[Agda] coinductively defined families

[Thorsten Altenkirch]

Using coinductive types as records I can write

    record Stream (A : Set) : Set where
        coinductive
        field
          hd : A
          tl : Stream A

and then use copatterns to define cons (after open Stream)

    _∷_ : {A : Set} → A → Stream A → Stream A
    hd (x ∷ xs) = x
    tl (x ∷ xs) = xs

Actually I wouldn't mind writing

    record Stream (A : Set) : Set where
      coinductive
      field
        hd : Stream A → A
        tl : Stream A → Stream A

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

    record Vec (A : Set) : ℕ → Set where
      coinductive
      field
        hd : ∀{n} → Vec A (suc n) → A
        tl : ∀{n} → Vec A (suc n) → Vec A n

and we can derive [] and cons by copatterns:

    [] : Vec A zero
    [] ()

    _∷_ : {A : Set} → A → Vec A n → Vec A (suc n)
    hd (x ∷ xs) = x
    tl (x ∷ xs) = xs

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. :-)

Maybe this has been discussed already? I haven't been able to go to AIMs for a while.
Thorsten





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