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R m -compatible if and only if the quasiorder is Ri | ∅-compatible for every i = 1, . . , n and ∅ | R j -compatible for every j = 1, . . , m. 4 Inductive height In this section we define the notion of inductive height. This notion arises when we characterise properness of a term t by a well-founded quasiorder on dom(t). First, we define the immediate subterm relation as follows. Definition 61 Let t be a term. Then, for every i = 1, . . , ar(t( )), we write t t/i. Moreover, we divide the relation into three kinds: D 1.

Proof: (If) For a proof by contradiction, assume t ∈ G \ G . Let p be a vicious path in t. If p is a functional vicious path, then it contradicts SCN. So, p contains only finitely many defined function symbols. We can thus find a prefix q of p such that p/q contains no defined function symbols. Then, by Lemma 27, t/p ∈ G and t/p contains a constructor vicious path p/q, contradicting DN. (Only If) Trivial. Lemma 33 If a semantically sorted system is CE, then the following conditions are equivalent: 1.

Only If) Trivial. 3. Algorithmicity 25 ∈ f ∈ A1 × · · · × An / B (a1 , . . , an ) 1 / bO − − denotational level /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o /o [f] ( a1 , . . , an ) 1 8 8 8/ s qq O O qqq q q q O O qqq q apply f O O q qq O O qqq infinitary q q q normalization qqqq f gggg gg gg g a1 operational level an Figure 7: Semantics of a defined function symbol Lemma 35 A semantically sorted system is strongly productive if and only if the system is WN, UN, CE and proper.