-- Hoogle documentation, generated by Haddock -- See Hoogle, http://www.haskell.org/hoogle/ -- | A numeric class hierarchy. -- -- This package provides alternative numeric classes over Prelude. -- -- The numeric class constellation looks somewhat like: -- -- --

Usage

-- --

--   >>> {-# LANGUAGE GHC2021 #-}
--   
--   >>> {-# LANGUAGE RebindableSyntax #-}
--   
--   >>> import NumHask.Prelude
--

-- -- See NumHask for a detailed overview. @package numhask @version 0.11.0.2 -- | Additive classes module NumHask.Algebra.Additive -- | or Addition -- -- For practical reasons, we begin the class tree with Additive. -- Starting with Associative and Unital, or using -- Semigroup and Monoid from base tends to confuse the -- interface once you start having to disinguish between (say) monoidal -- addition and monoidal multiplication. -- --

--   \a -> zero + a == a
--

-- --

--   \a -> a + zero == a
--

-- --

--   \a b c -> (a + b) + c == a + (b + c)
--

-- --

--   \a b -> a + b == b + a
--

-- -- By convention, (+) is regarded as commutative, but this is not -- universal, and the introduction of another symbol which means -- non-commutative addition seems a bit dogmatic. -- --

--   >>> zero + 1
--   1
--

-- --

--   >>> 1 + 1
--   2
--

class Additive a (+) :: Additive a => a -> a -> a zero :: Additive a => a infixl 6 + -- | Compute the sum of a Foldable. -- --

--   >>> sum [0..10]
--   55
--

sum :: (Additive a, Foldable f) => f a -> a -- | Compute the accumulating sum of a Traversable. -- --

--   >>> accsum [0..10]
--   [0,1,3,6,10,15,21,28,36,45,55]
--

accsum :: (Additive a, Traversable f) => f a -> f a -- | or Subtraction -- --

--   \a -> a - a == zero
--

-- --

--   \a -> negate a == zero - a
--

-- --

--   \a -> negate a + a == zero
--

-- --

--   \a -> a + negate a == zero
--

-- --

--   >>> negate 1
--   -1
--

-- --

--   >>> 1 - 2
--   -1
--

class (Additive a) => Subtractive a negate :: Subtractive a => a -> a (-) :: Subtractive a => a -> a -> a infixl 6 - instance NumHask.Algebra.Additive.Subtractive GHC.Types.Double instance NumHask.Algebra.Additive.Subtractive GHC.Types.Float instance NumHask.Algebra.Additive.Subtractive GHC.Types.Int instance NumHask.Algebra.Additive.Subtractive GHC.Num.Integer.Integer instance NumHask.Algebra.Additive.Subtractive GHC.Num.Natural.Natural instance NumHask.Algebra.Additive.Subtractive GHC.Int.Int8 instance NumHask.Algebra.Additive.Subtractive GHC.Int.Int16 instance NumHask.Algebra.Additive.Subtractive GHC.Int.Int32 instance NumHask.Algebra.Additive.Subtractive GHC.Int.Int64 instance NumHask.Algebra.Additive.Subtractive GHC.Types.Word instance NumHask.Algebra.Additive.Subtractive GHC.Word.Word8 instance NumHask.Algebra.Additive.Subtractive GHC.Word.Word16 instance NumHask.Algebra.Additive.Subtractive GHC.Word.Word32 instance NumHask.Algebra.Additive.Subtractive GHC.Word.Word64 instance NumHask.Algebra.Additive.Subtractive b => NumHask.Algebra.Additive.Subtractive (a -> b) instance NumHask.Algebra.Additive.Additive GHC.Types.Double instance NumHask.Algebra.Additive.Additive GHC.Types.Float instance NumHask.Algebra.Additive.Additive GHC.Types.Int instance NumHask.Algebra.Additive.Additive GHC.Num.Integer.Integer instance NumHask.Algebra.Additive.Additive GHC.Types.Bool instance NumHask.Algebra.Additive.Additive GHC.Num.Natural.Natural instance NumHask.Algebra.Additive.Additive GHC.Int.Int8 instance NumHask.Algebra.Additive.Additive GHC.Int.Int16 instance NumHask.Algebra.Additive.Additive GHC.Int.Int32 instance NumHask.Algebra.Additive.Additive GHC.Int.Int64 instance NumHask.Algebra.Additive.Additive GHC.Types.Word instance NumHask.Algebra.Additive.Additive GHC.Word.Word8 instance NumHask.Algebra.Additive.Additive GHC.Word.Word16 instance NumHask.Algebra.Additive.Additive GHC.Word.Word32 instance NumHask.Algebra.Additive.Additive GHC.Word.Word64 instance NumHask.Algebra.Additive.Additive b => NumHask.Algebra.Additive.Additive (a -> b) -- | The Group hierarchy module NumHask.Algebra.Group -- | A Magma is a tuple (T,magma) consisting of -- --

a type a, and
a function (magma) :: T -> T -> T

-- -- The mathematical laws for a magma are: -- --

magma is defined for all possible pairs of type T, and
magma is closed in the set of all possible values of type T

-- -- or, more tersly, -- --

--   ∀ a, b ∈ T: a ⊕ b ∈ T
--

-- -- These laws are true by construction in haskell: the type signature of -- ⊕ and the above mathematical laws are synonyms. class Magma a (⊕) :: Magma a => a -> a -> a infix 3 ⊕ -- | A Unital Magma is a magma with an identity element (the unit). -- --

--   unit ⊕ a = a
--   a ⊕ unit = a
--

class (Magma a) => Unital a unit :: Unital a => a -- | An Associative Magma -- --

--   (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c)
--

class (Magma a) => Associative a -- | A Commutative Magma is a Magma where the binary operation is -- commutative. -- --

--   a ⊕ b = b ⊕ a
--

class (Magma a) => Commutative a -- | An Absorbing is a Magma with an Absorbing Element -- --

--   a ⊕ absorb = absorb
--

class (Magma a) => Absorbing a absorb :: Absorbing a => a -- | An Invertible Magma -- --

--   ∀ a,b ∈ T: inv a ⊕ (a ⊕ b) = b = (b ⊕ a) ⊕ inv a
--

class (Magma a) => Invertible a inv :: Invertible a => a -> a -- | An Idempotent Magma is a magma where every element is -- Idempotent. -- --

--   a ⊕ a = a
--

class (Magma a) => Idempotent a -- | A Group is a Associative, Unital and Invertible Magma. type Group a = (Associative a, Unital a, Invertible a) -- | An Abelian Group is an Associative, Unital, Invertible and -- Commutative Magma . In other words, it is a Commutative Group type AbelianGroup a = (Associative a, Unital a, Invertible a, Commutative a) instance NumHask.Algebra.Group.Idempotent b => NumHask.Algebra.Group.Idempotent (a -> b) instance NumHask.Algebra.Group.Absorbing b => NumHask.Algebra.Group.Absorbing (a -> b) instance NumHask.Algebra.Group.Invertible b => NumHask.Algebra.Group.Invertible (a -> b) instance NumHask.Algebra.Group.Commutative b => NumHask.Algebra.Group.Commutative (a -> b) instance NumHask.Algebra.Group.Associative b => NumHask.Algebra.Group.Associative (a -> b) instance NumHask.Algebra.Group.Unital b => NumHask.Algebra.Group.Unital (a -> b) instance NumHask.Algebra.Group.Magma b => NumHask.Algebra.Group.Magma (a -> b) -- | Multiplicative classes module NumHask.Algebra.Multiplicative -- | or Multiplication -- -- For practical reasons, we begin the class tree with Additive -- and Multiplicative. Starting with Associative and -- Unital, or using Semigroup and Monoid from base -- tends to confuse the interface once you start having to disinguish -- between (say) monoidal addition and monoidal multiplication. -- --

--   \a -> one * a == a
--

-- --

--   \a -> a * one == a
--

-- --

--   \a b c -> (a * b) * c == a * (b * c)
--

-- -- By convention, (*) is regarded as not necessarily commutative, but -- this is not universal, and the introduction of another symbol which -- means commutative multiplication seems a bit dogmatic. -- --

--   >>> one * 2
--   2
--

-- --

--   >>> 2 * 3
--   6
--

class Multiplicative a (*) :: Multiplicative a => a -> a -> a one :: Multiplicative a => a infixl 7 * -- | Compute the product of a Foldable. -- --

--   >>> product [1..5]
--   120
--

product :: (Multiplicative a, Foldable f) => f a -> a -- | Compute the accumulating product of a Traversable. -- --

--   >>> accproduct [1..5]
--   [1,2,6,24,120]
--

accproduct :: (Multiplicative a, Traversable f) => f a -> f a -- | or Division -- -- Though unusual, the term Divisive usefully fits in with the grammer of -- other classes and avoids name clashes that occur with some popular -- libraries. -- --

--   \(a :: Double) -> a / a ~= one || a == zero
--

-- --

--   \(a :: Double) -> recip a ~= one / a || a == zero
--

-- --

--   \(a :: Double) -> recip a * a ~= one || a == zero
--

-- --

--   \(a :: Double) -> a * recip a ~= one || a == zero
--

-- --

--   >>> recip 2.0
--   0.5
--

-- --

--   >>> 1 / 2
--   0.5
--

class (Multiplicative a) => Divisive a recip :: Divisive a => a -> a (/) :: Divisive a => a -> a -> a infixl 7 / instance NumHask.Algebra.Multiplicative.Divisive GHC.Types.Double instance NumHask.Algebra.Multiplicative.Divisive GHC.Types.Float instance NumHask.Algebra.Multiplicative.Divisive b => NumHask.Algebra.Multiplicative.Divisive (a -> b) instance NumHask.Algebra.Multiplicative.Multiplicative GHC.Types.Double instance NumHask.Algebra.Multiplicative.Multiplicative GHC.Types.Float instance NumHask.Algebra.Multiplicative.Multiplicative GHC.Types.Int instance NumHask.Algebra.Multiplicative.Multiplicative GHC.Num.Integer.Integer instance NumHask.Algebra.Multiplicative.Multiplicative GHC.Types.Bool instance NumHask.Algebra.Multiplicative.Multiplicative GHC.Num.Natural.Natural instance NumHask.Algebra.Multiplicative.Multiplicative GHC.Int.Int8 instance NumHask.Algebra.Multiplicative.Multiplicative GHC.Int.Int16 instance NumHask.Algebra.Multiplicative.Multiplicative GHC.Int.Int32 instance NumHask.Algebra.Multiplicative.Multiplicative GHC.Int.Int64 instance NumHask.Algebra.Multiplicative.Multiplicative GHC.Types.Word instance NumHask.Algebra.Multiplicative.Multiplicative GHC.Word.Word8 instance NumHask.Algebra.Multiplicative.Multiplicative GHC.Word.Word16 instance NumHask.Algebra.Multiplicative.Multiplicative GHC.Word.Word32 instance NumHask.Algebra.Multiplicative.Multiplicative GHC.Word.Word64 instance NumHask.Algebra.Multiplicative.Multiplicative b => NumHask.Algebra.Multiplicative.Multiplicative (a -> b) -- | Ring classes module NumHask.Algebra.Ring -- | Distributive -- --

--   \a b c -> a * (b + c) == a * b + a * c
--

-- --

--   \a b c -> (a + b) * c == a * c + b * c
--

-- --

--   \a -> zero * a == zero
--

-- --

--   \a -> a * zero == zero
--

-- -- The sneaking in of the Absorption laws here glosses over the -- possibility that the multiplicative zero element does not have to -- correspond with the additive unital zero. type Distributive a = (Additive a, Multiplicative a) -- | A Ring is an abelian group under addition (Unital, -- Associative, Commutative, Invertible) and -- monoidal under multiplication (Unital, Associative), and -- where multiplication distributes over addition. -- --

--   \a -> zero + a == a
--   \a -> a + zero == a
--   \a b c -> (a + b) + c == a + (b + c)
--   \a b -> a + b == b + a
--   \a -> a - a == zero
--   \a -> negate a == zero - a
--   \a -> negate a + a == zero
--   \a -> a + negate a == zero
--   \a -> one * a == a
--   \a -> a * one == a
--   \a b c -> (a * b) * c == a * (b * c)
--   \a b c -> a * (b + c) == a * b + a * c
--   \a b c -> (a + b) * c == a * c + b * c
--   \a -> zero * a == zero
--   \a -> a * zero == zero
--

type Ring a = (Distributive a, Subtractive a) -- | A StarSemiring is a semiring with an additional unary operator -- (star) satisfying: -- --

--   \a -> star a == one + a * star a
--

class (Distributive a) => StarSemiring a star :: StarSemiring a => a -> a plus :: StarSemiring a => a -> a -- | A Kleene Algebra is a Star Semiring with idempotent addition. -- --

--   a * x + x = a ==> star a * x + x = x
--   x * a + x = a ==> x * star a + x = x
--

class (StarSemiring a, Idempotent a) => KleeneAlgebra a -- | Involutive Ring -- --

--   adj (a + b) ==> adj a + adj b
--   adj (a * b) ==> adj a * adj b
--   adj one ==> one
--   adj (adj a) ==> a
--

-- -- Note: elements for which adj a == a are called -- "self-adjoint". class (Distributive a) => InvolutiveRing a adj :: InvolutiveRing a => a -> a -- | Defining two requires adding the multiplicative unital to -- itself. In other words, the concept of two is a Ring one. -- --

--   >>> two
--   2
--

two :: (Multiplicative a, Additive a) => a instance NumHask.Algebra.Ring.InvolutiveRing GHC.Types.Double instance NumHask.Algebra.Ring.InvolutiveRing GHC.Types.Float instance NumHask.Algebra.Ring.InvolutiveRing GHC.Num.Integer.Integer instance NumHask.Algebra.Ring.InvolutiveRing GHC.Types.Int instance NumHask.Algebra.Ring.InvolutiveRing GHC.Num.Natural.Natural instance NumHask.Algebra.Ring.InvolutiveRing GHC.Int.Int8 instance NumHask.Algebra.Ring.InvolutiveRing GHC.Int.Int16 instance NumHask.Algebra.Ring.InvolutiveRing GHC.Int.Int32 instance NumHask.Algebra.Ring.InvolutiveRing GHC.Int.Int64 instance NumHask.Algebra.Ring.InvolutiveRing GHC.Types.Word instance NumHask.Algebra.Ring.InvolutiveRing GHC.Word.Word8 instance NumHask.Algebra.Ring.InvolutiveRing GHC.Word.Word16 instance NumHask.Algebra.Ring.InvolutiveRing GHC.Word.Word32 instance NumHask.Algebra.Ring.InvolutiveRing GHC.Word.Word64 -- | Algebra for Actions -- -- Convention: the |'s in the operators point towards the higher-kinded -- number, representing an operator or action into a structure. module NumHask.Algebra.Action -- | Additive Action -- --

--   m |+ zero == m
--

class (Additive (AdditiveScalar m)) => AdditiveAction m where { type AdditiveScalar m :: Type; } (|+) :: AdditiveAction m => m -> AdditiveScalar m -> m infixl 6 |+ -- | flipped additive action -- --

--   (+|) == flip (|+)
--   zero +| m = m
--

(+|) :: AdditiveAction m => AdditiveScalar m -> m -> m infixl 6 +| -- | Subtractive Action -- --

--   m |- zero = m
--

class (AdditiveAction m, Subtractive (AdditiveScalar m)) => SubtractiveAction m (|-) :: SubtractiveAction m => m -> AdditiveScalar m -> m infixl 6 |- -- | Subtraction with the scalar on the left -- --

--   (-|) == (+|) . negate
--   zero -| m = negate m
--

(-|) :: (AdditiveAction m, Subtractive m) => AdditiveScalar m -> m -> m infixl 6 -| -- | Multiplicative Action -- --

--   m |* one = m
--   m |* zero = zero
--

class (Multiplicative (Scalar m)) => MultiplicativeAction m where { type Scalar m :: Type; } (|*) :: MultiplicativeAction m => m -> Scalar m -> m infixl 7 |* -- | flipped multiplicative action -- --

--   (*|) == flip (|*)
--   one *| m = one
--   zero *| m = zero
--

(*|) :: MultiplicativeAction m => Scalar m -> m -> m infixl 7 *| -- | Divisive Action -- --

--   m |/ one = m
--

class (Divisive (Scalar m), MultiplicativeAction m) => DivisiveAction m (|/) :: DivisiveAction m => m -> Scalar m -> m infixl 7 |/ -- | left scalar division -- --

--   (/|) == (*|) . recip
--   one |/ m = recip m
--

(/|) :: (MultiplicativeAction m, Divisive m) => Scalar m -> m -> m -- | A Module -- --

--   a *| one == a
--   (a + b) *| c == (a *| c) + (b *| c)
--   c |* (a + b) == (c |* a) + (c |* b)
--   a *| zero == zero
--   a *| b == b |* a
--

type Module m = (Distributive (Scalar m), MultiplicativeAction m) -- | Integral classes module NumHask.Data.Integral -- | An Integral is anything that satisfies the law: -- --

--   \a b -> b == zero || b * (a `div` b) + (a `mod` b) == a
--

-- --

--   >>> 3 `divMod` 2
--   (1,1)
--

-- --

--   >>> (-3) `divMod` 2
--   (-2,1)
--

-- --

--   >>> (-3) `quotRem` 2
--   (-1,-1)
--

class (Distributive a) => Integral a div :: Integral a => a -> a -> a mod :: Integral a => a -> a -> a divMod :: Integral a => a -> a -> (a, a) quot :: Integral a => a -> a -> a rem :: Integral a => a -> a -> a quotRem :: Integral a => a -> a -> (a, a) infixl 7 `div` infixl 7 `mod` -- | toIntegral is kept separate from Integral to help with compatability -- issues. -- --

--   toIntegral a == a
--

class ToIntegral a b toIntegral :: ToIntegral a b => a -> b -- | Convert to an Int type ToInt a = ToIntegral a Int -- | Polymorphic version of fromInteger -- --

--   fromIntegral a == a
--

class FromIntegral a b fromIntegral :: FromIntegral a b => b -> a -- | Convert from an Int type FromInt a = FromIntegral a Int -- | fromInteger is special in two ways: -- --

numeric integral literals (like "42") are interpreted specifically -- as "fromInteger (42 :: GHC.Num.Integer)". The prelude version is used -- as default (or whatever fromInteger is in scope if RebindableSyntax is -- set).
The default rules in haskell2010 specify that constraints -- on fromInteger need to be in a form C v, where v is a -- Num or a subclass of Num.

-- -- So a type synonym such as type FromInteger a = FromIntegral a -- Integer doesn't work well with type defaulting; hence the need -- for a separate class. class FromInteger a fromInteger :: FromInteger a => Integer -> a -- |

--   >>> even 2
--   True
--

even :: (Eq a, Integral a) => a -> Bool -- |

--   >>> odd 3
--   True
--

odd :: (Eq a, Integral a) => a -> Bool -- | raise a number to an Integral power -- --

--   >>> 2 ^^ 3
--   8.0
--

-- --

--   >>> 2 ^^ (-2)
--   0.25
--

(^^) :: (Ord b, Divisive a, Subtractive b, Integral b) => a -> b -> a infixr 8 ^^ -- | raise a number to an Int power -- -- Note: This differs from (^) found in prelude which is a partial -- function (it errors on negative integrals). This is a monomorphic -- version of (^^) provided to help reduce ambiguous type noise in -- common usages. -- --

--   >>> 2 ^ 3
--   8.0
--

-- --

--   >>> 2 ^ (-2)
--   0.25
--

(^) :: Divisive a => a -> Int -> a infixr 8 ^ instance NumHask.Data.Integral.FromInteger GHC.Types.Double instance NumHask.Data.Integral.FromInteger GHC.Types.Float instance NumHask.Data.Integral.FromInteger GHC.Types.Int instance NumHask.Data.Integral.FromInteger GHC.Num.Integer.Integer instance NumHask.Data.Integral.FromInteger GHC.Num.Natural.Natural instance NumHask.Data.Integral.FromInteger GHC.Int.Int8 instance NumHask.Data.Integral.FromInteger GHC.Int.Int16 instance NumHask.Data.Integral.FromInteger GHC.Int.Int32 instance NumHask.Data.Integral.FromInteger GHC.Int.Int64 instance NumHask.Data.Integral.FromInteger GHC.Types.Word instance NumHask.Data.Integral.FromInteger GHC.Word.Word8 instance NumHask.Data.Integral.FromInteger GHC.Word.Word16 instance NumHask.Data.Integral.FromInteger GHC.Word.Word32 instance NumHask.Data.Integral.FromInteger GHC.Word.Word64 instance NumHask.Data.Integral.FromIntegral a b => NumHask.Data.Integral.FromIntegral (c -> a) b instance NumHask.Data.Integral.FromIntegral GHC.Types.Double GHC.Num.Integer.Integer instance NumHask.Data.Integral.FromIntegral GHC.Types.Float GHC.Num.Integer.Integer instance NumHask.Data.Integral.FromIntegral GHC.Types.Int GHC.Num.Integer.Integer instance NumHask.Data.Integral.FromIntegral GHC.Num.Integer.Integer GHC.Num.Integer.Integer instance NumHask.Data.Integral.FromIntegral GHC.Num.Natural.Natural GHC.Num.Integer.Integer instance NumHask.Data.Integral.FromIntegral GHC.Int.Int8 GHC.Num.Integer.Integer instance NumHask.Data.Integral.FromIntegral GHC.Int.Int16 GHC.Num.Integer.Integer instance NumHask.Data.Integral.FromIntegral GHC.Int.Int32 GHC.Num.Integer.Integer instance NumHask.Data.Integral.FromIntegral GHC.Int.Int64 GHC.Num.Integer.Integer instance NumHask.Data.Integral.FromIntegral GHC.Types.Word GHC.Num.Integer.Integer instance NumHask.Data.Integral.FromIntegral GHC.Word.Word8 GHC.Num.Integer.Integer instance NumHask.Data.Integral.FromIntegral GHC.Word.Word16 GHC.Num.Integer.Integer instance NumHask.Data.Integral.FromIntegral GHC.Word.Word32 GHC.Num.Integer.Integer instance NumHask.Data.Integral.FromIntegral GHC.Word.Word64 GHC.Num.Integer.Integer instance NumHask.Data.Integral.FromIntegral GHC.Types.Double GHC.Types.Int instance NumHask.Data.Integral.FromIntegral GHC.Types.Float GHC.Types.Int instance NumHask.Data.Integral.FromIntegral GHC.Types.Int GHC.Types.Int instance NumHask.Data.Integral.FromIntegral GHC.Num.Integer.Integer GHC.Types.Int instance NumHask.Data.Integral.FromIntegral GHC.Num.Natural.Natural GHC.Types.Int instance NumHask.Data.Integral.FromIntegral GHC.Int.Int8 GHC.Types.Int instance NumHask.Data.Integral.FromIntegral GHC.Int.Int16 GHC.Types.Int instance NumHask.Data.Integral.FromIntegral GHC.Int.Int32 GHC.Types.Int instance NumHask.Data.Integral.FromIntegral GHC.Int.Int64 GHC.Types.Int instance NumHask.Data.Integral.FromIntegral GHC.Types.Word GHC.Types.Int instance NumHask.Data.Integral.FromIntegral GHC.Word.Word8 GHC.Types.Int instance NumHask.Data.Integral.FromIntegral GHC.Word.Word16 GHC.Types.Int instance NumHask.Data.Integral.FromIntegral GHC.Word.Word32 GHC.Types.Int instance NumHask.Data.Integral.FromIntegral GHC.Word.Word64 GHC.Types.Int instance NumHask.Data.Integral.FromIntegral GHC.Num.Natural.Natural GHC.Num.Natural.Natural instance NumHask.Data.Integral.FromIntegral GHC.Int.Int8 GHC.Int.Int8 instance NumHask.Data.Integral.FromIntegral GHC.Int.Int16 GHC.Int.Int16 instance NumHask.Data.Integral.FromIntegral GHC.Int.Int32 GHC.Int.Int32 instance NumHask.Data.Integral.FromIntegral GHC.Int.Int64 GHC.Int.Int64 instance NumHask.Data.Integral.FromIntegral GHC.Types.Word GHC.Types.Word instance NumHask.Data.Integral.FromIntegral GHC.Word.Word8 GHC.Word.Word8 instance NumHask.Data.Integral.FromIntegral GHC.Word.Word16 GHC.Word.Word16 instance NumHask.Data.Integral.FromIntegral GHC.Word.Word32 GHC.Word.Word32 instance NumHask.Data.Integral.FromIntegral GHC.Word.Word64 GHC.Word.Word64 instance NumHask.Data.Integral.ToIntegral GHC.Num.Integer.Integer GHC.Num.Integer.Integer instance NumHask.Data.Integral.ToIntegral GHC.Types.Int GHC.Num.Integer.Integer instance NumHask.Data.Integral.ToIntegral GHC.Num.Natural.Natural GHC.Num.Integer.Integer instance NumHask.Data.Integral.ToIntegral GHC.Int.Int8 GHC.Num.Integer.Integer instance NumHask.Data.Integral.ToIntegral GHC.Int.Int16 GHC.Num.Integer.Integer instance NumHask.Data.Integral.ToIntegral GHC.Int.Int32 GHC.Num.Integer.Integer instance NumHask.Data.Integral.ToIntegral GHC.Int.Int64 GHC.Num.Integer.Integer instance NumHask.Data.Integral.ToIntegral GHC.Types.Word GHC.Num.Integer.Integer instance NumHask.Data.Integral.ToIntegral GHC.Word.Word8 GHC.Num.Integer.Integer instance NumHask.Data.Integral.ToIntegral GHC.Word.Word16 GHC.Num.Integer.Integer instance NumHask.Data.Integral.ToIntegral GHC.Word.Word32 GHC.Num.Integer.Integer instance NumHask.Data.Integral.ToIntegral GHC.Word.Word64 GHC.Num.Integer.Integer instance NumHask.Data.Integral.ToIntegral GHC.Types.Int GHC.Types.Int instance NumHask.Data.Integral.ToIntegral GHC.Num.Integer.Integer GHC.Types.Int instance NumHask.Data.Integral.ToIntegral GHC.Num.Natural.Natural GHC.Types.Int instance NumHask.Data.Integral.ToIntegral GHC.Int.Int8 GHC.Types.Int instance NumHask.Data.Integral.ToIntegral GHC.Int.Int16 GHC.Types.Int instance NumHask.Data.Integral.ToIntegral GHC.Int.Int32 GHC.Types.Int instance NumHask.Data.Integral.ToIntegral GHC.Int.Int64 GHC.Types.Int instance NumHask.Data.Integral.ToIntegral GHC.Types.Word GHC.Types.Int instance NumHask.Data.Integral.ToIntegral GHC.Word.Word8 GHC.Types.Int instance NumHask.Data.Integral.ToIntegral GHC.Word.Word16 GHC.Types.Int instance NumHask.Data.Integral.ToIntegral GHC.Word.Word32 GHC.Types.Int instance NumHask.Data.Integral.ToIntegral GHC.Word.Word64 GHC.Types.Int instance NumHask.Data.Integral.ToIntegral GHC.Num.Natural.Natural GHC.Num.Natural.Natural instance NumHask.Data.Integral.ToIntegral GHC.Int.Int8 GHC.Int.Int8 instance NumHask.Data.Integral.ToIntegral GHC.Int.Int16 GHC.Int.Int16 instance NumHask.Data.Integral.ToIntegral GHC.Int.Int32 GHC.Int.Int32 instance NumHask.Data.Integral.ToIntegral GHC.Int.Int64 GHC.Int.Int64 instance NumHask.Data.Integral.ToIntegral GHC.Types.Word GHC.Types.Word instance NumHask.Data.Integral.ToIntegral GHC.Word.Word8 GHC.Word.Word8 instance NumHask.Data.Integral.ToIntegral GHC.Word.Word16 GHC.Word.Word16 instance NumHask.Data.Integral.ToIntegral GHC.Word.Word32 GHC.Word.Word32 instance NumHask.Data.Integral.ToIntegral GHC.Word.Word64 GHC.Word.Word64 instance NumHask.Data.Integral.Integral GHC.Types.Int instance NumHask.Data.Integral.Integral GHC.Num.Integer.Integer instance NumHask.Data.Integral.Integral GHC.Num.Natural.Natural instance NumHask.Data.Integral.Integral GHC.Int.Int8 instance NumHask.Data.Integral.Integral GHC.Int.Int16 instance NumHask.Data.Integral.Integral GHC.Int.Int32 instance NumHask.Data.Integral.Integral GHC.Int.Int64 instance NumHask.Data.Integral.Integral GHC.Types.Word instance NumHask.Data.Integral.Integral GHC.Word.Word8 instance NumHask.Data.Integral.Integral GHC.Word.Word16 instance NumHask.Data.Integral.Integral GHC.Word.Word32 instance NumHask.Data.Integral.Integral GHC.Word.Word64 instance NumHask.Data.Integral.Integral b => NumHask.Data.Integral.Integral (a -> b) -- | Field classes module NumHask.Algebra.Field -- | A Field is a set on which addition, subtraction, -- multiplication, and division are defined. It is also assumed that -- multiplication is distributive over addition. -- -- A summary of the rules inherited from super-classes of Field: -- --

--   zero + a == a
--   a + zero == a
--   ((a + b) + c) (a + (b + c))
--   a + b == b + a
--   a - a == zero
--   negate a == zero - a
--   negate a + a == zero
--   a + negate a == zero
--   one * a == a
--   a * one == a
--   ((a * b) * c) == (a * (b * c))
--   (a * (b + c)) == (a * b + a * c)
--   ((a + b) * c) == (a * c + b * c)
--   a * zero == zero
--   zero * a == zero
--   a / a == one || a == zero
--   recip a == one / a || a == zero
--   recip a * a == one || a == zero
--   a * recip a == one || a == zero
--

type Field a = (Ring a, Divisive a) -- | A hyperbolic field class -- --

--   \a -> a < zero || (sqrt . (**2)) a == a
--

-- --

--   \a -> a < zero || (log . exp) a ~= a
--

-- --

--   \a b -> (b < zero) || a <= zero || a == 1 || abs (a ** logBase a b - b) < 10 * epsilon
--

class (Field a) => ExpField a exp :: ExpField a => a -> a log :: ExpField a => a -> a (**) :: ExpField a => a -> a -> a -- | log to the base of -- --

--   >>> logBase 2 8
--   2.9999999999999996
--

logBase :: ExpField a => a -> a -> a -- | square root -- --

--   >>> sqrt 4
--   2.0
--

sqrt :: ExpField a => a -> a -- | Quotienting of a Field into a Ring -- -- See Field of fractions -- --

--   \a -> a - one < floor a <= a <= ceiling a < a + one
--

class (Field a) => QuotientField a where { type Whole a :: Type; } properFraction :: QuotientField a => a -> (Whole a, a) -- | round to the nearest Int -- -- Exact ties are managed by rounding down ties if the whole component is -- even. -- --

--   >>> round (1.5 :: Double)
--   2
--

-- --

--   >>> round (2.5 :: Double)
--   2
--

round :: (QuotientField a, Eq (Whole a), Ring (Whole a)) => a -> Whole a -- | round to the nearest Int -- -- Exact ties are managed by rounding down ties if the whole component is -- even. -- --

--   >>> round (1.5 :: Double)
--   2
--

-- --

--   >>> round (2.5 :: Double)
--   2
--

round :: (QuotientField a, Integral (Whole a), Eq (Whole a), Ord a, Ring (Whole a)) => a -> Whole a -- | supply the next upper whole component -- --

--   >>> ceiling (1.001 :: Double)
--   2
--

ceiling :: (QuotientField a, Distributive (Whole a)) => a -> Whole a -- | supply the next upper whole component -- --

--   >>> ceiling (1.001 :: Double)
--   2
--

ceiling :: (QuotientField a, Ord a, Distributive (Whole a)) => a -> Whole a -- | supply the previous lower whole component -- --

--   >>> floor (1.001 :: Double)
--   1
--

floor :: (QuotientField a, Ring (Whole a)) => a -> Whole a -- | supply the previous lower whole component -- --

--   >>> floor (1.001 :: Double)
--   1
--

floor :: (QuotientField a, Ord a, Ring (Whole a)) => a -> Whole a -- | supply the whole component closest to zero -- --

--   >>> floor (-1.001 :: Double)
--   -2
--

-- --

--   >>> truncate (-1.001 :: Double)
--   -1
--

truncate :: (QuotientField a, Ring (Whole a)) => a -> Whole a -- | supply the whole component closest to zero -- --

--   >>> floor (-1.001 :: Double)
--   -2
--

-- --

--   >>> truncate (-1.001 :: Double)
--   -1
--

truncate :: (QuotientField a, Ord a, Ring (Whole a)) => a -> Whole a -- | infinity is defined for any Field. -- --

--   >>> one / zero + infinity
--   Infinity
--

-- --

--   >>> infinity + 1
--   Infinity
--

infinity :: Field a => a -- | negative infinity -- --

--   >>> negInfinity + infinity
--   NaN
--

negInfinity :: Field a => a -- | nan is defined as zero/zero -- -- but note the (social) law: -- --

--   >>> nan == zero / zero
--   False
--

nan :: Field a => a -- | Trigonometric Field -- -- The list of laws is quite long: trigonometric identities class (Field a) => TrigField a pi :: TrigField a => a sin :: TrigField a => a -> a cos :: TrigField a => a -> a tan :: TrigField a => a -> a asin :: TrigField a => a -> a acos :: TrigField a => a -> a atan :: TrigField a => a -> a atan2 :: TrigField a => a -> a -> a sinh :: TrigField a => a -> a cosh :: TrigField a => a -> a tanh :: TrigField a => a -> a asinh :: TrigField a => a -> a acosh :: TrigField a => a -> a atanh :: TrigField a => a -> a -- | A half is a Field because it requires addition, -- multiplication and division. -- --

--   >>> half :: Double
--   0.5
--

half :: Field a => a instance NumHask.Algebra.Field.TrigField GHC.Types.Double instance NumHask.Algebra.Field.TrigField GHC.Types.Float instance NumHask.Algebra.Field.TrigField b => NumHask.Algebra.Field.TrigField (a -> b) instance NumHask.Algebra.Field.QuotientField GHC.Types.Float instance NumHask.Algebra.Field.QuotientField GHC.Types.Double instance NumHask.Algebra.Field.ExpField GHC.Types.Double instance NumHask.Algebra.Field.ExpField GHC.Types.Float instance NumHask.Algebra.Field.ExpField b => NumHask.Algebra.Field.ExpField (a -> b) -- | Lattices module NumHask.Algebra.Lattice -- | A algebraic structure with element joins: See Semilattice -- --

--   Associativity: x \/ (y \/ z) == (x \/ y) \/ z
--   Commutativity: x \/ y == y \/ x
--   Idempotency:   x \/ x == x
--

class (Eq a) => JoinSemiLattice a (\/) :: JoinSemiLattice a => a -> a -> a infixr 5 \/ -- | The partial ordering induced by the join-semilattice structure joinLeq :: JoinSemiLattice a => a -> a -> Bool -- | The partial ordering induced by the join-semilattice structure (<\) :: JoinSemiLattice a => a -> a -> Bool infixr 6 <\ -- | A algebraic structure with element meets: See Semilattice -- --

--   Associativity: x /\ (y /\ z) == (x /\ y) /\ z
--   Commutativity: x /\ y == y /\ x
--   Idempotency:   x /\ x == x
--

class (Eq a) => MeetSemiLattice a (/\) :: MeetSemiLattice a => a -> a -> a infixr 6 /\ -- | The partial ordering induced by the meet-semilattice structure meetLeq :: MeetSemiLattice a => a -> a -> Bool -- | The partial ordering induced by the meet-semilattice structure ( a -> a -> Bool infixr 6 bottom for -- \/. -- --

--   Identity: x \/ bottom == x
--

class (JoinSemiLattice a) => BoundedJoinSemiLattice a bottom :: BoundedJoinSemiLattice a => a -- | A meet-semilattice with an identity element top for /\. -- --

--   Identity: x /\ top == x
--

class (MeetSemiLattice a) => BoundedMeetSemiLattice a top :: BoundedMeetSemiLattice a => a -- | The combination of two semi lattices makes a lattice if the absorption -- law holds: see Absorption Law and Lattice -- --

--   Absorption: a \/ (a /\ b) == a /\ (a \/ b) == a
--

type Lattice a = (JoinSemiLattice a, MeetSemiLattice a) -- | Lattices with both bounds type BoundedLattice a = (JoinSemiLattice a, MeetSemiLattice a, BoundedJoinSemiLattice a, BoundedMeetSemiLattice a) instance NumHask.Algebra.Lattice.BoundedMeetSemiLattice GHC.Types.Float instance NumHask.Algebra.Lattice.BoundedMeetSemiLattice GHC.Types.Double instance NumHask.Algebra.Lattice.BoundedMeetSemiLattice GHC.Types.Int instance NumHask.Algebra.Lattice.BoundedMeetSemiLattice GHC.Types.Bool instance NumHask.Algebra.Lattice.BoundedMeetSemiLattice GHC.Int.Int8 instance NumHask.Algebra.Lattice.BoundedMeetSemiLattice GHC.Int.Int16 instance NumHask.Algebra.Lattice.BoundedMeetSemiLattice GHC.Int.Int32 instance NumHask.Algebra.Lattice.BoundedMeetSemiLattice GHC.Int.Int64 instance NumHask.Algebra.Lattice.BoundedMeetSemiLattice GHC.Types.Word instance NumHask.Algebra.Lattice.BoundedMeetSemiLattice GHC.Word.Word8 instance NumHask.Algebra.Lattice.BoundedMeetSemiLattice GHC.Word.Word16 instance NumHask.Algebra.Lattice.BoundedMeetSemiLattice GHC.Word.Word32 instance NumHask.Algebra.Lattice.BoundedMeetSemiLattice GHC.Word.Word64 instance NumHask.Algebra.Lattice.BoundedJoinSemiLattice GHC.Types.Float instance NumHask.Algebra.Lattice.BoundedJoinSemiLattice GHC.Types.Double instance NumHask.Algebra.Lattice.BoundedJoinSemiLattice GHC.Types.Int instance NumHask.Algebra.Lattice.BoundedJoinSemiLattice GHC.Types.Bool instance NumHask.Algebra.Lattice.BoundedJoinSemiLattice GHC.Num.Natural.Natural instance NumHask.Algebra.Lattice.BoundedJoinSemiLattice GHC.Int.Int8 instance NumHask.Algebra.Lattice.BoundedJoinSemiLattice GHC.Int.Int16 instance NumHask.Algebra.Lattice.BoundedJoinSemiLattice GHC.Int.Int32 instance NumHask.Algebra.Lattice.BoundedJoinSemiLattice GHC.Int.Int64 instance NumHask.Algebra.Lattice.BoundedJoinSemiLattice GHC.Types.Word instance NumHask.Algebra.Lattice.BoundedJoinSemiLattice GHC.Word.Word8 instance NumHask.Algebra.Lattice.BoundedJoinSemiLattice GHC.Word.Word16 instance NumHask.Algebra.Lattice.BoundedJoinSemiLattice GHC.Word.Word32 instance NumHask.Algebra.Lattice.BoundedJoinSemiLattice GHC.Word.Word64 instance NumHask.Algebra.Lattice.MeetSemiLattice GHC.Types.Float instance NumHask.Algebra.Lattice.MeetSemiLattice GHC.Types.Double instance NumHask.Algebra.Lattice.MeetSemiLattice GHC.Types.Int instance NumHask.Algebra.Lattice.MeetSemiLattice GHC.Num.Integer.Integer instance NumHask.Algebra.Lattice.MeetSemiLattice GHC.Types.Bool instance NumHask.Algebra.Lattice.MeetSemiLattice GHC.Num.Natural.Natural instance NumHask.Algebra.Lattice.MeetSemiLattice GHC.Int.Int8 instance NumHask.Algebra.Lattice.MeetSemiLattice GHC.Int.Int16 instance NumHask.Algebra.Lattice.MeetSemiLattice GHC.Int.Int32 instance NumHask.Algebra.Lattice.MeetSemiLattice GHC.Int.Int64 instance NumHask.Algebra.Lattice.MeetSemiLattice GHC.Types.Word instance NumHask.Algebra.Lattice.MeetSemiLattice GHC.Word.Word8 instance NumHask.Algebra.Lattice.MeetSemiLattice GHC.Word.Word16 instance NumHask.Algebra.Lattice.MeetSemiLattice GHC.Word.Word32 instance NumHask.Algebra.Lattice.MeetSemiLattice GHC.Word.Word64 instance NumHask.Algebra.Lattice.JoinSemiLattice GHC.Types.Float instance NumHask.Algebra.Lattice.JoinSemiLattice GHC.Types.Double instance NumHask.Algebra.Lattice.JoinSemiLattice GHC.Types.Int instance NumHask.Algebra.Lattice.JoinSemiLattice GHC.Num.Integer.Integer instance NumHask.Algebra.Lattice.JoinSemiLattice GHC.Types.Bool instance NumHask.Algebra.Lattice.JoinSemiLattice GHC.Num.Natural.Natural instance NumHask.Algebra.Lattice.JoinSemiLattice GHC.Int.Int8 instance NumHask.Algebra.Lattice.JoinSemiLattice GHC.Int.Int16 instance NumHask.Algebra.Lattice.JoinSemiLattice GHC.Int.Int32 instance NumHask.Algebra.Lattice.JoinSemiLattice GHC.Int.Int64 instance NumHask.Algebra.Lattice.JoinSemiLattice GHC.Types.Word instance NumHask.Algebra.Lattice.JoinSemiLattice GHC.Word.Word8 instance NumHask.Algebra.Lattice.JoinSemiLattice GHC.Word.Word16 instance NumHask.Algebra.Lattice.JoinSemiLattice GHC.Word.Word32 instance NumHask.Algebra.Lattice.JoinSemiLattice GHC.Word.Word64 -- | Metric classes module NumHask.Algebra.Metric -- | Basis encapsulates the notion of magnitude (intuitively the -- quotienting of a higher-kinded number to a scalar one) and the basis -- on which the magnitude quotienting was performed. An instance needs to -- satisfy these laws: -- --

--   \a -> magnitude a >= zero
--   \a -> magnitude zero == zero
--   \a -> a == magnitude a *| basis a
--   \a -> magnitude (basis a) == one
--

-- -- The names chosen are meant to represent the spiritual idea of a basis -- rather than a specific mathematics. See -- https://en.wikipedia.org/wiki/Basis_(linear_algebra) & -- https://en.wikipedia.org/wiki/Norm_(mathematics) for some -- mathematical motivations. -- --

--   >>> magnitude (-0.5 :: Double)
--   0.5
--

-- --

--   >>> basis (-0.5 :: Double)
--   -1.0
--

class (Distributive (Mag a)) => Basis a where { type Mag a :: Type; type Base a :: Type; } -- | or length, or ||v|| magnitude :: Basis a => a -> Mag a -- | or direction, or v-hat basis :: Basis a => a -> Base a -- | Basis where the domain and magnitude codomain are the same. type Absolute a = (Basis a, Mag a ~ a) -- | Basis where the domain and basis codomain are the same. type Sign a = (Basis a, Base a ~ a) -- | Basis where the domain, magnitude codomain and basis codomain are the -- same. type EndoBased a = (Basis a, Mag a ~ a, Base a ~ a) -- | The absolute value of a number. -- --

--   \a -> abs a * signum a ~= a
--

-- --

--   >>> abs (-1)
--   1
--

abs :: Absolute a => a -> a -- | The sign of a number. -- --

--   >>> signum (-1)
--   -1
--

-- -- abs zero == zero, so any value for signum zero is -- ok. We choose lawful neutral: -- --

--   >>> signum zero == zero
--   True
--

signum :: Sign a => a -> a -- | Distance, which combines the Subtractive notion of difference, with -- Basis. -- --

--   distance a b >= zero
--   distance a a == zero
--   distance a b *| basis (a - b) == a - b
--

distance :: (Basis a, Subtractive a) => a -> a -> Mag a -- | Convert between a "co-ordinated" or "higher-kinded" number and a -- direction. -- --

--   ray . angle == basis
--   magnitude (ray x) == one
--

class (Distributive coord, Distributive (Dir coord)) => Direction coord where { type Dir coord :: Type; } angle :: Direction coord => coord -> Dir coord ray :: Direction coord => Dir coord -> coord -- | Something that has a magnitude and a direction, with both expressed as -- the same type. -- -- See Polar coordinate system data Polar a Polar :: a -> a -> Polar a [radial] :: Polar a -> a [azimuth] :: Polar a -> a -- | Convert a higher-kinded number that has direction, to a Polar polar :: (Dir (Base a) ~ Mag a, Basis a, Direction (Base a)) => a -> Polar (Mag a) -- | Convert a Polar to a (higher-kinded) number that has a direction. coord :: (Scalar m ~ Dir m, MultiplicativeAction m, Direction m) => Polar (Scalar m) -> m -- | A small number, especially useful for approximate equality. class (Eq a, Additive a) => Epsilon a epsilon :: Epsilon a => a -- | Note that the constraint is Lattice rather than Ord allowing broader -- usage. -- --

--   >>> nearZero (epsilon :: Double)
--   True
--

-- --

--   >>> nearZero (epsilon :: EuclideanPair Double)
--   True
--

nearZero :: (Epsilon a, Lattice a, Subtractive a) => a -> Bool -- | Approximate equality -- --

--   >>> aboutEqual zero (epsilon :: Double)
--   True
--

aboutEqual :: (Epsilon a, Lattice a, Subtractive a) => a -> a -> Bool -- | About equal operator. -- --

--   >>> (1.0 + epsilon) ~= (1.0 :: Double)
--   True
--

(~=) :: Epsilon a => (Lattice a, Subtractive a) => a -> a -> Bool infixl 4 ~= -- | Two dimensional cartesian coordinates. newtype EuclideanPair a EuclideanPair :: (a, a) -> EuclideanPair a [euclidPair] :: EuclideanPair a -> (a, a) instance GHC.Classes.Eq a => GHC.Classes.Eq (NumHask.Algebra.Metric.Polar a) instance GHC.Show.Show a => GHC.Show.Show (NumHask.Algebra.Metric.Polar a) instance GHC.Generics.Generic (NumHask.Algebra.Metric.Polar a) instance GHC.Show.Show a => GHC.Show.Show (NumHask.Algebra.Metric.EuclideanPair a) instance GHC.Classes.Eq a => GHC.Classes.Eq (NumHask.Algebra.Metric.EuclideanPair a) instance GHC.Generics.Generic (NumHask.Algebra.Metric.EuclideanPair a) instance GHC.Base.Functor NumHask.Algebra.Metric.EuclideanPair instance GHC.Base.Applicative NumHask.Algebra.Metric.EuclideanPair instance NumHask.Algebra.Additive.Additive a => NumHask.Algebra.Additive.Additive (NumHask.Algebra.Metric.EuclideanPair a) instance NumHask.Algebra.Additive.Subtractive a => NumHask.Algebra.Additive.Subtractive (NumHask.Algebra.Metric.EuclideanPair a) instance NumHask.Algebra.Multiplicative.Multiplicative a => NumHask.Algebra.Multiplicative.Multiplicative (NumHask.Algebra.Metric.EuclideanPair a) instance (NumHask.Algebra.Additive.Subtractive a, NumHask.Algebra.Multiplicative.Divisive a) => NumHask.Algebra.Multiplicative.Divisive (NumHask.Algebra.Metric.EuclideanPair a) instance NumHask.Algebra.Field.TrigField a => NumHask.Algebra.Metric.Direction (NumHask.Algebra.Metric.EuclideanPair a) instance (NumHask.Algebra.Field.ExpField a, GHC.Classes.Eq a) => NumHask.Algebra.Metric.Basis (NumHask.Algebra.Metric.EuclideanPair a) instance NumHask.Algebra.Metric.Epsilon a => NumHask.Algebra.Metric.Epsilon (NumHask.Algebra.Metric.EuclideanPair a) instance NumHask.Algebra.Lattice.JoinSemiLattice a => NumHask.Algebra.Lattice.JoinSemiLattice (NumHask.Algebra.Metric.EuclideanPair a) instance NumHask.Algebra.Lattice.MeetSemiLattice a => NumHask.Algebra.Lattice.MeetSemiLattice (NumHask.Algebra.Metric.EuclideanPair a) instance NumHask.Algebra.Lattice.BoundedJoinSemiLattice a => NumHask.Algebra.Lattice.BoundedJoinSemiLattice (NumHask.Algebra.Metric.EuclideanPair a) instance NumHask.Algebra.Lattice.BoundedMeetSemiLattice a => NumHask.Algebra.Lattice.BoundedMeetSemiLattice (NumHask.Algebra.Metric.EuclideanPair a) instance NumHask.Algebra.Multiplicative.Multiplicative a => NumHask.Algebra.Action.MultiplicativeAction (NumHask.Algebra.Metric.EuclideanPair a) instance NumHask.Algebra.Multiplicative.Divisive a => NumHask.Algebra.Action.DivisiveAction (NumHask.Algebra.Metric.EuclideanPair a) instance (GHC.Classes.Ord a, NumHask.Algebra.Field.TrigField a, NumHask.Algebra.Field.ExpField a) => NumHask.Algebra.Field.ExpField (NumHask.Algebra.Metric.EuclideanPair a) instance (GHC.Classes.Eq (NumHask.Algebra.Field.Whole a), NumHask.Algebra.Ring.Ring (NumHask.Algebra.Field.Whole a), NumHask.Algebra.Field.QuotientField a) => NumHask.Algebra.Field.QuotientField (NumHask.Algebra.Metric.EuclideanPair a) instance NumHask.Algebra.Metric.Epsilon GHC.Types.Double instance NumHask.Algebra.Metric.Epsilon GHC.Types.Float instance NumHask.Algebra.Metric.Epsilon GHC.Types.Int instance NumHask.Algebra.Metric.Epsilon GHC.Num.Integer.Integer instance NumHask.Algebra.Metric.Epsilon GHC.Int.Int8 instance NumHask.Algebra.Metric.Epsilon GHC.Int.Int16 instance NumHask.Algebra.Metric.Epsilon GHC.Int.Int32 instance NumHask.Algebra.Metric.Epsilon GHC.Int.Int64 instance NumHask.Algebra.Metric.Epsilon GHC.Types.Word instance NumHask.Algebra.Metric.Epsilon GHC.Word.Word8 instance NumHask.Algebra.Metric.Epsilon GHC.Word.Word16 instance NumHask.Algebra.Metric.Epsilon GHC.Word.Word32 instance NumHask.Algebra.Metric.Epsilon GHC.Word.Word64 instance (NumHask.Algebra.Additive.Additive a, NumHask.Algebra.Multiplicative.Multiplicative a) => NumHask.Algebra.Metric.Basis (NumHask.Algebra.Metric.Polar a) instance NumHask.Algebra.Metric.Basis GHC.Types.Double instance NumHask.Algebra.Metric.Basis GHC.Types.Float instance NumHask.Algebra.Metric.Basis GHC.Types.Int instance NumHask.Algebra.Metric.Basis GHC.Num.Integer.Integer instance NumHask.Algebra.Metric.Basis GHC.Num.Natural.Natural instance NumHask.Algebra.Metric.Basis GHC.Int.Int8 instance NumHask.Algebra.Metric.Basis GHC.Int.Int16 instance NumHask.Algebra.Metric.Basis GHC.Int.Int32 instance NumHask.Algebra.Metric.Basis GHC.Int.Int64 instance NumHask.Algebra.Metric.Basis GHC.Types.Word instance NumHask.Algebra.Metric.Basis GHC.Word.Word8 instance NumHask.Algebra.Metric.Basis GHC.Word.Word16 instance NumHask.Algebra.Metric.Basis GHC.Word.Word32 instance NumHask.Algebra.Metric.Basis GHC.Word.Word64 -- | Complex numbers. module NumHask.Data.Complex -- | The underlying representation is a newtype-wrapped tuple, compared -- with the base datatype. This was chosen to facilitate the use of -- DerivingVia. newtype Complex a Complex :: (a, a) -> Complex a [complexPair] :: Complex a -> (a, a) -- | Complex number constructor. (+:) :: a -> a -> Complex a infixl 6 +: -- | Extracts the real part of a complex number. realPart :: Complex a -> a -- | Extracts the imaginary part of a complex number. imagPart :: Complex a -> a instance (GHC.Classes.Ord a, NumHask.Algebra.Field.TrigField a, NumHask.Algebra.Field.ExpField a) => NumHask.Algebra.Field.ExpField (NumHask.Data.Complex.Complex a) instance NumHask.Algebra.Lattice.BoundedMeetSemiLattice a => NumHask.Algebra.Lattice.BoundedMeetSemiLattice (NumHask.Data.Complex.Complex a) instance NumHask.Algebra.Lattice.BoundedJoinSemiLattice a => NumHask.Algebra.Lattice.BoundedJoinSemiLattice (NumHask.Data.Complex.Complex a) instance NumHask.Algebra.Lattice.MeetSemiLattice a => NumHask.Algebra.Lattice.MeetSemiLattice (NumHask.Data.Complex.Complex a) instance NumHask.Algebra.Lattice.JoinSemiLattice a => NumHask.Algebra.Lattice.JoinSemiLattice (NumHask.Data.Complex.Complex a) instance NumHask.Algebra.Metric.Epsilon a => NumHask.Algebra.Metric.Epsilon (NumHask.Data.Complex.Complex a) instance NumHask.Algebra.Field.TrigField a => NumHask.Algebra.Metric.Direction (NumHask.Data.Complex.Complex a) instance (NumHask.Algebra.Field.ExpField a, GHC.Classes.Eq a) => NumHask.Algebra.Metric.Basis (NumHask.Data.Complex.Complex a) instance NumHask.Algebra.Additive.Subtractive a => NumHask.Algebra.Additive.Subtractive (NumHask.Data.Complex.Complex a) instance NumHask.Algebra.Additive.Additive a => NumHask.Algebra.Additive.Additive (NumHask.Data.Complex.Complex a) instance GHC.Base.Functor NumHask.Data.Complex.Complex instance GHC.Generics.Generic (NumHask.Data.Complex.Complex a) instance Data.Data.Data a => Data.Data.Data (NumHask.Data.Complex.Complex a) instance GHC.Read.Read a => GHC.Read.Read (NumHask.Data.Complex.Complex a) instance GHC.Show.Show a => GHC.Show.Show (NumHask.Data.Complex.Complex a) instance GHC.Classes.Eq a => GHC.Classes.Eq (NumHask.Data.Complex.Complex a) instance (NumHask.Algebra.Additive.Subtractive a, NumHask.Algebra.Multiplicative.Multiplicative a) => NumHask.Algebra.Multiplicative.Multiplicative (NumHask.Data.Complex.Complex a) instance (NumHask.Algebra.Additive.Subtractive a, NumHask.Algebra.Multiplicative.Divisive a) => NumHask.Algebra.Multiplicative.Divisive (NumHask.Data.Complex.Complex a) instance (NumHask.Algebra.Additive.Additive a, NumHask.Data.Integral.FromIntegral a b) => NumHask.Data.Integral.FromIntegral (NumHask.Data.Complex.Complex a) b instance (NumHask.Algebra.Ring.Distributive a, NumHask.Algebra.Additive.Subtractive a) => NumHask.Algebra.Ring.InvolutiveRing (NumHask.Data.Complex.Complex a) instance (GHC.Classes.Eq (NumHask.Algebra.Field.Whole a), NumHask.Algebra.Ring.Ring (NumHask.Algebra.Field.Whole a), NumHask.Algebra.Field.QuotientField a) => NumHask.Algebra.Field.QuotientField (NumHask.Data.Complex.Complex a) -- | Rational classes module NumHask.Data.Rational -- | A rational number data Ratio a (:%) :: !a -> !a -> Ratio a -- | Arbitrary-precision rational numbers, represented as a ratio of two -- Integer values. A rational number may be constructed using the -- % operator. type Rational = Ratio Integer -- | toRatio is equivalent to Real in base, but is polymorphic in -- the Integral type. -- --

--   >>> toRatio (3.1415927 :: Float) :: Ratio Integer
--   13176795 :% 4194304
--

class ToRatio a b toRatio :: ToRatio a b => a -> Ratio b -- | Fractional in base splits into fromRatio and Field -- --

--   >>> fromRatio (5 :% 2 :: Ratio Integer) :: Double
--   2.5
--

class FromRatio a b fromRatio :: FromRatio a b => Ratio b -> a -- | fromRational is special in two ways: -- --

numeric decimal literals (like "53.66") are interpreted as exactly -- "fromRational (53.66 :: GHC.Real.Ratio Integer)". The prelude version, -- GHC.Real.fromRational is used as default (or whatever is in scope if -- RebindableSyntax is set).
The default rules in haskell2010 specify that contraints on -- fromRational need to be in a form C v, where v is a -- Num or a subclass of Num.

-- -- So a type synonym of `type FromRational a = FromRatio a Integer` -- doesn't work well with type defaulting; hence the need for a separate -- class. class FromRational a fromRational :: FromRational a => Rational -> a -- | reduce normalises a ratio by dividing both numerator and -- denominator by their greatest common divisor. -- --

--   >>> reduce 72 60
--   6 :% 5
--

-- --

--   \a b -> reduce a b == a :% b || b == zero
--

reduce :: (Eq a, Subtractive a, EndoBased a, Integral a) => a -> a -> Ratio a -- | gcd x y is the non-negative factor of both x -- and y of which every common factor of x and -- y is also a factor; for example gcd 4 2 = 2, -- gcd (-4) 6 = 2, gcd 0 4 = 4. -- gcd 0 0 = 0. (That is, the common divisor -- that is "greatest" in the divisibility preordering.) -- -- Note: Since for signed fixed-width integer types, abs -- minBound < 0, the result may be negative if one of the -- arguments is minBound (and necessarily is if the other -- is 0 or minBound) for such types. -- --

--   >>> gcd 72 60
--   12
--

gcd :: (Eq a, EndoBased a, Integral a) => a -> a -> a instance GHC.Show.Show a => GHC.Show.Show (NumHask.Data.Rational.Ratio a) instance NumHask.Data.Rational.FromRational GHC.Types.Double instance NumHask.Data.Rational.FromRational GHC.Types.Float instance NumHask.Data.Rational.FromRational (NumHask.Data.Rational.Ratio GHC.Num.Integer.Integer) instance NumHask.Data.Rational.FromRatio GHC.Types.Double GHC.Num.Integer.Integer instance NumHask.Data.Rational.FromRatio GHC.Types.Float GHC.Num.Integer.Integer instance NumHask.Data.Rational.FromRatio GHC.Real.Rational GHC.Num.Integer.Integer instance NumHask.Data.Rational.ToRatio GHC.Types.Double GHC.Num.Integer.Integer instance NumHask.Data.Rational.ToRatio GHC.Types.Float GHC.Num.Integer.Integer instance NumHask.Data.Rational.ToRatio GHC.Real.Rational GHC.Num.Integer.Integer instance NumHask.Data.Rational.ToRatio (NumHask.Data.Rational.Ratio GHC.Num.Integer.Integer) GHC.Num.Integer.Integer instance NumHask.Data.Rational.ToRatio GHC.Types.Int GHC.Num.Integer.Integer instance NumHask.Data.Rational.ToRatio GHC.Num.Integer.Integer GHC.Num.Integer.Integer instance NumHask.Data.Rational.ToRatio GHC.Num.Natural.Natural GHC.Num.Integer.Integer instance NumHask.Data.Rational.ToRatio GHC.Int.Int8 GHC.Num.Integer.Integer instance NumHask.Data.Rational.ToRatio GHC.Int.Int16 GHC.Num.Integer.Integer instance NumHask.Data.Rational.ToRatio GHC.Int.Int32 GHC.Num.Integer.Integer instance NumHask.Data.Rational.ToRatio GHC.Int.Int64 GHC.Num.Integer.Integer instance NumHask.Data.Rational.ToRatio GHC.Types.Word GHC.Num.Integer.Integer instance NumHask.Data.Rational.ToRatio GHC.Word.Word8 GHC.Num.Integer.Integer instance NumHask.Data.Rational.ToRatio GHC.Word.Word16 GHC.Num.Integer.Integer instance NumHask.Data.Rational.ToRatio GHC.Word.Word32 GHC.Num.Integer.Integer instance NumHask.Data.Rational.ToRatio GHC.Word.Word64 GHC.Num.Integer.Integer instance (GHC.Classes.Eq a, NumHask.Algebra.Additive.Subtractive a, NumHask.Algebra.Metric.EndoBased a, NumHask.Algebra.Metric.Absolute a, NumHask.Data.Integral.Integral a) => GHC.Classes.Eq (NumHask.Data.Rational.Ratio a) instance (GHC.Classes.Ord a, NumHask.Data.Integral.Integral a, NumHask.Algebra.Metric.EndoBased a, NumHask.Algebra.Additive.Subtractive a) => GHC.Classes.Ord (NumHask.Data.Rational.Ratio a) instance (GHC.Classes.Ord a, NumHask.Algebra.Metric.EndoBased a, NumHask.Data.Integral.Integral a, NumHask.Algebra.Ring.Ring a) => NumHask.Algebra.Additive.Additive (NumHask.Data.Rational.Ratio a) instance (GHC.Classes.Ord a, NumHask.Algebra.Metric.EndoBased a, NumHask.Data.Integral.Integral a, NumHask.Algebra.Ring.Ring a) => NumHask.Algebra.Additive.Subtractive (NumHask.Data.Rational.Ratio a) instance (GHC.Classes.Ord a, NumHask.Algebra.Metric.EndoBased a, NumHask.Data.Integral.Integral a, NumHask.Algebra.Ring.Ring a) => NumHask.Algebra.Multiplicative.Multiplicative (NumHask.Data.Rational.Ratio a) instance (GHC.Classes.Ord a, NumHask.Algebra.Metric.EndoBased a, NumHask.Data.Integral.Integral a, NumHask.Algebra.Ring.Ring a) => NumHask.Algebra.Multiplicative.Divisive (NumHask.Data.Rational.Ratio a) instance (GHC.Classes.Ord a, NumHask.Algebra.Metric.EndoBased a, NumHask.Algebra.Metric.Absolute a, NumHask.Data.Integral.ToInt a, NumHask.Data.Integral.Integral a, NumHask.Algebra.Ring.Ring a) => NumHask.Algebra.Field.QuotientField (NumHask.Data.Rational.Ratio a) instance (GHC.Classes.Ord a, NumHask.Algebra.Metric.EndoBased a, NumHask.Data.Integral.Integral a, NumHask.Algebra.Ring.Ring a) => NumHask.Algebra.Metric.Basis (NumHask.Data.Rational.Ratio a) instance (GHC.Classes.Ord a, NumHask.Data.Integral.Integral a, NumHask.Algebra.Metric.EndoBased a, NumHask.Algebra.Additive.Subtractive a) => NumHask.Algebra.Lattice.JoinSemiLattice (NumHask.Data.Rational.Ratio a) instance (GHC.Classes.Ord a, NumHask.Data.Integral.Integral a, NumHask.Algebra.Metric.EndoBased a, NumHask.Algebra.Additive.Subtractive a) => NumHask.Algebra.Lattice.MeetSemiLattice (NumHask.Data.Rational.Ratio a) instance (GHC.Classes.Ord a, NumHask.Algebra.Metric.EndoBased a, NumHask.Data.Integral.Integral a, NumHask.Algebra.Ring.Ring a, NumHask.Algebra.Lattice.MeetSemiLattice a) => NumHask.Algebra.Metric.Epsilon (NumHask.Data.Rational.Ratio a) instance (NumHask.Data.Integral.FromIntegral a b, NumHask.Algebra.Multiplicative.Multiplicative a) => NumHask.Data.Integral.FromIntegral (NumHask.Data.Rational.Ratio a) b -- | Exceptions arising within numhask. module NumHask.Exception -- | A numhask exception. newtype NumHaskException NumHaskException :: String -> NumHaskException [errorMessage] :: NumHaskException -> String -- | Throw an exception. Exceptions may be thrown from purely functional -- code, but may only be caught within the IO monad. throw :: forall (r :: RuntimeRep) (a :: TYPE r) e. Exception e => e -> a instance GHC.Show.Show NumHask.Exception.NumHaskException instance GHC.Exception.Type.Exception NumHask.Exception.NumHaskException -- | Numeric classes. module NumHask -- | or Addition -- -- For practical reasons, we begin the class tree with Additive. -- Starting with Associative and Unital, or using -- Semigroup and Monoid from base tends to confuse the -- interface once you start having to disinguish between (say) monoidal -- addition and monoidal multiplication. -- --

--   \a -> zero + a == a
--

-- --

--   \a -> a + zero == a
--

-- --

--   \a b c -> (a + b) + c == a + (b + c)
--

-- --

--   \a b -> a + b == b + a
--

-- -- By convention, (+) is regarded as commutative, but this is not -- universal, and the introduction of another symbol which means -- non-commutative addition seems a bit dogmatic. -- --

--   >>> zero + 1
--   1
--

-- --

--   >>> 1 + 1
--   2
--

class Additive a (+) :: Additive a => a -> a -> a zero :: Additive a => a infixl 6 + -- | Compute the sum of a Foldable. -- --

--   >>> sum [0..10]
--   55
--

sum :: (Additive a, Foldable f) => f a -> a -- | Compute the accumulating sum of a Traversable. -- --

--   >>> accsum [0..10]
--   [0,1,3,6,10,15,21,28,36,45,55]
--

accsum :: (Additive a, Traversable f) => f a -> f a -- | or Subtraction -- --

--   \a -> a - a == zero
--

-- --

--   \a -> negate a == zero - a
--

-- --

--   \a -> negate a + a == zero
--

-- --

--   \a -> a + negate a == zero
--

-- --

--   >>> negate 1
--   -1
--

-- --

--   >>> 1 - 2
--   -1
--

class (Additive a) => Subtractive a negate :: Subtractive a => a -> a (-) :: Subtractive a => a -> a -> a infixl 6 - -- | or Multiplication -- -- For practical reasons, we begin the class tree with Additive -- and Multiplicative. Starting with Associative and -- Unital, or using Semigroup and Monoid from base -- tends to confuse the interface once you start having to disinguish -- between (say) monoidal addition and monoidal multiplication. -- --

--   \a -> one * a == a
--

-- --

--   \a -> a * one == a
--

-- --

--   \a b c -> (a * b) * c == a * (b * c)
--

--   >>> one * 2
--   2
--

-- --

--   >>> 2 * 3
--   6
--

class Multiplicative a (*) :: Multiplicative a => a -> a -> a one :: Multiplicative a => a infixl 7 * -- | Compute the product of a Foldable. -- --

--   >>> product [1..5]
--   120
--

product :: (Multiplicative a, Foldable f) => f a -> a -- | Compute the accumulating product of a Traversable. -- --

--   >>> accproduct [1..5]
--   [1,2,6,24,120]
--

--   \(a :: Double) -> a / a ~= one || a == zero
--

-- --

--   \(a :: Double) -> recip a ~= one / a || a == zero
--

-- --

--   \(a :: Double) -> recip a * a ~= one || a == zero
--

-- --

--   \(a :: Double) -> a * recip a ~= one || a == zero
--

-- --

--   >>> recip 2.0
--   0.5
--

-- --

--   >>> 1 / 2
--   0.5
--

class (Multiplicative a) => Divisive a recip :: Divisive a => a -> a (/) :: Divisive a => a -> a -> a infixl 7 / -- | Distributive -- --

--   \a b c -> a * (b + c) == a * b + a * c
--

-- --

--   \a b c -> (a + b) * c == a * c + b * c
--

-- --

--   \a -> zero * a == zero
--

-- --

--   \a -> a * zero == zero
--

--   \a -> zero + a == a
--   \a -> a + zero == a
--   \a b c -> (a + b) + c == a + (b + c)
--   \a b -> a + b == b + a
--   \a -> a - a == zero
--   \a -> negate a == zero - a
--   \a -> negate a + a == zero
--   \a -> a + negate a == zero
--   \a -> one * a == a
--   \a -> a * one == a
--   \a b c -> (a * b) * c == a * (b * c)
--   \a b c -> a * (b + c) == a * b + a * c
--   \a b c -> (a + b) * c == a * c + b * c
--   \a -> zero * a == zero
--   \a -> a * zero == zero
--

type Ring a = (Distributive a, Subtractive a) -- | A StarSemiring is a semiring with an additional unary operator -- (star) satisfying: -- --

--   \a -> star a == one + a * star a
--

class (Distributive a) => StarSemiring a star :: StarSemiring a => a -> a plus :: StarSemiring a => a -> a -- | A Kleene Algebra is a Star Semiring with idempotent addition. -- --

--   a * x + x = a ==> star a * x + x = x
--   x * a + x = a ==> x * star a + x = x
--

class (StarSemiring a, Idempotent a) => KleeneAlgebra a -- | Involutive Ring -- --

--   adj (a + b) ==> adj a + adj b
--   adj (a * b) ==> adj a * adj b
--   adj one ==> one
--   adj (adj a) ==> a
--

--   >>> two
--   2
--

two :: (Multiplicative a, Additive a) => a -- | A Field is a set on which addition, subtraction, -- multiplication, and division are defined. It is also assumed that -- multiplication is distributive over addition. -- -- A summary of the rules inherited from super-classes of Field: -- --

--   zero + a == a
--   a + zero == a
--   ((a + b) + c) (a + (b + c))
--   a + b == b + a
--   a - a == zero
--   negate a == zero - a
--   negate a + a == zero
--   a + negate a == zero
--   one * a == a
--   a * one == a
--   ((a * b) * c) == (a * (b * c))
--   (a * (b + c)) == (a * b + a * c)
--   ((a + b) * c) == (a * c + b * c)
--   a * zero == zero
--   zero * a == zero
--   a / a == one || a == zero
--   recip a == one / a || a == zero
--   recip a * a == one || a == zero
--   a * recip a == one || a == zero
--

type Field a = (Ring a, Divisive a) -- | A hyperbolic field class -- --

--   \a -> a < zero || (sqrt . (**2)) a == a
--

-- --

--   \a -> a < zero || (log . exp) a ~= a
--

-- --

--   \a b -> (b < zero) || a <= zero || a == 1 || abs (a ** logBase a b - b) < 10 * epsilon
--

class (Field a) => ExpField a exp :: ExpField a => a -> a log :: ExpField a => a -> a (**) :: ExpField a => a -> a -> a -- | log to the base of -- --

--   >>> logBase 2 8
--   2.9999999999999996
--

logBase :: ExpField a => a -> a -> a -- | square root -- --

--   >>> sqrt 4
--   2.0
--

sqrt :: ExpField a => a -> a -- | Quotienting of a Field into a Ring -- -- See Field of fractions -- --

--   \a -> a - one < floor a <= a <= ceiling a < a + one
--

--   >>> round (1.5 :: Double)
--   2
--

-- --

--   >>> round (2.5 :: Double)
--   2
--

round :: (QuotientField a, Eq (Whole a), Ring (Whole a)) => a -> Whole a -- | round to the nearest Int -- -- Exact ties are managed by rounding down ties if the whole component is -- even. -- --

--   >>> round (1.5 :: Double)
--   2
--

-- --

--   >>> round (2.5 :: Double)
--   2
--

round :: (QuotientField a, Integral (Whole a), Eq (Whole a), Ord a, Ring (Whole a)) => a -> Whole a -- | supply the next upper whole component -- --

--   >>> ceiling (1.001 :: Double)
--   2
--

ceiling :: (QuotientField a, Distributive (Whole a)) => a -> Whole a -- | supply the next upper whole component -- --

--   >>> ceiling (1.001 :: Double)
--   2
--

ceiling :: (QuotientField a, Ord a, Distributive (Whole a)) => a -> Whole a -- | supply the previous lower whole component -- --

--   >>> floor (1.001 :: Double)
--   1
--

floor :: (QuotientField a, Ring (Whole a)) => a -> Whole a -- | supply the previous lower whole component -- --

--   >>> floor (1.001 :: Double)
--   1
--

floor :: (QuotientField a, Ord a, Ring (Whole a)) => a -> Whole a -- | supply the whole component closest to zero -- --

--   >>> floor (-1.001 :: Double)
--   -2
--

-- --

--   >>> truncate (-1.001 :: Double)
--   -1
--

truncate :: (QuotientField a, Ring (Whole a)) => a -> Whole a -- | supply the whole component closest to zero -- --

--   >>> floor (-1.001 :: Double)
--   -2
--

-- --

--   >>> truncate (-1.001 :: Double)
--   -1
--

truncate :: (QuotientField a, Ord a, Ring (Whole a)) => a -> Whole a -- | Trigonometric Field -- -- The list of laws is quite long: trigonometric identities class (Field a) => TrigField a pi :: TrigField a => a sin :: TrigField a => a -> a cos :: TrigField a => a -> a tan :: TrigField a => a -> a asin :: TrigField a => a -> a acos :: TrigField a => a -> a atan :: TrigField a => a -> a atan2 :: TrigField a => a -> a -> a sinh :: TrigField a => a -> a cosh :: TrigField a => a -> a tanh :: TrigField a => a -> a asinh :: TrigField a => a -> a acosh :: TrigField a => a -> a atanh :: TrigField a => a -> a -- | infinity is defined for any Field. -- --

--   >>> one / zero + infinity
--   Infinity
--

-- --

--   >>> infinity + 1
--   Infinity
--

infinity :: Field a => a -- | negative infinity -- --

--   >>> negInfinity + infinity
--   NaN
--

negInfinity :: Field a => a -- | nan is defined as zero/zero -- -- but note the (social) law: -- --

--   >>> nan == zero / zero
--   False
--

nan :: Field a => a -- | A half is a Field because it requires addition, -- multiplication and division. -- --

--   >>> half :: Double
--   0.5
--

half :: Field a => a -- | A algebraic structure with element joins: See Semilattice -- --

--   Associativity: x \/ (y \/ z) == (x \/ y) \/ z
--   Commutativity: x \/ y == y \/ x
--   Idempotency:   x \/ x == x
--

--   Associativity: x /\ (y /\ z) == (x /\ y) /\ z
--   Commutativity: x /\ y == y /\ x
--   Idempotency:   x /\ x == x
--

--   Identity: x \/ bottom == x
--

class (JoinSemiLattice a) => BoundedJoinSemiLattice a bottom :: BoundedJoinSemiLattice a => a -- | A meet-semilattice with an identity element top for /\. -- --

--   Identity: x /\ top == x
--

class (MeetSemiLattice a) => BoundedMeetSemiLattice a top :: BoundedMeetSemiLattice a => a -- | Additive Action -- --

--   m |+ zero == m
--

class (Additive (AdditiveScalar m)) => AdditiveAction m where { type AdditiveScalar m :: Type; } (|+) :: AdditiveAction m => m -> AdditiveScalar m -> m infixl 6 |+ -- | flipped additive action -- --

--   (+|) == flip (|+)
--   zero +| m = m
--

(+|) :: AdditiveAction m => AdditiveScalar m -> m -> m infixl 6 +| -- | Subtractive Action -- --

--   m |- zero = m
--

--   (-|) == (+|) . negate
--   zero -| m = negate m
--

(-|) :: (AdditiveAction m, Subtractive m) => AdditiveScalar m -> m -> m infixl 6 -| -- | Multiplicative Action -- --

--   m |* one = m
--   m |* zero = zero
--

class (Multiplicative (Scalar m)) => MultiplicativeAction m where { type Scalar m :: Type; } (|*) :: MultiplicativeAction m => m -> Scalar m -> m infixl 7 |* -- | flipped multiplicative action -- --

--   (*|) == flip (|*)
--   one *| m = one
--   zero *| m = zero
--

(*|) :: MultiplicativeAction m => Scalar m -> m -> m infixl 7 *| -- | Divisive Action -- --

--   m |/ one = m
--

class (Divisive (Scalar m), MultiplicativeAction m) => DivisiveAction m (|/) :: DivisiveAction m => m -> Scalar m -> m infixl 7 |/ -- | left scalar division -- --

--   (/|) == (*|) . recip
--   one |/ m = recip m
--

(/|) :: (MultiplicativeAction m, Divisive m) => Scalar m -> m -> m -- | A Module -- --

--   a *| one == a
--   (a + b) *| c == (a *| c) + (b *| c)
--   c |* (a + b) == (c |* a) + (c |* b)
--   a *| zero == zero
--   a *| b == b |* a
--

type Module m = (Distributive (Scalar m), MultiplicativeAction m) -- | Basis encapsulates the notion of magnitude (intuitively the -- quotienting of a higher-kinded number to a scalar one) and the basis -- on which the magnitude quotienting was performed. An instance needs to -- satisfy these laws: -- --

--   \a -> magnitude a >= zero
--   \a -> magnitude zero == zero
--   \a -> a == magnitude a *| basis a
--   \a -> magnitude (basis a) == one
--

--   >>> magnitude (-0.5 :: Double)
--   0.5
--

-- --

--   >>> basis (-0.5 :: Double)
--   -1.0
--

--   \a -> abs a * signum a ~= a
--

-- --

--   >>> abs (-1)
--   1
--

abs :: Absolute a => a -> a -- | The sign of a number. -- --

--   >>> signum (-1)
--   -1
--

-- -- abs zero == zero, so any value for signum zero is -- ok. We choose lawful neutral: -- --

--   >>> signum zero == zero
--   True
--

signum :: Sign a => a -> a -- | Distance, which combines the Subtractive notion of difference, with -- Basis. -- --

--   distance a b >= zero
--   distance a a == zero
--   distance a b *| basis (a - b) == a - b
--

distance :: (Basis a, Subtractive a) => a -> a -> Mag a -- | Convert between a "co-ordinated" or "higher-kinded" number and a -- direction. -- --

--   ray . angle == basis
--   magnitude (ray x) == one
--

--   >>> aboutEqual zero (epsilon :: Double)
--   True
--

aboutEqual :: (Epsilon a, Lattice a, Subtractive a) => a -> a -> Bool -- | Note that the constraint is Lattice rather than Ord allowing broader -- usage. -- --

--   >>> nearZero (epsilon :: Double)
--   True
--

-- --

--   >>> nearZero (epsilon :: EuclideanPair Double)
--   True
--

nearZero :: (Epsilon a, Lattice a, Subtractive a) => a -> Bool -- | About equal operator. -- --

--   >>> (1.0 + epsilon) ~= (1.0 :: Double)
--   True
--

(~=) :: Epsilon a => (Lattice a, Subtractive a) => a -> a -> Bool infixl 4 ~= -- | The underlying representation is a newtype-wrapped tuple, compared -- with the base datatype. This was chosen to facilitate the use of -- DerivingVia. newtype Complex a Complex :: (a, a) -> Complex a [complexPair] :: Complex a -> (a, a) -- | Complex number constructor. (+:) :: a -> a -> Complex a infixl 6 +: -- | Extracts the real part of a complex number. realPart :: Complex a -> a -- | Extracts the imaginary part of a complex number. imagPart :: Complex a -> a -- | An Integral is anything that satisfies the law: -- --

--   \a b -> b == zero || b * (a `div` b) + (a `mod` b) == a
--

-- --

--   >>> 3 `divMod` 2
--   (1,1)
--

-- --

--   >>> (-3) `divMod` 2
--   (-2,1)
--

-- --

--   >>> (-3) `quotRem` 2
--   (-1,-1)
--

--   toIntegral a == a
--

class ToIntegral a b toIntegral :: ToIntegral a b => a -> b -- | Convert to an Int type ToInt a = ToIntegral a Int -- | Polymorphic version of fromInteger -- --

--   fromIntegral a == a
--

class FromIntegral a b fromIntegral :: FromIntegral a b => b -> a -- | fromInteger is special in two ways: -- --

numeric integral literals (like "42") are interpreted specifically -- as "fromInteger (42 :: GHC.Num.Integer)". The prelude version is used -- as default (or whatever fromInteger is in scope if RebindableSyntax is -- set).
The default rules in haskell2010 specify that constraints -- on fromInteger need to be in a form C v, where v is a -- Num or a subclass of Num.

--   >>> even 2
--   True
--

even :: (Eq a, Integral a) => a -> Bool -- |

--   >>> odd 3
--   True
--

odd :: (Eq a, Integral a) => a -> Bool -- | raise a number to an Integral power -- --

--   >>> 2 ^^ 3
--   8.0
--

-- --

--   >>> 2 ^^ (-2)
--   0.25
--

--   >>> 2 ^ 3
--   8.0
--

-- --

--   >>> 2 ^ (-2)
--   0.25
--

(^) :: Divisive a => a -> Int -> a infixr 8 ^ -- | A rational number data Ratio a (:%) :: !a -> !a -> Ratio a -- | Arbitrary-precision rational numbers, represented as a ratio of two -- Integer values. A rational number may be constructed using the -- % operator. type Rational = Ratio Integer -- | toRatio is equivalent to Real in base, but is polymorphic in -- the Integral type. -- --

--   >>> toRatio (3.1415927 :: Float) :: Ratio Integer
--   13176795 :% 4194304
--

class ToRatio a b toRatio :: ToRatio a b => a -> Ratio b -- | Fractional in base splits into fromRatio and Field -- --

--   >>> fromRatio (5 :% 2 :: Ratio Integer) :: Double
--   2.5
--

class FromRatio a b fromRatio :: FromRatio a b => Ratio b -> a -- | fromRational is special in two ways: -- --

numeric decimal literals (like "53.66") are interpreted as exactly -- "fromRational (53.66 :: GHC.Real.Ratio Integer)". The prelude version, -- GHC.Real.fromRational is used as default (or whatever is in scope if -- RebindableSyntax is set).
The default rules in haskell2010 specify that contraints on -- fromRational need to be in a form C v, where v is a -- Num or a subclass of Num.

--   >>> reduce 72 60
--   6 :% 5
--

-- --

--   \a b -> reduce a b == a :% b || b == zero
--

--   >>> gcd 72 60
--   12
--

gcd :: (Eq a, EndoBased a, Integral a) => a -> a -> a -- | A numhask exception. newtype NumHaskException NumHaskException :: String -> NumHaskException [errorMessage] :: NumHaskException -> String -- | Throw an exception. Exceptions may be thrown from purely functional -- code, but may only be caught within the IO monad. throw :: forall (r :: RuntimeRep) (a :: TYPE r) e. Exception e => e -> a -- | A prelude composed by overlaying numhask on Prelude, together with a -- few minor tweaks needed for RebindableSyntax. module NumHask.Prelude -- | RebindableSyntax splats this, and I'm not sure where it exists in GHC -- land ifThenElse :: Bool -> a -> a -> a -- | The fromList function constructs the structure l from -- the given list of Item l fromList :: IsList l => [Item l] -> l -- | The fromListN function takes the input list's length and -- potentially uses it to construct the structure l more -- efficiently compared to fromList. If the given number does not -- equal to the input list's length the behaviour of fromListN is -- not specified. -- --

--   fromListN (length xs) xs == fromList xs
--

fromListN :: IsList l => Int -> [Item l] -> l -- | Natural number -- -- Invariant: numbers <= 0xffffffffffffffff use the NS -- constructor data Natural pattern NatJ# :: BigNat -> Natural pattern NatS# :: Word# -> Natural -- | Application operator. This operator is redundant, since ordinary -- application (f x) means the same as (f $ x). -- However, $ has low, right-associative binding precedence, so it -- sometimes allows parentheses to be omitted; for example: -- --

--   f $ g $ h x  =  f (g (h x))
--

--   ((+) `on` f) x y = f x + f y
--

-- -- Typical usage: sortBy (compare `on` -- fst). -- -- Algebraic properties: -- --

(*) `on` id = (*) -- (if (*) ∉ {⊥, const
--   ⊥})

```
((*) `on` f) `on` g = (*) `on` (f . g)
```

flip on f . flip on g = flip on (g .
--   f)

on :: (b -> b -> c) -> (a -> b) -> a -> a -> c infixl 0 `on` -- | fix f is the least fixed point of the function -- f, i.e. the least defined x such that f x = -- x. -- -- For example, we can write the factorial function using direct -- recursion as -- --

--   >>> let fac n = if n <= 1 then 1 else n * fac (n-1) in fac 5
--   120
--

-- -- This uses the fact that Haskell’s let introduces recursive -- bindings. We can rewrite this definition using fix, -- --

--   >>> fix (\rec n -> if n <= 1 then 1 else n * rec (n-1)) 5
--   120
--

-- -- Instead of making a recursive call, we introduce a dummy parameter -- rec; when used within fix, this parameter then refers -- to fix’s argument, hence the recursion is reintroduced. fix :: (a -> a) -> a -- | & is a reverse application operator. This provides -- notational convenience. Its precedence is one higher than that of the -- forward application operator $, which allows & to be -- nested in $. -- --

--   >>> 5 & (+1) & show
--   "6"
--

(&) :: a -> (a -> b) -> b infixl 1 & -- | flip f takes its (first) two arguments in the reverse -- order of f. -- --

--   >>> flip (++) "hello" "world"
--   "worldhello"
--

flip :: (a -> b -> c) -> b -> a -> c -- | const x is a unary function which evaluates to x for -- all inputs. -- --

--   >>> const 42 "hello"
--   42
--

-- --

--   >>> map (const 42) [0..3]
--   [42,42,42,42]
--

const :: a -> b -> a -- | The Foldable class represents data structures that can be reduced to a -- summary value one element at a time. Strict left-associative folds are -- a good fit for space-efficient reduction, while lazy right-associative -- folds are a good fit for corecursive iteration, or for folds that -- short-circuit after processing an initial subsequence of the -- structure's elements. -- -- Instances can be derived automatically by enabling the -- DeriveFoldable extension. For example, a derived instance for -- a binary tree might be: -- --

--   {-# LANGUAGE DeriveFoldable #-}
--   data Tree a = Empty
--               | Leaf a
--               | Node (Tree a) a (Tree a)
--       deriving Foldable
--

-- -- A more detailed description can be found in the Overview -- section of Data.Foldable#overview. -- -- For the class laws see the Laws section of -- Data.Foldable#laws. class Foldable (t :: TYPE LiftedRep -> Type) -- | Given a structure with elements whose type is a Monoid, combine -- them via the monoid's (<>) operator. This fold -- is right-associative and lazy in the accumulator. When you need a -- strict left-associative fold, use foldMap' instead, with -- id as the map. -- --

Examples

-- -- Basic usage: -- --

--   >>> fold [[1, 2, 3], [4, 5], [6], []]
--   [1,2,3,4,5,6]
--

-- --

--   >>> fold $ Node (Leaf (Sum 1)) (Sum 3) (Leaf (Sum 5))
--   Sum {getSum = 9}
--

-- -- Folds of unbounded structures do not terminate when the monoid's -- (<>) operator is strict: -- --

--   >>> fold (repeat Nothing)
--   * Hangs forever *
--

-- -- Lazy corecursive folds of unbounded structures are fine: -- --

--   >>> take 12 $ fold $ map (\i -> [i..i+2]) [0..]
--   [0,1,2,1,2,3,2,3,4,3,4,5]
--   
--   >>> sum $ take 4000000 $ fold $ map (\i -> [i..i+2]) [0..]
--   2666668666666
--

fold :: (Foldable t, Monoid m) => t m -> m -- | Map each element of the structure into a monoid, and combine the -- results with (<>). This fold is -- right-associative and lazy in the accumulator. For strict -- left-associative folds consider foldMap' instead. -- --

Examples

-- -- Basic usage: -- --

--   >>> foldMap Sum [1, 3, 5]
--   Sum {getSum = 9}
--

-- --

--   >>> foldMap Product [1, 3, 5]
--   Product {getProduct = 15}
--

-- --

--   >>> foldMap (replicate 3) [1, 2, 3]
--   [1,1,1,2,2,2,3,3,3]
--

-- -- When a Monoid's (<>) is lazy in its second -- argument, foldMap can return a result even from an unbounded -- structure. For example, lazy accumulation enables -- Data.ByteString.Builder to efficiently serialise large data -- structures and produce the output incrementally: -- --

--   >>> import qualified Data.ByteString.Lazy as L
--   
--   >>> import qualified Data.ByteString.Builder as B
--   
--   >>> let bld :: Int -> B.Builder; bld i = B.intDec i <> B.word8 0x20
--   
--   >>> let lbs = B.toLazyByteString $ foldMap bld [0..]
--   
--   >>> L.take 64 lbs
--   "0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24"
--

foldMap :: (Foldable t, Monoid m) => (a -> m) -> t a -> m -- | A left-associative variant of foldMap that is strict in the -- accumulator. Use this method for strict reduction when partial results -- are merged via (<>). -- --

Examples

-- -- Define a Monoid over finite bit strings under xor. Use -- it to strictly compute the xor of a list of Int -- values. -- --

--   >>> :set -XGeneralizedNewtypeDeriving
--   
--   >>> import Data.Bits (Bits, FiniteBits, xor, zeroBits)
--   
--   >>> import Data.Foldable (foldMap')
--   
--   >>> import Numeric (showHex)
--   
--   >>> 
--   
--   >>> newtype X a = X a deriving (Eq, Bounded, Enum, Bits, FiniteBits)
--   
--   >>> instance Bits a => Semigroup (X a) where X a <> X b = X (a `xor` b)
--   
--   >>> instance Bits a => Monoid    (X a) where mempty     = X zeroBits
--   
--   >>> 
--   
--   >>> let bits :: [Int]; bits = [0xcafe, 0xfeed, 0xdeaf, 0xbeef, 0x5411]
--   
--   >>> (\ (X a) -> showString "0x" . showHex a $ "") $ foldMap' X bits
--   "0x42"
--

foldMap' :: (Foldable t, Monoid m) => (a -> m) -> t a -> m -- | Right-associative fold of a structure, lazy in the accumulator. -- -- In the case of lists, foldr, when applied to a binary operator, -- a starting value (typically the right-identity of the operator), and a -- list, reduces the list using the binary operator, from right to left: -- --

--   foldr f z [x1, x2, ..., xn] == x1 `f` (x2 `f` ... (xn `f` z)...)
--

-- -- Note that since the head of the resulting expression is produced by an -- application of the operator to the first element of the list, given an -- operator lazy in its right argument, foldr can produce a -- terminating expression from an unbounded list. -- -- For a general Foldable structure this should be semantically -- identical to, -- --

--   foldr f z = foldr f z . toList
--

-- --

Examples

-- -- Basic usage: -- --

--   >>> foldr (||) False [False, True, False]
--   True
--

-- --

--   >>> foldr (||) False []
--   False
--

-- --

--   >>> foldr (\c acc -> acc ++ [c]) "foo" ['a', 'b', 'c', 'd']
--   "foodcba"
--

-- --

Infinite structures

-- -- ⚠️ Applying foldr to infinite structures usually doesn't -- terminate. -- -- It may still terminate under one of the following conditions: -- --

the folding function is short-circuiting
the folding function is lazy on its second argument

-- --

Short-circuiting

-- -- (||) short-circuits on True values, so the -- following terminates because there is a True value finitely far -- from the left side: -- --

--   >>> foldr (||) False (True : repeat False)
--   True
--

-- -- But the following doesn't terminate: -- --

--   >>> foldr (||) False (repeat False ++ [True])
--   * Hangs forever *
--

-- --

Laziness in the second argument

-- -- Applying foldr to infinite structures terminates when the -- operator is lazy in its second argument (the initial accumulator is -- never used in this case, and so could be left undefined, but -- [] is more clear): -- --

--   >>> take 5 $ foldr (\i acc -> i : fmap (+3) acc) [] (repeat 1)
--   [1,4,7,10,13]
--

foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b -- | Right-associative fold of a structure, strict in the accumulator. This -- is rarely what you want. foldr' :: Foldable t => (a -> b -> b) -> b -> t a -> b -- | Left-associative fold of a structure, lazy in the accumulator. This is -- rarely what you want, but can work well for structures with efficient -- right-to-left sequencing and an operator that is lazy in its left -- argument. -- -- In the case of lists, foldl, when applied to a binary operator, -- a starting value (typically the left-identity of the operator), and a -- list, reduces the list using the binary operator, from left to right: -- --

--   foldl f z [x1, x2, ..., xn] == (...((z `f` x1) `f` x2) `f`...) `f` xn
--

-- -- Note that to produce the outermost application of the operator the -- entire input list must be traversed. Like all left-associative folds, -- foldl will diverge if given an infinite list. -- -- If you want an efficient strict left-fold, you probably want to use -- foldl' instead of foldl. The reason for this is that the -- latter does not force the inner results (e.g. z `f` x1 -- in the above example) before applying them to the operator (e.g. to -- (`f` x2)). This results in a thunk chain <math> -- elements long, which then must be evaluated from the outside-in. -- -- For a general Foldable structure this should be semantically -- identical to: -- --

--   foldl f z = foldl f z . toList
--

-- --

Examples

-- -- The first example is a strict fold, which in practice is best -- performed with foldl'. -- --

--   >>> foldl (+) 42 [1,2,3,4]
--   52
--

-- -- Though the result below is lazy, the input is reversed before -- prepending it to the initial accumulator, so corecursion begins only -- after traversing the entire input string. -- --

--   >>> foldl (\acc c -> c : acc) "abcd" "efgh"
--   "hgfeabcd"
--

-- -- A left fold of a structure that is infinite on the right cannot -- terminate, even when for any finite input the fold just returns the -- initial accumulator: -- --

--   >>> foldl (\a _ -> a) 0 $ repeat 1
--   * Hangs forever *
--

-- -- WARNING: When it comes to lists, you always want to use either -- foldl' or foldr instead. foldl :: Foldable t => (b -> a -> b) -> b -> t a -> b -- | Left-associative fold of a structure but with strict application of -- the operator. -- -- This ensures that each step of the fold is forced to Weak Head Normal -- Form before being applied, avoiding the collection of thunks that -- would otherwise occur. This is often what you want to strictly reduce -- a finite structure to a single strict result (e.g. sum). -- -- For a general Foldable structure this should be semantically -- identical to, -- --

--   foldl' f z = foldl' f z . toList
--

foldl' :: Foldable t => (b -> a -> b) -> b -> t a -> b -- | A variant of foldr that has no base case, and thus may only be -- applied to non-empty structures. -- -- This function is non-total and will raise a runtime exception if the -- structure happens to be empty. -- --

Examples

-- -- Basic usage: -- --

--   >>> foldr1 (+) [1..4]
--   10
--

-- --

--   >>> foldr1 (+) []
--   Exception: Prelude.foldr1: empty list
--

-- --

--   >>> foldr1 (+) Nothing
--   *** Exception: foldr1: empty structure
--

-- --

--   >>> foldr1 (-) [1..4]
--   -2
--

-- --

--   >>> foldr1 (&&) [True, False, True, True]
--   False
--

-- --

--   >>> foldr1 (||) [False, False, True, True]
--   True
--

-- --

--   >>> foldr1 (+) [1..]
--   * Hangs forever *
--

foldr1 :: Foldable t => (a -> a -> a) -> t a -> a -- | A variant of foldl that has no base case, and thus may only be -- applied to non-empty structures. -- -- This function is non-total and will raise a runtime exception if the -- structure happens to be empty. -- --

--   foldl1 f = foldl1 f . toList
--

-- --

Examples

-- -- Basic usage: -- --

--   >>> foldl1 (+) [1..4]
--   10
--

-- --

--   >>> foldl1 (+) []
--   *** Exception: Prelude.foldl1: empty list
--

-- --

--   >>> foldl1 (+) Nothing
--   *** Exception: foldl1: empty structure
--

-- --

--   >>> foldl1 (-) [1..4]
--   -8
--

-- --

--   >>> foldl1 (&&) [True, False, True, True]
--   False
--

-- --

--   >>> foldl1 (||) [False, False, True, True]
--   True
--

-- --

--   >>> foldl1 (+) [1..]
--   * Hangs forever *
--

foldl1 :: Foldable t => (a -> a -> a) -> t a -> a -- | List of elements of a structure, from left to right. If the entire -- list is intended to be reduced via a fold, just fold the structure -- directly bypassing the list. -- --

Examples

-- -- Basic usage: -- --

--   >>> toList Nothing
--   []
--

-- --

--   >>> toList (Just 42)
--   [42]
--

-- --

--   >>> toList (Left "foo")
--   []
--

-- --

--   >>> toList (Node (Leaf 5) 17 (Node Empty 12 (Leaf 8)))
--   [5,17,12,8]
--

-- -- For lists, toList is the identity: -- --

--   >>> toList [1, 2, 3]
--   [1,2,3]
--

toList :: Foldable t => t a -> [a] -- | Test whether the structure is empty. The default implementation is -- Left-associative and lazy in both the initial element and the -- accumulator. Thus optimised for structures where the first element can -- be accessed in constant time. Structures where this is not the case -- should have a non-default implementation. -- --

Examples

-- -- Basic usage: -- --

--   >>> null []
--   True
--

-- --

--   >>> null [1]
--   False
--

-- -- null is expected to terminate even for infinite structures. The -- default implementation terminates provided the structure is bounded on -- the left (there is a leftmost element). -- --

--   >>> null [1..]
--   False
--

null :: Foldable t => t a -> Bool -- | Returns the size/length of a finite structure as an Int. The -- default implementation just counts elements starting with the -- leftmost. Instances for structures that can compute the element count -- faster than via element-by-element counting, should provide a -- specialised implementation. -- --

Examples

-- -- Basic usage: -- --

--   >>> length []
--   0
--

-- --

--   >>> length ['a', 'b', 'c']
--   3
--   
--   >>> length [1..]
--   * Hangs forever *
--

length :: Foldable t => t a -> Int -- | Does the element occur in the structure? -- -- Note: elem is often used in infix form. -- --

Examples

-- -- Basic usage: -- --

--   >>> 3 `elem` []
--   False
--

-- --

--   >>> 3 `elem` [1,2]
--   False
--

-- --

--   >>> 3 `elem` [1,2,3,4,5]
--   True
--

-- -- For infinite structures, the default implementation of elem -- terminates if the sought-after value exists at a finite distance from -- the left side of the structure: -- --

--   >>> 3 `elem` [1..]
--   True
--

-- --

--   >>> 3 `elem` ([4..] ++ [3])
--   * Hangs forever *
--

elem :: (Foldable t, Eq a) => a -> t a -> Bool -- | The largest element of a non-empty structure. -- -- This function is non-total and will raise a runtime exception if the -- structure happens to be empty. A structure that supports random access -- and maintains its elements in order should provide a specialised -- implementation to return the maximum in faster than linear time. -- --

Examples

-- -- Basic usage: -- --

--   >>> maximum [1..10]
--   10
--

-- --

--   >>> maximum []
--   *** Exception: Prelude.maximum: empty list
--

-- --

--   >>> maximum Nothing
--   *** Exception: maximum: empty structure
--

-- -- WARNING: This function is partial for possibly-empty structures like -- lists. maximum :: (Foldable t, Ord a) => t a -> a -- | The least element of a non-empty structure. -- -- This function is non-total and will raise a runtime exception if the -- structure happens to be empty. A structure that supports random access -- and maintains its elements in order should provide a specialised -- implementation to return the minimum in faster than linear time. -- --

Examples

-- -- Basic usage: -- --

--   >>> minimum [1..10]
--   1
--

-- --

--   >>> minimum []
--   *** Exception: Prelude.minimum: empty list
--

-- --

--   >>> minimum Nothing
--   *** Exception: minimum: empty structure
--

-- -- WARNING: This function is partial for possibly-empty structures like -- lists. minimum :: (Foldable t, Ord a) => t a -> a infix 4 `elem` -- | Map each element of a structure to an Applicative action, -- evaluate these actions from left to right, and ignore the results. For -- a version that doesn't ignore the results see traverse. -- -- traverse_ is just like mapM_, but generalised to -- Applicative actions. -- --

Examples

-- -- Basic usage: -- --

--   >>> traverse_ print ["Hello", "world", "!"]
--   "Hello"
--   "world"
--   "!"
--

traverse_ :: (Foldable t, Applicative f) => (a -> f b) -> t a -> f () -- | Evaluate each monadic action in the structure from left to right, and -- ignore the results. For a version that doesn't ignore the results see -- sequence. -- -- sequence_ is just like sequenceA_, but specialised to -- monadic actions. sequence_ :: (Foldable t, Monad m) => t (m a) -> m () -- | Evaluate each action in the structure from left to right, and ignore -- the results. For a version that doesn't ignore the results see -- sequenceA. -- -- sequenceA_ is just like sequence_, but generalised to -- Applicative actions. -- --

Examples

-- -- Basic usage: -- --

--   >>> sequenceA_ [print "Hello", print "world", print "!"]
--   "Hello"
--   "world"
--   "!"
--

sequenceA_ :: (Foldable t, Applicative f) => t (f a) -> f () -- | or returns the disjunction of a container of Bools. For the -- result to be False, the container must be finite; True, -- however, results from a True value finitely far from the left -- end. -- --

Examples

-- -- Basic usage: -- --

--   >>> or []
--   False
--

-- --

--   >>> or [True]
--   True
--

-- --

--   >>> or [False]
--   False
--

-- --

--   >>> or [True, True, False]
--   True
--

-- --

--   >>> or (True : repeat False) -- Infinite list [True,False,False,False,...
--   True
--

-- --

--   >>> or (repeat False)
--   * Hangs forever *
--

or :: Foldable t => t Bool -> Bool -- | notElem is the negation of elem. -- --

Examples

-- -- Basic usage: -- --

--   >>> 3 `notElem` []
--   True
--

-- --

--   >>> 3 `notElem` [1,2]
--   True
--

-- --

--   >>> 3 `notElem` [1,2,3,4,5]
--   False
--

-- -- For infinite structures, notElem terminates if the value exists -- at a finite distance from the left side of the structure: -- --

--   >>> 3 `notElem` [1..]
--   False
--

-- --

--   >>> 3 `notElem` ([4..] ++ [3])
--   * Hangs forever *
--

notElem :: (Foldable t, Eq a) => a -> t a -> Bool infix 4 `notElem` -- | The sum of a collection of actions, generalizing concat. -- -- msum is just like asum, but specialised to -- MonadPlus. msum :: (Foldable t, MonadPlus m) => t (m a) -> m a -- | The least element of a non-empty structure with respect to the given -- comparison function. -- --

Examples

-- -- Basic usage: -- --

--   >>> minimumBy (compare `on` length) ["Hello", "World", "!", "Longest", "bar"]
--   "!"
--

-- -- WARNING: This function is partial for possibly-empty structures like -- lists. minimumBy :: Foldable t => (a -> a -> Ordering) -> t a -> a -- | The largest element of a non-empty structure with respect to the given -- comparison function. -- --

Examples

-- -- Basic usage: -- --

--   >>> maximumBy (compare `on` length) ["Hello", "World", "!", "Longest", "bar"]
--   "Longest"
--

-- -- WARNING: This function is partial for possibly-empty structures like -- lists. maximumBy :: Foldable t => (a -> a -> Ordering) -> t a -> a -- | Map each element of a structure to a monadic action, evaluate these -- actions from left to right, and ignore the results. For a version that -- doesn't ignore the results see mapM. -- -- mapM_ is just like traverse_, but specialised to monadic -- actions. mapM_ :: (Foldable t, Monad m) => (a -> m b) -> t a -> m () -- | for_ is traverse_ with its arguments flipped. For a -- version that doesn't ignore the results see for. This is -- forM_ generalised to Applicative actions. -- -- for_ is just like forM_, but generalised to -- Applicative actions. -- --

Examples

-- -- Basic usage: -- --

--   >>> for_ [1..4] print
--   1
--   2
--   3
--   4
--

for_ :: (Foldable t, Applicative f) => t a -> (a -> f b) -> f () -- | forM_ is mapM_ with its arguments flipped. For a version -- that doesn't ignore the results see forM. -- -- forM_ is just like for_, but specialised to monadic -- actions. forM_ :: (Foldable t, Monad m) => t a -> (a -> m b) -> m () -- | Right-to-left monadic fold over the elements of a structure. -- -- Given a structure t with elements (a, b, c, ..., x, -- y), the result of a fold with an operator function f is -- equivalent to: -- --

--   foldrM f z t = do
--       yy <- f y z
--       xx <- f x yy
--       ...
--       bb <- f b cc
--       aa <- f a bb
--       return aa -- Just @return z@ when the structure is empty
--

-- -- For a Monad m, given two functions f1 :: a -> m b -- and f2 :: b -> m c, their Kleisli composition (f1 -- >=> f2) :: a -> m c is defined by: -- --

--   (f1 >=> f2) a = f1 a >>= f2
--

-- -- Another way of thinking about foldrM is that it amounts to an -- application to z of a Kleisli composition: -- --

--   foldrM f z t = f y >=> f x >=> ... >=> f b >=> f a $ z
--

-- -- The monadic effects of foldrM are sequenced from right to -- left, and e.g. folds of infinite lists will diverge. -- -- If at some step the bind operator (>>=) -- short-circuits (as with, e.g., mzero in a MonadPlus), -- the evaluated effects will be from a tail of the element sequence. If -- you want to evaluate the monadic effects in left-to-right order, or -- perhaps be able to short-circuit after an initial sequence of -- elements, you'll need to use foldlM instead. -- -- If the monadic effects don't short-circuit, the outermost application -- of f is to the leftmost element a, so that, ignoring -- effects, the result looks like a right fold: -- --

--   a `f` (b `f` (c `f` (... (x `f` (y `f` z))))).
--

-- --

Examples

-- -- Basic usage: -- --

--   >>> let f i acc = do { print i ; return $ i : acc }
--   
--   >>> foldrM f [] [0..3]
--   3
--   2
--   1
--   0
--   [0,1,2,3]
--

foldrM :: (Foldable t, Monad m) => (a -> b -> m b) -> b -> t a -> m b -- | Left-to-right monadic fold over the elements of a structure. -- -- Given a structure t with elements (a, b, ..., w, x, -- y), the result of a fold with an operator function f is -- equivalent to: -- --

--   foldlM f z t = do
--       aa <- f z a
--       bb <- f aa b
--       ...
--       xx <- f ww x
--       yy <- f xx y
--       return yy -- Just @return z@ when the structure is empty
--

-- -- For a Monad m, given two functions f1 :: a -> m b -- and f2 :: b -> m c, their Kleisli composition (f1 -- >=> f2) :: a -> m c is defined by: -- --

--   (f1 >=> f2) a = f1 a >>= f2
--

-- -- Another way of thinking about foldlM is that it amounts to an -- application to z of a Kleisli composition: -- --

--   foldlM f z t =
--       flip f a >=> flip f b >=> ... >=> flip f x >=> flip f y $ z
--

-- -- The monadic effects of foldlM are sequenced from left to -- right. -- -- If at some step the bind operator (>>=) -- short-circuits (as with, e.g., mzero in a MonadPlus), -- the evaluated effects will be from an initial segment of the element -- sequence. If you want to evaluate the monadic effects in right-to-left -- order, or perhaps be able to short-circuit after processing a tail of -- the sequence of elements, you'll need to use foldrM instead. -- -- If the monadic effects don't short-circuit, the outermost application -- of f is to the rightmost element y, so that, -- ignoring effects, the result looks like a left fold: -- --

--   ((((z `f` a) `f` b) ... `f` w) `f` x) `f` y
--

-- --

Examples

-- -- Basic usage: -- --

--   >>> let f a e = do { print e ; return $ e : a }
--   
--   >>> foldlM f [] [0..3]
--   0
--   1
--   2
--   3
--   [3,2,1,0]
--

foldlM :: (Foldable t, Monad m) => (b -> a -> m b) -> b -> t a -> m b -- | The find function takes a predicate and a structure and returns -- the leftmost element of the structure matching the predicate, or -- Nothing if there is no such element. -- --

Examples

-- -- Basic usage: -- --

--   >>> find (> 42) [0, 5..]
--   Just 45
--

-- --

--   >>> find (> 12) [1..7]
--   Nothing
--

find :: Foldable t => (a -> Bool) -> t a -> Maybe a -- | Map a function over all the elements of a container and concatenate -- the resulting lists. -- --

Examples

-- -- Basic usage: -- --

--   >>> concatMap (take 3) [[1..], [10..], [100..], [1000..]]
--   [1,2,3,10,11,12,100,101,102,1000,1001,1002]
--

-- --

--   >>> concatMap (take 3) (Just [1..])
--   [1,2,3]
--

concatMap :: Foldable t => (a -> [b]) -> t a -> [b] -- | The concatenation of all the elements of a container of lists. -- --

Examples

-- -- Basic usage: -- --

--   >>> concat (Just [1, 2, 3])
--   [1,2,3]
--

-- --

--   >>> concat (Left 42)
--   []
--

-- --

--   >>> concat [[1, 2, 3], [4, 5], [6], []]
--   [1,2,3,4,5,6]
--

concat :: Foldable t => t [a] -> [a] -- | The sum of a collection of actions, generalizing concat. -- -- asum is just like msum, but generalised to -- Alternative. -- --

Examples

-- -- Basic usage: -- --

--   >>> asum [Just "Hello", Nothing, Just "World"]
--   Just "Hello"
--

asum :: (Foldable t, Alternative f) => t (f a) -> f a -- | Determines whether any element of the structure satisfies the -- predicate. -- --

Examples

-- -- Basic usage: -- --

--   >>> any (> 3) []
--   False
--

-- --

--   >>> any (> 3) [1,2]
--   False
--

-- --

--   >>> any (> 3) [1,2,3,4,5]
--   True
--

-- --

--   >>> any (> 3) [1..]
--   True
--

-- --

--   >>> any (> 3) [0, -1..]
--   * Hangs forever *
--

any :: Foldable t => (a -> Bool) -> t a -> Bool -- | and returns the conjunction of a container of Bools. For the -- result to be True, the container must be finite; False, -- however, results from a False value finitely far from the left -- end. -- --

Examples

-- -- Basic usage: -- --

--   >>> and []
--   True
--

-- --

--   >>> and [True]
--   True
--

-- --

--   >>> and [False]
--   False
--

-- --

--   >>> and [True, True, False]
--   False
--

-- --

--   >>> and (False : repeat True) -- Infinite list [False,True,True,True,...
--   False
--

-- --

--   >>> and (repeat True)
--   * Hangs forever *
--

and :: Foldable t => t Bool -> Bool -- | Determines whether all elements of the structure satisfy the -- predicate. -- --

Examples

-- -- Basic usage: -- --

--   >>> all (> 3) []
--   True
--

-- --

--   >>> all (> 3) [1,2]
--   False
--

-- --

--   >>> all (> 3) [1,2,3,4,5]
--   False
--

-- --

--   >>> all (> 3) [1..]
--   False
--

-- --

--   >>> all (> 3) [4..]
--   * Hangs forever *
--

all :: Foldable t => (a -> Bool) -> t a -> Bool -- | Append two lists, i.e., -- --

--   [x1, ..., xm] ++ [y1, ..., yn] == [x1, ..., xm, y1, ..., yn]
--   [x1, ..., xm] ++ [y1, ...] == [x1, ..., xm, y1, ...]
--

-- -- If the first list is not finite, the result is the first list. (++) :: [a] -> [a] -> [a] infixr 5 ++ -- | The value of seq a b is bottom if a is bottom, and -- otherwise equal to b. In other words, it evaluates the first -- argument a to weak head normal form (WHNF). seq is -- usually introduced to improve performance by avoiding unneeded -- laziness. -- -- A note on evaluation order: the expression seq a b does -- not guarantee that a will be evaluated before -- b. The only guarantee given by seq is that the both -- a and b will be evaluated before seq -- returns a value. In particular, this means that b may be -- evaluated before a. If you need to guarantee a specific order -- of evaluation, you must use the function pseq from the -- "parallel" package. seq :: forall {r :: RuntimeRep} a (b :: TYPE r). a -> b -> b infixr 0 `seq` -- | <math>. filter, applied to a predicate and a list, -- returns the list of those elements that satisfy the predicate; i.e., -- --

--   filter p xs = [ x | x <- xs, p x]
--

-- --

--   >>> filter odd [1, 2, 3]
--   [1,3]
--

filter :: (a -> Bool) -> [a] -> [a] -- | <math>. zip takes two lists and returns a list of -- corresponding pairs. -- --

--   >>> zip [1, 2] ['a', 'b']
--   [(1,'a'),(2,'b')]
--

-- -- If one input list is shorter than the other, excess elements of the -- longer list are discarded, even if one of the lists is infinite: -- --

--   >>> zip [1] ['a', 'b']
--   [(1,'a')]
--   
--   >>> zip [1, 2] ['a']
--   [(1,'a')]
--   
--   >>> zip [] [1..]
--   []
--   
--   >>> zip [1..] []
--   []
--

-- -- zip is right-lazy: -- --

--   >>> zip [] undefined
--   []
--   
--   >>> zip undefined []
--   *** Exception: Prelude.undefined
--   ...
--

-- -- zip is capable of list fusion, but it is restricted to its -- first list argument and its resulting list. zip :: [a] -> [b] -> [(a, b)] -- | The print function outputs a value of any printable type to the -- standard output device. Printable types are those that are instances -- of class Show; print converts values to strings for -- output using the show operation and adds a newline. -- -- For example, a program to print the first 20 integers and their powers -- of 2 could be written as: -- --

--   main = print ([(n, 2^n) | n <- [0..19]])
--

print :: Show a => a -> IO () -- | Extract the first component of a pair. fst :: (a, b) -> a -- | Extract the second component of a pair. snd :: (a, b) -> b -- | otherwise is defined as the value True. It helps to make -- guards more readable. eg. -- --

--   f x | x < 0     = ...
--       | otherwise = ...
--

otherwise :: Bool -- | <math>. map f xs is the list obtained by -- applying f to each element of xs, i.e., -- --

--   map f [x1, x2, ..., xn] == [f x1, f x2, ..., f xn]
--   map f [x1, x2, ...] == [f x1, f x2, ...]
--

-- --

--   >>> map (+1) [1, 2, 3]
--   [2,3,4]
--

map :: (a -> b) -> [a] -> [b] -- | Application operator. This operator is redundant, since ordinary -- application (f x) means the same as (f $ x). -- However, $ has low, right-associative binding precedence, so it -- sometimes allows parentheses to be omitted; for example: -- --

--   f $ g $ h x  =  f (g (h x))
--

-- -- It is also useful in higher-order situations, such as map -- ($ 0) xs, or zipWith ($) fs xs. -- -- Note that ($) is levity-polymorphic in its result -- type, so that foo $ True where foo :: Bool -> -- Int# is well-typed. ($) :: forall (r :: RuntimeRep) a (b :: TYPE r). (a -> b) -> a -> b infixr 0 $ -- | general coercion to fractional types realToFrac :: (Real a, Fractional b) => a -> b -- | The Bounded class is used to name the upper and lower limits of -- a type. Ord is not a superclass of Bounded since types -- that are not totally ordered may also have upper and lower bounds. -- -- The Bounded class may be derived for any enumeration type; -- minBound is the first constructor listed in the data -- declaration and maxBound is the last. Bounded may also -- be derived for single-constructor datatypes whose constituent types -- are in Bounded. class Bounded a minBound :: Bounded a => a maxBound :: Bounded a => a -- | Class Enum defines operations on sequentially ordered types. -- -- The enumFrom... methods are used in Haskell's translation of -- arithmetic sequences. -- -- Instances of Enum may be derived for any enumeration type -- (types whose constructors have no fields). The nullary constructors -- are assumed to be numbered left-to-right by fromEnum from -- 0 through n-1. See Chapter 10 of the Haskell -- Report for more details. -- -- For any type that is an instance of class Bounded as well as -- Enum, the following should hold: -- --

The calls succ maxBound and pred -- minBound should result in a runtime error.
fromEnum and toEnum should give a runtime error if -- the result value is not representable in the result type. For example, -- toEnum 7 :: Bool is an error.
enumFrom and enumFromThen should be defined with an -- implicit bound, thus:

-- --

--   enumFrom     x   = enumFromTo     x maxBound
--   enumFromThen x y = enumFromThenTo x y bound
--     where
--       bound | fromEnum y >= fromEnum x = maxBound
--             | otherwise                = minBound
--

class Enum a -- | the successor of a value. For numeric types, succ adds 1. succ :: Enum a => a -> a -- | the predecessor of a value. For numeric types, pred subtracts -- 1. pred :: Enum a => a -> a -- | Convert from an Int. toEnum :: Enum a => Int -> a -- | Convert to an Int. It is implementation-dependent what -- fromEnum returns when applied to a value that is too large to -- fit in an Int. fromEnum :: Enum a => a -> Int -- | Used in Haskell's translation of [n..] with [n..] = -- enumFrom n, a possible implementation being enumFrom n = n : -- enumFrom (succ n). For example: -- --

```
enumFrom 4 :: [Integer] = [4,5,6,7,...]
```

enumFrom 6 :: [Int] = [6,7,8,9,...,maxBound ::
--   Int]

enumFrom :: Enum a => a -> [a] -- | Used in Haskell's translation of [n,n'..] with [n,n'..] = -- enumFromThen n n', a possible implementation being -- enumFromThen n n' = n : n' : worker (f x) (f x n'), -- worker s v = v : worker s (s v), x = fromEnum n' - -- fromEnum n and f n y | n > 0 = f (n - 1) (succ y) | n < -- 0 = f (n + 1) (pred y) | otherwise = y For example: -- --

enumFromThen 4 6 :: [Integer] = [4,6,8,10...]

enumFromThen 6 2 :: [Int] = [6,2,-2,-6,...,minBound ::
--   Int]

enumFromThen :: Enum a => a -> a -> [a] -- | Used in Haskell's translation of [n..m] with [n..m] = -- enumFromTo n m, a possible implementation being enumFromTo n -- m | n <= m = n : enumFromTo (succ n) m | otherwise = []. For -- example: -- --

```
enumFromTo 6 10 :: [Int] = [6,7,8,9,10]
```
```
enumFromTo 42 1 :: [Integer] = []
```

enumFromTo :: Enum a => a -> a -> [a] -- | Used in Haskell's translation of [n,n'..m] with [n,n'..m] -- = enumFromThenTo n n' m, a possible implementation being -- enumFromThenTo n n' m = worker (f x) (c x) n m, x = -- fromEnum n' - fromEnum n, c x = bool (>=) ((x -- 0) f n y | n > 0 = f (n - 1) (succ y) | n < 0 = f (n + -- 1) (pred y) | otherwise = y and worker s c v m | c v m = v : -- worker s c (s v) m | otherwise = [] For example: -- --

enumFromThenTo 4 2 -6 :: [Integer] =
--   [4,2,0,-2,-4,-6]

```
enumFromThenTo 6 8 2 :: [Int] = []
```

enumFromThenTo :: Enum a => a -> a -> a -> [a] -- | The Eq class defines equality (==) and inequality -- (/=). All the basic datatypes exported by the Prelude -- are instances of Eq, and Eq may be derived for any -- datatype whose constituents are also instances of Eq. -- -- The Haskell Report defines no laws for Eq. However, instances -- are encouraged to follow these properties: -- --

Reflexivity x == x = True
Symmetry x == y = y == x
Transitivity if x == y && y == z = -- True, then x == z = True
Extensionality if x == y = True and -- f is a function whose return type is an instance of -- Eq, then f x == f y = True
Negation x /= y = not (x == -- y)

-- -- Minimal complete definition: either == or /=. class Eq a (==) :: Eq a => a -> a -> Bool (/=) :: Eq a => a -> a -> Bool infix 4 == infix 4 /= -- | Trigonometric and hyperbolic functions and related functions. -- -- The Haskell Report defines no laws for Floating. However, -- (+), (*) and exp are -- customarily expected to define an exponential field and have the -- following properties: -- --

exp (a + b) = exp a * exp b
exp (fromInteger 0) = fromInteger 1

class Fractional a => Floating a -- | Fractional numbers, supporting real division. -- -- The Haskell Report defines no laws for Fractional. However, -- (+) and (*) are customarily expected -- to define a division ring and have the following properties: -- --

recip gives the multiplicative inverse x -- * recip x = recip x * x = fromInteger 1

-- -- Note that it isn't customarily expected that a type instance of -- Fractional implement a field. However, all instances in -- base do. class Num a => Fractional a -- | The Monad class defines the basic operations over a -- monad, a concept from a branch of mathematics known as -- category theory. From the perspective of a Haskell programmer, -- however, it is best to think of a monad as an abstract datatype -- of actions. Haskell's do expressions provide a convenient -- syntax for writing monadic expressions. -- -- Instances of Monad should satisfy the following: -- --

Left identity return a >>= k = k -- a
Right identity m >>= return = -- m
Associativity m >>= (\x -> k x -- >>= h) = (m >>= k) >>= -- h

-- -- Furthermore, the Monad and Applicative operations should -- relate as follows: -- --

```
pure = return
```

m1 <*> m2 = m1 >>= (x1 -> m2
--   >>= (x2 -> return (x1 x2)))

-- -- The above laws imply: -- --

```
fmap f xs = xs >>= return .
--   f
```
```
(>>) = (*>)
```

-- -- and that pure and (<*>) satisfy the applicative -- functor laws. -- -- The instances of Monad for lists, Maybe and IO -- defined in the Prelude satisfy these laws. class Applicative m => Monad (m :: Type -> Type) -- | Sequentially compose two actions, passing any value produced by the -- first as an argument to the second. -- -- 'as >>= bs' can be understood as the do -- expression -- --

--   do a <- as
--      bs a
--

(>>=) :: Monad m => m a -> (a -> m b) -> m b -- | Sequentially compose two actions, discarding any value produced by the -- first, like sequencing operators (such as the semicolon) in imperative -- languages. -- -- 'as >> bs' can be understood as the do -- expression -- --

--   do as
--      bs
--

(>>) :: Monad m => m a -> m b -> m b -- | Inject a value into the monadic type. return :: Monad m => a -> m a infixl 1 >>= infixl 1 >> -- | A type f is a Functor if it provides a function fmap -- which, given any types a and b lets you apply any -- function from (a -> b) to turn an f a into an -- f b, preserving the structure of f. Furthermore -- f needs to adhere to the following: -- --

Identity fmap id == id
Composition fmap (f . g) == fmap f . -- fmap g

-- -- Note, that the second law follows from the free theorem of the type -- fmap and the first law, so you need only check that the former -- condition holds. class Functor (f :: Type -> Type) -- | fmap is used to apply a function of type (a -> b) -- to a value of type f a, where f is a functor, to produce a -- value of type f b. Note that for any type constructor with -- more than one parameter (e.g., Either), only the last type -- parameter can be modified with fmap (e.g., b in -- `Either a b`). -- -- Some type constructors with two parameters or more have a -- Bifunctor instance that allows both the last and the -- penultimate parameters to be mapped over. -- --

Examples

-- -- Convert from a Maybe Int to a Maybe String -- using show: -- --

--   >>> fmap show Nothing
--   Nothing
--   
--   >>> fmap show (Just 3)
--   Just "3"
--

-- -- Convert from an Either Int Int to an Either Int -- String using show: -- --

--   >>> fmap show (Left 17)
--   Left 17
--   
--   >>> fmap show (Right 17)
--   Right "17"
--

-- -- Double each element of a list: -- --

--   >>> fmap (*2) [1,2,3]
--   [2,4,6]
--

-- -- Apply even to the second element of a pair: -- --

--   >>> fmap even (2,2)
--   (2,True)
--

-- -- It may seem surprising that the function is only applied to the last -- element of the tuple compared to the list example above which applies -- it to every element in the list. To understand, remember that tuples -- are type constructors with multiple type parameters: a tuple of 3 -- elements (a,b,c) can also be written (,,) a b c and -- its Functor instance is defined for Functor ((,,) a -- b) (i.e., only the third parameter is free to be mapped over with -- fmap). -- -- It explains why fmap can be used with tuples containing -- values of different types as in the following example: -- --

--   >>> fmap even ("hello", 1.0, 4)
--   ("hello",1.0,True)
--

fmap :: Functor f => (a -> b) -> f a -> f b -- | Replace all locations in the input with the same value. The default -- definition is fmap . const, but this may be -- overridden with a more efficient version. (<$) :: Functor f => a -> f b -> f a infixl 4 <$ -- | Basic numeric class. -- -- The Haskell Report defines no laws for Num. However, -- (+) and (*) are customarily expected -- to define a ring and have the following properties: -- --

Associativity of (+) (x + y) + -- z = x + (y + z)
Commutativity of (+) x + y -- = y + x
fromInteger 0 is the additive -- identity x + fromInteger 0 = x
negate gives the additive inverse x + -- negate x = fromInteger 0
Associativity of (*) (x * y) * -- z = x * (y * z)
fromInteger 1 is the multiplicative -- identity x * fromInteger 1 = x and -- fromInteger 1 * x = x
Distributivity of (*) with respect to -- (+) a * (b + c) = (a * b) + (a * -- c) and (b + c) * a = (b * a) + (c * a)

-- -- Note that it isn't customarily expected that a type instance of -- both Num and Ord implement an ordered ring. Indeed, in -- base only Integer and Rational do. class Num a -- | The Ord class is used for totally ordered datatypes. -- -- Instances of Ord can be derived for any user-defined datatype -- whose constituent types are in Ord. The declared order of the -- constructors in the data declaration determines the ordering in -- derived Ord instances. The Ordering datatype allows a -- single comparison to determine the precise ordering of two objects. -- -- Ord, as defined by the Haskell report, implements a total order -- and has the following properties: -- --

Comparability x <= y || y <= x = -- True
Transitivity if x <= y && y <= -- z = True, then x <= z = True
Reflexivity x <= x = True
Antisymmetry if x <= y && y <= -- x = True, then x == y = True

-- -- The following operator interactions are expected to hold: -- --

x >= y = y <= x
x < y = x <= y && x /= y
x > y = y < x
x < y = compare x y == LT
x > y = compare x y == GT
x == y = compare x y == EQ
min x y == if x <= y then x else y = True
max x y == if x >= y then x else y = True

-- -- Note that (7.) and (8.) do not require min and -- max to return either of their arguments. The result is merely -- required to equal one of the arguments in terms of (==). -- -- Minimal complete definition: either compare or <=. -- Using compare can be more efficient for complex types. class Eq a => Ord a compare :: Ord a => a -> a -> Ordering (<) :: Ord a => a -> a -> Bool (<=) :: Ord a => a -> a -> Bool (>) :: Ord a => a -> a -> Bool (>=) :: Ord a => a -> a -> Bool max :: Ord a => a -> a -> a min :: Ord a => a -> a -> a infix 4 < infix 4 <= infix 4 > infix 4 >= -- | Parsing of Strings, producing values. -- -- Derived instances of Read make the following assumptions, which -- derived instances of Show obey: -- --

If the constructor is defined to be an infix operator, then the -- derived Read instance will parse only infix applications of the -- constructor (not the prefix form).
Associativity is not used to reduce the occurrence of parentheses, -- although precedence may be.
If the constructor is defined using record syntax, the derived -- Read will parse only the record-syntax form, and furthermore, -- the fields must be given in the same order as the original -- declaration.
The derived Read instance allows arbitrary Haskell -- whitespace between tokens of the input string. Extra parentheses are -- also allowed.

-- -- For example, given the declarations -- --

--   infixr 5 :^:
--   data Tree a =  Leaf a  |  Tree a :^: Tree a
--

-- -- the derived instance of Read in Haskell 2010 is equivalent to -- --

--   instance (Read a) => Read (Tree a) where
--   
--           readsPrec d r =  readParen (d > app_prec)
--                            (\r -> [(Leaf m,t) |
--                                    ("Leaf",s) <- lex r,
--                                    (m,t) <- readsPrec (app_prec+1) s]) r
--   
--                         ++ readParen (d > up_prec)
--                            (\r -> [(u:^:v,w) |
--                                    (u,s) <- readsPrec (up_prec+1) r,
--                                    (":^:",t) <- lex s,
--                                    (v,w) <- readsPrec (up_prec+1) t]) r
--   
--             where app_prec = 10
--                   up_prec = 5
--

-- -- Note that right-associativity of :^: is unused. -- -- The derived instance in GHC is equivalent to -- --

--   instance (Read a) => Read (Tree a) where
--   
--           readPrec = parens $ (prec app_prec $ do
--                                    Ident "Leaf" <- lexP
--                                    m <- step readPrec
--                                    return (Leaf m))
--   
--                        +++ (prec up_prec $ do
--                                    u <- step readPrec
--                                    Symbol ":^:" <- lexP
--                                    v <- step readPrec
--                                    return (u :^: v))
--   
--             where app_prec = 10
--                   up_prec = 5
--   
--           readListPrec = readListPrecDefault
--

-- -- Why do both readsPrec and readPrec exist, and why does -- GHC opt to implement readPrec in derived Read instances -- instead of readsPrec? The reason is that readsPrec is -- based on the ReadS type, and although ReadS is mentioned -- in the Haskell 2010 Report, it is not a very efficient parser data -- structure. -- -- readPrec, on the other hand, is based on a much more efficient -- ReadPrec datatype (a.k.a "new-style parsers"), but its -- definition relies on the use of the RankNTypes language -- extension. Therefore, readPrec (and its cousin, -- readListPrec) are marked as GHC-only. Nevertheless, it is -- recommended to use readPrec instead of readsPrec -- whenever possible for the efficiency improvements it brings. -- -- As mentioned above, derived Read instances in GHC will -- implement readPrec instead of readsPrec. The default -- implementations of readsPrec (and its cousin, readList) -- will simply use readPrec under the hood. If you are writing a -- Read instance by hand, it is recommended to write it like so: -- --

--   instance Read T where
--     readPrec     = ...
--     readListPrec = readListPrecDefault
--

class Read a -- | attempts to parse a value from the front of the string, returning a -- list of (parsed value, remaining string) pairs. If there is no -- successful parse, the returned list is empty. -- -- Derived instances of Read and Show satisfy the -- following: -- --

(x,"") is an element of (readsPrec d -- (showsPrec d x "")).

-- -- That is, readsPrec parses the string produced by -- showsPrec, and delivers the value that showsPrec started -- with. readsPrec :: Read a => Int -> ReadS a -- | The method readList is provided to allow the programmer to give -- a specialised way of parsing lists of values. For example, this is -- used by the predefined Read instance of the Char type, -- where values of type String should be are expected to use -- double quotes, rather than square brackets. readList :: Read a => ReadS [a] -- | Efficient, machine-independent access to the components of a -- floating-point number. class (RealFrac a, Floating a) => RealFloat a -- | a constant function, returning the radix of the representation (often -- 2) floatRadix :: RealFloat a => a -> Integer -- | a constant function, returning the number of digits of -- floatRadix in the significand floatDigits :: RealFloat a => a -> Int -- | a constant function, returning the lowest and highest values the -- exponent may assume floatRange :: RealFloat a => a -> (Int, Int) -- | The function decodeFloat applied to a real floating-point -- number returns the significand expressed as an Integer and an -- appropriately scaled exponent (an Int). If -- decodeFloat x yields (m,n), then x -- is equal in value to m*b^^n, where b is the -- floating-point radix, and furthermore, either m and -- n are both zero or else b^(d-1) <= abs m < -- b^d, where d is the value of floatDigits -- x. In particular, decodeFloat 0 = (0,0). If the -- type contains a negative zero, also decodeFloat (-0.0) = -- (0,0). The result of decodeFloat x is -- unspecified if either of isNaN x or -- isInfinite x is True. decodeFloat :: RealFloat a => a -> (Integer, Int) -- | encodeFloat performs the inverse of decodeFloat in the -- sense that for finite x with the exception of -0.0, -- uncurry encodeFloat (decodeFloat x) = x. -- encodeFloat m n is one of the two closest -- representable floating-point numbers to m*b^^n (or -- ±Infinity if overflow occurs); usually the closer, but if -- m contains too many bits, the result may be rounded in the -- wrong direction. encodeFloat :: RealFloat a => Integer -> Int -> a -- | exponent corresponds to the second component of -- decodeFloat. exponent 0 = 0 and for finite -- nonzero x, exponent x = snd (decodeFloat x) -- + floatDigits x. If x is a finite floating-point -- number, it is equal in value to significand x * b ^^ -- exponent x, where b is the floating-point radix. -- The behaviour is unspecified on infinite or NaN values. exponent :: RealFloat a => a -> Int -- | The first component of decodeFloat, scaled to lie in the open -- interval (-1,1), either 0.0 or of absolute -- value >= 1/b, where b is the floating-point -- radix. The behaviour is unspecified on infinite or NaN -- values. significand :: RealFloat a => a -> a -- | multiplies a floating-point number by an integer power of the radix scaleFloat :: RealFloat a => Int -> a -> a -- | True if the argument is an IEEE "not-a-number" (NaN) value isNaN :: RealFloat a => a -> Bool -- | True if the argument is an IEEE infinity or negative infinity isInfinite :: RealFloat a => a -> Bool -- | True if the argument is too small to be represented in -- normalized format isDenormalized :: RealFloat a => a -> Bool -- | True if the argument is an IEEE negative zero isNegativeZero :: RealFloat a => a -> Bool -- | True if the argument is an IEEE floating point number isIEEE :: RealFloat a => a -> Bool -- | Extracting components of fractions. class (Real a, Fractional a) => RealFrac a -- | Conversion of values to readable Strings. -- -- Derived instances of Show have the following properties, which -- are compatible with derived instances of Read: -- --

The result of show is a syntactically correct Haskell -- expression containing only constants, given the fixity declarations in -- force at the point where the type is declared. It contains only the -- constructor names defined in the data type, parentheses, and spaces. -- When labelled constructor fields are used, braces, commas, field -- names, and equal signs are also used.
If the constructor is defined to be an infix operator, then -- showsPrec will produce infix applications of the -- constructor.
the representation will be enclosed in parentheses if the -- precedence of the top-level constructor in x is less than -- d (associativity is ignored). Thus, if d is -- 0 then the result is never surrounded in parentheses; if -- d is 11 it is always surrounded in parentheses, -- unless it is an atomic expression.
If the constructor is defined using record syntax, then -- show will produce the record-syntax form, with the fields given -- in the same order as the original declaration.

-- -- For example, given the declarations -- --

--   infixr 5 :^:
--   data Tree a =  Leaf a  |  Tree a :^: Tree a
--

-- -- the derived instance of Show is equivalent to -- --

--   instance (Show a) => Show (Tree a) where
--   
--          showsPrec d (Leaf m) = showParen (d > app_prec) $
--               showString "Leaf " . showsPrec (app_prec+1) m
--            where app_prec = 10
--   
--          showsPrec d (u :^: v) = showParen (d > up_prec) $
--               showsPrec (up_prec+1) u .
--               showString " :^: "      .
--               showsPrec (up_prec+1) v
--            where up_prec = 5
--

-- -- Note that right-associativity of :^: is ignored. For example, -- --

show (Leaf 1 :^: Leaf 2 :^: Leaf 3) produces the -- string "Leaf 1 :^: (Leaf 2 :^: Leaf 3)".

class Show a -- | Convert a value to a readable String. -- -- showsPrec should satisfy the law -- --

--   showsPrec d x r ++ s  ==  showsPrec d x (r ++ s)
--

-- -- Derived instances of Read and Show satisfy the -- following: -- --

(x,"") is an element of (readsPrec d -- (showsPrec d x "")).

-- -- That is, readsPrec parses the string produced by -- showsPrec, and delivers the value that showsPrec started -- with. showsPrec :: Show a => Int -> a -> ShowS -- | A specialised variant of showsPrec, using precedence context -- zero, and returning an ordinary String. show :: Show a => a -> String -- | The method showList is provided to allow the programmer to give -- a specialised way of showing lists of values. For example, this is -- used by the predefined Show instance of the Char type, -- where values of type String should be shown in double quotes, -- rather than between square brackets. showList :: Show a => [a] -> ShowS -- | When a value is bound in do-notation, the pattern on the left -- hand side of <- might not match. In this case, this class -- provides a function to recover. -- -- A Monad without a MonadFail instance may only be used in -- conjunction with pattern that always match, such as newtypes, tuples, -- data types with only a single data constructor, and irrefutable -- patterns (~pat). -- -- Instances of MonadFail should satisfy the following law: -- fail s should be a left zero for >>=, -- --

--   fail s >>= f  =  fail s
--

-- -- If your Monad is also MonadPlus, a popular definition is -- --

--   fail _ = mzero
--

class Monad m => MonadFail (m :: Type -> Type) fail :: MonadFail m => String -> m a -- | A functor with application, providing operations to -- --

embed pure expressions (pure), and
sequence computations and combine their results (<*> -- and liftA2).

-- -- A minimal complete definition must include implementations of -- pure and of either <*> or liftA2. If it -- defines both, then they must behave the same as their default -- definitions: -- --

--   (<*>) = liftA2 id
--

-- --

--   liftA2 f x y = f <$> x <*> y
--

-- -- Further, any definition must satisfy the following: -- --

Identity
```
pure id <*> v =
--   v
```

Composition

pure (.) <*> u
--   <*> v <*> w = u <*> (v
--   <*> w)

Homomorphism
```
pure f <*>
--   pure x = pure (f x)
```
Interchange
```
u <*> pure y =
--   pure ($ y) <*> u
```

-- -- The other methods have the following default definitions, which may be -- overridden with equivalent specialized implementations: -- --

```
u *> v = (id <$ u)
--   <*> v
```
```
u <* v = liftA2 const u v
```

-- -- As a consequence of these laws, the Functor instance for -- f will satisfy -- --

```
fmap f x = pure f <*> x
```

-- -- It may be useful to note that supposing -- --

--   forall x y. p (q x y) = f x . g y
--

-- -- it follows from the above that -- --

--   liftA2 p (liftA2 q u v) = liftA2 f u . liftA2 g v
--

-- -- If f is also a Monad, it should satisfy -- --

```
pure = return
```

m1 <*> m2 = m1 >>= (x1 -> m2
--   >>= (x2 -> return (x1 x2)))

```
(*>) = (>>)
```

-- -- (which implies that pure and <*> satisfy the -- applicative functor laws). class Functor f => Applicative (f :: Type -> Type) -- | Lift a value. pure :: Applicative f => a -> f a -- | Sequential application. -- -- A few functors support an implementation of <*> that is -- more efficient than the default one. -- --

Example

-- -- Used in combination with (<$>), -- (<*>) can be used to build a record. -- --

--   >>> data MyState = MyState {arg1 :: Foo, arg2 :: Bar, arg3 :: Baz}
--

-- --

--   >>> produceFoo :: Applicative f => f Foo
--

-- --

--   >>> produceBar :: Applicative f => f Bar
--   
--   >>> produceBaz :: Applicative f => f Baz
--

-- --

--   >>> mkState :: Applicative f => f MyState
--   
--   >>> mkState = MyState <$> produceFoo <*> produceBar <*> produceBaz
--

(<*>) :: Applicative f => f (a -> b) -> f a -> f b -- | Sequence actions, discarding the value of the first argument. -- --

Examples

-- -- If used in conjunction with the Applicative instance for Maybe, -- you can chain Maybe computations, with a possible "early return" in -- case of Nothing. -- --

--   >>> Just 2 *> Just 3
--   Just 3
--

-- --

--   >>> Nothing *> Just 3
--   Nothing
--

-- -- Of course a more interesting use case would be to have effectful -- computations instead of just returning pure values. -- --

--   >>> import Data.Char
--   
--   >>> import Text.ParserCombinators.ReadP
--   
--   >>> let p = string "my name is " *> munch1 isAlpha <* eof
--   
--   >>> readP_to_S p "my name is Simon"
--   [("Simon","")]
--

(*>) :: Applicative f => f a -> f b -> f b -- | Sequence actions, discarding the value of the second argument. (<*) :: Applicative f => f a -> f b -> f a infixl 4 <* infixl 4 *> infixl 4 <*> -- | The Foldable class represents data structures that can be reduced to a -- summary value one element at a time. Strict left-associative folds are -- a good fit for space-efficient reduction, while lazy right-associative -- folds are a good fit for corecursive iteration, or for folds that -- short-circuit after processing an initial subsequence of the -- structure's elements. -- -- Instances can be derived automatically by enabling the -- DeriveFoldable extension. For example, a derived instance for -- a binary tree might be: -- --

--   {-# LANGUAGE DeriveFoldable #-}
--   data Tree a = Empty
--               | Leaf a
--               | Node (Tree a) a (Tree a)
--       deriving Foldable
--

-- -- A more detailed description can be found in the Overview -- section of Data.Foldable#overview. -- -- For the class laws see the Laws section of -- Data.Foldable#laws. class Foldable (t :: TYPE LiftedRep -> Type) -- | Map each element of the structure into a monoid, and combine the -- results with (<>). This fold is -- right-associative and lazy in the accumulator. For strict -- left-associative folds consider foldMap' instead. -- --

Examples

-- -- Basic usage: -- --

--   >>> foldMap Sum [1, 3, 5]
--   Sum {getSum = 9}
--

-- --

--   >>> foldMap Product [1, 3, 5]
--   Product {getProduct = 15}
--

-- --

--   >>> foldMap (replicate 3) [1, 2, 3]
--   [1,1,1,2,2,2,3,3,3]
--

--   >>> import qualified Data.ByteString.Lazy as L
--   
--   >>> import qualified Data.ByteString.Builder as B
--   
--   >>> let bld :: Int -> B.Builder; bld i = B.intDec i <> B.word8 0x20
--   
--   >>> let lbs = B.toLazyByteString $ foldMap bld [0..]
--   
--   >>> L.take 64 lbs
--   "0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24"
--

foldMap :: (Foldable t, Monoid m) => (a -> m) -> t a -> m -- | Right-associative fold of a structure, lazy in the accumulator. -- -- In the case of lists, foldr, when applied to a binary operator, -- a starting value (typically the right-identity of the operator), and a -- list, reduces the list using the binary operator, from right to left: -- --

--   foldr f z [x1, x2, ..., xn] == x1 `f` (x2 `f` ... (xn `f` z)...)
--

--   foldr f z = foldr f z . toList
--

-- --

Examples

-- -- Basic usage: -- --

--   >>> foldr (||) False [False, True, False]
--   True
--

-- --

--   >>> foldr (||) False []
--   False
--

-- --

--   >>> foldr (\c acc -> acc ++ [c]) "foo" ['a', 'b', 'c', 'd']
--   "foodcba"
--

-- --

Infinite structures

-- -- ⚠️ Applying foldr to infinite structures usually doesn't -- terminate. -- -- It may still terminate under one of the following conditions: -- --

the folding function is short-circuiting
the folding function is lazy on its second argument

-- --

Short-circuiting

-- -- (||) short-circuits on True values, so the -- following terminates because there is a True value finitely far -- from the left side: -- --

--   >>> foldr (||) False (True : repeat False)
--   True
--

-- -- But the following doesn't terminate: -- --

--   >>> foldr (||) False (repeat False ++ [True])
--   * Hangs forever *
--

-- --

Laziness in the second argument

--   >>> take 5 $ foldr (\i acc -> i : fmap (+3) acc) [] (repeat 1)
--   [1,4,7,10,13]
--

foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b -- | Left-associative fold of a structure, lazy in the accumulator. This is -- rarely what you want, but can work well for structures with efficient -- right-to-left sequencing and an operator that is lazy in its left -- argument. -- -- In the case of lists, foldl, when applied to a binary operator, -- a starting value (typically the left-identity of the operator), and a -- list, reduces the list using the binary operator, from left to right: -- --

--   foldl f z [x1, x2, ..., xn] == (...((z `f` x1) `f` x2) `f`...) `f` xn
--

--   foldl f z = foldl f z . toList
--

-- --

Examples

-- -- The first example is a strict fold, which in practice is best -- performed with foldl'. -- --

--   >>> foldl (+) 42 [1,2,3,4]
--   52
--

-- -- Though the result below is lazy, the input is reversed before -- prepending it to the initial accumulator, so corecursion begins only -- after traversing the entire input string. -- --

--   >>> foldl (\acc c -> c : acc) "abcd" "efgh"
--   "hgfeabcd"
--

-- -- A left fold of a structure that is infinite on the right cannot -- terminate, even when for any finite input the fold just returns the -- initial accumulator: -- --

--   >>> foldl (\a _ -> a) 0 $ repeat 1
--   * Hangs forever *
--

-- -- WARNING: When it comes to lists, you always want to use either -- foldl' or foldr instead. foldl :: Foldable t => (b -> a -> b) -> b -> t a -> b -- | A variant of foldr that has no base case, and thus may only be -- applied to non-empty structures. -- -- This function is non-total and will raise a runtime exception if the -- structure happens to be empty. -- --

Examples

-- -- Basic usage: -- --

--   >>> foldr1 (+) [1..4]
--   10
--

-- --

--   >>> foldr1 (+) []
--   Exception: Prelude.foldr1: empty list
--

-- --

--   >>> foldr1 (+) Nothing
--   *** Exception: foldr1: empty structure
--

-- --

--   >>> foldr1 (-) [1..4]
--   -2
--

-- --

--   >>> foldr1 (&&) [True, False, True, True]
--   False
--

-- --

--   >>> foldr1 (||) [False, False, True, True]
--   True
--

-- --

--   >>> foldr1 (+) [1..]
--   * Hangs forever *
--

--   foldl1 f = foldl1 f . toList
--

-- --

Examples

-- -- Basic usage: -- --

--   >>> foldl1 (+) [1..4]
--   10
--

-- --

--   >>> foldl1 (+) []
--   *** Exception: Prelude.foldl1: empty list
--

-- --

--   >>> foldl1 (+) Nothing
--   *** Exception: foldl1: empty structure
--

-- --

--   >>> foldl1 (-) [1..4]
--   -8
--

-- --

--   >>> foldl1 (&&) [True, False, True, True]
--   False
--

-- --

--   >>> foldl1 (||) [False, False, True, True]
--   True
--

-- --

--   >>> foldl1 (+) [1..]
--   * Hangs forever *
--

foldl1 :: Foldable t => (a -> a -> a) -> t a -> a -- | Test whether the structure is empty. The default implementation is -- Left-associative and lazy in both the initial element and the -- accumulator. Thus optimised for structures where the first element can -- be accessed in constant time. Structures where this is not the case -- should have a non-default implementation. -- --

Examples

-- -- Basic usage: -- --

--   >>> null []
--   True
--

-- --

--   >>> null [1]
--   False
--

-- -- null is expected to terminate even for infinite structures. The -- default implementation terminates provided the structure is bounded on -- the left (there is a leftmost element). -- --

--   >>> null [1..]
--   False
--

Examples

-- -- Basic usage: -- --

--   >>> length []
--   0
--

-- --

--   >>> length ['a', 'b', 'c']
--   3
--   
--   >>> length [1..]
--   * Hangs forever *
--

length :: Foldable t => t a -> Int -- | Does the element occur in the structure? -- -- Note: elem is often used in infix form. -- --

Examples

-- -- Basic usage: -- --

--   >>> 3 `elem` []
--   False
--

-- --

--   >>> 3 `elem` [1,2]
--   False
--

-- --

--   >>> 3 `elem` [1,2,3,4,5]
--   True
--

-- -- For infinite structures, the default implementation of elem -- terminates if the sought-after value exists at a finite distance from -- the left side of the structure: -- --

--   >>> 3 `elem` [1..]
--   True
--

-- --

--   >>> 3 `elem` ([4..] ++ [3])
--   * Hangs forever *
--

Examples

-- -- Basic usage: -- --

--   >>> maximum [1..10]
--   10
--

-- --

--   >>> maximum []
--   *** Exception: Prelude.maximum: empty list
--

-- --

--   >>> maximum Nothing
--   *** Exception: maximum: empty structure
--

Examples

-- -- Basic usage: -- --

--   >>> minimum [1..10]
--   1
--

-- --

--   >>> minimum []
--   *** Exception: Prelude.minimum: empty list
--

-- --

--   >>> minimum Nothing
--   *** Exception: minimum: empty structure
--

-- -- WARNING: This function is partial for possibly-empty structures like -- lists. minimum :: (Foldable t, Ord a) => t a -> a infix 4 `elem` -- | Functors representing data structures that can be transformed to -- structures of the same shape by performing an -- Applicative (or, therefore, Monad) action on each -- element from left to right. -- -- A more detailed description of what same shape means, the -- various methods, how traversals are constructed, and example advanced -- use-cases can be found in the Overview section of -- Data.Traversable#overview. -- -- For the class laws see the Laws section of -- Data.Traversable#laws. class (Functor t, Foldable t) => Traversable (t :: Type -> Type) -- | Map each element of a structure to an action, evaluate these actions -- from left to right, and collect the results. For a version that -- ignores the results see traverse_. -- --

Examples

-- -- Basic usage: -- -- In the first two examples we show each evaluated action mapping to the -- output structure. -- --

--   >>> traverse Just [1,2,3,4]
--   Just [1,2,3,4]
--

-- --

--   >>> traverse id [Right 1, Right 2, Right 3, Right 4]
--   Right [1,2,3,4]
--

-- -- In the next examples, we show that Nothing and Left -- values short circuit the created structure. -- --

--   >>> traverse (const Nothing) [1,2,3,4]
--   Nothing
--

-- --

--   >>> traverse (\x -> if odd x then Just x else Nothing)  [1,2,3,4]
--   Nothing
--

-- --

--   >>> traverse id [Right 1, Right 2, Right 3, Right 4, Left 0]
--   Left 0
--

traverse :: (Traversable t, Applicative f) => (a -> f b) -> t a -> f (t b) -- | Evaluate each action in the structure from left to right, and collect -- the results. For a version that ignores the results see -- sequenceA_. -- --

Examples

-- -- Basic usage: -- -- For the first two examples we show sequenceA fully evaluating a a -- structure and collecting the results. -- --

--   >>> sequenceA [Just 1, Just 2, Just 3]
--   Just [1,2,3]
--

-- --

--   >>> sequenceA [Right 1, Right 2, Right 3]
--   Right [1,2,3]
--

-- -- The next two example show Nothing and Just will short -- circuit the resulting structure if present in the input. For more -- context, check the Traversable instances for Either and -- Maybe. -- --

--   >>> sequenceA [Just 1, Just 2, Just 3, Nothing]
--   Nothing
--

-- --

--   >>> sequenceA [Right 1, Right 2, Right 3, Left 4]
--   Left 4
--

sequenceA :: (Traversable t, Applicative f) => t (f a) -> f (t a) -- | Map each element of a structure to a monadic action, evaluate these -- actions from left to right, and collect the results. For a version -- that ignores the results see mapM_. -- --

Examples

-- -- mapM is literally a traverse with a type signature -- restricted to Monad. Its implementation may be more efficient -- due to additional power of Monad. mapM :: (Traversable t, Monad m) => (a -> m b) -> t a -> m (t b) -- | Evaluate each monadic action in the structure from left to right, and -- collect the results. For a version that ignores the results see -- sequence_. -- --

Examples

-- -- Basic usage: -- -- The first two examples are instances where the input and and output of -- sequence are isomorphic. -- --

--   >>> sequence $ Right [1,2,3,4]
--   [Right 1,Right 2,Right 3,Right 4]
--

-- --

--   >>> sequence $ [Right 1,Right 2,Right 3,Right 4]
--   Right [1,2,3,4]
--

-- -- The following examples demonstrate short circuit behavior for -- sequence. -- --

--   >>> sequence $ Left [1,2,3,4]
--   Left [1,2,3,4]
--

-- --

--   >>> sequence $ [Left 0, Right 1,Right 2,Right 3,Right 4]
--   Left 0
--

sequence :: (Traversable t, Monad m) => t (m a) -> m (t a) -- | The class of semigroups (types with an associative binary operation). -- -- Instances should satisfy the following: -- --

Associativity x <> (y <> z) = -- (x <> y) <> z

class Semigroup a -- | An associative operation. -- --

--   >>> [1,2,3] <> [4,5,6]
--   [1,2,3,4,5,6]
--

(<>) :: Semigroup a => a -> a -> a infixr 6 <> -- | The class of monoids (types with an associative binary operation that -- has an identity). Instances should satisfy the following: -- --

Right identity x <> mempty = -- x
Left identity mempty <> x = -- x
Associativity x <> (y <> z) = -- (x <> y) <> z (Semigroup -- law)
Concatenation mconcat = foldr -- (<>) mempty

-- -- The method names refer to the monoid of lists under concatenation, but -- there are many other instances. -- -- Some types can be viewed as a monoid in more than one way, e.g. both -- addition and multiplication on numbers. In such cases we often define -- newtypes and make those instances of Monoid, e.g. -- Sum and Product. -- -- NOTE: Semigroup is a superclass of Monoid since -- base-4.11.0.0. class Semigroup a => Monoid a -- | Identity of mappend -- --

--   >>> "Hello world" <> mempty
--   "Hello world"
--

mempty :: Monoid a => a -- | An associative operation -- -- NOTE: This method is redundant and has the default -- implementation mappend = (<>) since -- base-4.11.0.0. Should it be implemented manually, since -- mappend is a synonym for (<>), it is expected that -- the two functions are defined the same way. In a future GHC release -- mappend will be removed from Monoid. mappend :: Monoid a => a -> a -> a -- | Fold a list using the monoid. -- -- For most types, the default definition for mconcat will be -- used, but the function is included in the class definition so that an -- optimized version can be provided for specific types. -- --

--   >>> mconcat ["Hello", " ", "Haskell", "!"]
--   "Hello Haskell!"
--

mconcat :: Monoid a => [a] -> a -- | A String is a list of characters. String constants in Haskell -- are values of type String. -- -- See Data.List for operations on lists. type String = [Char] -- | The character type Char is an enumeration whose values -- represent Unicode (or equivalently ISO/IEC 10646) code points (i.e. -- characters, see http://www.unicode.org/ for details). This set -- extends the ISO 8859-1 (Latin-1) character set (the first 256 -- characters), which is itself an extension of the ASCII character set -- (the first 128 characters). A character literal in Haskell has type -- Char. -- -- To convert a Char to or from the corresponding Int value -- defined by Unicode, use toEnum and fromEnum from the -- Enum class respectively (or equivalently ord and -- chr). data Char -- | Double-precision floating point numbers. It is desirable that this -- type be at least equal in range and precision to the IEEE -- double-precision type. data Double -- | Single-precision floating point numbers. It is desirable that this -- type be at least equal in range and precision to the IEEE -- single-precision type. data Float -- | A fixed-precision integer type with at least the range [-2^29 .. -- 2^29-1]. The exact range for a given implementation can be -- determined by using minBound and maxBound from the -- Bounded class. data Int -- | Arbitrary precision integers. In contrast with fixed-size integral -- types such as Int, the Integer type represents the -- entire infinite range of integers. -- -- Integers are stored in a kind of sign-magnitude form, hence do not -- expect two's complement form when using bit operations. -- -- If the value is small (fit into an Int), IS constructor -- is used. Otherwise Integer and IN constructors are used -- to store a BigNat representing respectively the positive or the -- negative value magnitude. -- -- Invariant: Integer and IN are used iff value doesn't fit -- in IS data Integer -- | The Maybe type encapsulates an optional value. A value of type -- Maybe a either contains a value of type a -- (represented as Just a), or it is empty (represented -- as Nothing). Using Maybe is a good way to deal with -- errors or exceptional cases without resorting to drastic measures such -- as error. -- -- The Maybe type is also a monad. It is a simple kind of error -- monad, where all errors are represented by Nothing. A richer -- error monad can be built using the Either type. data Maybe a Nothing :: Maybe a Just :: a -> Maybe a -- | Arbitrary-precision rational numbers, represented as a ratio of two -- Integer values. A rational number may be constructed using the -- % operator. type Rational = Ratio Integer -- | A value of type IO a is a computation which, when -- performed, does some I/O before returning a value of type a. -- -- There is really only one way to "perform" an I/O action: bind it to -- Main.main in your program. When your program is run, the I/O -- will be performed. It isn't possible to perform I/O from an arbitrary -- function, unless that function is itself in the IO monad and -- called at some point, directly or indirectly, from Main.main. -- -- IO is a monad, so IO actions can be combined using -- either the do-notation or the >> and >>= -- operations from the Monad class. data IO a -- | A Word is an unsigned integral type, with the same size as -- Int. data Word -- | The Either type represents values with two possibilities: a -- value of type Either a b is either Left -- a or Right b. -- -- The Either type is sometimes used to represent a value which is -- either correct or an error; by convention, the Left constructor -- is used to hold an error value and the Right constructor is -- used to hold a correct value (mnemonic: "right" also means "correct"). -- --

Examples

-- -- The type Either String Int is the type -- of values which can be either a String or an Int. The -- Left constructor can be used only on Strings, and the -- Right constructor can be used only on Ints: -- --

--   >>> let s = Left "foo" :: Either String Int
--   
--   >>> s
--   Left "foo"
--   
--   >>> let n = Right 3 :: Either String Int
--   
--   >>> n
--   Right 3
--   
--   >>> :type s
--   s :: Either String Int
--   
--   >>> :type n
--   n :: Either String Int
--

-- -- The fmap from our Functor instance will ignore -- Left values, but will apply the supplied function to values -- contained in a Right: -- --

--   >>> let s = Left "foo" :: Either String Int
--   
--   >>> let n = Right 3 :: Either String Int
--   
--   >>> fmap (*2) s
--   Left "foo"
--   
--   >>> fmap (*2) n
--   Right 6
--

-- -- The Monad instance for Either allows us to chain -- together multiple actions which may fail, and fail overall if any of -- the individual steps failed. First we'll write a function that can -- either parse an Int from a Char, or fail. -- --

--   >>> import Data.Char ( digitToInt, isDigit )
--   
--   >>> :{
--       let parseEither :: Char -> Either String Int
--           parseEither c
--             | isDigit c = Right (digitToInt c)
--             | otherwise = Left "parse error"
--   
--   >>> :}
--

-- -- The following should work, since both '1' and '2' -- can be parsed as Ints. -- --

--   >>> :{
--       let parseMultiple :: Either String Int
--           parseMultiple = do
--             x <- parseEither '1'
--             y <- parseEither '2'
--             return (x + y)
--   
--   >>> :}
--

-- --

--   >>> parseMultiple
--   Right 3
--

-- -- But the following should fail overall, since the first operation where -- we attempt to parse 'm' as an Int will fail: -- --

--   >>> :{
--       let parseMultiple :: Either String Int
--           parseMultiple = do
--             x <- parseEither 'm'
--             y <- parseEither '2'
--             return (x + y)
--   
--   >>> :}
--

-- --

--   >>> parseMultiple
--   Left "parse error"
--

data Either a b Left :: a -> Either a b Right :: b -> Either a b -- | The computation writeFile file str function writes the -- string str, to the file file. writeFile :: FilePath -> String -> IO () -- | The readLn function combines getLine and readIO. readLn :: Read a => IO a -- | The readIO function is similar to read except that it -- signals parse failure to the IO monad instead of terminating -- the program. readIO :: Read a => String -> IO a -- | The readFile function reads a file and returns the contents of -- the file as a string. The file is read lazily, on demand, as with -- getContents. readFile :: FilePath -> IO String -- | The same as putStr, but adds a newline character. putStrLn :: String -> IO () -- | Write a string to the standard output device (same as hPutStr -- stdout). putStr :: String -> IO () -- | Write a character to the standard output device (same as -- hPutChar stdout). putChar :: Char -> IO () -- | The interact function takes a function of type -- String->String as its argument. The entire input from the -- standard input device is passed to this function as its argument, and -- the resulting string is output on the standard output device. interact :: (String -> String) -> IO () -- | Read a line from the standard input device (same as hGetLine -- stdin). getLine :: IO String -- | The getContents operation returns all user input as a single -- string, which is read lazily as it is needed (same as -- hGetContents stdin). getContents :: IO String -- | Read a character from the standard input device (same as -- hGetChar stdin). getChar :: IO Char -- | The computation appendFile file str function appends -- the string str, to the file file. -- -- Note that writeFile and appendFile write a literal -- string to a file. To write a value of any printable type, as with -- print, use the show function to convert the value to a -- string first. -- --

--   main = appendFile "squares" (show [(x,x*x) | x <- [0,0.1..2]])
--

appendFile :: FilePath -> String -> IO () -- | Raise an IOException in the IO monad. ioError :: IOError -> IO a -- | File and directory names are values of type String, whose -- precise meaning is operating system dependent. Files can be opened, -- yielding a handle which can then be used to operate on the contents of -- that file. type FilePath = String -- | The Haskell 2010 type for exceptions in the IO monad. Any I/O -- operation may raise an IOException instead of returning a -- result. For a more general type of exception, including also those -- that arise in pure code, see Exception. -- -- In Haskell 2010, this is an opaque type. type IOError = IOException -- | Construct an IOException value with a string describing the -- error. The fail method of the IO instance of the -- Monad class raises a userError, thus: -- --

--   instance Monad IO where
--     ...
--     fail s = ioError (userError s)
--

userError :: String -> IOError -- | Evaluate each monadic action in the structure from left to right, and -- ignore the results. For a version that doesn't ignore the results see -- sequence. -- -- sequence_ is just like sequenceA_, but specialised to -- monadic actions. sequence_ :: (Foldable t, Monad m) => t (m a) -> m () -- | or returns the disjunction of a container of Bools. For the -- result to be False, the container must be finite; True, -- however, results from a True value finitely far from the left -- end. -- --

Examples

-- -- Basic usage: -- --

--   >>> or []
--   False
--

-- --

--   >>> or [True]
--   True
--

-- --

--   >>> or [False]
--   False
--

-- --

--   >>> or [True, True, False]
--   True
--

-- --

--   >>> or (True : repeat False) -- Infinite list [True,False,False,False,...
--   True
--

-- --

--   >>> or (repeat False)
--   * Hangs forever *
--

or :: Foldable t => t Bool -> Bool -- | notElem is the negation of elem. -- --

Examples

-- -- Basic usage: -- --

--   >>> 3 `notElem` []
--   True
--

-- --

--   >>> 3 `notElem` [1,2]
--   True
--

-- --

--   >>> 3 `notElem` [1,2,3,4,5]
--   False
--

-- -- For infinite structures, notElem terminates if the value exists -- at a finite distance from the left side of the structure: -- --

--   >>> 3 `notElem` [1..]
--   False
--

-- --

--   >>> 3 `notElem` ([4..] ++ [3])
--   * Hangs forever *
--

notElem :: (Foldable t, Eq a) => a -> t a -> Bool infix 4 `notElem` -- | Map each element of a structure to a monadic action, evaluate these -- actions from left to right, and ignore the results. For a version that -- doesn't ignore the results see mapM. -- -- mapM_ is just like traverse_, but specialised to monadic -- actions. mapM_ :: (Foldable t, Monad m) => (a -> m b) -> t a -> m () -- | Map a function over all the elements of a container and concatenate -- the resulting lists. -- --

Examples

-- -- Basic usage: -- --

--   >>> concatMap (take 3) [[1..], [10..], [100..], [1000..]]
--   [1,2,3,10,11,12,100,101,102,1000,1001,1002]
--

-- --

--   >>> concatMap (take 3) (Just [1..])
--   [1,2,3]
--

concatMap :: Foldable t => (a -> [b]) -> t a -> [b] -- | The concatenation of all the elements of a container of lists. -- --

Examples

-- -- Basic usage: -- --

--   >>> concat (Just [1, 2, 3])
--   [1,2,3]
--

-- --

--   >>> concat (Left 42)
--   []
--

-- --

--   >>> concat [[1, 2, 3], [4, 5], [6], []]
--   [1,2,3,4,5,6]
--

concat :: Foldable t => t [a] -> [a] -- | Determines whether any element of the structure satisfies the -- predicate. -- --

Examples

-- -- Basic usage: -- --

--   >>> any (> 3) []
--   False
--

-- --

--   >>> any (> 3) [1,2]
--   False
--

-- --

--   >>> any (> 3) [1,2,3,4,5]
--   True
--

-- --

--   >>> any (> 3) [1..]
--   True
--

-- --

--   >>> any (> 3) [0, -1..]
--   * Hangs forever *
--

Examples

-- -- Basic usage: -- --

--   >>> and []
--   True
--

-- --

--   >>> and [True]
--   True
--

-- --

--   >>> and [False]
--   False
--

-- --

--   >>> and [True, True, False]
--   False
--

-- --

--   >>> and (False : repeat True) -- Infinite list [False,True,True,True,...
--   False
--

-- --

--   >>> and (repeat True)
--   * Hangs forever *
--

and :: Foldable t => t Bool -> Bool -- | Determines whether all elements of the structure satisfy the -- predicate. -- --

Examples

-- -- Basic usage: -- --

--   >>> all (> 3) []
--   True
--

-- --

--   >>> all (> 3) [1,2]
--   False
--

-- --

--   >>> all (> 3) [1,2,3,4,5]
--   False
--

-- --

--   >>> all (> 3) [1..]
--   False
--

-- --

--   >>> all (> 3) [4..]
--   * Hangs forever *
--

all :: Foldable t => (a -> Bool) -> t a -> Bool -- | words breaks a string up into a list of words, which were -- delimited by white space. -- --

--   >>> words "Lorem ipsum\ndolor"
--   ["Lorem","ipsum","dolor"]
--

words :: String -> [String] -- | unwords is an inverse operation to words. It joins words -- with separating spaces. -- --

--   >>> unwords ["Lorem", "ipsum", "dolor"]
--   "Lorem ipsum dolor"
--

unwords :: [String] -> String -- | unlines is an inverse operation to lines. It joins -- lines, after appending a terminating newline to each. -- --

--   >>> unlines ["Hello", "World", "!"]
--   "Hello\nWorld\n!\n"
--

unlines :: [String] -> String -- | lines breaks a string up into a list of strings at newline -- characters. The resulting strings do not contain newlines. -- -- Note that after splitting the string at newline characters, the last -- part of the string is considered a line even if it doesn't end with a -- newline. For example, -- --

--   >>> lines ""
--   []
--

-- --

--   >>> lines "\n"
--   [""]
--

-- --

--   >>> lines "one"
--   ["one"]
--

-- --

--   >>> lines "one\n"
--   ["one"]
--

-- --

--   >>> lines "one\n\n"
--   ["one",""]
--

-- --

--   >>> lines "one\ntwo"
--   ["one","two"]
--

-- --

--   >>> lines "one\ntwo\n"
--   ["one","two"]
--

-- -- Thus lines s contains at least as many elements as -- newlines in s. lines :: String -> [String] -- | equivalent to readsPrec with a precedence of 0. reads :: Read a => ReadS a -- | The read function reads input from a string, which must be -- completely consumed by the input process. read fails with an -- error if the parse is unsuccessful, and it is therefore -- discouraged from being used in real applications. Use readMaybe -- or readEither for safe alternatives. -- --

--   >>> read "123" :: Int
--   123
--

-- --

--   >>> read "hello" :: Int
--   *** Exception: Prelude.read: no parse
--

read :: Read a => String -> a -- | Case analysis for the Either type. If the value is -- Left a, apply the first function to a; if it -- is Right b, apply the second function to b. -- --

Examples

-- -- We create two values of type Either String -- Int, one using the Left constructor and another -- using the Right constructor. Then we apply "either" the -- length function (if we have a String) or the "times-two" -- function (if we have an Int): -- --

--   >>> let s = Left "foo" :: Either String Int
--   
--   >>> let n = Right 3 :: Either String Int
--   
--   >>> either length (*2) s
--   3
--   
--   >>> either length (*2) n
--   6
--

either :: (a -> c) -> (b -> c) -> Either a b -> c -- | readParen True p parses what p parses, -- but surrounded with parentheses. -- -- readParen False p parses what p -- parses, but optionally surrounded with parentheses. readParen :: Bool -> ReadS a -> ReadS a -- | The lex function reads a single lexeme from the input, -- discarding initial white space, and returning the characters that -- constitute the lexeme. If the input string contains only white space, -- lex returns a single successful `lexeme' consisting of the -- empty string. (Thus lex "" = [("","")].) If there is -- no legal lexeme at the beginning of the input string, lex fails -- (i.e. returns []). -- -- This lexer is not completely faithful to the Haskell lexical syntax in -- the following respects: -- --

Qualified names are not handled properly
Octal and hexadecimal numerics are not recognized as a single -- token
Comments are not treated properly

lex :: ReadS String -- | A parser for a type a, represented as a function that takes a -- String and returns a list of possible parses as -- (a,String) pairs. -- -- Note that this kind of backtracking parser is very inefficient; -- reading a large structure may be quite slow (cf ReadP). type ReadS a = String -> [(a, String)] -- | lcm x y is the smallest positive integer that both -- x and y divide. lcm :: Integral a => a -> a -> a -- | The shows functions return a function that prepends the -- output String to an existing String. This allows -- constant-time concatenation of results using function composition. type ShowS = String -> String -- | equivalent to showsPrec with a precedence of 0. shows :: Show a => a -> ShowS -- | utility function converting a String to a show function that -- simply prepends the string unchanged. showString :: String -> ShowS -- | utility function that surrounds the inner show function with -- parentheses when the Bool parameter is True. showParen :: Bool -> ShowS -> ShowS -- | utility function converting a Char to a show function that -- simply prepends the character unchanged. showChar :: Char -> ShowS -- | The zipWith3 function takes a function which combines three -- elements, as well as three lists and returns a list of the function -- applied to corresponding elements, analogous to zipWith. It is -- capable of list fusion, but it is restricted to its first list -- argument and its resulting list. -- --

--   zipWith3 (,,) xs ys zs == zip3 xs ys zs
--   zipWith3 f [x1,x2,x3..] [y1,y2,y3..] [z1,z2,z3..] == [f x1 y1 z1, f x2 y2 z2, f x3 y3 z3..]
--

zipWith3 :: (a -> b -> c -> d) -> [a] -> [b] -> [c] -> [d] -- | <math>. zipWith generalises zip by zipping with -- the function given as the first argument, instead of a tupling -- function. -- --

--   zipWith (,) xs ys == zip xs ys
--   zipWith f [x1,x2,x3..] [y1,y2,y3..] == [f x1 y1, f x2 y2, f x3 y3..]
--

-- -- For example, zipWith (+) is applied to two lists to -- produce the list of corresponding sums: -- --

--   >>> zipWith (+) [1, 2, 3] [4, 5, 6]
--   [5,7,9]
--

-- -- zipWith is right-lazy: -- --

--   >>> let f = undefined
--   
--   >>> zipWith f [] undefined
--   []
--

-- -- zipWith is capable of list fusion, but it is restricted to its -- first list argument and its resulting list. zipWith :: (a -> b -> c) -> [a] -> [b] -> [c] -- | zip3 takes three lists and returns a list of triples, analogous -- to zip. It is capable of list fusion, but it is restricted to -- its first list argument and its resulting list. zip3 :: [a] -> [b] -> [c] -> [(a, b, c)] -- | The unzip3 function takes a list of triples and returns three -- lists, analogous to unzip. -- --

--   >>> unzip3 []
--   ([],[],[])
--   
--   >>> unzip3 [(1, 'a', True), (2, 'b', False)]
--   ([1,2],"ab",[True,False])
--

unzip3 :: [(a, b, c)] -> ([a], [b], [c]) -- | unzip transforms a list of pairs into a list of first -- components and a list of second components. -- --

--   >>> unzip []
--   ([],[])
--   
--   >>> unzip [(1, 'a'), (2, 'b')]
--   ([1,2],"ab")
--

unzip :: [(a, b)] -> ([a], [b]) -- | takeWhile, applied to a predicate p and a list -- xs, returns the longest prefix (possibly empty) of -- xs of elements that satisfy p. -- --

--   >>> takeWhile (< 3) [1,2,3,4,1,2,3,4]
--   [1,2]
--   
--   >>> takeWhile (< 9) [1,2,3]
--   [1,2,3]
--   
--   >>> takeWhile (< 0) [1,2,3]
--   []
--

takeWhile :: (a -> Bool) -> [a] -> [a] -- | take n, applied to a list xs, returns the -- prefix of xs of length n, or xs itself if -- n >= length xs. -- --

--   >>> take 5 "Hello World!"
--   "Hello"
--   
--   >>> take 3 [1,2,3,4,5]
--   [1,2,3]
--   
--   >>> take 3 [1,2]
--   [1,2]
--   
--   >>> take 3 []
--   []
--   
--   >>> take (-1) [1,2]
--   []
--   
--   >>> take 0 [1,2]
--   []
--

-- -- It is an instance of the more general genericTake, in which -- n may be of any integral type. take :: Int -> [a] -> [a] -- | <math>. Extract the elements after the head of a list, which -- must be non-empty. -- --

--   >>> tail [1, 2, 3]
--   [2,3]
--   
--   >>> tail [1]
--   []
--   
--   >>> tail []
--   *** Exception: Prelude.tail: empty list
--

tail :: [a] -> [a] -- | splitAt n xs returns a tuple where first element is -- xs prefix of length n and second element is the -- remainder of the list: -- --

--   >>> splitAt 6 "Hello World!"
--   ("Hello ","World!")
--   
--   >>> splitAt 3 [1,2,3,4,5]
--   ([1,2,3],[4,5])
--   
--   >>> splitAt 1 [1,2,3]
--   ([1],[2,3])
--   
--   >>> splitAt 3 [1,2,3]
--   ([1,2,3],[])
--   
--   >>> splitAt 4 [1,2,3]
--   ([1,2,3],[])
--   
--   >>> splitAt 0 [1,2,3]
--   ([],[1,2,3])
--   
--   >>> splitAt (-1) [1,2,3]
--   ([],[1,2,3])
--

-- -- It is equivalent to (take n xs, drop n xs) when -- n is not _|_ (splitAt _|_ xs = _|_). -- splitAt is an instance of the more general -- genericSplitAt, in which n may be of any integral -- type. splitAt :: Int -> [a] -> ([a], [a]) -- | span, applied to a predicate p and a list xs, -- returns a tuple where first element is longest prefix (possibly empty) -- of xs of elements that satisfy p and second element -- is the remainder of the list: -- --

--   >>> span (< 3) [1,2,3,4,1,2,3,4]
--   ([1,2],[3,4,1,2,3,4])
--   
--   >>> span (< 9) [1,2,3]
--   ([1,2,3],[])
--   
--   >>> span (< 0) [1,2,3]
--   ([],[1,2,3])
--

-- -- span p xs is equivalent to (takeWhile p xs, -- dropWhile p xs) span :: (a -> Bool) -> [a] -> ([a], [a]) -- | <math>. scanr1 is a variant of scanr that has no -- starting value argument. -- --

--   >>> scanr1 (+) [1..4]
--   [10,9,7,4]
--   
--   >>> scanr1 (+) []
--   []
--   
--   >>> scanr1 (-) [1..4]
--   [-2,3,-1,4]
--   
--   >>> scanr1 (&&) [True, False, True, True]
--   [False,False,True,True]
--   
--   >>> scanr1 (||) [True, True, False, False]
--   [True,True,False,False]
--   
--   >>> force $ scanr1 (+) [1..]
--   *** Exception: stack overflow
--

scanr1 :: (a -> a -> a) -> [a] -> [a] -- | <math>. scanr is the right-to-left dual of scanl. -- Note that the order of parameters on the accumulating function are -- reversed compared to scanl. Also note that -- --

--   head (scanr f z xs) == foldr f z xs.
--

-- --

--   >>> scanr (+) 0 [1..4]
--   [10,9,7,4,0]
--   
--   >>> scanr (+) 42 []
--   [42]
--   
--   >>> scanr (-) 100 [1..4]
--   [98,-97,99,-96,100]
--   
--   >>> scanr (\nextChar reversedString -> nextChar : reversedString) "foo" ['a', 'b', 'c', 'd']
--   ["abcdfoo","bcdfoo","cdfoo","dfoo","foo"]
--   
--   >>> force $ scanr (+) 0 [1..]
--   *** Exception: stack overflow
--

scanr :: (a -> b -> b) -> b -> [a] -> [b] -- | <math>. scanl1 is a variant of scanl that has no -- starting value argument: -- --

--   scanl1 f [x1, x2, ...] == [x1, x1 `f` x2, ...]
--

-- --

--   >>> scanl1 (+) [1..4]
--   [1,3,6,10]
--   
--   >>> scanl1 (+) []
--   []
--   
--   >>> scanl1 (-) [1..4]
--   [1,-1,-4,-8]
--   
--   >>> scanl1 (&&) [True, False, True, True]
--   [True,False,False,False]
--   
--   >>> scanl1 (||) [False, False, True, True]
--   [False,False,True,True]
--   
--   >>> scanl1 (+) [1..]
--   * Hangs forever *
--

scanl1 :: (a -> a -> a) -> [a] -> [a] -- | <math>. scanl is similar to foldl, but returns a -- list of successive reduced values from the left: -- --

--   scanl f z [x1, x2, ...] == [z, z `f` x1, (z `f` x1) `f` x2, ...]
--

-- -- Note that -- --

--   last (scanl f z xs) == foldl f z xs
--

-- --

--   >>> scanl (+) 0 [1..4]
--   [0,1,3,6,10]
--   
--   >>> scanl (+) 42 []
--   [42]
--   
--   >>> scanl (-) 100 [1..4]
--   [100,99,97,94,90]
--   
--   >>> scanl (\reversedString nextChar -> nextChar : reversedString) "foo" ['a', 'b', 'c', 'd']
--   ["foo","afoo","bafoo","cbafoo","dcbafoo"]
--   
--   >>> scanl (+) 0 [1..]
--   * Hangs forever *
--

scanl :: (b -> a -> b) -> b -> [a] -> [b] -- | reverse xs returns the elements of xs in -- reverse order. xs must be finite. -- --

--   >>> reverse []
--   []
--   
--   >>> reverse [42]
--   [42]
--   
--   >>> reverse [2,5,7]
--   [7,5,2]
--   
--   >>> reverse [1..]
--   * Hangs forever *
--

reverse :: [a] -> [a] -- | replicate n x is a list of length n with -- x the value of every element. It is an instance of the more -- general genericReplicate, in which n may be of any -- integral type. -- --

--   >>> replicate 0 True
--   []
--   
--   >>> replicate (-1) True
--   []
--   
--   >>> replicate 4 True
--   [True,True,True,True]
--

replicate :: Int -> a -> [a] -- | repeat x is an infinite list, with x the -- value of every element. -- --

--   >>> take 20 $ repeat 17
--   [17,17,17,17,17,17,17,17,17...
--

repeat :: a -> [a] -- | <math>. lookup key assocs looks up a key in an -- association list. -- --

--   >>> lookup 2 []
--   Nothing
--   
--   >>> lookup 2 [(1, "first")]
--   Nothing
--   
--   >>> lookup 2 [(1, "first"), (2, "second"), (3, "third")]
--   Just "second"
--

lookup :: Eq a => a -> [(a, b)] -> Maybe b -- | <math>. Extract the last element of a list, which must be finite -- and non-empty. -- --

--   >>> last [1, 2, 3]
--   3
--   
--   >>> last [1..]
--   * Hangs forever *
--   
--   >>> last []
--   *** Exception: Prelude.last: empty list
--

last :: [a] -> a -- | iterate f x returns an infinite list of repeated -- applications of f to x: -- --

--   iterate f x == [x, f x, f (f x), ...]
--

-- -- Note that iterate is lazy, potentially leading to thunk -- build-up if the consumer doesn't force each iterate. See -- iterate' for a strict variant of this function. -- --

--   >>> take 10 $ iterate not True
--   [True,False,True,False...
--   
--   >>> take 10 $ iterate (+3) 42
--   [42,45,48,51,54,57,60,63...
--

iterate :: (a -> a) -> a -> [a] -- | <math>. Return all the elements of a list except the last one. -- The list must be non-empty. -- --

--   >>> init [1, 2, 3]
--   [1,2]
--   
--   >>> init [1]
--   []
--   
--   >>> init []
--   *** Exception: Prelude.init: empty list
--

init :: [a] -> [a] -- | <math>. Extract the first element of a list, which must be -- non-empty. -- --

--   >>> head [1, 2, 3]
--   1
--   
--   >>> head [1..]
--   1
--   
--   >>> head []
--   *** Exception: Prelude.head: empty list
--

head :: [a] -> a -- | dropWhile p xs returns the suffix remaining after -- takeWhile p xs. -- --

--   >>> dropWhile (< 3) [1,2,3,4,5,1,2,3]
--   [3,4,5,1,2,3]
--   
--   >>> dropWhile (< 9) [1,2,3]
--   []
--   
--   >>> dropWhile (< 0) [1,2,3]
--   [1,2,3]
--

dropWhile :: (a -> Bool) -> [a] -> [a] -- | drop n xs returns the suffix of xs after the -- first n elements, or [] if n >= length -- xs. -- --

--   >>> drop 6 "Hello World!"
--   "World!"
--   
--   >>> drop 3 [1,2,3,4,5]
--   [4,5]
--   
--   >>> drop 3 [1,2]
--   []
--   
--   >>> drop 3 []
--   []
--   
--   >>> drop (-1) [1,2]
--   [1,2]
--   
--   >>> drop 0 [1,2]
--   [1,2]
--

-- -- It is an instance of the more general genericDrop, in which -- n may be of any integral type. drop :: Int -> [a] -> [a] -- | cycle ties a finite list into a circular one, or equivalently, -- the infinite repetition of the original list. It is the identity on -- infinite lists. -- --

--   >>> cycle []
--   *** Exception: Prelude.cycle: empty list
--   
--   >>> take 20 $ cycle [42]
--   [42,42,42,42,42,42,42,42,42,42...
--   
--   >>> take 20 $ cycle [2, 5, 7]
--   [2,5,7,2,5,7,2,5,7,2,5,7...
--

cycle :: [a] -> [a] -- | break, applied to a predicate p and a list -- xs, returns a tuple where first element is longest prefix -- (possibly empty) of xs of elements that do not satisfy -- p and second element is the remainder of the list: -- --

--   >>> break (> 3) [1,2,3,4,1,2,3,4]
--   ([1,2,3],[4,1,2,3,4])
--   
--   >>> break (< 9) [1,2,3]
--   ([],[1,2,3])
--   
--   >>> break (> 9) [1,2,3]
--   ([1,2,3],[])
--

-- -- break p is equivalent to span (not . -- p). break :: (a -> Bool) -> [a] -> ([a], [a]) -- | List index (subscript) operator, starting from 0. It is an instance of -- the more general genericIndex, which takes an index of any -- integral type. -- --

--   >>> ['a', 'b', 'c'] !! 0
--   'a'
--   
--   >>> ['a', 'b', 'c'] !! 2
--   'c'
--   
--   >>> ['a', 'b', 'c'] !! 3
--   *** Exception: Prelude.!!: index too large
--   
--   >>> ['a', 'b', 'c'] !! (-1)
--   *** Exception: Prelude.!!: negative index
--

(!!) :: [a] -> Int -> a infixl 9 !! -- | The maybe function takes a default value, a function, and a -- Maybe value. If the Maybe value is Nothing, the -- function returns the default value. Otherwise, it applies the function -- to the value inside the Just and returns the result. -- --

Examples

-- -- Basic usage: -- --

--   >>> maybe False odd (Just 3)
--   True
--

-- --

--   >>> maybe False odd Nothing
--   False
--

-- -- Read an integer from a string using readMaybe. If we succeed, -- return twice the integer; that is, apply (*2) to it. If -- instead we fail to parse an integer, return 0 by default: -- --

--   >>> import Text.Read ( readMaybe )
--   
--   >>> maybe 0 (*2) (readMaybe "5")
--   10
--   
--   >>> maybe 0 (*2) (readMaybe "")
--   0
--

-- -- Apply show to a Maybe Int. If we have Just n, -- we want to show the underlying Int n. But if we have -- Nothing, we return the empty string instead of (for example) -- "Nothing": -- --

--   >>> maybe "" show (Just 5)
--   "5"
--   
--   >>> maybe "" show Nothing
--   ""
--

maybe :: b -> (a -> b) -> Maybe a -> b -- | An infix synonym for fmap. -- -- The name of this operator is an allusion to $. Note the -- similarities between their types: -- --

--    ($)  ::              (a -> b) ->   a ->   b
--   (<$>) :: Functor f => (a -> b) -> f a -> f b
--

-- -- Whereas $ is function application, <$> is function -- application lifted over a Functor. -- --

Examples

-- -- Convert from a Maybe Int to a Maybe -- String using show: -- --

--   >>> show <$> Nothing
--   Nothing
--   
--   >>> show <$> Just 3
--   Just "3"
--

-- -- Convert from an Either Int Int to an -- Either Int String using show: -- --

--   >>> show <$> Left 17
--   Left 17
--   
--   >>> show <$> Right 17
--   Right "17"
--

-- -- Double each element of a list: -- --

--   >>> (*2) <$> [1,2,3]
--   [2,4,6]
--

-- -- Apply even to the second element of a pair: -- --

--   >>> even <$> (2,2)
--   (2,True)
--

(<$>) :: Functor f => (a -> b) -> f a -> f b infixl 4 <$> -- | uncurry converts a curried function to a function on pairs. -- --

Examples

-- --

--   >>> uncurry (+) (1,2)
--   3
--

-- --

--   >>> uncurry ($) (show, 1)
--   "1"
--

-- --

--   >>> map (uncurry max) [(1,2), (3,4), (6,8)]
--   [2,4,8]
--

uncurry :: (a -> b -> c) -> (a, b) -> c -- | curry converts an uncurried function to a curried function. -- --

Examples

-- --

--   >>> curry fst 1 2
--   1
--

curry :: ((a, b) -> c) -> a -> b -> c -- | until p f yields the result of applying f -- until p holds. until :: (a -> Bool) -> (a -> a) -> a -> a -- | flip f takes its (first) two arguments in the reverse -- order of f. -- --

--   >>> flip (++) "hello" "world"
--   "worldhello"
--

flip :: (a -> b -> c) -> b -> a -> c -- | const x is a unary function which evaluates to x for -- all inputs. -- --

--   >>> const 42 "hello"
--   42
--

-- --

--   >>> map (const 42) [0..3]
--   [42,42,42,42]
--

const :: a -> b -> a -- | asTypeOf is a type-restricted version of const. It is -- usually used as an infix operator, and its typing forces its first -- argument (which is usually overloaded) to have the same type as the -- second. asTypeOf :: a -> a -> a -- | Same as >>=, but with the arguments interchanged. (=<<) :: Monad m => (a -> m b) -> m a -> m b infixr 1 =<< -- | Strict (call-by-value) application operator. It takes a function and -- an argument, evaluates the argument to weak head normal form (WHNF), -- then calls the function with that value. ($!) :: forall (r :: RuntimeRep) a (b :: TYPE r). (a -> b) -> a -> b infixr 0 $! -- | A special case of error. It is expected that compilers will -- recognize this and insert error messages which are more appropriate to -- the context in which undefined appears. undefined :: forall (r :: RuntimeRep) (a :: TYPE r). HasCallStack => a -- | A variant of error that does not produce a stack trace. errorWithoutStackTrace :: forall (r :: RuntimeRep) (a :: TYPE r). [Char] -> a -- | error stops execution and displays an error message. error :: forall (r :: RuntimeRep) (a :: TYPE r). HasCallStack => [Char] -> a -- | Boolean "and", lazy in the second argument (&&) :: Bool -> Bool -> Bool infixr 3 && -- | Boolean "not" not :: Bool -> Bool -- | Boolean "or", lazy in the second argument (||) :: Bool -> Bool -> Bool infixr 2 ||