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---------------------------------------------------------------------------- --- The standard prelude of Curry. --- All top-level functions defined in this module --- are always available in any Curry program. --- --- @category general ---------------------------------------------------------------------------- {-# OPTIONS_CYMAKE -Wno-incomplete-patterns -Wno-overlapping #-} module Prelude where -- Lines beginning with "--++" are part of the prelude -- but cannot parsed by the compiler -- Infix operator declarations: infixl 9 !! infixr 9 . infixl 7 *, `div`, `mod`, `quot`, `rem` infixl 6 +, - -- infixr 5 : -- declared together with list infixr 5 ++ infix 4 =:=, ==, /=, <, >, <=, >=, =:<= infix 4 `elem`, `notElem` infixr 3 && infixr 2 || infixl 1 >>, >>= infixr 0 $, $!, $!!, $#, $##, `seq`, &, &>, ? -- externally defined types for numbers and characters data Int data Float data Char type String = [Char] -- Some standard combinators: --- Function composition. (.) :: (b -> c) -> (a -> b) -> (a -> c) f . g = \x -> f (g x) --- Identity function. id :: a -> a id x = x --- Constant function. const :: a -> _ -> a const x _ = x --- Converts an uncurried function to a curried function. curry :: ((a,b) -> c) -> a -> b -> c curry f a b = f (a,b) --- Converts an curried function to a function on pairs. uncurry :: (a -> b -> c) -> (a,b) -> c uncurry f (a,b) = f a b --- (flip f) is identical to f but with the order of arguments reversed. flip :: (a -> b -> c) -> b -> a -> c flip f x y = f y x --- Repeats application of a function until a predicate holds. until :: (a -> Bool) -> (a -> a) -> a -> a until p f x = if p x then x else until p f (f x) --- Evaluates the first argument to head normal form (which could also --- be a free variable) and returns the second argument. seq :: _ -> a -> a x `seq` y = const y $! x --- Evaluates the argument to head normal form and returns it. --- Suspends until the result is bound to a non-variable term. ensureNotFree :: a -> a ensureNotFree external --- Evaluates the argument to spine form and returns it. --- Suspends until the result is bound to a non-variable spine. ensureSpine :: [a] -> [a] ensureSpine l = ensureList (ensureNotFree l) where ensureList [] = [] ensureList (x:xs) = x : ensureSpine xs --- Right-associative application. ($) :: (a -> b) -> a -> b f $ x = f x --- Right-associative application with strict evaluation of its argument --- to head normal form. ($!) :: (a -> b) -> a -> b ($!) external --- Right-associative application with strict evaluation of its argument --- to normal form. ($!!) :: (a -> b) -> a -> b ($!!) external --- Right-associative application with strict evaluation of its argument --- to a non-variable term. ($#) :: (a -> b) -> a -> b f $# x = f $! (ensureNotFree x) --- Right-associative application with strict evaluation of its argument --- to ground normal form. ($##) :: (a -> b) -> a -> b ($##) external --- Aborts the execution with an error message. error :: String -> _ error x = prim_error $## x prim_error :: String -> _ prim_error external --- A non-reducible polymorphic function. --- It is useful to express a failure in a search branch of the execution. --- It could be defined by: `failed = head []` failed :: _ failed external -- Boolean values -- already defined as builtin, since it is required for if-then-else data Bool = False | True --- Sequential conjunction on Booleans. (&&) :: Bool -> Bool -> Bool True && x = x False && _ = False --- Sequential disjunction on Booleans. (||) :: Bool -> Bool -> Bool True || _ = True False || x = x --- Negation on Booleans. not :: Bool -> Bool not True = False not False = True --- Useful name for the last condition in a sequence of conditional equations. otherwise :: Bool otherwise = True --- The standard conditional. It suspends if the condition is a free variable. if_then_else :: Bool -> a -> a -> a if_then_else b t f = case b of True -> t False -> f --- Enforce a Boolean condition to be true. --- The computation fails if the argument evaluates to `False`. solve :: Bool -> Bool solve True = True --- Conditional expression. --- An expression like `(c &> e)` is evaluated by evaluating the first --- argument to `True` and then evaluating `e`. --- The expression has no value if the condition does not evaluate to `True`. (&>) :: Bool -> a -> a True &> x = x --- Equality on finite ground data terms. (==) :: a -> a -> Bool (==) external --- Disequality. (/=) :: a -> a -> Bool x /= y = not (x==y) --- The equational constraint. --- `(e1 =:= e2)` is satisfiable if both sides `e1` and `e2` can be --- reduced to a unifiable data term (i.e., a term without defined --- function symbols). (=:=) :: a -> a -> Bool (=:=) external --- Concurrent conjunction. --- An expression like `(c1 & c2)` is evaluated by evaluating --- the `c1` and `c2` in a concurrent manner. (&) :: Bool -> Bool -> Bool (&) external --- Ordering type. Useful as a result of comparison functions. data Ordering = LT | EQ | GT --- Comparison of arbitrary ground data terms. --- Data constructors are compared in the order of their definition --- in the datatype declarations and recursively in the arguments. compare :: a -> a -> Ordering compare x y | x == y = EQ | x <= y = LT | otherwise = GT --- Less-than on ground data terms. (<) :: a -> a -> Bool x < y = not (y <= x) --- Greater-than on ground data terms. (>) :: a -> a -> Bool x > y = not (x <= y) --- Less-or-equal on ground data terms. (<=) :: a -> a -> Bool (<=) external --- Greater-or-equal on ground data terms. (>=) :: a -> a -> Bool x >= y = not (x < y) --- Maximum of ground data terms. max :: a -> a -> a max x y = if x >= y then x else y --- Minimum of ground data terms. min :: a -> a -> a min x y = if x <= y then x else y -- Pairs --++ data (a,b) = (a,b) --- Selects the first component of a pair. fst :: (a,_) -> a fst (x,_) = x --- Selects the second component of a pair. snd :: (_,b) -> b snd (_,y) = y -- Unit type --++ data () = () -- Lists --++ data [a] = [] | a : [a] --- Computes the first element of a list. head :: [a] -> a head (x:_) = x --- Computes the remaining elements of a list. tail :: [a] -> [a] tail (_:xs) = xs --- Is a list empty? null :: [_] -> Bool null [] = True null (_:_) = False --- Concatenates two lists. --- Since it is flexible, it could be also used to split a list --- into two sublists etc. (++) :: [a] -> [a] -> [a] [] ++ ys = ys (x:xs) ++ ys = x : xs++ys --- Computes the length of a list. length :: [_] -> Int length xs = len xs 0 where len [] n = n len (_:ys) n = let np1 = n + 1 in len ys $!! np1 --length [] = 0 --length (_:xs) = 1 + length xs --- List index (subscript) operator, head has index 0. (!!) :: [a] -> Int -> a (x:xs) !! n | n==0 = x | n>0 = xs !! (n-1) --- Map a function on all elements of a list. map :: (a->b) -> [a] -> [b] map _ [] = [] map f (x:xs) = f x : map f xs --- Accumulates all list elements by applying a binary operator from --- left to right. Thus, --- --- foldl f z [x1,x2,...,xn] = (...((z `f` x1) `f` x2) ...) `f` xn foldl :: (a -> b -> a) -> a -> [b] -> a foldl _ z [] = z foldl f z (x:xs) = foldl f (f z x) xs --- Accumulates a non-empty list from left to right. foldl1 :: (a -> a -> a) -> [a] -> a foldl1 f (x:xs) = foldl f x xs --- Accumulates all list elements by applying a binary operator from --- right to left. Thus, --- --- foldr f z [x1,x2,...,xn] = (x1 `f` (x2 `f` ... (xn `f` z)...)) foldr :: (a->b->b) -> b -> [a] -> b foldr _ z [] = z foldr f z (x:xs) = f x (foldr f z xs) --- Accumulates a non-empty list from right to left: foldr1 :: (a -> a -> a) -> [a] -> a foldr1 _ [x] = x foldr1 f (x:xs@(_:_)) = f x (foldr1 f xs) --- Filters all elements satisfying a given predicate in a list. filter :: (a -> Bool) -> [a] -> [a] filter _ [] = [] filter p (x:xs) = if p x then x : filter p xs else filter p xs --- Joins two lists into one list of pairs. If one input list is shorter than --- the other, the additional elements of the longer list are discarded. zip :: [a] -> [b] -> [(a,b)] zip [] _ = [] zip (_:_) [] = [] zip (x:xs) (y:ys) = (x,y) : zip xs ys --- Joins three lists into one list of triples. If one input list is shorter --- than the other, the additional elements of the longer lists are discarded. zip3 :: [a] -> [b] -> [c] -> [(a,b,c)] zip3 [] _ _ = [] zip3 (_:_) [] _ = [] zip3 (_:_) (_:_) [] = [] zip3 (x:xs) (y:ys) (z:zs) = (x,y,z) : zip3 xs ys zs --- Joins two lists into one list by applying a combination function to --- corresponding pairs of elements. Thus `zip = zipWith (,)` zipWith :: (a->b->c) -> [a] -> [b] -> [c] zipWith _ [] _ = [] zipWith _ (_:_) [] = [] zipWith f (x:xs) (y:ys) = f x y : zipWith f xs ys --- Joins three lists into one list by applying a combination function to --- corresponding triples of elements. Thus `zip3 = zipWith3 (,,)` zipWith3 :: (a->b->c->d) -> [a] -> [b] -> [c] -> [d] zipWith3 _ [] _ _ = [] zipWith3 _ (_:_) [] _ = [] zipWith3 _ (_:_) (_:_) [] = [] zipWith3 f (x:xs) (y:ys) (z:zs) = f x y z : zipWith3 f xs ys zs --- Transforms a list of pairs into a pair of lists. unzip :: [(a,b)] -> ([a],[b]) unzip [] = ([],[]) unzip ((x,y):ps) = (x:xs,y:ys) where (xs,ys) = unzip ps --- Transforms a list of triples into a triple of lists. unzip3 :: [(a,b,c)] -> ([a],[b],[c]) unzip3 [] = ([],[],[]) unzip3 ((x,y,z):ts) = (x:xs,y:ys,z:zs) where (xs,ys,zs) = unzip3 ts --- Concatenates a list of lists into one list. concat :: [[a]] -> [a] concat l = foldr (++) [] l --- Maps a function from elements to lists and merges the result into one list. concatMap :: (a -> [b]) -> [a] -> [b] concatMap f = concat . map f --- Infinite list of repeated applications of a function f to an element x. --- Thus, `iterate f x = [x, f x, f (f x),...]` iterate :: (a -> a) -> a -> [a] iterate f x = x : iterate f (f x) --- Infinite list where all elements have the same value. --- Thus, `repeat x = [x, x, x,...]` repeat :: a -> [a] repeat x = x : repeat x --- List of length n where all elements have the same value. replicate :: Int -> a -> [a] replicate n x = take n (repeat x) --- Returns prefix of length n. take :: Int -> [a] -> [a] take n l = if n<=0 then [] else takep n l where takep _ [] = [] takep m (x:xs) = x : take (m-1) xs --- Returns suffix without first n elements. drop :: Int -> [a] -> [a] drop n xs = if n<=0 then xs else case xs of [] -> [] (_:ys) -> drop (n-1) ys --- (splitAt n xs) is equivalent to (take n xs, drop n xs) splitAt :: Int -> [a] -> ([a],[a]) splitAt n l = if n<=0 then ([],l) else splitAtp n l where splitAtp _ [] = ([],[]) splitAtp m (x:xs) = let (ys,zs) = splitAt (m-1) xs in (x:ys,zs) --- Returns longest prefix with elements satisfying a predicate. takeWhile :: (a -> Bool) -> [a] -> [a] takeWhile _ [] = [] takeWhile p (x:xs) = if p x then x : takeWhile p xs else [] --- Returns suffix without takeWhile prefix. dropWhile :: (a -> Bool) -> [a] -> [a] dropWhile _ [] = [] dropWhile p (x:xs) = if p x then dropWhile p xs else x:xs --- (span p xs) is equivalent to (takeWhile p xs, dropWhile p xs) span :: (a -> Bool) -> [a] -> ([a],[a]) span _ [] = ([],[]) span p (x:xs) | p x = let (ys,zs) = span p xs in (x:ys, zs) | otherwise = ([],x:xs) --- (break p xs) is equivalent to (takeWhile (not.p) xs, dropWhile (not.p) xs). --- Thus, it breaks a list at the first occurrence of an element satisfying p. break :: (a -> Bool) -> [a] -> ([a],[a]) break p = span (not . p) --- Breaks a string into a list of lines where a line is terminated at a --- newline character. The resulting lines do not contain newline characters. lines :: String -> [String] lines [] = [] lines (x:xs) = let (l,xs_l) = splitline (x:xs) in l : lines xs_l where splitline [] = ([],[]) splitline (c:cs) = if c=='\n' then ([],cs) else let (ds,es) = splitline cs in (c:ds,es) --- Concatenates a list of strings with terminating newlines. unlines :: [String] -> String unlines ls = concatMap (++"\n") ls --- Breaks a string into a list of words where the words are delimited by --- white spaces. words :: String -> [String] words s = let s1 = dropWhile isSpace s in if s1=="" then [] else let (w,s2) = break isSpace s1 in w : words s2 where isSpace c = c == ' ' || c == '\t' || c == '\n' || c == '\r' --- Concatenates a list of strings with a blank between two strings. unwords :: [String] -> String unwords ws = if ws==[] then [] else foldr1 (\w s -> w ++ ' ':s) ws --- Reverses the order of all elements in a list. reverse :: [a] -> [a] reverse = foldl (flip (:)) [] --- Computes the conjunction of a Boolean list. and :: [Bool] -> Bool and = foldr (&&) True --- Computes the disjunction of a Boolean list. or :: [Bool] -> Bool or = foldr (||) False --- Is there an element in a list satisfying a given predicate? any :: (a -> Bool) -> [a] -> Bool any p = or . map p --- Is a given predicate satisfied by all elements in a list? all :: (a -> Bool) -> [a] -> Bool all p = and . map p --- Element of a list? elem :: a -> [a] -> Bool elem x = any (x==) --- Not element of a list? notElem :: a -> [a] -> Bool notElem x = all (x/=) --- Looks up a key in an association list. lookup :: a -> [(a,b)] -> Maybe b lookup _ [] = Nothing lookup k ((x,y):xys) | k==x = Just y | otherwise = lookup k xys --- Generates an infinite sequence of ascending integers. enumFrom :: Int -> [Int] -- [n..] enumFrom n = n : enumFrom (n+1) --- Generates an infinite sequence of integers with a particular in/decrement. enumFromThen :: Int -> Int -> [Int] -- [n1,n2..] enumFromThen n1 n2 = iterate ((n2-n1)+) n1 --- Generates a sequence of ascending integers. enumFromTo :: Int -> Int -> [Int] -- [n..m] enumFromTo n m = if n>m then [] else n : enumFromTo (n+1) m --- Generates a sequence of integers with a particular in/decrement. enumFromThenTo :: Int -> Int -> Int -> [Int] -- [n1,n2..m] enumFromThenTo n1 n2 m = takeWhile p (enumFromThen n1 n2) where p x | n2 >= n1 = (x <= m) | otherwise = (x >= m) --- Converts a character into its ASCII value. ord :: Char -> Int ord c = prim_ord $# c prim_ord :: Char -> Int prim_ord external --- Converts a Unicode value into a character, fails iff the value is out of bounds chr :: Int -> Char chr n | n >= 0 = prim_chr $# n -- chr n | n < 0 || n > 1114111 = failed -- | otherwise = prim_chr $# n prim_chr :: Int -> Char prim_chr external -- Types of primitive arithmetic functions and predicates --- Adds two integers. (+) :: Int -> Int -> Int (+) external --- Subtracts two integers. (-) :: Int -> Int -> Int (-) external --- Multiplies two integers. (*) :: Int -> Int -> Int (*) external --- Integer division. The value is the integer quotient of its arguments --- and always truncated towards negative infinity. --- Thus, the value of <code>13 `div` 5</code> is <code>2</code>, --- and the value of <code>-15 `div` 4</code> is <code>-4</code>. div :: Int -> Int -> Int div external --- Integer remainder. The value is the remainder of the integer division and --- it obeys the rule <code>x `mod` y = x - y * (x `div` y)</code>. --- Thus, the value of <code>13 `mod` 5</code> is <code>3</code>, --- and the value of <code>-15 `mod` 4</code> is <code>-3</code>. mod :: Int -> Int -> Int mod external --- Returns an integer (quotient,remainder) pair. --- The value is the integer quotient of its arguments --- and always truncated towards negative infinity. divMod :: Int -> Int -> (Int, Int) divMod external --- Integer division. The value is the integer quotient of its arguments --- and always truncated towards zero. --- Thus, the value of <code>13 `quot` 5</code> is <code>2</code>, --- and the value of <code>-15 `quot` 4</code> is <code>-3</code>. quot :: Int -> Int -> Int quot external --- Integer remainder. The value is the remainder of the integer division and --- it obeys the rule <code>x `rem` y = x - y * (x `quot` y)</code>. --- Thus, the value of <code>13 `rem` 5</code> is <code>3</code>, --- and the value of <code>-15 `rem` 4</code> is <code>-3</code>. rem :: Int -> Int -> Int rem external --- Returns an integer (quotient,remainder) pair. --- The value is the integer quotient of its arguments --- and always truncated towards zero. quotRem :: Int -> Int -> (Int, Int) quotRem external --- Unary minus. Usually written as "- e". negate :: Int -> Int negate x = 0 - x --- Unary minus on Floats. Usually written as "-e". negateFloat :: Float -> Float negateFloat external -- Constraints (included for backward compatibility) type Success = Bool --- The always satisfiable constraint. success :: Success success = True -- Maybe type data Maybe a = Nothing | Just a maybe :: b -> (a -> b) -> Maybe a -> b maybe n _ Nothing = n maybe _ f (Just x) = f x -- Either type data Either a b = Left a | Right b either :: (a -> c) -> (b -> c) -> Either a b -> c either f _ (Left x) = f x either _ g (Right x) = g x -- Monadic IO data IO _ -- conceptually: World -> (a,World) --- Sequential composition of actions. --- @param a - An action --- @param fa - A function from a value into an action --- @return An action that first performs a (yielding result r) --- and then performs (fa r) (>>=) :: IO a -> (a -> IO b) -> IO b (>>=) external --- The empty action that directly returns its argument. return :: a -> IO a return external --- Sequential composition of actions. --- @param a1 - An action --- @param a2 - An action --- @return An action that first performs a1 and then a2 (>>) :: IO _ -> IO b -> IO b a >> b = a >>= (\_ -> b) --- The empty action that returns nothing. done :: IO () done = return () --- An action that puts its character argument on standard output. putChar :: Char -> IO () putChar c = prim_putChar $## c prim_putChar :: Char -> IO () prim_putChar external --- An action that reads a character from standard output and returns it. getChar :: IO Char getChar external --- An action that (lazily) reads a file and returns its contents. readFile :: String -> IO String readFile f = prim_readFile $## f prim_readFile :: String -> IO String prim_readFile external --- An action that writes a file. --- @param filename - The name of the file to be written. --- @param contents - The contents to be written to the file. writeFile :: String -> String -> IO () writeFile f s = (prim_writeFile $## f) $## s prim_writeFile :: String -> String -> IO () prim_writeFile external --- An action that appends a string to a file. --- It behaves like writeFile if the file does not exist. --- @param filename - The name of the file to be written. --- @param contents - The contents to be appended to the file. appendFile :: String -> String -> IO () appendFile f s = (prim_appendFile $## f) $## s prim_appendFile :: String -> String -> IO () prim_appendFile external --- Action to print a string on stdout. putStr :: String -> IO () putStr [] = done putStr (c:cs) = putChar c >> putStr cs --- Action to print a string with a newline on stdout. putStrLn :: String -> IO () putStrLn cs = putStr cs >> putChar '\n' --- Action to read a line from stdin. getLine :: IO String getLine = do c <- getChar if c=='\n' then return [] else do cs <- getLine return (c:cs) ---------------------------------------------------------------------------- -- Error handling in the I/O monad: --- The (abstract) type of error values. --- Currently, it distinguishes between general IO errors, --- user-generated errors (see 'userError'), failures and non-determinism --- errors during IO computations. These errors can be caught by 'catch' --- and shown by 'showError'. --- Each error contains a string shortly explaining the error. --- This type might be extended in the future to distinguish --- further error situations. data IOError = IOError String -- normal IO error | UserError String -- user-specified error | FailError String -- failing computation | NondetError String -- non-deterministic computation --- A user error value is created by providing a description of the --- error situation as a string. userError :: String -> IOError userError s = UserError s --- Raises an I/O exception with a given error value. ioError :: IOError -> IO _ ioError err = prim_ioError $## err prim_ioError :: IOError -> IO _ prim_ioError external --- Shows an error values as a string. showError :: IOError -> String showError (IOError s) = "i/o error: " ++ s showError (UserError s) = "user error: " ++ s showError (FailError s) = "fail error: " ++ s showError (NondetError s) = "nondet error: " ++ s --- Catches a possible error or failure during the execution of an --- I/O action. `(catch act errfun)` executes the I/O action --- `act`. If an exception or failure occurs --- during this I/O action, the function `errfun` is applied --- to the error value. catch :: IO a -> (IOError -> IO a) -> IO a catch external ---------------------------------------------------------------------------- --- Converts an arbitrary term into an external string representation. show :: _ -> String show x = prim_show $## x prim_show :: _ -> String prim_show external --- Converts a term into a string and prints it. print :: _ -> IO () print t = putStrLn (show t) --- Solves a constraint as an I/O action. --- Note: the constraint should be always solvable in a deterministic way doSolve :: Bool -> IO () doSolve b | b = done -- IO monad auxiliary functions: --- Executes a sequence of I/O actions and collects all results in a list. sequenceIO :: [IO a] -> IO [a] sequenceIO [] = return [] sequenceIO (c:cs) = do x <- c xs <- sequenceIO cs return (x:xs) --- Executes a sequence of I/O actions and ignores the results. sequenceIO_ :: [IO _] -> IO () sequenceIO_ = foldr (>>) done --- Maps an I/O action function on a list of elements. --- The results of all I/O actions are collected in a list. mapIO :: (a -> IO b) -> [a] -> IO [b] mapIO f = sequenceIO . map f --- Maps an I/O action function on a list of elements. --- The results of all I/O actions are ignored. mapIO_ :: (a -> IO _) -> [a] -> IO () mapIO_ f = sequenceIO_ . map f --- Folds a list of elements using an binary I/O action and a value --- for the empty list. foldIO :: (a -> b -> IO a) -> a -> [b] -> IO a foldIO _ a [] = return a foldIO f a (x:xs) = f a x >>= \fax -> foldIO f fax xs --- Apply a pure function to the result of an I/O action. liftIO :: (a -> b) -> IO a -> IO b liftIO f m = m >>= return . f --- Like `mapIO`, but with flipped arguments. --- --- This can be useful if the definition of the function is longer --- than those of the list, like in --- --- forIO [1..10] $ \n -> do --- ... forIO :: [a] -> (a -> IO b) -> IO [b] forIO xs f = mapIO f xs --- Like `mapIO_`, but with flipped arguments. --- --- This can be useful if the definition of the function is longer --- than those of the list, like in --- --- forIO_ [1..10] $ \n -> do --- ... forIO_ :: [a] -> (a -> IO b) -> IO () forIO_ xs f = mapIO_ f xs --- Performs an `IO` action unless the condition is met. unless :: Bool -> IO () -> IO () unless p act = if p then done else act --- Performs an `IO` action when the condition is met. when :: Bool -> IO () -> IO () when p act = if p then act else done ---------------------------------------------------------------- -- Non-determinism and free variables: --- Non-deterministic choice _par excellence_. --- The value of `x ? y` is either `x` or `y`. --- @param x - The right argument. --- @param y - The left argument. --- @return either `x` or `y` non-deterministically. (?) :: a -> a -> a x ? _ = x _ ? y = y -- Returns non-deterministically any element of a list. anyOf :: [a] -> a anyOf = foldr1 (?) --- Evaluates to a fresh free variable. unknown :: _ unknown = let x free in x ---------------------------------------------------------------- --- Identity type synonym used to mark deterministic operations. type DET a = a --- Identity function used by the partial evaluator --- to mark expressions to be partially evaluated. PEVAL :: a -> a PEVAL x = x --- Evaluates the argument to normal form and returns it. normalForm :: a -> a normalForm x = id $!! x --- Evaluates the argument to ground normal form and returns it. --- Suspends as long as the normal form of the argument is not ground. groundNormalForm :: a -> a groundNormalForm x = id $## x -- Only for internal use: -- Representation of higher-order applications in FlatCurry. apply :: (a -> b) -> a -> b apply external -- Only for internal use: -- Representation of conditional rules in FlatCurry. cond :: Bool -> a -> a cond external --- Non-strict equational constraint. Used to implement functional patterns. (=:<=) :: a -> a -> Bool (=:<=) external -- the end of the standard prelude |