%**#### Figure 5

\ecaption{Standard Haskell Classes}
\label{standard-classes}
\end{figure}
Default class method declarations (Section~\ref{classes}) are provided
for many of the methods in standard classes. A comment with each
@class@ declaration in Chapter~\ref{stdprelude} specifies the
smallest collection of method definitions that, together with the
default declarations, provide a reasonable definition for all the
class methods. If there is no such comment, then all class methods
must be given to fully specify an instance.
\subsubsection{The Eq Class}
\indexclass{Eq}
\indextt{==}
\indextt{/=}
\bprog
@
class Eq a where
(==), (/=) :: a -> a -> Bool
x /= y = not (x == y)
x == y = not (x /= y)
@
\eprog
The @Eq@ class provides equality (@==@) and inequality (@/=@) methods.
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@.
This declaration gives default method declarations for both @/=@ and @==@,
each being defined in terms of the other. If an instance declaration
for @Eq@ defines neither @==@ nor @/=@, then both will loop.
If one is defined, the default method for the other will make use of
the one that is defined. If both are defined, neither default method is used.
\subsubsection{The Ord Class}
\indexclass{Ord}
\indextt{<}
\indextt{<=}
\indextt{>}
\indextt{>=}
\indextt{compare}
\indextt{max}
\indextt{min}
\bprog
@
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 = y
| otherwise = x
min x y | x <= y = x
| otherwise = y
@
\eprog
The @Ord@ class is used for totally ordered datatypes. All basic
datatypes
except for functions, @IO@, and @IOError@, 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 default declarations allow a user to create an @Ord@ instance
either with a type-specific @compare@ function or with type-specific
@==@ and @<=@ functions.
\subsubsection{The Read and Show Classes}
\indexsynonym{ReadS}
\indexsynonym{ShowS}
\indexclass{Read}
\indexclass{Show}
\indextt{show}
\indextt{readsPrec}
\indextt{showsPrec}
\indextt{readList}
\indextt{showList}
\bprog
@
type ReadS a = String -> [(a,String)]
type ShowS = String -> String
class Read a where
readsPrec :: Int -> ReadS a
readList :: ReadS [a]
-- ... default decl for readList given in Prelude
class Show a where
showsPrec :: Int -> a -> ShowS
show :: a -> String
showList :: [a] -> ShowS
showsPrec _ x s = show x ++ s
show x = showsPrec 0 x ""
-- ... default decl for showList given in Prelude
@
\eprog
The @Read@ and @Show@ classes are used to convert values to
or from strings.
The @Int@ argument to @showsPrec@ and @readsPrec@ gives the operator
precedence of the enclosing context (see Section~\ref{derived-text}).
@showsPrec@ and @showList@ return a @String@-to-@String@
function, to allow constant-time concatenation of its results using function
composition.
A specialised variant, @show@, is also provided, which
uses precedence context zero, and returns an ordinary @String@.
The method @showList@ is provided to allow the programmer to
give a specialised way of showing lists of values. This is particularly
useful for the @Char@ type, where values of type @String@ should be
shown in double quotes, rather than between square brackets.
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@.
All @Prelude@ types, except function types and @IO@ types,
are instances of @Show@ and @Read@.
(If desired, a programmer can easily make functions and @IO@ types
into (vacuous) instances of @Show@, by providing an instance declaration.)
\indextt{reads}
\indextt{shows}
\indextt{read}
For convenience, the Prelude provides the following auxiliary
functions:
\bprog
@
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"
@
\eprog
\indextt{shows}\indextt{reads}\indextt{show}\indextt{read}
@shows@ and @reads@ use a default precedence of 0. The @read@ function reads
input from a string, which must be completely consumed by the input
process.
\indextt{lex}
The function @lex :: ReadS String@, used by @read@, is also part of the Prelude.
It 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 @[]@).
\subsubsection{The Enum Class}
\label{enum-class}
\indexclass{Enum}
\indextt{toEnum}
\indextt{fromEnum}
\indextt{enumFrom}
\indextt{enumFromThen}
\indextt{enumFromTo}
\indextt{enumFromThenTo}
\bprog
@
class Enum a where
succ, pred :: a -> a
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]
-- Default declarations given in Prelude
@
\eprog
Class @Enum@ defines operations on sequentially ordered types.
The functions @succ@ and @pred@ return the successor and predecessor,
respectively, of a value.
The functions @fromEnum@ and @toEnum@ map values from a type in
@Enum@ to and from @Int@.
The @enumFrom@... methods are used when translating arithmetic
sequences (Section~\ref{arithmetic-sequences}).
Instances of @Enum@ may be derived for any enumeration type (types
whose constructors have no fields); see Chapter~\ref{derived-appendix}.
For any type that is an instance of class @Bounded@ as well as @Enum@, the following
should hold:
\begin{itemize}
\item The calls @succ maxBound@ and @pred minBound@ should result in
a runtime error.
\item @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.
\item @enumFrom@ and @enumFromThen@ should be defined with
an implicit bound, thus:
\bprog
@
enumFrom x = enumFromTo x maxBound
enumFromThen x y = enumFromThenTo x y bound
where
bound | fromEnum y >= fromEnum x = maxBound
| otherwise = minBound
@
\eprog
\end{itemize}
The following @Prelude@ types are instances of @Enum@:
\begin{itemize}
\item Enumeration types: @()@, @Bool@, and @Ordering@. The
semantics of these instances is given by Chapter~\ref{derived-appendix}.
For example, @[LT..]@ is the list @[LT,EQ,GT]@.
\item @Char@: the instance is given in Chapter~\ref{stdprelude}, based
on the primitive functions that convert between a @Char@ and an @Int@.
For example, @enumFromTo 'a' 'z'@ denotes
the list of lowercase letters in alphabetical order.
\item Numeric types: @Int@, @Integer@, @Float@, @Double@. The semantics
of these instances is given next.
\end{itemize}
For all four numeric types, @succ@ adds 1, and @pred@ subtracts 1.
The conversions @fromEnum@ and @toEnum@ convert between the type and @Int@.
In the case of @Float@ and @Double@, the digits after the decimal point may be lost.
It is implementation-dependent what @fromEnum@ returns when applied to
a value that is too large to fit in an @Int@.
For the types @Int@ and @Integer@, the enumeration functions
have the following meaning:
\begin{itemize}
\item The sequence "@enumFrom@~e_1" is the list "@[@e_1@,@e_1+1@,@e_1+2@,@...@]@".
\item The sequence "@enumFromThen@~e_1~e_2" is the list "@[@e_1@,@e_1+i@,@e_1+2i@,@...@]@",
where the increment, "i", is "e_2-e_1". The increment may be zero or negative.
If the increment is zero, all the list elements are the same.
\item The sequence "@enumFromTo@~e_1~e_3" is
the list "@[@e_1@,@e_1+1@,@e_1+2@,@...e_3@]@".
The list is empty if "e_1 > e_3".
\item The sequence "@enumFromThenTo@~e_1~e_2~e_3"
is the list "@[@e_1@,@e_1+i@,@e_1+2i@,@...e_3@]@",
where the increment, "i", is "e_2-e_1". If the increment
is positive or zero, the list terminates when the next element would
be greater than "e_3"; the list is empty if "e_1 > e_3".
If the increment is negative, the list terminates when the next element would
be less than "e_3"; the list is empty if "e1 < e_3".
\end{itemize}
For @Float@ and @Double@, the semantics of the @enumFrom@ family is
given by the rules for @Int@ above, except that the list terminates
when the elements become greater than "e_3+i/2" for positive increment
"i", or when they become less than "e_3+i/2" for negative "i".
For all four of these Prelude numeric types, all of the @enumFrom@
family of functions are strict in all their arguments.
\subsubsection{The Functor Class}
\indexclass{Functor}
\indextt{fmap}
\index{functor}
\bprog
@
class Functor f where
fmap :: (a -> b) -> f a -> f b
@
\eprog
The @Functor@
class is used for types that can be mapped over. Lists, @IO@, and
@Maybe@ are in this class.
Instances of @Functor@ should satisfy the following laws:
\[\ba{lcl}
@fmap id@&=&@id@\\
@fmap (f . g)@&=&@fmap f . fmap g@\\
\ea\]\ifhtml{\\}{}
All instances of @Functor@ defined in the Prelude satisfy these laws.
\subsubsection{The Monad Class}
\label{monad-class}
\indexclass{Monad}
\indextt{return}
\indextt{fail}
\indextt{>>}
\indextt{>>=}
\index{monad}
\bprog
@
class Monad m where
(>>=) :: m a -> (a -> m b) -> m b
(>>) :: m a -> m b -> m b
return :: a -> m a
fail :: String -> m a
m >> k = m >>= \_ -> k
fail s = error s
@
\eprog
The @Monad@ class defines the basic operations over a {\em monad}.
See Chapter~\ref{io} for more information about monads.
``@do@'' expressions provide a convenient syntax for writing
monadic expressions (see Section~\ref{do-expressions}).
The @fail@ method is invoked on pattern-match failure in a @do@
expression.
In the Prelude, lists,
@Maybe@, and @IO@ are all instances of @Monad@.
The @fail@ method for lists returns the empty list @[]@,
for @Maybe@ returns @Nothing@, and for @IO@ raises a user
exception in the IO monad (see Section~\ref{io-exceptions}).
Instances of @Monad@ should satisfy the following laws:
\[\ba{lcl}
@return a >>= k@&=&@k a@ \\
@m >>= return@&=&@m@ \\
@m >>= (\x -> k x >>= h)@&=&@(m >>= k) >>= h@\\
\ea\]\ifhtml{\\}{}
Instances of both @Monad@ and @Functor@ should additionally satisfy the law:
\[\ba{lcl}
@fmap f xs@&=&@xs >>= return . f@\\
\ea\]\ifhtml{\\}{}
All instances of @Monad@ defined in the Prelude satisfy these laws.
The Prelude provides the following auxiliary functions:
\bprog
@
sequence :: 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 ()
(=<<) :: Monad m => (a -> m b) -> m a -> m b
@
\eprog
\indextt{sequence}
\index{sequence{\char '137}@@{\tt sequence{\char '137}}}
\index{mapM{\char '137}@@{\tt mapM{\char '137}}}
\indextt{mapM}
\indextt{=<<}
\subsubsection{The Bounded Class}
\indexclass{Bounded}
\indextt{minBound}
\indextt{maxBound}
@
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@.
\subsection{Numbers}
\label{numbers}
\index{number}
\Haskell{} provides several kinds of numbers; the numeric
types and the operations upon them have been heavily influenced by
%Common Lisp \cite{steele:common-lisp} and Scheme \cite{RRRRS}.
Common Lisp and Scheme.
Numeric function names and operators are usually overloaded, using
several type classes with an inclusion relation shown in
Figure~\ref{standard-classes}.
The class @Num@\indexclass{Num} of numeric
types is a subclass of @Eq@\indexclass{Eq}, since all numbers may be
compared for equality; its subclass @Real@\indexclass{Real} is also a
subclass of @Ord@\indexclass{Ord}, since the other comparison operations
apply to all but complex numbers (defined in the @Complex@ library).
The class @Integral@\indexclass{Integral} contains integers of both
limited and unlimited range; the class
@Fractional@\indexclass{Fractional} contains all non-integral types; and
the class @Floating@\indexclass{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 @Ratio@
library.
The default floating point operations defined by the \Haskell{}
Prelude do not
conform to current language independent arithmetic (LIA) standards. These
standards require considerably more complexity in the numeric
structure and have thus been relegated to a library. Some, but not
all, aspects of the IEEE floating point standard have been
accounted for in Prelude class @RealFloat@.
The standard numeric types are listed in Table~\ref{standard-numeric-types}.
The finite-precision integer type @Int@\indextycon{Int} covers at
least the range
"[ - 2^{29}, 2^{29} - 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.
%(Figure~\ref{basic-numeric-2}, page~\pageref{basic-numeric-2}) define the limits of
%@Int@\indextycon{Int} in each implementation.
@Float@\indextycon{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@\indextycon{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
("\bot", semantically), a truncated value, or a special value such as
infinity, indefinite, etc.
\begin{table}[tb]
\[
\bto{|l|l|l|}
\hline
\multicolumn{1}{|c|}{Type} &
\multicolumn{1}{c|}{Class} &
\multicolumn{1}{c|}{Description} \\ \hline
@Integer@ & @Integral@ & Arbitrary-precision integers \\
@Int@ & @Integral@ & Fixed-precision integers \\
@(Integral a) => Ratio a@ & @RealFrac@ & Rational numbers \\
@Float@ & @RealFloat@ & Real floating-point, single precision \\
@Double@ & @RealFloat@ & Real floating-point, double precision \\
@(RealFloat a) => Complex a@ & @Floating@ & Complex floating-point \\
\hline
\eto
\]
%**