(*  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, balanced trees, 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 I: 'a -> 'a

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

  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

  (*exceptions*)

  exception EXCEPTION of exn * string

  val do_transform_failure: bool ref

  val transform_failure: (exn -> exn) -> ('a -> 'b) -> 'a -> 'b

  datatype 'a result = Result of 'a | Exn of exn

  val capture: ('a -> 'b) -> 'a -> 'b result

  val release: 'a result -> 'a

  val get_result: 'a result -> 'a option

  val get_exn: 'a result -> exn option

  (*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: '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 gcd: IntInf.int * IntInf.int -> IntInf.int

  val lcm: IntInf.int * IntInf.int -> IntInf.int

  val inc: int ref -> int

  val dec: int ref -> int

  val upto: int * int -> int list

  val downto: int * int -> int list

  val downto0: int list * int -> bool

  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 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 gen_submultiset: ('a * 'b -> bool) -> 'a list * 'b list -> bool

  (*lists as tables -- see also Pure/General/alist.ML*)

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

  val merge_lists: ''a list -> ''a list -> ''a list

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

  (*balanced trees*)

  exception Balance

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

  val access_bal: ('a -> 'a) * ('a -> 'a) * 'a -> int -> int -> 'a

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

  (*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

  (*current directory*)

  val cd: string -> unit

  val pwd: unit -> string

  (*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

  val scanwords: (string -> bool) -> string list -> string list

  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

  val seq: ('a -> unit) -> 'a list -> unit

end;

structure Library: LIBRARY =

struct

(* functions *)

fun I x = x;

fun K x = fn _ => 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 n f x =

  let fun rep (0, x) = x

        | rep (n, x) = rep (n - 1, f x)

  in rep (n, x) end;

(* exceptions *)

val do_transform_failure = ref true;

fun transform_failure exn f x =

  if ! do_transform_failure then

    f x handle Interrupt => raise Interrupt | e => raise exn e

  else f x;

exception EXCEPTION of exn * string;

datatype 'a result =

  Result of 'a |

  Exn of exn;

fun capture f x = Result (f x) handle e => Exn e;

fun release (Result y) = y

  | release (Exn e) = raise e;

fun get_result (Result x) = SOME x

  | get_result _ = NONE;

fun get_exn (Exn exn) = SOME exn

  | get_exn _ = NONE;

(* 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);

(*apply function to components*)

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 flag value f x =

  let

    val orig_value = ! flag;

    val _ = flag := value;

    val result = capture f x;

    val _ = flag := orig_value;

  in release result end;

(** 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 n xs = unfold_rev n dest xs;

(*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*)

fun find_first _ [] = NONE

  | find_first pred (x :: xs) =

      if pred x then SOME x else find_first pred xs;

(*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);

(*like Lisp's MAPC -- seq proc [x1, ..., xn] evaluates

  (proc x1; ...; proc xn) for side effects*)

val seq = List.app;

(*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 =

  let

    fun fold_aux [] [] z = z

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

      | fold_aux _ _ _ = raise UnequalLengths;

  in fold_aux end;

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 gcd (x, y) =

  let fun gxd x y : IntInf.int =

    if y = IntInf.fromInt 0 then x else gxd y (x mod y)

  in if x < y then gxd y x else gxd x y end;

fun lcm (x, y) = (x * y) div gcd (x, y);

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);

(*predicate: downto0 (is, n) <=> is = [n, n - 1, ..., 0]*)

fun downto0 (i :: is, n) = i = n andalso downto0 (is, n - 1)

  | downto0 ([], n) = n = ~1;

(* 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 ^ "_" ^ Int.toString 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 (replicate d " " @ 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 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 gen_submultiset _  ([],    _)  = true

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

      member eq ys x  andalso  gen_submultiset eq (xs, remove1 eq x ys);

(** association lists -- legacy operations **)

fun gen_merge_lists _ xs [] = xs

  | gen_merge_lists _ [] ys = ys

  | gen_merge_lists eq xs ys = xs @ filter_out (member eq xs) ys;

fun merge_lists xs ys = gen_merge_lists (op =) xs ys;

fun merge_alists xs = gen_merge_lists (eq_fst (op =)) xs;

(** balanced trees **)

exception Balance;      (*indicates non-positive argument to balancing fun*)

(*balanced folding; avoids deep nesting*)

fun fold_bal f [x] = x

  | fold_bal f [] = raise Balance

  | fold_bal f xs =

      let val (ps, qs) = chop (length xs div 2) xs

      in  f (fold_bal f ps, fold_bal f qs)  end;

(*construct something of the form f(...g(...(x)...)) for balanced access*)

fun access_bal (f, g, x) n i =

  let fun acc n i =     (*1<=i<=n*)

          if n=1 then x else

          let val n2 = n div 2

          in  if i<=n2 then f (acc n2 i)

                       else g (acc (n-n2) (i-n2))

          end

  in  if 1<=i andalso i<=n then acc n i else raise Balance  end;

(*construct ALL such accesses; could try harder to share recursive calls!*)

fun accesses_bal (f, g, x) n =

  let fun acc n =

          if n=1 then [x] else

          let val n2 = n div 2

              val acc2 = acc n2

          in  if n-n2=n2 then map f acc2 @ map g acc2

                         else map f acc2 @ map g (acc (n-n2)) end

  in  if 1<=n then acc n else raise Balance  end;

(** orders **)

fun is_equal EQUAL = true

  | is_equal _ = false;

fun rev_order LESS = GREATER