sanctuary-def

sanctuary-def is a run-time type system for JavaScript. It facilitates the definition of curried JavaScript functions that are explicit about the number of arguments to which they may be applied and the types of those arguments.

It is conventional to import the package as $:

const $ = require ('sanctuary-def');

The next step is to define an environment. An environment is an array of types. env is an environment containing all the built-in JavaScript types. It may be used as the basis for environments that include custom types in addition to the built-in types:

//    Integer :: Type
const Integer = '...';

//    NonZeroInteger :: Type
const NonZeroInteger = '...';

//    env :: Array Type
const env = $.env.concat ([Integer, NonZeroInteger]);

Type constructors such as List :: Type -> Type cannot be included in an environment as they're not of the correct type. One could, though, use a type constructor to define a fixed number of concrete types:

//    env :: Array Type
const env = $.env.concat ([
  List ($.Number),                // :: Type
  List ($.String),                // :: Type
  List (List ($.Number)),         // :: Type
  List (List ($.String)),         // :: Type
  List (List (List ($.Number))),  // :: Type
  List (List (List ($.String))),  // :: Type
]);

Not only would this be tedious, but one could never enumerate all possible types as there are infinitely many. Instead, one should use Unknown:

//    env :: Array Type
const env = $.env.concat ([List ($.Unknown)]);

The next step is to define a def function for the environment using $.create:

//    def :: String -> StrMap (Array TypeClass) -> Array Type -> Function -> Function
const def = $.create ({checkTypes: true, env});

The checkTypes option determines whether type checking is enabled. This allows one to only pay the performance cost of run-time type checking during development. For example:

//    def :: String -> StrMap (Array TypeClass) -> Array Type -> Function -> Function
const def = $.create ({
  checkTypes: process.env.NODE_ENV === 'development',
  env,
});

def is a function for defining functions. For example:

//    add :: Number -> Number -> Number
const add =
def ('add')                           // name
    ({})                              // type-class constraints
    ([$.Number, $.Number, $.Number])  // input and output types
    (x => y => x + y);                // implementation

[$.Number, $.Number, $.Number] specifies that add takes two arguments of type Number, one at a time, and returns a value of type Number.

Applying add to two arguments, one at a time, gives the expected result:

add (2) (2);
// => 4

Applying add to multiple arguments at once results in an exception being thrown:

add (2, 2, 2);
// ! TypeError: ‘add’ applied to the wrong number of arguments
//
//   add :: Number -> Number -> Number
//          ^^^^^^
//            1
//
//   Expected one argument but received three arguments:
//
//     - 2
//     - 2
//     - 2

Applying add to one argument produces a function awaiting the remaining argument. This is known as partial application. Partial application allows more specific functions to be defined in terms of more general ones:

//    inc :: Number -> Number
const inc = add (1);

inc (7);
// => 8

JavaScript's implicit type coercion often obfuscates the source of type errors. Consider the following function:

//    _add :: Number -> Number -> Number
const _add = x => y => x + y;

The type signature indicates that _add takes arguments of type Number, but this is not enforced. This allows type errors to be silently ignored:

_add ('2') ('2');
// => '22'

add, on the other hand, throws if applied to arguments of the wrong types:

add ('2') ('2');
// ! TypeError: Invalid value
//
//   add :: Number -> Number -> Number
//          ^^^^^^
//            1
//
//   1)  "2" :: String
//
//   The value at position 1 is not a member of ‘Number’.

Type checking is performed as arguments are provided (rather than once all arguments have been provided), so type errors are reported early:

add ('X');
// ! TypeError: Invalid value
//
//   add :: Number -> Number -> Number
//          ^^^^^^
//            1
//
//   1)  "X" :: String
//
//   The value at position 1 is not a member of ‘Number’.

Types

Conceptually, a type is a set of values. One can think of a value of type Type as a function of type Any -> Boolean that tests values for membership in the set (though this is an oversimplification).

Unknown :: Type

Type used to represent missing type information. The type of [], for example, is Array ???.

May be used with type constructors when defining environments. Given a type constructor List :: Type -> Type, one could use List ($.Unknown) to include an infinite number of types in an environment:

• List Number
• List String
• List (List Number)
• List (List String)
• List (List (List Number))
• List (List (List String))
• ...

Void :: Type

Uninhabited type.

May be used to convey that a type parameter of an algebraic data type will not be used. For example, a future of type Future Void String will never be rejected.

Any :: Type

Type comprising every JavaScript value.

AnyFunction :: Type

Type comprising every Function value.

Arguments :: Type

Type comprising every arguments object.

Array :: Type -⁠> Type

Constructor for homogeneous Array types.

Array0 :: Type

Type whose sole member is [].

Array1 :: Type -⁠> Type

Constructor for singleton Array types.

Array2 :: Type -⁠> Type -⁠> Type

Constructor for heterogeneous Array types of length 2. ['foo', true] is a member of Array2 String Boolean.

Boolean :: Type

Type comprising true and false.

Buffer :: Type

Type comprising every Buffer object.

Date :: Type

Type comprising every Date value.

ValidDate :: Type

Type comprising every Date value except new Date (NaN).

Descending :: Type -⁠> Type

Descending type constructor.

Either :: Type -⁠> Type -⁠> Type

Either type constructor.

Error :: Type

Type comprising every Error value, including values of more specific constructors such as SyntaxError and TypeError.

Fn :: Type -⁠> Type -⁠> Type

Binary type constructor for unary function types. $.Fn (I) (O) represents I -> O, the type of functions that take a value of type I and return a value of type O.

Function :: NonEmpty (Array Type) -⁠> Type

Constructor for Function types.

Examples:

• $.Function ([$.Date, $.String]) represents the Date -> String type; and
• $.Function ([a, b, a]) represents the (a, b) -> a type.

HtmlElement :: Type

Type comprising every HTML element.

Identity :: Type -⁠> Type

Identity type constructor.

JsMap :: Type -⁠> Type -⁠> Type

Constructor for native Map types. $.JsMap ($.Number) ($.String), for example, is the type comprising every native Map whose keys are numbers and whose values are strings.

JsSet :: Type -⁠> Type

Constructor for native Set types. $.JsSet ($.Number), for example, is the type comprising every native Set whose values are numbers.

Maybe :: Type -⁠> Type

Maybe type constructor.

Module :: Type

Type comprising every ES module.

NonEmpty :: Type -⁠> Type

Constructor for non-empty types. $.NonEmpty ($.String), for example, is the type comprising every String value except ''.

The given type must satisfy the Monoid and Setoid specifications.

Null :: Type

Type whose sole member is null.

Nullable :: Type -⁠> Type

Constructor for types that include null as a member.

Number :: Type

Type comprising every primitive Number value (including NaN).

PositiveNumber :: Type

Type comprising every Number value greater than zero.

NegativeNumber :: Type

Type comprising every Number value less than zero.

ValidNumber :: Type

Type comprising every Number value except NaN.

NonZeroValidNumber :: Type

Type comprising every ValidNumber value except 0 and -0.

FiniteNumber :: Type

Type comprising every ValidNumber value except Infinity and -Infinity.

NonZeroFiniteNumber :: Type

Type comprising every FiniteNumber value except 0 and -0.

PositiveFiniteNumber :: Type

Type comprising every FiniteNumber value greater than zero.

NegativeFiniteNumber :: Type

Type comprising every FiniteNumber value less than zero.

Integer :: Type

Type comprising every integer in the range [Number.MIN_SAFE_INTEGER .. Number.MAX_SAFE_INTEGER].

NonZeroInteger :: Type

Type comprising every Integer value except 0 and -0.

NonNegativeInteger :: Type

Type comprising every non-negative Integer value (including -0). Also known as the set of natural numbers under ISO 80000-2:2009.

PositiveInteger :: Type

Type comprising every Integer value greater than zero.

NegativeInteger :: Type

Type comprising every Integer value less than zero.

Object :: Type

Type comprising every "plain" Object value. Specifically, values created via:

• object literal syntax;
• Object.create; or
• the new operator in conjunction with Object or a custom constructor function.

Pair :: Type -⁠> Type -⁠> Type

Pair type constructor.

RegExp :: Type

Type comprising every RegExp value.

GlobalRegExp :: Type

Type comprising every RegExp value whose global flag is true.

See also NonGlobalRegExp.

NonGlobalRegExp :: Type

Type comprising every RegExp value whose global flag is false.

See also GlobalRegExp.

StrMap :: Type -⁠> Type

Constructor for homogeneous Object types.

{foo: 1, bar: 2, baz: 3}, for example, is a member of StrMap Number; {foo: 1, bar: 2, baz: 'XXX'} is not.

String :: Type

Type comprising every primitive String value.

RegexFlags :: Type

Type comprising the canonical RegExp flags:

• ''
• 'g'
• 'i'
• 'm'
• 'gi'
• 'gm'
• 'im'
• 'gim'

Symbol :: Type

Type comprising every Symbol value.

Type :: Type

Type comprising every Type value.

TypeClass :: Type

Type comprising every TypeClass value.

Undefined :: Type

Type whose sole member is undefined.

env :: Array Type

An array of types:

• AnyFunction
• Arguments
• Array (Unknown)
• Array2 (Unknown) (Unknown)
• Boolean
• Buffer
• Date
• Descending (Unknown)
• Either (Unknown) (Unknown)
• Error
• Fn (Unknown) (Unknown)
• HtmlElement
• Identity (Unknown)
• JsMap (Unknown) (Unknown)
• JsSet (Unknown)
• Maybe (Unknown)
• Module
• Null
• Number
• Object
• Pair (Unknown) (Unknown)
• RegExp
• StrMap (Unknown)
• String
• Symbol
• Type
• TypeClass
• Undefined

test :: Array Type -⁠> Type -⁠> a -⁠> Boolean

Takes an environment, a type, and any value. Returns true if the value is a member of the type; false otherwise.

The environment is only significant if the type contains type variables.

Type constructors

sanctuary-def provides several functions for defining types.

NullaryType :: String -⁠> String -⁠> Array Type -⁠> (Any -⁠> Boolean) -⁠> Type

Type constructor for types with no type variables (such as Number).

To define a nullary type t one must provide:

• the name of t (exposed as t.name);

• the documentation URL of t (exposed as t.url);

• an array of supertypes (exposed as t.supertypes); and

• a predicate that accepts any value that is a member of every one of the given supertypes, and returns true if (and only if) the value is a member of t.

For example:

//    Integer :: Type
const Integer = $.NullaryType
  ('Integer')
  ('http://example.com/my-package#Integer')
  ([])
  (x => typeof x === 'number' &&
        Math.floor (x) === x &&
        x >= Number.MIN_SAFE_INTEGER &&
        x <= Number.MAX_SAFE_INTEGER);

//    NonZeroInteger :: Type
const NonZeroInteger = $.NullaryType
  ('NonZeroInteger')
  ('http://example.com/my-package#NonZeroInteger')
  ([Integer])
  (x => x !== 0);

//    rem :: Integer -> NonZeroInteger -> Integer
const rem =
def ('rem')
    ({})
    ([Integer, NonZeroInteger, Integer])
    (x => y => x % y);

rem (42) (5);
// => 2

rem (0.5);
// ! TypeError: Invalid value
//
//   rem :: Integer -> NonZeroInteger -> Integer
//          ^^^^^^^
//             1
//
//   1)  0.5 :: Number
//
//   The value at position 1 is not a member of ‘Integer’.
//
//   See http://example.com/my-package#Integer for information about the Integer type.

rem (42) (0);
// ! TypeError: Invalid value
//
//   rem :: Integer -> NonZeroInteger -> Integer
//                     ^^^^^^^^^^^^^^
//                           1
//
//   1)  0 :: Number
//
//   The value at position 1 is not a member of ‘NonZeroInteger’.
//
//   See http://example.com/my-package#NonZeroInteger for information about the NonZeroInteger type.

UnaryType :: Foldable f => String -⁠> String -⁠> Array Type -⁠> (Any -⁠> Boolean) -⁠> (t a -⁠> f a) -⁠> Type -⁠> Type

Type constructor for types with one type variable (such as Array).

To define a unary type t a one must provide:

• the name of t (exposed as t.name);

• the documentation URL of t (exposed as t.url);

• an array of supertypes (exposed as t.supertypes);

• a predicate that accepts any value that is a member of every one of the given supertypes, and returns true if (and only if) the value is a member of t x for some type x;

• a function that takes any value of type t a and returns the values of type a contained in the t; and

• the type of a.

For example:

const show = require ('sanctuary-show');
const type = require ('sanctuary-type-identifiers');

//    maybeTypeIdent :: String
const maybeTypeIdent = 'my-package/Maybe';

//    Maybe :: Type -> Type
const Maybe = $.UnaryType
  ('Maybe')
  ('http://example.com/my-package#Maybe')
  ([])
  (x => type (x) === maybeTypeIdent)
  (maybe => maybe.isJust ? [maybe.value] : []);

//    Nothing :: Maybe a
const Nothing = {
  'isJust': false,
  'isNothing': true,
  '@@type': maybeTypeIdent,
  '@@show': () => 'Nothing',
};

//    Just :: a -> Maybe a
const Just = x => ({
  'isJust': true,
  'isNothing': false,
  '@@type': maybeTypeIdent,
  '@@show': () => `Just (${show (x)})`,
  'value': x,
});

//    fromMaybe :: a -> Maybe a -> a
const fromMaybe =
def ('fromMaybe')
    ({})
    ([a, Maybe (a), a])
    (x => m => m.isJust ? m.value : x);

fromMaybe (0) (Just (42));
// => 42

fromMaybe (0) (Nothing);
// => 0

fromMaybe (0) (Just ('XXX'));
// ! TypeError: Type-variable constraint violation
//
//   fromMaybe :: a -> Maybe a -> a
//                ^          ^
//                1          2
//
//   1)  0 :: Number
//
//   2)  "XXX" :: String
//
//   Since there is no type of which all the above values are members, the type-variable constraint has been violated.

BinaryType :: Foldable f => String -⁠> String -⁠> Array Type -⁠> (Any -⁠> Boolean) -⁠> (t a b -⁠> f a) -⁠> (t a b -⁠> f b) -⁠> Type -⁠> Type -⁠> Type

Type constructor for types with two type variables (such as Array2).

To define a binary type t a b one must provide:

• the name of t (exposed as t.name);

• the documentation URL of t (exposed as t.url);

• an array of supertypes (exposed as t.supertypes);

• a predicate that accepts any value that is a member of every one of the given supertypes, and returns true if (and only if) the value is a member of t x y for some types x and y;

• a function that takes any value of type t a b and returns the values of type a contained in the t;

• a function that takes any value of type t a b and returns the values of type b contained in the t;

• the type of a; and

• the type of b.

For example:

const type = require ('sanctuary-type-identifiers');

//    pairTypeIdent :: String
const pairTypeIdent = 'my-package/Pair';

//    $Pair :: Type -> Type -> Type
const $Pair = $.BinaryType
  ('Pair')
  ('http://example.com/my-package#Pair')
  ([])
  (x => type (x) === pairTypeIdent)
  (({fst}) => [fst])
  (({snd}) => [snd]);

//    Pair :: a -> b -> Pair a b
const Pair =
def ('Pair')
    ({})
    ([a, b, $Pair (a) (b)])
    (fst => snd => ({
       'fst': fst,
       'snd': snd,
       '@@type': pairTypeIdent,
       '@@show': () => `Pair (${show (fst)}) (${show (snd)})`,
     }));

//    Rank :: Type
const Rank = $.NullaryType
  ('Rank')
  ('http://example.com/my-package#Rank')
  ([$.String])
  (x => /^(A|2|3|4|5|6|7|8|9|10|J|Q|K)$/.test (x));

//    Suit :: Type
const Suit = $.NullaryType
  ('Suit')
  ('http://example.com/my-package#Suit')
  ([$.String])
  (x => /^[\u2660\u2663\u2665\u2666]$/.test (x));

//    Card :: Type
const Card = $Pair (Rank) (Suit);

//    showCard :: Card -> String
const showCard =
def ('showCard')
    ({})
    ([Card, $.String])
    (card => card.fst + card.snd);

showCard (Pair ('A') ('♠'));
// => 'A♠'

showCard (Pair ('X') ('♠'));
// ! TypeError: Invalid value
//
//   showCard :: Pair Rank Suit -> String
//                    ^^^^
//                     1
//
//   1)  "X" :: String
//
//   The value at position 1 is not a member of ‘Rank’.
//
//   See http://example.com/my-package#Rank for information about the Rank type.

EnumType :: String -⁠> String -⁠> Array Any -⁠> Type

Type constructor for enumerated types (such as RegexFlags).

To define an enumerated type t one must provide:

• the name of t (exposed as t.name);

• the documentation URL of t (exposed as t.url); and

• an array of distinct values.

For example:

//    Denomination :: Type
const Denomination = $.EnumType
  ('Denomination')
  ('http://example.com/my-package#Denomination')
  ([10, 20, 50, 100, 200]);

RecordType :: StrMap Type -⁠> Type

RecordType is used to construct anonymous record types. The type definition specifies the name and type of each required field. A field is an enumerable property (either an own property or an inherited property).

To define an anonymous record type one must provide:

• an object mapping field name to type.

For example:

//    Point :: Type
const Point = $.RecordType ({x: $.FiniteNumber, y: $.FiniteNumber});

//    dist :: Point -> Point -> FiniteNumber
const dist =
def ('dist')
    ({})
    ([Point, Point, $.FiniteNumber])
    (p => q => Math.sqrt (Math.pow (p.x - q.x, 2) +
                          Math.pow (p.y - q.y, 2)));

dist ({x: 0, y: 0}) ({x: 3, y: 4});
// => 5

dist ({x: 0, y: 0}) ({x: 3, y: 4, color: 'red'});
// => 5

dist ({x: 0, y: 0}) ({x: NaN, y: NaN});
// ! TypeError: Invalid value
//
//   dist :: { x :: FiniteNumber, y :: FiniteNumber } -> { x :: FiniteNumber, y :: FiniteNumber } -> FiniteNumber
//                                                              ^^^^^^^^^^^^
//                                                                   1
//
//   1)  NaN :: Number
//
//   The value at position 1 is not a member of ‘FiniteNumber’.

dist (0);
// ! TypeError: Invalid value
//
//   dist :: { x :: FiniteNumber, y :: FiniteNumber } -> { x :: FiniteNumber, y :: FiniteNumber } -> FiniteNumber
//           ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
//                              1
//
//   1)  0 :: Number
//
//   The value at position 1 is not a member of ‘{ x :: FiniteNumber, y :: FiniteNumber }’.

NamedRecordType :: NonEmpty String -⁠> String -⁠> Array Type -⁠> StrMap Type -⁠> Type

NamedRecordType is used to construct named record types. The type definition specifies the name and type of each required field. A field is an enumerable property (either an own property or an inherited property).

To define a named record type t one must provide:

• the name of t (exposed as t.name);

• the documentation URL of t (exposed as t.url);

• an array of supertypes (exposed as t.supertypes); and

• an object mapping field name to type.

For example:

//    Circle :: Type
const Circle = $.NamedRecordType
  ('my-package/Circle')
  ('http://example.com/my-package#Circle')
  ([])
  ({radius: $.PositiveFiniteNumber});

//    Cylinder :: Type
const Cylinder = $.NamedRecordType
  ('Cylinder')
  ('http://example.com/my-package#Cylinder')
  ([Circle])
  ({height: $.PositiveFiniteNumber});

//    volume :: Cylinder -> PositiveFiniteNumber
const volume =
def ('volume')
    ({})
    ([Cylinder, $.FiniteNumber])
    (cyl => Math.PI * cyl.radius * cyl.radius * cyl.height);

volume ({radius: 2, height: 10});
// => 125.66370614359172

volume ({radius: 2});
// ! TypeError: Invalid value
//
//   volume :: Cylinder -> FiniteNumber
//             ^^^^^^^^
//                1
//
//   1)  {"radius": 2} :: Object, StrMap Number
//
//   The value at position 1 is not a member of ‘Cylinder’.
//
//   See http://example.com/my-package#Cylinder for information about the Cylinder type.

TypeVariable :: String -⁠> Type

Polymorphism is powerful. Not being able to define a function for all types would be very limiting indeed: one couldn't even define the identity function!

Before defining a polymorphic function one must define one or more type variables:

const a = $.TypeVariable ('a');
const b = $.TypeVariable ('b');

//    id :: a -> a
const id = def ('id') ({}) ([a, a]) (x => x);

id (42);
// => 42

id (null);
// => null

The same type variable may be used in multiple positions, creating a constraint:

//    cmp :: a -> a -> Number
const cmp =
def ('cmp')
    ({})
    ([a, a, $.Number])
    (x => y => x < y ? -1 : x > y ? 1 : 0);

cmp (42) (42);
// => 0

cmp ('a') ('z');
// => -1

cmp ('z') ('a');
// => 1

cmp (0) ('1');
// ! TypeError: Type-variable constraint violation
//
//   cmp :: a -> a -> Number
//          ^    ^
//          1    2
//
//   1)  0 :: Number
//
//   2)  "1" :: String
//
//   Since there is no type of which all the above values are members, the type-variable constraint has been violated.

UnaryTypeVariable :: String -⁠> Type -⁠> Type

Combines UnaryType and TypeVariable.

To define a unary type variable t a one must provide:

• a name (conventionally matching ^[a-z]$); and

• the type of a.

Consider the type of a generalized map:

map :: Functor f => (a -> b) -> f a -> f b

f is a unary type variable. With two (nullary) type variables, one unary type variable, and one type class it's possible to define a fully polymorphic map function:

const $ = require ('sanctuary-def');
const Z = require ('sanctuary-type-classes');

const a = $.TypeVariable ('a');
const b = $.TypeVariable ('b');
const f = $.UnaryTypeVariable ('f');

//    map :: Functor f => (a -> b) -> f a -> f b
const map =
def ('map')
    ({f: [Z.Functor]})
    ([$.Function ([a, b]), f (a), f (b)])
    (f => functor => Z.map (f, functor));

Whereas a regular type variable is fully resolved (a might become Array (Array String), for example), a unary type variable defers to its type argument, which may itself be a type variable. The type argument corresponds to the type argument of a unary type or the second type argument of a binary type. The second type argument of Map k v, for example, is v. One could replace Functor => f with Map k or with Map Integer, but not with Map.

This shallow inspection makes it possible to constrain a value's "outer" and "inner" types independently.

BinaryTypeVariable :: String -⁠> Type -⁠> Type -⁠> Type

Combines BinaryType and TypeVariable.

To define a binary type variable t a b one must provide:

• a name (conventionally matching ^[a-z]$);

• the type of a; and

• the type of b.

The more detailed explanation of UnaryTypeVariable also applies to BinaryTypeVariable.

Thunk :: Type -⁠> Type

$.Thunk (T) is shorthand for $.Function ([T]), the type comprising every nullary function (thunk) that returns a value of type T.

Predicate :: Type -⁠> Type

$.Predicate (T) is shorthand for $.Fn (T) ($.Boolean), the type comprising every predicate function that takes a value of type T.

Type classes

One can trivially define a function of type String -> String -> String that concatenates two strings. This is overly restrictive, though, since other types support concatenation (Array a, for example).

One could use a type variable to define a polymorphic "concat" function:

//    _concat :: a -> a -> a
const _concat =
def ('_concat')
    ({})
    ([a, a, a])
    (x => y => x.concat (y));

_concat ('fizz') ('buzz');
// => 'fizzbuzz'

_concat ([1, 2]) ([3, 4]);
// => [1, 2, 3, 4]

_concat ([1, 2]) ('buzz');
// ! TypeError: Type-variable constraint violation
//
//   _concat :: a -> a -> a
//              ^    ^
//              1    2
//
//   1)  [1, 2] :: Array Number
//
//   2)  "buzz" :: String
//
//   Since there is no type of which all the above values are members, the type-variable constraint has been violated.

The type of _concat is misleading: it suggests that it can operate on any two values of any one type. In fact there's an implicit constraint, since the type must support concatenation (in mathematical terms, the type must have a semigroup). Violating this implicit constraint results in a run-time error in the implementation:

_concat (null) (null);
// ! TypeError: Cannot read property 'concat' of null

The solution is to constrain a by first defining a TypeClass value, then specifying the constraint in the definition of the "concat" function:

const Z = require ('sanctuary-type-classes');

//    Semigroup :: TypeClass
const Semigroup = Z.TypeClass (
  'my-package/Semigroup',
  'http://example.com/my-package#Semigroup',
  [],
  x => x != null && typeof x.concat === 'function'
);

//    concat :: Semigroup a => a -> a -> a
const concat =
def ('concat')
    ({a: [Semigroup]})
    ([a, a, a])
    (x => y => x.concat (y));

concat ([1, 2]) ([3, 4]);
// => [1, 2, 3, 4]

concat (null) (null);
// ! TypeError: Type-class constraint violation
//
//   concat :: Semigroup a => a -> a -> a
//             ^^^^^^^^^^^    ^
//                            1
//
//   1)  null :: Null
//
//   ‘concat’ requires ‘a’ to satisfy the Semigroup type-class constraint; the value at position 1 does not.
//
//   See http://example.com/my-package#Semigroup for information about the my-package/Semigroup type class.

Multiple constraints may be placed on a type variable by including multiple TypeClass values in the array (e.g. {a: [Foo, Bar, Baz]}).