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By Ariya Isihara.

ISBN-10: 9086594425

ISBN-13: 9789086594429

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2. Quasiorders 41 Lemma 55 (induction principle) On a set A with a well-founded quasiorder , we can perform generalized mathematical induction. That is, if a subset B ⊆ A satisfies the induction clause ∀a ∈ A. (∀b ∈ A. b ≺ a ⇒ b ∈ B) ⇒ a ∈ B then B = A. Proof: For a proof by contradiction, we assume that B satisfies the above induction clause and B = A. We will construct an infinite descending chain a0 a1 a2 . . such that ai ∈ A \ C. Let C = A \ B. Since B ⊆ A and B = A, we have C = ∅ and thus we can find a ∈ C.

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).

Examples 37 5. Countable ordinals, of sort ORD. The constructors O: () → ORD S: ORD → ORD L : STREAMORD → ORD have the output sort ORD. This is an inductive sort, but the construction of objects depends on a coinductive sort STREAMORD . I ORD := O | S(ORD) | L(STREAMORD ) 6. Coinductive natural numbers, of the coinductive sort CONAT, representing N ∪ {∞} with ∞ + n = n + ∞ = ∞ for all n ∈ N ∪ {∞}. These numbers can be implemented by the constructors 0˙ : () → CONAT and s˙ : CONAT → CONAT. C CONAT := 0˙ | s˙ (CONAT) Though this sort does not appear in the running examples, we present this to contrast with the inductive sort NAT.

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Algorithmic term rewriting systems by Ariya Isihara.

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