Algorithmic term rewriting systems by Ariya Isihara.

Algorithmic term rewriting systems by Ariya Isihara.

By Ariya Isihara.

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Tn ), where f ∈ D and t1 , . . , tn ∈ V, forms a redex, viz. the rules cover all the patterns of constructor contexts. Lemma 31 A semantically sorted system is CR if the system is WN and UN. Lemma 32 A semantically sorted system is proper if and only if the system is DN and SCN. 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.

Technical preliminaries exponentiation. Note that the case distinction for exponentiation is a bit awkward because of 00 ; one might think of letting 00 = 1 for a simpler implementation, however, that system would not be able to compute 0ω = 0 correctly. For example, 2 · ω is computed as follows. 2 · ω ≡ 2 · omega → → → 2 · L( 0 : 1 : 2 : 3 : . . ) → L(mulL ( 2 , 0 : 1 : 2 : 3 : . . )) → → → L(( 2 · 0 ) : ( 2 · 1 ) : ( 2 · 2 ) : ( 2 · 3 ) : . . ) → → → L( 0 : 2 : 4 : 6 : . . ) Since sup{0, 2, 4, 6, .

Is not proper, but s˙ s˙ s˙ . . is. This s˙ s˙ s˙ . . represents ∞. e. e. the rewrite rules. Hamming numbers Following most of tutorials on functional programming, we first implement the enumeration of the Hamming numbers as STREAMNAT . A Hamming number is a natural number whose prime factors consists of 2, 3, and 5 only. We intend to enumerate these numbers in ascending order. Ham → s 0 : mrg(mrg(Ham2, Ham3), Ham5) Ham2 → sca(Ham, s s 0) Ham3 → sca(Ham, s s s 0) Ham5 → sca(Ham, s s s s s 0) will provide the enumeration of the Hamming numbers with Ham, where the function mrg enumerates the union of a pair of enumerated streams of natural numbers; the function sca computes the scalar multiple, or pointwise multiplication, of a stream by a natural number.

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