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
93 import Prelude hiding (lookup,filter,foldr,foldl,null,map)
97 import qualified Data.List as List
98 import Data.Monoid (Monoid(..))
104 import List (nub,sort)
105 import qualified List
108 #if __GLASGOW_HASKELL__
110 import Data.Generics.Basics
111 import Data.Generics.Instances
114 #if __GLASGOW_HASKELL__ >= 503
116 import GHC.Exts ( Word(..), Int(..), shiftRL# )
117 #elif __GLASGOW_HASKELL__
119 import GlaExts ( Word(..), Int(..), shiftRL# )
124 infixl 9 \\{-This comment teaches CPP correct behaviour -}
126 -- A "Nat" is a natural machine word (an unsigned Int)
129 natFromInt :: Int -> Nat
130 natFromInt i = fromIntegral i
132 intFromNat :: Nat -> Int
133 intFromNat w = fromIntegral w
135 shiftRL :: Nat -> Int -> Nat
136 #if __GLASGOW_HASKELL__
137 {--------------------------------------------------------------------
138 GHC: use unboxing to get @shiftRL@ inlined.
139 --------------------------------------------------------------------}
140 shiftRL (W# x) (I# i)
143 shiftRL x i = shiftR x i
146 {--------------------------------------------------------------------
148 --------------------------------------------------------------------}
149 -- | /O(n+m)/. See 'difference'.
150 (\\) :: IntSet -> IntSet -> IntSet
151 m1 \\ m2 = difference m1 m2
153 {--------------------------------------------------------------------
155 --------------------------------------------------------------------}
156 -- | A set of integers.
158 | Tip {-# UNPACK #-} !Int
159 | Bin {-# UNPACK #-} !Prefix {-# UNPACK #-} !Mask !IntSet !IntSet
164 instance Monoid IntSet where
169 #if __GLASGOW_HASKELL__
171 {--------------------------------------------------------------------
173 --------------------------------------------------------------------}
175 -- This instance preserves data abstraction at the cost of inefficiency.
176 -- We omit reflection services for the sake of data abstraction.
178 instance Data IntSet where
179 gfoldl f z is = z fromList `f` (toList is)
180 toConstr _ = error "toConstr"
181 gunfold _ _ = error "gunfold"
182 dataTypeOf _ = mkNorepType "Data.IntSet.IntSet"
186 {--------------------------------------------------------------------
188 --------------------------------------------------------------------}
189 -- | /O(1)/. Is the set empty?
190 null :: IntSet -> Bool
194 -- | /O(n)/. Cardinality of the set.
195 size :: IntSet -> Int
198 Bin p m l r -> size l + size r
202 -- | /O(min(n,W))/. Is the value a member of the set?
203 member :: Int -> IntSet -> Bool
207 | nomatch x p m -> False
208 | zero x m -> member x l
209 | otherwise -> member x r
213 -- | /O(log n)/. Is the element not in the set?
214 notMember :: Int -> IntSet -> Bool
215 notMember k = not . member k
217 -- 'lookup' is used by 'intersection' for left-biasing
218 lookup :: Int -> IntSet -> Maybe Int
220 = let nk = natFromInt k in seq nk (lookupN nk t)
222 lookupN :: Nat -> IntSet -> Maybe Int
226 | zeroN k (natFromInt m) -> lookupN k l
227 | otherwise -> lookupN k r
229 | (k == natFromInt kx) -> Just kx
230 | otherwise -> Nothing
233 {--------------------------------------------------------------------
235 --------------------------------------------------------------------}
236 -- | /O(1)/. The empty set.
241 -- | /O(1)/. A set of one element.
242 singleton :: Int -> IntSet
246 {--------------------------------------------------------------------
248 --------------------------------------------------------------------}
249 -- | /O(min(n,W))/. Add a value to the set. When the value is already
250 -- an element of the set, it is replaced by the new one, ie. 'insert'
252 insert :: Int -> IntSet -> IntSet
256 | nomatch x p m -> join x (Tip x) p t
257 | zero x m -> Bin p m (insert x l) r
258 | otherwise -> Bin p m l (insert x r)
261 | otherwise -> join x (Tip x) y t
264 -- right-biased insertion, used by 'union'
265 insertR :: Int -> IntSet -> IntSet
269 | nomatch x p m -> join x (Tip x) p t
270 | zero x m -> Bin p m (insert x l) r
271 | otherwise -> Bin p m l (insert x r)
274 | otherwise -> join x (Tip x) y t
277 -- | /O(min(n,W))/. Delete a value in the set. Returns the
278 -- original set when the value was not present.
279 delete :: Int -> IntSet -> IntSet
284 | zero x m -> bin p m (delete x l) r
285 | otherwise -> bin p m l (delete x r)
292 {--------------------------------------------------------------------
294 --------------------------------------------------------------------}
295 -- | The union of a list of sets.
296 unions :: [IntSet] -> IntSet
298 = foldlStrict union empty xs
301 -- | /O(n+m)/. The union of two sets.
302 union :: IntSet -> IntSet -> IntSet
303 union t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
304 | shorter m1 m2 = union1
305 | shorter m2 m1 = union2
306 | p1 == p2 = Bin p1 m1 (union l1 l2) (union r1 r2)
307 | otherwise = join p1 t1 p2 t2
309 union1 | nomatch p2 p1 m1 = join p1 t1 p2 t2
310 | zero p2 m1 = Bin p1 m1 (union l1 t2) r1
311 | otherwise = Bin p1 m1 l1 (union r1 t2)
313 union2 | nomatch p1 p2 m2 = join p1 t1 p2 t2
314 | zero p1 m2 = Bin p2 m2 (union t1 l2) r2
315 | otherwise = Bin p2 m2 l2 (union t1 r2)
317 union (Tip x) t = insert x t
318 union t (Tip x) = insertR x t -- right bias
323 {--------------------------------------------------------------------
325 --------------------------------------------------------------------}
326 -- | /O(n+m)/. Difference between two sets.
327 difference :: IntSet -> IntSet -> IntSet
328 difference t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
329 | shorter m1 m2 = difference1
330 | shorter m2 m1 = difference2
331 | p1 == p2 = bin p1 m1 (difference l1 l2) (difference r1 r2)
334 difference1 | nomatch p2 p1 m1 = t1
335 | zero p2 m1 = bin p1 m1 (difference l1 t2) r1
336 | otherwise = bin p1 m1 l1 (difference r1 t2)
338 difference2 | nomatch p1 p2 m2 = t1
339 | zero p1 m2 = difference t1 l2
340 | otherwise = difference t1 r2
342 difference t1@(Tip x) t2
346 difference Nil t = Nil
347 difference t (Tip x) = delete x t
352 {--------------------------------------------------------------------
354 --------------------------------------------------------------------}
355 -- | /O(n+m)/. The intersection of two sets.
356 intersection :: IntSet -> IntSet -> IntSet
357 intersection t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
358 | shorter m1 m2 = intersection1
359 | shorter m2 m1 = intersection2
360 | p1 == p2 = bin p1 m1 (intersection l1 l2) (intersection r1 r2)
363 intersection1 | nomatch p2 p1 m1 = Nil
364 | zero p2 m1 = intersection l1 t2
365 | otherwise = intersection r1 t2
367 intersection2 | nomatch p1 p2 m2 = Nil
368 | zero p1 m2 = intersection t1 l2
369 | otherwise = intersection t1 r2
371 intersection t1@(Tip x) t2
374 intersection t (Tip x)
378 intersection Nil t = Nil
379 intersection t Nil = Nil
383 {--------------------------------------------------------------------
385 --------------------------------------------------------------------}
386 -- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal).
387 isProperSubsetOf :: IntSet -> IntSet -> Bool
388 isProperSubsetOf t1 t2
389 = case subsetCmp t1 t2 of
393 subsetCmp t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
395 | shorter m2 m1 = subsetCmpLt
396 | p1 == p2 = subsetCmpEq
397 | otherwise = GT -- disjoint
399 subsetCmpLt | nomatch p1 p2 m2 = GT
400 | zero p1 m2 = subsetCmp t1 l2
401 | otherwise = subsetCmp t1 r2
402 subsetCmpEq = case (subsetCmp l1 l2, subsetCmp r1 r2) of
408 subsetCmp (Bin p m l r) t = GT
409 subsetCmp (Tip x) (Tip y)
411 | otherwise = GT -- disjoint
414 | otherwise = GT -- disjoint
415 subsetCmp Nil Nil = EQ
418 -- | /O(n+m)/. Is this a subset?
419 -- @(s1 `isSubsetOf` s2)@ tells whether @s1@ is a subset of @s2@.
421 isSubsetOf :: IntSet -> IntSet -> Bool
422 isSubsetOf t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
423 | shorter m1 m2 = False
424 | shorter m2 m1 = match p1 p2 m2 && (if zero p1 m2 then isSubsetOf t1 l2
425 else isSubsetOf t1 r2)
426 | otherwise = (p1==p2) && isSubsetOf l1 l2 && isSubsetOf r1 r2
427 isSubsetOf (Bin p m l r) t = False
428 isSubsetOf (Tip x) t = member x t
429 isSubsetOf Nil t = True
432 {--------------------------------------------------------------------
434 --------------------------------------------------------------------}
435 -- | /O(n)/. Filter all elements that satisfy some predicate.
436 filter :: (Int -> Bool) -> IntSet -> IntSet
440 -> bin p m (filter pred l) (filter pred r)
446 -- | /O(n)/. partition the set according to some predicate.
447 partition :: (Int -> Bool) -> IntSet -> (IntSet,IntSet)
451 -> let (l1,l2) = partition pred l
452 (r1,r2) = partition pred r
453 in (bin p m l1 r1, bin p m l2 r2)
456 | otherwise -> (Nil,t)
460 -- | /O(log n)/. The expression (@'split' x set@) is a pair @(set1,set2)@
461 -- where all elements in @set1@ are lower than @x@ and all elements in
462 -- @set2@ larger than @x@.
464 -- > split 3 (fromList [1..5]) == (fromList [1,2], fromList [3,4])
465 split :: Int -> IntSet -> (IntSet,IntSet)
469 | m < 0 -> if x >= 0 then let (lt,gt) = split' x l in (union r lt, gt)
470 else let (lt,gt) = split' x r in (lt, union gt l)
471 -- handle negative numbers.
472 | otherwise -> split' x t
476 | otherwise -> (Nil,Nil)
479 split' :: Int -> IntSet -> (IntSet,IntSet)
483 | match x p m -> if zero x m then let (lt,gt) = split' x l in (lt,union gt r)
484 else let (lt,gt) = split' x r in (union l lt,gt)
485 | otherwise -> if x < p then (Nil, t)
490 | otherwise -> (Nil,Nil)
493 -- | /O(log n)/. Performs a 'split' but also returns whether the pivot
494 -- element was found in the original set.
495 splitMember :: Int -> IntSet -> (IntSet,Bool,IntSet)
499 | m < 0 -> if x >= 0 then let (lt,found,gt) = splitMember' x l in (union r lt, found, gt)
500 else let (lt,found,gt) = splitMember' x r in (lt, found, union gt l)
501 -- handle negative numbers.
502 | otherwise -> splitMember' x t
504 | x>y -> (t,False,Nil)
505 | x<y -> (Nil,False,t)
506 | otherwise -> (Nil,True,Nil)
507 Nil -> (Nil,False,Nil)
509 splitMember' :: Int -> IntSet -> (IntSet,Bool,IntSet)
513 | match x p m -> if zero x m then let (lt,found,gt) = splitMember x l in (lt,found,union gt r)
514 else let (lt,found,gt) = splitMember x r in (union l lt,found,gt)
515 | otherwise -> if x < p then (Nil, False, t)
518 | x>y -> (t,False,Nil)
519 | x<y -> (Nil,False,t)
520 | otherwise -> (Nil,True,Nil)
521 Nil -> (Nil,False,Nil)
523 {----------------------------------------------------------------------
525 ----------------------------------------------------------------------}
527 -- | /O(n*min(n,W))/.
528 -- @'map' f s@ is the set obtained by applying @f@ to each element of @s@.
530 -- It's worth noting that the size of the result may be smaller if,
531 -- for some @(x,y)@, @x \/= y && f x == f y@
533 map :: (Int->Int) -> IntSet -> IntSet
534 map f = fromList . List.map f . toList
536 {--------------------------------------------------------------------
538 --------------------------------------------------------------------}
539 -- | /O(n)/. Fold over the elements of a set in an unspecified order.
541 -- > sum set == fold (+) 0 set
542 -- > elems set == fold (:) [] set
543 fold :: (Int -> b -> b) -> b -> IntSet -> b
546 Bin 0 m l r | m < 0 -> foldr f (foldr f z l) r
547 -- put negative numbers before.
548 Bin p m l r -> foldr f z t
552 foldr :: (Int -> b -> b) -> b -> IntSet -> b
555 Bin p m l r -> foldr f (foldr f z r) l
559 {--------------------------------------------------------------------
561 --------------------------------------------------------------------}
562 -- | /O(n)/. The elements of a set. (For sets, this is equivalent to toList)
563 elems :: IntSet -> [Int]
567 {--------------------------------------------------------------------
569 --------------------------------------------------------------------}
570 -- | /O(n)/. Convert the set to a list of elements.
571 toList :: IntSet -> [Int]
575 -- | /O(n)/. Convert the set to an ascending list of elements.
576 toAscList :: IntSet -> [Int]
577 toAscList t = toList t
579 -- | /O(n*min(n,W))/. Create a set from a list of integers.
580 fromList :: [Int] -> IntSet
582 = foldlStrict ins empty xs
586 -- | /O(n*min(n,W))/. Build a set from an ascending list of elements.
587 fromAscList :: [Int] -> IntSet
591 -- | /O(n*min(n,W))/. Build a set from an ascending list of distinct elements.
592 fromDistinctAscList :: [Int] -> IntSet
593 fromDistinctAscList xs
597 {--------------------------------------------------------------------
599 --------------------------------------------------------------------}
600 instance Eq IntSet where
601 t1 == t2 = equal t1 t2
602 t1 /= t2 = nequal t1 t2
604 equal :: IntSet -> IntSet -> Bool
605 equal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
606 = (m1 == m2) && (p1 == p2) && (equal l1 l2) && (equal r1 r2)
607 equal (Tip x) (Tip y)
612 nequal :: IntSet -> IntSet -> Bool
613 nequal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
614 = (m1 /= m2) || (p1 /= p2) || (nequal l1 l2) || (nequal r1 r2)
615 nequal (Tip x) (Tip y)
617 nequal Nil Nil = False
620 {--------------------------------------------------------------------
622 --------------------------------------------------------------------}
624 instance Ord IntSet where
625 compare s1 s2 = compare (toAscList s1) (toAscList s2)
626 -- tentative implementation. See if more efficient exists.
628 {--------------------------------------------------------------------
630 --------------------------------------------------------------------}
631 instance Show IntSet where
632 showsPrec p xs = showParen (p > 10) $
633 showString "fromList " . shows (toList xs)
635 showSet :: [Int] -> ShowS
639 = showChar '{' . shows x . showTail xs
641 showTail [] = showChar '}'
642 showTail (x:xs) = showChar ',' . shows x . showTail xs
644 {--------------------------------------------------------------------
646 --------------------------------------------------------------------}
647 instance Read IntSet where
648 #ifdef __GLASGOW_HASKELL__
649 readPrec = parens $ prec 10 $ do
650 Ident "fromList" <- lexP
654 readListPrec = readListPrecDefault
656 readsPrec p = readParen (p > 10) $ \ r -> do
657 ("fromList",s) <- lex r
659 return (fromList xs,t)
662 {--------------------------------------------------------------------
664 --------------------------------------------------------------------}
666 #include "Typeable.h"
667 INSTANCE_TYPEABLE0(IntSet,intSetTc,"IntSet")
669 {--------------------------------------------------------------------
671 --------------------------------------------------------------------}
672 -- | /O(n)/. Show the tree that implements the set. The tree is shown
673 -- in a compressed, hanging format.
674 showTree :: IntSet -> String
676 = showTreeWith True False s
679 {- | /O(n)/. The expression (@'showTreeWith' hang wide map@) shows
680 the tree that implements the set. If @hang@ is
681 'True', a /hanging/ tree is shown otherwise a rotated tree is shown. If
682 @wide@ is 'True', an extra wide version is shown.
684 showTreeWith :: Bool -> Bool -> IntSet -> String
685 showTreeWith hang wide t
686 | hang = (showsTreeHang wide [] t) ""
687 | otherwise = (showsTree wide [] [] t) ""
689 showsTree :: Bool -> [String] -> [String] -> IntSet -> ShowS
690 showsTree wide lbars rbars t
693 -> showsTree wide (withBar rbars) (withEmpty rbars) r .
694 showWide wide rbars .
695 showsBars lbars . showString (showBin p m) . showString "\n" .
696 showWide wide lbars .
697 showsTree wide (withEmpty lbars) (withBar lbars) l
699 -> showsBars lbars . showString " " . shows x . showString "\n"
700 Nil -> showsBars lbars . showString "|\n"
702 showsTreeHang :: Bool -> [String] -> IntSet -> ShowS
703 showsTreeHang wide bars t
706 -> showsBars bars . showString (showBin p m) . showString "\n" .
708 showsTreeHang wide (withBar bars) l .
710 showsTreeHang wide (withEmpty bars) r
712 -> showsBars bars . showString " " . shows x . showString "\n"
713 Nil -> showsBars bars . showString "|\n"
716 = "*" -- ++ show (p,m)
719 | wide = showString (concat (reverse bars)) . showString "|\n"
722 showsBars :: [String] -> ShowS
726 _ -> showString (concat (reverse (tail bars))) . showString node
729 withBar bars = "| ":bars
730 withEmpty bars = " ":bars
733 {--------------------------------------------------------------------
735 --------------------------------------------------------------------}
736 {--------------------------------------------------------------------
738 --------------------------------------------------------------------}
739 join :: Prefix -> IntSet -> Prefix -> IntSet -> IntSet
741 | zero p1 m = Bin p m t1 t2
742 | otherwise = Bin p m t2 t1
747 {--------------------------------------------------------------------
748 @bin@ assures that we never have empty trees within a tree.
749 --------------------------------------------------------------------}
750 bin :: Prefix -> Mask -> IntSet -> IntSet -> IntSet
753 bin p m l r = Bin p m l r
756 {--------------------------------------------------------------------
757 Endian independent bit twiddling
758 --------------------------------------------------------------------}
759 zero :: Int -> Mask -> Bool
761 = (natFromInt i) .&. (natFromInt m) == 0
763 nomatch,match :: Int -> Prefix -> Mask -> Bool
770 mask :: Int -> Mask -> Prefix
772 = maskW (natFromInt i) (natFromInt m)
774 zeroN :: Nat -> Nat -> Bool
775 zeroN i m = (i .&. m) == 0
777 {--------------------------------------------------------------------
778 Big endian operations
779 --------------------------------------------------------------------}
780 maskW :: Nat -> Nat -> Prefix
782 = intFromNat (i .&. (complement (m-1) `xor` m))
784 shorter :: Mask -> Mask -> Bool
786 = (natFromInt m1) > (natFromInt m2)
788 branchMask :: Prefix -> Prefix -> Mask
790 = intFromNat (highestBitMask (natFromInt p1 `xor` natFromInt p2))
792 {----------------------------------------------------------------------
793 Finding the highest bit (mask) in a word [x] can be done efficiently in
795 * convert to a floating point value and the mantissa tells us the
796 [log2(x)] that corresponds with the highest bit position. The mantissa
797 is retrieved either via the standard C function [frexp] or by some bit
798 twiddling on IEEE compatible numbers (float). Note that one needs to
799 use at least [double] precision for an accurate mantissa of 32 bit
801 * use bit twiddling, a logarithmic sequence of bitwise or's and shifts (bit).
802 * use processor specific assembler instruction (asm).
804 The most portable way would be [bit], but is it efficient enough?
805 I have measured the cycle counts of the different methods on an AMD
806 Athlon-XP 1800 (~ Pentium III 1.8Ghz) using the RDTSC instruction:
808 highestBitMask: method cycles
815 highestBit: method cycles
822 Wow, the bit twiddling is on today's RISC like machines even faster
823 than a single CISC instruction (BSR)!
824 ----------------------------------------------------------------------}
826 {----------------------------------------------------------------------
827 [highestBitMask] returns a word where only the highest bit is set.
828 It is found by first setting all bits in lower positions than the
829 highest bit and than taking an exclusive or with the original value.
830 Allthough the function may look expensive, GHC compiles this into
831 excellent C code that subsequently compiled into highly efficient
832 machine code. The algorithm is derived from Jorg Arndt's FXT library.
833 ----------------------------------------------------------------------}
834 highestBitMask :: Nat -> Nat
836 = case (x .|. shiftRL x 1) of
837 x -> case (x .|. shiftRL x 2) of
838 x -> case (x .|. shiftRL x 4) of
839 x -> case (x .|. shiftRL x 8) of
840 x -> case (x .|. shiftRL x 16) of
841 x -> case (x .|. shiftRL x 32) of -- for 64 bit platforms
842 x -> (x `xor` (shiftRL x 1))
845 {--------------------------------------------------------------------
847 --------------------------------------------------------------------}
851 (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)
855 {--------------------------------------------------------------------
857 --------------------------------------------------------------------}
858 testTree :: [Int] -> IntSet
859 testTree xs = fromList xs
860 test1 = testTree [1..20]
861 test2 = testTree [30,29..10]
862 test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]
864 {--------------------------------------------------------------------
866 --------------------------------------------------------------------}
871 { configMaxTest = 500
872 , configMaxFail = 5000
873 , configSize = \n -> (div n 2 + 3)
874 , configEvery = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
878 {--------------------------------------------------------------------
879 Arbitrary, reasonably balanced trees
880 --------------------------------------------------------------------}
881 instance Arbitrary IntSet where
882 arbitrary = do{ xs <- arbitrary
883 ; return (fromList xs)
887 {--------------------------------------------------------------------
888 Single, Insert, Delete
889 --------------------------------------------------------------------}
890 prop_Single :: Int -> Bool
892 = (insert x empty == singleton x)
894 prop_InsertDelete :: Int -> IntSet -> Property
895 prop_InsertDelete k t
896 = not (member k t) ==> delete k (insert k t) == t
899 {--------------------------------------------------------------------
901 --------------------------------------------------------------------}
902 prop_UnionInsert :: Int -> IntSet -> Bool
904 = union t (singleton x) == insert x t
906 prop_UnionAssoc :: IntSet -> IntSet -> IntSet -> Bool
907 prop_UnionAssoc t1 t2 t3
908 = union t1 (union t2 t3) == union (union t1 t2) t3
910 prop_UnionComm :: IntSet -> IntSet -> Bool
912 = (union t1 t2 == union t2 t1)
914 prop_Diff :: [Int] -> [Int] -> Bool
916 = toAscList (difference (fromList xs) (fromList ys))
917 == List.sort ((List.\\) (nub xs) (nub ys))
919 prop_Int :: [Int] -> [Int] -> Bool
921 = toAscList (intersection (fromList xs) (fromList ys))
922 == List.sort (nub ((List.intersect) (xs) (ys)))
924 {--------------------------------------------------------------------
926 --------------------------------------------------------------------}
928 = forAll (choose (5,100)) $ \n ->
930 in fromAscList xs == fromList xs
932 prop_List :: [Int] -> Bool
934 = (sort (nub xs) == toAscList (fromList xs))