src/Pure/library.ML
 changeset 233 efd6b4bb14dd parent 205 0dd3a0a264cd child 245 faf3de36fdb1
```--- a/src/Pure/library.ML	Tue Jan 18 16:58:41 1994 +0100
+++ b/src/Pure/library.ML	Wed Jan 19 14:10:54 1994 +0100
@@ -1,354 +1,42 @@
(*  Title:      Pure/library.ML
ID:         \$Id\$
-    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

-Basic library: booleans, lists, pairs, input/output, etc.
+Basic library: functions, options, pairs, booleans, lists, integers,
+strings, lists as sets, association lists, generic tables, balanced trees,
+input / output, timing, filenames, misc functions.
*)

-(**** Booleans: operators for combining predicates ****)
-
-infix orf;
-fun p orf q = fn x => p x orelse q x ;
-
-infix andf;
-fun p andf q = fn x => p x andalso q x ;
+(** functions **)

-fun notf p x = not (p x) ;
+(*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;

-fun orl [] = false
-  | orl (x::l) =  x  orelse  orl l;
-
-fun andl [] = true
-  | andl (x::l) =  x  andalso  andl l;
+(*combine two functions forming the union of their domains*)
+infix orelf;
+fun f orelf g = fn x => f x handle Match => g x;

-(*exists pred [x1,...,xn] ======>  pred(x1)  orelse  ...  orelse  pred(xn)*)
-fun exists (pred: 'a -> bool) : 'a list -> bool =
-  let fun boolf [] = false
-        | boolf (x::l) = (pred x) orelse boolf l
-  in boolf end;
+(*application of (infix) operator to its left or right argument*)
+fun apl (x, f) y = f (x, y);
+fun apr (f, y) x = f (x, y);

-(*forall pred [x1,...,xn] ======>  pred(x1)  andalso  ...  andalso  pred(xn)*)
-fun forall (pred: 'a -> bool) : 'a list -> bool =
-  let fun boolf [] = true
-        | boolf (x::l) = (pred x) andalso (boolf l)
-  in boolf end;
+(*functional for pairs*)
+fun pairself f (x, y) = (f x, f y);

-
-(** curried equality **)
-
-fun equal x y = (x = y);
-
-fun not_equal x y = x <> y;
+(*function exponentiation: f(...(f x)...) with n applications of f*)
+fun funpow n f x =
+  let fun rep (0, x) = x
+        | rep (n, x) = rep (n - 1, f x)
+  in rep (n, x) end;

-(*** Lists ***)
-
-exception LIST of string;
-
-(*discriminator and selectors for lists. *)
-fun null   []   = true
-  | null (_::_) = false;
-
-fun hd   []   = raise LIST "hd"
-  | hd (a::_) = a;
-
-fun tl   []   = raise LIST "tl"
-  | tl (_::l) = l;
-
-
-(*curried cons and reverse cons*)
-
-fun cons x xs = x :: xs;
-
-fun rcons xs x = x :: xs;
-
-
-(*curried functions for pairing and reversed pairing*)
-fun pair x y = (x,y);
-fun rpair x y = (y,x);
-
-fun fst(x,y) = x and snd(x,y) = y;
-
-(*Handy combinators*)
-fun curry f x y = f(x,y);
-fun uncurry f(x,y) = f x y;
-fun I x = x  and  K x y = x;
-
-(*Combine two functions forming the union of their domains*)
-infix orelf;
-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);
-
-(*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);
-
-fun square (n: int) = n*n;
-
-fun fact 0 = 1
-  | fact n = n * fact(n-1);
-
-
-(*The following versions of fold are designed to fit nicely with infixes.*)
-
-(*  (op @) (e, [x1,...,xn])  ======>   ((e @ x1) @ x2) ... @ xn
-    for operators that associate to the left.  TAIL RECURSIVE*)
-fun foldl (f: 'a * 'b -> 'a) : 'a * 'b list -> 'a =
-  let fun itl (e, [])  = e
-        | itl (e, a::l) = itl (f(e,a), l)
-  in  itl end;
-
-(*  (op @) ([x1,...,xn], e)  ======>   x1 @ (x2 ... @ (xn @ e))
-    for operators that associate to the right.  Not tail recursive.*)
-fun foldr f (l,e) =
-  let fun itr [] = e
-        | itr (a::l) = f(a, itr l)
-  in  itr l  end;
-
-(*  (op @) [x1,...,xn]  ======>   x1 @ (x2 ..(x[n-1]. @ xn))
-    for n>0, operators that associate to the right.  Not tail recursive.*)
-fun foldr1 f l =
-  let fun itr [x] = x
-        | itr (x::l) = f(x, itr l)
-  in  itr l  end;
-
-
-(*Length of a list.  Should unquestionably be a standard function*)
-local fun length1 (n, [ ])  = n   (*TAIL RECURSIVE*)
-        | length1 (n, x::l) = length1 (n+1, l)
-in  fun length l = length1 (0,l) end;
-
-
-(*Take the first n elements from l.*)
-fun take (n, []) = []
-  | take (n, x::xs) = if n>0 then x::take(n-1,xs)
-                      else  [];
-
-(*Drop the first n elements from l.*)
-fun drop (_, [])    = []
-  | drop (n, x::xs) = if n>0 then drop (n-1, xs)
-                             else x::xs;
-
-(*Return nth element of l, 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::l => x;
-
-
-(*Last element of a list*)
-fun last_elem [] = raise LIST "last_elem"
-  | last_elem [x] = x
-  | last_elem (_ :: xs) = last_elem xs;
-
-
-(*make the list [from, from+1, ..., to]*)
-infix upto;
-fun from upto to =
-    if from>to then []  else  from :: ((from+1) upto to);
-
-(*make the list [from, from-1, ..., to]*)
-infix downto;
-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;
-
-(*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::l) = (proc x;  seqf l)
-  in  seqf end;
-
-
-
-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;
-
-
-(*** Input/Output ***)
-
-fun prs s = output(std_out,s);
-fun writeln s = prs (s ^ "\n");
-
-(*Print error message and abort to top level*)
-exception ERROR;
-fun error msg = (writeln msg; raise ERROR);
-fun sys_error msg = (writeln "-- System Error --"; error msg);
-
-fun assert p msg = if p then () else error msg;
-fun deny p msg = if p then error msg else ();
-
-(*For the "test" target in Makefiles -- signifies successful termination*)
-fun maketest msg =
-    (writeln msg;
-     output(open_out "test", "Test examples ran successfully\n"));
-
-(*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""));
-
-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*)
-val is_blank : string -> bool = fn
-     " " => true  |  "\t" => true  |  "\n" => true  |  _ => false;
-
-val is_letdig = is_quasi_letter orf is_digit;
-
-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;
-
-
-(*** Timing ***)
-
-(*Unconditional timing function*)
-val timeit = cond_timeit true;
-
-(*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 ^ " ****")));
-
-
-(*** Misc functions ***)
-
-(*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;
-
-(*Combine two lists forming a list of pairs:
-  [x1,...,xn] ~~ [y1,...,yn]  ======>   [(x1,y1), ..., (xn,yn)] *)
-infix ~~;
-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);
-
-(*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;
-
-(*Flatten a list of lists to a list.*)
-fun flat (ls: 'c list list) : 'c list = foldr (op @) (ls,[]);
-
-
-(*** polymorphic set operations ***)
-
-(*membership in a list*)
-infix mem;
-fun x mem []  =  false
-  | x mem (y::l)  =  (x=y) orelse (x mem l);
-
-(*insertion into list if not already there*)
-infix ins;
-fun x ins xs = if x mem xs then  xs   else  x::xs;
-
-(*union of sets represented as lists: no repetitions*)
-infix union;
-fun   xs    union [] = xs
-  |   []    union ys = ys
-  | (x::xs) union ys = xs union (x ins ys);
-
-infix inter;
-fun   []    inter ys = []
-  | (x::xs) inter ys = if x mem ys then x::(xs inter ys)
-                                   else     xs inter ys;
-
-infix subset;
-fun   []    subset ys = true
-  | (x::xs) subset ys = x mem ys   andalso  xs subset ys;
-
-(*removing an element from a list WITHOUT duplicates*)
-infix \;
-fun (y::ys) \ x = if x=y then ys else y::(ys \ x)
-  |   []    \ x = [];
-
-infix \\;
-val op \\ = foldl (op \);
-
-(*** option stuff ***)
+(** options **)

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

@@ -363,131 +51,656 @@
fun is_none (Some _) = false
| is_none None = true;

+fun apsome f (Some x) = Some (f x)
+  | apsome _ None = None;

-(*** Association lists ***)
+
+
+(** 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 *)
+
+infix orf;
+fun p orf q = fn x => p x orelse q x;
+
+infix andf;
+fun p andf q = fn x => p x andalso q x;
+
+fun notf p x = not (p x);

-(*Association list lookup*)
-fun assoc ([], key) = None
-  | assoc ((keyi,xi)::pairs, key) =
-      if key=keyi then Some xi  else assoc (pairs,key);
+
+(* predicates on lists *)
+
+fun orl [] = false
+  | orl (x :: xs) = x orelse orl xs;
+
+fun andl [] = true
+  | andl (x :: xs) = x andalso andl xs;
+
+(*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;

-fun assocs ps x = case assoc(ps,x) of None => [] | Some(ys) => ys;
+
+
+(** 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*)

-(*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;
+(*  (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;

-(*Copy the list preserving elements that satisfy the predicate*)
-fun filter (pred: 'a->bool) : 'a list -> 'a list =
+(*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;
+        | filt (x :: xs) = if pred x then x :: filt xs else filt xs
+  in filt end;

fun filter_out f = filter (not o f);

-(*** List operations, generalized to an arbitrary equality function "eq"
-     -- so what good are equality types?? ***)
+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);
+
+
+(* lists of pairs *)
+
+(*combine two lists forming a list of pairs:
+  [x1, ..., xn] ~~ [y1, ..., yn]  ===>  [(x1, y1), ..., (xn, yn)]*)
+infix ~~;
+fun [] ~~ [] = []
+  | (x :: xs) ~~ (y :: ys) = (x, y) :: (xs ~~ ys)
+  | _ ~~ _ = raise LIST "~~";
+
+(*combine two lists*)
+fun map2 _ ([], []) = []
+  | map2 f (x :: xs, y :: ys) = (f (x, y) :: map2 f (xs, ys))
+  | map2 _ _ = raise LIST "map2";
+
+(*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 *)
+
+infix prefix;
+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;
+fun dec i = i := ! i - 1;
+
+
+fun square (n:int) = n * n;
+
+fun fact 0 = 1
+  | fact n = n * fact (n - 1);
+
+
+(* lists of integers *)
+
+(*make the list [from, from + 1, ..., to]*)
+infix upto;
+fun from upto to =
+  if from > to then [] else from :: ((from + 1) upto to);
+
+(*make the list [from, from - 1, ..., to]*)
+infix downto;
+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;
+
+
+(* operations on integer lists *)

-(*removing an element from a list -- possibly WITH duplicates*)
-fun gen_rem eq (xs,y) = filter_out (fn x => eq(x,y)) xs;
+fun sum [] = 0
+  | sum (n :: ns) = n + sum ns;
+
+fun max [m:int] = m
+  | max (m :: n :: ns) = if m > n then max (m :: ns) else max (n :: ns)
+  | max [] = raise LIST "max";
+
+fun min [m:int] = m
+  | min (m :: n :: ns) = if m < n then min (m :: ns) else min (n :: ns)
+  | min [] = raise LIST "min";
+
+
+(* 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*)
+val is_blank : string -> bool =
+  fn " " => true | "\t" => true | "\n" => true | _ => false;
+
+val is_letdig = is_quasi_letter orf is_digit;
+
+
+(*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;
+
+
+(*simple quoting (does not escape special chars)*)
+fun quote s = "\"" ^ s ^ "\"";
+
+(*space_implode "..." (explode "hello"); gives "h...e...l...l...o"*)
+fun space_implode a bs = implode (separate a bs);
+
+(*concatenate messages, one per line, into a string*)
+val cat_lines = implode o (map (apr (op ^, "\n")));
+
+
+
+(** lists as sets **)
+
+(*membership in a list*)
+infix mem;
+fun x mem [] = false
+  | x mem (y :: ys) = x = y orelse x mem 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);
+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*)
+infix ins;
+fun x ins xs = if x mem 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;
+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*)
+infix union;
+fun xs union [] = xs
+  | [] union ys = ys
+  | (x :: xs) union ys = xs union (x ins 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));
+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*)
+infix inter;
+fun [] inter ys = []
+  | (x :: xs) inter ys =
+      if x mem ys then x :: (xs inter ys) else xs inter ys;
+
+
+(*subset*)
+infix subset;
+fun [] subset ys = true
+  | (x :: xs) subset ys = x mem ys andalso xs subset ys;
+
+fun gen_subset eq (xs, ys) = forall (fn x => gen_mem eq (x, ys)) xs;
+
+
+(*removing an element from a list WITHOUT duplicates*)
+infix \;
+fun (y :: ys) \ x = if x = y then ys else y :: (ys \ x)
+  | [] \ x = [];
+
+infix \\;
+val op \\ = foldl (op \);

-(*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);
+(*removing an element from a list -- possibly WITH duplicates*)
+fun gen_rem eq (xs, y) = filter_out (fn x => eq (x, y)) xs;
+
+val gen_rems = foldl o gen_rem;
+
+
+(*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;

-(** Finding list elements and duplicates **)
+    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;
+
+val distinct = gen_distinct (op =);
+
+
+(*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;
+
+
+
+(** association lists **)

-(* 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;
+(*association list lookup*)
+fun assoc ([], key) = None
+  | assoc ((keyi, xi) :: pairs, key) =
+      if key = keyi then Some xi else assoc (pairs, key);
+
+fun assocs ps x =
+  (case assoc (ps, x) of
+    None => []
+  | Some ys => ys);
+
+(*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;
+
+
+
+(** generic tables **)

-(*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;
+(*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;

-fun distinct1 (seen, []) = rev seen
-  | distinct1 (seen, x::xs) =
-      if x mem seen then distinct1 (seen, xs)
-    		    else distinct1 (x::seen, xs);
+(*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;

-(*Makes a list of the distinct members of the input*)
-fun distinct xs = distinct1([],xs);
+
+(*lists as tables*)
+val extend_list = generic_extend (op =) I I;
+val merge_lists = generic_merge (op =) I I;
+

-(*Use the keyfun to make a list of (x,key) pairs.*)
+(** 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;
+
+
+
+(** input / output **)
+
+fun prs s = output (std_out, s);
+fun writeln s = prs (s ^ "\n");
+
+
+(*print error message and abort to top level*)
+exception ERROR;
+fun error msg = (writeln msg; raise ERROR);
+fun sys_error msg = (writeln "*** System Error ***"; error msg);
+
+fun assert p msg = if p then () else error msg;
+fun deny p msg = if p then error msg else ();
+
+
+(* FIXME close file (?) *)
+(*for the "test" target in Makefiles -- signifies successful termination*)
+fun maketest msg =
+  (writeln msg; output (open_out "test", "Test examples ran successfully\n"));
+
+
+(*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*)
+val timeit = cond_timeit true;
+
+(*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 ^ " ****")));
+
+
+
+(** filenames **)
+
+(*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;
+
+
+
+(** 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;
+  let fun keypair x = (x, keyfun x)
+  in map keypair end;

-(*Given a list of (x,key) pairs and a searchkey
+(*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;
-
-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;
+  | 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. *)
+  Preserves order of elements in both lists.*)
fun partition (pred: 'a->bool) (ys: 'a list) : ('a list * 'a list) =
-	  | 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;
+          | 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
+          | 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 ]
+(*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
+  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
+            let val (ns, rest) = partition (p k) xs;
+            in  ns :: part(k+1)rest  end
in  part i end;

-(*Insertion sort.  Stable (does not reorder equal elements)
-  'less' is less-than test on type 'a. *)
-fun sort (less: 'a*'a -> bool) =
+(* 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;
+        | 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;
@@ -496,146 +709,60 @@
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 zs) (ys,ys)
-          fun step(u,us) = (u, if x mem us then zs union us else us)
-      in (x,zs) :: map step qs end;
-
-(*** Converting integers to strings, generating identifiers, etc. ***)
+(* transitive closure (not Warshall's algorithm) *)

-(*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)
+fun transitive_closure [] = []
+  | transitive_closure ((x, ys)::ps) =
+      let val qs = transitive_closure ps
+          val zs = foldl (fn (zs, y) => assocs qs y union zs) (ys, ys)
+          fun step(u, us) = (u, if x mem us then zs union us else us)
+      in (x, zs) :: map step qs end;

-(*Expands a number into a string of characters starting from "zerochar"
- example: radixstring(2,"0", 8)  gives  "1000" *)
-  let val offset = ord(zerochar);
-      fun chrof n = chr(offset+n)
-  in  implode (map chrof (radixpand (base,num)))  end;

-fun string_of_int n =
-
-val print_int = prs o string_of_int;
+(* 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 a = ord "a" and z = ord "z" and A = ord "A" and Z = ord "Z"
+  and k0 = ord "0" and k9 = ord "9"
in

(*Increment a list of letters like a reversed base 26 number.
-  If head is "z",  bumps chars in tail.
+  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. *)
+  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
+        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;
+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),"")));
-
-
-(*** Operations on integer lists ***)
-
-fun sum [] = 0
-  | sum (n::ns) = n + sum ns;
-
-fun max[m : int]  = m
-  | max(m::n::ns) = if m>n  then  max(m::ns)  else  max(n::ns)
-  | max []        = raise LIST "max";
-
-fun min[m : int]  = m
-  | min(m::n::ns) = if m<n  then  min(m::ns)  else  min(n::ns)
-  | min []        = raise LIST "min";
-
-
-(*** Lexical scanning ***)
-
-(* [x1,...,xi,...,xn]  --->  ([x1,...,x(i-1)], [xi,..., xn])
-   where xi is the first element that does not satisfy the predicate*)
-fun take_prefix (pred : 'a -> bool)  (xs: 'a list) : 'a list * 'a list =
-  let fun take (rxs, []) = (rev rxs, [])
-	| take (rxs, x::xs) =
-	    if  pred x  then  take(x::rxs, xs)  else  (rev rxs, x::xs)
-  in  take([],xs)  end;
-
-(* [x1,...,xi,...,xn]  --->  ([x1,...,xi], [x(i+1),..., xn])
-   where xi is the last element that does not satisfy the predicate*)
-fun take_suffix _ [] = ([], [])
-  | take_suffix pred (x :: xs) =
-      (case take_suffix pred xs of
-        ([], sffx) => if pred x then ([], x :: sffx) else ([x], sffx)
-      | (prfx, sffx) => (x :: prfx, sffx));
+fun bump_string s : string = implode (rev (bump_list(rev(explode s), "")));

-infix prefix;
-fun [] prefix _ = true
-  | (x::xs) prefix (y::ys) = (x=y) andalso (xs prefix ys)
-  | _ prefix _ = false;
-
-(* [x1, x2, ..., xn] ---> [x1, s, x2, s, ..., s, xn] *)
-fun separate s (x :: (xs as _ :: _)) = x :: s :: separate s xs
-  | separate _ xs = xs;
+(* lexical scanning *)

-(*space_implode "..." (explode "hello");  gives  "h...e...l...l...o" *)
-fun space_implode a bs = implode (separate a bs);
-
-(*simple quoting (does not escape special chars) *)
-fun quote s = "\"" ^ s ^ "\"";
-
-(*Concatenate messages, one per line, into a string*)
-val cat_lines = implode o (map (apr(op^,"\n")));
-
-(*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 =
+(*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;
-
-
-(*** Operations on filenames ***)
+        | 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;

-(*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;```