(*  Title:      Pure/library.ML

    ID:         $Id$

    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

    Author:     Markus Wenzel, TU Munich

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

rational numbers, strings, lists as sets, association lists, generic

tables, balanced trees, orders, current directory, misc.

*)

infix |> |>> |>>> ~~ \ \\ ins ins_string ins_int orf andf prefix upto downto

  mem mem_int mem_string union union_int union_string inter inter_int

  inter_string subset subset_int subset_string;

infix 3 oo ooo oooo;

signature BASIC_LIBRARY =

sig

  (*functions*)

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

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

  val I: 'a -> 'a

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

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

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

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

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

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

  val funpow: int -> ('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

  (*stamps*)

  type stamp

  val stamp: unit -> stamp

  (*options*)

  val is_none: 'a option -> bool

  (*old options -- invalidated*)

  datatype invalid = None of invalid

  exception OPTION of invalid

  exception ERROR

  val try: ('a -> 'b) -> 'a -> 'b option

  val can: ('a -> 'b) -> 'a -> bool

  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

  (*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 * 'b) * (''a * 'c) -> bool

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

  val conditional: bool -> (unit -> unit) -> unit

  (*lists*)

  exception UnequalLengths

  val cons: 'a -> 'a list -> 'a list

  val single: 'a -> 'a list

  val append: 'a list -> 'a list -> 'a list

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

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

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

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

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

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

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

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

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

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

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

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

  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_first: ('a -> 'b option) -> 'a list -> 'b option

  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_out: ('a -> bool) -> 'a list -> 'a list

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

  val exists2: ('a * 'b -> bool) -> 'a list * 'b list -> bool

  val forall2: ('a * 'b -> bool) -> 'a list * 'b list -> bool

  val seq2: ('a * 'b -> unit) -> 'a list * 'b list -> unit

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

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

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

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

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

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

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

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

  (*integers*)

  val gcd: int * int -> int

  val lcm: int * int -> 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 string_of_indexname: string * int -> string

    (* CB: note alternative Syntax.string_of_vname has nicer syntax *)

  val read_radixint: int * string list -> int * string list

  val read_int: string list -> int * string list

  val oct_char: string -> string

  (*rational numbers*)

  type rat

  exception RAT of string

  val rep_rat: rat -> int * int

  val ratadd: rat * rat -> rat

  val ratmul: rat * rat -> rat

  val ratinv: rat -> rat

  val int_ratdiv: int * int -> rat

  val ratneg: rat -> rat

  val rat_of_int: int -> rat

  (*strings*)

  val nth_elem_string: int * string -> string

  val foldl_string: ('a * string -> 'a) -> 'a * string -> 'a

  val exists_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 suffix: string -> string -> string

  val unsuffix: string -> string -> string

  val unprefix: string -> string -> string

  val replicate_string: int -> string -> string

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

  (*lists as sets*)

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

  val mem_int: int * int list -> bool

  val mem_string: string * string list -> bool

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

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

  val ins_int: int * int list -> int list

  val ins_string: string * string list -> string list

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

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

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

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

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

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

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

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

  val distinct: ''a list -> ''a list

  val findrep: ''a list -> ''a list

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

  val duplicates: ''a list -> ''a list

  (*association lists*)

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

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

  val assoc_string: (string * 'a) list * string -> 'a option

  val assoc_string_int: ((string * int) * 'a) list * (string * int) -> 'a option

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

  val assoc2: (''a * (''b * 'c) list) list * (''a * ''b) -> 'c option

  val gen_assoc: ('a * 'b -> bool) -> ('b * 'c) list * 'a -> 'c option

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

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

  (*lists as tables*)

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

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

  val merge_lists: ''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

  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 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 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_strings: string list -> string list

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

  val unique_strings: string list -> string 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 make_keylist: ('a -> 'b) -> 'a list -> ('a * 'b) list

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

  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

end;

signature LIBRARY =

sig

  include BASIC_LIBRARY

  val the: 'a option -> 'a

  val if_none: 'a option -> 'a -> 'a

  val is_some: 'a option -> bool

  val apsome: ('a -> 'b) -> 'a option -> 'b option

  val rev_append: 'a list -> 'a list -> 'a list

  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 nth_elem: int * 'a list -> 'a

  val last_elem: 'a list -> 'a

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

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

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

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

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

end;

structure Library: LIBRARY =

struct

(** functions **)

(*handy combinators*)

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

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

fun I x = x;

fun K x y = x;

(*reverse apply*)

fun (x |> f) = f x;

fun ((x, y) |>> f) = (f x, y);

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

(*application of (infix) operator to its left or right argument*)

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

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

(*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;

(*concatenation: 2 and 3 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);

(** stamps **)

type stamp = unit ref;

val stamp: unit -> stamp = ref;

(** options **)

val the = valOf;

fun if_none opt a = getOpt (opt, a);

val is_some = isSome;

fun is_none opt = not (isSome opt);

val apsome = Option.map;

(*invalidate former constructors to prevent accidental use as match-all patterns!*)

datatype invalid = None of invalid;

exception OPTION of invalid;

(* exception handling *)

exception ERROR;

fun try f x = SOME (f x)

  handle Interrupt => raise Interrupt | ERROR => raise ERROR | _ => NONE;

fun can f x = is_some (try f x);

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;

(** 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 ((x1, _), (x2, _)) = x1 = x2;

fun eq_snd ((_, y1), (_, y2)) = 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 **)

(* equality *)

fun equal x y = x = y;

fun not_equal x y = x <> y;

(* operators for combining predicates *)

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

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

(* predicates on lists *)

(*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, handling exceptions*)

fun setmp flag value f x =

  let

    val orig_value = ! flag;

    fun return y = (flag := orig_value; y);

  in

    flag := value;

    return (f x handle exn => (return (); raise exn))

  end;

(* conditional execution *)

fun conditional b f = if b then f () else ();

(** lists **)

exception UnequalLengths;

fun cons x xs = x :: xs;

fun single x = [x];

fun append xs ys = xs @ ys;

fun rev_append xs ys = List.revAppend (xs, ys);

fun apply [] x = x

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

(* fold *)

fun fold _ [] y = y

  | fold f (x :: xs) y = fold f xs (f x y);

fun fold_rev _ [] y = y

  | fold_rev f (x :: xs) y = f x (fold_rev f xs y);

fun foldl_map _ (x, []) = (x, [])

  | foldl_map f (x, y :: ys) =

      let

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

        val (x'', ys') = foldl_map f (x', ys);

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

(*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 l =

  let fun itr [x] = x

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

  in  itr l  end;

fun foldln f xs e = fst (foldl (fn ((e,i), x) => (f (x,i) e, i+1)) ((e,1),xs));

(* basic list functions *)

(*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 splitAt(n,[]) = ([],[])

  | splitAt(n,xs as x::ys) =

      if n>0 then let val (ps,qs) = splitAt(n-1,ys) in (x::ps,qs) end

      else ([],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 EXCEPTION if list too short*)

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

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

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

  | map_nth_elem _ _ [] = raise Subscript;

(*last element of a list*)

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);

(*update nth element*)

fun nth_update x n_xs =

    (case splitAt n_xs of

      (_,[]) => raise Subscript

    | (prfx, _ :: sffx') => prfx @ (x :: sffx'))

(*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);

(*flatten a list of lists to a list*)

val flat = List.concat;

fun unflat (xs :: xss) ys =

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

      in ps :: unflat xss qs end

  | unflat [] [] = []

  | unflat _ _ = raise UnequalLengths;

(*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 mapfilter = 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 exists2 _ ([], []) = false

  | exists2 pred (x :: xs, y :: ys) = pred (x, y) orelse exists2 pred (xs, ys)

  | exists2 _ _ = raise UnequalLengths;

fun forall2 _ ([], []) = true

  | forall2 pred (x :: xs, y :: ys) = pred (x, y) andalso forall2 pred (xs, ys)

  | forall2 _ _ = raise UnequalLengths;

fun seq2 _ ([], []) = ()

  | seq2 f (x :: xs, y :: ys) = (f (x, y); seq2 f (xs, ys))

  | seq2 _ _ = 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;

fun equal_lists eq (xs, ys) = length xs = length ys andalso forall2 eq (xs, ys);

(* prefixes, suffixes *)

fun [] prefix _ = true

  | (x :: xs) prefix (y :: ys) = x = y andalso (xs prefix ys)

  | _ prefix _ = 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;

(* [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 suffixes1 xs = map rev (prefixes1 (rev xs));

(** integers **)

fun gcd(x,y) =

  let fun gxd x y =

    if y = 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 (from upto to) =

  if from > to then [] else from :: ((from + 1) upto to);

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

fun (from downto to) =

  if from < to then [] else from :: ((from - 1) downto to);

(*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 string_of_indexname (a,0) = a

  | string_of_indexname (a,i) = a ^ "_" ^ Int.toString i;

(* read integers *)

fun read_radixint (radix: int, cs) : int * string list =

  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(radix*num + ord c - zero, cs)

              else (num, c::cs)

  in  scan(0,cs)  end;

fun read_int cs = read_radixint (10, cs);

fun oct_char s = chr (#1 (read_radixint (8, explode s)));

(** strings **)

(*functions tuned for strings, avoiding explode*)

fun nth_elem_string (i, str) =

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

    SOME s => s

  | NONE => raise Subscript);

fun foldl_string f (x0, str) =

  let

    val n = size str;

    fun fold (x, i) =

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

  in fold (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;

(*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 str =

      let

        fun expl chs =

          (case take_prefix (not_equal sep) chs of

            (cs, []) => [implode cs]

          | (cs, _ :: cs') => implode cs :: expl cs');

      in expl (explode str) end;

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;

(*untabify*)

fun untabify chs =

  let

    val tab_width = 8;

    fun untab (_, "\n") = (0, ["\n"])

      | untab (pos, "\t") =

          let val d = tab_width - (pos mod tab_width) in (pos + d, replicate d " ") end

      | untab (pos, c) = (pos + 1, [c]);

  in

    if not (exists (equal "\t") chs) then chs

    else flat (#2 (foldl_map untab (0, chs)))

  end;

(*append suffix*)

fun suffix sfx s = s ^ sfx;

(*remove suffix*)

fun unsuffix sfx s =

  let

    val cs = explode s;

    val prfx_len = size s - size sfx;

  in

    if prfx_len >= 0 andalso implode (drop (prfx_len, cs)) = sfx then

      implode (take (prfx_len, cs))

    else raise Fail "unsuffix"

  end;

(*remove prefix*)

fun unprefix prfx s =

  let val (prfx', sfx) = splitAt (size prfx, explode s)

  in

    if implode prfx' = prfx then implode sfx

    else raise Fail "unprefix"

  end;

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 **)

(*membership in a list*)

fun x mem [] = false

  | x mem (y :: ys) = x = y orelse x mem ys;

(*membership in a list, optimized version for ints*)

fun (x:int) mem_int [] = false

  | x mem_int (y :: ys) = x = y orelse x mem_int ys;

(*membership in a list, optimized version for strings*)

fun (x:string) mem_string [] = false

  | x mem_string (y :: ys) = x = y orelse x mem_string ys;

(*generalized membership test*)

fun gen_mem eq (x, []) = false

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

(*insert and remove*)

fun insert eq x xs = if gen_mem eq (x, xs) then xs else x :: xs;

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

(*insertion into list if not already there*)

fun (x ins xs) = if x mem xs then xs else x :: xs;

(*insertion into list, optimized version for ints*)

fun (x ins_int xs) = if x mem_int xs then xs else x :: xs;

(*insertion into list, optimized version for strings*)

fun (x ins_string xs) = if x mem_string xs then xs else x :: xs;

(*generalized insertion*)

fun gen_ins eq (x, xs) = insert eq x xs;

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

fun xs union [] = xs

  | [] union ys = ys

  | (x :: xs) union ys = xs union (x ins 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 (x ins_int 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 (x ins_string ys);

(*generalized union*)

fun gen_union eq (xs, []) = xs

  | gen_union eq ([], ys) = ys

  | gen_union eq (x :: xs, ys) = gen_union eq (xs, gen_ins 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 gen_mem eq (x,ys) 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 (fn x => gen_mem eq (x, 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);

(*removing an element from a list -- possibly WITH duplicates*)

fun gen_rem eq (xs, y) = filter_out (fn x => eq (x, y)) xs;

fun gen_rems eq (xs, ys) = filter_out (fn x => gen_mem eq (x, ys)) xs;

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

  first of equal elements*)

fun gen_distinct eq lst =

  let

    val memb = gen_mem eq;

    fun dist (rev_seen, []) = rev rev_seen

      | dist (rev_seen, x :: xs) =

          if memb (x, rev_seen) then dist (rev_seen, xs)

          else dist (x :: rev_seen, xs);

  in

    dist ([], lst)

  end;

fun distinct l = gen_distinct (op =) l;

(*returns the tail beginning with the first repeated element, or []*)

fun findrep [] = []

  | findrep (x :: xs) = if x mem xs then x :: xs else findrep xs;

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

  order, takes first of equal elements*)

fun gen_duplicates eq lst =

  let

    val memb = gen_mem eq;

    fun dups (rev_dups, []) = rev rev_dups

      | dups (rev_dups, x :: xs) =

          if memb (x, rev_dups) orelse not (memb (x, xs)) then

            dups (rev_dups, xs)

          else dups (x :: rev_dups, xs);

  in

    dups ([], lst)

  end;

fun duplicates l = gen_duplicates (op =) l;

(** association lists **)

(*association list lookup*)

fun assoc ([], key) = NONE

  | assoc ((keyi, xi) :: pairs, key) =

      if key = keyi then SOME xi else assoc (pairs, key);

(*association list lookup, optimized version for ints*)

fun assoc_int ([], (key:int)) = NONE

  | assoc_int ((keyi, xi) :: pairs, key) =

      if key = keyi then SOME xi else assoc_int (pairs, key);