# Record Types¶

Records are types for grouping values together. They generalise the dependent product type by providing named fields and (optional) further components.

## Example: the Pair type constructor¶

Record types can be declared using the `record`

keyword

```
record Pair (A B : Set) : Set where
field
fst : A
snd : B
```

This defines a new type constructor `Pair : Set → Set → Set`

and two
projection functions

```
Pair.fst : {A B : Set} → Pair A B → A
Pair.snd : {A B : Set} → Pair A B → B
```

Note that the parameters `A`

and `B`

are implicit arguments to the
projection functions.

Elements of record types can be defined using a record expression

```
p23 : Pair Nat Nat
p23 = record { fst = 2; snd = 3 }
```

or using copatterns. Copatterns may be used prefix

```
p34 : Pair Nat Nat
Pair.fst p34 = 3
Pair.snd p34 = 4
```

suffix (in which case they are written prefixed with a dot)

```
p56 : Pair Nat Nat
p56 .Pair.fst = 5
p56 .Pair.snd = 6
```

or using an anonymous copattern-matching lambda (you may only use the suffix form of copatterns in this case)

```
p78 : Pair Nat Nat
p78 = λ where
.Pair.fst → 7
.Pair.snd → 8
```

If you use the `constructor`

keyword, you can also use the named
constructor to define elements of the record type:

```
record Pair (A B : Set) : Set where
constructor _,_
field
fst : A
snd : B
p45 : Pair Nat Nat
p45 = 4 , 5
```

In this sense, record types behave much like single constructor datatypes (but see Eta-expansion below).

## Declaring, constructing and decomposing records¶

### Declaring record types¶

The general form of a record declaration is as follows:

```
record <recordname> <parameters> : Set <level> where
<directives>
constructor <constructorname>
field
<fieldname1> : <type1>
<fieldname2> : <type2>
-- ...
<declarations>
```

All the components are optional, and can be given in any order. In particular, fields can be given in more than one block, interspersed with other declarations. Each field is a component of the record. Types of later fields can depend on earlier fields.

The directives available are `eta-equality`

, `no-eta-equality`

,
`pattern`

(see Eta-expansion), `inductive`

and `co-inductive`

(see
Recursive records).

### Constructing record values¶

Record values are constructed by giving a value for each record field:

```
record { <fieldname1> = <term1> ; <fieldname2> = <term2> ; ... }
```

where the types of the terms match the types of the fields. If a
constructor `<constructorname>`

has been declared for the record,
this can also be written

```
<constructorname> <term1> <term2> ...
```

For named definitions, this can also be expressed using copatterns:

```
<named-def> : <recordname> <parameters>
<recordname>.<fieldname1> <named-def> = <term1>
<recordname>.<fieldname2> <named-def> = <term2>
...
```

Records can also be constructed by updating other records.

#### Building records from modules¶

The `record { <fields> }`

syntax also accepts module names. Fields
are defined using the corresponding definitions from the given module.
For instance assuming this record type R and module M:

```
record R : Set where
field
x : X
y : Y
z : Z
module M where
x = ...
y = ...
r : R
r = record { M; z = ... }
```

This construction supports any combination of explicit field definitions and applied modules. If a field is both given explicitly and available in one of the modules, then the explicit one takes precedence. If a field is available in more than one module then this is ambiguous and therefore rejected. As a consequence the order of assignments does not matter.

The modules can be both applied to arguments and have import directives such as hiding, using, and renaming. Here is a contrived example building on the example above:

```
module M2 (a : A) where
w = ...
z = ...
r2 : A → R
r2 a = record { M hiding (y); M2 a renaming (w to y) }
```

### Decomposing record values¶

With the field name, we can project the corresponding component out of a record value. It is also possible to pattern match against inductive records:

```
sum : Pair Nat Nat → Nat
sum (x , y) = x + y
```

Or, using a let binding record pattern:

```
sum' : Pair Nat Nat → Nat
sum' p = let (x , y) = p in x + y
```

Note

Naming the constructor is not required to enable pattern matching against record values. Record expressions can appear as patterns.

### Record update¶

Assume that we have a record type and a corresponding value:

```
record MyRecord : Set where
field
a b c : Nat
old : MyRecord
old = record { a = 1; b = 2; c = 3 }
```

Then we can update (some of) the record value’s fields in the following way:

```
new : MyRecord
new = record old { a = 0; c = 5 }
```

Here `new`

normalises to `record { a = 0; b = 2; c = 5 }`

. Any
expression yielding a value of type `MyRecord`

can be used instead of
`old`

. Using that records can be built from module names, together with the fact that
all records define a module, this can also be
written as

```
new' : MyRecord
new' = record { MyRecord old; a = 0; c = 5}
```

Record updating is not allowed to change types: the resulting value must have the same type as the original one, including the record parameters. Thus, the type of a record update can be inferred if the type of the original record can be inferred.

The record update syntax is expanded before type checking. When the expression

```
record old { upd-fields }
```

is checked against a record type `R`

, it is expanded to

```
let r = old in record { new-fields }
```

where `old`

is required to have type `R`

and `new-fields`

is defined as
follows: for each field `x`

in `R`

,

if

`x = e`

is contained in`upd-fields`

then`x = e`

is included in`new-fields`

, and otherwiseif

`x`

is an explicit field then`x = R.x r`

is included in`new-fields`

, andif

`x`

is an implicit or instance field, then it is omitted from`new-fields`

.

The reason for treating implicit and instance fields specially is to allow code like the following:

```
data Vec (A : Set) : Nat → Set where
[] : Vec A zero
_∷_ : ∀{n} → A → Vec A n → Vec A (suc n)
record R : Set where
field
{length} : Nat
vec : Vec Nat length
-- More fields ...
xs : R
xs = record { vec = 0 ∷ 1 ∷ 2 ∷ [] }
ys = record xs { vec = 0 ∷ [] }
```

Without the special treatment the last expression would need to
include a new binding for `length`

(for instance `length = _`

).

## Record modules¶

Along with a new type, a record declaration also defines a module with the same name, parameterised over an element of the record type containing the projection functions. This allows records to be “opened”, bringing the fields into scope. For instance

```
swap : {A B : Set} → Pair A B → Pair B A
swap p = snd , fst
where open Pair p
```

In the example, the record module `Pair`

has the shape

```
module Pair {A B : Set} (p : Pair A B) where
fst : A
snd : B
```

Note

This is not quite right: The projection functions take the parameters as erased arguments. However, the parameters are not erased in the module telescope if they were not erased to start with.

It’s possible to add arbitrary definitions to the record module, by defining them inside the record declaration

```
record Functor (F : Set → Set) : Set₁ where
field
fmap : ∀ {A B} → (A → B) → F A → F B
_<$_ : ∀ {A B} → A → F B → F A
x <$ fb = fmap (λ _ → x) fb
```

Note

In general new definitions need to appear after the field declarations, but
simple non-recursive function definitions without pattern matching can be
interleaved with the fields. The reason for this restriction is that the
type of the record constructor needs to be expressible using let-expressions.
In the example below `D₁`

can only contain declarations for which the
generated type of `mkR`

is well-formed.

```
record R Γ : Setᵢ where
constructor mkR
field f₁ : A₁
D₁
field f₂ : A₂
mkR : ∀ {Γ} (f₁ : A₁) (let D₁) (f₂ : A₂) → R Γ
```

## Eta-expansion¶

The eta (η) rule for a record type

```
record R : Set where
field
a : A
b : B
c : C
```

states that every `x : R`

is definitionally equal to ```
record { a =
R.a x ; b = R.b x ; c = R.c x }
```

. By default, all (inductive) record
types enjoy η-equality if the positivity checker has confirmed it is
safe to have it. The keywords `eta-equality`

/`no-eta-equality`

enable/disable η rules for the record type being declared.

## Recursive records¶

Recursive records need to be declared as either inductive or coinductive.

```
record Tree (A : Set) : Set where
inductive
constructor tree
field
elem : A
subtrees : List (Tree A)
record Stream (A : Set) : Set where
coinductive
constructor _::_
field
head : A
tail : Stream A
```

Inductive records have `eta-equality`

on by default, while
`no-eta-equality`

is the default for coinductive records. In fact,
the `eta-equality`

and `coinductive`

directives are not allowed
together, since this can easily make Agda loop. This can be overridden
at your own risk by using the pragma `ETA`

instead.

It is possible to pattern match on inductive records, but not on coinductive ones.

However, inductive records without η-equality do not support both matching on
the record constructor and construction of record elements by
copattern matching. It has been discovered that the combination of
both leads to loss of subject reduction, i.e., reduction does not
preserve typing. For records without η, matching on the record
constructor is off by default and construction by copattern matching
is on. If you want the converse, you can add the record directive
`pattern`

:

```
record HereditaryList : Set where
inductive
no-eta-equality
pattern
field sublists : List HereditaryList
pred : HereditaryList → List HereditaryList
pred record{ sublists = ts } = ts
```

If both `eta-equality`

and `pattern`

are given for a record types,
Agda will alert the user of a redundant `pattern`

directive.
However, if η is inferred but not declared explicitly, Agda will just
ignore a redundant `pattern`

directive; this is because the default
can be changed globally by option `--no-eta-equality`

.

## Instance fields¶

Instance fields, that is record fields marked with `{{ }}`

can be used to
model “superclass” dependencies. For example:

```
record Eq (A : Set) : Set where
field
_==_ : A → A → Bool
open Eq {{...}}
```

```
record Ord (A : Set) : Set where
field
_<_ : A → A → Bool
{{eqA}} : Eq A
open Ord {{...}} hiding (eqA)
```

Now anytime you have a function taking an `Ord A`

argument the `Eq A`

instance
is also available by virtue of η-expansion. So this works as you would expect:

```
_≤_ : {A : Set} {{OrdA : Ord A}} → A → A → Bool
x ≤ y = (x == y) || (x < y)
```

There is a problem however if you have multiple record arguments with conflicting
instance fields. For instance, suppose we also have a `Num`

record with an `Eq`

field

```
record Num (A : Set) : Set where
field
fromNat : Nat → A
{{eqA}} : Eq A
open Num {{...}} hiding (eqA)
_≤3 : {A : Set} {{OrdA : Ord A}} {{NumA : Num A}} → A → Bool
x ≤3 = (x == fromNat 3) || (x < fromNat 3)
```

Here the `Eq A`

argument to `_==_`

is not resolved since there are two conflicting
candidates: `Ord.eqA OrdA`

and `Num.eqA NumA`

. To solve this problem you can declare
instance fields as *overlappable* using the `overlap`

keyword:

```
record Ord (A : Set) : Set where
field
_<_ : A → A → Bool
overlap {{eqA}} : Eq A
open Ord {{...}} hiding (eqA)
record Num (A : Set) : Set where
field
fromNat : Nat → A
overlap {{eqA}} : Eq A
open Num {{...}} hiding (eqA)
_≤3 : {A : Set} {{OrdA : Ord A}} {{NumA : Num A}} → A → Bool
x ≤3 = (x == fromNat 3) || (x < fromNat 3)
```

Whenever there are multiple valid candidates for an instance goal, if **all** candidates
are overlappable, the goal is solved by the left-most candidate. In the example above
that means that the `Eq A`

goal is solved by the instance from the `Ord`

argument.

Clauses for instance fields can be omitted when defining values of record
types. For instance we can define `Nat`

instances for `Eq`

, `Ord`

and
`Num`

as follows, leaving out cases for the `eqA`

fields:

```
instance
EqNat : Eq Nat
_==_ {{EqNat}} = Agda.Builtin.Nat._==_
OrdNat : Ord Nat
_<_ {{OrdNat}} = Agda.Builtin.Nat._<_
NumNat : Num Nat
fromNat {{NumNat}} n = n
```