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<div>Using coinductive types as records I can write</div>
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<div> record Stream (A : Set) : Set where</div>
<div> coinductive</div>
<div> field</div>
<div> hd : A</div>
<div> tl : Stream A</div>
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<div>and then use copatterns to define cons (after open Stream)</div>
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<div> _∷_ : {A : Set} → A → Stream A → Stream A</div>
<div> hd (x ∷ xs) = x</div>
<div> tl (x ∷ xs) = xs</div>
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<div>Actually I wouldn't mind writing</div>
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<div>
<div> record Stream (A : Set) : Set where</div>
<div> coinductive</div>
<div> field</div>
<div> hd : Stream A → A</div>
<div> tl : Stream A → Stream A</div>
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<div>as in inductive definitions we also write the codomain even though we know what it has to be. However, this is more interesting for families because we should be able to write</div>
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<div> record Vec (A : Set) : ℕ → Set where</div>
<div> coinductive</div>
<div> field</div>
<div> hd : ∀{n} → Vec A (suc n) → A</div>
<div> tl : ∀{n} → Vec A (suc n) → Vec A n</div>
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<div>and we can derive [] and cons by copatterns:</div>
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<div> [] : Vec A zero</div>
<div> [] ()</div>
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<div> _∷_ : {A : Set} → A → Vec A n → Vec A (suc n)</div>
<div> hd (x ∷ xs) = x</div>
<div> tl (x ∷ xs) = xs</div>
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<div>here [] is defined as a trivial copattern (no destructor applies). Actually in this case the inductive and the coinductive vectors are isomorphic. A more interesting use case would be to define coinductive vectors indexed by conatural numbers. And I have
others. :-)</div>
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<div>Maybe this has been discussed already? I haven't been able to go to AIMs for a while.</div>
<div>Thorsten</div>
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