`abs' takes the absolute value of a number - e.g. abs (-3) is 3, abs 3.5 is 3.5

> abs :: num->num > abs x = -x,`and' applied to a list of truthvalues, takes their logical conjunction.ifx<0 > = x,otherwise

> and :: [bool]->bool > and = foldr (&) True`arctan' is the trigonometric function, inverse tangent. It returns a result in the range -pi/2 to pi/2. See also `sin', `cos'.

> arctan :: num->num ||defined internally`bool' is the type comprising the two truthvalues.

bool ::= False | True ||primitive to Miranda`char' is the type comprising ascii characters (e.g. 'a', '\n').

char ::`cjustify' applied to a number and a string, centre justifies the string in a field of the specified width. See also `ljustify', `rjustify', `spaces'.type||primitive to Miranda

> cjustify :: num->[char]->[char] > cjustify n s = spaces lmargin++s++spaces rmargin >`code' applied to a character returns the integer which is its ascii code. E.g.where> margin = n - # s > lmargin = margindiv2 > rmargin = margin - lmargin

code 'a' = 97.See also `decode'.

> code :: char->num ||defined internally`concat' applied to a list of lists, joins them all together into a single list with `++'. E.g.

concat [[1,2],[],[3,4]] = [1,2,3,4].

> concat :: [[*]]->[*] > concat = foldr (++) []`const' is a combinator for creating constant-valued functions. E.g. (const 3) is the function that always returns 3.

> const :: *->**->* > const x y = x`converse' is a combinator for inverting the order of arguments of a two-argument function.

> converse :: (*->**->***)->**->*->*** > converse f a b = f b a`cos' is the trigonometric cosine function, argument in radians.

> cos :: num->num ||defined internally`decode' applied to a number between 0 and 255 returns the corresponding value of type char.

> decode :: num->char ||defined internally`digit' is a predicate on characters. True if the character is a digit. See also `letter'.

> digit :: char->bool > digit x = '0'<=x<='9'`drop' applied to a number and a list returns the list with that many elements removed from the front. If the list has less than the required number of elements, `drop' returns []. Example

drop 2 [1,2,3,4] = [3,4]See also `take'.

> drop :: num->[*]->[*] ||defined internally, as below

drop (n+1) (a:x) = drop n x drop n x = x,`dropwhile' applied to a predicate and a list, removes elements from the front of the list while the predicate is satisfied. Example:ifinteger n = error "drop applied to fractional number",otherwise

dropwhile digit "123gone" = "gone"See also `takewhile'.

> dropwhile :: (*->bool)->[*]->[*] > dropwhile f [] = [] > dropwhile f (a:x) = dropwhile f x,`e' is a transcendental number, the base of natural logarithms.iff a > = a:x,otherwise

> e :: num > e = exp 1`entier' when applied to a number returns its integer part, meaning the largest integer not exceeding it. E.g.

entier 1.0 = 1 entier 3.5 = 3 entier (-3.5) = -4.Notice that for Miranda the number `1' and the number `1.0' are different values - for example they yield different results under the `integer' test. However `1=1.0' is True, because of the automatic conversion from integer to float.

> entier :: num->num ||defined internally

A useful fact about `entier', which relates it to the operators **div** and
**mod**, is that the following law holds for any integers a, b with b~=0 and
a/b within the range for which integers can be represented exactly as
fractional numbers

a`error' applied to a string creates an error value with the associated message. Error values are all equivalent to the undefined value - any attempt to access the value causes the program to terminate and print the string as a diagnostic.divb = entier (a/b)

> error :: [char]->* ||defined internally`exp' is the exponential function on real numbers. See also `log'.

> exp :: num->num ||defined internally`filemode' applied to a string representing the pathname of a UNIX file, returns a string of length four giving the access permissions of the current process to the file. The permissions are encoded as (in this order) "drwx", any permission not granted is replaced by a '-' character. If there is no file at pathname p, filemode p returns the empty string. Example

member (filemode f) 'w'tests f for write permission. See also `getenv', `read', `system'.

> filemode :: [char]->[char] ||defined internally`filter' applied to a predicate and a list, returns a list containing only those elements that satisfy the predicate. Example

filter (>5) [3,7,2,8,1,17] = [7,8,17]

> filter :: (*->bool)->[*]->[*] > filter f x = [a | a<-x; f a]`foldl' folds up a list, using a given binary operator and a given start value, in a left associative way. Example:

foldl op r [a,b,c] = (((r $op a) $op b) $op c)But note that in order to run in constant space, foldl forces `op' to evaluate its first parameter. See the definitions of `product', `reverse', `sum' for examples of its use. See also `foldr'.

> foldl :: (*->**->*)->*->[**]->* ||defined internally, as below foldl op r [] = r foldl op r (a:x) = strict (foldl op) (op r a) x where strict f x = seq x (f x)WARNING - this definition of foldl differs from that in older versions of Miranda. The one here is the same as that in Bird and Wadler (1988). The old definition had the two args of `op' reversed. That is:-

old_foldl op r = new_foldl (converse op) rthe function `converse' has been added to the standard environment.

`foldl1' folds left over non-empty lists. See the definitions of `max', `min' for examples of its use.

> foldl1 :: (*->*->*)->[*]->* ||defined internally, as below foldl1 op (a:x) = foldl op a x foldl1 op [] = error "foldl1 applied to []"`foldr' folds up a list, using a given binary operator and a given start value, in a right associative way. Example:

foldr op r [a,b,c] = a $op (b $op (c $op r))See the definitions of `and', `concat', `or', for examples of its use.

> foldr :: (*->**->**)->**->[*]->** ||defined internally, as below foldr op r [] = r foldr op r (a:x) = op a (foldr op r x)`foldr1' folds right over non-empty lists.

> foldr1 :: (*->*->*)->[*]->* > foldr1 op [a] = a > foldr1 op (a:b:x) = op a (foldr1 op (b:x)) > foldr1 op [] = error "foldr1 applied to []"`force' applied to any data structure, returns it, but forces a check that every part of the structure is defined. Example

hd(force x)returns the hd of x, but fully evaluates x first (so x must be finite). See also `seq'. Notice in particular the idiom `seq (force a) b' which returns `b' but only after fully evaluating `a'.

> force :: *->* ||defined internally`fst' returns the first component of a pair. See also `snd'.

> fst :: (*,**)->* > fst (a,b) = a`getenv' looks up a string in the user's UNIX environment. Example

getenv "HOME"returns the pathname of your home directory. [If you want to see what else is in your UNIX environment, say `printenv' as a UNIX command.]

> getenv :: [char]->[char] ||defined internally`hd' applied to a non empty list, returns its first element. It is an error to apply `hd' to the empty list, []. See also `tl'.

> hd :: [*]->* > hd (a:x) = a > hd [] = error "hd []"`hugenum' is the largest fractional number that can exist in this implementation (should be around 1e308 for IEEE standard 64 bit floating point). See also `tinynum'.

> hugenum :: num ||defined internally`id' is the identity function - applied to any object it returns it.

> id :: *->* > id x = x`index' applied to a (finite or infinite) list, returns a list of its legal subscript values, in ascending order. E.g. index "hippopotamus" is [0,1,2,3,4,5,6,7,8,9,10,11].

> index :: [*]->[num] > index x = f 0 x >`init' is dual to `tl', it returns a list without its last component. Examplewhere> f n [] = [] > f n (a:x) = n:f(n+1)x

init [1,2,3,4] = [1,2,3].See also `last'. [Note, by the `dual' of a list processing function we mean the function which does the same job in a world where all lists have been reversed.]

> init :: [*]->[*] > init (a:x) = [],`integer' is a predicate on numbers. True if and only if the number is not fractional.ifx=[] > = a:init x,otherwise> init [] = error "init []"

> integer :: num->bool ||defined internally`iterate' - iterate f x returns the infinite list [x, f x, f(f x), ... ] Example, iterate (2*) 1 yields a list of the powers of 2. In the definition note use of ", .." to express a recurrence relation (see manual section 13/2).

> iterate :: (*->*)->*->[*] > iterate f x = [y | y<-x, f y ..]`last' applied to a non empty list returns its last element. This function is the dual of `hd'. Note that for any non-empty list x

(init x ++ [last x]) = x

> last :: [*]->* ||defined internally, as below

last x = x!(#x-1)`lay' applied to a list of strings, joins them together after appending a newline character to each string. Example

lay ["hello","world"] = "hello\nworld\n"Used to format output thus,

lay(map show x)as a top level expression, causes the elements of the list x to be printed one per line. See also `layn', `lines'.

> lay :: [[char]]->[char] > lay [] = [] > lay (a:x) = a++"\n"++lay x`layn' is similar to `lay', but produces output with numbered lines.

> layn :: [[char]]->[char] > layn x = f 1 x >'letter' is a predicate on characters. True if the character is a letter.where> f n [] = [] > f n (a:x) = rjustify 4 (show n) ++") "++a++"\n"++f (n+1) x

> letter :: char->bool > letter c = 'a'<=c<='z' \/ 'A'<=c<='Z'`limit' applied to a list of values, returns the first value which is the same as its successor. Useful in testing for convergence. For example the following Miranda expression computes the square root of 2 by the Newton-Raphson method

limit [x | x<-2, 0.5*(x + 2/x).. ]

> limit :: [*]->* > limit (a:b:x) = a,`lines' applied to a list of characters containing newlines, returns a list of lists, by breaking the original into lines. The newline characters are removed from the result. Example, `lines' applied toifa=b > = limit (b:x),otherwise> limit other = error "incorrect use of limit"

"hello world\nit's me,\neric\n"returns ["hello world","it's me","eric"]. Note that `lines' treats newline as a terminator, not a separator (although it will tolerate a missing '\n' on the last line).

> lines :: [char]->[[char]] > lines [] = [] > lines (a:x) = []:lines x,Note that the inverse of `lines' is the function `lay', in that applying `lay' to the output of `lines' will restore the original string (except that a final newline will be added, if missing in the original string).ifa='\n' > = (a:x1):xrest,otherwise>where> (x1:xrest) = lines x,ifx~=[] > = []:[],otherwise> ||this handles missing '\n' on last line

`ljustify' applied to a number and a string, left justifies the string in a field of the specified width.

> ljustify :: num->[char]->[char] > ljustify n s = s++spaces(n - # s)`log' applied to a number returns its natural logarithm (i.e. logarithm to the base `e'). It is the inverse of the exponential function, `exp'. See also log10. Note that the log functions use a different algorithm when applied to integer arguments (rather than just converting to float first) so it is possible to take log, or log10, of very large integers.

> log :: num->num ||defined internally`log10' applied to a number returns its logarithm to the base 10.

> log10 :: num->num ||defined internally`map' applied to a function and a list returns a copy of the list in which the given function has been applied to every element.

> map :: (*->**)->[*]->[**] > map f x = [f a | a<-x]`map2' is similar to `map', but takes a function of two arguments, and maps it along two argument lists. We could also define `map3', `map4' etc., but they are much less often needed.

> map2 :: (*->**->***)->[*]->[**]->[***] > map2 f x y = [f a b | (a,b)<-zip2 x y]Note: the Bird and Wadler function `zipwith' is just an uncurried version of `map2', that is `zipwith f (x,y)' means `map2 f x y'.

`max' applied to a list returns the largest element under the built in ordering of `>'. Examples

max [1,2,12,-6,5] = 12 max "hippopotamus" = 'u'See also `min', `sort'.

> max :: [*]->* > max = foldl1 max2`max2' applied to two values of the same type returns the larger under the built in ordering of '>'. See also `min2'.

> max2 :: *->*->* > max2 a b = a,`member' applied to a list and a value returns True or False as the value is or not present in the list.ifa>=b > = b,otherwise

> member :: [*]->*->bool > member x a = or (map (=a) x)`merge' applied to two sorted lists merges them to produce a single sorted result. Used to define `sort', see later.

> merge :: [*]->[*]->[*] ||defined internally, as below

merge [] y = y merge (a:x) [] = a:x merge (a:x) (b:y) = a:merge x (b:y),`min' applied to a list returns its least member under `<'.ifa<=b = b:merge (a:x) y,otherwise

> min :: [*]->* > min = foldl1 min2`min2' applied to two values of the same type returns the smaller under the built in ordering of '<'.

> min2 :: *->*->* > min2 a b = b,`mkset' applied to a list returns a copy of the list from which any duplicated elements have been removed. A list without duplications can be used to represent a set, whence the name. Works even on infinite list, but (beware) takes a time quadratic in the number of elements processed.ifa>b > = a,otherwise

> mkset :: [*]->[*] > mkset [] = [] > mkset (a:x) = a:filter (~=a) (mkset x)`neg' is a function of one numeric argument, with the same action as the unary `-' operator.

> neg :: num->num > neg x = -x`num' is the type comprising both integer and fractional numbers (such as 42, -12.73e8).

num ::`numval' converts a numeric string to the corresponding number - can cope with sign, decimal point and scale factor (uses same rules as Miranda compiler). Strings containing inappropriate characters cause an error (exception - leading white space is harmless).type||primitive to Miranda

> numval :: [char]->num ||defined internally`or' applied to a list of truthvalues, takes their logical disjunction.

> or :: [bool]->bool > or = foldr (\/) False`pi' is the well known real number (the ratio of the circumference of a circle to its diameter).

> pi :: num > pi = 4*arctan 1`postfix' takes an element and a list and adds the element to the end of the list. This is the dual of the prefix operator, `:'.

> postfix :: *->[*]->[*] > postfix a x = x ++ [a]`product' applied to list of numbers returns their product. See also `sum'.

> product :: [num]->num > product = foldl (*) 1`read' returns the contents of file with a given pathname. Provides an interface to the UNIX filing system. If the file is empty `read' returns [], but if the file does not exist, or lacks read permission, `read' causes an error. See also `filemode', `getenv'.

> read :: [char]->[char] ||defined internally

`rep' applied to a number and a value, returns a list containing the specified number of instances of the value. (The name is short for `replicate'.) Example

rep 6 'o' = "oooooo"See also `repeat'.

> rep :: num->*->[*] > rep n x = take n (repeat x)`repeat' applied to a value returns an infinite list, all of whose elements are the given value.

> repeat :: *->[*] > repeat x = xs >`reverse' applied to any finite list returns a list of the same elements in reverse order.wherexs = x:xs

> reverse :: [*]->[*] > reverse = foldl (converse(:)) []`rjustify' applied to a number and a string, right justifies the string in a field of the specified width.

> rjustify :: num->[char]->[char] > rjustify n s = spaces(n - # s)++s`scan op r' applies `foldl op r' to every initial segment of a list. For example `scan (+) 0 x' computes running sums.

> scan :: (*->**->*)->*->[**]->[*] > scan op = g >There is another way to explain `scan', which makes it clearer why it is useful. Let s0 be the initial state of an automaton, and f::state->input->state, its state transition function - then `scan f s0' is a function that takes a list of inputs for the automaton and returns the resulting list of states, starting with s0.where> g r = (r:). rest >where> rest [] = [] > rest (a:x) = g (op r a) x

`seq' applied to two values, returns the second but checks that the first value is not completely undefined. Sometimes needed, e.g. to ensure correct synchronisation in interactive programs.

> seq :: *->**->** ||defined internally

`shownum' applied to a number returns as a string a standard print representation for it. A special case of the operator `show'. Applied to fractional numbers `shownum' gives 12 decimal places (less any trailing zeros), using a format appropriate to the size of number. For more detailed control over number format see `showfloat', `showscaled'.

> shownum :: num->[char] ||defined internally,`showfloat p x' returns as a string the number x printed in floating point format, that is in the form "digits.digits", where the integer p (>=0) gives the number of digits after the decimal point.

> showfloat :: num->num->[char] ||defined internally,`showscaled p x' returns as a string the number x printed in scientific format, that is in the form "n.nnnnnne[+/-]nn", where the integer p (>=0) gives the number of digits required after the decimal point.

> showscaled :: num->num->[char] ||defined internally,`sin' is the trigonometric sine function, argument in radians.

> sin :: num->num ||defined internally`snd' returns the second component of a pair.

> snd :: (*,**)->** > snd (a,b) = b`sort' applied to any finite list sorts the elements of the list into ascending order on the built in '<' relation. Note that you cannot sort a list of functions. Example

sort "hippopotamus" = "ahimoopppstu"The following definition uses merge-sort, which has n log n worst-case behaviour.

> sort :: [*]->[*] > sort x = x,`spaces' applied to a number returns a list of that many spaces.ifn<=1 > = merge (sort(take n2 x)) (sort(drop n2 x)),otherwise>where> n = # x > n2 = ndiv2

> spaces :: num->[char] > spaces n = rep n ' '`sqrt' is the square root function on (integer or fractional) numbers. The result is always fractional.

> sqrt :: num->num ||defined internally`subtract' is a name for (converse) infix minus. Needed because you cannot form postsections in `-'. (See manual entry on `sections'.) Example

subtract 3is the function that subtracts 3.

> subtract :: num->num->num > subtract x y = y - x`sum' applied to list of numbers returns their sum.

> sum :: [num]->num > sum = foldl (+) 0`sys_message' is an algebraic type containing a family of constructors used to control output to UNIX files. See manual section on Output to UNIX files for details.

> sys_message ::= Stdout [char] | Stderr [char] | Tofile [char] [char] | > Closefile [char] | Appendfile [char] | System [char] | > Exit num`system' applied to a string causes the string to be executed as a UNIX shell command (by `sh'). The result returned is a 3-tuple, comprising the standard_output, error_output, and exit_status respectively, resulting from the execution of the UNIX command. See manual section on Input from UNIX files etc for more details.

> system :: [char]->([char],[char],num) ||defined internally`take' applied to a number and a list returns the specified number of elements from the front of the list. If the list has less than the required number of elements, `take' returns as many as it can get. Examples

take 2 [1,2,3,4] = [1,2] take 7 "girls" = "girls"

> take :: num->[*]->[*] ||defined internally, as below

take (n+1) (a:x) = a : take n x take n x = [],`takewhile' applied to a predicate and a list, takes elements from the front of the list while the predicate is satisfied. Example: takewhile digit "123gone" = "123"ifinteger n = error "take applied to fractional number",otherwise

> takewhile :: (*->bool)->[*]->[*] > takewhile f [] = [] > takewhile f (a:x) = a:takewhile f x,`tinynum' is the smallest positive fractional number that can be distinguished from zero in this implementation (should be around 1e-324 for IEEE standard 64 bit floating point).iff a > = [],otherwise

> tinynum :: num ||defined internally`tl' applied to a non empty list returns the list without its first element. Example, tl "snow" is "now".

> tl :: [*]->[*] > tl (a:x) = x > tl [] = error "tl []"`transpose' applied to a list of lists, returns their transpose (in the sense of matrix transpose - rows and columns are interchanged). Example

transpose [[1,2,3],[4,5,6]] = [[1,4],[2,5],[3,6]]The following definition is slightly more subtle than is at first sight necessary, in order to deal correctly with `upper triangular' matrices. Example

transpose [[1,2,3],[4,5],[6]] = [[1,4,6],[2,5],[3]]

> transpose :: [[*]]->[[*]] > transpose x = [],It might be thought that this function belongs in a specialised library of matrix handling functions, but it has been found useful as a general purpose list processing function, whence its inclusion in the standard environment.ifx'=[] > = map hd x':transpose(map tl x'),otherwise>where> x' = takewhile (~=[]) x

`undef' is a name for the completely undefined value. Any attempt access it results in an error message. Note that `undef' belongs to every type.

> undef :: * > undef = error "undefined"`until' applied to a predicate, a function and a value, returns the result of applying the function to the value the smallest number of times necessary to satisfy the predicate. Example

until (>1000) (2*) 1 = 1024

> until :: (*->bool)->(*->*)->*->* > until f g x = x,`zip2' applied to two lists returns a list of pairs, formed by tupling together corresponding elements of the given lists. Exampleiff x > = until f g (g x),otherwise

zip2 [0..3] "type" = [(0,'t'),(1,'y'),(2,'p'),(3,'e')]This function is often useful in list comprehensions, where it provides an idiom for accessing two generators simultaneously. For example the following expression returns the scalar product of x and y (x,y::[num])

sum [ a*b | (a,b) <- zip2 x y ]

> zip2 :: [*]->[**]->[(*,**)] ||defined internally, as below zip2 (a:x) (b:y) = (a,b):zip2 x y zip2 x y = []Note that if the lists being zipped are of different lengths, the length of the result is that of the shortest list (this holds for zip2 and all the following zip functions).

The function `zip3' is analogous but takes three lists and returns a list of 3-tuples. Similarly for `zip4', `zip5', `zip6' - zip functions above zip6 are not provided in the standard environment.

> zip3 (a:x) (b:y) (c:z) = (a,b,c):zip3 x y z > zip3 x y z = [] > zip4 (a:w) (b:x) (c:y) (d:z) = (a,b,c,d):zip4 w x y z > zip4 w x y z = [] > zip5 (a:v) (b:w) (c:x) (d:y) (e:z) = (a,b,c,d,e):zip5 v w x y z > zip5 v w x y z = [] > zip6 (a:u)(b:v)(c:w)(d:x)(e:y)(f:z) = (a,b,c,d,e,f):zip6 u v w x y z > zip6 u v w x y z = []The following is included for compatibility with Bird and Wadler (1988). The normal Miranda style is to use the curried form `zip2'.

> zip :: ([*],[**])->[(*,**)] > zip (x,y) = zip2 x yEnd of definitions of the standard environment