The Haskell 1.4 Report
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6  Predefined Types and Classes

The Haskell Prelude contains predefined classes, types, and functions that are implicitly imported into every Haskell program. In this section, we describe the types and classes found in the Prelude. Most functions are not described in detail here as they can easily be understood from their definitions as given in Appendix A. Other predefined types such as arrays, complex numbers, and rationals are defined in the Haskell Library Report.

6.1  Standard Haskell Types

These types are defined by the Haskell Prelude. Numeric types are described in Section 6.3. When appropriate, the Haskell definition of the type is given. Some definitions may not be completely valid on syntactic grounds but they faithfully convey the meaning of the underlying type.

6.1.1  Booleans



data  Bool  =  False | True deriving 
                             (Read, Show, Eq, Ord, Enum, Bounded)

The boolean type Bool is an enumeration. The basic boolean functions are && (and), || (or), and not. The name otherwise is defined as True to make guarded expressions more readable.

6.1.2  Characters and Strings

The character type Char is an enumeration and consists of 16 bit values, conforming to the Unicode standard [10]. The lexical syntax for characters is defined in Section 2.5; character literals are nullary constructors in the datatype Char. Type Char is an instance of the classes Read, Show, Eq, Ord, Enum, and Bounded. The toEnum and fromEnum functions, standard functions over bounded enumerations, map characters onto Int values in the range [ 0 , 216-1 ].

Note that ASCII control characters each have several representations in character literals: numeric escapes, ASCII mnemonic escapes, and the \^X notation. In addition, there are the following equivalences: \a and \BEL, \b and \BS, \f and \FF, \r and \CR, \t and \HT, \v and \VT, and \n and \LF.

A string is a list of characters:

type  String  =  [Char]

Strings may be abbreviated using the lexical syntax described in Section 2.5. For example, "A string" abbreviates

[ 'A',' ','s','t','r', 'i','n','g']

6.1.3  Lists



data  [a]  =  [] | a : [a]  deriving (Eq, Ord)

Lists are an algebraic datatype of two constructors, although with special syntax, as described in Section 3.7. The first constructor is the null list, written `[]' ("nil"), and the second is `:' ("cons"). The module PreludeList (see Appendix A.1) defines many standard list functions. Arithmetic sequences and list comprehensions, two convenient syntaxes for special kinds of lists, are described in Sections 3.10 and 3.11, respectively. Lists are an instance of classes Read, Show, Eq, Ord, Monad, MonadZero, and MonadPlus.

6.1.4  Tuples

Tuples are algebraic datatypes with special syntax, as defined in Section 3.8. Each tuple type has a single constructor. There is no upper bound on the size of a tuple. However, some Haskell implementations may restrict the size of tuples and limit the instances associated with larger tuples. The Prelude and libraries define tuple functions such as zip for tuples up to a size of 7. All tuples are instances of Eq, Ord, Bounded, Read, and Show. Classes defined in the libraries may also supply instances for tuple types. The constructor for a tuple is written by omitting the expressions surrounding the commas: thus (x,y) and (,) x y produce the same value. The following functions are defined for pairs (2-tuples): fst, snd, curry, and uncurry. Similar functions are not predefined for larger tuples.

6.1.5  The Unit Datatype



data  () = () deriving (Eq, Ord, Bounded, Enum, Read, Show)

The unit datatype () has one non-_|_ member, the nullary constructor (). See also Section 3.9.

6.1.6  The Void Datatype



data Void

The Void has no constructors; only _|_ is an instance of this type.

6.1.7  Function Types

Functions are an abstract type: no constructors directly create functional values. Functions are an instance of the Show class but not Read. The following simple functions are found the Prelude: id, const, (.), flip, ($), and until.

6.1.8  The IO and IOError Types

The IO type serves as a tag for operations (actions) that interact with the outside world. The IO type is abstract: no constructors are visible to the user. IO is an instance of the Monad and Show classes. Section 7 describes I/O operations.

IOError is an abstract type representing errors raised by I/O operations. It is an instance of Show and Eq. Values of this type are constructed by the various I/O functions and are not presented in any further detail in this report. The Library Report contains many other I/O functions.

6.1.9  Other Types



data  Maybe a     =  Nothing | Just a deriving (Eq, Ord, Read, Show)
data  Either a b  =  Left a | Right b deriving (Eq, Ord, Read, Show)
data  Ordering    =  LT | EQ | GT deriving
                                  (Eq, Ord, Bounded, Enum, Read, Show)

The Maybe type is an instance of classes Functor, Monad, MonadZero and MonadPlus. The Ordering type is used by compare in the class Ord. The functions maybe and either are found in the Prelude.

6.2  Standard Haskell Classes

Figure 5 shows the hierarchy of Haskell classes defined in the Prelude and the Prelude types that are instances of these classes. The Void type is not mentioned in this figure since it is not a member of any class.
Diagram of standard Haskell classes

Figure 5

Standard Haskell Classes

6.2.1  The Eq Class



class  Eq a  where
        (==), (/=)  ::  a -> a -> Bool
        x /= y   =  not (x == y)

All basic datatypes except for functions and IO are instances of this class. Instances of Eq can be derived for any user-defined datatype whose constituents are also instances of Eq.

6.2.2  The Ord Class



class  (Eq a) => Ord a  where
        compare                 :: a -> a -> Ordering
        (<), (<=), (>=), (>)    :: a -> a -> Bool
        max, min :: a -> a -> a

    compare x y
    | x == y    = EQ
    | x <= y    = LT
    | otherwise = GT

    x <= y              = compare x y /= GT
    x <  y = compare x y == LT
    x >= y = compare x y /= LT
    x >  y = compare x y == GT

-- note that (min x y, max x y) = (x,y) or (y,x)
    max x y | x >= y =  x
    | otherwise =  y
    min x y | x <  y =  x
    | otherwise =  y

The Ord class is used for totally ordered datatypes. All basic datatypes except for functions and IO are instances of this class. 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. The defaults allow a user to create an Ord instance either with a type-specific compare function or with type-specific == and <= functions.

6.2.3  The Read and Show Classes



type  ReadS a = String -> [(a,String)]
type  ShowS   = String -> String

class  Read a  where
    readsPrec :: Int -> ReadS a
    readList  :: ReadS [a]

class  Show a  where
    showsPrec :: Int -> a -> ShowS
    showList  :: [a] -> ShowS

The Read and Show classes are used to convert values to or from strings. Derived instances of Read and Show replicate the style in which a constructor is declared: infix constructors and field names are used on input and output. Strings produced by showsPrec are usually readable by readsPrec. Functions and the IO type are not in Read.

For convenience, the Prelude provides the following auxiliary functions:

reads          :: (Read a) => ReadS a
reads =  readsPrec 0

shows       :: (Show a) => a -> ShowS
shows =  showsPrec 0

read       :: (Read a) => String -> a
read s       =  case [x | (x,t) <- reads s, ("","") <- lex t] of
[x] -> x
[]  -> error "PreludeText.read: no parse"
_   -> error "PreludeText.read: ambiguous parse"

show       :: (Show a) => a -> String
show x       =  shows x ""

shows and reads use a default precedence of 0. The show function returns a String instead of a ShowS; the read function reads input from a string, which must be completely consumed by the input process. The lex function used by read is also part of the Prelude.

6.2.4  The Enum Class



class  (Ord a) => Enum a where
    toEnum              :: Int -> a
    fromEnum            :: a -> Int
    enumFrom :: a -> [a] -- [n..]
    enumFromThen :: a -> a -> [a] -- [n,n'..]
    enumFromTo :: a -> a -> [a] -- [n..m]
    enumFromThenTo :: a -> a -> a -> [a] -- [n,n'..m]

    enumFromTo n m      =  takeWhile (<= m) (enumFrom n)
    enumFromThenTo n n' m
                        =  takeWhile (if n' >= n then (<= m) else (>= m))
                                     (enumFromThen n n')

Class Enum defines operations on sequentially ordered types. The toEnum and fromEnum functions map values from a type in Enum onto Int. These functions are not meaningful for all instances of Enum: floating point values or Integer may not be mapped onto an Int. An runtime error occurs if either toEnum or fromEnum is given a value not mappable to the result type. Instances of Enum may be derived for any enumeration type (types whose constructors have no fields). There are also Enum instances for floats.

6.2.5  Monadic Classes



class  Functor f  where
    map     :: (a -> b) -> (f a -> f b)

class  Monad m  where
    (>>=)   :: m a -> (a -> m b) -> m b
    (>>)    :: m a -> m b -> m b
    return  :: a -> m a

class  (Monad m) => MonadZero m  where
    zero    :: m a

class  (MonadZero m) => MonadPlus m  where
   (++)         :: m a -> m a -> m a

These classes define the basic monadic operations. See Section 7 for more information about monads. The monadic classes serve to organize a set of operations common to a number of related types. These types are all container types: that is, they contain a value or values of another type. (To be precise, types in these classes must have kind *->*.) In the Prelude, lists, Maybe, and IO are all predefined container types.

The Functor class is used for types that can be mapped over. Lists, IO, and Maybe are in this class. The IO type, Maybe, and lists are instances of Monad. The do syntax provides a more readable notation for the operators in Monad. Both lists and Maybe are instances of the MonadZero class. The MonadPlus class provides a `monadic addition' operator: ++. In the Prelude, Maybe and lists are in this class. For lists, ++ defines concatenation. For Maybe, the ++ function returns the first non-empty value (if any).

Instances of these classes should satisfy the following laws:

map id=id
map (f . g)=map f . map g
map f xs=xs >>= return . f
return a >>= k=k a
m >>= return=m
m >>= (\x -> k x >>= h)=(m >>= k) >>= h
m >> zero=zero
zero >>= m=zero
m ++ zero=m
zero ++ m=m

All instances defined in the Prelude satisfy these laws.

The Prelude provides the following auxiliary functions:

accumulate      :: Monad m => [m a] -> m [a] 
sequence        :: Monad m => [m a] -> m () 
mapM            :: Monad m => (a -> m b) -> [a] -> m [b]
mapM_           :: Monad m => (a -> m b) -> [a] -> m ()
guard           :: MonadZero m => Bool -> m ()

6.2.6  The Bounded Class


class  Bounded a  where
    minBound, maxBound :: a

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 types Int, Char, Bool, (), Ordering, and all tuples are instances of Bounded. 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.

6.2.7  The Eval Class



class Eval a where
  strict      :: (a -> b) -> a -> b
  seq         :: a -> b -> b
  strict f x  =  x `seq` f x

Class Eval is a special class for which no instances may be explicitly defined. An Eval instance is implicitly derived for every datatype. Functions as well as all other built-in types are in Eval. (As a consequence, _|_ is not the same as \x ->  _|_ since seq can be used to distinguish them.)

The functions seq and strict are defined by the equations:

seq _|_b = _|_
seq a b = b, if a /=_|_
strict f x = seq x (f x)

These functions are usually introduced to improve performance by avoiding unneeded laziness. Strict datatypes (see Section 4.2.1) are defined in terms of the strict function. This class explicitly marks functions and types that employ polymorphic strictness.

The Eval instance for a type T with a constructor C implicitly derived by the compiler is:

instance Eval T where
  x `seq` y = case x of
                C -> y
                _ -> y  -- catches any other constructors in T

The case is used to force evaluation of the first argument to `seq` before returning the second argument. The constructor mentioned by seq is arbitrary: any constructor from T can be used.

6.3  Numbers

Haskell provides several kinds of numbers; the numeric types and the operations upon them have been heavily influenced by Common Lisp and Scheme. Numeric function names and operators are usually overloaded, using several type classes with an inclusion relation shown in Figure 5. The class Num of numeric types is a subclass of Eq, since all numbers may be compared for equality; its subclass Real is also a subclass of Ord, since the other comparison operations apply to all but complex numbers (defined in the Complex library). The class Integral contains both fixed- and arbitrary-precision integers; the class Fractional contains all non-integral types; and the class Floating contains all floating-point types, both real and complex.

The Prelude defines only the most basic numeric types: fixed sized integers (Int), arbitrary precision integers (Integer), single precision floating (Float), and double precision floating (Double). Other numeric types such as rationals and complex numbers are defined in libraries. In particular, the type Rational is a ratio of two Integer values, as defined in the Rational library.

The default floating point operations defined by the Haskell Prelude do not conform to current language independent arithmetic (LIA) standards. These standards require considerable more complexity in the numeric structure and have thus been relegated to a library. Some, but not all, aspects of the IEEE standard floating point standard have been accounted for in class RealFloat.

Table 3 lists the standard numeric types. The type Int covers at least the range [ - 229, 229 - 1]. As Int is an instance of the Bounded class, maxBound and minBound can be used to determine the exact Int range defined by an implementation. Float is implementation-defined; it is desirable that this type be at least equal in range and precision to the IEEE single-precision type. Similarly, Double should cover IEEE double-precision. The results of exceptional conditions (such as overflow or underflow) on the fixed-precision numeric types are undefined; an implementation may choose error (_|_, semantically), a truncated value, or a special value such as infinity, indefinite, etc.

Type Class Description
Integer Integral Arbitrary-precision integers
Int Integral Fixed-precision integers
(Integral a) => Ratio a RealFrac Rational numbers
Float RealFloat Real floating-point, single precision
Double RealFloat Real floating-point, double precision
(RealFloat a) => Complex a Floating Complex floating-point

Table 2

Standard Numeric Types

The standard numeric classes and other numeric functions defined in the Prelude are shown in Figures 6--7. Figure 5 shows the class dependencies and built-in types that are instances of the numeric classes.


class  (Eq a, Show a, Eval a) => Num a  where
    (+), (-), (*) :: a -> a -> a
    negate :: a -> a
    abs, signum :: a -> a
    fromInteger :: Integer -> a

class  (Num a, Ord a) => Real a  where
    toRational ::  a -> Rational

class  (Real a, Enum a) => Integral a  where
    quot, rem, div, mod :: a -> a -> a
    quotRem, divMod :: a -> a -> (a,a)
    toInteger :: a -> Integer

class  (Num a) => Fractional a  where
    (/) :: a -> a -> a
    recip :: a -> a
    fromRational :: Rational -> a

class  (Fractional a) => Floating a  where
    pi :: a
    exp, log, sqrt :: a -> a
    (**), logBase :: a -> a -> a
    sin, cos, tan :: a -> a
    asin, acos, atan :: a -> a
    sinh, cosh, tanh :: a -> a
    asinh, acosh, atanh :: a -> a

Figure 6

Standard Numeric Classes and Related Operations, Part 1


class  (Real a, Fractional a) => RealFrac a  where
    properFraction :: (Integral b) => a -> (b,a)
    truncate, round :: (Integral b) => a -> b
    ceiling, floor :: (Integral b) => a -> b

class  (RealFrac a, Floating a) => RealFloat a  where
    floatRadix :: a -> Integer
    floatDigits :: a -> Int
    floatRange :: a -> (Int,Int)
    decodeFloat :: a -> (Integer,Int)
    encodeFloat :: Integer -> Int -> a
    exponent :: a -> Int
    significand :: a -> a
    scaleFloat :: Int -> a -> a
    isNaN, isInfinite, isDenormalized, isNegativeZero, isIEEE 
                        :: a -> Bool

fromIntegral :: (Integral a, Num b) => a -> b
gcd, lcm :: (Integral a) => a -> a-> a
(^) :: (Num a, Integral b) => a -> b -> a
(^^) :: (Fractional a, Integral b) => a -> b -> a

fromRealFrac         :: (RealFrac a, Fractional b) => a -> b

atan2 :: (RealFloat a) => a -> a -> a

Figure 7

Standard Numeric Classes and Related Operations, Part 2

6.3.1  Numeric Literals

The syntax of numeric literals is given in Section 2.4. An integer literal represents the application of the function fromInteger to the appropriate value of type Integer. Similarly, a floating literal stands for an application of fromRational to a value of type Rational (that is, Ratio Integer). Given the typings:

fromInteger  :: (Num a) => Integer -> a
fromRational :: (Fractional a) => Rational -> a

integer and floating literals have the typings (Num a) => a and (Fractional a) => a, respectively. Numeric literals are defined in this indirect way so that they may be interpreted as values of any appropriate numeric type. See Section 4.3.4 for a discussion of overloading ambiguity.

6.3.2  Arithmetic and Number-Theoretic Operations

The infix class methods (+), (*), (-), and the unary function negate (which can also be written as a prefix minus sign; see section 3.4) apply to all numbers. The class methods quot, rem, div, and mod apply only to integral numbers, while the class method (/) applies only to fractional ones. The quot, rem, div, and mod class methods satisfy these laws:

(x `quot` y)*y + (x `rem` y) == x
(x `div`  y)*y + (x `mod` y) == x

`quot` is integer division truncated toward zero, while the result of `div` is truncated toward negative infinity. The quotRem class method takes a dividend and a divisor as arguments and returns a (quotient, remainder) pair; divMod is defined similarly:

quotRem x y  =  (x `quot` y, x `rem` y)
divMod  x y  =  (x `div` y, x `mod` y)

Also available on integral numbers are the even and odd predicates:

even x     =  x `rem` 2 == 0
odd     =  not . even

Finally, there are the greatest common divisor and least common multiple functions: gcd x y is the greatest integer that divides both x and y. lcm x y is the smallest positive integer that both x and y divide.

6.3.3  Exponentiation and Logarithms

The one-argument exponential function exp and the logarithm function log act on floating-point numbers and use base e. logBase a x returns the logarithm of x in base a. sqrt returns the principal square root of a floating-point number. There are three two-argument exponentiation operations: (^) raises any number to a nonnegative integer power, (^^) raises a fractional number to any integer power, and (**) takes two floating-point arguments. The value of x^0 or x^^0 is 1 for any x, including zero; 0**y is undefined.

6.3.4  Magnitude and Sign

A number has a magnitude and a sign. The functions abs and signum apply to any number and satisfy the law:

abs x * signum x == x

For real numbers, these functions are defined by:

abs x    | x >= 0  = x
         | x <  0  = -x

signum x | x >  0  = 1
         | x == 0  = 0
         | x <  0  = -1

6.3.5  Trigonometric Functions

The circular and hyperbolic sine, cosine, and tangent functions and their inverses are provided for floating-point numbers. A version of arctangent taking two real floating-point arguments is also provided: For real floating x and y, atan2 y x differs from atan (y/x) in that its range is ( -p, p] rather than (- p/ 2 , p/ 2 ) (because the signs of the arguments provide quadrant information), and that it is defined when x is zero.

The precise definition of the above functions is as in Common Lisp, which in turn follows Penfield's proposal for APL [7]. See these references for discussions of branch cuts, discontinuities, and implementation.

6.3.6  Coercions and Component Extraction

The ceiling, floor, truncate, and round functions each take a real fractional argument and return an integral result. ceiling x returns the least integer not less than x, and floor x, the greatest integer not greater than x. truncate x yields the integer nearest x between 0 and x, inclusive. round x returns the nearest integer to x, the even integer if x is equidistant between two integers.

The function properFraction takes a real fractional number x and returns a pair comprising x as a proper fraction: an integral number with the same sign as x and a fraction with the same type and sign as x and with absolute value less than 1. The ceiling, floor, truncate, and round functions can be defined in terms of this one.

Two functions convert numbers to type Rational: toRational returns the rational equivalent of its real argument with full precision; approxRational takes two real fractional arguments x and e and returns the simplest rational number within e of x, where a rational p/q in reduced form is simpler than another p' / q' if |p| <=|p'| and q <=q' . Every real interval contains a unique simplest rational; in particular, note that 0/1 is the simplest rational of all.

The class methods of class RealFloat allow efficient, machine-independent access to the components of a floating-point number. The functions floatRadix, floatDigits, and floatRange give the parameters of a floating-point type: the radix of the representation, the number of digits of this radix in the significand, and the lowest and highest values the exponent may assume, respectively. 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 mbn, where b is the floating-point radix, and furthermore, either m and n are both zero or else bd-1<=m<bd, where d is the value of floatDigits x. encodeFloat performs the inverse of this transformation. The functions significand and exponent together provide the same information as decodeFloat, but rather than an Integer, significand x yields a value of the same type as x, scaled to lie in the open interval (-1,1). exponent 0 is zero. scaleFloat multiplies a floating-point number by an integer power of the radix.

The functions isNaN, isInfinite, isDenormalized, isNegativeZero, and isIEEE all support numbers represented using the IEEE standard. For non-IEEE floating point numbers, these may all return false.

Also available are the following coercion functions:

fromIntegral :: (Integral a, Num b) => a -> b
fromRealFrac :: (RealFrac a, Fractional b) => a -> b

The Haskell 1.4 Report
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March 27, 1997