# Instance Arguments¶

Instance arguments are a special kind of implicit arguments that get solved by a special instance resolution algorithm, rather than by the unification algorithm used for normal implicit arguments. Instance arguments are the Agda equivalent of Haskell type class constraints and can be used for many of the same purposes.

An instance argument will be resolved if its type is a *named type*
(i.e. a data type or record type) or a *variable type* (i.e. a
previously bound variable of type Set ℓ), and a unique *instance* of
the required type can be built from declared
instances and the current context.

## Usage¶

Instance arguments are enclosed in double curly braces `{{ }}`

, e.g. `{{x : T}}`

.
Alternatively they can be enclosed, with proper spacing, e.g. `⦃ x : T ⦄`

, in the
unicode braces `⦃ ⦄`

(`U+2983`

and `U+2984`

, which can be typed as
`\{{`

and `\}}`

in the Emacs mode).

For instance, given a function `_==_`

```
_==_ : {A : Set} {{eqA : Eq A}} → A → A → Bool
```

for some suitable type `Eq`

, you might define

```
elem : {A : Set} {{eqA : Eq A}} → A → List A → Bool
elem x (y ∷ xs) = x == y || elem x xs
elem x [] = false
```

Here the instance argument to `_==_`

is solved by the corresponding argument
to `elem`

. Just like ordinary implicit arguments, instance arguments can be
given explicitly. The above definition is equivalent to

```
elem : {A : Set} {{eqA : Eq A}} → A → List A → Bool
elem {{eqA}} x (y ∷ xs) = _==_ {{eqA}} x y || elem {{eqA}} x xs
elem x [] = false
```

A very useful function that exploits this is the function `it`

which lets you
apply instance resolution to solve an arbitrary goal:

```
it : ∀ {a} {A : Set a} {{_ : A}} → A
it {{x}} = x
```

Note that instance arguments in types are always named, but the name can be `_`

:

```
_==_ : {A : Set} → {{Eq A}} → A → A → Bool -- INVALID
```

```
_==_ : {A : Set} {{_ : Eq A}} → A → A → Bool -- VALID
```

### Defining type classes¶

The type of an instance argument should have the form `{Γ} → C vs`

,
where `C`

is a postulated name, a bound variable, or the name of a
data or record type, and `{Γ}`

denotes an arbitrary number of
implicit or instance arguments (see Dependent instances below
for an example where `{Γ}`

is non-empty).

Instances with explicit arguments are also accepted but will not be considered as instances because the value of the explicit arguments cannot be derived automatically. Having such an instance has no effect and thus raises a warning.

Instance arguments whose types end in any other type are currently also accepted but cannot be resolved by instance search, so they must be given by hand. For this reason it is not recommended to use such instance arguments. Doing so will also raise a warning.

Other than that there are no requirements on the type of an instance argument. In particular, there is no special declaration to say that a type is a “type class”. Instead, Haskell-style type classes are usually defined as record types. For instance,

```
record Monoid {a} (A : Set a) : Set a where
field
mempty : A
_<>_ : A → A → A
```

In order to make the fields of the record available as functions taking instance arguments you can use the special module application

```
open Monoid {{...}} public
```

This will bring into scope

```
mempty : ∀ {a} {A : Set a} {{_ : Monoid A}} → A
_<>_ : ∀ {a} {A : Set a} {{_ : Monoid A}} → A → A → A
```

Superclass dependencies can be implemented using Instance fields.

See Module application and Record modules for details about how
the module application is desugared. If defined by hand, `mempty`

would be

```
mempty : ∀ {a} {A : Set a} {{_ : Monoid A}} → A
mempty {{mon}} = Monoid.mempty mon
```

Although record types are a natural fit for Haskell-style type classes, you can use instance arguments with data types to good effect. See the Examples below.

### Declaring instances¶

As seen above, instance arguments in the context are available when solving
instance arguments, but you also need to be able to
define top-level instances for concrete types. This is done using the
`instance`

keyword, which starts a block in
which each definition is marked as an instance available for instance
resolution. For example, an instance `Monoid (List A)`

can be defined as

```
instance
ListMonoid : ∀ {a} {A : Set a} → Monoid (List A)
ListMonoid = record { mempty = []; _<>_ = _++_ }
```

Or equivalently, using copatterns:

```
instance
ListMonoid : ∀ {a} {A : Set a} → Monoid (List A)
mempty {{ListMonoid}} = []
_<>_ {{ListMonoid}} xs ys = xs ++ ys
```

Top-level instances must target a named type (`Monoid`

in this case), and
cannot be declared for types in the context.

You can define local instances in let-expressions in the same way as a top-level instance. For example:

```
mconcat : ∀ {a} {A : Set a} {{_ : Monoid A}} → List A → A
mconcat [] = mempty
mconcat (x ∷ xs) = x <> mconcat xs
sum : List Nat → Nat
sum xs =
let instance
NatMonoid : Monoid Nat
NatMonoid = record { mempty = 0; _<>_ = _+_ }
in mconcat xs
```

Instances can have instance arguments themselves, which will be filled in recursively during instance resolution. For instance,

```
record Eq {a} (A : Set a) : Set a where
field
_==_ : A → A → Bool
open Eq {{...}} public
instance
eqList : ∀ {a} {A : Set a} {{_ : Eq A}} → Eq (List A)
_==_ {{eqList}} [] [] = true
_==_ {{eqList}} (x ∷ xs) (y ∷ ys) = x == y && xs == ys
_==_ {{eqList}} _ _ = false
eqNat : Eq Nat
_==_ {{eqNat}} = natEquals
ex : Bool
ex = (1 ∷ 2 ∷ 3 ∷ []) == (1 ∷ 2 ∷ []) -- false
```

Note the two calls to `_==_`

in the right-hand side of the second clause. The
first uses the `Eq A`

instance and the second uses a recursive call to
`eqList`

. In the example `ex`

, instance resolution, needing a value of type ```
Eq
(List Nat)
```

, will try to use the `eqList`

instance and find that it needs an
instance argument of type `Eq Nat`

, it will then solve that with `eqNat`

and return the solution `eqList {{eqNat}}`

.

Note

At the moment there is no termination check on instances, so it is possible
to construct non-sensical instances like
`loop : ∀ {a} {A : Set a} {{_ : Eq A}} → Eq A`

.
To prevent looping in cases like this, the search depth of instance search
is limited, and once the maximum depth is reached, a type error will be
thrown. You can set the maximum depth using the `--instance-search-depth`

flag.

### Restricting instance search¶

To restrict an instance to the current module, you can mark it as private. For instance,

```
record Default (A : Set) : Set where
field default : A
open Default {{...}} public
module M where
private
instance
defaultNat : Default Nat
defaultNat .default = 6
test₁ : Nat
test₁ = default
_ : test₁ ≡ 6
_ = refl
open M
instance
defaultNat : Default Nat
defaultNat .default = 42
test₂ : Nat
test₂ = default
_ : test₂ ≡ 42
_ = refl
```

#### Constructor instances¶

Although instance arguments are most commonly used for record types,
mimicking Haskell-style type classes, they can also be used with data
types. In this case you often want the constructors to be instances,
which is achieved by declaring them inside an `instance`

block. Constructors can only be declared as instances if all their
arguments are implicit or instance arguments. See
Instance resolution below for the details.

A simple example of a constructor that can be made an instance is the reflexivity constructor of the equality type:

```
data _≡_ {a} {A : Set a} (x : A) : A → Set a where
instance refl : x ≡ x
```

This allows trivial equality proofs to be inferred by instance resolution,
which can make working with functions that have preconditions less of a burden.
As an example, here is how one could use this to define a function that takes a
natural number and gives back a `Fin n`

(the type of naturals smaller than
`n`

):

```
data Fin : Nat → Set where
zero : ∀ {n} → Fin (suc n)
suc : ∀ {n} → Fin n → Fin (suc n)
mkFin : ∀ {n} (m : Nat) {{_ : suc m - n ≡ 0}} → Fin n
mkFin {zero} m {{}}
mkFin {suc n} zero = zero
mkFin {suc n} (suc m) = suc (mkFin m)
five : Fin 6
five = mkFin 5 -- OK
```

In the first clause of `mkFin`

we use an absurd pattern to discharge the impossible assumption ```
suc m ≡
0
```

. See the next section for
another example of constructor instances.

Record fields can also be declared instances, with the effect that the corresponding projection function is considered a top-level instance.

#### Overlapping instances¶

By default, Agda does not allow overlapping instances. Two instances are defined to overlap if they could both solve the instance goal when given appropriate solutions for their recursive (instance) arguments.

For example, in code below, the instances zero and suc overlap for the goal ex₁, because either one of them can be used to solve the goal when given appropriate arguments, hence instance search fails.

```
infix 4 _∈_
data _∈_ {A : Set} (x : A) : List A → Set where
instance
zero : ∀ {xs} → x ∈ x ∷ xs
suc : ∀ {y xs} {{_ : x ∈ xs}} → x ∈ y ∷ xs
ex₁ : 1 ∈ 1 ∷ 2 ∷ 3 ∷ 4 ∷ []
ex₁ = it -- overlapping instances
```

Overlapping instances can be enabled via the `--overlapping-instances`

flag. Be aware that enabling this flag might lead to an exponential
slowdown in instance resolution and possibly (apparent) looping
behaviour.

### Examples¶

#### Dependent instances¶

Consider a variant on the `Eq`

class where the equality function produces a
proof in the case the arguments are equal:

```
record Eq {a} (A : Set a) : Set a where
field
_==_ : (x y : A) → Maybe (x ≡ y)
open Eq {{...}} public
```

A simple boolean-valued equality function is problematic for types with dependencies, like the Σ-type

```
data Σ {a b} (A : Set a) (B : A → Set b) : Set (a ⊔ b) where
_,_ : (x : A) → B x → Σ A B
```

since given two pairs `x , y`

and `x₁ , y₁`

, the types of the second
components `y`

and `y₁`

can be completely different and not admit an
equality test. Only when `x`

and `x₁`

are *really equal* can we hope to
compare `y`

and `y₁`

. Having the equality function return a proof means
that we are guaranteed that when `x`

and `x₁`

compare equal, they really
are equal, and comparing `y`

and `y₁`

makes sense.

An `Eq`

instance for `Σ`

can be defined as follows:

```
instance
eqΣ : ∀ {a b} {A : Set a} {B : A → Set b} {{_ : Eq A}} {{_ : ∀ {x} → Eq (B x)}} → Eq (Σ A B)
_==_ {{eqΣ}} (x , y) (x₁ , y₁) with x == x₁
_==_ {{eqΣ}} (x , y) (x₁ , y₁) | nothing = nothing
_==_ {{eqΣ}} (x , y) (.x , y₁) | just refl with y == y₁
_==_ {{eqΣ}} (x , y) (.x , y₁) | just refl | nothing = nothing
_==_ {{eqΣ}} (x , y) (.x , .y) | just refl | just refl = just refl
```

Note that the instance argument for `B`

states that there should be
an `Eq`

instance for `B x`

, for any `x : A`

. The argument `x`

must be implicit, indicating that it needs to be inferred by
unification whenever the `B`

instance is used. See
Instance resolution below for more details.

## Instance resolution¶

Given a goal that should be solved using instance resolution we proceed in the following four stages:

- Verify the goal
- First we check that the goal type has the right shape to be solved
by instance resolution. It should be of the form
`{Γ} → C vs`

, where the target type`C`

is a variable from the context or the name of a data or record type, and`{Γ}`

denotes a telescope of implicit or instance arguments. If this is not the case instance resolution fails with an error message[1]. - Find candidates
- In the second stage we compute a set of
*candidates*. Let-bound variables and top-level definitions in scope are candidates if they are defined in an`instance`

block. Lambda-bound variables, i.e. variables bound in lambdas, function types, left-hand sides, or module parameters, are candidates if they are bound as instance arguments using`{{ }}`

. Only candidates of type`{Δ} → C us`

, where`C`

is the target type computed in the previous stage and`{Δ}`

only contains implicit or instance arguments, are considered. - Check the candidates
We attempt to use each candidate in turn to build an instance of the goal type

`{Γ} → C vs`

. First we extend the current context by`{Γ}`

. Then, given a candidate`c : {Δ} → A`

we generate fresh metavariables`αs : {Δ}`

for the arguments of`c`

, with ordinary metavariables for implicit arguments, and instance metavariables, solved by a recursive call to instance resolution, for instance arguments.Next we unify

`A[Δ := αs]`

with`C vs`

and apply instance resolution to the instance metavariables in`αs`

. Both unification and instance resolution have three possible outcomes:*yes*,*no*, or*maybe*. In case we get a*no*answer from any of them, the current candidate is discarded, otherwise we return the potential solution`λ {Γ} → c αs`

.- Compute the result
From the previous stage we get a list of potential solutions. If the list is empty we fail with an error saying that no instance for

`C vs`

could be found (*no*). If there is a single solution we use it to solve the goal (*yes*), and if there are multiple solutions we check if they are all equal. If they are, we solve the goal with one of them (*yes*), but if they are not, we postpone instance resolution (*maybe*), hoping that some of the*maybes*will turn into*nos*once we know more about the involved metavariables.If there are left-over instance problems at the end of type checking, the corresponding metavariables are printed in the Emacs status buffer together with their types and source location. The candidates that gave rise to potential solutions can be printed with the show constraints command (

`C-c C-=`

).

[1] | Instance goal verification is buggy at the moment. See issue #1322. |