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Description

Specification for interoperability of common algebraic structures in JavaScript

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Fantasy Land Specification

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(aka "Algebraic JavaScript Specification")

This project specifies interoperability of common algebraic structures:

General

An algebra is a set of values, a set of operators that it is closed under and some laws it must obey.

Each Fantasy Land algebra is a separate specification. An algebra may have dependencies on other algebras which must be implemented.

Terminology

  1. "value" is any JavaScript value, including any which have the structures defined below.
  2. "equivalent" is an appropriate definition of equivalence for the given value. The definition should ensure that the two values can be safely swapped out in a program that respects abstractions. For example:
    • Two lists are equivalent if they are equivalent at all indices.
    • Two plain old JavaScript objects, interpreted as dictionaries, are equivalent when they are equivalent for all keys.
    • Two promises are equivalent when they yield equivalent values.
    • Two functions are equivalent if they yield equivalent outputs for equivalent inputs.

Type signature notation

The type signature notation used in this document is described below:1

  • ::
    "is a member of".
    • e :: t
      can be read as: "the expression
      e
      is a member of type
      t
      ".
    • true :: Boolean
      - "
      true
      is a member of type
      Boolean
      ".
    • 42 :: Integer, Number
      - "
      42
      is a member of the
      Integer
      and
      Number
      types".
  • New types can be created via type constructors.
    • Type constructors can take zero or more type arguments.
    • Array
      is a type constructor which takes one type argument.
    • Array String
      is the type of all arrays of strings. Each of the following has type
      Array String
      :
      []
      ,
      ['foo', 'bar', 'baz']
      .
    • Array (Array String)
      is the type of all arrays of arrays of strings. Each of the following has type
      Array (Array String)
      :
      []
      ,
      [ [], []
      ]
      ,
      [ [], ['foo'], ['bar', 'baz'] ]
      .
  • Lowercase letters stand for type variables.
    • Type variables can take any type unless they have been restricted by means of type constraints (see fat arrow below).
  • ->
    (arrow) Function type constructor.
    • ->
      is an infix type constructor that takes two type arguments where left argument is the input type and the right argument is the output type.
    • ->
      's input type can be a grouping of types to create the type of a function which accepts zero or more arguments. The syntax is:
      () -> 
      , where
       comprises zero
      or more comma–space (
      ,
      )-separated type representations and parens may be omitted for unary functions.
    • String -> Array String
      is a type satisfied by functions which take a
      String
      and return an
      Array String
      .
    • String -> Array String -> Array String
      is a type satisfied by functions which take a
      String
      and return a function which takes an
      Array String
      and returns an
      Array String
      .
    • (String, Array String) -> Array String
      is a type satisfied by functions which take a
      String
      and an
      Array String
      as arguments and return an
      Array String
      .
    • () -> Number
      is a type satisfied by functions which do not take arguments and return a
      Number
      .
  • ~>
    (squiggly arrow) Method type constructor.
    • When a function is a property of an Object, it is called a method. All methods have an implicit parameter type - the type of which they are a property.
    • a ~> a -> a
      is a type satisfied by methods on Objects of type
      a
      which take a type
      a
      as an argument and return a value of type
      a
      .
  • =>
    (fat arrow) Expresses constraints on type variables.
    • In
      a ~> a -> a
      (see squiggly arrow above),
      a
      can be of any type.
      Semigroup a => a ~> a -> a
      adds a constraint such that the type
      a
      must now satisfy the
      Semigroup
      typeclass. To satisfy a typeclass means to lawfully implement all functions/methods specified by that typeclass.

For example:

fantasy-land/traverse :: Applicative f, Traversable t => t a ~> (TypeRep f, a -> f b) -> f (t b)
'-------------------'    '--------------------------'    '-'    '-------------------'    '-----'
 '                        '                               '      '                        '
 '                        ' - type constraints            '      ' - argument types       ' - return type
 '                                                        '
 '- method name                                           ' - method target type

  1. See the Types section in Sanctuary's docs for more info.

Type representatives

Certain behaviours are defined from the perspective of a member of a type. Other behaviours do not require a member. Thus certain algebras require a type to provide a value-level representative (with certain properties). The Identity type, for example, could provide

Id
as its type representative:
Id :: TypeRep Identity
.

If a type provides a type representative, each member of the type must have a

constructor
property which is a reference to the type representative.

Algebras

Setoid

  1. a['fantasy-land/equals'](a) === true
    (reflexivity)
  2. a['fantasy-land/equals'](b) === b['fantasy-land/equals'](a)
    (symmetry)
  3. If
    a['fantasy-land/equals'](b)
    and
    b['fantasy-land/equals'](c)
    , then
    a['fantasy-land/equals'](c)
    (transitivity)

fantasy-land/equals
method

fantasy-land/equals :: Setoid a => a ~> a -> Boolean

A value which has a Setoid must provide a

fantasy-land/equals
method. The
fantasy-land/equals
method takes one argument:
a['fantasy-land/equals'](b)
  1. b
    must be a value of the same Setoid
    1. If
      b
      is not the same Setoid, behaviour of
      fantasy-land/equals
      is unspecified (returning
      false
      is recommended).
  2. fantasy-land/equals
    must return a boolean (
    true
    or
    false
    ).

Ord

A value that implements the Ord specification must also implement the Setoid specification.

  1. a['fantasy-land/lte'](b)
    or
    b['fantasy-land/lte'](a)
    (totality)
  2. If
    a['fantasy-land/lte'](b)
    and
    b['fantasy-land/lte'](a)
    , then
    a['fantasy-land/equals'](b)
    (antisymmetry)
  3. If
    a['fantasy-land/lte'](b)
    and
    b['fantasy-land/lte'](c)
    , then
    a['fantasy-land/lte'](c)
    (transitivity)

fantasy-land/lte
method

fantasy-land/lte :: Ord a => a ~> a -> Boolean

A value which has an Ord must provide a

fantasy-land/lte
method. The
fantasy-land/lte
method takes one argument:
 a['fantasy-land/lte'](b)
  1. b
    must be a value of the same Ord
    1. If
      b
      is not the same Ord, behaviour of
      fantasy-land/lte
      is unspecified (returning
      false
      is recommended).
  2. fantasy-land/lte
    must return a boolean (
    true
    or
    false
    ).

Semigroupoid

  1. a['fantasy-land/compose'](b)['fantasy-land/compose'](c) === a['fantasy-land/compose'](b['fantasy-land/compose'](c))
    (associativity)

fantasy-land/compose
method

fantasy-land/compose :: Semigroupoid c => c i j ~> c j k -> c i k

A value which has a Semigroupoid must provide a

fantasy-land/compose
method. The
fantasy-land/compose
method takes one argument:
a['fantasy-land/compose'](b)
  1. b
    must be a value of the same Semigroupoid
    1. If
      b
      is not the same semigroupoid, behaviour of
      fantasy-land/compose
      is unspecified.
  2. fantasy-land/compose
    must return a value of the same Semigroupoid.

Category

A value that implements the Category specification must also implement the Semigroupoid specification.

  1. a['fantasy-land/compose'](C['fantasy-land/id']())
    is equivalent to
    a
    (right identity)
  2. C['fantasy-land/id']()['fantasy-land/compose'](a)
    is equivalent to
    a
    (left identity)

fantasy-land/id
method

fantasy-land/id :: Category c => () -> c a a

A value which has a Category must provide a

fantasy-land/id
function on its type representative:
C['fantasy-land/id']()

Given a value

c
, one can access its type representative via the
constructor
property:
c.constructor['fantasy-land/id']()
  1. fantasy-land/id
    must return a value of the same Category

Semigroup

  1. a['fantasy-land/concat'](b)['fantasy-land/concat'](c)
    is equivalent to
    a['fantasy-land/concat'](b['fantasy-land/concat'](c))
    (associativity)

fantasy-land/concat
method

fantasy-land/concat :: Semigroup a => a ~> a -> a

A value which has a Semigroup must provide a

fantasy-land/concat
method. The
fantasy-land/concat
method takes one argument:
s['fantasy-land/concat'](b)
  1. b
    must be a value of the same Semigroup
    1. If
      b
      is not the same semigroup, behaviour of
      fantasy-land/concat
      is unspecified.
  2. fantasy-land/concat
    must return a value of the same Semigroup.

Monoid

A value that implements the Monoid specification must also implement the Semigroup specification.

  1. m['fantasy-land/concat'](M['fantasy-land/empty']())
    is equivalent to
    m
    (right identity)
  2. M['fantasy-land/empty']()['fantasy-land/concat'](m)
    is equivalent to
    m
    (left identity)

fantasy-land/empty
method

fantasy-land/empty :: Monoid m => () -> m

A value which has a Monoid must provide a

fantasy-land/empty
function on its type representative:
M['fantasy-land/empty']()

Given a value

m
, one can access its type representative via the
constructor
property:
m.constructor['fantasy-land/empty']()
  1. fantasy-land/empty
    must return a value of the same Monoid

Group

A value that implements the Group specification must also implement the Monoid specification.

  1. g['fantasy-land/concat'](g['fantasy-land/invert']())
    is equivalent to
    g.constructor['fantasy-land/empty']()
    (right inverse)
  2. g['fantasy-land/invert']()['fantasy-land/concat'](g)
    is equivalent to
    g.constructor['fantasy-land/empty']()
    (left inverse)

fantasy-land/invert
method

fantasy-land/invert :: Group g => g ~> () -> g

A value which has a Group must provide a

fantasy-land/invert
method. The
fantasy-land/invert
method takes no arguments:
g['fantasy-land/invert']()
  1. fantasy-land/invert
    must return a value of the same Group.

Filterable

  1. v['fantasy-land/filter'](x => p(x) && q(x))
    is equivalent to
    v['fantasy-land/filter'](p)['fantasy-land/filter'](q)
    (distributivity)
  2. v['fantasy-land/filter'](x => true)
    is equivalent to
    v
    (identity)
  3. v['fantasy-land/filter'](x => false)
    is equivalent to
    w['fantasy-land/filter'](x => false)
    if
    v
    and
    w
    are values of the same Filterable (annihilation)

fantasy-land/filter
method

fantasy-land/filter :: Filterable f => f a ~> (a -> Boolean) -> f a

A value which has a Filterable must provide a

fantasy-land/filter
method. The
fantasy-land/filter
method takes one argument:
v['fantasy-land/filter'](p)
  1. p
    must be a function.
    1. If
      p
      is not a function, the behaviour of
      fantasy-land/filter
      is unspecified.
    2. p
      must return either
      true
      or
      false
      . If it returns any other value, the behaviour of
      fantasy-land/filter
      is unspecified.
  2. fantasy-land/filter
    must return a value of the same Filterable.

Functor

  1. u['fantasy-land/map'](a => a)
    is equivalent to
    u
    (identity)
  2. u['fantasy-land/map'](x => f(g(x)))
    is equivalent to
    u['fantasy-land/map'](g)['fantasy-land/map'](f)
    (composition)

fantasy-land/map
method

fantasy-land/map :: Functor f => f a ~> (a -> b) -> f b

A value which has a Functor must provide a

fantasy-land/map
method. The
fantasy-land/map
method takes one argument:
u['fantasy-land/map'](f)
  1. f
    must be a function,
    1. If
      f
      is not a function, the behaviour of
      fantasy-land/map
      is unspecified.
    2. f
      can return any value.
    3. No parts of
      f
      's return value should be checked.
  2. fantasy-land/map
    must return a value of the same Functor

Contravariant

  1. u['fantasy-land/contramap'](a => a)
    is equivalent to
    u
    (identity)
  2. u['fantasy-land/contramap'](x => f(g(x)))
    is equivalent to
    u['fantasy-land/contramap'](f)['fantasy-land/contramap'](g)
    (composition)

fantasy-land/contramap
method

fantasy-land/contramap :: Contravariant f => f a ~> (b -> a) -> f b

A value which has a Contravariant must provide a

fantasy-land/contramap
method. The
fantasy-land/contramap
method takes one argument:
u['fantasy-land/contramap'](f)
  1. f
    must be a function,
    1. If
      f
      is not a function, the behaviour of
      fantasy-land/contramap
      is unspecified.
    2. f
      can return any value.
    3. No parts of
      f
      's return value should be checked.
  2. fantasy-land/contramap
    must return a value of the same Contravariant

Apply

A value that implements the Apply specification must also implement the Functor specification.

  1. v['fantasy-land/ap'](u['fantasy-land/ap'](a['fantasy-land/map'](f => g => x => f(g(x)))))
    is equivalent to
    v['fantasy-land/ap'](u)['fantasy-land/ap'](a)
    (composition)

fantasy-land/ap
method

fantasy-land/ap :: Apply f => f a ~> f (a -> b) -> f b

A value which has an Apply must provide a

fantasy-land/ap
method. The
fantasy-land/ap
method takes one argument:
a['fantasy-land/ap'](b)
  1. b
    must be an Apply of a function
    1. If
      b
      does not represent a function, the behaviour of
      fantasy-land/ap
      is unspecified.
    2. b
      must be same Apply as
      a
      .
  2. a
    must be an Apply of any value
  3. fantasy-land/ap
    must apply the function in Apply
    b
    to the value in Apply
    a
    1. No parts of return value of that function should be checked.
  4. The

    Apply
    returned by
    fantasy-land/ap
    must be the same as
    a
    and
    b

Applicative

A value that implements the Applicative specification must also implement the Apply specification.

  1. v['fantasy-land/ap'](A['fantasy-land/of'](x => x))
    is equivalent to
    v
    (identity)
  2. A['fantasy-land/of'](x)['fantasy-land/ap'](A['fantasy-land/of'](f))
    is equivalent to
    A['fantasy-land/of'](f(x))
    (homomorphism)
  3. A['fantasy-land/of'](y)['fantasy-land/ap'](u)
    is equivalent to
    u['fantasy-land/ap'](A['fantasy-land/of'](f => f(y)))
    (interchange)

fantasy-land/of
method

fantasy-land/of :: Applicative f => a -> f a

A value which has an Applicative must provide a

fantasy-land/of
function on its type representative. The
fantasy-land/of
function takes one argument:
F['fantasy-land/of'](a)

Given a value

f
, one can access its type representative via the
constructor
property:
f.constructor['fantasy-land/of'](a)
  1. fantasy-land/of
    must provide a value of the same Applicative
    1. No parts of
      a
      should be checked

Alt

A value that implements the Alt specification must also implement the Functor specification.

  1. a['fantasy-land/alt'](b)['fantasy-land/alt'](c)
    is equivalent to
    a['fantasy-land/alt'](b['fantasy-land/alt'](c))
    (associativity)
  2. a['fantasy-land/alt'](b)['fantasy-land/map'](f)
    is equivalent to
    a['fantasy-land/map'](f)['fantasy-land/alt'](b['fantasy-land/map'](f))
    (distributivity)

fantasy-land/alt
method

fantasy-land/alt :: Alt f => f a ~> f a -> f a

A value which has a Alt must provide a

fantasy-land/alt
method. The
fantasy-land/alt
method takes one argument:
a['fantasy-land/alt'](b)
  1. b
    must be a value of the same Alt
    1. If
      b
      is not the same Alt, behaviour of
      fantasy-land/alt
      is unspecified.
    2. a
      and
      b
      can contain any value of same type.
    3. No parts of
      a
      's and
      b
      's containing value should be checked.
  2. fantasy-land/alt
    must return a value of the same Alt.

Plus

A value that implements the Plus specification must also implement the Alt specification.

  1. x['fantasy-land/alt'](A['fantasy-land/zero']())
    is equivalent to
    x
    (right identity)
  2. A['fantasy-land/zero']()['fantasy-land/alt'](x)
    is equivalent to
    x
    (left identity)
  3. A['fantasy-land/zero']()['fantasy-land/map'](f)
    is equivalent to
    A['fantasy-land/zero']()
    (annihilation)

fantasy-land/zero
method

fantasy-land/zero :: Plus f => () -> f a

A value which has a Plus must provide a

fantasy-land/zero
function on its type representative:
A['fantasy-land/zero']()

Given a value

x
, one can access its type representative via the
constructor
property:
x.constructor['fantasy-land/zero']()
  1. fantasy-land/zero
    must return a value of the same Plus

Alternative

A value that implements the Alternative specification must also implement the Applicative and Plus specifications.

  1. x['fantasy-land/ap'](f['fantasy-land/alt'](g))
    is equivalent to
    x['fantasy-land/ap'](f)['fantasy-land/alt'](x['fantasy-land/ap'](g))
    (distributivity)
  2. x['fantasy-land/ap'](A['fantasy-land/zero']())
    is equivalent to
    A['fantasy-land/zero']()
    (annihilation)

Foldable

  1. u['fantasy-land/reduce']
    is equivalent to
    u['fantasy-land/reduce']((acc, x) => acc.concat([x]), []).reduce

fantasy-land/reduce
method

fantasy-land/reduce :: Foldable f => f a ~> ((b, a) -> b, b) -> b

A value which has a Foldable must provide a

fantasy-land/reduce
method. The
fantasy-land/reduce
method takes two arguments:
u['fantasy-land/reduce'](f, x)
  1. f
    must be a binary function
    1. if
      f
      is not a function, the behaviour of
      fantasy-land/reduce
      is unspecified.
    2. The first argument to
      f
      must be the same type as
      x
      .
    3. f
      must return a value of the same type as
      x
      .
    4. No parts of
      f
      's return value should be checked.
  2. x
    is the initial accumulator value for the reduction
    1. No parts of
      x
      should be checked.

Traversable

A value that implements the Traversable specification must also implement the Functor and Foldable specifications.

  1. t(u['fantasy-land/traverse'](F, x => x))
    is equivalent to
    u['fantasy-land/traverse'](G, t)
    for any
    t
    such that
    t(a)['fantasy-land/map'](f)
    is equivalent to
    t(a['fantasy-land/map'](f))
    (naturality)
  2. u['fantasy-land/traverse'](F, F['fantasy-land/of'])
    is equivalent to
    F['fantasy-land/of'](u)
    for any Applicative
    F
    (identity)
  3. u['fantasy-land/traverse'](Compose, x => new Compose(x))
    is equivalent to
    new Compose(u['fantasy-land/traverse'](F, x => x)['fantasy-land/map'](x => x['fantasy-land/traverse'](G, x => x)))
    for
    Compose
    defined below and any Applicatives
    F
    and
    G
    (composition)
function Compose(c) {
  this.c = c;
}

Compose['fantasy-land/of'] = function(x) { return new Compose(F'fantasy-land/of'); };

Compose.prototype['fantasy-land/ap'] = function(f) { return new Compose(this.c['fantasy-land/ap'](f.c['fantasy-land/map'](u => y => y'fantasy-land/ap'))); };

Compose.prototype['fantasy-land/map'] = function(f) { return new Compose(this.c['fantasy-land/map'](y => y'fantasy-land/map')); };

fantasy-land/traverse
method

fantasy-land/traverse :: Applicative f, Traversable t => t a ~> (TypeRep f, a -> f b) -> f (t b)

A value which has a Traversable must provide a

fantasy-land/traverse
method. The
fantasy-land/traverse
method takes two arguments:
u['fantasy-land/traverse'](A, f)
  1. A
    must be the type representative of an Applicative.
  2. f
    must be a function which returns a value
    1. If
      f
      is not a function, the behaviour of
      fantasy-land/traverse
      is unspecified.
    2. f
      must return a value of the type represented by
      A
      .
  3. fantasy-land/traverse
    must return a value of the type represented by
    A
    .

Chain

A value that implements the Chain specification must also implement the Apply specification.

  1. m['fantasy-land/chain'](f)['fantasy-land/chain'](g)
    is equivalent to
    m['fantasy-land/chain'](x => f(x)['fantasy-land/chain'](g))
    (associativity)

fantasy-land/chain
method

fantasy-land/chain :: Chain m => m a ~> (a -> m b) -> m b

A value which has a Chain must provide a

fantasy-land/chain
method. The
fantasy-land/chain
method takes one argument:
m['fantasy-land/chain'](f)
  1. f
    must be a function which returns a value
    1. If
      f
      is not a function, the behaviour of
      fantasy-land/chain
      is unspecified.
    2. f
      must return a value of the same Chain
  2. fantasy-land/chain
    must return a value of the same Chain

ChainRec

A value that implements the ChainRec specification must also implement the Chain specification.

  1. M['fantasy-land/chainRec']((next, done, v) => p(v) ? d(v)['fantasy-land/map'](done) : n(v)['fantasy-land/map'](next), i)
    is equivalent to
    (function step(v) { return p(v) ? d(v) : n(v)['fantasy-land/chain'](step); }(i))
    (equivalence)
  2. Stack usage of
    M['fantasy-land/chainRec'](f, i)
    must be at most a constant multiple of the stack usage of
    f
    itself.

fantasy-land/chainRec
method

fantasy-land/chainRec :: ChainRec m => ((a -> c, b -> c, a) -> m c, a) -> m b

A Type which has a ChainRec must provide a

fantasy-land/chainRec
function on its type representative. The
fantasy-land/chainRec
function takes two arguments:
M['fantasy-land/chainRec'](f, i)

Given a value

m
, one can access its type representative via the
constructor
property:
m.constructor['fantasy-land/chainRec'](f, i)
  1. f
    must be a function which returns a value
    1. If
      f
      is not a function, the behaviour of
      fantasy-land/chainRec
      is unspecified.
    2. f
      takes three arguments
      next
      ,
      done
      ,
      value
      1. next
        is a function which takes one argument of same type as
        i
        and can return any value
      2. done
        is a function which takes one argument and returns the same type as the return value of
        next
      3. value
        is some value of the same type as
        i
    3. f
      must return a value of the same ChainRec which contains a value returned from either
      done
      or
      next
  2. fantasy-land/chainRec
    must return a value of the same ChainRec which contains a value of same type as argument of
    done

Monad

A value that implements the Monad specification must also implement the Applicative and Chain specifications.

  1. M['fantasy-land/of'](a)['fantasy-land/chain'](f)
    is equivalent to
    f(a)
    (left identity)
  2. m['fantasy-land/chain'](M['fantasy-land/of'])
    is equivalent to
    m
    (right identity)

Extend

A value that implements the Extend specification must also implement the Functor specification.

  1. w['fantasy-land/extend'](g)['fantasy-land/extend'](f)
    is equivalent to
    w['fantasy-land/extend'](_w => f(_w['fantasy-land/extend'](g)))

fantasy-land/extend
method

fantasy-land/extend :: Extend w => w a ~> (w a -> b) -> w b

An Extend must provide a

fantasy-land/extend
method. The
fantasy-land/extend
method takes one argument:
 w['fantasy-land/extend'](f)
  1. f
    must be a function which returns a value
    1. If
      f
      is not a function, the behaviour of
      fantasy-land/extend
      is unspecified.
    2. f
      must return a value of type
      v
      , for some variable
      v
      contained in
      w
      .
    3. No parts of
      f
      's return value should be checked.
  2. fantasy-land/extend
    must return a value of the same Extend.

Comonad

A value that implements the Comonad specification must also implement the Extend specification.

  1. w['fantasy-land/extend'](_w => _w['fantasy-land/extract']())
    is equivalent to
    w
    (left identity)
  2. w['fantasy-land/extend'](f)['fantasy-land/extract']()
    is equivalent to
    f(w)
    (right identity)

fantasy-land/extract
method

fantasy-land/extract :: Comonad w => w a ~> () -> a

A value which has a Comonad must provide a

fantasy-land/extract
method on itself. The
fantasy-land/extract
method takes no arguments:
w['fantasy-land/extract']()
  1. fantasy-land/extract
    must return a value of type
    v
    , for some variable
    v
    contained in
    w
    .
    1. v
      must have the same type that
      f
      returns in
      fantasy-land/extend
      .

Bifunctor

A value that implements the Bifunctor specification must also implement the Functor specification.

  1. p['fantasy-land/bimap'](a => a, b => b)
    is equivalent to
    p
    (identity)
  2. p['fantasy-land/bimap'](a => f(g(a)), b => h(i(b))
    is equivalent to
    p['fantasy-land/bimap'](g, i)['fantasy-land/bimap'](f, h)
    (composition)

fantasy-land/bimap
method

fantasy-land/bimap :: Bifunctor f => f a c ~> (a -> b, c -> d) -> f b d

A value which has a Bifunctor must provide a

fantasy-land/bimap
method. The
fantasy-land/bimap
method takes two arguments:
c['fantasy-land/bimap'](f, g)
  1. f
    must be a function which returns a value
    1. If
      f
      is not a function, the behaviour of
      fantasy-land/bimap
      is unspecified.
    2. f
      can return any value.
    3. No parts of
      f
      's return value should be checked.
  2. g
    must be a function which returns a value
    1. If
      g
      is not a function, the behaviour of
      fantasy-land/bimap
      is unspecified.
    2. g
      can return any value.
    3. No parts of
      g
      's return value should be checked.
  3. fantasy-land/bimap
    must return a value of the same Bifunctor.

Profunctor

A value that implements the Profunctor specification must also implement the Functor specification.

  1. p['fantasy-land/promap'](a => a, b => b)
    is equivalent to
    p
    (identity)
  2. p['fantasy-land/promap'](a => f(g(a)), b => h(i(b)))
    is equivalent to
    p['fantasy-land/promap'](f, i)['fantasy-land/promap'](g, h)
    (composition)

fantasy-land/promap
method

fantasy-land/promap :: Profunctor p => p b c ~> (a -> b, c -> d) -> p a d

A value which has a Profunctor must provide a

fantasy-land/promap
method.

The

fantasy-land/promap
method takes two arguments:
c['fantasy-land/promap'](f, g)
  1. f
    must be a function which returns a value
    1. If
      f
      is not a function, the behaviour of
      fantasy-land/promap
      is unspecified.
    2. f
      can return any value.
    3. No parts of
      f
      's return value should be checked.
  2. g
    must be a function which returns a value
    1. If
      g
      is not a function, the behaviour of
      fantasy-land/promap
      is unspecified.
    2. g
      can return any value.
    3. No parts of
      g
      's return value should be checked.
  3. fantasy-land/promap
    must return a value of the same Profunctor

Derivations

When creating data types which satisfy multiple algebras, authors may choose to implement certain methods then derive the remaining methods. Derivations:

  • fantasy-land/equals
    may be derived from
    fantasy-land/lte
    :

    function equals(other) { return this['fantasy-land/lte'](other) && other['fantasy-land/lte'](this); }
    
  • fantasy-land/map
    may be derived from
    fantasy-land/ap
    and
    fantasy-land/of
    :

    function map(f) { return this['fantasy-land/ap'](this.constructor['fantasy-land/of'](f)); }
    
  • fantasy-land/map
    may be derived from
    fantasy-land/chain
    and
    fantasy-land/of
    :

    function map(f) { return this['fantasy-land/chain'](a => this.constructor['fantasy-land/of'](f(a))); }
    
  • fantasy-land/map
    may be derived from
    fantasy-land/bimap
    :

    function map(f) { return this['fantasy-land/bimap'](a => a, f); }
    
  • fantasy-land/map
    may be derived from
    fantasy-land/promap
    :

    function map(f) { return this['fantasy-land/promap'](a => a, f); }
    
  • fantasy-land/ap
    may be derived from
    fantasy-land/chain
    :

    function ap(m) { return m['fantasy-land/chain'](f => this['fantasy-land/map'](f)); }
    
  • fantasy-land/reduce
    may be derived as follows:

    function reduce(f, acc) {
      function Const(value) {
        this.value = value;
      }
      Const['fantasy-land/of'] = function(_) {
        return new Const(acc);
      };
      Const.prototype['fantasy-land/map'] = function(_) {
        return this;
      };
      Const.prototype['fantasy-land/ap'] = function(b) {
        return new Const(f(b.value, this.value));
      };
      return this['fantasy-land/traverse'](x => new Const(x), Const['fantasy-land/of']).value;
    }
    
  • fantasy-land/map
    may be derived as follows:

    function map(f) {
      function Id(value) {
        this.value = value;
      }
      Id['fantasy-land/of'] = function(x) {
        return new Id(x);
      };
      Id.prototype['fantasy-land/map'] = function(f) {
        return new Id(f(this.value));
      };
      Id.prototype['fantasy-land/ap'] = function(b) {
        return new Id(this.value(b.value));
      };
      return this['fantasy-land/traverse'](x => Id['fantasy-land/of'](f(x)), Id['fantasy-land/of']).value;
    }
    
  • fantasy-land/filter
    may be derived from
    fantasy-land/of
    ,
    fantasy-land/chain
    , and
    fantasy-land/zero
    :

    function filter(pred) {
      var F = this.constructor;
      return this['fantasy-land/chain'](x => pred(x) ? F['fantasy-land/of'](x) : F['fantasy-land/zero']());
    }
    
  • fantasy-land/filter
    may be derived from
    fantasy-land/concat
    ,
    fantasy-land/of
    ,
    fantasy-land/zero
    , and
    fantasy-land/reduce
    :

    function filter(pred) {
      var F = this.constructor;
      return this['fantasy-land/reduce']((f, x) => pred(x) ? f['fantasy-land/concat'](F['fantasy-land/of'](x)) : f, F['fantasy-land/zero']());
    }
    

If a data type provides a method which could be derived, its behaviour must be equivalent to that of the derivation (or derivations).

Notes

  1. If there's more than a single way to implement the methods and laws, the implementation should choose one and provide wrappers for other uses.
  2. It's discouraged to overload the specified methods. It can easily result in broken and buggy behaviour.
  3. It is recommended to throw an exception on unspecified behaviour.
  4. An
    Identity
    container which implements many of the methods is provided by sanctuary-identity.

Alternatives

There also exists Static Land Specification with exactly the same ideas as Fantasy Land but based on static methods instead of instance methods.

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