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+
+
+--------------------------------------------------
+-- The class of ordered sets
+-- as described in section 2.2.1
+
+-- and an example implementation, 
+-- derived from the implementation in appendix A.1
+
+
+module OrdSet (OrdSet(..), Set) where
+
+import List (intersperse)
+
+
+--------------------------------------------------
+-- the class of ordered sets
+
+class OrdSet m where
+  emptySet  :: Ord a => m a
+  unitSet   :: Ord a => a -> m a
+  isEmpty   :: Ord a => m a -> Bool
+  elemSet   :: Ord a => a -> m a -> Bool
+  (<++>)    :: Ord a => m a -> m a -> m a
+  (<\\>)    :: Ord a => m a -> m a -> m a
+  plusMinus :: Ord a => m a -> m a -> (m a, m a)
+  union     :: Ord a => [m a] -> m a
+  makeSet   :: Ord a => [a] -> m a
+  elems     :: Ord a => m a -> [a]
+  ordSet    :: Ord a => [a] -> m a
+  limit     :: Ord a => (a -> m a) -> m a -> m a
+
+  xs <++> ys      = fst (plusMinus xs ys)
+  xs <\\> ys      = snd (plusMinus xs ys)
+  plusMinus xs ys = (xs <++> ys, xs <\\> ys)
+
+  union []   = emptySet
+  union [xs] = xs
+  union xyss = union xss <++> union yss
+    where (xss, yss)       = split xyss
+	  split (x:y:xyss) = let (xs, ys) = split xyss in (x:xs, y:ys)
+	  split xs         = (xs, [])
+
+  makeSet xs = union (map unitSet xs)
+
+  limit more start = limit' (start, start)
+    where limit' (old, new)
+	      | isEmpty new' = old
+	      | otherwise    = limit' (plusMinus new' old)
+	    where new'       = union (map more (elems new))
+
+
+--------------------------------------------------
+-- sets as ordered lists, 
+-- paired with a binary tree
+
+data Set a = Set [a] (TreeSet a)
+
+instance Eq a => Eq (Set a) where
+  Set xs _ == Set ys _ = xs == ys
+
+instance Ord a => Ord (Set a) where
+  compare (Set xs _) (Set ys _) = compare xs ys
+
+instance Show a => Show (Set a) where
+  show (Set xs _) = "{" ++ concat (intersperse "," (map show xs)) ++ "}"
+
+instance OrdSet Set where
+  emptySet  = Set []  (makeTree [])
+  unitSet a = Set [a] (makeTree [a])
+
+  isEmpty   (Set xs _) = null xs
+  elemSet a (Set _ xt) = elemTree a xt
+
+  plusMinus (Set xs _) (Set ys _) = (Set ps (makeTree ps), Set ms (makeTree ms))
+    where (ps, ms) = plm xs ys
+	  plm [] ys = (ys, [])
+	  plm xs [] = (xs, xs)
+	  plm xs@(x:xs') ys@(y:ys') = case compare x y of
+				        LT -> let (ps, ms) = plm xs' ys  in (x:ps, x:ms)
+					GT -> let (ps, ms) = plm xs  ys' in (y:ps,   ms)
+					EQ -> let (ps, ms) = plm xs' ys' in (x:ps,   ms)
+
+  elems (Set xs _) = xs
+  ordSet xs        = Set xs (makeTree xs)
+
+
+--------------------------------------------------
+-- binary search trees
+-- for logarithmic lookup time
+
+data TreeSet a = Nil | Node (TreeSet a) a (TreeSet a)
+
+makeTree xs = tree
+  where (tree,[])       = sl2bst (length xs) xs
+	sl2bst 0 xs     = (Nil, xs)
+	sl2bst 1 (a:xs) = (Node Nil a Nil, xs)
+	sl2bst n xs     = (Node ltree a rtree, zs)
+          where llen    = (n-1) `div` 2
+                rlen    = n - 1 - llen
+                (ltree, a:ys) = sl2bst llen xs
+                (rtree, zs)   = sl2bst rlen ys
+
+elemTree a Nil = False
+elemTree a (Node ltree x rtree) 
+    = case compare a x of
+        LT -> elemTree a ltree 
+        GT -> elemTree a rtree 
+        EQ -> True
+
+