Coinduction

The corecursive definitions below are accepted if the option --guardedness is active:

{-# OPTIONS --guardedness #-}

(An alternative approach is to use Sized Types.)

Coinductive Records

It is possible to define the type of infinite lists (or streams) of elements of some type A as follows,

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

As opposed to inductive record types, we have to introduce the keyword coinductive before defining the fields that constitute the record.

It is interesting to note that is not necessary to give an explicit constructor to the record type Stream A.

We can as well define bisimilarity (equivalence) of a pair of Stream A as a coinductive record.

record _≈_ {A : Set} (xs : Stream A) (ys : Stream A) : Set where
  coinductive
  field
    hd-≈ : hd xs ≡ hd ys
    tl-≈ : tl xs ≈ tl ys

Using copatterns we can define a pair of functions on Stream such that one returns a Stream with the elements in the even positions and the other the elements in odd positions.

even :  {A}  Stream A  Stream A
hd (even x) = hd x
tl (even x) = even (tl (tl x))

odd :  {A}  Stream A  Stream A
odd x = even (tl x)

split :  {A}  Stream A  Stream A × Stream A
split xs = even xs , odd xs

And merge a pair of Stream by interleaving their elements.

merge :  {A}  Stream A × Stream A  Stream A
hd (merge (fst , snd)) = hd fst
tl (merge (fst , snd)) = merge (snd , tl fst)

Finally, we can prove that split is the left inverse of merge.

merge-split-id :  {A} (xs : Stream A)  merge (split xs) ≈ xs
hd-≈ (merge-split-id _)  = refl
tl-≈ (merge-split-id xs) = merge-split-id (tl xs)

Old Coinduction

Note

This is the old way of coinduction support in Agda. You are advised to use Coinductive Records instead.

To use coinduction it is recommended that you import the module Coinduction from the standard library. Coinductive types can then be defined by labelling coinductive occurrences using the delay operator :

data Coℕ : Set where
   zero : Coℕ
   suc  : ∞ Coℕ  Coℕ

The type A can be seen as a suspended computation of type A. It comes with delay and force functions:

♯_ :  {a} {A : Set a}  A  ∞ A
  :  {a} {A : Set a}  ∞ A  A

Values of coinductive types can be constructed using corecursion, which does not need to terminate, but has to be productive. As an approximation to productivity the termination checker requires that corecursive definitions are guarded by coinductive constructors. As an example the infinite “natural number” can be defined as follows:

inf : Coℕ
inf = suc (♯ inf)

The check for guarded corecursion is integrated with the check for size-change termination, thus allowing interesting combinations of inductive and coinductive types. We can for instance define the type of stream processors, along with some functions:

-- Infinite streams.

data Stream (A : Set) : Set where
  _∷_ : (x : A) (xs :(Stream A))  Stream A

-- A stream processor SP A B consumes elements of A and produces
-- elements of B. It can only consume a finite number of A’s before
-- producing a B.

data SP (A B : Set) : Set where
  get : (f : A  SP A B)  SP A B
  put : (b : B) (sp :(SP A B))  SP A B

-- The function eat is defined by an outer corecursion into Stream B
-- and an inner recursion on SP A B.

eat :  {A B}  SP A B  Stream A  Stream B
eat (get f)    (a ∷ as) = eat (f a) (♭ as)
eat (put b sp) as       = b ∷ ♯ eat (♭ sp) as

-- Composition of stream processors.

_∘_ :  {A B C}  SP B C  SP A B  SP A C
get f₁    ∘ put x sp₂ = f₁ x ∘ ♭ sp₂
put x sp₁ ∘ sp₂       = put x ((♭ sp₁ ∘ sp₂))
sp₁       ∘ get f₂    = get (λ x  sp₁ ∘ f₂ x)

It is also possible to define “coinductive families”. It is recommended not to use the delay constructor (♯_) in a constructor’s index expressions. The following definition of equality between coinductive “natural numbers” is discouraged:

data _≈’_ : Coℕ  Coℕ  Set where
  zero : zero ≈’ zero
  suc  :  {m n} (m ≈’ n)  suc (♯ m) ≈’ suc (♯ n)

The recommended definition is the following one:

data _≈_ : Coℕ  Coℕ  Set where
  zero : zero ≈ zero
  suc  :  {m n} (♭ m ≈ ♭ n)  suc m ≈ suc n