src/Pure/library.ML
author haftmann
Tue Sep 06 08:30:43 2005 +0200 (2005-09-06)
changeset 17271 2756a73f63a5
parent 17257 0ab67cb765da
child 17313 7d97f60293ae
permissions -rw-r--r--
introduced some new-style AList operations
     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 rational numbers, strings, lists as sets, association lists, generic
     8 tables, balanced trees, orders, current directory, misc.
     9 *)
    10 
    11 infix 1 |> |-> ||> ||>> |>> |>>> #> #->;
    12 infix 2 ?;
    13 infix 3 o oo ooo oooo;
    14 
    15 infix 4 ~~ upto downto;
    16 infix orf andf prefix \ \\ ins ins_string ins_int 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 I: 'a -> 'a
    23   val K: 'a -> 'b -> 'a
    24   val curry: ('a * 'b -> 'c) -> 'a -> 'b -> 'c
    25   val uncurry: ('a -> 'b -> 'c) -> 'a * 'b -> 'c
    26   val |> : 'a * ('a -> 'b) -> 'b
    27   val |-> : ('c * 'a) * ('c -> 'a -> 'b) -> 'b
    28   val ||> : ('c * 'a) * ('a -> 'b) -> 'c * 'b
    29   val ||>> : ('c * 'a) * ('a -> 'd * 'b) -> ('c * 'd) * 'b
    30   val |>> : ('a * 'c) * ('a -> 'b) -> 'b * 'c
    31   val |>>> : ('a * 'c) * ('a -> 'b * 'd) -> 'b * ('c * 'd)
    32   val #> : ('a -> 'b) * ('b -> 'c) -> 'a -> 'c
    33   val #-> : ('a -> 'c * 'b) * ('c -> 'b -> 'd) -> 'a -> 'd
    34   val ? : ('a -> bool) * ('a -> 'a) -> 'a -> 'a
    35   val ` : ('b -> 'a) -> 'b -> 'a * 'b
    36   val tap: ('b -> 'a) -> 'b -> 'b
    37   val oo: ('a -> 'b) * ('c -> 'd -> 'a) -> 'c -> 'd -> 'b
    38   val ooo: ('a -> 'b) * ('c -> 'd -> 'e -> 'a) -> 'c -> 'd -> 'e -> 'b
    39   val oooo: ('a -> 'b) * ('c -> 'd -> 'e -> 'f -> 'a) -> 'c -> 'd -> 'e -> 'f -> 'b
    40   val funpow: int -> ('a -> 'a) -> 'a -> 'a
    41   val apl: 'a * ('a * 'b -> 'c) -> 'b -> 'c
    42   val apr: ('a * 'b -> 'c) * 'b -> 'a -> 'c
    43 
    44   (*old options -- invalidated*)
    45   datatype invalid = None of invalid
    46   exception OPTION of invalid
    47 
    48   (*options*)
    49   val the: 'a option -> 'a
    50   val these: 'a list option -> 'a list
    51   val if_none: 'a option -> 'a -> 'a
    52   val is_some: 'a option -> bool
    53   val is_none: 'a option -> bool
    54 
    55   exception ERROR
    56   val try: ('a -> 'b) -> 'a -> 'b option
    57   val can: ('a -> 'b) -> 'a -> bool
    58   datatype 'a result = Result of 'a | Exn of exn
    59   val capture: ('a -> 'b) -> 'a -> 'b result
    60   val release: 'a result -> 'a
    61   val get_result: 'a result -> 'a option
    62   val get_exn: 'a result -> exn option
    63 
    64   (*pairs*)
    65   val pair: 'a -> 'b -> 'a * 'b
    66   val rpair: 'a -> 'b -> 'b * 'a
    67   val fst: 'a * 'b -> 'a
    68   val snd: 'a * 'b -> 'b
    69   val eq_fst: (''a * 'b) * (''a * 'c) -> bool
    70   val eq_snd: ('a * ''b) * ('c * ''b) -> bool
    71   val swap: 'a * 'b -> 'b * 'a
    72   val apfst: ('a -> 'b) -> 'a * 'c -> 'b * 'c
    73   val apsnd: ('a -> 'b) -> 'c * 'a -> 'c * 'b
    74   val pairself: ('a -> 'b) -> 'a * 'a -> 'b * 'b
    75 
    76   (*booleans*)
    77   val equal: ''a -> ''a -> bool
    78   val not_equal: ''a -> ''a -> bool
    79   val orf: ('a -> bool) * ('a -> bool) -> 'a -> bool
    80   val andf: ('a -> bool) * ('a -> bool) -> 'a -> bool
    81   val exists: ('a -> bool) -> 'a list -> bool
    82   val forall: ('a -> bool) -> 'a list -> bool
    83   val set: bool ref -> bool
    84   val reset: bool ref -> bool
    85   val toggle: bool ref -> bool
    86   val change: 'a ref -> ('a -> 'a) -> unit
    87   val setmp: 'a ref -> 'a -> ('b -> 'c) -> 'b -> 'c
    88   val conditional: bool -> (unit -> unit) -> unit
    89 
    90   (*lists*)
    91   exception UnequalLengths
    92   val cons: 'a -> 'a list -> 'a list
    93   val single: 'a -> 'a list
    94   val append: 'a list -> 'a list -> 'a list
    95   val apply: ('a -> 'a) list -> 'a -> 'a
    96   val fold: ('a -> 'b -> 'b) -> 'a list -> 'b -> 'b
    97   val fold_rev: ('a -> 'b -> 'b) -> 'a list -> 'b -> 'b
    98   val fold_map: ('a -> 'b -> 'c * 'b) -> 'a list -> 'b -> 'c list * 'b
    99   val foldl_map: ('a * 'b -> 'a * 'c) -> 'a * 'b list -> 'a * 'c list
   100   val foldr1: ('a * 'a -> 'a) -> 'a list -> 'a
   101   val foldln: ('a * int -> 'b -> 'b) -> 'a list -> 'b -> 'b
   102   val unflat: 'a list list -> 'b list -> 'b list list
   103   val splitAt: int * 'a list -> 'a list * 'a list
   104   val dropwhile: ('a -> bool) -> 'a list -> 'a list
   105   val map_nth_elem: int -> ('a -> 'a) -> 'a list -> 'a list
   106   val split_last: 'a list -> 'a list * 'a
   107   val nth_update: 'a -> int * 'a list -> 'a list
   108   val find_index: ('a -> bool) -> 'a list -> int
   109   val find_index_eq: ''a -> ''a list -> int
   110   val find_first: ('a -> bool) -> 'a list -> 'a option
   111   val get_first: ('a -> 'b option) -> 'a list -> 'b option
   112   val separate: 'a -> 'a list -> 'a list
   113   val replicate: int -> 'a -> 'a list
   114   val multiply: 'a list * 'a list list -> 'a list list
   115   val product: 'a list -> 'b list -> ('a * 'b) list
   116   val filter: ('a -> bool) -> 'a list -> 'a list
   117   val filter_out: ('a -> bool) -> 'a list -> 'a list
   118   val map2: ('a * 'b -> 'c) -> 'a list * 'b list -> 'c list
   119   val exists2: ('a * 'b -> bool) -> 'a list * 'b list -> bool
   120   val forall2: ('a * 'b -> bool) -> 'a list * 'b list -> bool
   121   val seq2: ('a * 'b -> unit) -> 'a list * 'b list -> unit
   122   val ~~ : 'a list * 'b list -> ('a * 'b) list
   123   val split_list: ('a * 'b) list -> 'a list * 'b list
   124   val equal_lists: ('a * 'b -> bool) -> 'a list * 'b list -> bool
   125   val prefix: ''a list * ''a list -> bool
   126   val take_prefix: ('a -> bool) -> 'a list -> 'a list * 'a list
   127   val take_suffix: ('a -> bool) -> 'a list -> 'a list * 'a list
   128   val prefixes1: 'a list -> 'a list list
   129   val suffixes1: 'a list -> 'a list list
   130 
   131   (*integers*)
   132   val gcd: IntInf.int * IntInf.int -> IntInf.int
   133   val lcm: IntInf.int * IntInf.int -> IntInf.int
   134   val inc: int ref -> int
   135   val dec: int ref -> int
   136   val upto: int * int -> int list
   137   val downto: int * int -> int list
   138   val downto0: int list * int -> bool
   139   val radixpand: int * int -> int list
   140   val radixstring: int * string * int -> string
   141   val string_of_int: int -> string
   142   val string_of_indexname: string * int -> string
   143   val read_radixint: int * string list -> int * string list
   144   val read_int: string list -> int * string list
   145   val oct_char: string -> string
   146 
   147   (*rational numbers*)
   148   type rat
   149   exception RAT of string
   150   val rep_rat: rat -> IntInf.int * IntInf.int
   151   val ratge0: rat -> bool
   152   val ratgt0: rat -> bool
   153   val ratle: rat * rat -> bool
   154   val ratlt: rat * rat -> bool
   155   val ratadd: rat * rat -> rat
   156   val ratmul: rat * rat -> rat
   157   val ratinv: rat -> rat
   158   val int_ratdiv: IntInf.int * IntInf.int -> rat
   159   val ratneg: rat -> rat
   160   val rat_of_int: int -> rat
   161   val rat_of_intinf: IntInf.int -> rat
   162   val rat0: rat
   163 
   164   (*strings*)
   165   val nth_elem_string: int * string -> string
   166   val fold_string: (string -> 'a -> 'a) -> string -> 'a -> 'a
   167   val exists_string: (string -> bool) -> string -> bool
   168   val forall_string: (string -> bool) -> string -> bool
   169   val enclose: string -> string -> string -> string
   170   val unenclose: string -> string
   171   val quote: string -> string
   172   val space_implode: string -> string list -> string
   173   val commas: string list -> string
   174   val commas_quote: string list -> string
   175   val cat_lines: string list -> string
   176   val space_explode: string -> string -> string list
   177   val split_lines: string -> string list
   178   val prefix_lines: string -> string -> string
   179   val untabify: string list -> string list
   180   val suffix: string -> string -> string
   181   val unsuffix: string -> string -> string
   182   val unprefix: string -> string -> string
   183   val replicate_string: int -> string -> string
   184   val translate_string: (string -> string) -> string -> string
   185 
   186   (*lists as sets -- see also Pure/General/ord_list.ML*)
   187   val mem: ''a * ''a list -> bool
   188   val mem_int: int * int list -> bool
   189   val mem_string: string * string list -> bool
   190   val gen_mem: ('a * 'b -> bool) -> 'a * 'b list -> bool
   191   val ins: ''a * ''a list -> ''a list
   192   val ins_int: int * int list -> int list
   193   val ins_string: string * string list -> string list
   194   val gen_ins: ('a * 'a -> bool) -> 'a * 'a list -> 'a list
   195   val member: ('b * 'a -> bool) -> 'a list -> 'b -> bool
   196   val insert: ('a * 'a -> bool) -> 'a -> 'a list -> 'a list
   197   val remove: ('b * 'a -> bool) -> 'b -> 'a list -> 'a list
   198   val union: ''a list * ''a list -> ''a list
   199   val union_int: int list * int list -> int list
   200   val union_string: string list * string list -> string list
   201   val gen_union: ('a * 'a -> bool) -> 'a list * 'a list -> 'a list
   202   val gen_inter: ('a * 'b -> bool) -> 'a list * 'b list -> 'a list
   203   val inter: ''a list * ''a list -> ''a list
   204   val inter_int: int list * int list -> int list
   205   val inter_string: string list * string list -> string list
   206   val subset: ''a list * ''a list -> bool
   207   val subset_int: int list * int list -> bool
   208   val subset_string: string list * string list -> bool
   209   val eq_set: ''a list * ''a list -> bool
   210   val eq_set_string: string list * string list -> bool
   211   val gen_subset: ('a * 'b -> bool) -> 'a list * 'b list -> bool
   212   val \ : ''a list * ''a -> ''a list
   213   val \\ : ''a list * ''a list -> ''a list
   214   val gen_rem: ('a * 'b -> bool) -> 'a list * 'b -> 'a list
   215   val gen_rems: ('a * 'b -> bool) -> 'a list * 'b list -> 'a list
   216   val gen_distinct: ('a * 'a -> bool) -> 'a list -> 'a list
   217   val distinct: ''a list -> ''a list
   218   val findrep: ''a list -> ''a list
   219   val gen_duplicates: ('a * 'a -> bool) -> 'a list -> 'a list
   220   val duplicates: ''a list -> ''a list
   221   val has_duplicates: ('a * 'a -> bool) -> 'a list -> bool
   222 
   223   (*association lists -- see also Pure/General/alist.ML*)
   224   val assoc: (''a * 'b) list * ''a -> 'b option
   225   val assoc_int: (int * 'a) list * int -> 'a option
   226   val assoc_string: (string * 'a) list * string -> 'a option
   227   val assoc_string_int: ((string * int) * 'a) list * (string * int) -> 'a option
   228   val assocs: (''a * 'b list) list -> ''a -> 'b list
   229   val gen_assoc: ('a * 'b -> bool) -> ('b * 'c) list * 'a -> 'c option
   230   val overwrite: (''a * 'b) list * (''a * 'b) -> (''a * 'b) list
   231   val gen_overwrite: ('a * 'a -> bool) -> ('a * 'b) list * ('a * 'b) -> ('a * 'b) list
   232 
   233   (*lists as tables*)
   234   val gen_merge_lists: ('a * 'a -> bool) -> 'a list -> 'a list -> 'a list
   235   val gen_merge_lists': ('a * 'a -> bool) -> 'a list -> 'a list -> 'a list
   236   val merge_lists: ''a list -> ''a list -> ''a list
   237   val merge_lists': ''a list -> ''a list -> ''a list
   238   val merge_alists: (''a * 'b) list -> (''a * 'b) list -> (''a * 'b) list
   239   val merge_alists': (''a * 'b) list -> (''a * 'b) list -> (''a * 'b) list
   240 
   241   (*balanced trees*)
   242   exception Balance
   243   val fold_bal: ('a * 'a -> 'a) -> 'a list -> 'a
   244   val access_bal: ('a -> 'a) * ('a -> 'a) * 'a -> int -> int -> 'a
   245   val accesses_bal: ('a -> 'a) * ('a -> 'a) * 'a -> int -> 'a list
   246 
   247   (*orders*)
   248   val rev_order: order -> order
   249   val make_ord: ('a * 'a -> bool) -> 'a * 'a -> order
   250   val int_ord: int * int -> order
   251   val string_ord: string * string -> order
   252   val fast_string_ord: string * string -> order
   253   val option_ord: ('a * 'b -> order) -> 'a option * 'b option -> order
   254   val prod_ord: ('a * 'b -> order) -> ('c * 'd -> order) -> ('a * 'c) * ('b * 'd) -> order
   255   val dict_ord: ('a * 'b -> order) -> 'a list * 'b list -> order
   256   val list_ord: ('a * 'b -> order) -> 'a list * 'b list -> order
   257   val sort: ('a * 'a -> order) -> 'a list -> 'a list
   258   val sort_strings: string list -> string list
   259   val sort_wrt: ('a -> string) -> 'a list -> 'a list
   260   val unique_strings: string list -> string list
   261 
   262   (*random numbers*)
   263   exception RANDOM
   264   val random: unit -> real
   265   val random_range: int -> int -> int
   266   val one_of: 'a list -> 'a
   267   val frequency: (int * 'a) list -> 'a
   268 
   269   (*current directory*)
   270   val cd: string -> unit
   271   val pwd: unit -> string
   272 
   273   (*misc*)
   274   val make_keylist: ('a -> 'b) -> 'a list -> ('a * 'b) list
   275   val keyfilter: ('a * ''b) list -> ''b -> 'a list
   276   val partition_eq: ('a * 'a -> bool) -> 'a list -> 'a list list
   277   val partition_list: (int -> 'a -> bool) -> int -> int -> 'a list -> 'a list list
   278   val gensym: string -> string
   279   val scanwords: (string -> bool) -> string list -> string list
   280   type stamp
   281   val stamp: unit -> stamp
   282   type serial
   283   val serial: unit -> serial
   284   structure Object: sig type T end
   285 end;
   286 
   287 signature LIBRARY =
   288 sig
   289   include BASIC_LIBRARY
   290   val foldl: ('a * 'b -> 'a) -> 'a * 'b list -> 'a
   291   val foldr: ('a * 'b -> 'b) -> 'a list * 'b -> 'b
   292   val take: int * 'a list -> 'a list
   293   val drop: int * 'a list -> 'a list
   294   val nth_elem: int * 'a list -> 'a
   295   val last_elem: 'a list -> 'a
   296   val flat: 'a list list -> 'a list
   297   val seq: ('a -> unit) -> 'a list -> unit
   298   val partition: ('a -> bool) -> 'a list -> 'a list * 'a list
   299   val mapfilter: ('a -> 'b option) -> 'a list -> 'b list
   300 end;
   301 
   302 structure Library: LIBRARY =
   303 struct
   304 
   305 
   306 (** functions **)
   307 
   308 fun I x = x;
   309 fun K x = fn _ => x;
   310 fun curry f x y = f (x, y);
   311 fun uncurry f (x, y) = f x y;
   312 
   313 (*reverse application and structured results*)
   314 fun x |> f = f x;
   315 fun (x, y) |-> f = f x y;
   316 fun (x, y) |>> f = (f x, y);
   317 fun (x, y) ||> f = (x, f y);
   318 fun (x, y) |>>> f = let val (x', z) = f x in (x', (y, z)) end;
   319 fun (x, y) ||>> f = let val (z, y') = f y in ((x, z), y') end;
   320 
   321 (*reverse composition*)
   322 fun f #> g = g o f;
   323 fun f #-> g = uncurry g o f;
   324 
   325 (*conditional application*)
   326 fun b ? f = fn x => if b x then f x else x
   327 
   328 (*view results*)
   329 fun `f = fn x => (f x, x);
   330 fun tap f x = (f x; x);
   331 
   332 (*composition with multiple args*)
   333 fun (f oo g) x y = f (g x y);
   334 fun (f ooo g) x y z = f (g x y z);
   335 fun (f oooo g) x y z w = f (g x y z w);
   336 
   337 (*function exponentiation: f(...(f x)...) with n applications of f*)
   338 fun funpow n f x =
   339   let fun rep (0, x) = x
   340         | rep (n, x) = rep (n - 1, f x)
   341   in rep (n, x) end;
   342 
   343 (*application of (infix) operator to its left or right argument*)
   344 fun apl (x, f) y = f (x, y);
   345 fun apr (f, y) x = f (x, y);
   346 
   347 
   348 (** options **)
   349 
   350 (*invalidate former constructors to prevent accidental use as match-all patterns!*)
   351 datatype invalid = None of invalid;
   352 exception OPTION of invalid;
   353 
   354 val the = Option.valOf;
   355 fun these (SOME x) = x
   356   | these _ = []
   357 
   358 (*strict!*)
   359 fun if_none NONE y = y
   360   | if_none (SOME x) _ = x;
   361 
   362 fun is_some (SOME _) = true
   363   | is_some NONE = false;
   364 
   365 fun is_none (SOME _) = false
   366   | is_none NONE = true;
   367 
   368 
   369 (* exception handling *)
   370 
   371 exception ERROR;
   372 
   373 fun try f x = SOME (f x)
   374   handle Interrupt => raise Interrupt | ERROR => raise ERROR | _ => NONE;
   375 
   376 fun can f x = is_some (try f x);
   377 
   378 
   379 datatype 'a result =
   380   Result of 'a |
   381   Exn of exn;
   382 
   383 fun capture f x = Result (f x) handle e => Exn e;
   384 
   385 fun release (Result y) = y
   386   | release (Exn e) = raise e;
   387 
   388 fun get_result (Result x) = SOME x
   389   | get_result _ = NONE;
   390 
   391 fun get_exn (Exn exn) = SOME exn
   392   | get_exn _ = NONE;
   393 
   394 
   395 
   396 (** pairs **)
   397 
   398 fun pair x y = (x, y);
   399 fun rpair x y = (y, x);
   400 
   401 fun fst (x, y) = x;
   402 fun snd (x, y) = y;
   403 
   404 fun eq_fst ((x1, _), (x2, _)) = x1 = x2;
   405 fun eq_snd ((_, y1), (_, y2)) = y1 = y2;
   406 
   407 fun swap (x, y) = (y, x);
   408 
   409 (*apply function to components*)
   410 fun apfst f (x, y) = (f x, y);
   411 fun apsnd f (x, y) = (x, f y);
   412 fun pairself f (x, y) = (f x, f y);
   413 
   414 
   415 
   416 (** booleans **)
   417 
   418 (* equality *)
   419 
   420 fun equal x y = x = y;
   421 fun not_equal x y = x <> y;
   422 
   423 
   424 (* operators for combining predicates *)
   425 
   426 fun p orf q = fn x => p x orelse q x;
   427 fun p andf q = fn x => p x andalso q x;
   428 
   429 
   430 (* predicates on lists *)
   431 
   432 (*exists pred [x1, ..., xn] ===> pred x1 orelse ... orelse pred xn*)
   433 fun exists (pred: 'a -> bool) : 'a list -> bool =
   434   let fun boolf [] = false
   435         | boolf (x :: xs) = pred x orelse boolf xs
   436   in boolf end;
   437 
   438 (*forall pred [x1, ..., xn] ===> pred x1 andalso ... andalso pred xn*)
   439 fun forall (pred: 'a -> bool) : 'a list -> bool =
   440   let fun boolf [] = true
   441         | boolf (x :: xs) = pred x andalso boolf xs
   442   in boolf end;
   443 
   444 
   445 (* flags *)
   446 
   447 fun set flag = (flag := true; true);
   448 fun reset flag = (flag := false; false);
   449 fun toggle flag = (flag := not (! flag); ! flag);
   450 
   451 fun change r f = r := f (! r);
   452 
   453 (*temporarily set flag, handling exceptions*)
   454 fun setmp flag value f x =
   455   let
   456     val orig_value = ! flag;
   457     fun return y = (flag := orig_value; y);
   458   in
   459     flag := value;
   460     return (f x handle exn => (return (); raise exn))
   461   end;
   462 
   463 
   464 (* conditional execution *)
   465 
   466 fun conditional b f = if b then f () else ();
   467 
   468 
   469 
   470 (** lists **)
   471 
   472 exception UnequalLengths;
   473 
   474 fun cons x xs = x :: xs;
   475 fun single x = [x];
   476 
   477 fun append xs ys = xs @ ys;
   478 
   479 fun apply [] x = x
   480   | apply (f :: fs) x = apply fs (f x);
   481 
   482 
   483 (* fold *)
   484 
   485 fun fold f =
   486   let
   487     fun fold_aux [] y = y
   488       | fold_aux (x :: xs) y = fold_aux xs (f x y);
   489   in fold_aux end;
   490 
   491 fun fold_rev f =
   492   let
   493     fun fold_aux [] y = y
   494       | fold_aux (x :: xs) y = f x (fold_aux xs y);
   495   in fold_aux end;
   496 
   497 fun fold_map f =
   498   let
   499     fun fold_aux [] y = ([], y)
   500       | fold_aux (x :: xs) y =
   501           let
   502             val (x', y') = f x y;
   503             val (xs', y'') = fold_aux xs y';
   504           in (x' :: xs', y'') end;
   505   in fold_aux end;
   506 
   507 (*the following versions of fold are designed to fit nicely with infixes*)
   508 
   509 (*  (op @) (e, [x1, ..., xn])  ===>  ((e @ x1) @ x2) ... @ xn
   510     for operators that associate to the left (TAIL RECURSIVE)*)
   511 fun foldl (f: 'a * 'b -> 'a) : 'a * 'b list -> 'a =
   512   let fun itl (e, [])  = e
   513         | itl (e, a::l) = itl (f(e, a), l)
   514   in  itl end;
   515 
   516 (*  (op @) ([x1, ..., xn], e)  ===>   x1 @ (x2 ... @ (xn @ e))
   517     for operators that associate to the right (not tail recursive)*)
   518 fun foldr f (l, e) =
   519   let fun itr [] = e
   520         | itr (a::l) = f(a, itr l)
   521   in  itr l  end;
   522 
   523 (*  (op @) [x1, ..., xn]  ===>   x1 @ (x2 ... @ (x[n-1] @ xn))
   524     for n > 0, operators that associate to the right (not tail recursive)*)
   525 fun foldr1 f l =
   526   let fun itr [x] = x
   527         | itr (x::l) = f(x, itr l)
   528   in  itr l  end;
   529 
   530 fun foldln f xs e = fst (foldl (fn ((e,i), x) => (f (x,i) e, i+1)) ((e,1),xs));
   531 
   532 fun foldl_map f =
   533   let
   534     fun fold_aux (x, []) = (x, [])
   535       | fold_aux (x, y :: ys) =
   536           let
   537             val (x', y') = f (x, y);
   538             val (x'', ys') = fold_aux (x', ys);
   539           in (x'', y' :: ys') end;
   540   in fold_aux end;
   541 
   542 
   543 (* basic list functions *)
   544 
   545 (*take the first n elements from a list*)
   546 fun take (n, []) = []
   547   | take (n, x :: xs) =
   548       if n > 0 then x :: take (n - 1, xs) else [];
   549 
   550 (*drop the first n elements from a list*)
   551 fun drop (n, []) = []
   552   | drop (n, x :: xs) =
   553       if n > 0 then drop (n - 1, xs) else x :: xs;
   554 
   555 fun splitAt(n,[]) = ([],[])
   556   | splitAt(n,xs as x::ys) =
   557       if n>0 then let val (ps,qs) = splitAt(n-1,ys) in (x::ps,qs) end
   558       else ([],xs)
   559 
   560 fun dropwhile P [] = []
   561   | dropwhile P (ys as x::xs) = if P x then dropwhile P xs else ys;
   562 
   563 (*return nth element of a list, where 0 designates the first element;
   564   raise EXCEPTION if list too short*)
   565 fun nth_elem (i,xs) = List.nth(xs,i);
   566 
   567 fun map_nth_elem 0 f (x :: xs) = f x :: xs
   568   | map_nth_elem n f (x :: xs) = x :: map_nth_elem (n - 1) f xs
   569   | map_nth_elem _ _ [] = raise Subscript;
   570 
   571 (*last element of a list*)
   572 val last_elem = List.last;
   573 
   574 (*rear decomposition*)
   575 fun split_last [] = raise Empty
   576   | split_last [x] = ([], x)
   577   | split_last (x :: xs) = apfst (cons x) (split_last xs);
   578 
   579 (*update nth element*)
   580 fun nth_update x n_xs =
   581     (case splitAt n_xs of
   582       (_,[]) => raise Subscript
   583     | (prfx, _ :: sffx') => prfx @ (x :: sffx'))
   584 
   585 (*find the position of an element in a list*)
   586 fun find_index pred =
   587   let fun find _ [] = ~1
   588         | find n (x :: xs) = if pred x then n else find (n + 1) xs;
   589   in find 0 end;
   590 
   591 fun find_index_eq x = find_index (equal x);
   592 
   593 (*find first element satisfying predicate*)
   594 fun find_first _ [] = NONE
   595   | find_first pred (x :: xs) =
   596       if pred x then SOME x else find_first pred xs;
   597 
   598 (*get first element by lookup function*)
   599 fun get_first _ [] = NONE
   600   | get_first f (x :: xs) =
   601       (case f x of
   602         NONE => get_first f xs
   603       | some => some);
   604 
   605 (*flatten a list of lists to a list*)
   606 val flat = List.concat;
   607 
   608 fun unflat (xs :: xss) ys =
   609       let val (ps,qs) = splitAt(length xs,ys)
   610       in ps :: unflat xss qs end
   611   | unflat [] [] = []
   612   | unflat _ _ = raise UnequalLengths;
   613 
   614 (*like Lisp's MAPC -- seq proc [x1, ..., xn] evaluates
   615   (proc x1; ...; proc xn) for side effects*)
   616 val seq = List.app;
   617 
   618 (*separate s [x1, x2, ..., xn]  ===>  [x1, s, x2, s, ..., s, xn]*)
   619 fun separate s (x :: (xs as _ :: _)) = x :: s :: separate s xs
   620   | separate _ xs = xs;
   621 
   622 (*make the list [x, x, ..., x] of length n*)
   623 fun replicate n (x: 'a) : 'a list =
   624   let fun rep (0, xs) = xs
   625         | rep (n, xs) = rep (n - 1, x :: xs)
   626   in
   627     if n < 0 then raise Subscript
   628     else rep (n, [])
   629   end;
   630 
   631 fun translate_string f = String.translate (f o String.str);
   632 
   633 (*multiply [a, b, c, ...] * [xs, ys, zs, ...]*)
   634 fun multiply ([], _) = []
   635   | multiply (x :: xs, yss) = map (cons x) yss @ multiply (xs, yss);
   636 
   637 (*direct product*)
   638 fun product _ [] = []
   639   | product [] _ = []
   640   | product (x :: xs) ys = map (pair x) ys @ product xs ys;
   641 
   642 
   643 (* filter *)
   644 
   645 (*copy the list preserving elements that satisfy the predicate*)
   646 val filter = List.filter;
   647 fun filter_out f = filter (not o f);
   648 val mapfilter = List.mapPartial;
   649 
   650 
   651 (* lists of pairs *)
   652 
   653 exception UnequalLengths;
   654 
   655 fun map2 _ ([], []) = []
   656   | map2 f (x :: xs, y :: ys) = f (x, y) :: map2 f (xs, ys)
   657   | map2 _ _ = raise UnequalLengths;
   658 
   659 fun exists2 _ ([], []) = false
   660   | exists2 pred (x :: xs, y :: ys) = pred (x, y) orelse exists2 pred (xs, ys)
   661   | exists2 _ _ = raise UnequalLengths;
   662 
   663 fun forall2 _ ([], []) = true
   664   | forall2 pred (x :: xs, y :: ys) = pred (x, y) andalso forall2 pred (xs, ys)
   665   | forall2 _ _ = raise UnequalLengths;
   666 
   667 fun seq2 _ ([], []) = ()
   668   | seq2 f (x :: xs, y :: ys) = (f (x, y); seq2 f (xs, ys))
   669   | seq2 _ _ = raise UnequalLengths;
   670 
   671 (*combine two lists forming a list of pairs:
   672   [x1, ..., xn] ~~ [y1, ..., yn]  ===>  [(x1, y1), ..., (xn, yn)]*)
   673 fun [] ~~ [] = []
   674   | (x :: xs) ~~ (y :: ys) = (x, y) :: (xs ~~ ys)
   675   | _ ~~ _ = raise UnequalLengths;
   676 
   677 (*inverse of ~~; the old 'split':
   678   [(x1, y1), ..., (xn, yn)]  ===>  ([x1, ..., xn], [y1, ..., yn])*)
   679 val split_list = ListPair.unzip;
   680 
   681 fun equal_lists eq (xs, ys) = length xs = length ys andalso forall2 eq (xs, ys);
   682 
   683 
   684 (* prefixes, suffixes *)
   685 
   686 fun [] prefix _ = true
   687   | (x :: xs) prefix (y :: ys) = x = y andalso (xs prefix ys)
   688   | _ prefix _ = false;
   689 
   690 (* [x1, ..., xi, ..., xn]  --->  ([x1, ..., x(i-1)], [xi, ..., xn])
   691    where xi is the first element that does not satisfy the predicate*)
   692 fun take_prefix (pred : 'a -> bool)  (xs: 'a list) : 'a list * 'a list =
   693   let fun take (rxs, []) = (rev rxs, [])
   694         | take (rxs, x :: xs) =
   695             if  pred x  then  take(x :: rxs, xs)  else  (rev rxs, x :: xs)
   696   in  take([], xs)  end;
   697 
   698 (* [x1, ..., xi, ..., xn]  --->  ([x1, ..., xi], [x(i+1), ..., xn])
   699    where xi is the last element that does not satisfy the predicate*)
   700 fun take_suffix _ [] = ([], [])
   701   | take_suffix pred (x :: xs) =
   702       (case take_suffix pred xs of
   703         ([], sffx) => if pred x then ([], x :: sffx) else ([x], sffx)
   704       | (prfx, sffx) => (x :: prfx, sffx));
   705 
   706 fun prefixes1 [] = []
   707   | prefixes1 (x :: xs) = map (cons x) ([] :: prefixes1 xs);
   708 
   709 fun suffixes1 xs = map rev (prefixes1 (rev xs));
   710 
   711 
   712 
   713 (** integers **)
   714 
   715 fun gcd (x, y) =
   716   let fun gxd x y : IntInf.int =
   717     if y = 0 then x else gxd y (x mod y)
   718   in if x < y then gxd y x else gxd x y end;
   719 
   720 fun lcm (x, y) = (x * y) div gcd (x, y);
   721 
   722 fun inc i = (i := ! i + 1; ! i);
   723 fun dec i = (i := ! i - 1; ! i);
   724 
   725 
   726 (* lists of integers *)
   727 
   728 (*make the list [from, from + 1, ..., to]*)
   729 fun (from upto to) =
   730   if from > to then [] else from :: ((from + 1) upto to);
   731 
   732 (*make the list [from, from - 1, ..., to]*)
   733 fun (from downto to) =
   734   if from < to then [] else from :: ((from - 1) downto to);
   735 
   736 (*predicate: downto0 (is, n) <=> is = [n, n - 1, ..., 0]*)
   737 fun downto0 (i :: is, n) = i = n andalso downto0 (is, n - 1)
   738   | downto0 ([], n) = n = ~1;
   739 
   740 
   741 (* convert integers to strings *)
   742 
   743 (*expand the number in the given base;
   744   example: radixpand (2, 8) gives [1, 0, 0, 0]*)
   745 fun radixpand (base, num) : int list =
   746   let
   747     fun radix (n, tail) =
   748       if n < base then n :: tail
   749       else radix (n div base, (n mod base) :: tail)
   750   in radix (num, []) end;
   751 
   752 (*expands a number into a string of characters starting from "zerochar";
   753   example: radixstring (2, "0", 8) gives "1000"*)
   754 fun radixstring (base, zerochar, num) =
   755   let val offset = ord zerochar;
   756       fun chrof n = chr (offset + n)
   757   in implode (map chrof (radixpand (base, num))) end;
   758 
   759 
   760 val string_of_int = Int.toString;
   761 
   762 fun string_of_indexname (a,0) = a
   763   | string_of_indexname (a,i) = a ^ "_" ^ Int.toString i;
   764 
   765 
   766 (* read integers *)
   767 
   768 fun read_radixint (radix: int, cs) : int * string list =
   769   let val zero = ord"0"
   770       val limit = zero+radix
   771       fun scan (num,[]) = (num,[])
   772         | scan (num, c::cs) =
   773               if  zero <= ord c  andalso  ord c < limit
   774               then scan(radix*num + ord c - zero, cs)
   775               else (num, c::cs)
   776   in  scan(0,cs)  end;
   777 
   778 fun read_int cs = read_radixint (10, cs);
   779 
   780 fun oct_char s = chr (#1 (read_radixint (8, explode s)));
   781 
   782 
   783 
   784 (** strings **)
   785 
   786 (* functions tuned for strings, avoiding explode *)
   787 
   788 fun nth_elem_string (i, str) =
   789   (case try String.substring (str, i, 1) of
   790     SOME s => s
   791   | NONE => raise Subscript);
   792 
   793 fun fold_string f str x0 =
   794   let
   795     val n = size str;
   796     fun iter (x, i) =
   797       if i < n then iter (f (String.substring (str, i, 1)) x, i + 1) else x;
   798   in iter (x0, 0) end;
   799 
   800 fun exists_string pred str =
   801   let
   802     val n = size str;
   803     fun ex i = i < n andalso (pred (String.substring (str, i, 1)) orelse ex (i + 1));
   804   in ex 0 end;
   805 
   806 fun forall_string pred = not o exists_string (not o pred);
   807 
   808 (*enclose in brackets*)
   809 fun enclose lpar rpar str = lpar ^ str ^ rpar;
   810 fun unenclose str = String.substring (str, 1, size str - 2);
   811 
   812 (*simple quoting (does not escape special chars)*)
   813 val quote = enclose "\"" "\"";
   814 
   815 (*space_implode "..." (explode "hello") = "h...e...l...l...o"*)
   816 fun space_implode a bs = implode (separate a bs);
   817 
   818 val commas = space_implode ", ";
   819 val commas_quote = commas o map quote;
   820 
   821 (*concatenate messages, one per line, into a string*)
   822 val cat_lines = space_implode "\n";
   823 
   824 (*space_explode "." "h.e..l.lo" = ["h", "e", "", "l", "lo"]*)
   825 fun space_explode _ "" = []
   826   | space_explode sep str =
   827       let
   828         fun expl chs =
   829           (case take_prefix (not_equal sep) chs of
   830             (cs, []) => [implode cs]
   831           | (cs, _ :: cs') => implode cs :: expl cs');
   832       in expl (explode str) end;
   833 
   834 val split_lines = space_explode "\n";
   835 
   836 fun prefix_lines "" txt = txt
   837   | prefix_lines prfx txt = txt |> split_lines |> map (fn s => prfx ^ s) |> cat_lines;
   838 
   839 fun untabify chs =
   840   let
   841     val tab_width = 8;
   842 
   843     fun untab (_, "\n") = (0, ["\n"])
   844       | untab (pos, "\t") =
   845           let val d = tab_width - (pos mod tab_width) in (pos + d, replicate d " ") end
   846       | untab (pos, c) = (pos + 1, [c]);
   847   in
   848     if not (exists (equal "\t") chs) then chs
   849     else flat (#2 (foldl_map untab (0, chs)))
   850   end;
   851 
   852 fun suffix sffx s = s ^ sffx;
   853 
   854 fun unsuffix sffx s =
   855   if String.isSuffix sffx s then String.substring (s, 0, size s - size sffx)
   856   else raise Fail "unsuffix";
   857 
   858 fun unprefix prfx s =
   859   if String.isPrefix prfx s then String.substring (s, size prfx, size s - size prfx)
   860   else raise Fail "unprefix";
   861 
   862 fun replicate_string 0 _ = ""
   863   | replicate_string 1 a = a
   864   | replicate_string k a =
   865       if k mod 2 = 0 then replicate_string (k div 2) (a ^ a)
   866       else replicate_string (k div 2) (a ^ a) ^ a;
   867 
   868 
   869 
   870 (** lists as sets -- see also Pure/General/ord_list.ML **)
   871 
   872 (*membership in a list*)
   873 fun x mem [] = false
   874   | x mem (y :: ys) = x = y orelse x mem ys;
   875 
   876 (*membership in a list, optimized version for ints*)
   877 fun (x:int) mem_int [] = false
   878   | x mem_int (y :: ys) = x = y orelse x mem_int ys;
   879 
   880 (*membership in a list, optimized version for strings*)
   881 fun (x:string) mem_string [] = false
   882   | x mem_string (y :: ys) = x = y orelse x mem_string ys;
   883 
   884 (*generalized membership test*)
   885 fun gen_mem eq (x, []) = false
   886   | gen_mem eq (x, y :: ys) = eq (x, y) orelse gen_mem eq (x, ys);
   887 
   888 (*member, insert, and remove -- with canonical argument order*)
   889 fun member eq xs x = gen_mem eq (x, xs);
   890 fun insert eq x xs = if gen_mem eq (x, xs) then xs else x :: xs;
   891 fun remove eq x xs = if gen_mem eq (x, xs) then filter_out (fn y => eq (x, y)) xs else xs;
   892 
   893 (*insertion into list if not already there*)
   894 fun (x ins xs) = if x mem xs then xs else x :: xs;
   895 
   896 (*insertion into list, optimized version for ints*)
   897 fun (x ins_int xs) = if x mem_int xs then xs else x :: xs;
   898 
   899 (*insertion into list, optimized version for strings*)
   900 fun (x ins_string xs) = if x mem_string xs then xs else x :: xs;
   901 
   902 (*generalized insertion*)
   903 fun gen_ins eq (x, xs) = insert eq x xs;
   904 
   905 (*union of sets represented as lists: no repetitions*)
   906 fun xs union [] = xs
   907   | [] union ys = ys
   908   | (x :: xs) union ys = xs union (x ins ys);
   909 
   910 (*union of sets, optimized version for ints*)
   911 fun (xs:int list) union_int [] = xs
   912   | [] union_int ys = ys
   913   | (x :: xs) union_int ys = xs union_int (x ins_int ys);
   914 
   915 (*union of sets, optimized version for strings*)
   916 fun (xs:string list) union_string [] = xs
   917   | [] union_string ys = ys
   918   | (x :: xs) union_string ys = xs union_string (x ins_string ys);
   919 
   920 (*generalized union*)
   921 fun gen_union eq (xs, []) = xs
   922   | gen_union eq ([], ys) = ys
   923   | gen_union eq (x :: xs, ys) = gen_union eq (xs, gen_ins eq (x, ys));
   924 
   925 
   926 (*intersection*)
   927 fun [] inter ys = []
   928   | (x :: xs) inter ys =
   929       if x mem ys then x :: (xs inter ys) else xs inter ys;
   930 
   931 (*intersection, optimized version for ints*)
   932 fun ([]:int list) inter_int ys = []
   933   | (x :: xs) inter_int ys =
   934       if x mem_int ys then x :: (xs inter_int ys) else xs inter_int ys;
   935 
   936 (*intersection, optimized version for strings *)
   937 fun ([]:string list) inter_string ys = []
   938   | (x :: xs) inter_string ys =
   939       if x mem_string ys then x :: (xs inter_string ys) else xs inter_string ys;
   940 
   941 (*generalized intersection*)
   942 fun gen_inter eq ([], ys) = []
   943   | gen_inter eq (x::xs, ys) =
   944       if gen_mem eq (x,ys) then x :: gen_inter eq (xs, ys)
   945                            else      gen_inter eq (xs, ys);
   946 
   947 
   948 (*subset*)
   949 fun [] subset ys = true
   950   | (x :: xs) subset ys = x mem ys andalso xs subset ys;
   951 
   952 (*subset, optimized version for ints*)
   953 fun ([]: int list) subset_int ys = true
   954   | (x :: xs) subset_int ys = x mem_int ys andalso xs subset_int ys;
   955 
   956 (*subset, optimized version for strings*)
   957 fun ([]: string list) subset_string ys = true
   958   | (x :: xs) subset_string ys = x mem_string ys andalso xs subset_string ys;
   959 
   960 (*set equality*)
   961 fun eq_set (xs, ys) =
   962   xs = ys orelse (xs subset ys andalso ys subset xs);
   963 
   964 (*set equality for strings*)
   965 fun eq_set_string ((xs: string list), ys) =
   966   xs = ys orelse (xs subset_string ys andalso ys subset_string xs);
   967 
   968 fun gen_subset eq (xs, ys) = forall (fn x => gen_mem eq (x, ys)) xs;
   969 
   970 
   971 (*removing an element from a list WITHOUT duplicates*)
   972 fun (y :: ys) \ x = if x = y then ys else y :: (ys \ x)
   973   | [] \ x = [];
   974 
   975 fun ys \\ xs = foldl (op \) (ys,xs);
   976 
   977 (*removing an element from a list -- possibly WITH duplicates*)
   978 fun gen_rem eq (xs, y) = filter_out (fn x => eq (x, y)) xs;
   979 fun gen_rems eq (xs, ys) = filter_out (fn x => gen_mem eq (x, ys)) xs;
   980 
   981 (*makes a list of the distinct members of the input; preserves order, takes
   982   first of equal elements*)
   983 fun gen_distinct eq lst =
   984   let
   985     val memb = gen_mem eq;
   986 
   987     fun dist (rev_seen, []) = rev rev_seen
   988       | dist (rev_seen, x :: xs) =
   989           if memb (x, rev_seen) then dist (rev_seen, xs)
   990           else dist (x :: rev_seen, xs);
   991   in
   992     dist ([], lst)
   993   end;
   994 
   995 fun distinct l = gen_distinct (op =) l;
   996 
   997 (*returns the tail beginning with the first repeated element, or []*)
   998 fun findrep [] = []
   999   | findrep (x :: xs) = if x mem xs then x :: xs else findrep xs;
  1000 
  1001 
  1002 (*returns a list containing all repeated elements exactly once; preserves
  1003   order, takes first of equal elements*)
  1004 fun gen_duplicates eq lst =
  1005   let
  1006     val memb = gen_mem eq;
  1007 
  1008     fun dups (rev_dups, []) = rev rev_dups
  1009       | dups (rev_dups, x :: xs) =
  1010           if memb (x, rev_dups) orelse not (memb (x, xs)) then
  1011             dups (rev_dups, xs)
  1012           else dups (x :: rev_dups, xs);
  1013   in
  1014     dups ([], lst)
  1015   end;
  1016 
  1017 fun duplicates l = gen_duplicates (op =) l;
  1018 
  1019 fun has_duplicates eq =
  1020   let
  1021     fun dups [] = false
  1022       | dups (x :: xs) = member eq xs x orelse dups xs;
  1023   in dups end;
  1024 
  1025 
  1026 
  1027 (** association lists **)
  1028 
  1029 (*association list lookup*)
  1030 fun assoc ([], key) = NONE
  1031   | assoc ((keyi, xi) :: pairs, key) =
  1032       if key = keyi then SOME xi else assoc (pairs, key);
  1033 
  1034 (*association list lookup, optimized version for ints*)
  1035 fun assoc_int ([], (key:int)) = NONE
  1036   | assoc_int ((keyi, xi) :: pairs, key) =
  1037       if key = keyi then SOME xi else assoc_int (pairs, key);
  1038 
  1039 (*association list lookup, optimized version for strings*)
  1040 fun assoc_string ([], (key:string)) = NONE
  1041   | assoc_string ((keyi, xi) :: pairs, key) =
  1042       if key = keyi then SOME xi else assoc_string (pairs, key);
  1043 
  1044 (*association list lookup, optimized version for string*ints*)
  1045 fun assoc_string_int ([], (key:string*int)) = NONE
  1046   | assoc_string_int ((keyi, xi) :: pairs, key) =
  1047       if key = keyi then SOME xi else assoc_string_int (pairs, key);
  1048 
  1049 fun assocs z = curry (these o assoc) z
  1050 
  1051 (*generalized association list lookup*)
  1052 fun gen_assoc eq ([], key) = NONE
  1053   | gen_assoc eq ((keyi, xi) :: pairs, key) =
  1054       if eq (key, keyi) then SOME xi else gen_assoc eq (pairs, key);
  1055 
  1056 (*association list update*)
  1057 fun overwrite (al, p as (key, _)) =
  1058   let fun over ((q as (keyi, _)) :: pairs) =
  1059             if keyi = key then p :: pairs else q :: (over pairs)
  1060         | over [] = [p]
  1061   in over al end;
  1062 
  1063 fun gen_overwrite eq (al, p as (key, _)) =
  1064   let fun over ((q as (keyi, _)) :: pairs) =
  1065             if eq (keyi, key) then p :: pairs else q :: (over pairs)
  1066         | over [] = [p]
  1067   in over al end;
  1068 
  1069 
  1070 (* lists as tables *)
  1071 
  1072 fun gen_merge_lists _ xs [] = xs
  1073   | gen_merge_lists _ [] ys = ys
  1074   | gen_merge_lists eq xs ys = xs @ gen_rems eq (ys, xs);
  1075 
  1076 fun gen_merge_lists' _ xs [] = xs
  1077   | gen_merge_lists' _ [] ys = ys
  1078   | gen_merge_lists' eq xs ys = gen_rems eq (ys, xs) @ xs;
  1079 
  1080 fun merge_lists xs ys = gen_merge_lists (op =) xs ys;
  1081 fun merge_lists' xs ys = gen_merge_lists' (op =) xs ys;
  1082 fun merge_alists al = gen_merge_lists eq_fst al;
  1083 fun merge_alists' al = gen_merge_lists' eq_fst al;
  1084 
  1085 
  1086 
  1087 (** balanced trees **)
  1088 
  1089 exception Balance;      (*indicates non-positive argument to balancing fun*)
  1090 
  1091 (*balanced folding; avoids deep nesting*)
  1092 fun fold_bal f [x] = x
  1093   | fold_bal f [] = raise Balance
  1094   | fold_bal f xs =
  1095       let val (ps,qs) = splitAt(length xs div 2, xs)
  1096       in  f (fold_bal f ps, fold_bal f qs)  end;
  1097 
  1098 (*construct something of the form f(...g(...(x)...)) for balanced access*)
  1099 fun access_bal (f, g, x) n i =
  1100   let fun acc n i =     (*1<=i<=n*)
  1101           if n=1 then x else
  1102           let val n2 = n div 2
  1103           in  if i<=n2 then f (acc n2 i)
  1104                        else g (acc (n-n2) (i-n2))
  1105           end
  1106   in  if 1<=i andalso i<=n then acc n i else raise Balance  end;
  1107 
  1108 (*construct ALL such accesses; could try harder to share recursive calls!*)
  1109 fun accesses_bal (f, g, x) n =
  1110   let fun acc n =
  1111           if n=1 then [x] else
  1112           let val n2 = n div 2
  1113               val acc2 = acc n2
  1114           in  if n-n2=n2 then map f acc2 @ map g acc2
  1115                          else map f acc2 @ map g (acc (n-n2)) end
  1116   in  if 1<=n then acc n else raise Balance  end;
  1117 
  1118 
  1119 
  1120 (** orders **)
  1121 
  1122 fun rev_order LESS = GREATER
  1123   | rev_order EQUAL = EQUAL
  1124   | rev_order GREATER = LESS;
  1125 
  1126 (*assume rel is a linear strict order*)
  1127 fun make_ord rel (x, y) =
  1128   if rel (x, y) then LESS
  1129   else if rel (y, x) then GREATER
  1130   else EQUAL;
  1131 
  1132 val int_ord = Int.compare;
  1133 val string_ord = String.compare;
  1134 
  1135 fun fast_string_ord (s1, s2) =
  1136   (case int_ord (size s1, size s2) of EQUAL => string_ord (s1, s2) | ord => ord);
  1137 
  1138 fun option_ord ord (SOME x, SOME y) = ord (x, y)
  1139   | option_ord _ (NONE, NONE) = EQUAL
  1140   | option_ord _ (NONE, SOME _) = LESS
  1141   | option_ord _ (SOME _, NONE) = GREATER;
  1142 
  1143 (*lexicographic product*)
  1144 fun prod_ord a_ord b_ord ((x, y), (x', y')) =
  1145   (case a_ord (x, x') of EQUAL => b_ord (y, y') | ord => ord);
  1146 
  1147 (*dictionary order -- in general NOT well-founded!*)
  1148 fun dict_ord elem_ord (x :: xs, y :: ys) =
  1149       (case elem_ord (x, y) of EQUAL => dict_ord elem_ord (xs, ys) | ord => ord)
  1150   | dict_ord _ ([], []) = EQUAL
  1151   | dict_ord _ ([], _ :: _) = LESS
  1152   | dict_ord _ (_ :: _, []) = GREATER;
  1153 
  1154 (*lexicographic product of lists*)
  1155 fun list_ord elem_ord (xs, ys) =
  1156   (case int_ord (length xs, length ys) of EQUAL => dict_ord elem_ord (xs, ys) | ord => ord);
  1157 
  1158 
  1159 (* sorting *)
  1160 
  1161 (*quicksort (stable, i.e. does not reorder equal elements)*)
  1162 fun sort ord =
  1163   let
  1164     fun qsort [] = []
  1165       | qsort (xs as [_]) = xs
  1166       | qsort (xs as [x, y]) = if ord (x, y) = GREATER then [y, x] else xs
  1167       | qsort xs =
  1168           let val (lts, eqs, gts) = part (nth_elem (length xs div 2, xs)) xs
  1169           in qsort lts @ eqs @ qsort gts end
  1170     and part _ [] = ([], [], [])
  1171       | part pivot (x :: xs) = add (ord (x, pivot)) x (part pivot xs)
  1172     and add LESS x (lts, eqs, gts) = (x :: lts, eqs, gts)
  1173       | add EQUAL x (lts, eqs, gts) = (lts, x :: eqs, gts)
  1174       | add GREATER x (lts, eqs, gts) = (lts, eqs, x :: gts);
  1175   in qsort end;
  1176 
  1177 (*sort strings*)
  1178 val sort_strings = sort string_ord;
  1179 fun sort_wrt sel xs = sort (string_ord o pairself sel) xs;
  1180 
  1181 fun unique_strings ([]: string list) = []
  1182   | unique_strings [x] = [x]
  1183   | unique_strings (x :: y :: ys) =
  1184       if x = y then unique_strings (y :: ys)
  1185       else x :: unique_strings (y :: ys);
  1186 
  1187 
  1188 (** random numbers **)
  1189 
  1190 exception RANDOM;
  1191 
  1192 fun rmod x y = x - y * Real.realFloor (x / y);
  1193 
  1194 local
  1195   val a = 16807.0;
  1196   val m = 2147483647.0;
  1197   val random_seed = ref 1.0;
  1198 in
  1199 
  1200 fun random () =
  1201   let val r = rmod (a * !random_seed) m
  1202   in (random_seed := r; r) end;
  1203 
  1204 end;
  1205 
  1206 fun random_range l h =
  1207   if h < l orelse l < 0 then raise RANDOM
  1208   else l + Real.floor (rmod (random ()) (real (h - l + 1)));
  1209 
  1210 fun one_of xs = nth_elem (random_range 0 (length xs - 1), xs);
  1211 
  1212 fun frequency xs =
  1213   let
  1214     val sum = foldl op + (0, map fst xs);
  1215     fun pick n ((k, x) :: xs) =
  1216       if n <= k then x else pick (n - k) xs
  1217   in pick (random_range 1 sum) xs end;
  1218 
  1219 
  1220 (** current directory **)
  1221 
  1222 val cd = OS.FileSys.chDir;
  1223 val pwd = OS.FileSys.getDir;
  1224 
  1225 
  1226 
  1227 (** rational numbers **)
  1228 (* Keep them normalized! *)
  1229 
  1230 datatype rat = Rat of bool * IntInf.int * IntInf.int
  1231 
  1232 exception RAT of string;
  1233 
  1234 fun rep_rat(Rat(a,p,q)) = (if a then p else ~p,q)
  1235 
  1236 fun ratnorm(a,p,q) = if p=0 then Rat(a,0,1) else
  1237   let val absp = abs p
  1238       val m = gcd(absp,q)
  1239   in Rat(a = (0 <= p), absp div m, q div m) end;
  1240 
  1241 fun ratcommon(p,q,r,s) =
  1242   let val den = lcm(q,s)
  1243   in (p*(den div q), r*(den div s), den) end
  1244 
  1245 fun ratge0(Rat(a,p,q)) = a;
  1246 fun ratgt0(Rat(a,p,q)) = a andalso p > 0;
  1247 
  1248 fun ratle(Rat(a,p,q),Rat(b,r,s)) =
  1249   not a andalso b orelse
  1250   a = b andalso
  1251     let val (p,r,_) = ratcommon(p,q,r,s)
  1252     in if a then p <= r else r <= p end
  1253 
  1254 fun ratlt(Rat(a,p,q),Rat(b,r,s)) =
  1255   not a andalso b orelse
  1256   a = b andalso
  1257     let val (p,r,_) = ratcommon(p,q,r,s)
  1258     in if a then p < r else r < p end
  1259 
  1260 fun ratadd(Rat(a,p,q),Rat(b,r,s)) =
  1261   let val (p,r,den) = ratcommon(p,q,r,s)
  1262       val num = (if a then p else ~p) + (if b then r else ~r)
  1263   in ratnorm(true,num,den) end;
  1264 
  1265 fun ratmul(Rat(a,p,q),Rat(b,r,s)) = ratnorm(a=b,p*r,q*s)
  1266 
  1267 fun ratinv(Rat(a,p,q)) = if p=0 then raise RAT "ratinv" else Rat(a,q,p)
  1268 
  1269 fun int_ratdiv(p,q) =
  1270   if q=0 then raise RAT "int_ratdiv" else ratnorm(0<=q, p, abs q)
  1271 
  1272 fun ratneg(Rat(b,p,q)) = Rat(not b,p,q);
  1273 
  1274 fun rat_of_intinf i = if i < 0 then Rat(false,abs i,1) else Rat(true,i,1);
  1275 
  1276 fun rat_of_int i = rat_of_intinf (IntInf.fromInt i);
  1277 
  1278 val rat0 = rat_of_int 0; 
  1279 
  1280 (** misc **)
  1281 
  1282 (*use the keyfun to make a list of (x, key) pairs*)
  1283 fun make_keylist (keyfun: 'a->'b) : 'a list -> ('a * 'b) list =
  1284   let fun keypair x = (x, keyfun x)
  1285   in map keypair end;
  1286 
  1287 (*given a list of (x, key) pairs and a searchkey
  1288   return the list of xs from each pair whose key equals searchkey*)
  1289 fun keyfilter [] searchkey = []
  1290   | keyfilter ((x, key) :: pairs) searchkey =
  1291       if key = searchkey then x :: keyfilter pairs searchkey
  1292       else keyfilter pairs searchkey;
  1293 
  1294 
  1295 (*Partition list into elements that satisfy predicate and those that don't.
  1296   Preserves order of elements in both lists.*)
  1297 val partition = List.partition;
  1298 
  1299 fun partition_eq (eq:'a * 'a -> bool) =
  1300   let
  1301     fun part [] = []
  1302       | part (x :: ys) =
  1303           let val (xs, xs') = partition (fn y => eq (x, y)) ys
  1304           in (x::xs)::(part xs') end
  1305   in part end;
  1306 
  1307 
  1308 (*Partition a list into buckets  [ bi, b(i+1), ..., bj ]
  1309    putting x in bk if p(k)(x) holds.  Preserve order of elements if possible.*)
  1310 fun partition_list p i j =
  1311   let fun part k xs =
  1312             if k>j then
  1313               (case xs of [] => []
  1314                          | _ => raise Fail "partition_list")
  1315             else
  1316             let val (ns, rest) = partition (p k) xs;
  1317             in  ns :: part(k+1)rest  end
  1318   in  part i end;
  1319 
  1320 
  1321 (* generating identifiers *)
  1322 
  1323 (** Freshly generated identifiers; supplied prefix MUST start with a letter **)
  1324 local
  1325 (*Maps 0-63 to A-Z, a-z, 0-9 or _ or ' for generating random identifiers*)
  1326 fun char i =      if i<26 then chr (ord "A" + i)
  1327              else if i<52 then chr (ord "a" + i - 26)
  1328              else if i<62 then chr (ord"0" + i - 52)
  1329              else if i=62 then "_"
  1330              else  (*i=63*)    "'";
  1331 
  1332 val charVec = Vector.tabulate (64, char);
  1333 
  1334 fun newid n =
  1335   let
  1336   in  implode (map (fn i => Vector.sub(charVec,i)) (radixpand (64,n)))  end;
  1337 
  1338 val seedr = ref 0;
  1339 
  1340 in
  1341 
  1342 fun gensym pre = pre ^ (#1(newid (!seedr), inc seedr));
  1343 
  1344 end;
  1345 
  1346 
  1347 (* lexical scanning *)
  1348 
  1349 (*scan a list of characters into "words" composed of "letters" (recognized by
  1350   is_let) and separated by any number of non-"letters"*)
  1351 fun scanwords is_let cs =
  1352   let fun scan1 [] = []
  1353         | scan1 cs =
  1354             let val (lets, rest) = take_prefix is_let cs
  1355             in implode lets :: scanwords is_let rest end;
  1356   in scan1 (#2 (take_prefix (not o is_let) cs)) end;
  1357 
  1358 
  1359 (* stamps and serial numbers *)
  1360 
  1361 type stamp = unit ref;
  1362 val stamp: unit -> stamp = ref;
  1363 
  1364 type serial = int;
  1365 local val count = ref 0
  1366 in fun serial () = inc count end;
  1367 
  1368 
  1369 (* generic objects *)
  1370 
  1371 (*note that the builtin exception datatype may be extended by new
  1372   constructors at any time*)
  1373 structure Object = struct type T = exn end;
  1374 
  1375 end;
  1376 
  1377 structure BasicLibrary: BASIC_LIBRARY = Library;
  1378 open BasicLibrary;