[Agda] Question about colists, musical notation vs coinductive records, and all that

[Eric Finster]

Hi Bill,

I would have written this:

  record Colist (A : Set) : Set where
    constructor hd
    coinductive
    field

      Hd : Maybe (A × Colist A)

  open Colist

  iterate1 : ∀ {a} → (a → Maybe a) → a → Colist a
  Hd (iterate1 {A} f a) = Just (a , loop (f a))

    where loop : Maybe A → Colist A
          loop Nothing = hd Nothing
          loop (Just a') = iterate1 f a'

Is that what you had in mind?

Best,

Eric

On Wed, Apr 29, 2020 at 6:58 PM William Harrison
<william.lawrence.harrison at gmail.com> wrote:
>
> Hi-
>
> I have a question about the best way to handle colists — aka, potentially infinite lists as opposed to streams — in Agda. I give some examples below in Haskell and Agda, and I’ve also attached *.hs and *.agda files with complete, stand-alone definitions.
>
> In Haskell, the built-in iterate function always produces an infinite list:
>
> iterate :: (a -> a) -> a -> [a]
> iterate f a =  a : iterate f (f a)
>
>
> This can be represented in Agda using a Stream (defined as a coinductive record):
>
> iterate : ∀ {a} → (a → a) → a → Stream a
> hd (iterate f a) = a
> tl (iterate f a) = iterate f (f a)
>
>
> Now, back in Haskell, we can define a *potentially* infinite list by introducing Maybe in the codomain of f:
>
> iterate1 :: (a -> Maybe a) -> a -> [a]
> iterate1 f a = a : case f a of
>                         Just a1 -> iterate1 f a1
>                         Nothing -> []
>
>
> Using the musical notation for coinduction in Agda, I can get something similar:
>
> data Colist (A : Set) : Set where
>   []  : Colist A
>   _∷_ : A → (∞ (Colist A)) → Colist A
>
> iterate1 : ∀ {a} → (a → Maybe a) → a → Colist a
> iterate1 f a = a ∷ ♯ (case f a of
>                        λ { (just a₁) → (iterate1 f a₁)
>                          ; nothing   → []
>                          })
>
>
> My question is, how do I use coinductive records to define Colist rather than musical notation. Is there a standard approach? That isn’t clear. I'm supposing that musical notation should be avoided.
>
> What I’m really trying to define is analogous functions to iterate for what I’ll call the transcript data type, defined below in Haskell and Agda:
>
> data T e a = e :<< T e a | V a — a "transcript"; like a list with an "answer"
>
> data T (e : Set) (a : Set) : Set where
>    V    : a → T e a
>    _<<_ : e → ∞ (T e a) → T e a
>
>
> All these definitions are given in the attached files.
>
> Thanks!
> Bill
>
>
>
>
>
> _______________________________________________
> Agda mailing list
> Agda at lists.chalmers.se
> https://lists.chalmers.se/mailman/listinfo/agda