By Ariya Isihara.

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**Example text**

I LISTS := [ ]S | consS fin (S, LISTS ) (S ∈ S) For convenience, we write x; y to denote consS fin (x, y), for every S. S Moreover, we write [ ] to mean any [ ] when the sort is clear from the context. For example, T; F; [ ] stands for consBOOL (T, consBOOL (F, [ ]BOOL )) fin fin and 0; s 0; s s 0; [ ] stands for NAT NAT NAT consNAT ))) fin (0, consfin (s 0, consfin (s s 0, [ ] 4. Streams, of sort STREAMS , representing infinite sequences of objects of sort S, for any sort S. Streams of S are implemented by the constructor consS : S × STREAMS → STREAMS For each sort S, the sort STREAMS is a coinductive sort.

The constructors 0: () → NAT s : NAT → NAT have the output sort NAT. This is an inductive sort. I NAT := 0 | s(NAT) 3. Lists, of sort LISTS , representing finite sequences of objects of sort S, for any sort S. Lists of S are implemented by the empty list [ ]S : () → LISTS and the concatenated list of an object and a list, consS fin : S × LISTS → LISTS Each LISTS is an inductive sort. I LISTS := [ ]S | consS fin (S, LISTS ) (S ∈ S) For convenience, we write x; y to denote consS fin (x, y), for every S.

2. e. (G ∩ N) ⊆ V. Proof: (1⇒2) For a proof by contradiction, assume s ∈ (G ∩ N) \ V. We will coinductively construct a path ps in s, which forms a functional vicious path, contradicting CN. Let A = {q ∈ dom(s) | s(q) ∈ D}. Since s ∈ V, we have A = ∅. Moreover, we have A = { }; otherwise we have s ∈ N by CE, which contradicts the supposition. Choose a q ∈ A \ { } and let s ≡ s/q. Note that s ∈ N \ V. Now, let ps = q · ps . Observe that ps certainly forms a functional vicious path. (2⇒1) Trivial.