src/Pure/library.ML
author haftmann
Wed Dec 05 14:16:05 2007 +0100 (2007-12-05)
changeset 25538 58e8ba3b792b
parent 25224 6d4d26e878f4
child 25549 7040555f20c7
permissions -rw-r--r--
map_product and fold_product
     1 (*  Title:      Pure/library.ML
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Author:     Markus Wenzel, TU Muenchen
     5 
     6 Basic library: functions, options, pairs, booleans, lists, integers,
     7 strings, lists as sets, orders, current directory, misc.
     8 
     9 See also General/basics.ML for the most fundamental concepts.
    10 *)
    11 
    12 infix 1 |>>>
    13 infix 2 ?
    14 infix 3 o oo ooo oooo
    15 infix 4 ~~ upto downto
    16 infix orf andf \ \\ mem mem_int mem_string union union_int
    17   union_string inter inter_int inter_string subset subset_int subset_string
    18 
    19 signature BASIC_LIBRARY =
    20 sig
    21   (*functions*)
    22   val undefined: 'a -> 'b
    23   val I: 'a -> 'a
    24   val K: 'a -> 'b -> 'a
    25   val curry: ('a * 'b -> 'c) -> 'a -> 'b -> 'c
    26   val uncurry: ('a -> 'b -> 'c) -> 'a * 'b -> 'c
    27   val |>>> : ('a * 'c) * ('a -> 'b * 'd) -> 'b * ('c * 'd)
    28   val ? : bool * ('a -> 'a) -> 'a -> 'a
    29   val oo: ('a -> 'b) * ('c -> 'd -> 'a) -> 'c -> 'd -> 'b
    30   val ooo: ('a -> 'b) * ('c -> 'd -> 'e -> 'a) -> 'c -> 'd -> 'e -> 'b
    31   val oooo: ('a -> 'b) * ('c -> 'd -> 'e -> 'f -> 'a) -> 'c -> 'd -> 'e -> 'f -> 'b
    32   val funpow: int -> ('a -> 'a) -> 'a -> 'a
    33 
    34   (*errors*)
    35   exception SYS_ERROR of string
    36   val sys_error: string -> 'a
    37   exception ERROR of string
    38   val error: string -> 'a
    39   val cat_error: string -> string -> 'a
    40   val assert_all: ('a -> bool) -> 'a list -> ('a -> string) -> unit
    41 
    42   (*pairs*)
    43   val pair: 'a -> 'b -> 'a * 'b
    44   val rpair: 'a -> 'b -> 'b * 'a
    45   val fst: 'a * 'b -> 'a
    46   val snd: 'a * 'b -> 'b
    47   val eq_fst: ('a * 'c -> bool) -> ('a * 'b) * ('c * 'd) -> bool
    48   val eq_snd: ('b * 'd -> bool) -> ('a * 'b) * ('c * 'd) -> bool
    49   val eq_pair: ('a * 'c -> bool) -> ('b * 'd -> bool) -> ('a * 'b) * ('c * 'd) -> bool
    50   val swap: 'a * 'b -> 'b * 'a
    51   val apfst: ('a -> 'b) -> 'a * 'c -> 'b * 'c
    52   val apsnd: ('a -> 'b) -> 'c * 'a -> 'c * 'b
    53   val pairself: ('a -> 'b) -> 'a * 'a -> 'b * 'b
    54 
    55   (*booleans*)
    56   val equal: ''a -> ''a -> bool
    57   val not_equal: ''a -> ''a -> bool
    58   val orf: ('a -> bool) * ('a -> bool) -> 'a -> bool
    59   val andf: ('a -> bool) * ('a -> bool) -> 'a -> bool
    60   val exists: ('a -> bool) -> 'a list -> bool
    61   val forall: ('a -> bool) -> 'a list -> bool
    62   val set: bool ref -> bool
    63   val reset: bool ref -> bool
    64   val toggle: bool ref -> bool
    65   val change: 'a ref -> ('a -> 'a) -> unit
    66   val setmp_noncritical: 'a ref -> 'a -> ('b -> 'c) -> 'b -> 'c
    67   val setmp: 'a ref -> 'a -> ('b -> 'c) -> 'b -> 'c
    68 
    69   (*lists*)
    70   exception UnequalLengths
    71   val single: 'a -> 'a list
    72   val the_single: 'a list -> 'a
    73   val singleton: ('a list -> 'b list) -> 'a -> 'b
    74   val yield_singleton: ('a list -> 'c -> 'b list * 'c) -> 'a -> 'c -> 'b * 'c
    75   val apply: ('a -> 'a) list -> 'a -> 'a
    76   val perhaps_apply: ('a -> 'a option) list -> 'a -> 'a option
    77   val perhaps_loop: ('a -> 'a option) -> 'a -> 'a option
    78   val foldr1: ('a * 'a -> 'a) -> 'a list -> 'a
    79   val foldl_map: ('a * 'b -> 'a * 'c) -> 'a * 'b list -> 'a * 'c list
    80   val flat: 'a list list -> 'a list
    81   val unflat: 'a list list -> 'b list -> 'b list list
    82   val burrow: ('a list -> 'b list) -> 'a list list -> 'b list list
    83   val burrow_options: ('a list -> 'b list) -> 'a option list -> 'b option list
    84   val fold_burrow: ('a list -> 'c -> 'b list * 'd) -> 'a list list -> 'c -> 'b list list * 'd
    85   val maps: ('a -> 'b list) -> 'a list -> 'b list
    86   val chop: int -> 'a list -> 'a list * 'a list
    87   val dropwhile: ('a -> bool) -> 'a list -> 'a list
    88   val nth: 'a list -> int -> 'a
    89   val nth_map: int -> ('a -> 'a) -> 'a list -> 'a list
    90   val nth_drop: int -> 'a list -> 'a list
    91   val nth_list: 'a list list -> int -> 'a list
    92   val map_index: (int * 'a -> 'b) -> 'a list -> 'b list
    93   val fold_index: (int * 'a -> 'b -> 'b) -> 'a list -> 'b -> 'b
    94   val split_last: 'a list -> 'a list * 'a
    95   val find_index: ('a -> bool) -> 'a list -> int
    96   val find_index_eq: ''a -> ''a list -> int
    97   val find_first: ('a -> bool) -> 'a list -> 'a option
    98   val get_index: ('a -> 'b option) -> 'a list -> (int * 'b) option
    99   val get_first: ('a -> 'b option) -> 'a list -> 'b option
   100   val eq_list: ('a * 'b -> bool) -> 'a list * 'b list -> bool
   101   val map2: ('a -> 'b -> 'c) -> 'a list -> 'b list -> 'c list
   102   val fold2: ('a -> 'b -> 'c -> 'c) -> 'a list -> 'b list -> 'c -> 'c
   103   val map_product: ('a -> 'b -> 'c) -> 'a list -> 'b list -> 'c list
   104   val fold_product: ('a -> 'b -> 'c -> 'c) -> 'a list -> 'b list -> 'c -> 'c
   105   val zip_options: 'a list -> 'b option list -> ('a * 'b) list
   106   val ~~ : 'a list * 'b list -> ('a * 'b) list
   107   val split_list: ('a * 'b) list -> 'a list * 'b list
   108   val separate: 'a -> 'a list -> 'a list
   109   val replicate: int -> 'a -> 'a list
   110   val multiply: 'a list -> 'a list list -> 'a list list
   111   val filter: ('a -> bool) -> 'a list -> 'a list
   112   val filter_out: ('a -> bool) -> 'a list -> 'a list
   113   val map_filter: ('a -> 'b option) -> 'a list -> 'b list
   114   val is_prefix: ('a * 'a -> bool) -> 'a list -> 'a list -> bool
   115   val take_prefix: ('a -> bool) -> 'a list -> 'a list * 'a list
   116   val chop_prefix: ('a * 'b -> bool) -> 'a list * 'b list -> 'a list * ('a list * 'b list)
   117   val take_suffix: ('a -> bool) -> 'a list -> 'a list * 'a list
   118   val prefixes1: 'a list -> 'a list list
   119   val prefixes: 'a list -> 'a list list
   120   val suffixes1: 'a list -> 'a list list
   121   val suffixes: 'a list -> 'a list list
   122 
   123   (*integers*)
   124   val inc: int ref -> int
   125   val dec: int ref -> int
   126   val upto: int * int -> int list
   127   val downto: int * int -> int list
   128   val radixpand: int * int -> int list
   129   val radixstring: int * string * int -> string
   130   val string_of_int: int -> string
   131   val signed_string_of_int: int -> string
   132   val string_of_indexname: string * int -> string
   133   val read_radix_int: int -> string list -> int * string list
   134   val read_int: string list -> int * string list
   135   val oct_char: string -> string
   136 
   137   (*strings*)
   138   val nth_string: string -> int -> string
   139   val fold_string: (string -> 'a -> 'a) -> string -> 'a -> 'a
   140   val exists_string: (string -> bool) -> string -> bool
   141   val forall_string: (string -> bool) -> string -> bool
   142   val enclose: string -> string -> string -> string
   143   val unenclose: string -> string
   144   val quote: string -> string
   145   val space_implode: string -> string list -> string
   146   val commas: string list -> string
   147   val commas_quote: string list -> string
   148   val cat_lines: string list -> string
   149   val space_explode: string -> string -> string list
   150   val split_lines: string -> string list
   151   val prefix_lines: string -> string -> string
   152   val untabify: string list -> string list
   153   val prefix: string -> string -> string
   154   val suffix: string -> string -> string
   155   val unprefix: string -> string -> string
   156   val unsuffix: string -> string -> string
   157   val replicate_string: int -> string -> string
   158   val translate_string: (string -> string) -> string -> string
   159 
   160   (*lists as sets -- see also Pure/General/ord_list.ML*)
   161   val member: ('b * 'a -> bool) -> 'a list -> 'b -> bool
   162   val insert: ('a * 'a -> bool) -> 'a -> 'a list -> 'a list
   163   val remove: ('b * 'a -> bool) -> 'b -> 'a list -> 'a list
   164   val update: ('a * 'a -> bool) -> 'a -> 'a list -> 'a list
   165   val subtract: ('b * 'a -> bool) -> 'b list -> 'a list -> 'a list
   166   val merge: ('a * 'a -> bool) -> 'a list * 'a list -> 'a list
   167   val mem: ''a * ''a list -> bool
   168   val mem_int: int * int list -> bool
   169   val mem_string: string * string list -> bool
   170   val union: ''a list * ''a list -> ''a list
   171   val union_int: int list * int list -> int list
   172   val union_string: string list * string list -> string list
   173   val gen_union: ('a * 'a -> bool) -> 'a list * 'a list -> 'a list
   174   val gen_inter: ('a * 'b -> bool) -> 'a list * 'b list -> 'a list
   175   val inter: ''a list * ''a list -> ''a list
   176   val inter_int: int list * int list -> int list
   177   val inter_string: string list * string list -> string list
   178   val subset: ''a list * ''a list -> bool
   179   val subset_int: int list * int list -> bool
   180   val subset_string: string list * string list -> bool
   181   val eq_set: ''a list * ''a list -> bool
   182   val eq_set_string: string list * string list -> bool
   183   val gen_subset: ('a * 'b -> bool) -> 'a list * 'b list -> bool
   184   val gen_eq_set: ('a * 'b -> bool) -> 'a list * 'b list -> bool
   185   val \ : ''a list * ''a -> ''a list
   186   val \\ : ''a list * ''a list -> ''a list
   187   val distinct: ('a * 'a -> bool) -> 'a list -> 'a list
   188   val duplicates: ('a * 'a -> bool) -> 'a list -> 'a list
   189   val has_duplicates: ('a * 'a -> bool) -> 'a list -> bool
   190 
   191   (*lists as multisets*)
   192   val remove1: ('b * 'a -> bool) -> 'b -> 'a list -> 'a list
   193   val submultiset: ('a * 'b -> bool) -> 'a list * 'b list -> bool
   194 
   195   (*orders*)
   196   val is_equal: order -> bool
   197   val rev_order: order -> order
   198   val make_ord: ('a * 'a -> bool) -> 'a * 'a -> order
   199   val bool_ord: bool * bool -> order
   200   val int_ord: int * int -> order
   201   val string_ord: string * string -> order
   202   val fast_string_ord: string * string -> order
   203   val option_ord: ('a * 'b -> order) -> 'a option * 'b option -> order
   204   val prod_ord: ('a * 'b -> order) -> ('c * 'd -> order) -> ('a * 'c) * ('b * 'd) -> order
   205   val dict_ord: ('a * 'b -> order) -> 'a list * 'b list -> order
   206   val list_ord: ('a * 'b -> order) -> 'a list * 'b list -> order
   207   val sort: ('a * 'a -> order) -> 'a list -> 'a list
   208   val sort_distinct: ('a * 'a -> order) -> 'a list -> 'a list
   209   val sort_strings: string list -> string list
   210   val sort_wrt: ('a -> string) -> 'a list -> 'a list
   211 
   212   (*random numbers*)
   213   exception RANDOM
   214   val random: unit -> real
   215   val random_range: int -> int -> int
   216   val one_of: 'a list -> 'a
   217   val frequency: (int * 'a) list -> 'a
   218 
   219   (*misc*)
   220   val divide_and_conquer: ('a -> 'a list * ('b list -> 'b)) -> 'a -> 'b
   221   val partition_eq: ('a * 'a -> bool) -> 'a list -> 'a list list
   222   val partition_list: (int -> 'a -> bool) -> int -> int -> 'a list -> 'a list list
   223   val gensym: string -> string
   224   type stamp
   225   val stamp: unit -> stamp
   226   type serial
   227   val serial: unit -> serial
   228   val serial_string: unit -> string
   229   structure Object: sig type T end
   230 end;
   231 
   232 signature LIBRARY =
   233 sig
   234   include BASIC_LIBRARY
   235   val foldl: ('a * 'b -> 'a) -> 'a * 'b list -> 'a
   236   val foldr: ('a * 'b -> 'b) -> 'a list * 'b -> 'b
   237   val take: int * 'a list -> 'a list
   238   val drop: int * 'a list -> 'a list
   239   val last_elem: 'a list -> 'a
   240 end;
   241 
   242 structure Library: LIBRARY =
   243 struct
   244 
   245 (* functions *)
   246 
   247 fun undefined _ = raise Match;
   248 
   249 fun I x = x;
   250 fun K x = fn _ => x;
   251 fun curry f x y = f (x, y);
   252 fun uncurry f (x, y) = f x y;
   253 
   254 (*application and structured results -- old version*)
   255 fun (x, y) |>>> f = let val (x', z) = f x in (x', (y, z)) end;
   256 
   257 (*conditional application*)
   258 fun b ? f = fn x => if b then f x else x;
   259 
   260 (*composition with multiple args*)
   261 fun (f oo g) x y = f (g x y);
   262 fun (f ooo g) x y z = f (g x y z);
   263 fun (f oooo g) x y z w = f (g x y z w);
   264 
   265 (*function exponentiation: f(...(f x)...) with n applications of f*)
   266 fun funpow (0: int) _ x = x
   267   | funpow n f x = funpow (n - 1) f (f x);
   268 
   269 
   270 (* errors *)
   271 
   272 exception SYS_ERROR of string;
   273 fun sys_error msg = raise SYS_ERROR msg;
   274 
   275 exception ERROR of string;
   276 fun error msg = raise ERROR msg;
   277 
   278 fun cat_error "" msg = error msg
   279   | cat_error msg1 msg2 = error (msg1 ^ "\n" ^ msg2);
   280 
   281 fun assert_all pred list msg =
   282   let
   283     fun ass [] = ()
   284       | ass (x :: xs) = if pred x then ass xs else error (msg x);
   285   in ass list end;
   286 
   287 
   288 (* pairs *)
   289 
   290 fun pair x y = (x, y);
   291 fun rpair x y = (y, x);
   292 
   293 fun fst (x, y) = x;
   294 fun snd (x, y) = y;
   295 
   296 fun eq_fst eq ((x1, _), (x2, _)) = eq (x1, x2);
   297 fun eq_snd eq ((_, y1), (_, y2)) = eq (y1, y2);
   298 fun eq_pair eqx eqy ((x1, y1), (x2, y2)) = eqx (x1, x2) andalso eqy (y1, y2);
   299 
   300 fun swap (x, y) = (y, x);
   301 
   302 fun apfst f (x, y) = (f x, y);
   303 fun apsnd f (x, y) = (x, f y);
   304 fun pairself f (x, y) = (f x, f y);
   305 
   306 
   307 (* booleans *)
   308 
   309 (*polymorphic equality*)
   310 fun equal x y = x = y;
   311 fun not_equal x y = x <> y;
   312 
   313 (*combining predicates*)
   314 fun p orf q = fn x => p x orelse q x;
   315 fun p andf q = fn x => p x andalso q x;
   316 
   317 (*exists pred [x1, ..., xn] ===> pred x1 orelse ... orelse pred xn*)
   318 fun exists (pred: 'a -> bool) : 'a list -> bool =
   319   let fun boolf [] = false
   320         | boolf (x :: xs) = pred x orelse boolf xs
   321   in boolf end;
   322 
   323 (*forall pred [x1, ..., xn] ===> pred x1 andalso ... andalso pred xn*)
   324 fun forall (pred: 'a -> bool) : 'a list -> bool =
   325   let fun boolf [] = true
   326         | boolf (x :: xs) = pred x andalso boolf xs
   327   in boolf end;
   328 
   329 
   330 (* flags *)
   331 
   332 fun set flag = (flag := true; true);
   333 fun reset flag = (flag := false; false);
   334 fun toggle flag = (flag := not (! flag); ! flag);
   335 
   336 fun change r f = r := f (! r);
   337 
   338 (*temporarily set flag during execution*)
   339 fun setmp_noncritical flag value f x =
   340   let
   341     val orig_value = ! flag;
   342     val _ = flag := value;
   343     val result = Exn.capture f x;
   344     val _ = flag := orig_value;
   345   in Exn.release result end;
   346 
   347 fun setmp flag value f x = NAMED_CRITICAL "setmp" (fn () => setmp_noncritical flag value f x);
   348 
   349 
   350 
   351 (** lists **)
   352 
   353 exception UnequalLengths;
   354 
   355 fun single x = [x];
   356 
   357 fun the_single [x] = x
   358   | the_single _ = raise Empty;
   359 
   360 fun singleton f x = the_single (f [x]);
   361 
   362 fun yield_singleton f x = f [x] #>> the_single;
   363 
   364 fun apply [] x = x
   365   | apply (f :: fs) x = apply fs (f x);
   366 
   367 fun perhaps_apply funs arg =
   368   let
   369     fun app [] res = res
   370       | app (f :: fs) (changed, x) =
   371           (case f x of
   372             NONE => app fs (changed, x)
   373           | SOME x' => app fs (true, x'));
   374   in (case app funs (false, arg) of (false, _) => NONE | (true, arg') => SOME arg') end;
   375 
   376 fun perhaps_loop f arg =
   377   let
   378     fun loop (changed, x) =
   379       (case f x of
   380         NONE => (changed, x)
   381       | SOME x' => loop (true, x'));
   382   in (case loop (false, arg) of (false, _) => NONE | (true, arg') => SOME arg') end;
   383 
   384 
   385 (* fold -- old versions *)
   386 
   387 (*the following versions of fold are designed to fit nicely with infixes*)
   388 
   389 (*  (op @) (e, [x1, ..., xn])  ===>  ((e @ x1) @ x2) ... @ xn
   390     for operators that associate to the left (TAIL RECURSIVE)*)
   391 fun foldl (f: 'a * 'b -> 'a) : 'a * 'b list -> 'a =
   392   let fun itl (e, [])  = e
   393         | itl (e, a::l) = itl (f(e, a), l)
   394   in  itl end;
   395 
   396 (*  (op @) ([x1, ..., xn], e)  ===>   x1 @ (x2 ... @ (xn @ e))
   397     for operators that associate to the right (not tail recursive)*)
   398 fun foldr f (l, e) =
   399   let fun itr [] = e
   400         | itr (a::l) = f(a, itr l)
   401   in  itr l  end;
   402 
   403 (*  (op @) [x1, ..., xn]  ===>   x1 @ (x2 ... @ (x[n-1] @ xn))
   404     for n > 0, operators that associate to the right (not tail recursive)*)
   405 fun foldr1 f [] = raise Empty
   406   | foldr1 f l =
   407       let fun itr [x] = x
   408             | itr (x::l) = f(x, itr l)
   409       in  itr l  end;
   410 
   411 fun foldl_map f =
   412   let
   413     fun fold_aux (x, []) = (x, [])
   414       | fold_aux (x, y :: ys) =
   415           let
   416             val (x', y') = f (x, y);
   417             val (x'', ys') = fold_aux (x', ys);
   418           in (x'', y' :: ys') end;
   419   in fold_aux end;
   420 
   421 
   422 (* basic list functions *)
   423 
   424 fun eq_list eq (list1, list2) =
   425   let
   426     fun eq_lst (x :: xs, y :: ys) = eq (x, y) andalso eq_lst (xs, ys)
   427       | eq_lst _ = true;
   428   in length list1 = length list2 andalso eq_lst (list1, list2) end;
   429 
   430 fun maps f [] = []
   431   | maps f (x :: xs) = f x @ maps f xs;
   432 
   433 (*copy the list preserving elements that satisfy the predicate*)
   434 val filter = List.filter;
   435 fun filter_out f = filter (not o f);
   436 val map_filter = List.mapPartial;
   437 
   438 fun chop (0: int) xs = ([], xs)
   439   | chop _ [] = ([], [])
   440   | chop n (x :: xs) = chop (n - 1) xs |>> cons x;
   441 
   442 (*take the first n elements from a list*)
   443 fun take (n: int, []) = []
   444   | take (n, x :: xs) =
   445       if n > 0 then x :: take (n - 1, xs) else [];
   446 
   447 (*drop the first n elements from a list*)
   448 fun drop (n: int, []) = []
   449   | drop (n, x :: xs) =
   450       if n > 0 then drop (n - 1, xs) else x :: xs;
   451 
   452 fun dropwhile P [] = []
   453   | dropwhile P (ys as x::xs) = if P x then dropwhile P xs else ys;
   454 
   455 (*return nth element of a list, where 0 designates the first element;
   456   raise Subscript if list too short*)
   457 fun nth xs i = List.nth (xs, i);
   458 
   459 fun nth_list xss i = nth xss i handle Subscript => [];
   460 
   461 fun nth_map 0 f (x :: xs) = f x :: xs
   462   | nth_map n f (x :: xs) = x :: nth_map (n - 1) f xs
   463   | nth_map (_: int) _ [] = raise Subscript;
   464 
   465 fun nth_drop n xs =
   466   List.take (xs, n) @ List.drop (xs, n + 1);
   467 
   468 fun map_index f =
   469   let
   470     fun mapp (_: int) [] = []
   471       | mapp i (x :: xs) = f (i, x) :: mapp (i + 1) xs
   472   in mapp 0 end;
   473 
   474 fun fold_index f =
   475   let
   476     fun fold_aux (_: int) [] y = y
   477       | fold_aux i (x :: xs) y = fold_aux (i + 1) xs (f (i, x) y);
   478   in fold_aux 0 end;
   479 
   480 val last_elem = List.last;
   481 
   482 (*rear decomposition*)
   483 fun split_last [] = raise Empty
   484   | split_last [x] = ([], x)
   485   | split_last (x :: xs) = apfst (cons x) (split_last xs);
   486 
   487 (*find the position of an element in a list*)
   488 fun find_index pred =
   489   let fun find (_: int) [] = ~1
   490         | find n (x :: xs) = if pred x then n else find (n + 1) xs;
   491   in find 0 end;
   492 
   493 fun find_index_eq x = find_index (equal x);
   494 
   495 (*find first element satisfying predicate*)
   496 val find_first = List.find;
   497 
   498 (*get first element by lookup function*)
   499 fun get_first _ [] = NONE
   500   | get_first f (x :: xs) =
   501       (case f x of
   502         NONE => get_first f xs
   503       | some => some);
   504 
   505 fun get_index f =
   506   let
   507     fun get (_: int) [] = NONE
   508       | get i (x :: xs) =
   509           case f x
   510            of NONE => get (i + 1) xs
   511             | SOME y => SOME (i, y)
   512   in get 0 end;
   513 
   514 val flat = List.concat;
   515 
   516 fun unflat (xs :: xss) ys =
   517       let val (ps, qs) = chop (length xs) ys
   518       in ps :: unflat xss qs end
   519   | unflat [] [] = []
   520   | unflat _ _ = raise UnequalLengths;
   521 
   522 fun burrow f xss = unflat xss (f (flat xss));
   523 
   524 fun burrow_options f os = map (try hd) (burrow f (map the_list os));
   525 
   526 fun fold_burrow f xss s =
   527   apfst (unflat xss) (f (flat xss) s);
   528 
   529 (*separate s [x1, x2, ..., xn]  ===>  [x1, s, x2, s, ..., s, xn]*)
   530 fun separate s (x :: (xs as _ :: _)) = x :: s :: separate s xs
   531   | separate _ xs = xs;
   532 
   533 (*make the list [x, x, ..., x] of length n*)
   534 fun replicate (n: int) x =
   535   let fun rep (0, xs) = xs
   536         | rep (n, xs) = rep (n - 1, x :: xs)
   537   in
   538     if n < 0 then raise Subscript
   539     else rep (n, [])
   540   end;
   541 
   542 fun translate_string f = String.translate (f o String.str);
   543 
   544 (*multiply [a, b, c, ...] * [xs, ys, zs, ...]*)
   545 fun multiply [] _ = []
   546   | multiply (x :: xs) yss = map (cons x) yss @ multiply xs yss;
   547 
   548 (*direct product*)
   549 fun map_product f _ [] = []
   550   | map_product f [] _ = []
   551   | map_product f (x :: xs) ys = map (f x) ys @ map_product f xs ys;
   552 
   553 fun fold_product f _ [] z = z
   554   | fold_product f [] _ z = z
   555   | fold_product f (x :: xs) ys z = z |> fold (f x) ys |> fold_product f xs ys;
   556 
   557 (*lists of pairs*)
   558 
   559 exception UnequalLengths;
   560 
   561 fun map2 _ [] [] = []
   562   | map2 f (x :: xs) (y :: ys) = f x y :: map2 f xs ys
   563   | map2 _ _ _ = raise UnequalLengths;
   564 
   565 fun fold2 f [] [] z = z
   566   | fold2 f (x :: xs) (y :: ys) z = fold2 f xs ys (f x y z)
   567   | fold2 f _ _ _ = raise UnequalLengths;
   568 
   569 fun zip_options (x :: xs) (SOME y :: ys) = (x, y) :: zip_options xs ys
   570   | zip_options (_ :: xs) (NONE :: ys) = zip_options xs ys
   571   | zip_options _ [] = []
   572   | zip_options [] _ = raise UnequalLengths;
   573 
   574 (*combine two lists forming a list of pairs:
   575   [x1, ..., xn] ~~ [y1, ..., yn]  ===>  [(x1, y1), ..., (xn, yn)]*)
   576 fun [] ~~ [] = []
   577   | (x :: xs) ~~ (y :: ys) = (x, y) :: (xs ~~ ys)
   578   | _ ~~ _ = raise UnequalLengths;
   579 
   580 (*inverse of ~~; the old 'split':
   581   [(x1, y1), ..., (xn, yn)]  ===>  ([x1, ..., xn], [y1, ..., yn])*)
   582 val split_list = ListPair.unzip;
   583 
   584 
   585 (* prefixes, suffixes *)
   586 
   587 fun is_prefix _ [] _ = true
   588   | is_prefix eq (x :: xs) (y :: ys) = eq (x, y) andalso is_prefix eq xs ys
   589   | is_prefix eq _ _ = false;
   590 
   591 (* [x1, ..., xi, ..., xn]  --->  ([x1, ..., x(i-1)], [xi, ..., xn])
   592    where xi is the first element that does not satisfy the predicate*)
   593 fun take_prefix (pred : 'a -> bool)  (xs: 'a list) : 'a list * 'a list =
   594   let fun take (rxs, []) = (rev rxs, [])
   595         | take (rxs, x :: xs) =
   596             if  pred x  then  take(x :: rxs, xs)  else  (rev rxs, x :: xs)
   597   in  take([], xs)  end;
   598 
   599 fun chop_prefix eq ([], ys) = ([], ([], ys))
   600   | chop_prefix eq (xs, []) = ([], (xs, []))
   601   | chop_prefix eq (xs as x :: xs', ys as y :: ys') =
   602       if eq (x, y) then
   603         let val (ps', xys'') = chop_prefix eq (xs', ys')
   604         in (x :: ps', xys'') end
   605       else ([], (xs, ys));
   606 
   607 (* [x1, ..., xi, ..., xn]  --->  ([x1, ..., xi], [x(i+1), ..., xn])
   608    where xi is the last element that does not satisfy the predicate*)
   609 fun take_suffix _ [] = ([], [])
   610   | take_suffix pred (x :: xs) =
   611       (case take_suffix pred xs of
   612         ([], sffx) => if pred x then ([], x :: sffx) else ([x], sffx)
   613       | (prfx, sffx) => (x :: prfx, sffx));
   614 
   615 fun prefixes1 [] = []
   616   | prefixes1 (x :: xs) = map (cons x) ([] :: prefixes1 xs);
   617 
   618 fun prefixes xs = [] :: prefixes1 xs;
   619 
   620 fun suffixes1 xs = map rev (prefixes1 (rev xs));
   621 fun suffixes xs = [] :: suffixes1 xs;
   622 
   623 
   624 
   625 (** integers **)
   626 
   627 fun inc i = (i := ! i + (1: int); ! i);
   628 fun dec i = (i := ! i - (1: int); ! i);
   629 
   630 
   631 (* lists of integers *)
   632 
   633 (*make the list [from, from + 1, ..., to]*)
   634 fun ((i: int) upto j) =
   635   if i > j then [] else i :: (i + 1 upto j);
   636 
   637 (*make the list [from, from - 1, ..., to]*)
   638 fun ((i: int) downto j) =
   639   if i < j then [] else i :: (i - 1 downto j);
   640 
   641 
   642 (* convert integers to strings *)
   643 
   644 (*expand the number in the given base;
   645   example: radixpand (2, 8) gives [1, 0, 0, 0]*)
   646 fun radixpand (base, num) : int list =
   647   let
   648     fun radix (n, tail) =
   649       if n < base then n :: tail
   650       else radix (n div base, (n mod base) :: tail)
   651   in radix (num, []) end;
   652 
   653 (*expands a number into a string of characters starting from "zerochar";
   654   example: radixstring (2, "0", 8) gives "1000"*)
   655 fun radixstring (base, zerochar, num) =
   656   let val offset = ord zerochar;
   657       fun chrof n = chr (offset + n)
   658   in implode (map chrof (radixpand (base, num))) end;
   659 
   660 
   661 val string_of_int = Int.toString;
   662 
   663 fun signed_string_of_int i =
   664   if i < 0 then "-" ^ string_of_int (~ i) else string_of_int i;
   665 
   666 fun string_of_indexname (a, 0) = a
   667   | string_of_indexname (a, i) = a ^ "_" ^ string_of_int i;
   668 
   669 
   670 (* read integers *)
   671 
   672 fun read_radix_int radix cs =
   673   let
   674     val zero = ord "0";
   675     val limit = zero + radix;
   676     fun scan (num, []) = (num, [])
   677       | scan (num, c :: cs) =
   678         if zero <= ord c andalso ord c < limit then
   679           scan (radix * num + (ord c - zero), cs)
   680         else (num, c :: cs);
   681   in scan (0, cs) end;
   682 
   683 val read_int = read_radix_int 10;
   684 
   685 fun oct_char s = chr (#1 (read_radix_int 8 (explode s)));
   686 
   687 
   688 
   689 (** strings **)
   690 
   691 (* functions tuned for strings, avoiding explode *)
   692 
   693 fun nth_string str i =
   694   (case try String.substring (str, i, 1) of
   695     SOME s => s
   696   | NONE => raise Subscript);
   697 
   698 fun fold_string f str x0 =
   699   let
   700     val n = size str;
   701     fun iter (x, i) =
   702       if i < n then iter (f (String.substring (str, i, 1)) x, i + 1) else x;
   703   in iter (x0, 0) end;
   704 
   705 fun exists_string pred str =
   706   let
   707     val n = size str;
   708     fun ex i = i < n andalso (pred (String.substring (str, i, 1)) orelse ex (i + 1));
   709   in ex 0 end;
   710 
   711 fun forall_string pred = not o exists_string (not o pred);
   712 
   713 (*enclose in brackets*)
   714 fun enclose lpar rpar str = lpar ^ str ^ rpar;
   715 fun unenclose str = String.substring (str, 1, size str - 2);
   716 
   717 (*simple quoting (does not escape special chars)*)
   718 val quote = enclose "\"" "\"";
   719 
   720 (*space_implode "..." (explode "hello") = "h...e...l...l...o"*)
   721 fun space_implode a bs = implode (separate a bs);
   722 
   723 val commas = space_implode ", ";
   724 val commas_quote = commas o map quote;
   725 
   726 (*concatenate messages, one per line, into a string*)
   727 val cat_lines = space_implode "\n";
   728 
   729 (*space_explode "." "h.e..l.lo" = ["h", "e", "", "l", "lo"]*)
   730 fun space_explode _ "" = []
   731   | space_explode sep s = String.fields (fn c => str c = sep) s;
   732 
   733 val split_lines = space_explode "\n";
   734 
   735 fun prefix_lines "" txt = txt
   736   | prefix_lines prfx txt = txt |> split_lines |> map (fn s => prfx ^ s) |> cat_lines;
   737 
   738 fun untabify chs =
   739   let
   740     val tab_width = 8;
   741 
   742     fun untab pos [] ys = rev ys
   743       | untab pos ("\n" :: xs) ys = untab 0 xs ("\n" :: ys)
   744       | untab pos ("\t" :: xs) ys =
   745           let val d = tab_width - (pos mod tab_width)
   746           in untab (pos + d) xs (funpow d (cons " ") ys) end
   747       | untab pos (c :: xs) ys = untab (pos + 1) xs (c :: ys);
   748   in
   749     if not (exists (fn c => c = "\t") chs) then chs
   750     else untab 0 chs []
   751   end;
   752 
   753 fun prefix prfx s = prfx ^ s;
   754 fun suffix sffx s = s ^ sffx;
   755 
   756 fun unprefix prfx s =
   757   if String.isPrefix prfx s then String.substring (s, size prfx, size s - size prfx)
   758   else raise Fail "unprefix";
   759 
   760 fun unsuffix sffx s =
   761   if String.isSuffix sffx s then String.substring (s, 0, size s - size sffx)
   762   else raise Fail "unsuffix";
   763 
   764 fun replicate_string (0: int) _ = ""
   765   | replicate_string 1 a = a
   766   | replicate_string k a =
   767       if k mod 2 = 0 then replicate_string (k div 2) (a ^ a)
   768       else replicate_string (k div 2) (a ^ a) ^ a;
   769 
   770 
   771 
   772 (** lists as sets -- see also Pure/General/ord_list.ML **)
   773 
   774 (*canonical member, insert, remove*)
   775 fun member eq list x =
   776   let
   777     fun memb [] = false
   778       | memb (y :: ys) = eq (x, y) orelse memb ys;
   779   in memb list end;
   780 
   781 fun insert eq x xs = if member eq xs x then xs else x :: xs;
   782 fun remove eq x xs = if member eq xs x then filter_out (fn y => eq (x, y)) xs else xs;
   783 fun update eq x xs = cons x (remove eq x xs);
   784 
   785 fun subtract eq = fold (remove eq);
   786 
   787 fun merge _ ([], ys) = ys
   788   | merge eq (xs, ys) = fold_rev (insert eq) ys xs;
   789 
   790 (*old-style infixes*)
   791 fun x mem xs = member (op =) xs x;
   792 fun (x: int) mem_int xs = member (op =) xs x;
   793 fun (x: string) mem_string xs = member (op =) xs x;
   794 
   795 
   796 (*union of sets represented as lists: no repetitions*)
   797 fun xs union [] = xs
   798   | [] union ys = ys
   799   | (x :: xs) union ys = xs union (insert (op =) x ys);
   800 
   801 (*union of sets, optimized version for ints*)
   802 fun (xs:int list) union_int [] = xs
   803   | [] union_int ys = ys
   804   | (x :: xs) union_int ys = xs union_int (insert (op =) x ys);
   805 
   806 (*union of sets, optimized version for strings*)
   807 fun (xs:string list) union_string [] = xs
   808   | [] union_string ys = ys
   809   | (x :: xs) union_string ys = xs union_string (insert (op =) x ys);
   810 
   811 (*generalized union*)
   812 fun gen_union eq (xs, []) = xs
   813   | gen_union eq ([], ys) = ys
   814   | gen_union eq (x :: xs, ys) = gen_union eq (xs, insert eq x ys);
   815 
   816 
   817 (*intersection*)
   818 fun [] inter ys = []
   819   | (x :: xs) inter ys =
   820       if x mem ys then x :: (xs inter ys) else xs inter ys;
   821 
   822 (*intersection, optimized version for ints*)
   823 fun ([]:int list) inter_int ys = []
   824   | (x :: xs) inter_int ys =
   825       if x mem_int ys then x :: (xs inter_int ys) else xs inter_int ys;
   826 
   827 (*intersection, optimized version for strings *)
   828 fun ([]:string list) inter_string ys = []
   829   | (x :: xs) inter_string ys =
   830       if x mem_string ys then x :: (xs inter_string ys) else xs inter_string ys;
   831 
   832 (*generalized intersection*)
   833 fun gen_inter eq ([], ys) = []
   834   | gen_inter eq (x::xs, ys) =
   835       if member eq ys x then x :: gen_inter eq (xs, ys)
   836       else gen_inter eq (xs, ys);
   837 
   838 
   839 (*subset*)
   840 fun [] subset ys = true
   841   | (x :: xs) subset ys = x mem ys andalso xs subset ys;
   842 
   843 (*subset, optimized version for ints*)
   844 fun ([]: int list) subset_int ys = true
   845   | (x :: xs) subset_int ys = x mem_int ys andalso xs subset_int ys;
   846 
   847 (*subset, optimized version for strings*)
   848 fun ([]: string list) subset_string ys = true
   849   | (x :: xs) subset_string ys = x mem_string ys andalso xs subset_string ys;
   850 
   851 (*set equality*)
   852 fun eq_set (xs, ys) =
   853   xs = ys orelse (xs subset ys andalso ys subset xs);
   854 
   855 (*set equality for strings*)
   856 fun eq_set_string ((xs: string list), ys) =
   857   xs = ys orelse (xs subset_string ys andalso ys subset_string xs);
   858 
   859 fun gen_subset eq (xs, ys) = forall (member eq ys) xs;
   860 
   861 fun gen_eq_set eq (xs, ys) =
   862   eq_list eq (xs, ys) orelse
   863     (gen_subset eq (xs, ys) andalso gen_subset (eq o swap) (ys, xs));
   864 
   865 
   866 (*removing an element from a list WITHOUT duplicates*)
   867 fun (y :: ys) \ x = if x = y then ys else y :: (ys \ x)
   868   | [] \ x = [];
   869 fun ys \\ xs = foldl (op \) (ys,xs);
   870 
   871 
   872 (*makes a list of the distinct members of the input; preserves order, takes
   873   first of equal elements*)
   874 fun distinct eq lst =
   875   let
   876     fun dist (rev_seen, []) = rev rev_seen
   877       | dist (rev_seen, x :: xs) =
   878           if member eq rev_seen x then dist (rev_seen, xs)
   879           else dist (x :: rev_seen, xs);
   880   in dist ([], lst) end;
   881 
   882 (*returns a list containing all repeated elements exactly once; preserves
   883   order, takes first of equal elements*)
   884 fun duplicates eq lst =
   885   let
   886     fun dups (rev_dups, []) = rev rev_dups
   887       | dups (rev_dups, x :: xs) =
   888           if member eq rev_dups x orelse not (member eq xs x) then
   889             dups (rev_dups, xs)
   890           else dups (x :: rev_dups, xs);
   891   in dups ([], lst) end;
   892 
   893 fun has_duplicates eq =
   894   let
   895     fun dups [] = false
   896       | dups (x :: xs) = member eq xs x orelse dups xs;
   897   in dups end;
   898 
   899 
   900 
   901 (** lists as multisets **)
   902 
   903 fun remove1 _ _ [] = raise Empty
   904   | remove1 eq y (x::xs) = if eq (y, x) then xs else x :: remove1 eq y xs;
   905 
   906 fun submultiset _ ([], _)  = true
   907   | submultiset eq (x :: xs, ys) = member eq ys x andalso submultiset eq (xs, remove1 eq x ys);
   908 
   909 
   910 
   911 (** orders **)
   912 
   913 fun is_equal EQUAL = true
   914   | is_equal _ = false;
   915 
   916 fun rev_order LESS = GREATER
   917   | rev_order EQUAL = EQUAL
   918   | rev_order GREATER = LESS;
   919 
   920 (*assume rel is a linear strict order*)
   921 fun make_ord rel (x, y) =
   922   if rel (x, y) then LESS
   923   else if rel (y, x) then GREATER
   924   else EQUAL;
   925 
   926 fun bool_ord (false, true) = LESS
   927   | bool_ord (true, false) = GREATER
   928   | bool_ord _ = EQUAL;
   929 
   930 val int_ord = Int.compare;
   931 val string_ord = String.compare;
   932 
   933 fun fast_string_ord (s1, s2) =
   934   (case int_ord (size s1, size s2) of EQUAL => string_ord (s1, s2) | ord => ord);
   935 
   936 fun option_ord ord (SOME x, SOME y) = ord (x, y)
   937   | option_ord _ (NONE, NONE) = EQUAL
   938   | option_ord _ (NONE, SOME _) = LESS
   939   | option_ord _ (SOME _, NONE) = GREATER;
   940 
   941 (*lexicographic product*)
   942 fun prod_ord a_ord b_ord ((x, y), (x', y')) =
   943   (case a_ord (x, x') of EQUAL => b_ord (y, y') | ord => ord);
   944 
   945 (*dictionary order -- in general NOT well-founded!*)
   946 fun dict_ord elem_ord (x :: xs, y :: ys) =
   947       (case elem_ord (x, y) of EQUAL => dict_ord elem_ord (xs, ys) | ord => ord)
   948   | dict_ord _ ([], []) = EQUAL
   949   | dict_ord _ ([], _ :: _) = LESS
   950   | dict_ord _ (_ :: _, []) = GREATER;
   951 
   952 (*lexicographic product of lists*)
   953 fun list_ord elem_ord (xs, ys) =
   954   (case int_ord (length xs, length ys) of EQUAL => dict_ord elem_ord (xs, ys) | ord => ord);
   955 
   956 
   957 (* sorting *)
   958 
   959 (*quicksort -- stable, i.e. does not reorder equal elements*)
   960 fun quicksort unique ord =
   961   let
   962     fun qsort [] = []
   963       | qsort (xs as [_]) = xs
   964       | qsort (xs as [x, y]) =
   965           (case ord (x, y) of
   966             LESS => xs
   967           | EQUAL => if unique then [x] else xs
   968           | GREATER => [y, x])
   969       | qsort xs =
   970           let val (lts, eqs, gts) = part (nth xs (length xs div 2)) xs
   971           in qsort lts @ eqs @ qsort gts end
   972     and part _ [] = ([], [], [])
   973       | part pivot (x :: xs) = add (ord (x, pivot)) x (part pivot xs)
   974     and add LESS x (lts, eqs, gts) = (x :: lts, eqs, gts)
   975       | add EQUAL x (lts, [], gts) = (lts, [x], gts)
   976       | add EQUAL x (res as (lts, eqs, gts)) = if unique then res else (lts, x :: eqs, gts)
   977       | add GREATER x (lts, eqs, gts) = (lts, eqs, x :: gts);
   978   in qsort end;
   979 
   980 fun sort ord = quicksort false ord;
   981 fun sort_distinct ord = quicksort true ord;
   982 
   983 val sort_strings = sort string_ord;
   984 fun sort_wrt sel xs = sort (string_ord o pairself sel) xs;
   985 
   986 
   987 
   988 (** random numbers **)
   989 
   990 exception RANDOM;
   991 
   992 fun rmod x y = x - y * Real.realFloor (x / y);
   993 
   994 local
   995   val a = 16807.0;
   996   val m = 2147483647.0;
   997   val random_seed = ref 1.0;
   998 in
   999 
  1000 fun random () = CRITICAL (fn () =>
  1001   let val r = rmod (a * ! random_seed) m
  1002   in (random_seed := r; r) end);
  1003 
  1004 end;
  1005 
  1006 fun random_range l h =
  1007   if h < l orelse l < 0 then raise RANDOM
  1008   else l + Real.floor (rmod (random ()) (real (h - l + 1)));
  1009 
  1010 fun one_of xs = nth xs (random_range 0 (length xs - 1));
  1011 
  1012 fun frequency xs =
  1013   let
  1014     val sum = foldl op + (0, map fst xs);
  1015     fun pick n ((k: int, x) :: xs) =
  1016       if n <= k then x else pick (n - k) xs
  1017   in pick (random_range 1 sum) xs end;
  1018 
  1019 
  1020 
  1021 (** misc **)
  1022 
  1023 fun divide_and_conquer decomp x =
  1024   let val (ys, recomb) = decomp x
  1025   in recomb (map (divide_and_conquer decomp) ys) end;
  1026 
  1027 
  1028 (*Partition a list into buckets  [ bi, b(i+1), ..., bj ]
  1029    putting x in bk if p(k)(x) holds.  Preserve order of elements if possible.*)
  1030 fun partition_list p i j =
  1031   let fun part k xs =
  1032             if k>j then
  1033               (case xs of [] => []
  1034                          | _ => raise Fail "partition_list")
  1035             else
  1036             let val (ns, rest) = List.partition (p k) xs;
  1037             in  ns :: part(k+1)rest  end
  1038   in  part (i: int) end;
  1039 
  1040 fun partition_eq (eq:'a * 'a -> bool) =
  1041   let
  1042     fun part [] = []
  1043       | part (x :: ys) =
  1044           let val (xs, xs') = List.partition (fn y => eq (x, y)) ys
  1045           in (x::xs)::(part xs') end
  1046   in part end;
  1047 
  1048 
  1049 
  1050 (* generating identifiers *)
  1051 
  1052 (** Freshly generated identifiers; supplied prefix MUST start with a letter **)
  1053 local
  1054 (*Maps 0-61 to A-Z, a-z, 0-9; exclude _ or ' to avoid clash with internal/unusual indentifiers*)
  1055 fun gensym_char i =
  1056   if i<26 then chr (ord "A" + i)
  1057   else if i<52 then chr (ord "a" + i - 26)
  1058   else chr (ord "0" + i - 52);
  1059 
  1060 val char_vec = Vector.tabulate (62, gensym_char);
  1061 fun newid n = implode (map (fn i => Vector.sub (char_vec, i)) (radixpand (62, n)));
  1062 
  1063 val gensym_seed = ref (0: int);
  1064 
  1065 in
  1066   fun gensym pre = pre ^ newid (NAMED_CRITICAL "gensym" (fn () => inc gensym_seed));
  1067 end;
  1068 
  1069 
  1070 (* stamps and serial numbers *)
  1071 
  1072 type stamp = unit ref;
  1073 val stamp: unit -> stamp = ref;
  1074 
  1075 type serial = int;
  1076 local val count = ref (0: int)
  1077 in fun serial () = NAMED_CRITICAL "serial" (fn () => inc count) end;
  1078 
  1079 val serial_string = string_of_int o serial;
  1080 
  1081 
  1082 (* generic objects *)
  1083 
  1084 (*note that the builtin exception datatype may be extended by new
  1085   constructors at any time*)
  1086 structure Object = struct type T = exn end;
  1087 
  1088 end;
  1089 
  1090 structure BasicLibrary: BASIC_LIBRARY = Library;
  1091 open BasicLibrary;