src/Pure/library.ML

+(*  Title: 	library
+    ID:         $Id$
+    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1992  University of Cambridge
+
+Basic library: booleans, lists, pairs, input/output, etc.
+*)
+
+
+(**** 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 ;
+
+fun notf p x = not (p x) ;
+
+fun orl [] = false
+  | orl (x::l) =  x  orelse  orl l;
+
+fun andl [] = true
+  | andl (x::l) =  x  andalso  andl l;
+
+(*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;
+
+(*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;
+
+
+(*** 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 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;
+
+
+(*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;
+
+
+(*** Balanced folding; access to 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 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 ***)
+
+datatype 'a option = None | Some of 'a;
+
+exception OPTION of string;
+
+fun the (Some x) = x
+  | the None = raise OPTION "the";
+
+fun is_some (Some _) = true
+  | is_some None = false;
+
+fun is_none (Some _) = false
+  | is_none None = true;
+
+
+(*** Association lists ***)
+
+(*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;
+
+(*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;
+
+(*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);
+
+
+(*** List operations, generalized to an arbitrary equality function "eq"
+     -- so what good are equality types?? ***)
+
+(*removing an element from a list -- possibly WITH duplicates*)
+fun gen_rem eq (xs,y) = filter_out (fn x => eq(x,y)) xs;
+
+(*generalized membership test*)
+fun gen_mem eq (x, [])     =  false
+  | gen_mem eq (x, y::ys)  =  eq(x,y) orelse gen_mem eq (x,ys);
+
+(*generalized insertion*)
+fun gen_ins eq (x,xs) = if gen_mem eq (x,xs) then  xs   else  x::xs;
+
+(*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));
+
+(*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);
+
+(** Finding list elements and duplicates **)
+
+(* 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;
+
+(*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;
+
+fun distinct1 (seen, []) = rev seen
+  | distinct1 (seen, x::xs) =
+      if x mem seen then distinct1 (seen, xs)
+    		    else distinct1 (x::seen, xs);
+
+(*Makes a list of the distinct members of the input*)
+fun distinct xs = distinct1([],xs);
+
+
+(*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;
+
+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;
+
+
+(*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;
+
+
+(*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;
+
+(*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. ***)
+
+(*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);
+
+val print_int = prs o string_of_int;
+
+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")
+in
+
+(*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),"")));
+
+
+(*** 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;
+
+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;
+
+(*space_implode "..." (explode "hello");  gives  "h...e...l...l...o" *)
+fun space_implode a bs = implode (separate a bs); 
+
+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 = 
+  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;