[Agda] coinductively defined families

[Thorsten Altenkirch]

One problem with it is that your proposed syntax would break all existing code with records in it.

Yes, I just thought about that. One would need to write "destructor" instead of "field". if one wants to include the domain.

Thorsten

From: Jesper Cockx <Jesper at sikanda.be<mailto:Jesper at sikanda.be>>
Date: Wednesday, 7 March 2018 at 14:20
To: Thorsten Altenkirch <psztxa at exmail.nottingham.ac.uk<mailto:psztxa at exmail.nottingham.ac.uk>>
Cc: Agda mailing list <agda at lists.chalmers.se<mailto:agda at lists.chalmers.se>>
Subject: Re: [Agda] coinductively defined families

Probably you know you can already write this:

open import Agda.Builtin.Nat
open import Agda.Builtin.Equality

record Vec (A : Set) (n : Nat) : Set where
  inductive
  field
    hd : ∀ {m} → n ≡ suc m → A
    tl : ∀ {m} → n ≡ suc m → Vec A m
open Vec

[] : ∀ {A} → Vec A 0
[] .hd ()
[] .tl ()

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

but I agree that having syntax for indexed records would be a nice thing to have. One problem with it is that your proposed syntax would break all existing code with records in it.

I was actually thinking recently of going in the opposite direction and making indexed datatypes behave more like records, so you could have projections and eta-laws when there's only a single possible constructor for the given indices (as would be the case for vectors).

-- Jesper


On Wed, Mar 7, 2018 at 2:58 PM, Thorsten Altenkirch <Thorsten.Altenkirch at nottingham.ac.uk<mailto:Thorsten.Altenkirch at nottingham.ac.uk>> wrote:
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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