(*  Title:      Pure/library.ML

     ID:         $Id$

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     Copyright   1992  University of Cambridge

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

 strings, lists as sets, association lists, generic tables, balanced trees,

 orders, input / output, timing, filenames, misc functions.

 *)

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

       mem mem_int mem_string union union_int union_string  

       inter inter_int inter_string subset subset_int subset_string subdir_of;

 structure Library =

 struct

 (** functions **)

 (*handy combinators*)

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

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

 fun I x = x;

 fun K x y = x;

 (*reverse apply*)

 fun (x |> f) = f x;

 (*combine two functions forming the union of their domains*)

 fun (f orelf g) = fn x => f x handle Match => g x;

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

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

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

 (*functional for pairs*)

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

 (*function exponentiation: f(...(f x)...) with n applications of f*)

 fun funpow n f x =

   let fun rep (0, x) = x

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

   in rep (n, x) end;

 (** stamps **)

 type stamp = unit ref;

 val stamp: unit -> stamp = ref;

 (** options **)

 datatype 'a option = None | Some of 'a;

 exception OPTION of string;

 fun the (Some x) = x

   | the None = raise OPTION "the";

 fun if_none None y = y

   | if_none (Some x) _ = x;

 fun is_some (Some _) = true

   | is_some None = false;

 fun is_none (Some _) = false

   | is_none None = true;

 fun apsome f (Some x) = Some (f x)

   | apsome _ None = None;

 (** pairs **)

 fun pair x y = (x, y);

 fun rpair x y = (y, x);

 fun fst (x, y) = x;

 fun snd (x, y) = y;

 fun eq_fst ((x1, _), (x2, _)) = x1 = x2;

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

 fun swap (x, y) = (y, x);

 (*apply the function to a component of a pair*)

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

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

 (** booleans **)

 (* equality *)

 fun equal x y = x = y;

 fun not_equal x y = x <> y;

 (* operators for combining predicates *)

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

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

 fun notf p x = not (p x);

 (* predicates on lists *)

 fun orl [] = false

   | orl (x :: xs) = x orelse orl xs;

 fun andl [] = true

   | andl (x :: xs) = x andalso andl xs;

 (*Several object-logics declare theories named List or Option, hiding the

   eponymous basis library structures.*)

 structure List_ = List

 and       Option_ = Option;

 (*exists pred [x1, ..., xn] ===> pred x1 orelse ... orelse pred xn*)

 fun exists (pred: 'a -> bool) : 'a list -> bool =

   let fun boolf [] = false

         | boolf (x :: xs) = pred x orelse boolf xs

   in boolf end;

 (*forall pred [x1, ..., xn] ===> pred x1 andalso ... andalso pred xn*)

 fun forall (pred: 'a -> bool) : 'a list -> bool =

   let fun boolf [] = true

         | boolf (x :: xs) = pred x andalso boolf xs

   in boolf end;

 (* flags *)

 fun set flag = (flag := true; true);

 fun reset flag = (flag := false; false);

 fun toggle flag = (flag := not (! flag); ! flag);

 fun setmp flag value f x =

   let

     val orig_value = ! flag;

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

   in

     flag := value;

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

   end;

 (** lists **)

 exception LIST of string;

 fun null [] = true

   | null (_ :: _) = false;

 fun hd [] = raise LIST "hd"

   | hd (x :: _) = x;

 fun tl [] = raise LIST "tl"

   | tl (_ :: xs) = xs;

 fun cons x xs = x :: xs;

 (* fold *)

 (*the following versions of fold are designed to fit nicely with infixes*)

 (*  (op @) (e, [x1, ..., xn])  ===>  ((e @ x1) @ x2) ... @ xn

     for operators that associate to the left (TAIL RECURSIVE)*)

 fun foldl (f: 'a * 'b -> 'a) : 'a * 'b list -> 'a =

   let fun itl (e, [])  = e

         | itl (e, a::l) = itl (f(e, a), l)

   in  itl end;

 (*  (op @) ([x1, ..., xn], e)  ===>   x1 @ (x2 ... @ (xn @ e))

     for operators that associate to the right (not tail recursive)*)

 fun foldr f (l, e) =

   let fun itr [] = e

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

   in  itr l  end;

 (*  (op @) [x1, ..., xn]  ===>   x1 @ (x2 ... @ (x[n-1] @ xn))

     for n > 0, operators that associate to the right (not tail recursive)*)

 fun foldr1 f l =

   let fun itr [x] = x                       (* FIXME [] case: elim warn (?) *)

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

   in  itr l  end;

 (* basic list functions *)

 (*length of a list, should unquestionably be a standard function*)

 local fun length1 (n, [])  = n   (*TAIL RECURSIVE*)

         | length1 (n, x :: xs) = length1 (n + 1, xs)

 in  fun length l = length1 (0, l) end;

 (*take the first n elements from a list*)

 fun take (n, []) = []

   | take (n, x :: xs) =

       if n > 0 then x :: take (n - 1, xs) else [];

 (*drop the first n elements from a list*)

 fun drop (n, []) = []

   | drop (n, x :: xs) =

       if n > 0 then drop (n - 1, xs) else x :: xs;

 (*return nth element of a list, where 0 designates the first element;

   raise EXCEPTION if list too short*)

 fun nth_elem NL =

   (case drop NL of

     [] => raise LIST "nth_elem"

   | x :: _ => x);

 (*last element of a list*)

 fun last_elem [] = raise LIST "last_elem"

   | last_elem [x] = x

   | last_elem (_ :: xs) = last_elem xs;

 (*find the position of an element in a list*)

 fun find (x, ys) =

   let fun f (y :: ys, i) = if x = y then i else f (ys, i + 1)

         | f (_, _) = raise LIST "find"

   in f (ys, 0) end;

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

 fun flat (ls: 'c list list) : 'c list = foldr (op @) (ls, []);

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

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

 fun seq (proc: 'a -> unit) : 'a list -> unit =

   let fun seqf [] = ()

         | seqf (x :: xs) = (proc x; seqf xs)

   in seqf end;

 (*separate s [x1, x2, ..., xn]  ===>  [x1, s, x2, s, ..., s, xn]*)

 fun separate s (x :: (xs as _ :: _)) = x :: s :: separate s xs

   | separate _ xs = xs;

 (*make the list [x, x, ..., x] of length n*)

 fun replicate n (x: 'a) : 'a list =

   let fun rep (0, xs) = xs

         | rep (n, xs) = rep (n - 1, x :: xs)

   in

     if n < 0 then raise LIST "replicate"

     else rep (n, [])

   end;

 (* filter *)

 (*copy the list preserving elements that satisfy the predicate*)

 fun filter (pred: 'a->bool) : 'a list -> 'a list =

   let fun filt [] = []

         | filt (x :: xs) = if pred x then x :: filt xs else filt xs

   in filt end;

 fun filter_out f = filter (not o f);

 fun mapfilter (f: 'a -> 'b option) ([]: 'a list) = [] : 'b list

   | mapfilter f (x :: xs) =

       (case f x of

         None => mapfilter f xs

       | Some y => y :: mapfilter f xs);

 fun find_first _ [] = None

   | find_first pred (x :: xs) =

       if pred x then Some x else find_first pred xs;

 (* lists of pairs *)

 fun map2 _ ([], []) = []

   | map2 f (x :: xs, y :: ys) = (f (x, y) :: map2 f (xs, ys))

   | map2 _ _ = raise LIST "map2";

 fun exists2 _ ([], []) = false

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

   | exists2 _ _ = raise LIST "exists2";

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

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

   | forall2 _ _ = raise LIST "forall2";

 (*combine two lists forming a list of pairs:

   [x1, ..., xn] ~~ [y1, ..., yn]  ===>  [(x1, y1), ..., (xn, yn)]*)

 fun [] ~~ [] = []

   | (x :: xs) ~~ (y :: ys) = (x, y) :: (xs ~~ ys)

   | _ ~~ _ = raise LIST "~~";

 (*inverse of ~~; the old 'split':

   [(x1, y1), ..., (xn, yn)]  ===>  ([x1, ..., xn], [y1, ..., yn])*)

 fun split_list (l: ('a * 'b) list) = (map #1 l, map #2 l);

 (* prefixes, suffixes *)

 fun [] prefix _ = true

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

   | _ prefix _ = false;

 (* [x1, ..., xi, ..., xn]  --->  ([x1, ..., x(i-1)], [xi, ..., xn])

    where xi is the first element that does not satisfy the predicate*)

 fun take_prefix (pred : 'a -> bool)  (xs: 'a list) : 'a list * 'a list =

   let fun take (rxs, []) = (rev rxs, [])

         | take (rxs, x :: xs) =

             if  pred x  then  take(x :: rxs, xs)  else  (rev rxs, x :: xs)

   in  take([], xs)  end;

 (* [x1, ..., xi, ..., xn]  --->  ([x1, ..., xi], [x(i+1), ..., xn])

    where xi is the last element that does not satisfy the predicate*)

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

   | take_suffix pred (x :: xs) =

       (case take_suffix pred xs of

         ([], sffx) => if pred x then ([], x :: sffx) else ([x], sffx)

       | (prfx, sffx) => (x :: prfx, sffx));

 (** integers **)

 fun inc i = (i := ! i + 1; ! i);

 fun dec i = (i := ! i - 1; ! i);

 (* lists of integers *)

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

 fun (from upto to) =

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

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

 fun (from downto to) =

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

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

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

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

 (* convert integers to strings *)

 (*expand the number in the given base;

   example: radixpand (2, 8) gives [1, 0, 0, 0]*)

 fun radixpand (base, num) : int list =

   let

     fun radix (n, tail) =

       if n < base then n :: tail

       else radix (n div base, (n mod base) :: tail)

   in radix (num, []) end;

 (*expands a number into a string of characters starting from "zerochar";

   example: radixstring (2, "0", 8) gives "1000"*)

 fun radixstring (base, zerochar, num) =

   let val offset = ord zerochar;

       fun chrof n = chr (offset + n)

   in implode (map chrof (radixpand (base, num))) end;

 fun string_of_int n =

   if n < 0 then "~" ^ radixstring (10, "0", ~n) else radixstring (10, "0", n);

 (** strings **)

 fun is_letter ch =

   ord "A" <= ord ch andalso ord ch <= ord "Z" orelse

   ord "a" <= ord ch andalso ord ch <= ord "z";

 fun is_digit ch =

   ord "0" <= ord ch andalso ord ch <= ord "9";

 (*letter or _ or prime (')*)

 fun is_quasi_letter "_" = true

   | is_quasi_letter "'" = true

   | is_quasi_letter ch = is_letter ch;

 (*white space: blanks, tabs, newlines, formfeeds*)

 val is_blank : string -> bool =

   fn " " => true | "\t" => true | "\n" => true | "\^L" => true | "160" => true

     | _ => false;

 val is_letdig = is_quasi_letter orf is_digit;

 (*printable chars*)

 fun is_printable c = ord c > ord " " andalso ord c <= ord "~";

 (*lower all chars of string*)

 val to_lower =

   let

     fun lower ch =

       if ch >= "A" andalso ch <= "Z" then

         chr (ord ch - ord "A" + ord "a")

       else ch;

   in implode o (map lower) o explode end;

 (*enclose in brackets*)

 fun enclose lpar rpar str = lpar ^ str ^ rpar;

 (*simple quoting (does not escape special chars)*)

 val quote = enclose "\"" "\"";

 (*space_implode "..." (explode "hello"); gives "h...e...l...l...o"*)

 fun space_implode a bs = implode (separate a bs);

 val commas = space_implode ", ";

 val commas_quote = commas o map quote;

 (*concatenate messages, one per line, into a string*)

 val cat_lines = space_implode "\n";

 (*space_explode "." "h.e..l.lo"; gives ["h", "e", "l", "lo"]*)

 fun space_explode sep s =

   let fun divide [] "" = []

         | divide [] part = [part]

         | divide (c::s) part =

             if c = sep then

               (if part = "" then divide s "" else part :: divide s "")

             else divide s (part ^ c)

   in divide (explode s) "" end;

 (** lists as sets **)

 (*membership in a list*)

 fun x mem [] = false

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

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

 fun (x:int) mem_int [] = false

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

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

 fun (x:string) mem_string [] = false

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

 (*generalized membership test*)

 fun gen_mem eq (x, []) = false

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

 (*insertion into list if not already there*)

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

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

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

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

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

 (*generalized insertion*)

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

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

 fun xs union [] = xs

   | [] union ys = ys

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

 (*union of sets, optimized version for ints*)

 fun (xs:int list) union_int [] = xs

   | [] union_int ys = ys

   | (x :: xs) union_int ys = xs union_int (x ins_int ys);

 (*union of sets, optimized version for strings*)

 fun (xs:string list) union_string [] = xs

   | [] union_string ys = ys

   | (x :: xs) union_string ys = xs union_string (x ins_string ys);

 (*generalized union*)

 fun gen_union eq (xs, []) = xs

   | gen_union eq ([], ys) = ys

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

 (*intersection*)

 fun [] inter ys = []

   | (x :: xs) inter ys =

       if x mem ys then x :: (xs inter ys) else xs inter ys;

 (*intersection, optimized version for ints*)

 fun ([]:int list) inter_int ys = []

   | (x :: xs) inter_int ys =

       if x mem_int ys then x :: (xs inter_int ys) else xs inter_int ys;

 (*intersection, optimized version for strings *)

 fun ([]:string list) inter_string ys = []

   | (x :: xs) inter_string ys =

       if x mem_string ys then x :: (xs inter_string ys) else xs inter_string ys;

 (*subset*)

 fun [] subset ys = true

   | (x :: xs) subset ys = x mem ys andalso xs subset ys;

 (*subset, optimized version for ints*)

 fun ([]:int list) subset_int ys = true

   | (x :: xs) subset_int ys = x mem_int ys andalso xs subset_int ys;

 (*subset, optimized version for strings*)

 fun ([]:string list) subset_string ys = true

   | (x :: xs) subset_string ys = x mem_string ys andalso xs subset_string ys;

 (*set equality for strings*)

 fun eq_set_string ((xs:string list), ys) =

   xs = ys orelse (xs subset_string ys andalso ys subset_string xs);

 fun gen_subset eq (xs, ys) = forall (fn x => gen_mem eq (x, ys)) xs;

 (*removing an element from a list WITHOUT duplicates*)

 fun (y :: ys) \ x = if x = y then ys else y :: (ys \ x)

   | [] \ x = [];

 fun ys \\ xs = foldl (op \) (ys,xs);

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

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

 fun gen_rems eq = foldl (gen_rem eq);

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

   first of equal elements*)

 fun gen_distinct eq lst =

   let

     val memb = gen_mem eq;

     fun dist (rev_seen, []) = rev rev_seen

       | dist (rev_seen, x :: xs) =

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

           else dist (x :: rev_seen, xs);

   in

     dist ([], lst)

   end;

 fun distinct l = gen_distinct (op =) l;

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

 fun findrep [] = []

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

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

   order, takes first of equal elements*)

 fun gen_duplicates eq lst =

   let

     val memb = gen_mem eq;

     fun dups (rev_dups, []) = rev rev_dups

       | dups (rev_dups, x :: xs) =

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

             dups (rev_dups, xs)

           else dups (x :: rev_dups, xs);

   in

     dups ([], lst)

   end;

 fun duplicates l = gen_duplicates (op =) l;

 (** association lists **)

 (*association list lookup*)

 fun assoc ([], key) = None

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

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

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

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

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

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

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

 fun assoc_string ([], (key:string)) = None

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

       if key = keyi then Some xi else assoc_string (pairs, key);

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

 fun assoc_string_int ([], (key:string*int)) = None

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

       if key = keyi then Some xi else assoc_string_int (pairs, key);

 fun assocs ps x =

   (case assoc (ps, x) of

     None => []

   | Some ys => ys);

 (*two-fold association list lookup*)

 fun assoc2 (aal, (key1, key2)) =

   (case assoc (aal, key1) of

     Some al => assoc (al, key2)

   | None => None);

 (*generalized association list lookup*)

 fun gen_assoc eq ([], key) = None

   | gen_assoc eq ((keyi, xi) :: pairs, key) =

       if eq (key, keyi) then Some xi else gen_assoc eq (pairs, key);

 (*association list update*)

 fun overwrite (al, p as (key, _)) =

   let fun over ((q as (keyi, _)) :: pairs) =

             if keyi = key then p :: pairs else q :: (over pairs)

         | over [] = [p]

   in over al end;

 fun gen_overwrite eq (al, p as (key, _)) =

   let fun over ((q as (keyi, _)) :: pairs) =

             if eq (keyi, key) then p :: pairs else q :: (over pairs)

         | over [] = [p]

   in over al end;

 (** generic tables **)

 (*Tables are supposed to be 'efficient' encodings of lists of elements distinct

   wrt. an equality "eq". The extend and merge operations below are optimized

   for long-term space efficiency.*)

 (*append (new) elements to a table*)

 fun generic_extend _ _ _ tab [] = tab

   | generic_extend eq dest_tab mk_tab tab1 lst2 =

       let

         val lst1 = dest_tab tab1;

         val new_lst2 = gen_rems eq (lst2, lst1);

       in

         if null new_lst2 then tab1

         else mk_tab (lst1 @ new_lst2)

       end;

 (*append (new) elements of 2nd table to 1st table*)

 fun generic_merge eq dest_tab mk_tab tab1 tab2 =

   let

     val lst1 = dest_tab tab1;

     val lst2 = dest_tab tab2;

     val new_lst2 = gen_rems eq (lst2, lst1);

   in

     if null new_lst2 then tab1

     else if gen_subset eq (lst1, lst2) then tab2

     else mk_tab (lst1 @ new_lst2)

   end;

 (*lists as tables*)

 fun extend_list tab = generic_extend (op =) I I tab;

 fun merge_lists tab = generic_merge (op =) I I tab;

 fun merge_rev_lists xs [] = xs

   | merge_rev_lists [] ys = ys

   | merge_rev_lists xs (y :: ys) =

       (if y mem xs then I else cons y) (merge_rev_lists xs ys);

 (** balanced trees **)

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

 (*balanced folding; avoids deep nesting*)

 fun fold_bal f [x] = x

   | fold_bal f [] = raise Balance

   | fold_bal f xs =

       let val k = length xs div 2

       in  f (fold_bal f (take(k, xs)),

              fold_bal f (drop(k, xs)))

       end;

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

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

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

           if n=1 then x else

           let val n2 = n div 2

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

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

           end

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

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

 fun accesses_bal (f, g, x) n =

   let fun acc n =

           if n=1 then [x] else

           let val n2 = n div 2

               val acc2 = acc n2

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

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

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

 (** orders **)

 datatype order = LESS | EQUAL | GREATER;

 fun intord (i, j: int) =

   if i < j then LESS

   else if i = j then EQUAL

   else GREATER;

 fun stringord (a, b: string) =

   if a < b then LESS

   else if a = b then EQUAL

   else GREATER;

 (** input / output **)

 val cd = OS.FileSys.chDir;

 val pwd = OS.FileSys.getDir;

 val prs_fn = ref(fn s => TextIO.output (TextIO.stdOut, s));

 fun prs s = !prs_fn s;

 fun writeln s = prs (s ^ "\n");

 (* TextIO.output to LaTeX / xdvi *)

 fun latex s = 

         execute ( "( cd /tmp ; echo \"" ^ s ^

         "\" | isa2latex -s > $$.tex ; latex $$.tex ; xdvi $$.dvi ; rm $$.* ) > /dev/null &" ) ;

 (*print warning*)

 val warning_fn = ref(fn s => TextIO.output (TextIO.stdOut, s ^ "\n"));

 fun warning s = !warning_fn ("Warning: " ^ s);

 (*print error message and abort to top level*)

 val error_fn = ref(fn s => TextIO.output 

 		            (TextIO.stdOut, "\n*** " ^ s ^ "\n\n"));

 exception ERROR;

 fun error msg = (!error_fn msg;

 		 TextIO.flushOut TextIO.stdOut; 

 		 raise ERROR);

 fun sys_error msg = (!error_fn "*** SYSTEM ERROR ***"; error msg);

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

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

 (*Assert pred for every member of l, generating a message if pred fails*)

 fun assert_all pred l msg_fn = 

   let fun asl [] = ()

         | asl (x::xs) = if pred x then asl xs

                         else error (msg_fn x)

   in  asl l  end;

 (*for the "test" target in Makefiles -- signifies successful termination*)

 fun maketest msg =

   (writeln msg; 

    let val os = TextIO.openOut "test" 

    in  TextIO.output (os, "Test examples ran successfully\n");

        TextIO.closeOut os

    end);

 (*print a list surrounded by the brackets lpar and rpar, with comma separator

   print nothing for empty list*)

 fun print_list (lpar, rpar, pre: 'a -> unit) (l : 'a list) =

   let fun prec x = (prs ","; pre x)

   in

     (case l of

       [] => ()

     | x::l => (prs lpar; pre x; seq prec l; prs rpar))

   end;

 (*print a list of items separated by newlines*)

 fun print_list_ln (pre: 'a -> unit) : 'a list -> unit =

   seq (fn x => (pre x; writeln ""));

 val print_int = prs o string_of_int;

 (** timing **)

 (*unconditional timing function*)

 fun timeit x = cond_timeit true x;

 (*timed application function*)

 fun timeap f x = timeit (fn () => f x);

 (*timed "use" function, printing filenames*)

 fun time_use fname = timeit (fn () =>

   (writeln ("\n**** Starting " ^ fname ^ " ****"); use fname;

    writeln ("\n**** Finished " ^ fname ^ " ****")));

 (*For Makefiles: use the file, but exit with error code if errors found.*)

 fun exit_use fname = use fname handle _ => exit 1;

 (** filenames and paths **)

 (*Convert UNIX filename of the form "path/file" to "path/" and "file";

   if filename contains no slash, then it returns "" and "file"*)

 val split_filename =

   (pairself implode) o take_suffix (not_equal "/") o explode;

 val base_name = #2 o split_filename;

 (*Merge splitted filename (path and file);

   if path does not end with one a slash is appended*)

 fun tack_on "" name = name

   | tack_on path name =

       if last_elem (explode path) = "/" then path ^ name

       else path ^ "/" ^ name;

 (*Remove the extension of a filename, i.e. the part after the last '.'*)

 val remove_ext = implode o #1 o take_suffix (not_equal ".") o explode;

 (*Make relative path to reach an absolute location from a different one*)

 fun relative_path cur_path dest_path =

   let (*Remove common beginning of both paths and make relative path*)

       fun mk_relative [] [] = []

         | mk_relative [] ds = ds

         | mk_relative cs [] = map (fn _ => "..") cs

         | mk_relative (c::cs) (d::ds) =

             if c = d then mk_relative cs ds

             else ".." :: map (fn _ => "..") cs @ (d::ds);

   in if cur_path = "" orelse hd (explode cur_path) <> "/" orelse

         dest_path = "" orelse hd (explode dest_path) <> "/" then

        error "Relative or empty path passed to relative_path"

      else ();

      space_implode "/" (mk_relative (space_explode "/" cur_path)

                                     (space_explode "/" dest_path))

   end;

 (*Determine if absolute path1 is a subdirectory of absolute path2*)

 fun path1 subdir_of path2 =

   if hd (explode path1) <> "/" orelse hd (explode path2) <> "/" then

     error "Relative or empty path passed to subdir_of"

   else (space_explode "/" path2) prefix (space_explode "/" path1);

 fun absolute_path cwd file =

   let fun rm_points [] result = rev result

         | rm_points (".."::ds) result = rm_points ds (tl result)

         | rm_points ("."::ds) result = rm_points ds result

         | rm_points (d::ds) result = rm_points ds (d::result);

   in if file = "" then ""

      else if hd (explode file) = "/" then file

      else "/" ^ space_implode "/"

                   (rm_points (space_explode "/" (tack_on cwd file)) [])

   end;

 (** misc functions **)

 (*use the keyfun to make a list of (x, key) pairs*)

 fun make_keylist (keyfun: 'a->'b) : 'a list -> ('a * 'b) list =

   let fun keypair x = (x, keyfun x)

   in map keypair end;

 (*given a list of (x, key) pairs and a searchkey

   return the list of xs from each pair whose key equals searchkey*)

 fun keyfilter [] searchkey = []

   | keyfilter ((x, key) :: pairs) searchkey =

       if key = searchkey then x :: keyfilter pairs searchkey

       else keyfilter pairs searchkey;

 (*Partition list into elements that satisfy predicate and those that don't.

   Preserves order of elements in both lists.*)

 fun partition (pred: 'a->bool) (ys: 'a list) : ('a list * 'a list) =

     let fun part ([], answer) = answer

           | part (x::xs, (ys, ns)) = if pred(x)

             then  part (xs, (x::ys, ns))

             else  part (xs, (ys, x::ns))

     in  part (rev ys, ([], []))  end;

 fun partition_eq (eq:'a * 'a -> bool) =

     let fun part [] = []

           | part (x::ys) = let val (xs, xs') = partition (apl(x, eq)) ys

                            in (x::xs)::(part xs') end

     in part end;

 (*Partition a list into buckets  [ bi, b(i+1), ..., bj ]

    putting x in bk if p(k)(x) holds.  Preserve order of elements if possible.*)

 fun partition_list p i j =

   let fun part k xs =

             if k>j then

               (case xs of [] => []

                          | _ => raise LIST "partition_list")

             else

             let val (ns, rest) = partition (p k) xs;

             in  ns :: part(k+1)rest  end

   in  part i end;

 (* sorting *)

 (*insertion sort; stable (does not reorder equal elements)

   'less' is less-than test on type 'a*)

 fun sort (less: 'a*'a -> bool) =

   let fun insert (x, []) = [x]

         | insert (x, y::ys) =

               if less(y, x) then y :: insert (x, ys) else x::y::ys;

       fun sort1 [] = []

         | sort1 (x::xs) = insert (x, sort1 xs)

   in  sort1  end;

 (*sort strings*)

 val sort_strings = sort (op <= : string * string -> bool);

 (* transitive closure (not Warshall's algorithm) *)

 fun transitive_closure [] = []

   | transitive_closure ((x, ys)::ps) =

       let val qs = transitive_closure ps

           val zs = foldl (fn (zs, y) => assocs qs y union_string zs) (ys, ys)

           fun step(u, us) = (u, if x mem_string us then zs union_string us 

                                 else us)

       in (x, zs) :: map step qs end;

 (* generating identifiers *)

 local

   val a = ord "a" and z = ord "z" and A = ord "A" and Z = ord "Z"

   and k0 = ord "0" and k9 = ord "9"

   val seedr = ref 0;

 in

 (*Maps 0-63 to A-Z, a-z, 0-9 or _ or ' for generating random identifiers*)

 fun newid n = 

   let fun char i =

                if i<26 then chr (A+i)

           else if i<52 then chr (a+i-26)

           else if i<62 then chr (k0+i-52)

           else if i=62 then "_"

           else  (*i=63*)    "'"

   in  implode (map char (radixpand (64,n)))  end;

 (*Freshly generated identifiers with given prefix; MUST start with a letter*)

 fun gensym pre = pre ^ 

                  (#1(newid (!seedr), 

                      seedr := 1+ !seedr))

 (*Increment a list of letters like a reversed base 26 number.

   If head is "z", bumps chars in tail.

   Digits are incremented as if they were integers.

   "_" and "'" are not changed.

   For making variants of identifiers.*)

 fun bump_int_list(c::cs) = if c="9" then "0" :: bump_int_list cs else

         if k0 <= ord(c) andalso ord(c) < k9 then chr(ord(c)+1) :: cs

         else "1" :: c :: cs

   | bump_int_list([]) = error("bump_int_list: not an identifier");

 fun bump_list([], d) = [d]

   | bump_list(["'"], d) = [d, "'"]

   | bump_list("z"::cs, _) = "a" :: bump_list(cs, "a")

   | bump_list("Z"::cs, _) = "A" :: bump_list(cs, "A")

   | bump_list("9"::cs, _) = "0" :: bump_int_list cs

   | bump_list(c::cs, _) = let val k = ord(c)

         in if (a <= k andalso k < z) orelse (A <= k andalso k < Z) orelse

               (k0 <= k andalso k < k9) then chr(k+1) :: cs else

            if c="'" orelse c="_" then c :: bump_list(cs, "") else

                 error("bump_list: not legal in identifier: " ^

                         implode(rev(c::cs)))

         end;

 end;

 fun bump_string s : string = implode (rev (bump_list(rev(explode s), "")));

 (* lexical scanning *)

 (*scan a list of characters into "words" composed of "letters" (recognized by

   is_let) and separated by any number of non-"letters"*)

 fun scanwords is_let cs =

   let fun scan1 [] = []

         | scan1 cs =

             let val (lets, rest) = take_prefix is_let cs

             in implode lets :: scanwords is_let rest end;

   in scan1 (#2 (take_prefix (not o is_let) cs)) end;

 end;

 (*Variable-branching trees: for proof terms*)

 datatype 'a mtree = Join of 'a * 'a mtree list;

 open Library;