src/Pure/library.ML

removed redundant lemmas

(*  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, association lists, generic
tables, balanced trees, orders, current directory, misc.
*)

infix 1 |> |-> ||> ||>> |>> |>>> #> #->;
infix 2 ?;
infix 3 o oo ooo oooo;

infix 4 ~~ upto downto;
infix orf andf \ \\ ins ins_string ins_int 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 * ('a -> 'b) -> 'b
  val |-> : ('c * 'a) * ('c -> 'a -> 'b) -> 'b
  val ||> : ('c * 'a) * ('a -> 'b) -> 'c * 'b
  val ||>> : ('c * 'a) * ('a -> 'd * 'b) -> ('c * 'd) * 'b
  val |>> : ('a * 'c) * ('a -> 'b) -> 'b * 'c
  val |>>> : ('a * 'c) * ('a -> 'b * 'd) -> 'b * ('c * 'd)
  val #> : ('a -> 'b) * ('b -> 'c) -> 'a -> 'c
  val #-> : ('a -> 'c * 'b) * ('c -> 'b -> 'd) -> 'a -> 'd
  val ? : ('a -> bool) * ('a -> 'a) -> 'a -> 'a
  val ` : ('b -> 'a) -> 'b -> 'a * 'b
  val tap: ('b -> 'a) -> 'b -> 'b
  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

  (*old options -- invalidated*)
  datatype invalid = None of invalid
  exception OPTION of invalid

  (*options*)
  val the: 'a option -> 'a
  val these: 'a list option -> 'a list
  val the_default: 'a -> 'a option -> 'a
  val the_list: 'a option -> 'a list
  val if_none: 'a option -> 'a -> 'a
  val is_some: 'a option -> bool
  val is_none: 'a option -> bool
  val perhaps: ('a -> 'a option) -> 'a -> 'a

  (*exceptions*)
  val try: ('a -> 'b) -> 'a -> 'b option
  val can: ('a -> 'b) -> 'a -> bool
  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 ERROR of string
  val error: string -> 'a
  val cat_error: string -> string -> 'a
  val sys_error: string -> 'a
  val assert: bool -> string -> unit
  val deny: bool -> string -> unit
  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 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 fold_map: ('a -> 'b -> 'c * 'b) -> 'a list -> 'b -> 'c list * 'b
  val foldr1: ('a * 'a -> 'a) -> 'a list -> 'a
  val foldl_map: ('a * 'b -> 'a * 'c) -> 'a * 'b list -> 'a * 'c 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 splitAt: int * 'a list -> 'a list * 'a list
  val dropwhile: ('a -> bool) -> 'a list -> 'a list
  val nth: 'a list -> int -> 'a
  val nth_update: int * 'a -> 'a list -> 'a list
  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_first: ('a -> 'b option) -> 'a list -> 'b option
  val map2: ('a -> 'b -> 'c) -> 'a list -> 'b list -> 'c list
  val fold2: ('a -> 'b -> 'c -> 'c) -> 'a list -> 'b list -> 'c -> 'c
  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 equal_lists: ('a * 'b -> bool) -> 'a list * 'b list -> bool
  val is_prefix: ('a * 'a -> bool) -> '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: 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 string_of_indexname: string * int -> string
  val read_radixint: int * string list -> 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 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 ins: ''a * ''a list -> ''a list
  val ins_int: int * int list -> int list
  val ins_string: string * string list -> string 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 duplicates: ('a * 'a -> bool) -> 'a list -> 'a list
  val has_duplicates: ('a * 'a -> bool) -> 'a 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 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 is_equal: order -> bool
  val rev_order: order -> order
  val make_ord: ('a * 'a -> bool) -> 'a * 'a -> order
  val eq_ord: ('a -> order) -> 'a -> bool
  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 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
  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 flat: 'a list list -> 'a list
  val seq: ('a -> unit) -> 'a list -> unit
  val partition: ('a -> bool) -> 'a list -> 'a list * 'a list
  val mapfilter: ('a -> 'b option) -> 'a list -> 'b list
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;

(*reverse application and structured results*)
fun x |> f = f x;
fun (x, y) |-> f = f x y;
fun (x, y) |>> f = (f x, y);
fun (x, y) ||> f = (x, f y);
fun (x, y) |>>> f = let val (x', z) = f x in (x', (y, z)) end;
fun (x, y) ||>> f = let val (z, y') = f y in ((x, z), y') end;

(*reverse composition*)
fun f #> g = g o f;
fun f #-> g = uncurry g o f;

(*conditional application*)
fun b ? f = fn x => if b x then f x else x;

(*view results*)
fun `f = fn x => (f x, x);
fun tap f = fn x => (f x; 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;


(** options **)

(*invalidate former constructors to prevent accidental use as match-all patterns!*)
datatype invalid = None of invalid;
exception OPTION of invalid;

val the = Option.valOf;

fun these (SOME x) = x
  | these _ = [];

fun the_default x (SOME y) = y
  | the_default x _ = x;

fun the_list (SOME x) = [x]
  | the_list _ = []

(*strict!*)
fun if_none NONE y = y
  | if_none (SOME x) _ = x;

fun is_some (SOME _) = true
  | is_some NONE = false;

fun is_none (SOME _) = false
  | is_none NONE = true;

fun perhaps f x = the_default x (f x);


(* 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;


fun try f x = SOME (f x)
  handle Interrupt => raise Interrupt | _ => 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;


(* errors *)

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 sys_error msg = error ("## SYSTEM ERROR ##\n" ^ msg);

fun assert p msg = if p then () else error msg;
fun deny p msg = if p then error msg else ();

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 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 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;


(* 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 apply [] x = x
  | apply (f :: fs) x = apply fs (f x);


(* fold *)

fun fold f =
  let
    fun fold_aux [] y = y
      | fold_aux (x :: xs) y = fold_aux xs (f x y);
  in fold_aux end;

fun fold_rev f =
  let
    fun fold_aux [] y = y
      | fold_aux (x :: xs) y = f x (fold_aux xs y);
  in fold_aux end;

fun fold_map f =
  let
    fun fold_aux [] y = ([], y)
      | fold_aux (x :: xs) y =
          let
            val (x', y') = f x y;
            val (xs', y'') = fold_aux xs y';
          in (x' :: xs', y'') end;
  in fold_aux 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 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;

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 *)

(*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 Subscript if list too short*)
fun nth xs i = List.nth (xs, i);

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

(*update nth element*)
fun nth_update (n, x) xs =
    (case splitAt (n, xs) of
      (_, []) => raise Subscript
    | (prfx, _ :: sffx') => prfx @ (x :: sffx'))

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;

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);

(*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;

fun burrow f xss =
  unflat xss ((f o 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 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 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;


(*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) =
  let
    fun eq' [] [] = true
      | eq' (x :: xs) (y :: ys) = eq (x, y) andalso eq' xs ys
  in length xs = length ys andalso eq' xs ys end;


(* 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;

(* [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 : IntInf.int =
    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_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 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;

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;

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 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;

fun x ins xs = insert (op =) x xs;
fun (x:int) ins_int xs = insert (op =) x xs;
fun (x:string) ins_string xs = insert (op =) 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, 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;


(*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 (member eq 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
    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;

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 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;



(** association lists **)

(* lists as tables -- legacy operations *)

fun gen_merge_lists _ xs [] = xs
  | gen_merge_lists _ [] ys = ys
  | gen_merge_lists eq xs ys = xs @ gen_rems eq (ys, xs);

fun gen_merge_lists' _ xs [] = xs
  | gen_merge_lists' _ [] ys = ys
  | gen_merge_lists' eq xs ys = gen_rems eq (ys, xs) @ xs;

fun merge_lists xs ys = gen_merge_lists (op =) xs ys;
fun merge_lists' xs ys = gen_merge_lists' (op =) xs ys;
fun merge_alists al = gen_merge_lists (eq_fst (op =)) al;
fun merge_alists' al = gen_merge_lists' (eq_fst (op =)) al;



(** 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) = splitAt(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
  | 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;

fun eq_ord ord xy =
  case ord xy
   of EQUAL => true
    | _ => false;

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 () =
  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;


(** current directory **)

val cd = OS.FileSys.chDir;
val pwd = OS.FileSys.getDir;


(** misc **)

(*Partition list into elements that satisfy predicate and those that don't.
  Preserves order of elements in both lists.*)
val partition = List.partition;

fun partition_eq (eq:'a * 'a -> bool) =
  let
    fun part [] = []
      | part (x :: ys) =
          let val (xs, xs') = partition (fn y => eq (x, y)) ys
          in (x::xs)::(part xs') end
  in part 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) = partition (p k) xs;
            in  ns :: part(k+1)rest  end
  in  part i end;


(* generating identifiers *)

(** Freshly generated identifiers; supplied prefix MUST start with a letter **)
local
(*Maps 0-63 to A-Z, a-z, 0-9 or _ or ' for generating random identifiers*)
fun char i =      if i<26 then chr (ord "A" + i)
             else if i<52 then chr (ord "a" + i - 26)
             else if i<62 then chr (ord"0" + i - 52)
             else if i=62 then "_"
             else  (*i=63*)    "'";

val charVec = Vector.tabulate (64, char);

fun newid n =
  let
  in  implode (map (fn i => Vector.sub(charVec,i)) (radixpand (64,n)))  end;

val seedr = ref 0;

in

fun gensym pre = pre ^ (#1(newid (!seedr), inc seedr));

end;


(* lexical scanning *)

(*scan a list of characters into "words" composed of "letters" (recognized by
  is_let) and separated by any number of non-"letters"*)
fun scanwords is_let cs =
  let fun scan1 [] = []
        | scan1 cs =
            let val (lets, rest) = take_prefix is_let cs
            in implode lets :: scanwords is_let rest end;
  in scan1 (#2 (take_prefix (not o is_let) cs)) 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 () = inc count end;


(* 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;