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What could this strange type be isomorphic to?
module WeirdIso where
open import Coinduction
open import Data.Nat
open import Data.Stream
open import Relation.Binary.PropositionalEquality
data Weird : Set where
A : (x : Weird) Weird
B : (x : ∞ Weird) Weird
data _≈W_ : Weird → Weird → Set where
eqA : {x y} (p : x ≈W y) A x ≈W A y
eqB : {x y} (p : ∞ (♭ x ≈W ♭ y)) B x ≈W B y
to : Weird Stream ℕ
to (A x) with to x
to (A x) | y ∷ ys = suc y ∷ ys
to (B x) = zero ∷ ♯ to (♭ x)
from : Stream ℕ Weird
from (zero ∷ xs) = B (♯ from (♭ xs))
from (suc x ∷ xs) = A (from (x ∷ xs))
to∘from≈id : x to (from x) ≈ x
to∘from≈id (zero ∷ xs) = refl ∷ ♯ to∘from≈id (♭ xs)
to∘from≈id (suc x ∷ xs) with to (from (x ∷ xs)) | to∘from≈id (x ∷ xs)
to∘from≈id (suc x ∷ xs) | ._ | p ∷ ps = cong suc p ∷ ps
from∘to≈id : x from (to x) ≈W x
from∘to≈id (A x) with to x | from∘to≈id x
from∘to≈id (A x) | y ∷ ys | q = eqA q
from∘to≈id (B x) = eqB (♯ from∘to≈id (♭ x))

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commented Nov 23, 2016

What's the TL;DR. answer to the title question? Weird seems neither data nor codata since it mixes recursion and corecursion...

EDIT: Sorry thinko, the types of to and from have the answer...


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commented May 26, 2017

@Blaisorblade that's right, it's Stream ℕ! There are a lot of fun things one can do with mixed data/codata. See also my gist on infinite prime numbers that uses a fancier type to express that there's an infinite stream of primes with finite non-primes between them.

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