Run-time type system for JavaScript
npm install sanctuary-defsanctuary-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 $:
``javascript`
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:
`javascript
// 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:
`javascript`
// 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][]:
`javascript`
// env :: Array Type
const env = $.env.concat ([List ($.Unknown)]);
The next step is to define a def function for the environment using$.create:
`javascript`
// 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:
`javascript`
// def :: String -> StrMap (Array TypeClass) -> Array Type -> Function -> Function
const def = $.create ({
checkTypes: process.env.NODE_ENV === 'development',
env,
});
def is a function for defining functions. For example:
`javascript`
// add :: Number -> Number -> Number
const add =
def ('add') // name
({}) // type-class constraints
([$.Number, $.Number, $.Number]) // input and output types
(x => y => x + y); // implementation
[$.Number, $.Number, $.Number] specifies that add takes two argumentsNumber
of type , one at a time, and returns a value of type Number.
Applying add to two arguments, one at a time, gives the expected result:
`javascript`
add (2) (2);
// => 4
Applying add to multiple arguments at once results in an exception being
thrown:
`javascript`
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:
`javascript
// 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:
`javascript`
// _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:
`javascript`
_add ('2') ('2');
// => '22'
add, on the other hand, throws if applied to arguments of the wrong
types:
`javascript`
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:
`javascript`
add ('X');
// ! TypeError: Invalid value
//
// add :: Number -> Number -> Number
// ^^^^^^
// 1
//
// 1) "X" :: String
//
// The value at position 1 is not a member of ‘Number’.
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 [],Array ???
for example, is .
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
-
-
-
-
-
-
#### 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][arguments] object.
Constructor for homogeneous Array types.
#### Array0 :: Type
Type whose sole member is [].
Constructor for singleton Array types.
#### Array2 :: Type -> Type -> Type
Constructor for heterogeneous Array types of length 2. ['foo', true] isArray2 String Boolean
a member of .
#### 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 , the type of functions that take a value ofI
type 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
- 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.
Constructor for native Set types. $.JsSet ($.Number), for example,
is the type comprising every native Set whose values are numbers.
[Maybe][] type constructor.
#### Module :: Type
Type comprising every ES module.
#### NonEmpty :: Type -> Type
Constructor for non-empty types. $.NonEmpty ($.String), for example, isString
the type comprising every [][] 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).
Type comprising every [Number][] value greater than zero.
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][min] .. [Number.MAX_SAFE_INTEGER][max]].
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.
Type comprising every [Integer][] value greater than zero.
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][]; ornew
- the 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][].
Type comprising every [RegExp][] value whose global flag is false.
See also [GlobalRegExp][].
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:
- ''
-
-
-
-
-
-
-
#### 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 valuefalse
is a member of the type; otherwise.
The environment is only significant if the type contains
[type variables][].
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 valuet
is a member of .
For example:
`javascript
// 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.
`
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 valuet x
is a member of for some type x;
- a function that takes any value of type t a and returns the valuesa
of type contained in the t; and
- the type of a.
For example:
`javascript
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.
`
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 valuet x y
is a member of for some types x and y;
- a function that takes any value of type t a b and returns thea
values of type contained in the t;
- a function that takes any value of type t a b and returns theb
values of type contained in the t;
- the type of a; and
- the type of b.
For example:
`javascript
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:
`javascript`
// 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:
`javascript
// 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:
`javascript
// 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:
`javascript
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:
`javascript
// 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:
`haskell`
map :: Functor f => (a -> b) -> f a -> f b
f is a unary type variable. With two (nullary) type variables, onemap
unary type variable, and one [type class][] it's possible to define a
fully polymorphic function:
`javascript
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 becomeArray (Array String), for example), a unary type variable defers toMap k v
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 , forv
example, is . One could replace Functor => f with Map k or withMap 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 toBinaryTypeVariable.
$.Thunk (T) is shorthand for $.Function ([T]), the type comprisingT
every nullary function (thunk) that returns a value of type .
#### Predicate :: Type -> Type
$.Predicate (T) is shorthand for $.Fn (T) ($.Boolean), the typeT
comprising every predicate function that takes a value of type .
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 (, for example).
One could use a type variable to define a polymorphic "concat" function:
`javascript
// _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][semigroup]
terms, the type must have a [semigroup][FL:Semigroup]). Violating this
implicit constraint results in a run-time error in the implementation:
`javascript`
_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:
`javascript
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]}).
[Buffer]: https://nodejs.org/api/buffer.html#buffer_buffer
[Descending]: https://github.com/sanctuary-js/sanctuary-descending/tree/v2.1.0
[Either]: https://github.com/sanctuary-js/sanctuary-either/tree/v2.1.0
[FL:Semigroup]: https://github.com/fantasyland/fantasy-land#semigroup
[HTML element]: https://developer.mozilla.org/en-US/docs/Web/HTML/Element
[Identity]: https://github.com/sanctuary-js/sanctuary-identity/tree/v2.1.0
[Maybe]: https://github.com/sanctuary-js/sanctuary-maybe/tree/v2.1.0
[Monoid]: https://github.com/fantasyland/fantasy-land#monoid
[Pair]: https://github.com/sanctuary-js/sanctuary-pair/tree/v2.1.0
[Setoid]: https://github.com/fantasyland/fantasy-land#setoid
[Unknown]: #Unknown
[Array]: #ArrayArray2
[]: #Array2BinaryType
[]: #BinaryTypeDate
[]: #DateFiniteNumber
[]: #FiniteNumberGlobalRegExp
[]: #GlobalRegExpInteger
[]: #IntegerNonGlobalRegExp
[]: #NonGlobalRegExpNumber
[]: #NumberObject.create
[]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Object/createRegExp
[]: #RegExpRegexFlags
[]: #RegexFlagsString
[]: #StringSyntaxError
[]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/SyntaxErrorTypeClass
[]: https://github.com/sanctuary-js/sanctuary-type-classes#TypeClassTypeError
[]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/TypeErrorTypeVariable
[]: #TypeVariableUnaryType
[]: #UnaryTypeUnaryTypeVariable
[]: #UnaryTypeVariableUnknown
[]: #UnknownValidNumber
[]: #ValidNumberenv`]: #env
[
[arguments]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Functions/arguments
[enumerated types]: https://en.wikipedia.org/wiki/Enumerated_type
[max]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Number/MAX_SAFE_INTEGER
[min]: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Number/MIN_SAFE_INTEGER
[semigroup]: https://en.wikipedia.org/wiki/Semigroup
[type class]: #type-classes
[type variables]: #TypeVariable
[types]: #types