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Prelude.purs
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Prelude.purs
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module Prelude
( Unit(..), unit
, ($), (#)
, flip
, const
, asTypeOf
, otherwise
, Semigroupoid, compose, (<<<), (>>>)
, Category, id
, Functor, map, (<$>), (<#>), void
, Apply, apply, (<*>)
, Applicative, pure, liftA1
, Bind, bind, (>>=)
, Monad, return, liftM1, ap
, Semigroup, append, (<>), (++)
, Semiring, add, zero, mul, one, (+), (*)
, ModuloSemiring, div, mod, (/)
, Ring, sub, negate, (-)
, Num
, DivisionRing
, Eq, eq, (==), (/=)
, Ordering(..), Ord, compare, (<), (>), (<=), (>=)
, Bounded, top, bottom
, Lattice, sup, inf, (||), (&&)
, BoundedLattice
, ComplementedLattice, not
, DistributiveLattice
, BooleanAlgebra
, Show, show
) where
-- | The `Unit` type has a single inhabitant, called `unit`. It represents
-- | values with no computational content.
-- |
-- | `Unit` is often used, wrapped in a monadic type constructor, as the
-- | return type of a computation where only
-- | the _effects_ are important.
newtype Unit = Unit {}
-- | `unit` is the sole inhabitant of the `Unit` type.
unit :: Unit
unit = Unit {}
infixr 0 $
infixl 1 #
-- | Applies a function to its argument.
-- |
-- | ```purescript
-- | length $ groupBy productCategory $ filter isInStock $ products
-- | ```
-- |
-- | is equivalent to:
-- |
-- | ```purescript
-- | length (groupBy productCategory (filter isInStock products))
-- | ```
-- |
-- | `($)` is different from [`(#)`](#-2) because it is right-infix instead of
-- | left: `a $ b $ c $ d x = a $ (b $ (c $ (d $ x))) = a (b (c (d x)))`
($) :: forall a b. (a -> b) -> a -> b
($) f x = f x
-- | Applies an argument to a function.
-- |
-- | ```purescript
-- | products # filter isInStock # groupBy productCategory # length
-- | ```
-- |
-- | is equivalent to:
-- |
-- | ```purescript
-- | length (groupBy productCategory (filter isInStock products))
-- | ```
-- |
-- | `(#)` is different from [`($)`](#-1) because it is left-infix instead of
-- | right: `x # a # b # c # d = (((x # a) # b) # c) # d = d (c (b (a x)))`
(#) :: forall a b. a -> (a -> b) -> b
(#) x f = f x
-- | Flips the order of the arguments to a function of two arguments.
-- |
-- | ```purescript
-- | flip const 1 2 = const 2 1 = 2
-- | ```
flip :: forall a b c. (a -> b -> c) -> b -> a -> c
flip f b a = f a b
-- | Returns its first argument and ignores its second.
-- |
-- | ```purescript
-- | const 1 "hello" = 1
-- | ```
const :: forall a b. a -> b -> a
const a _ = a
-- | This function returns its first argument, and can be used to assert type
-- | equalities. This can be useful when types are otherwise ambiguous.
-- |
-- | ```purescript
-- | main = print $ [] `asTypeOf` [0]
-- | ```
-- |
-- | If instead, we had written `main = print []`, the type of the argument
-- | `[]` would have been ambiguous, resulting in a compile-time error.
asTypeOf :: forall a. a -> a -> a
asTypeOf x _ = x
-- | An alias for `true`, which can be useful in guard clauses:
-- |
-- | ```purescript
-- | max x y | x >= y = x
-- | | otherwise = y
-- | ```
otherwise :: Boolean
otherwise = true
infixr 9 >>>
infixr 9 <<<
-- | A `Semigroupoid` is similar to a [`Category`](#category) but does not
-- | require an identity element `id`, just composable morphisms.
-- |
-- | `Semigroupoid`s must satisfy the following law:
-- |
-- | - Associativity: `p <<< (q <<< r) = (p <<< q) <<< r`
-- |
-- | One example of a `Semigroupoid` is the function type constructor `(->)`,
-- | with `(<<<)` defined as function composition.
class Semigroupoid a where
compose :: forall b c d. a c d -> a b c -> a b d
instance semigroupoidFn :: Semigroupoid (->) where
compose f g x = f (g x)
(<<<) :: forall a b c d. (Semigroupoid a) => a c d -> a b c -> a b d
(<<<) = compose
-- | Forwards composition, or `(<<<)` with its arguments reversed.
(>>>) :: forall a b c d. (Semigroupoid a) => a b c -> a c d -> a b d
(>>>) f g = g <<< f
-- | `Category`s consist of objects and composable morphisms between them, and
-- | as such are [`Semigroupoids`](#semigroupoid), but unlike `semigroupoids`
-- | must have an identity element.
-- |
-- | Instances must satisfy the following law in addition to the
-- | `Semigroupoid` law:
-- |
-- | - Identity: `id <<< p = p <<< id = p`
class (Semigroupoid a) <= Category a where
id :: forall t. a t t
instance categoryFn :: Category (->) where
id x = x
infixl 4 <$>
infixl 1 <#>
-- | A `Functor` is a type constructor which supports a mapping operation
-- | `(<$>)`.
-- |
-- | `(<$>)` can be used to turn functions `a -> b` into functions
-- | `f a -> f b` whose argument and return types use the type constructor `f`
-- | to represent some computational context.
-- |
-- | Instances must satisfy the following laws:
-- |
-- | - Identity: `(<$>) id = id`
-- | - Composition: `(<$>) (f <<< g) = (f <$>) <<< (g <$>)`
class Functor f where
map :: forall a b. (a -> b) -> f a -> f b
instance functorFn :: Functor ((->) r) where
map = compose
instance functorArray :: Functor Array where
map = arrayMap
foreign import arrayMap
"""
function arrayMap(f) {
return function (arr) {
var l = arr.length;
var result = new Array(l);
for (var i = 0; i < l; i++) {
result[i] = f(arr[i]);
}
return result;
};
}
""" :: forall a b. (a -> b) -> Array a -> Array b
(<$>) :: forall f a b. (Functor f) => (a -> b) -> f a -> f b
(<$>) = map
-- | `(<#>)` is `(<$>)` with its arguments reversed. For example:
-- |
-- | ```purescript
-- | [1, 2, 3] <#> \n -> n * n
-- | ```
(<#>) :: forall f a b. (Functor f) => f a -> (a -> b) -> f b
(<#>) fa f = f <$> fa
-- | The `void` function is used to ignore the type wrapped by a
-- | [`Functor`](#functor), replacing it with `Unit` and keeping only the type
-- | information provided by the type constructor itself.
-- |
-- | `void` is often useful when using `do` notation to change the return type
-- | of a monadic computation:
-- |
-- | ```purescript
-- | main = forE 1 10 \n -> void do
-- | print n
-- | print (n * n)
-- | ```
void :: forall f a. (Functor f) => f a -> f Unit
void fa = const unit <$> fa
infixl 4 <*>
-- | The `Apply` class provides the `(<*>)` which is used to apply a function
-- | to an argument under a type constructor.
-- |
-- | `Apply` can be used to lift functions of two or more arguments to work on
-- | values wrapped with the type constructor `f`. It might also be understood
-- | in terms of the `lift2` function:
-- |
-- | ```purescript
-- | lift2 :: forall f a b c. (Apply f) => (a -> b -> c) -> f a -> f b -> f c
-- | lift2 f a b = f <$> a <*> b
-- | ```
-- |
-- | `(<*>)` is recovered from `lift2` as `lift2 ($)`. That is, `(<*>)` lifts
-- | the function application operator `($)` to arguments wrapped with the
-- | type constructor `f`.
-- |
-- | Instances must satisfy the following law in addition to the `Functor`
-- | laws:
-- |
-- | - Associative composition: `(<<<) <$> f <*> g <*> h = f <*> (g <*> h)`
-- |
-- | Formally, `Apply` represents a strong lax semi-monoidal endofunctor.
class (Functor f) <= Apply f where
apply :: forall a b. f (a -> b) -> f a -> f b
instance applyFn :: Apply ((->) r) where
apply f g x = f x (g x)
instance applyArray :: Apply Array where
apply = ap
(<*>) :: forall f a b. (Apply f) => f (a -> b) -> f a -> f b
(<*>) = apply
-- | The `Applicative` type class extends the [`Apply`](#apply) type class
-- | with a `pure` function, which can be used to create values of type `f a`
-- | from values of type `a`.
-- |
-- | Where [`Apply`](#apply) provides the ability to lift functions of two or
-- | more arguments to functions whose arguments are wrapped using `f`, and
-- | [`Functor`](#functor) provides the ability to lift functions of one
-- | argument, `pure` can be seen as the function which lifts functions of
-- | _zero_ arguments. That is, `Applicative` functors support a lifting
-- | operation for any number of function arguments.
-- |
-- | Instances must satisfy the following laws in addition to the `Apply`
-- | laws:
-- |
-- | - Identity: `(pure id) <*> v = v`
-- | - Composition: `(pure <<<) <*> f <*> g <*> h = f <*> (g <*> h)`
-- | - Homomorphism: `(pure f) <*> (pure x) = pure (f x)`
-- | - Interchange: `u <*> (pure y) = (pure ($ y)) <*> u`
class (Apply f) <= Applicative f where
pure :: forall a. a -> f a
instance applicativeFn :: Applicative ((->) r) where
pure = const
instance applicativeArray :: Applicative Array where
pure x = [x]
-- | `return` is an alias for `pure`.
return :: forall m a. (Applicative m) => a -> m a
return = pure
-- | `liftA1` provides a default implementation of `(<$>)` for any
-- | [`Applicative`](#applicative) functor, without using `(<$>)` as provided
-- | by the [`Functor`](#functor)-[`Applicative`](#applicative) superclass
-- | relationship.
-- |
-- | `liftA1` can therefore be used to write [`Functor`](#functor) instances
-- | as follows:
-- |
-- | ```purescript
-- | instance functorF :: Functor F where
-- | map = liftA1
-- | ```
liftA1 :: forall f a b. (Applicative f) => (a -> b) -> f a -> f b
liftA1 f a = pure f <*> a
infixl 1 >>=
-- | The `Bind` type class extends the [`Apply`](#apply) type class with a
-- | "bind" operation `(>>=)` which composes computations in sequence, using
-- | the return value of one computation to determine the next computation.
-- |
-- | The `>>=` operator can also be expressed using `do` notation, as follows:
-- |
-- | ```purescript
-- | x >>= f = do y <- x
-- | f y
-- | ```
-- |
-- | where the function argument of `f` is given the name `y`.
-- |
-- | Instances must satisfy the following law in addition to the `Apply`
-- | laws:
-- |
-- | - Associativity: `(x >>= f) >>= g = x >>= (\k => f k >>= g)`
-- |
-- | Associativity tells us that we can regroup operations which use `do`
-- | notation so that we can unambiguously write, for example:
-- |
-- | ```purescript
-- | do x <- m1
-- | y <- m2 x
-- | m3 x y
-- | ```
class (Apply m) <= Bind m where
bind :: forall a b. m a -> (a -> m b) -> m b
instance bindFn :: Bind ((->) r) where
bind m f x = f (m x) x
instance bindArray :: Bind Array where
bind = arrayBind
foreign import arrayBind
"""
function arrayBind (arr) {
return function (f) {
var result = [];
for (var i = 0, l = arr.length; i < l; i++) {
Array.prototype.push.apply(result, f(arr[i]));
}
return result;
};
}
""" :: forall a b. Array a -> (a -> Array b) -> Array b
(>>=) :: forall m a b. (Bind m) => m a -> (a -> m b) -> m b
(>>=) = bind
-- | The `Monad` type class combines the operations of the `Bind` and
-- | `Applicative` type classes. Therefore, `Monad` instances represent type
-- | constructors which support sequential composition, and also lifting of
-- | functions of arbitrary arity.
-- |
-- | Instances must satisfy the following laws in addition to the
-- | `Applicative` and `Bind` laws:
-- |
-- | - Left Identity: `pure x >>= f = f x`
-- | - Right Identity: `x >>= pure = x`
class (Applicative m, Bind m) <= Monad m
instance monadFn :: Monad ((->) r)
instance monadArray :: Monad Array
-- | `liftM1` provides a default implementation of `(<$>)` for any
-- | [`Monad`](#monad), without using `(<$>)` as provided by the
-- | [`Functor`](#functor)-[`Monad`](#monad) superclass relationship.
-- |
-- | `liftM1` can therefore be used to write [`Functor`](#functor) instances
-- | as follows:
-- |
-- | ```purescript
-- | instance functorF :: Functor F where
-- | map = liftM1
-- | ```
liftM1 :: forall m a b. (Monad m) => (a -> b) -> m a -> m b
liftM1 f a = do
a' <- a
return (f a')
-- | `ap` provides a default implementation of `(<*>)` for any
-- | [`Monad`](#monad), without using `(<*>)` as provided by the
-- | [`Apply`](#apply)-[`Monad`](#monad) superclass relationship.
-- |
-- | `ap` can therefore be used to write [`Apply`](#apply) instances as
-- | follows:
-- |
-- | ```purescript
-- | instance applyF :: Apply F where
-- | apply = ap
-- | ```
ap :: forall m a b. (Monad m) => m (a -> b) -> m a -> m b
ap f a = do
f' <- f
a' <- a
return (f' a')
infixr 5 <>
infixr 5 ++
-- | The `Semigroup` type class identifies an associative operation on a type.
-- |
-- | Instances are required to satisfy the following law:
-- |
-- | - Associativity: `(x <> y) <> z = x <> (y <> z)`
-- |
-- | One example of a `Semigroup` is `String`, with `(<>)` defined as string
-- | concatenation.
class Semigroup a where
append :: a -> a -> a
-- | `(<>)` is an alias for `append`.
(<>) :: forall s. (Semigroup s) => s -> s -> s
(<>) = append
-- | `(++)` is an alias for `append`.
(++) :: forall s. (Semigroup s) => s -> s -> s
(++) = append
instance semigroupString :: Semigroup String where
append = concatString
instance semigroupUnit :: Semigroup Unit where
append _ _ = unit
instance semigroupFn :: (Semigroup s') => Semigroup (s -> s') where
append f g = \x -> f x <> g x
instance semigroupOrdering :: Semigroup Ordering where
append LT _ = LT
append GT _ = GT
append EQ y = y
instance semigroupArray :: Semigroup (Array a) where
append = concatArray
foreign import concatString
"""
function concatString(s1) {
return function(s2) {
return s1 + s2;
};
}
""" :: String -> String -> String
foreign import concatArray
"""
function concatArray (xs) {
return function (ys) {
return xs.concat(ys);
};
}
""" :: forall a. Array a -> Array a -> Array a
infixl 6 +
infixl 7 *
-- | The `Semiring` class is for types that support an addition and
-- | multiplication operation.
-- |
-- | Instances must satisfy the following laws:
-- |
-- | - Commutative monoid under addition:
-- | - Associativity: `(a + b) + c = a + (b + c)`
-- | - Identity: `zero + a = a + zero = a`
-- | - Commutative: `a + b = b + a`
-- | - Monoid under multiplication:
-- | - Associativity: `(a * b) * c = a * (b * c)`
-- | - Identity: `one * a = a * one = a`
-- | - Multiplication distributes over addition:
-- | - Left distributivity: `a * (b + c) = (a * b) + (a * c)`
-- | - Right distributivity: `(a + b) * c = (a * c) + (b * c)`
-- | - Annihiliation: `zero * a = a * zero = zero`
class Semiring a where
add :: a -> a -> a
zero :: a
mul :: a -> a -> a
one :: a
instance semiringInt :: Semiring Int where
add = intAdd
zero = 0
mul = intMul
one = 1
instance semiringNumber :: Semiring Number where
add = numAdd
zero = 0.0
mul = numMul
one = 1.0
instance semiringUnit :: Semiring Unit where
add _ _ = unit
zero = unit
mul _ _ = unit
one = unit
(+) :: forall a. (Semiring a) => a -> a -> a
(+) = add
(*) :: forall a. (Semiring a) => a -> a -> a
(*) = mul
infixl 6 -
-- | The `Ring` class is for types that support addition, multiplication,
-- | and subtraction operations.
-- |
-- | Instances must satisfy the following law in addition to the `Semiring`
-- | laws:
-- |
-- | - Additive inverse: `a + (-a) = (-a) + a = zero`
class (Semiring a) <= Ring a where
sub :: a -> a -> a
instance ringInt :: Ring Int where
sub = intSub
instance ringNumber :: Ring Number where
sub = numSub
instance ringUnit :: Ring Unit where
sub _ _ = unit
(-) :: forall a. (Ring a) => a -> a -> a
(-) = sub
negate :: forall a. (Ring a) => a -> a
negate a = zero - a
infixl 7 /
-- | The `ModuloSemiring` class is for types that support addition,
-- | multiplication, division, and modulo (division remainder) operations.
-- |
-- | Instances must satisfy the following law in addition to the `Semiring`
-- | laws:
-- |
-- | - Remainder: `a / b * b + (a `mod` b) = a`
class (Semiring a) <= ModuloSemiring a where
div :: a -> a -> a
mod :: a -> a -> a
instance moduloSemiringInt :: ModuloSemiring Int where
div = intDiv
mod = intMod
instance moduloSemiringNumber :: ModuloSemiring Number where
div = numDiv
mod _ _ = 0.0
instance moduloSemiringUnit :: ModuloSemiring Unit where
div _ _ = unit
mod _ _ = unit
(/) :: forall a. (ModuloSemiring a) => a -> a -> a
(/) = div
-- | A `Ring` where every nonzero element has a multiplicative inverse.
-- |
-- | Instances must satisfy the following law in addition to the `Ring` and
-- | `ModuloSemiring` laws:
-- |
-- | - Multiplicative inverse: `(one / x) * x = one`
-- |
-- | As a consequence of this ```a `mod` b = zero``` as no divide operation
-- | will have a remainder.
class (Ring a, ModuloSemiring a) <= DivisionRing a
instance divisionRingNumber :: DivisionRing Number
instance divisionRingUnit :: DivisionRing Unit
-- | The `Num` class is for types that are commutative fields.
-- |
-- | Instances must satisfy the following law in addition to the
-- | `DivisionRing` laws:
-- |
-- | - Commutative multiplication: `a * b = b * a`
class (DivisionRing a) <= Num a
instance numNumber :: Num Number
instance numUnit :: Num Unit
foreign import intAdd
"""
function intAdd(x) {
return function(y) {
return (x + y)|0;
};
}
""" :: Int -> Int -> Int
foreign import intMul
"""
function intMul(x) {
return function(y) {
return (x * y)|0;
};
}
""" :: Int -> Int -> Int
foreign import intDiv
"""
function intDiv(x) {
return function(y) {
return (x / y)|0;
};
}
""" :: Int -> Int -> Int
foreign import intMod
"""
function intMod(x) {
return function(y) {
return x % y;
};
}
""" :: Int -> Int -> Int
foreign import intSub
"""
function intSub(x) {
return function(y) {
return (x - y)|0;
};
}
""" :: Int -> Int -> Int
foreign import numAdd
"""
function numAdd(n1) {
return function(n2) {
return n1 + n2;
};
}
""" :: Number -> Number -> Number
foreign import numMul
"""
function numMul(n1) {
return function(n2) {
return n1 * n2;
};
}
""" :: Number -> Number -> Number
foreign import numDiv
"""
function numDiv(n1) {
return function(n2) {
return n1 / n2;
};
}
""" :: Number -> Number -> Number
foreign import numSub
"""
function numSub(n1) {
return function(n2) {
return n1 - n2;
};
}
""" :: Number -> Number -> Number
infix 4 ==
infix 4 /=
-- | The `Eq` type class represents types which support decidable equality.
-- |
-- | `Eq` instances should satisfy the following laws:
-- |
-- | - Reflexivity: `x == x = true`
-- | - Symmetry: `x == y = y == x`
-- | - Transitivity: if `x == y` and `y == z` then `x == z`
-- | - Negation: `x /= y = not (x == y)`
class Eq a where
eq :: a -> a -> Boolean
(==) :: forall a. (Eq a) => a -> a -> Boolean
(==) = eq
(/=) :: forall a. (Eq a) => a -> a -> Boolean
(/=) x y = not (x == y)
instance eqBoolean :: Eq Boolean where
eq = refEq
instance eqInt :: Eq Int where
eq = refEq
instance eqNumber :: Eq Number where
eq = refEq
instance eqChar :: Eq Char where
eq = refEq
instance eqString :: Eq String where
eq = refEq
instance eqUnit :: Eq Unit where
eq _ _ = true
instance eqArray :: (Eq a) => Eq (Array a) where
eq = eqArrayImpl (==)
instance eqOrdering :: Eq Ordering where
eq LT LT = true
eq GT GT = true
eq EQ EQ = true
eq _ _ = false
foreign import refEq
"""
function refEq(r1) {
return function(r2) {
return r1 === r2;
};
}
""" :: forall a. a -> a -> Boolean
foreign import refIneq
"""
function refIneq(r1) {
return function(r2) {
return r1 !== r2;
};
}
""" :: forall a. a -> a -> Boolean
foreign import eqArrayImpl
"""
function eqArrayImpl(f) {
return function(xs) {
return function(ys) {
if (xs.length !== ys.length) return false;
for (var i = 0; i < xs.length; i++) {
if (!f(xs[i])(ys[i])) return false;
}
return true;
};
};
}
""" :: forall a. (a -> a -> Boolean) -> Array a -> Array a -> Boolean
-- | The `Ordering` data type represents the three possible outcomes of
-- | comparing two values:
-- |
-- | `LT` - The first value is _less than_ the second.
-- | `GT` - The first value is _greater than_ the second.
-- | `EQ` - The first value is _equal to_ or _incomparable to_ the second.
data Ordering = LT | GT | EQ
-- | The `Ord` type class represents types which support comparisons.
-- |
-- | `Ord` instances should satisfy the laws of _partially orderings_:
-- |
-- | - Reflexivity: `a <= a`
-- | - Antisymmetry: if `a <= b` and `b <= a` then `a = b`
-- | - Transitivity: if `a <= b` and `b <= c` then `a <= c`
class (Eq a) <= Ord a where
compare :: a -> a -> Ordering
instance ordBoolean :: Ord Boolean where
compare = unsafeCompare
instance ordInt :: Ord Int where
compare = unsafeCompare
instance ordNumber :: Ord Number where
compare = unsafeCompare
instance ordString :: Ord String where
compare = unsafeCompare
instance ordChar :: Ord Char where
compare = unsafeCompare
instance ordUnit :: Ord Unit where
compare _ _ = EQ
instance ordArray :: (Ord a) => Ord (Array a) where
compare xs ys = compare 0 $ ordArrayImpl (\x y -> case compare x y of
EQ -> 0
LT -> 1
GT -> -1) xs ys
foreign import ordArrayImpl """
function ordArrayImpl(f) {
return function (xs) {
return function (ys) {
var i = 0;
var xlen = xs.length;
var ylen = ys.length;
while (i < xlen && i < ylen) {
var x = xs[i];
var y = ys[i];
var o = f(x)(y);
if (o !== 0) {
return o;
}
i++;
}
if (xlen == ylen) {
return 0;
} else if (xlen > ylen) {
return -1;
} else {
return 1;
}
};
};
}
""" :: forall a. (a -> a -> Int) -> Array a -> Array a -> Int
instance ordOrdering :: Ord Ordering where
compare LT LT = EQ
compare EQ EQ = EQ
compare GT GT = EQ
compare LT _ = LT
compare EQ LT = GT
compare EQ GT = LT
compare GT _ = GT
infixl 4 <
infixl 4 >
infixl 4 <=
infixl 4 >=
-- | Test whether one value is _strictly less than_ another.
(<) :: forall a. (Ord a) => a -> a -> Boolean
(<) a1 a2 = case a1 `compare` a2 of
LT -> true
_ -> false
-- | Test whether one value is _strictly greater than_ another.
(>) :: forall a. (Ord a) => a -> a -> Boolean
(>) a1 a2 = case a1 `compare` a2 of
GT -> true
_ -> false
-- | Test whether one value is _non-strictly less than_ another.
(<=) :: forall a. (Ord a) => a -> a -> Boolean
(<=) a1 a2 = case a1 `compare` a2 of
GT -> false
_ -> true
-- | Test whether one value is _non-strictly greater than_ another.
(>=) :: forall a. (Ord a) => a -> a -> Boolean
(>=) a1 a2 = case a1 `compare` a2 of
LT -> false
_ -> true
unsafeCompare :: forall a. a -> a -> Ordering
unsafeCompare = unsafeCompareImpl LT EQ GT
foreign import unsafeCompareImpl
"""
function unsafeCompareImpl(lt) {
return function(eq) {
return function(gt) {
return function(x) {
return function(y) {
return x < y ? lt : x > y ? gt : eq;
};
};
};
};
}
""" :: forall a. Ordering -> Ordering -> Ordering -> a -> a -> Ordering
-- | The `Bounded` type class represents types that are finite partially
-- | ordered sets.
-- |
-- | Instances should satisfy the following law in addition to the `Ord` laws:
-- |
-- | - Ordering: `bottom <= a <= top`
class (Ord a) <= Bounded a where
top :: a
bottom :: a
instance boundedBoolean :: Bounded Boolean where
top = true
bottom = false
instance boundedUnit :: Bounded Unit where
top = unit
bottom = unit
instance boundedOrdering :: Bounded Ordering where
top = GT
bottom = LT
instance boundedInt :: Bounded Int where
top = 2147483647
bottom = -2147483648
-- | The `Lattice` type class represents types that are partially ordered
-- | sets with a supremum (`sup` or `||`) and infimum (`inf` or `&&`).
-- |
-- | Instances should satisfy the following laws in addition to the `Ord`
-- | laws:
-- |
-- | - Supremum:
-- | - `a || b >= a`
-- | - `a || b >= b`
-- | - Infimum:
-- | - `a && b <= a`
-- | - `a && b <= b`
-- | - Associativity:
-- | - `a || (b || c) = (a || b) || c`
-- | - `a && (b && c) = (a && b) && c`
-- | - Commutativity:
-- | - `a || b = b || a`
-- | - `a && b = b && a`
-- | - Absorption:
-- | - `a || (a && b) = a`
-- | - `a && (a || b) = a`
-- | - Idempotent:
-- | - `a || a = a`
-- | - `a && a = a`
class (Ord a) <= Lattice a where
sup :: a -> a -> a
inf :: a -> a -> a
instance latticeBoolean :: Lattice Boolean where
sup = boolOr
inf = boolAnd
instance latticeUnit :: Lattice Unit where
sup _ _ = unit
inf _ _ = unit
infixr 2 ||
infixr 3 &&
-- | The `sup` operator.
(||) :: forall a. (Lattice a) => a -> a -> a
(||) = sup
-- | The `inf` operator.
(&&) :: forall a. (Lattice a) => a -> a -> a
(&&) = inf
-- | The `BoundedLattice` type class represents types that are finite
-- | lattices.
-- |
-- | Instances should satisfy the following law in addition to the `Lattice`
-- | and `Bounded` laws:
-- |
-- | - Identity:
-- | - `a || bottom = a`
-- | - `a && top = a`
-- | - Annihiliation:
-- | - `a || top = top`
-- | - `a && bottom = bottom`
class (Bounded a, Lattice a) <= BoundedLattice a
instance boundedLatticeBoolean :: BoundedLattice Boolean
instance boundedLatticeUnit :: BoundedLattice Unit
-- | The `ComplementedLattice` type class represents types that are lattices
-- | where every member is also uniquely complemented.
-- |
-- | Instances should satisfy the following law in addition to the
-- | `BoundedLattice` laws:
-- |
-- | - Complemented:
-- | - `not a || a == top`
-- | - `not a && a == bottom`
-- | - Double negation:
-- | - `not <<< not == id`
class (BoundedLattice a) <= ComplementedLattice a where
not :: a -> a
instance complementedLatticeBoolean :: ComplementedLattice Boolean where
not = boolNot
instance complementedLatticeUnit :: ComplementedLattice Unit where
not _ = unit
-- | The `DistributiveLattice` type class represents types that are lattices
-- | where the `&&` and `||` distribute over each other.
-- |
-- | Instances should satisfy the following law in addition to the `Lattice`
-- | laws:
-- |
-- | - Distributivity: `x && (y || z) = (x && y) || (x && z)`
class (Lattice a) <= DistributiveLattice a
instance distributiveLatticeBoolean :: DistributiveLattice Boolean