(*  Title:      Pure/library.ML

     ID:         $Id$

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     Author:     Markus Wenzel, TU Muenchen

 Basic library: functions, options, pairs, booleans, lists, integers,

 strings, lists as sets, orders, current directory, misc.

 See also General/basics.ML for the most fundamental concepts.

 *)

 infix 1 |>>>

 infix 2 ?

 infix 3 o oo ooo oooo

 infix 4 ~~ upto downto

 infix orf andf \ \\ mem mem_int mem_string union union_int

   union_string inter inter_int inter_string subset subset_int subset_string

 signature BASIC_LIBRARY =

 sig

   (*functions*)

   val undefined: 'a -> 'b

   val I: 'a -> 'a

   val K: 'a -> 'b -> 'a

   val flip: ('a -> 'b -> 'c) -> 'b -> 'a -> 'c

   val curry: ('a * 'b -> 'c) -> 'a -> 'b -> 'c

   val uncurry: ('a -> 'b -> 'c) -> 'a * 'b -> 'c

   val |>>> : ('a * 'c) * ('a -> 'b * 'd) -> 'b * ('c * 'd)

   val ? : bool * ('a -> 'a) -> 'a -> 'a

   val oo: ('a -> 'b) * ('c -> 'd -> 'a) -> 'c -> 'd -> 'b

   val ooo: ('a -> 'b) * ('c -> 'd -> 'e -> 'a) -> 'c -> 'd -> 'e -> 'b

   val oooo: ('a -> 'b) * ('c -> 'd -> 'e -> 'f -> 'a) -> 'c -> 'd -> 'e -> 'f -> 'b

   val funpow: int -> ('a -> 'a) -> 'a -> 'a

   (*errors*)

   exception SYS_ERROR of string

   val sys_error: string -> 'a

   exception ERROR of string

   val error: string -> 'a

   val cat_error: string -> string -> 'a

   val assert_all: ('a -> bool) -> 'a list -> ('a -> string) -> unit

   (*pairs*)

   val pair: 'a -> 'b -> 'a * 'b

   val rpair: 'a -> 'b -> 'b * 'a

   val fst: 'a * 'b -> 'a

   val snd: 'a * 'b -> 'b

   val eq_fst: ('a * 'c -> bool) -> ('a * 'b) * ('c * 'd) -> bool

   val eq_snd: ('b * 'd -> bool) -> ('a * 'b) * ('c * 'd) -> bool

   val eq_pair: ('a * 'c -> bool) -> ('b * 'd -> bool) -> ('a * 'b) * ('c * 'd) -> bool

   val swap: 'a * 'b -> 'b * 'a

   val apfst: ('a -> 'b) -> 'a * 'c -> 'b * 'c

   val apsnd: ('a -> 'b) -> 'c * 'a -> 'c * 'b

   val pairself: ('a -> 'b) -> 'a * 'a -> 'b * 'b

   (*booleans*)

   val equal: ''a -> ''a -> bool

   val not_equal: ''a -> ''a -> bool

   val orf: ('a -> bool) * ('a -> bool) -> 'a -> bool

   val andf: ('a -> bool) * ('a -> bool) -> 'a -> bool

   val exists: ('a -> bool) -> 'a list -> bool

   val forall: ('a -> bool) -> 'a list -> bool

   val set: bool ref -> bool

   val reset: bool ref -> bool

   val toggle: bool ref -> bool

   val change: 'a ref -> ('a -> 'a) -> unit

   val setmp_noncritical: 'a ref -> 'a -> ('b -> 'c) -> 'b -> 'c

   val setmp: 'a ref -> 'a -> ('b -> 'c) -> 'b -> 'c

   (*lists*)

   exception UnequalLengths

   val single: 'a -> 'a list

   val the_single: 'a list -> 'a

   val singleton: ('a list -> 'b list) -> 'a -> 'b

   val apply: ('a -> 'a) list -> 'a -> 'a

   val foldr1: ('a * 'a -> 'a) -> 'a list -> 'a

   val foldl_map: ('a * 'b -> 'a * 'c) -> 'a * 'b list -> 'a * 'c list

   val flat: 'a list list -> 'a list

   val unflat: 'a list list -> 'b list -> 'b list list

   val burrow: ('a list -> 'b list) -> 'a list list -> 'b list list

   val fold_burrow: ('a list -> 'c -> 'b list * 'd) -> 'a list list -> 'c -> 'b list list * 'd

   val maps: ('a -> 'b list) -> 'a list -> 'b list

   val chop: int -> 'a list -> 'a list * 'a list

   val dropwhile: ('a -> bool) -> 'a list -> 'a list

   val nth: 'a list -> int -> 'a

   val nth_map: int -> ('a -> 'a) -> 'a list -> 'a list

   val nth_list: 'a list list -> int -> 'a list

   val map_index: (int * 'a -> 'b) -> 'a list -> 'b list

   val fold_index: (int * 'a -> 'b -> 'b) -> 'a list -> 'b -> 'b

   val split_last: 'a list -> 'a list * 'a

   val find_index: ('a -> bool) -> 'a list -> int

   val find_index_eq: ''a -> ''a list -> int

   val find_first: ('a -> bool) -> 'a list -> 'a option

   val get_index: ('a -> 'b option) -> 'a list -> (int * 'b) option

   val get_first: ('a -> 'b option) -> 'a list -> 'b option

   val eq_list: ('a * 'b -> bool) -> 'a list * 'b list -> bool

   val map2: ('a -> 'b -> 'c) -> 'a list -> 'b list -> 'c list

   val fold2: ('a -> 'b -> 'c -> 'c) -> 'a list -> 'b list -> 'c -> 'c

   val zip_options: 'a list -> 'b option list -> ('a * 'b) list

   val ~~ : 'a list * 'b list -> ('a * 'b) list

   val split_list: ('a * 'b) list -> 'a list * 'b list

   val separate: 'a -> 'a list -> 'a list

   val replicate: int -> 'a -> 'a list

   val multiply: 'a list -> 'a list list -> 'a list list

   val product: 'a list -> 'b list -> ('a * 'b) list

   val filter: ('a -> bool) -> 'a list -> 'a list

   val filter_out: ('a -> bool) -> 'a list -> 'a list

   val map_filter: ('a -> 'b option) -> 'a list -> 'b list

   val is_prefix: ('a * 'a -> bool) -> 'a list -> 'a list -> bool

   val take_prefix: ('a -> bool) -> 'a list -> 'a list * 'a list

   val chop_prefix: ('a * 'b -> bool) -> 'a list * 'b list -> 'a list * ('a list * 'b list)

   val take_suffix: ('a -> bool) -> 'a list -> 'a list * 'a list

   val prefixes1: 'a list -> 'a list list

   val prefixes: 'a list -> 'a list list

   val suffixes1: 'a list -> 'a list list

   val suffixes: 'a list -> 'a list list

   (*integers*)

   val inc: int ref -> int

   val dec: int ref -> int

   val upto: int * int -> int list

   val downto: int * int -> int list

   val radixpand: int * int -> int list

   val radixstring: int * string * int -> string

   val string_of_int: int -> string

   val signed_string_of_int: int -> string

   val string_of_indexname: string * int -> string

   val read_intinf: int -> string list -> IntInf.int * string list

   val read_int: string list -> int * string list

   val oct_char: string -> string

   (*strings*)

   val nth_string: string -> int -> string

   val fold_string: (string -> 'a -> 'a) -> string -> 'a -> 'a

   val exists_string: (string -> bool) -> string -> bool

   val forall_string: (string -> bool) -> string -> bool

   val enclose: string -> string -> string -> string

   val unenclose: string -> string

   val quote: string -> string

   val space_implode: string -> string list -> string

   val commas: string list -> string

   val commas_quote: string list -> string

   val cat_lines: string list -> string

   val space_explode: string -> string -> string list

   val split_lines: string -> string list

   val prefix_lines: string -> string -> string

   val untabify: string list -> string list

   val prefix: string -> string -> string

   val suffix: string -> string -> string

   val unprefix: string -> string -> string

   val unsuffix: string -> string -> string

   val replicate_string: int -> string -> string

   val translate_string: (string -> string) -> string -> string

   (*lists as sets -- see also Pure/General/ord_list.ML*)

   val member: ('b * 'a -> bool) -> 'a list -> 'b -> bool

   val insert: ('a * 'a -> bool) -> 'a -> 'a list -> 'a list

   val remove: ('b * 'a -> bool) -> 'b -> 'a list -> 'a list

   val update: ('a * 'a -> bool) -> 'a -> 'a list -> 'a list

   val subtract: ('b * 'a -> bool) -> 'b list -> 'a list -> 'a list

   val merge: ('a * 'a -> bool) -> 'a list * 'a list -> 'a list

   val mem: ''a * ''a list -> bool

   val mem_int: int * int list -> bool

   val mem_string: string * string list -> bool

   val union: ''a list * ''a list -> ''a list

   val union_int: int list * int list -> int list

   val union_string: string list * string list -> string list

   val gen_union: ('a * 'a -> bool) -> 'a list * 'a list -> 'a list

   val gen_inter: ('a * 'b -> bool) -> 'a list * 'b list -> 'a list

   val inter: ''a list * ''a list -> ''a list

   val inter_int: int list * int list -> int list

   val inter_string: string list * string list -> string list

   val subset: ''a list * ''a list -> bool

   val subset_int: int list * int list -> bool

   val subset_string: string list * string list -> bool

   val eq_set: ''a list * ''a list -> bool

   val eq_set_string: string list * string list -> bool

   val gen_subset: ('a * 'b -> bool) -> 'a list * 'b list -> bool

   val gen_eq_set: ('a * 'b -> bool) -> 'a list * 'b list -> bool

   val \ : ''a list * ''a -> ''a list

   val \\ : ''a list * ''a list -> ''a list

   val distinct: ('a * 'a -> bool) -> 'a list -> 'a list

   val duplicates: ('a * 'a -> bool) -> 'a list -> 'a list

   val has_duplicates: ('a * 'a -> bool) -> 'a list -> bool

   (*lists as multisets*)

   val remove1: ('b * 'a -> bool) -> 'b -> 'a list -> 'a list

   val submultiset: ('a * 'b -> bool) -> 'a list * 'b list -> bool

   (*orders*)

   val is_equal: order -> bool

   val rev_order: order -> order

   val make_ord: ('a * 'a -> bool) -> 'a * 'a -> order

   val int_ord: int * int -> order

   val string_ord: string * string -> order

   val fast_string_ord: string * string -> order

   val option_ord: ('a * 'b -> order) -> 'a option * 'b option -> order

   val prod_ord: ('a * 'b -> order) -> ('c * 'd -> order) -> ('a * 'c) * ('b * 'd) -> order

   val dict_ord: ('a * 'b -> order) -> 'a list * 'b list -> order

   val list_ord: ('a * 'b -> order) -> 'a list * 'b list -> order

   val sort: ('a * 'a -> order) -> 'a list -> 'a list

   val sort_distinct: ('a * 'a -> order) -> 'a list -> 'a list

   val sort_strings: string list -> string list

   val sort_wrt: ('a -> string) -> 'a list -> 'a list

   (*random numbers*)

   exception RANDOM

   val random: unit -> real

   val random_range: int -> int -> int

   val one_of: 'a list -> 'a

   val frequency: (int * 'a) list -> 'a

   (*misc*)

   val divide_and_conquer: ('a -> 'a list * ('b list -> 'b)) -> 'a -> 'b

   val partition_eq: ('a * 'a -> bool) -> 'a list -> 'a list list

   val partition_list: (int -> 'a -> bool) -> int -> int -> 'a list -> 'a list list

   val gensym: string -> string

   type stamp

   val stamp: unit -> stamp

   type serial

   val serial: unit -> serial

   val serial_string: unit -> string

   structure Object: sig type T end

 end;

 signature LIBRARY =

 sig

   include BASIC_LIBRARY

   val foldl: ('a * 'b -> 'a) -> 'a * 'b list -> 'a

   val foldr: ('a * 'b -> 'b) -> 'a list * 'b -> 'b

   val take: int * 'a list -> 'a list

   val drop: int * 'a list -> 'a list

   val last_elem: 'a list -> 'a

 end;

 structure Library: LIBRARY =

 struct

 (* functions *)

 fun undefined _ = raise Match;

 fun I x = x;

 fun K x = fn _ => x;

 fun flip f x y = f y x;

 fun curry f x y = f (x, y);

 fun uncurry f (x, y) = f x y;

 (*application and structured results -- old version*)

 fun (x, y) |>>> f = let val (x', z) = f x in (x', (y, z)) end;

 (*conditional application*)

 fun b ? f = fn x => if b then f x else x;

 (*composition with multiple args*)

 fun (f oo g) x y = f (g x y);

 fun (f ooo g) x y z = f (g x y z);

 fun (f oooo g) x y z w = f (g x y z w);

 (*function exponentiation: f(...(f x)...) with n applications of f*)

 fun funpow 0 _ x = x

   | funpow n f x = funpow (n - 1) f (f x);

 (* errors *)

 exception SYS_ERROR of string;

 fun sys_error msg = raise SYS_ERROR msg;

 exception ERROR of string;

 fun error msg = raise ERROR msg;

 fun cat_error "" msg = error msg

   | cat_error msg1 msg2 = error (msg1 ^ "\n" ^ msg2);

 fun assert_all pred list msg =

   let

     fun ass [] = ()

       | ass (x :: xs) = if pred x then ass xs else error (msg x);

   in ass list end;

 (* pairs *)

 fun pair x y = (x, y);

 fun rpair x y = (y, x);

 fun fst (x, y) = x;

 fun snd (x, y) = y;

 fun eq_fst eq ((x1, _), (x2, _)) = eq (x1, x2);

 fun eq_snd eq ((_, y1), (_, y2)) = eq (y1, y2);

 fun eq_pair eqx eqy ((x1, y1), (x2, y2)) = eqx (x1, x2) andalso eqy (y1, y2);

 fun swap (x, y) = (y, x);

 fun apfst f (x, y) = (f x, y);

 fun apsnd f (x, y) = (x, f y);

 fun pairself f (x, y) = (f x, f y);

 (* booleans *)

 (*polymorphic equality*)

 fun equal x y = x = y;

 fun not_equal x y = x <> y;

 (*combining predicates*)

 fun p orf q = fn x => p x orelse q x;

 fun p andf q = fn x => p x andalso q x;

 (*exists pred [x1, ..., xn] ===> pred x1 orelse ... orelse pred xn*)

 fun exists (pred: 'a -> bool) : 'a list -> bool =

   let fun boolf [] = false

         | boolf (x :: xs) = pred x orelse boolf xs

   in boolf end;

 (*forall pred [x1, ..., xn] ===> pred x1 andalso ... andalso pred xn*)

 fun forall (pred: 'a -> bool) : 'a list -> bool =

   let fun boolf [] = true

         | boolf (x :: xs) = pred x andalso boolf xs

   in boolf end;

 (* flags *)

 fun set flag = (flag := true; true);

 fun reset flag = (flag := false; false);

 fun toggle flag = (flag := not (! flag); ! flag);

 fun change r f = r := f (! r);

 (*temporarily set flag during execution*)

 fun setmp_noncritical flag value f x =

   let

     val orig_value = ! flag;

     val _ = flag := value;

     val result = Exn.capture f x;

     val _ = flag := orig_value;

   in Exn.release result end;

 fun setmp flag value f x = NAMED_CRITICAL "setmp" (fn () => setmp_noncritical flag value f x);

 (** lists **)

 exception UnequalLengths;

 fun single x = [x];

 fun the_single [x] = x

   | the_single _ = raise Empty;

 fun singleton f x = the_single (f [x]);

 fun apply [] x = x

   | apply (f :: fs) x = apply fs (f x);

 (* fold -- old versions *)

 (*the following versions of fold are designed to fit nicely with infixes*)

 (*  (op @) (e, [x1, ..., xn])  ===>  ((e @ x1) @ x2) ... @ xn

     for operators that associate to the left (TAIL RECURSIVE)*)

 fun foldl (f: 'a * 'b -> 'a) : 'a * 'b list -> 'a =

   let fun itl (e, [])  = e

         | itl (e, a::l) = itl (f(e, a), l)

   in  itl end;

 (*  (op @) ([x1, ..., xn], e)  ===>   x1 @ (x2 ... @ (xn @ e))

     for operators that associate to the right (not tail recursive)*)

 fun foldr f (l, e) =

   let fun itr [] = e

         | itr (a::l) = f(a, itr l)

   in  itr l  end;

 (*  (op @) [x1, ..., xn]  ===>   x1 @ (x2 ... @ (x[n-1] @ xn))

     for n > 0, operators that associate to the right (not tail recursive)*)

 fun foldr1 f [] = raise Empty

   | foldr1 f l =

       let fun itr [x] = x

             | itr (x::l) = f(x, itr l)

       in  itr l  end;

 fun foldl_map f =

   let

     fun fold_aux (x, []) = (x, [])

       | fold_aux (x, y :: ys) =

           let

             val (x', y') = f (x, y);

             val (x'', ys') = fold_aux (x', ys);

           in (x'', y' :: ys') end;

   in fold_aux end;

 (* basic list functions *)

 fun eq_list eq (list1, list2) =

   let

     fun eq_lst (x :: xs, y :: ys) = eq (x, y) andalso eq_lst (xs, ys)

       | eq_lst _ = true;

   in length list1 = length list2 andalso eq_lst (list1, list2) end;

 fun maps f [] = []

   | maps f (x :: xs) = f x @ maps f xs;

 fun chop 0 xs = ([], xs)

   | chop _ [] = ([], [])

   | chop n (x :: xs) = chop (n - 1) xs |>> cons x;

 (*take the first n elements from a list*)

 fun take (n, []) = []

   | take (n, x :: xs) =

       if n > 0 then x :: take (n - 1, xs) else [];

 (*drop the first n elements from a list*)

 fun drop (n, []) = []

   | drop (n, x :: xs) =

       if n > 0 then drop (n - 1, xs) else x :: xs;

 fun dropwhile P [] = []

   | dropwhile P (ys as x::xs) = if P x then dropwhile P xs else ys;

 (*return nth element of a list, where 0 designates the first element;

   raise Subscript if list too short*)

 fun nth xs i = List.nth (xs, i);

 fun nth_list xss i = nth xss i handle Subscript => [];

 fun nth_map 0 f (x :: xs) = f x :: xs

   | nth_map n f (x :: xs) = x :: nth_map (n - 1) f xs

   | nth_map _ _ [] = raise Subscript;

 fun map_index f =

   let

     fun mapp _ [] = []

       | mapp i (x :: xs) = f (i, x) :: mapp (i+1) xs

   in mapp 0 end;

 fun fold_index f =

   let

     fun fold_aux _ [] y = y

       | fold_aux i (x :: xs) y = fold_aux (i+1) xs (f (i, x) y);

   in fold_aux 0 end;

 val last_elem = List.last;

 (*rear decomposition*)

 fun split_last [] = raise Empty

   | split_last [x] = ([], x)

   | split_last (x :: xs) = apfst (cons x) (split_last xs);

 (*find the position of an element in a list*)

 fun find_index pred =

   let fun find _ [] = ~1

         | find n (x :: xs) = if pred x then n else find (n + 1) xs;

   in find 0 end;

 fun find_index_eq x = find_index (equal x);

 (*find first element satisfying predicate*)

 val find_first = List.find;

 (*get first element by lookup function*)

 fun get_first _ [] = NONE

   | get_first f (x :: xs) =

       (case f x of

         NONE => get_first f xs

       | some => some);

 fun get_index f =

   let

     fun get _ [] = NONE

       | get i (x :: xs) =

           case f x

            of NONE => get (i + 1) xs

             | SOME y => SOME (i, y)

   in get 0 end;

 val flat = List.concat;

 fun unflat (xs :: xss) ys =

       let val (ps, qs) = chop (length xs) ys

       in ps :: unflat xss qs end

   | unflat [] [] = []

   | unflat _ _ = raise UnequalLengths;

 fun burrow f xss = unflat xss (f (flat xss));

 fun fold_burrow f xss s =

   apfst (unflat xss) (f (flat xss) s);

 (*separate s [x1, x2, ..., xn]  ===>  [x1, s, x2, s, ..., s, xn]*)

 fun separate s (x :: (xs as _ :: _)) = x :: s :: separate s xs

   | separate _ xs = xs;

 (*make the list [x, x, ..., x] of length n*)

 fun replicate n (x: 'a) : 'a list =

   let fun rep (0, xs) = xs

         | rep (n, xs) = rep (n - 1, x :: xs)

   in

     if n < 0 then raise Subscript

     else rep (n, [])

   end;

 fun translate_string f = String.translate (f o String.str);

 (*multiply [a, b, c, ...] * [xs, ys, zs, ...]*)

 fun multiply [] _ = []

   | multiply (x :: xs) yss = map (cons x) yss @ multiply xs yss;

 (*direct product*)

 fun product _ [] = []

   | product [] _ = []

   | product (x :: xs) ys = map (pair x) ys @ product xs ys;

 (* filter *)

 (*copy the list preserving elements that satisfy the predicate*)

 val filter = List.filter;

 fun filter_out f = filter (not o f);

 val map_filter = List.mapPartial;

 (* lists of pairs *)

 exception UnequalLengths;

 fun map2 _ [] [] = []

   | map2 f (x :: xs) (y :: ys) = f x y :: map2 f xs ys

   | map2 _ _ _ = raise UnequalLengths;

 fun fold2 f [] [] z = z

   | fold2 f (x :: xs) (y :: ys) z = fold2 f xs ys (f x y z)

   | fold2 f _ _ _ = raise UnequalLengths;

 fun zip_options (x :: xs) (SOME y :: ys) = (x, y) :: zip_options xs ys

   | zip_options (_ :: xs) (NONE :: ys) = zip_options xs ys

   | zip_options _ [] = []

   | zip_options [] _ = raise UnequalLengths;

 (*combine two lists forming a list of pairs:

   [x1, ..., xn] ~~ [y1, ..., yn]  ===>  [(x1, y1), ..., (xn, yn)]*)

 fun [] ~~ [] = []

   | (x :: xs) ~~ (y :: ys) = (x, y) :: (xs ~~ ys)

   | _ ~~ _ = raise UnequalLengths;

 (*inverse of ~~; the old 'split':

   [(x1, y1), ..., (xn, yn)]  ===>  ([x1, ..., xn], [y1, ..., yn])*)

 val split_list = ListPair.unzip;

 (* prefixes, suffixes *)

 fun is_prefix _ [] _ = true

   | is_prefix eq (x :: xs) (y :: ys) = eq (x, y) andalso is_prefix eq xs ys

   | is_prefix eq _ _ = false;

 (* [x1, ..., xi, ..., xn]  --->  ([x1, ..., x(i-1)], [xi, ..., xn])

    where xi is the first element that does not satisfy the predicate*)

 fun take_prefix (pred : 'a -> bool)  (xs: 'a list) : 'a list * 'a list =

   let fun take (rxs, []) = (rev rxs, [])

         | take (rxs, x :: xs) =

             if  pred x  then  take(x :: rxs, xs)  else  (rev rxs, x :: xs)

   in  take([], xs)  end;

 fun chop_prefix eq ([], ys) = ([], ([], ys))

   | chop_prefix eq (xs, []) = ([], (xs, []))

   | chop_prefix eq (xs as x :: xs', ys as y :: ys') =

       if eq (x, y) then

         let val (ps', xys'') = chop_prefix eq (xs', ys')

         in (x :: ps', xys'') end

       else ([], (xs, ys));

 (* [x1, ..., xi, ..., xn]  --->  ([x1, ..., xi], [x(i+1), ..., xn])

    where xi is the last element that does not satisfy the predicate*)

 fun take_suffix _ [] = ([], [])

   | take_suffix pred (x :: xs) =

       (case take_suffix pred xs of

         ([], sffx) => if pred x then ([], x :: sffx) else ([x], sffx)

       | (prfx, sffx) => (x :: prfx, sffx));

 fun prefixes1 [] = []

   | prefixes1 (x :: xs) = map (cons x) ([] :: prefixes1 xs);

 fun prefixes xs = [] :: prefixes1 xs;

 fun suffixes1 xs = map rev (prefixes1 (rev xs));

 fun suffixes xs = [] :: suffixes1 xs;

 (** integers **)

 fun inc i = (i := ! i + 1; ! i);

 fun dec i = (i := ! i - 1; ! i);

 (* lists of integers *)

 (*make the list [from, from + 1, ..., to]*)

 fun (i upto j) =

   if i > j then [] else i :: (i + 1 upto j);

 (*make the list [from, from - 1, ..., to]*)

 fun (i downto j) =

   if i < j then [] else i :: (i - 1 downto j);

 (* convert integers to strings *)

 (*expand the number in the given base;

   example: radixpand (2, 8) gives [1, 0, 0, 0]*)

 fun radixpand (base, num) : int list =

   let

     fun radix (n, tail) =

       if n < base then n :: tail

       else radix (n div base, (n mod base) :: tail)

   in radix (num, []) end;

 (*expands a number into a string of characters starting from "zerochar";

   example: radixstring (2, "0", 8) gives "1000"*)

 fun radixstring (base, zerochar, num) =

   let val offset = ord zerochar;

       fun chrof n = chr (offset + n)

   in implode (map chrof (radixpand (base, num))) end;

 val string_of_int = Int.toString;

 fun signed_string_of_int i =

   if i < 0 then "-" ^ string_of_int (~ i) else string_of_int i;

 fun string_of_indexname (a, 0) = a

   | string_of_indexname (a, i) = a ^ "_" ^ string_of_int i;

 (* read integers *)

 fun read_intinf radix cs =

   let

     val zero = ord "0";

     val limit = zero + radix;

     fun scan (num, []) = (num, [])

       | scan (num, c :: cs) =

         if zero <= ord c andalso ord c < limit then

           scan (IntInf.fromInt radix * num + IntInf.fromInt (ord c - zero), cs)

         else (num, c :: cs);

   in scan (IntInf.fromInt 0, cs) end;

 fun read_int cs = apfst IntInf.toInt (read_intinf 10 cs);

 fun oct_char s = chr (IntInf.toInt (#1 (read_intinf 8 (explode s))));

 (** strings **)

 (* functions tuned for strings, avoiding explode *)

 fun nth_string str i =

   (case try String.substring (str, i, 1) of

     SOME s => s

   | NONE => raise Subscript);

 fun fold_string f str x0 =

   let

     val n = size str;

     fun iter (x, i) =

       if i < n then iter (f (String.substring (str, i, 1)) x, i + 1) else x;

   in iter (x0, 0) end;

 fun exists_string pred str =

   let

     val n = size str;

     fun ex i = i < n andalso (pred (String.substring (str, i, 1)) orelse ex (i + 1));

   in ex 0 end;

 fun forall_string pred = not o exists_string (not o pred);

 (*enclose in brackets*)

 fun enclose lpar rpar str = lpar ^ str ^ rpar;

 fun unenclose str = String.substring (str, 1, size str - 2);

 (*simple quoting (does not escape special chars)*)

 val quote = enclose "\"" "\"";

 (*space_implode "..." (explode "hello") = "h...e...l...l...o"*)

 fun space_implode a bs = implode (separate a bs);

 val commas = space_implode ", ";

 val commas_quote = commas o map quote;

 (*concatenate messages, one per line, into a string*)

 val cat_lines = space_implode "\n";

 (*space_explode "." "h.e..l.lo" = ["h", "e", "", "l", "lo"]*)

 fun space_explode _ "" = []

   | space_explode sep s = String.fields (fn c => str c = sep) s;

 val split_lines = space_explode "\n";

 fun prefix_lines "" txt = txt

   | prefix_lines prfx txt = txt |> split_lines |> map (fn s => prfx ^ s) |> cat_lines;

 fun untabify chs =

   let

     val tab_width = 8;

     fun untab pos [] ys = rev ys

       | untab pos ("\n" :: xs) ys = untab 0 xs ("\n" :: ys)

       | untab pos ("\t" :: xs) ys =

           let val d = tab_width - (pos mod tab_width)

           in untab (pos + d) xs (funpow d (cons " ") ys) end

       | untab pos (c :: xs) ys = untab (pos + 1) xs (c :: ys);

   in

     if not (exists (fn c => c = "\t") chs) then chs

     else untab 0 chs []

   end;

 fun prefix prfx s = prfx ^ s;

 fun suffix sffx s = s ^ sffx;

 fun unprefix prfx s =

   if String.isPrefix prfx s then String.substring (s, size prfx, size s - size prfx)

   else raise Fail "unprefix";

 fun unsuffix sffx s =

   if String.isSuffix sffx s then String.substring (s, 0, size s - size sffx)

   else raise Fail "unsuffix";

 fun replicate_string 0 _ = ""

   | replicate_string 1 a = a

   | replicate_string k a =

       if k mod 2 = 0 then replicate_string (k div 2) (a ^ a)

       else replicate_string (k div 2) (a ^ a) ^ a;

 (** lists as sets -- see also Pure/General/ord_list.ML **)

 (*canonical member, insert, remove*)

 fun member eq list x =

   let

     fun memb [] = false

       | memb (y :: ys) = eq (x, y) orelse memb ys;

   in memb list end;

 fun insert eq x xs = if member eq xs x then xs else x :: xs;

 fun remove eq x xs = if member eq xs x then filter_out (fn y => eq (x, y)) xs else xs;

 fun update eq x xs = cons x (remove eq x xs);

 fun subtract eq = fold (remove eq);

 fun merge _ ([], ys) = ys

   | merge eq (xs, ys) = fold_rev (insert eq) ys xs;

 (*old-style infixes*)

 fun x mem xs = member (op =) xs x;

 fun (x: int) mem_int xs = member (op =) xs x;

 fun (x: string) mem_string xs = member (op =) xs x;

 (*union of sets represented as lists: no repetitions*)

 fun xs union [] = xs

   | [] union ys = ys

   | (x :: xs) union ys = xs union (insert (op =) x ys);

 (*union of sets, optimized version for ints*)

 fun (xs:int list) union_int [] = xs

   | [] union_int ys = ys

   | (x :: xs) union_int ys = xs union_int (insert (op =) x ys);

 (*union of sets, optimized version for strings*)

 fun (xs:string list) union_string [] = xs

   | [] union_string ys = ys

   | (x :: xs) union_string ys = xs union_string (insert (op =) x ys);

 (*generalized union*)

 fun gen_union eq (xs, []) = xs

   | gen_union eq ([], ys) = ys

   | gen_union eq (x :: xs, ys) = gen_union eq (xs, insert eq x ys);

 (*intersection*)

 fun [] inter ys = []

   | (x :: xs) inter ys =

       if x mem ys then x :: (xs inter ys) else xs inter ys;

 (*intersection, optimized version for ints*)

 fun ([]:int list) inter_int ys = []

   | (x :: xs) inter_int ys =

       if x mem_int ys then x :: (xs inter_int ys) else xs inter_int ys;

 (*intersection, optimized version for strings *)

 fun ([]:string list) inter_string ys = []

   | (x :: xs) inter_string ys =

       if x mem_string ys then x :: (xs inter_string ys) else xs inter_string ys;

 (*generalized intersection*)

 fun gen_inter eq ([], ys) = []

   | gen_inter eq (x::xs, ys) =

       if member eq ys x then x :: gen_inter eq (xs, ys)

       else gen_inter eq (xs, ys);

 (*subset*)

 fun [] subset ys = true

   | (x :: xs) subset ys = x mem ys andalso xs subset ys;

 (*subset, optimized version for ints*)

 fun ([]: int list) subset_int ys = true

   | (x :: xs) subset_int ys = x mem_int ys andalso xs subset_int ys;

 (*subset, optimized version for strings*)

 fun ([]: string list) subset_string ys = true

   | (x :: xs) subset_string ys = x mem_string ys andalso xs subset_string ys;

 (*set equality*)

 fun eq_set (xs, ys) =

   xs = ys orelse (xs subset ys andalso ys subset xs);

 (*set equality for strings*)

 fun eq_set_string ((xs: string list), ys) =

   xs = ys orelse (xs subset_string ys andalso ys subset_string xs);

 fun gen_subset eq (xs, ys) = forall (member eq ys) xs;

 fun gen_eq_set eq (xs, ys) =

   eq_list eq (xs, ys) orelse

     (gen_subset eq (xs, ys) andalso gen_subset (eq o swap) (ys, xs));

 (*removing an element from a list WITHOUT duplicates*)

 fun (y :: ys) \ x = if x = y then ys else y :: (ys \ x)

   | [] \ x = [];

 fun ys \\ xs = foldl (op \) (ys,xs);

 (*makes a list of the distinct members of the input; preserves order, takes

   first of equal elements*)

 fun distinct eq lst =

   let

     fun dist (rev_seen, []) = rev rev_seen

       | dist (rev_seen, x :: xs) =

           if member eq rev_seen x then dist (rev_seen, xs)

           else dist (x :: rev_seen, xs);

   in dist ([], lst) end;

 (*returns a list containing all repeated elements exactly once; preserves

   order, takes first of equal elements*)

 fun duplicates eq lst =

   let

     fun dups (rev_dups, []) = rev rev_dups

       | dups (rev_dups, x :: xs) =

           if member eq rev_dups x orelse not (member eq xs x) then

             dups (rev_dups, xs)

           else dups (x :: rev_dups, xs);

   in dups ([], lst) end;

 fun has_duplicates eq =

   let

     fun dups [] = false

       | dups (x :: xs) = member eq xs x orelse dups xs;

   in dups end;

 (** lists as multisets **)

 fun remove1 _ _ [] = raise Empty

   | remove1 eq y (x::xs) = if eq (y, x) then xs else x :: remove1 eq y xs;

 fun submultiset _ ([], _)  = true

   | submultiset eq (x :: xs, ys) = member eq ys x andalso submultiset eq (xs, remove1 eq x ys);

 (** orders **)

 fun is_equal EQUAL = true

   | is_equal _ = false;

 fun rev_order LESS = GREATER

   | rev_order EQUAL = EQUAL

   | rev_order GREATER = LESS;

 (*assume rel is a linear strict order*)

 fun make_ord rel (x, y) =

   if rel (x, y) then LESS

   else if rel (y, x) then GREATER

   else EQUAL;

 val int_ord = Int.compare;

 val string_ord = String.compare;

 fun fast_string_ord (s1, s2) =

   (case int_ord (size s1, size s2) of EQUAL => string_ord (s1, s2) | ord => ord);

 fun option_ord ord (SOME x, SOME y) = ord (x, y)

   | option_ord _ (NONE, NONE) = EQUAL

   | option_ord _ (NONE, SOME _) = LESS

   | option_ord _ (SOME _, NONE) = GREATER;

 (*lexicographic product*)

 fun prod_ord a_ord b_ord ((x, y), (x', y')) =

   (case a_ord (x, x') of EQUAL => b_ord (y, y') | ord => ord);

 (*dictionary order -- in general NOT well-founded!*)

 fun dict_ord elem_ord (x :: xs, y :: ys) =

       (case elem_ord (x, y) of EQUAL => dict_ord elem_ord (xs, ys) | ord => ord)

   | dict_ord _ ([], []) = EQUAL

   | dict_ord _ ([], _ :: _) = LESS

   | dict_ord _ (_ :: _, []) = GREATER;

 (*lexicographic product of lists*)

 fun list_ord elem_ord (xs, ys) =

   (case int_ord (length xs, length ys) of EQUAL => dict_ord elem_ord (xs, ys) | ord => ord);

 (* sorting *)

 (*quicksort -- stable, i.e. does not reorder equal elements*)

 fun quicksort unique ord =

   let

     fun qsort [] = []

       | qsort (xs as [_]) = xs

       | qsort (xs as [x, y]) =

           (case ord (x, y) of

             LESS => xs

           | EQUAL => if unique then [x] else xs

           | GREATER => [y, x])

       | qsort xs =

           let val (lts, eqs, gts) = part (nth xs (length xs div 2)) xs

           in qsort lts @ eqs @ qsort gts end

     and part _ [] = ([], [], [])

       | part pivot (x :: xs) = add (ord (x, pivot)) x (part pivot xs)

     and add LESS x (lts, eqs, gts) = (x :: lts, eqs, gts)

       | add EQUAL x (lts, [], gts) = (lts, [x], gts)

       | add EQUAL x (res as (lts, eqs, gts)) = if unique then res else (lts, x :: eqs, gts)

       | add GREATER x (lts, eqs, gts) = (lts, eqs, x :: gts);

   in qsort end;

 fun sort ord = quicksort false ord;

 fun sort_distinct ord = quicksort true ord;

 val sort_strings = sort string_ord;

 fun sort_wrt sel xs = sort (string_ord o pairself sel) xs;

 (** random numbers **)

 exception RANDOM;

 fun rmod x y = x - y * Real.realFloor (x / y);

 local

   val a = 16807.0;

   val m = 2147483647.0;

   val random_seed = ref 1.0;

 in

 fun random () = CRITICAL (fn () =>

   let val r = rmod (a * ! random_seed) m

   in (random_seed := r; r) end);

 end;

 fun random_range l h =

   if h < l orelse l < 0 then raise RANDOM

   else l + Real.floor (rmod (random ()) (real (h - l + 1)));

 fun one_of xs = nth xs (random_range 0 (length xs - 1));

 fun frequency xs =

   let

     val sum = foldl op + (0, map fst xs);

     fun pick n ((k: int, x) :: xs) =

       if n <= k then x else pick (n - k) xs

   in pick (random_range 1 sum) xs end;

 (** misc **)

 fun divide_and_conquer decomp x =

   let val (ys, recomb) = decomp x

   in recomb (map (divide_and_conquer decomp) ys) end;

 (*Partition a list into buckets  [ bi, b(i+1), ..., bj ]

    putting x in bk if p(k)(x) holds.  Preserve order of elements if possible.*)

 fun partition_list p i j =

   let fun part k xs =

             if k>j then

               (case xs of [] => []

                          | _ => raise Fail "partition_list")

             else

             let val (ns, rest) = List.partition (p k) xs;

             in  ns :: part(k+1)rest  end

   in  part i end;

 fun partition_eq (eq:'a * 'a -> bool) =

   let

     fun part [] = []

       | part (x :: ys) =

           let val (xs, xs') = List.partition (fn y => eq (x, y)) ys

           in (x::xs)::(part xs') end

   in part end;

 (* generating identifiers *)

 (** Freshly generated identifiers; supplied prefix MUST start with a letter **)

 local

 (*Maps 0-61 to A-Z, a-z, 0-9; exclude _ or ' to avoid clash with internal/unusual indentifiers*)

 fun gensym_char i =

   if i<26 then chr (ord "A" + i)

   else if i<52 then chr (ord "a" + i - 26)

   else chr (ord "0" + i - 52);

 val char_vec = Vector.tabulate (62, gensym_char);

 fun newid n = implode (map (fn i => Vector.sub (char_vec, i)) (radixpand (62, n)));

 val gensym_seed = ref 0;

 in

   fun gensym pre = pre ^ newid (NAMED_CRITICAL "gensym" (fn () => inc gensym_seed));

 end;

 (* stamps and serial numbers *)

 type stamp = unit ref;

 val stamp: unit -> stamp = ref;

 type serial = int;

 local val count = ref 0

 in fun serial () = NAMED_CRITICAL "serial" (fn () => inc count) end;

 val serial_string = string_of_int o serial;

 (* generic objects *)

 (*note that the builtin exception datatype may be extended by new

   constructors at any time*)

 structure Object = struct type T = exn end;

 end;

 structure BasicLibrary: BASIC_LIBRARY = Library;

 open BasicLibrary;