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(* Title: library
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1992 University of Cambridge
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Basic library: booleans, lists, pairs, input/output, etc.
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*)
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(**** Booleans: operators for combining predicates ****)
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infix orf;
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fun p orf q = fn x => p x orelse q x ;
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infix andf;
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fun p andf q = fn x => p x andalso q x ;
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fun notf p x = not (p x) ;
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fun orl [] = false
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| orl (x::l) = x orelse orl l;
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fun andl [] = true
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| andl (x::l) = x andalso andl l;
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(*exists pred [x1,...,xn] ======> pred(x1) orelse ... orelse pred(xn)*)
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fun exists (pred: 'a -> bool) : 'a list -> bool =
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let fun boolf [] = false
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| boolf (x::l) = (pred x) orelse boolf l
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in boolf end;
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(*forall pred [x1,...,xn] ======> pred(x1) andalso ... andalso pred(xn)*)
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fun forall (pred: 'a -> bool) : 'a list -> bool =
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let fun boolf [] = true
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| boolf (x::l) = (pred x) andalso (boolf l)
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in boolf end;
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(*** Lists ***)
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exception LIST of string;
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(*discriminator and selectors for lists. *)
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fun null [] = true
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| null (_::_) = false;
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fun hd [] = raise LIST "hd"
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| hd (a::_) = a;
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fun tl [] = raise LIST "tl"
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| tl (_::l) = l;
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(*curried functions for pairing and reversed pairing*)
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fun pair x y = (x,y);
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fun rpair x y = (y,x);
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fun fst(x,y) = x and snd(x,y) = y;
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(*Handy combinators*)
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fun curry f x y = f(x,y);
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fun uncurry f(x,y) = f x y;
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fun I x = x and K x y = x;
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(*Combine two functions forming the union of their domains*)
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infix orelf;
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fun f orelf g = fn x => f x handle Match=> g x;
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(*Application of (infix) operator to its left or right argument*)
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fun apl (x,f) y = f(x,y);
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fun apr (f,y) x = f(x,y);
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(*functional for pairs*)
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fun pairself f (x,y) = (f x, f y);
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(*Apply the function to a component of a pair*)
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fun apfst f (x, y) = (f x, y);
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fun apsnd f (x, y) = (x, f y);
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fun square (n: int) = n*n;
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fun fact 0 = 1
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| fact n = n * fact(n-1);
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(*The following versions of fold are designed to fit nicely with infixes.*)
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(* (op @) (e, [x1,...,xn]) ======> ((e @ x1) @ x2) ... @ xn
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for operators that associate to the left. TAIL RECURSIVE*)
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fun foldl (f: 'a * 'b -> 'a) : 'a * 'b list -> 'a =
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let fun itl (e, []) = e
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| itl (e, a::l) = itl (f(e,a), l)
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in itl end;
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(* (op @) ([x1,...,xn], e) ======> x1 @ (x2 ... @ (xn @ e))
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for operators that associate to the right. Not tail recursive.*)
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fun foldr f (l,e) =
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let fun itr [] = e
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| itr (a::l) = f(a, itr l)
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in itr l end;
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(* (op @) [x1,...,xn] ======> x1 @ (x2 ..(x[n-1]. @ xn))
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for n>0, operators that associate to the right. Not tail recursive.*)
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fun foldr1 f l =
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let fun itr [x] = x
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| itr (x::l) = f(x, itr l)
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in itr l end;
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(*Length of a list. Should unquestionably be a standard function*)
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local fun length1 (n, [ ]) = n (*TAIL RECURSIVE*)
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| length1 (n, x::l) = length1 (n+1, l)
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in fun length l = length1 (0,l) end;
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(*Take the first n elements from l.*)
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fun take (n, []) = []
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| take (n, x::xs) = if n>0 then x::take(n-1,xs)
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else [];
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(*Drop the first n elements from l.*)
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fun drop (_, []) = []
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| drop (n, x::xs) = if n>0 then drop (n-1, xs)
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else x::xs;
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(*Return nth element of l, where 0 designates the first element;
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raise EXCEPTION if list too short.*)
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fun nth_elem NL = case (drop NL) of
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[] => raise LIST "nth_elem"
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| x::l => x;
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(*make the list [from, from+1, ..., to]*)
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infix upto;
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fun from upto to =
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if from>to then [] else from :: ((from+1) upto to);
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(*make the list [from, from-1, ..., to]*)
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infix downto;
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fun from downto to =
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if from<to then [] else from :: ((from-1) downto to);
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(* predicate: downto0(is,n) <=> is = [n,n-1,...,0] *)
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fun downto0(i::is,n) = i=n andalso downto0(is,n-1)
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| downto0([],n) = n = ~1;
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(*Like Lisp's MAPC -- seq proc [x1,...,xn] evaluates
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proc(x1); ... ; proc(xn) for side effects.*)
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fun seq (proc: 'a -> unit) : 'a list -> unit =
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let fun seqf [] = ()
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| seqf (x::l) = (proc x; seqf l)
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in seqf end;
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(*** Balanced folding; access to balanced trees ***)
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exception Balance; (*indicates non-positive argument to balancing fun*)
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(*Balanced folding; avoids deep nesting*)
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fun fold_bal f [x] = x
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| fold_bal f [] = raise Balance
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| fold_bal f xs =
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let val k = length xs div 2
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in f (fold_bal f (take(k,xs)),
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fold_bal f (drop(k,xs)))
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end;
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(*Construct something of the form f(...g(...(x)...)) for balanced access*)
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fun access_bal (f,g,x) n i =
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let fun acc n i = (* 1<=i<=n*)
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if n=1 then x else
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let val n2 = n div 2
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in if i<=n2 then f (acc n2 i)
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else g (acc (n-n2) (i-n2))
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end
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in if 1<=i andalso i<=n then acc n i else raise Balance end;
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(*Construct ALL such accesses; could try harder to share recursive calls!*)
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fun accesses_bal (f,g,x) n =
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let fun acc n =
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if n=1 then [x] else
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let val n2 = n div 2
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val acc2 = acc n2
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in if n-n2=n2 then map f acc2 @ map g acc2
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else map f acc2 @ map g (acc (n-n2)) end
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in if 1<=n then acc n else raise Balance end;
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(*** Input/Output ***)
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fun prs s = output(std_out,s);
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fun writeln s = prs (s ^ "\n");
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(*Print error message and abort to top level*)
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exception ERROR;
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fun error (msg) = (writeln msg; raise ERROR);
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fun assert p msg = if p then () else error msg;
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fun deny p msg = if p then error msg else ();
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(*For the "test" target in Makefiles -- signifies successful termination*)
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fun maketest msg =
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(writeln msg;
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output(open_out "test", "Test examples ran successfully\n"));
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(*print a list surrounded by the brackets lpar and rpar, with comma separator
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print nothing for empty list*)
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fun print_list (lpar, rpar, pre: 'a -> unit) (l : 'a list) =
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let fun prec(x) = (prs","; pre(x))
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in case l of
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[] => ()
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| x::l => (prs lpar; pre x; seq prec l; prs rpar)
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end;
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(*print a list of items separated by newlines*)
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fun print_list_ln (pre: 'a -> unit) : 'a list -> unit =
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seq (fn x => (pre x; writeln""));
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fun is_letter ch =
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(ord"A" <= ord ch) andalso (ord ch <= ord"Z") orelse
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(ord"a" <= ord ch) andalso (ord ch <= ord"z");
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fun is_digit ch =
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(ord"0" <= ord ch) andalso (ord ch <= ord"9");
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(*letter or _ or prime (') *)
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fun is_quasi_letter "_" = true
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| is_quasi_letter "'" = true
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| is_quasi_letter ch = is_letter ch;
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(*white space: blanks, tabs, newlines*)
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val is_blank : string -> bool = fn
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" " => true | "\t" => true | "\n" => true | _ => false;
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val is_letdig = is_quasi_letter orf is_digit;
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val to_lower =
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let
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fun lower ch =
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if ch >= "A" andalso ch <= "Z" then
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chr (ord ch - ord "A" + ord "a")
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else ch;
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in
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implode o (map lower) o explode
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end;
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(*** Timing ***)
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(*Unconditional timing function*)
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val timeit = cond_timeit true;
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(*Timed application function*)
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fun timeap f x = timeit(fn()=> f x);
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(*Timed "use" function, printing filenames*)
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fun time_use fname = timeit(fn()=>
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(writeln("\n**** Starting " ^ fname ^ " ****"); use fname;
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writeln("\n**** Finished " ^ fname ^ " ****")));
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(*** Misc functions ***)
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(*Function exponentiation: f(...(f x)...) with n applications of f *)
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fun funpow n f x =
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let fun rep (0,x) = x
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| rep (n,x) = rep (n-1, f x)
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in rep (n,x) end;
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(*Combine two lists forming a list of pairs:
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[x1,...,xn] ~~ [y1,...,yn] ======> [(x1,y1), ..., (xn,yn)] *)
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infix ~~;
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fun [] ~~ [] = []
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| (x::xs) ~~ (y::ys) = (x,y) :: (xs ~~ ys)
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| _ ~~ _ = raise LIST "~~";
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(*Inverse of ~~; the old 'split'.
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[(x1,y1), ..., (xn,yn)] ======> ( [x1,...,xn] , [y1,...,yn] ) *)
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fun split_list (l: ('a*'b)list) = (map #1 l, map #2 l);
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(*make the list [x; x; ...; x] of length n*)
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fun replicate n (x: 'a) : 'a list =
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let fun rep (0,xs) = xs
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| rep (n,xs) = rep(n-1, x::xs)
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in if n<0 then raise LIST "replicate"
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else rep (n,[])
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end;
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(*Flatten a list of lists to a list.*)
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fun flat (ls: 'c list list) : 'c list = foldr (op @) (ls,[]);
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(*** polymorphic set operations ***)
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(*membership in a list*)
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infix mem;
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fun x mem [] = false
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| x mem (y::l) = (x=y) orelse (x mem l);
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(*insertion into list if not already there*)
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infix ins;
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fun x ins xs = if x mem xs then xs else x::xs;
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(*union of sets represented as lists: no repetitions*)
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infix union;
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fun xs union [] = xs
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| [] union ys = ys
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| (x::xs) union ys = xs union (x ins ys);
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infix inter;
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fun [] inter ys = []
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| (x::xs) inter ys = if x mem ys then x::(xs inter ys)
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else xs inter ys;
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infix subset;
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fun [] subset ys = true
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| (x::xs) subset ys = x mem ys andalso xs subset ys;
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(*removing an element from a list WITHOUT duplicates*)
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infix \;
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fun (y::ys) \ x = if x=y then ys else y::(ys \ x)
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| [] \ x = [];
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infix \\;
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val op \\ = foldl (op \);
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(*** option stuff ***)
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datatype 'a option = None | Some of 'a;
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exception OPTION of string;
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fun the (Some x) = x
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| the None = raise OPTION "the";
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fun is_some (Some _) = true
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| is_some None = false;
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fun is_none (Some _) = false
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| is_none None = true;
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(*** Association lists ***)
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(*Association list lookup*)
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fun assoc ([], key) = None
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| assoc ((keyi,xi)::pairs, key) =
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if key=keyi then Some xi else assoc (pairs,key);
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fun assocs ps x = case assoc(ps,x) of None => [] | Some(ys) => ys;
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(*Association list update*)
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fun overwrite(al,p as (key,_)) =
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let fun over((q as (keyi,_))::pairs) =
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if keyi=key then p::pairs else q::(over pairs)
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| over[] = [p]
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in over al end;
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(*Copy the list preserving elements that satisfy the predicate*)
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fun filter (pred: 'a->bool) : 'a list -> 'a list =
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let fun filt [] = []
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| filt (x::xs) = if pred(x) then x :: filt xs else filt xs
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in filt end;
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fun filter_out f = filter (not o f);
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(*** List operations, generalized to an arbitrary equality function "eq"
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-- so what good are equality types?? ***)
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(*removing an element from a list -- possibly WITH duplicates*)
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fun gen_rem eq (xs,y) = filter_out (fn x => eq(x,y)) xs;
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(*generalized membership test*)
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fun gen_mem eq (x, []) = false
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| gen_mem eq (x, y::ys) = eq(x,y) orelse gen_mem eq (x,ys);
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(*generalized insertion*)
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fun gen_ins eq (x,xs) = if gen_mem eq (x,xs) then xs else x::xs;
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(*generalized union*)
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fun gen_union eq (xs,[]) = xs
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| gen_union eq ([],ys) = ys
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| gen_union eq (x::xs,ys) = gen_union eq (xs, gen_ins eq (x,ys));
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(*Generalized association list lookup*)
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fun gen_assoc eq ([], key) = None
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| gen_assoc eq ((keyi,xi)::pairs, key) =
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if eq(key,keyi) then Some xi else gen_assoc eq (pairs,key);
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(** Finding list elements and duplicates **)
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(* find the position of an element in a list *)
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fun find(x,ys) =
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let fun f(y::ys,i) = if x=y then i else f(ys,i+1)
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| f(_,_) = raise LIST "find"
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in f(ys,0) end;
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401 |
(*Returns the tail beginning with the first repeated element, or []. *)
|
|
402 |
fun findrep [] = []
|
|
403 |
| findrep (x::xs) = if x mem xs then x::xs else findrep xs;
|
|
404 |
|
|
405 |
fun distinct1 (seen, []) = rev seen
|
|
406 |
| distinct1 (seen, x::xs) =
|
|
407 |
if x mem seen then distinct1 (seen, xs)
|
|
408 |
else distinct1 (x::seen, xs);
|
|
409 |
|
|
410 |
(*Makes a list of the distinct members of the input*)
|
|
411 |
fun distinct xs = distinct1([],xs);
|
|
412 |
|
|
413 |
|
|
414 |
(*Use the keyfun to make a list of (x,key) pairs.*)
|
|
415 |
fun make_keylist (keyfun: 'a->'b) : 'a list -> ('a * 'b) list =
|
|
416 |
let fun keypair x = (x, keyfun x)
|
|
417 |
in map keypair end;
|
|
418 |
|
|
419 |
(*Given a list of (x,key) pairs and a searchkey
|
|
420 |
return the list of xs from each pair whose key equals searchkey*)
|
|
421 |
fun keyfilter [] searchkey = []
|
|
422 |
| keyfilter ((x,key)::pairs) searchkey =
|
|
423 |
if key=searchkey then x :: keyfilter pairs searchkey
|
|
424 |
else keyfilter pairs searchkey;
|
|
425 |
|
|
426 |
fun mapfilter (f: 'a -> 'b option) ([]: 'a list) = [] : 'b list
|
|
427 |
| mapfilter f (x::xs) =
|
|
428 |
case (f x) of
|
|
429 |
None => mapfilter f xs
|
|
430 |
| Some y => y :: mapfilter f xs;
|
|
431 |
|
|
432 |
|
|
433 |
(*Partition list into elements that satisfy predicate and those that don't.
|
|
434 |
Preserves order of elements in both lists. *)
|
|
435 |
fun partition (pred: 'a->bool) (ys: 'a list) : ('a list * 'a list) =
|
|
436 |
let fun part ([], answer) = answer
|
|
437 |
| part (x::xs, (ys, ns)) = if pred(x)
|
|
438 |
then part (xs, (x::ys, ns))
|
|
439 |
else part (xs, (ys, x::ns))
|
|
440 |
in part (rev ys, ([],[])) end;
|
|
441 |
|
|
442 |
|
|
443 |
fun partition_eq (eq:'a * 'a -> bool) =
|
|
444 |
let fun part [] = []
|
|
445 |
| part (x::ys) = let val (xs,xs') = partition (apl(x,eq)) ys
|
|
446 |
in (x::xs)::(part xs') end
|
|
447 |
in part end;
|
|
448 |
|
|
449 |
|
|
450 |
(*Partition a list into buckets [ bi, b(i+1),...,bj ]
|
|
451 |
putting x in bk if p(k)(x) holds. Preserve order of elements if possible.*)
|
|
452 |
fun partition_list p i j =
|
|
453 |
let fun part k xs =
|
|
454 |
if k>j then
|
|
455 |
(case xs of [] => []
|
|
456 |
| _ => raise LIST "partition_list")
|
|
457 |
else
|
|
458 |
let val (ns,rest) = partition (p k) xs;
|
|
459 |
in ns :: part(k+1)rest end
|
|
460 |
in part i end;
|
|
461 |
|
|
462 |
|
|
463 |
(*Insertion sort. Stable (does not reorder equal elements)
|
|
464 |
'less' is less-than test on type 'a. *)
|
|
465 |
fun sort (less: 'a*'a -> bool) =
|
|
466 |
let fun insert (x, []) = [x]
|
|
467 |
| insert (x, y::ys) =
|
|
468 |
if less(y,x) then y :: insert (x,ys) else x::y::ys;
|
|
469 |
fun sort1 [] = []
|
|
470 |
| sort1 (x::xs) = insert (x, sort1 xs)
|
|
471 |
in sort1 end;
|
|
472 |
|
|
473 |
(*Transitive Closure. Not Warshall's algorithm*)
|
|
474 |
fun transitive_closure [] = []
|
|
475 |
| transitive_closure ((x,ys)::ps) =
|
|
476 |
let val qs = transitive_closure ps
|
|
477 |
val zs = foldl (fn (zs,y) => assocs qs y union zs) (ys,ys)
|
|
478 |
fun step(u,us) = (u, if x mem us then zs union us else us)
|
|
479 |
in (x,zs) :: map step qs end;
|
|
480 |
|
|
481 |
(*** Converting integers to strings, generating identifiers, etc. ***)
|
|
482 |
|
|
483 |
(*Expand the number in the given base
|
|
484 |
example: radixpand(2, 8) gives [1, 0, 0, 0] *)
|
|
485 |
fun radixpand (base,num) : int list =
|
|
486 |
let fun radix (n,tail) =
|
|
487 |
if n<base then n :: tail
|
|
488 |
else radix (n div base, (n mod base) :: tail)
|
|
489 |
in radix (num,[]) end;
|
|
490 |
|
|
491 |
(*Expands a number into a string of characters starting from "zerochar"
|
|
492 |
example: radixstring(2,"0", 8) gives "1000" *)
|
|
493 |
fun radixstring (base,zerochar,num) =
|
|
494 |
let val offset = ord(zerochar);
|
|
495 |
fun chrof n = chr(offset+n)
|
|
496 |
in implode (map chrof (radixpand (base,num))) end;
|
|
497 |
|
|
498 |
fun string_of_int n =
|
|
499 |
if n < 0 then "~" ^ radixstring(10,"0",~n) else radixstring(10,"0",n);
|
|
500 |
|
|
501 |
val print_int = prs o string_of_int;
|
|
502 |
|
|
503 |
local
|
|
504 |
val a = ord("a") and z = ord("z") and A = ord("A") and Z = ord("Z")
|
|
505 |
and k0 = ord("0") and k9 = ord("9")
|
|
506 |
in
|
|
507 |
|
|
508 |
(*Increment a list of letters like a reversed base 26 number.
|
|
509 |
If head is "z", bumps chars in tail.
|
|
510 |
Digits are incremented as if they were integers.
|
|
511 |
"_" and "'" are not changed.
|
|
512 |
For making variants of identifiers. *)
|
|
513 |
|
|
514 |
fun bump_int_list(c::cs) = if c="9" then "0" :: bump_int_list cs else
|
|
515 |
if k0 <= ord(c) andalso ord(c) < k9 then chr(ord(c)+1) :: cs
|
|
516 |
else "1" :: c :: cs
|
|
517 |
| bump_int_list([]) = error("bump_int_list: not an identifier");
|
|
518 |
|
|
519 |
fun bump_list([],d) = [d]
|
|
520 |
| bump_list(["'"],d) = [d,"'"]
|
|
521 |
| bump_list("z"::cs,_) = "a" :: bump_list(cs,"a")
|
|
522 |
| bump_list("Z"::cs,_) = "A" :: bump_list(cs,"A")
|
|
523 |
| bump_list("9"::cs,_) = "0" :: bump_int_list cs
|
|
524 |
| bump_list(c::cs,_) = let val k = ord(c)
|
|
525 |
in if (a <= k andalso k < z) orelse (A <= k andalso k < Z) orelse
|
|
526 |
(k0 <= k andalso k < k9) then chr(k+1) :: cs else
|
|
527 |
if c="'" orelse c="_" then c :: bump_list(cs,"") else
|
|
528 |
error("bump_list: not legal in identifier: " ^
|
|
529 |
implode(rev(c::cs)))
|
|
530 |
end;
|
|
531 |
|
|
532 |
end;
|
|
533 |
|
|
534 |
fun bump_string s : string = implode (rev (bump_list(rev(explode s),"")));
|
|
535 |
|
|
536 |
|
|
537 |
(*** Operations on integer lists ***)
|
|
538 |
|
|
539 |
fun sum [] = 0
|
|
540 |
| sum (n::ns) = n + sum ns;
|
|
541 |
|
|
542 |
fun max[m : int] = m
|
|
543 |
| max(m::n::ns) = if m>n then max(m::ns) else max(n::ns)
|
|
544 |
| max [] = raise LIST "max";
|
|
545 |
|
|
546 |
fun min[m : int] = m
|
|
547 |
| min(m::n::ns) = if m<n then min(m::ns) else min(n::ns)
|
|
548 |
| min [] = raise LIST "min";
|
|
549 |
|
|
550 |
|
|
551 |
(*** Lexical scanning ***)
|
|
552 |
|
|
553 |
(* [x1,...,xi,...,xn] ---> ([x1,...,x(i-1)], [xi,..., xn])
|
|
554 |
where xi is the first element that does not satisfy the predicate*)
|
|
555 |
fun take_prefix (pred : 'a -> bool) (xs: 'a list) : 'a list * 'a list =
|
|
556 |
let fun take (rxs, []) = (rev rxs, [])
|
|
557 |
| take (rxs, x::xs) =
|
|
558 |
if pred x then take(x::rxs, xs) else (rev rxs, x::xs)
|
|
559 |
in take([],xs) end;
|
|
560 |
|
|
561 |
infix prefix;
|
|
562 |
fun [] prefix _ = true
|
|
563 |
| (x::xs) prefix (y::ys) = (x=y) andalso (xs prefix ys)
|
|
564 |
| _ prefix _ = false;
|
|
565 |
|
|
566 |
(* [x1, x2, ..., xn] ---> [x1, s, x2, s, ..., s, xn] *)
|
|
567 |
fun separate s (x :: (xs as _ :: _)) = x :: s :: separate s xs
|
|
568 |
| separate _ xs = xs;
|
|
569 |
|
|
570 |
(*space_implode "..." (explode "hello"); gives "h...e...l...l...o" *)
|
|
571 |
fun space_implode a bs = implode (separate a bs);
|
|
572 |
|
|
573 |
fun quote s = "\"" ^ s ^ "\"";
|
|
574 |
|
|
575 |
(*Concatenate messages, one per line, into a string*)
|
|
576 |
val cat_lines = implode o (map (apr(op^,"\n")));
|
|
577 |
|
|
578 |
(*Scan a list of characters into "words" composed of "letters" (recognized
|
|
579 |
by is_let) and separated by any number of non-"letters".*)
|
|
580 |
fun scanwords is_let cs =
|
|
581 |
let fun scan1 [] = []
|
|
582 |
| scan1 cs =
|
|
583 |
let val (lets, rest) = take_prefix is_let cs
|
|
584 |
in implode lets :: scanwords is_let rest end;
|
|
585 |
in scan1 (#2 (take_prefix (not o is_let) cs)) end;
|