of a list.

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