1 {-# OPTIONS -cpp -fglasgow-exts #-}
2 -----------------------------------------------------------------------------
4 -- Module : Data.IntSet
5 -- Copyright : (c) Daan Leijen 2002
7 -- Maintainer : libraries@haskell.org
8 -- Stability : provisional
9 -- Portability : portable
11 -- An efficient implementation of integer sets.
13 -- This module is intended to be imported @qualified@, to avoid name
14 -- clashes with "Prelude" functions. eg.
16 -- > import Data.IntSet as Set
18 -- The implementation is based on /big-endian patricia trees/. This data
19 -- structure performs especially well on binary operations like 'union'
20 -- and 'intersection'. However, my benchmarks show that it is also
21 -- (much) faster on insertions and deletions when compared to a generic
22 -- size-balanced set implementation (see "Data.Set").
24 -- * Chris Okasaki and Andy Gill, \"/Fast Mergeable Integer Maps/\",
25 -- Workshop on ML, September 1998, pages 77-86,
26 -- <http://www.cse.ogi.edu/~andy/pub/finite.htm>
28 -- * D.R. Morrison, \"/PATRICIA -- Practical Algorithm To Retrieve
29 -- Information Coded In Alphanumeric/\", Journal of the ACM, 15(4),
30 -- October 1968, pages 514-534.
32 -- Many operations have a worst-case complexity of /O(min(n,W))/.
33 -- This means that the operation can become linear in the number of
34 -- elements with a maximum of /W/ -- the number of bits in an 'Int'
36 -----------------------------------------------------------------------------
40 IntSet -- instance Eq,Show
92 import Prelude hiding (lookup,filter,foldr,foldl,null,map)
96 import qualified Data.List as List
97 import Data.Monoid (Monoid(..))
103 import List (nub,sort)
104 import qualified List
107 #if __GLASGOW_HASKELL__
109 import Data.Generics.Basics
110 import Data.Generics.Instances
113 #if __GLASGOW_HASKELL__ >= 503
115 import GHC.Exts ( Word(..), Int(..), shiftRL# )
116 #elif __GLASGOW_HASKELL__
118 import GlaExts ( Word(..), Int(..), shiftRL# )
123 infixl 9 \\{-This comment teaches CPP correct behaviour -}
125 -- A "Nat" is a natural machine word (an unsigned Int)
128 natFromInt :: Int -> Nat
129 natFromInt i = fromIntegral i
131 intFromNat :: Nat -> Int
132 intFromNat w = fromIntegral w
134 shiftRL :: Nat -> Int -> Nat
135 #if __GLASGOW_HASKELL__
136 {--------------------------------------------------------------------
137 GHC: use unboxing to get @shiftRL@ inlined.
138 --------------------------------------------------------------------}
139 shiftRL (W# x) (I# i)
142 shiftRL x i = shiftR x i
145 {--------------------------------------------------------------------
147 --------------------------------------------------------------------}
148 -- | /O(n+m)/. See 'difference'.
149 (\\) :: IntSet -> IntSet -> IntSet
150 m1 \\ m2 = difference m1 m2
152 {--------------------------------------------------------------------
154 --------------------------------------------------------------------}
155 -- | A set of integers.
157 | Tip {-# UNPACK #-} !Int
158 | Bin {-# UNPACK #-} !Prefix {-# UNPACK #-} !Mask !IntSet !IntSet
163 instance Monoid IntSet where
168 #if __GLASGOW_HASKELL__
170 {--------------------------------------------------------------------
172 --------------------------------------------------------------------}
174 -- This instance preserves data abstraction at the cost of inefficiency.
175 -- We omit reflection services for the sake of data abstraction.
177 instance Data IntSet where
178 gfoldl f z is = z fromList `f` (toList is)
179 toConstr _ = error "toConstr"
180 gunfold _ _ = error "gunfold"
181 dataTypeOf _ = mkNorepType "Data.IntSet.IntSet"
185 {--------------------------------------------------------------------
187 --------------------------------------------------------------------}
188 -- | /O(1)/. Is the set empty?
189 null :: IntSet -> Bool
193 -- | /O(n)/. Cardinality of the set.
194 size :: IntSet -> Int
197 Bin p m l r -> size l + size r
201 -- | /O(min(n,W))/. Is the value a member of the set?
202 member :: Int -> IntSet -> Bool
206 | nomatch x p m -> False
207 | zero x m -> member x l
208 | otherwise -> member x r
212 -- 'lookup' is used by 'intersection' for left-biasing
213 lookup :: Int -> IntSet -> Maybe Int
215 = let nk = natFromInt k in seq nk (lookupN nk t)
217 lookupN :: Nat -> IntSet -> Maybe Int
221 | zeroN k (natFromInt m) -> lookupN k l
222 | otherwise -> lookupN k r
224 | (k == natFromInt kx) -> Just kx
225 | otherwise -> Nothing
228 {--------------------------------------------------------------------
230 --------------------------------------------------------------------}
231 -- | /O(1)/. The empty set.
236 -- | /O(1)/. A set of one element.
237 singleton :: Int -> IntSet
241 {--------------------------------------------------------------------
243 --------------------------------------------------------------------}
244 -- | /O(min(n,W))/. Add a value to the set. When the value is already
245 -- an element of the set, it is replaced by the new one, ie. 'insert'
247 insert :: Int -> IntSet -> IntSet
251 | nomatch x p m -> join x (Tip x) p t
252 | zero x m -> Bin p m (insert x l) r
253 | otherwise -> Bin p m l (insert x r)
256 | otherwise -> join x (Tip x) y t
259 -- right-biased insertion, used by 'union'
260 insertR :: Int -> IntSet -> IntSet
264 | nomatch x p m -> join x (Tip x) p t
265 | zero x m -> Bin p m (insert x l) r
266 | otherwise -> Bin p m l (insert x r)
269 | otherwise -> join x (Tip x) y t
272 -- | /O(min(n,W))/. Delete a value in the set. Returns the
273 -- original set when the value was not present.
274 delete :: Int -> IntSet -> IntSet
279 | zero x m -> bin p m (delete x l) r
280 | otherwise -> bin p m l (delete x r)
287 {--------------------------------------------------------------------
289 --------------------------------------------------------------------}
290 -- | The union of a list of sets.
291 unions :: [IntSet] -> IntSet
293 = foldlStrict union empty xs
296 -- | /O(n+m)/. The union of two sets.
297 union :: IntSet -> IntSet -> IntSet
298 union t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
299 | shorter m1 m2 = union1
300 | shorter m2 m1 = union2
301 | p1 == p2 = Bin p1 m1 (union l1 l2) (union r1 r2)
302 | otherwise = join p1 t1 p2 t2
304 union1 | nomatch p2 p1 m1 = join p1 t1 p2 t2
305 | zero p2 m1 = Bin p1 m1 (union l1 t2) r1
306 | otherwise = Bin p1 m1 l1 (union r1 t2)
308 union2 | nomatch p1 p2 m2 = join p1 t1 p2 t2
309 | zero p1 m2 = Bin p2 m2 (union t1 l2) r2
310 | otherwise = Bin p2 m2 l2 (union t1 r2)
312 union (Tip x) t = insert x t
313 union t (Tip x) = insertR x t -- right bias
318 {--------------------------------------------------------------------
320 --------------------------------------------------------------------}
321 -- | /O(n+m)/. Difference between two sets.
322 difference :: IntSet -> IntSet -> IntSet
323 difference t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
324 | shorter m1 m2 = difference1
325 | shorter m2 m1 = difference2
326 | p1 == p2 = bin p1 m1 (difference l1 l2) (difference r1 r2)
329 difference1 | nomatch p2 p1 m1 = t1
330 | zero p2 m1 = bin p1 m1 (difference l1 t2) r1
331 | otherwise = bin p1 m1 l1 (difference r1 t2)
333 difference2 | nomatch p1 p2 m2 = t1
334 | zero p1 m2 = difference t1 l2
335 | otherwise = difference t1 r2
337 difference t1@(Tip x) t2
341 difference Nil t = Nil
342 difference t (Tip x) = delete x t
347 {--------------------------------------------------------------------
349 --------------------------------------------------------------------}
350 -- | /O(n+m)/. The intersection of two sets.
351 intersection :: IntSet -> IntSet -> IntSet
352 intersection t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
353 | shorter m1 m2 = intersection1
354 | shorter m2 m1 = intersection2
355 | p1 == p2 = bin p1 m1 (intersection l1 l2) (intersection r1 r2)
358 intersection1 | nomatch p2 p1 m1 = Nil
359 | zero p2 m1 = intersection l1 t2
360 | otherwise = intersection r1 t2
362 intersection2 | nomatch p1 p2 m2 = Nil
363 | zero p1 m2 = intersection t1 l2
364 | otherwise = intersection t1 r2
366 intersection t1@(Tip x) t2
369 intersection t (Tip x)
373 intersection Nil t = Nil
374 intersection t Nil = Nil
378 {--------------------------------------------------------------------
380 --------------------------------------------------------------------}
381 -- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal).
382 isProperSubsetOf :: IntSet -> IntSet -> Bool
383 isProperSubsetOf t1 t2
384 = case subsetCmp t1 t2 of
388 subsetCmp t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
390 | shorter m2 m1 = subsetCmpLt
391 | p1 == p2 = subsetCmpEq
392 | otherwise = GT -- disjoint
394 subsetCmpLt | nomatch p1 p2 m2 = GT
395 | zero p1 m2 = subsetCmp t1 l2
396 | otherwise = subsetCmp t1 r2
397 subsetCmpEq = case (subsetCmp l1 l2, subsetCmp r1 r2) of
403 subsetCmp (Bin p m l r) t = GT
404 subsetCmp (Tip x) (Tip y)
406 | otherwise = GT -- disjoint
409 | otherwise = GT -- disjoint
410 subsetCmp Nil Nil = EQ
413 -- | /O(n+m)/. Is this a subset?
414 -- @(s1 `isSubsetOf` s2)@ tells whether @s1@ is a subset of @s2@.
416 isSubsetOf :: IntSet -> IntSet -> Bool
417 isSubsetOf t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
418 | shorter m1 m2 = False
419 | shorter m2 m1 = match p1 p2 m2 && (if zero p1 m2 then isSubsetOf t1 l2
420 else isSubsetOf t1 r2)
421 | otherwise = (p1==p2) && isSubsetOf l1 l2 && isSubsetOf r1 r2
422 isSubsetOf (Bin p m l r) t = False
423 isSubsetOf (Tip x) t = member x t
424 isSubsetOf Nil t = True
427 {--------------------------------------------------------------------
429 --------------------------------------------------------------------}
430 -- | /O(n)/. Filter all elements that satisfy some predicate.
431 filter :: (Int -> Bool) -> IntSet -> IntSet
435 -> bin p m (filter pred l) (filter pred r)
441 -- | /O(n)/. partition the set according to some predicate.
442 partition :: (Int -> Bool) -> IntSet -> (IntSet,IntSet)
446 -> let (l1,l2) = partition pred l
447 (r1,r2) = partition pred r
448 in (bin p m l1 r1, bin p m l2 r2)
451 | otherwise -> (Nil,t)
455 -- | /O(log n)/. The expression (@'split' x set@) is a pair @(set1,set2)@
456 -- where all elements in @set1@ are lower than @x@ and all elements in
457 -- @set2@ larger than @x@.
459 -- > split 3 (fromList [1..5]) == (fromList [1,2], fromList [3,4])
460 split :: Int -> IntSet -> (IntSet,IntSet)
464 | m < 0 -> if x >= 0 then let (lt,gt) = split' x l in (union r lt, gt)
465 else let (lt,gt) = split' x r in (lt, union gt l)
466 -- handle negative numbers.
467 | otherwise -> split' x t
471 | otherwise -> (Nil,Nil)
474 split' :: Int -> IntSet -> (IntSet,IntSet)
478 | match x p m -> if zero x m then let (lt,gt) = split' x l in (lt,union gt r)
479 else let (lt,gt) = split' x r in (union l lt,gt)
480 | otherwise -> if x < p then (Nil, t)
485 | otherwise -> (Nil,Nil)
488 -- | /O(log n)/. Performs a 'split' but also returns whether the pivot
489 -- element was found in the original set.
490 splitMember :: Int -> IntSet -> (IntSet,Bool,IntSet)
494 | m < 0 -> if x >= 0 then let (lt,found,gt) = splitMember' x l in (union r lt, found, gt)
495 else let (lt,found,gt) = splitMember' x r in (lt, found, union gt l)
496 -- handle negative numbers.
497 | otherwise -> splitMember' x t
499 | x>y -> (t,False,Nil)
500 | x<y -> (Nil,False,t)
501 | otherwise -> (Nil,True,Nil)
502 Nil -> (Nil,False,Nil)
504 splitMember' :: Int -> IntSet -> (IntSet,Bool,IntSet)
508 | match x p m -> if zero x m then let (lt,found,gt) = splitMember x l in (lt,found,union gt r)
509 else let (lt,found,gt) = splitMember x r in (union l lt,found,gt)
510 | otherwise -> if x < p then (Nil, False, t)
513 | x>y -> (t,False,Nil)
514 | x<y -> (Nil,False,t)
515 | otherwise -> (Nil,True,Nil)
516 Nil -> (Nil,False,Nil)
518 {----------------------------------------------------------------------
520 ----------------------------------------------------------------------}
522 -- | /O(n*min(n,W))/.
523 -- @'map' f s@ is the set obtained by applying @f@ to each element of @s@.
525 -- It's worth noting that the size of the result may be smaller if,
526 -- for some @(x,y)@, @x \/= y && f x == f y@
528 map :: (Int->Int) -> IntSet -> IntSet
529 map f = fromList . List.map f . toList
531 {--------------------------------------------------------------------
533 --------------------------------------------------------------------}
534 -- | /O(n)/. Fold over the elements of a set in an unspecified order.
536 -- > sum set == fold (+) 0 set
537 -- > elems set == fold (:) [] set
538 fold :: (Int -> b -> b) -> b -> IntSet -> b
541 Bin 0 m l r | m < 0 -> foldr f (foldr f z l) r
542 -- put negative numbers before.
543 Bin p m l r -> foldr f z t
547 foldr :: (Int -> b -> b) -> b -> IntSet -> b
550 Bin p m l r -> foldr f (foldr f z r) l
554 {--------------------------------------------------------------------
556 --------------------------------------------------------------------}
557 -- | /O(n)/. The elements of a set. (For sets, this is equivalent to toList)
558 elems :: IntSet -> [Int]
562 {--------------------------------------------------------------------
564 --------------------------------------------------------------------}
565 -- | /O(n)/. Convert the set to a list of elements.
566 toList :: IntSet -> [Int]
570 -- | /O(n)/. Convert the set to an ascending list of elements.
571 toAscList :: IntSet -> [Int]
572 toAscList t = toList t
574 -- | /O(n*min(n,W))/. Create a set from a list of integers.
575 fromList :: [Int] -> IntSet
577 = foldlStrict ins empty xs
581 -- | /O(n*min(n,W))/. Build a set from an ascending list of elements.
582 fromAscList :: [Int] -> IntSet
586 -- | /O(n*min(n,W))/. Build a set from an ascending list of distinct elements.
587 fromDistinctAscList :: [Int] -> IntSet
588 fromDistinctAscList xs
592 {--------------------------------------------------------------------
594 --------------------------------------------------------------------}
595 instance Eq IntSet where
596 t1 == t2 = equal t1 t2
597 t1 /= t2 = nequal t1 t2
599 equal :: IntSet -> IntSet -> Bool
600 equal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
601 = (m1 == m2) && (p1 == p2) && (equal l1 l2) && (equal r1 r2)
602 equal (Tip x) (Tip y)
607 nequal :: IntSet -> IntSet -> Bool
608 nequal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
609 = (m1 /= m2) || (p1 /= p2) || (nequal l1 l2) || (nequal r1 r2)
610 nequal (Tip x) (Tip y)
612 nequal Nil Nil = False
615 {--------------------------------------------------------------------
617 --------------------------------------------------------------------}
619 instance Ord IntSet where
620 compare s1 s2 = compare (toAscList s1) (toAscList s2)
621 -- tentative implementation. See if more efficient exists.
623 {--------------------------------------------------------------------
625 --------------------------------------------------------------------}
626 instance Show IntSet where
627 showsPrec p xs = showParen (p > 10) $
628 showString "fromList " . shows (toList xs)
630 showSet :: [Int] -> ShowS
634 = showChar '{' . shows x . showTail xs
636 showTail [] = showChar '}'
637 showTail (x:xs) = showChar ',' . shows x . showTail xs
639 {--------------------------------------------------------------------
641 --------------------------------------------------------------------}
642 instance Read IntSet where
643 #ifdef __GLASGOW_HASKELL__
644 readPrec = parens $ prec 10 $ do
645 Ident "fromList" <- lexP
649 readListPrec = readListPrecDefault
651 readsPrec p = readParen (p > 10) $ \ r -> do
652 ("fromList",s) <- lex r
654 return (fromList xs,t)
657 {--------------------------------------------------------------------
659 --------------------------------------------------------------------}
661 #include "Typeable.h"
662 INSTANCE_TYPEABLE0(IntSet,intSetTc,"IntSet")
664 {--------------------------------------------------------------------
666 --------------------------------------------------------------------}
667 -- | /O(n)/. Show the tree that implements the set. The tree is shown
668 -- in a compressed, hanging format.
669 showTree :: IntSet -> String
671 = showTreeWith True False s
674 {- | /O(n)/. The expression (@'showTreeWith' hang wide map@) shows
675 the tree that implements the set. If @hang@ is
676 'True', a /hanging/ tree is shown otherwise a rotated tree is shown. If
677 @wide@ is 'True', an extra wide version is shown.
679 showTreeWith :: Bool -> Bool -> IntSet -> String
680 showTreeWith hang wide t
681 | hang = (showsTreeHang wide [] t) ""
682 | otherwise = (showsTree wide [] [] t) ""
684 showsTree :: Bool -> [String] -> [String] -> IntSet -> ShowS
685 showsTree wide lbars rbars t
688 -> showsTree wide (withBar rbars) (withEmpty rbars) r .
689 showWide wide rbars .
690 showsBars lbars . showString (showBin p m) . showString "\n" .
691 showWide wide lbars .
692 showsTree wide (withEmpty lbars) (withBar lbars) l
694 -> showsBars lbars . showString " " . shows x . showString "\n"
695 Nil -> showsBars lbars . showString "|\n"
697 showsTreeHang :: Bool -> [String] -> IntSet -> ShowS
698 showsTreeHang wide bars t
701 -> showsBars bars . showString (showBin p m) . showString "\n" .
703 showsTreeHang wide (withBar bars) l .
705 showsTreeHang wide (withEmpty bars) r
707 -> showsBars bars . showString " " . shows x . showString "\n"
708 Nil -> showsBars bars . showString "|\n"
711 = "*" -- ++ show (p,m)
714 | wide = showString (concat (reverse bars)) . showString "|\n"
717 showsBars :: [String] -> ShowS
721 _ -> showString (concat (reverse (tail bars))) . showString node
724 withBar bars = "| ":bars
725 withEmpty bars = " ":bars
728 {--------------------------------------------------------------------
730 --------------------------------------------------------------------}
731 {--------------------------------------------------------------------
733 --------------------------------------------------------------------}
734 join :: Prefix -> IntSet -> Prefix -> IntSet -> IntSet
736 | zero p1 m = Bin p m t1 t2
737 | otherwise = Bin p m t2 t1
742 {--------------------------------------------------------------------
743 @bin@ assures that we never have empty trees within a tree.
744 --------------------------------------------------------------------}
745 bin :: Prefix -> Mask -> IntSet -> IntSet -> IntSet
748 bin p m l r = Bin p m l r
751 {--------------------------------------------------------------------
752 Endian independent bit twiddling
753 --------------------------------------------------------------------}
754 zero :: Int -> Mask -> Bool
756 = (natFromInt i) .&. (natFromInt m) == 0
758 nomatch,match :: Int -> Prefix -> Mask -> Bool
765 mask :: Int -> Mask -> Prefix
767 = maskW (natFromInt i) (natFromInt m)
769 zeroN :: Nat -> Nat -> Bool
770 zeroN i m = (i .&. m) == 0
772 {--------------------------------------------------------------------
773 Big endian operations
774 --------------------------------------------------------------------}
775 maskW :: Nat -> Nat -> Prefix
777 = intFromNat (i .&. (complement (m-1) `xor` m))
779 shorter :: Mask -> Mask -> Bool
781 = (natFromInt m1) > (natFromInt m2)
783 branchMask :: Prefix -> Prefix -> Mask
785 = intFromNat (highestBitMask (natFromInt p1 `xor` natFromInt p2))
787 {----------------------------------------------------------------------
788 Finding the highest bit (mask) in a word [x] can be done efficiently in
790 * convert to a floating point value and the mantissa tells us the
791 [log2(x)] that corresponds with the highest bit position. The mantissa
792 is retrieved either via the standard C function [frexp] or by some bit
793 twiddling on IEEE compatible numbers (float). Note that one needs to
794 use at least [double] precision for an accurate mantissa of 32 bit
796 * use bit twiddling, a logarithmic sequence of bitwise or's and shifts (bit).
797 * use processor specific assembler instruction (asm).
799 The most portable way would be [bit], but is it efficient enough?
800 I have measured the cycle counts of the different methods on an AMD
801 Athlon-XP 1800 (~ Pentium III 1.8Ghz) using the RDTSC instruction:
803 highestBitMask: method cycles
810 highestBit: method cycles
817 Wow, the bit twiddling is on today's RISC like machines even faster
818 than a single CISC instruction (BSR)!
819 ----------------------------------------------------------------------}
821 {----------------------------------------------------------------------
822 [highestBitMask] returns a word where only the highest bit is set.
823 It is found by first setting all bits in lower positions than the
824 highest bit and than taking an exclusive or with the original value.
825 Allthough the function may look expensive, GHC compiles this into
826 excellent C code that subsequently compiled into highly efficient
827 machine code. The algorithm is derived from Jorg Arndt's FXT library.
828 ----------------------------------------------------------------------}
829 highestBitMask :: Nat -> Nat
831 = case (x .|. shiftRL x 1) of
832 x -> case (x .|. shiftRL x 2) of
833 x -> case (x .|. shiftRL x 4) of
834 x -> case (x .|. shiftRL x 8) of
835 x -> case (x .|. shiftRL x 16) of
836 x -> case (x .|. shiftRL x 32) of -- for 64 bit platforms
837 x -> (x `xor` (shiftRL x 1))
840 {--------------------------------------------------------------------
842 --------------------------------------------------------------------}
846 (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)
850 {--------------------------------------------------------------------
852 --------------------------------------------------------------------}
853 testTree :: [Int] -> IntSet
854 testTree xs = fromList xs
855 test1 = testTree [1..20]
856 test2 = testTree [30,29..10]
857 test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]
859 {--------------------------------------------------------------------
861 --------------------------------------------------------------------}
866 { configMaxTest = 500
867 , configMaxFail = 5000
868 , configSize = \n -> (div n 2 + 3)
869 , configEvery = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
873 {--------------------------------------------------------------------
874 Arbitrary, reasonably balanced trees
875 --------------------------------------------------------------------}
876 instance Arbitrary IntSet where
877 arbitrary = do{ xs <- arbitrary
878 ; return (fromList xs)
882 {--------------------------------------------------------------------
883 Single, Insert, Delete
884 --------------------------------------------------------------------}
885 prop_Single :: Int -> Bool
887 = (insert x empty == singleton x)
889 prop_InsertDelete :: Int -> IntSet -> Property
890 prop_InsertDelete k t
891 = not (member k t) ==> delete k (insert k t) == t
894 {--------------------------------------------------------------------
896 --------------------------------------------------------------------}
897 prop_UnionInsert :: Int -> IntSet -> Bool
899 = union t (singleton x) == insert x t
901 prop_UnionAssoc :: IntSet -> IntSet -> IntSet -> Bool
902 prop_UnionAssoc t1 t2 t3
903 = union t1 (union t2 t3) == union (union t1 t2) t3
905 prop_UnionComm :: IntSet -> IntSet -> Bool
907 = (union t1 t2 == union t2 t1)
909 prop_Diff :: [Int] -> [Int] -> Bool
911 = toAscList (difference (fromList xs) (fromList ys))
912 == List.sort ((List.\\) (nub xs) (nub ys))
914 prop_Int :: [Int] -> [Int] -> Bool
916 = toAscList (intersection (fromList xs) (fromList ys))
917 == List.sort (nub ((List.intersect) (xs) (ys)))
919 {--------------------------------------------------------------------
921 --------------------------------------------------------------------}
923 = forAll (choose (5,100)) $ \n ->
925 in fromAscList xs == fromList xs
927 prop_List :: [Int] -> Bool
929 = (sort (nub xs) == toAscList (fromList xs))