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 -- Since many function names (but not the type name) clash with
14 -- "Prelude" names, this module is usually imported @qualified@, e.g.
16 -- > import Data.IntSet (IntSet)
17 -- > import qualified Data.IntSet as IntSet
19 -- The implementation is based on /big-endian patricia trees/. This data
20 -- structure performs especially well on binary operations like 'union'
21 -- and 'intersection'. However, my benchmarks show that it is also
22 -- (much) faster on insertions and deletions when compared to a generic
23 -- size-balanced set implementation (see "Data.Set").
25 -- * Chris Okasaki and Andy Gill, \"/Fast Mergeable Integer Maps/\",
26 -- Workshop on ML, September 1998, pages 77-86,
27 -- <http://www.cse.ogi.edu/~andy/pub/finite.htm>
29 -- * D.R. Morrison, \"/PATRICIA -- Practical Algorithm To Retrieve
30 -- Information Coded In Alphanumeric/\", Journal of the ACM, 15(4),
31 -- October 1968, pages 514-534.
33 -- Many operations have a worst-case complexity of /O(min(n,W))/.
34 -- This means that the operation can become linear in the number of
35 -- elements with a maximum of /W/ -- the number of bits in an 'Int'
37 -----------------------------------------------------------------------------
41 IntSet -- instance Eq,Show
94 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 (Data(..), mkNorepType)
111 import Data.Generics.Instances ()
114 #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 -- | /O(log n)/. Is the element not in the set?
213 notMember :: Int -> IntSet -> Bool
214 notMember k = not . member k
216 -- 'lookup' is used by 'intersection' for left-biasing
217 lookup :: Int -> IntSet -> Maybe Int
219 = let nk = natFromInt k in seq nk (lookupN nk t)
221 lookupN :: Nat -> IntSet -> Maybe Int
225 | zeroN k (natFromInt m) -> lookupN k l
226 | otherwise -> lookupN k r
228 | (k == natFromInt kx) -> Just kx
229 | otherwise -> Nothing
232 {--------------------------------------------------------------------
234 --------------------------------------------------------------------}
235 -- | /O(1)/. The empty set.
240 -- | /O(1)/. A set of one element.
241 singleton :: Int -> IntSet
245 {--------------------------------------------------------------------
247 --------------------------------------------------------------------}
248 -- | /O(min(n,W))/. Add a value to the set. When the value is already
249 -- an element of the set, it is replaced by the new one, ie. 'insert'
251 insert :: Int -> IntSet -> IntSet
255 | nomatch x p m -> join x (Tip x) p t
256 | zero x m -> Bin p m (insert x l) r
257 | otherwise -> Bin p m l (insert x r)
260 | otherwise -> join x (Tip x) y t
263 -- right-biased insertion, used by 'union'
264 insertR :: Int -> IntSet -> IntSet
268 | nomatch x p m -> join x (Tip x) p t
269 | zero x m -> Bin p m (insert x l) r
270 | otherwise -> Bin p m l (insert x r)
273 | otherwise -> join x (Tip x) y t
276 -- | /O(min(n,W))/. Delete a value in the set. Returns the
277 -- original set when the value was not present.
278 delete :: Int -> IntSet -> IntSet
283 | zero x m -> bin p m (delete x l) r
284 | otherwise -> bin p m l (delete x r)
291 {--------------------------------------------------------------------
293 --------------------------------------------------------------------}
294 -- | The union of a list of sets.
295 unions :: [IntSet] -> IntSet
297 = foldlStrict union empty xs
300 -- | /O(n+m)/. The union of two sets.
301 union :: IntSet -> IntSet -> IntSet
302 union t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
303 | shorter m1 m2 = union1
304 | shorter m2 m1 = union2
305 | p1 == p2 = Bin p1 m1 (union l1 l2) (union r1 r2)
306 | otherwise = join p1 t1 p2 t2
308 union1 | nomatch p2 p1 m1 = join p1 t1 p2 t2
309 | zero p2 m1 = Bin p1 m1 (union l1 t2) r1
310 | otherwise = Bin p1 m1 l1 (union r1 t2)
312 union2 | nomatch p1 p2 m2 = join p1 t1 p2 t2
313 | zero p1 m2 = Bin p2 m2 (union t1 l2) r2
314 | otherwise = Bin p2 m2 l2 (union t1 r2)
316 union (Tip x) t = insert x t
317 union t (Tip x) = insertR x t -- right bias
322 {--------------------------------------------------------------------
324 --------------------------------------------------------------------}
325 -- | /O(n+m)/. Difference between two sets.
326 difference :: IntSet -> IntSet -> IntSet
327 difference t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
328 | shorter m1 m2 = difference1
329 | shorter m2 m1 = difference2
330 | p1 == p2 = bin p1 m1 (difference l1 l2) (difference r1 r2)
333 difference1 | nomatch p2 p1 m1 = t1
334 | zero p2 m1 = bin p1 m1 (difference l1 t2) r1
335 | otherwise = bin p1 m1 l1 (difference r1 t2)
337 difference2 | nomatch p1 p2 m2 = t1
338 | zero p1 m2 = difference t1 l2
339 | otherwise = difference t1 r2
341 difference t1@(Tip x) t2
345 difference Nil t = Nil
346 difference t (Tip x) = delete x t
351 {--------------------------------------------------------------------
353 --------------------------------------------------------------------}
354 -- | /O(n+m)/. The intersection of two sets.
355 intersection :: IntSet -> IntSet -> IntSet
356 intersection t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
357 | shorter m1 m2 = intersection1
358 | shorter m2 m1 = intersection2
359 | p1 == p2 = bin p1 m1 (intersection l1 l2) (intersection r1 r2)
362 intersection1 | nomatch p2 p1 m1 = Nil
363 | zero p2 m1 = intersection l1 t2
364 | otherwise = intersection r1 t2
366 intersection2 | nomatch p1 p2 m2 = Nil
367 | zero p1 m2 = intersection t1 l2
368 | otherwise = intersection t1 r2
370 intersection t1@(Tip x) t2
373 intersection t (Tip x)
377 intersection Nil t = Nil
378 intersection t Nil = Nil
382 {--------------------------------------------------------------------
384 --------------------------------------------------------------------}
385 -- | /O(n+m)/. Is this a proper subset? (ie. a subset but not equal).
386 isProperSubsetOf :: IntSet -> IntSet -> Bool
387 isProperSubsetOf t1 t2
388 = case subsetCmp t1 t2 of
392 subsetCmp t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
394 | shorter m2 m1 = subsetCmpLt
395 | p1 == p2 = subsetCmpEq
396 | otherwise = GT -- disjoint
398 subsetCmpLt | nomatch p1 p2 m2 = GT
399 | zero p1 m2 = subsetCmp t1 l2
400 | otherwise = subsetCmp t1 r2
401 subsetCmpEq = case (subsetCmp l1 l2, subsetCmp r1 r2) of
407 subsetCmp (Bin p m l r) t = GT
408 subsetCmp (Tip x) (Tip y)
410 | otherwise = GT -- disjoint
413 | otherwise = GT -- disjoint
414 subsetCmp Nil Nil = EQ
417 -- | /O(n+m)/. Is this a subset?
418 -- @(s1 `isSubsetOf` s2)@ tells whether @s1@ is a subset of @s2@.
420 isSubsetOf :: IntSet -> IntSet -> Bool
421 isSubsetOf t1@(Bin p1 m1 l1 r1) t2@(Bin p2 m2 l2 r2)
422 | shorter m1 m2 = False
423 | shorter m2 m1 = match p1 p2 m2 && (if zero p1 m2 then isSubsetOf t1 l2
424 else isSubsetOf t1 r2)
425 | otherwise = (p1==p2) && isSubsetOf l1 l2 && isSubsetOf r1 r2
426 isSubsetOf (Bin p m l r) t = False
427 isSubsetOf (Tip x) t = member x t
428 isSubsetOf Nil t = True
431 {--------------------------------------------------------------------
433 --------------------------------------------------------------------}
434 -- | /O(n)/. Filter all elements that satisfy some predicate.
435 filter :: (Int -> Bool) -> IntSet -> IntSet
439 -> bin p m (filter pred l) (filter pred r)
445 -- | /O(n)/. partition the set according to some predicate.
446 partition :: (Int -> Bool) -> IntSet -> (IntSet,IntSet)
450 -> let (l1,l2) = partition pred l
451 (r1,r2) = partition pred r
452 in (bin p m l1 r1, bin p m l2 r2)
455 | otherwise -> (Nil,t)
459 -- | /O(log n)/. The expression (@'split' x set@) is a pair @(set1,set2)@
460 -- where all elements in @set1@ are lower than @x@ and all elements in
461 -- @set2@ larger than @x@.
463 -- > split 3 (fromList [1..5]) == (fromList [1,2], fromList [3,4])
464 split :: Int -> IntSet -> (IntSet,IntSet)
468 | m < 0 -> if x >= 0 then let (lt,gt) = split' x l in (union r lt, gt)
469 else let (lt,gt) = split' x r in (lt, union gt l)
470 -- handle negative numbers.
471 | otherwise -> split' x t
475 | otherwise -> (Nil,Nil)
478 split' :: Int -> IntSet -> (IntSet,IntSet)
482 | match x p m -> if zero x m then let (lt,gt) = split' x l in (lt,union gt r)
483 else let (lt,gt) = split' x r in (union l lt,gt)
484 | otherwise -> if x < p then (Nil, t)
489 | otherwise -> (Nil,Nil)
492 -- | /O(log n)/. Performs a 'split' but also returns whether the pivot
493 -- element was found in the original set.
494 splitMember :: Int -> IntSet -> (IntSet,Bool,IntSet)
498 | m < 0 -> if x >= 0 then let (lt,found,gt) = splitMember' x l in (union r lt, found, gt)
499 else let (lt,found,gt) = splitMember' x r in (lt, found, union gt l)
500 -- handle negative numbers.
501 | otherwise -> splitMember' x t
503 | x>y -> (t,False,Nil)
504 | x<y -> (Nil,False,t)
505 | otherwise -> (Nil,True,Nil)
506 Nil -> (Nil,False,Nil)
508 splitMember' :: Int -> IntSet -> (IntSet,Bool,IntSet)
512 | match x p m -> if zero x m then let (lt,found,gt) = splitMember x l in (lt,found,union gt r)
513 else let (lt,found,gt) = splitMember x r in (union l lt,found,gt)
514 | otherwise -> if x < p then (Nil, False, t)
517 | x>y -> (t,False,Nil)
518 | x<y -> (Nil,False,t)
519 | otherwise -> (Nil,True,Nil)
520 Nil -> (Nil,False,Nil)
522 {----------------------------------------------------------------------
524 ----------------------------------------------------------------------}
526 -- | /O(n*min(n,W))/.
527 -- @'map' f s@ is the set obtained by applying @f@ to each element of @s@.
529 -- It's worth noting that the size of the result may be smaller if,
530 -- for some @(x,y)@, @x \/= y && f x == f y@
532 map :: (Int->Int) -> IntSet -> IntSet
533 map f = fromList . List.map f . toList
535 {--------------------------------------------------------------------
537 --------------------------------------------------------------------}
538 -- | /O(n)/. Fold over the elements of a set in an unspecified order.
540 -- > sum set == fold (+) 0 set
541 -- > elems set == fold (:) [] set
542 fold :: (Int -> b -> b) -> b -> IntSet -> b
545 Bin 0 m l r | m < 0 -> foldr f (foldr f z l) r
546 -- put negative numbers before.
547 Bin p m l r -> foldr f z t
551 foldr :: (Int -> b -> b) -> b -> IntSet -> b
554 Bin p m l r -> foldr f (foldr f z r) l
558 {--------------------------------------------------------------------
560 --------------------------------------------------------------------}
561 -- | /O(n)/. The elements of a set. (For sets, this is equivalent to toList)
562 elems :: IntSet -> [Int]
566 {--------------------------------------------------------------------
568 --------------------------------------------------------------------}
569 -- | /O(n)/. Convert the set to a list of elements.
570 toList :: IntSet -> [Int]
574 -- | /O(n)/. Convert the set to an ascending list of elements.
575 toAscList :: IntSet -> [Int]
576 toAscList t = toList t
578 -- | /O(n*min(n,W))/. Create a set from a list of integers.
579 fromList :: [Int] -> IntSet
581 = foldlStrict ins empty xs
585 -- | /O(n*min(n,W))/. Build a set from an ascending list of elements.
586 fromAscList :: [Int] -> IntSet
590 -- | /O(n*min(n,W))/. Build a set from an ascending list of distinct elements.
591 fromDistinctAscList :: [Int] -> IntSet
592 fromDistinctAscList xs
596 {--------------------------------------------------------------------
598 --------------------------------------------------------------------}
599 instance Eq IntSet where
600 t1 == t2 = equal t1 t2
601 t1 /= t2 = nequal t1 t2
603 equal :: IntSet -> IntSet -> Bool
604 equal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
605 = (m1 == m2) && (p1 == p2) && (equal l1 l2) && (equal r1 r2)
606 equal (Tip x) (Tip y)
611 nequal :: IntSet -> IntSet -> Bool
612 nequal (Bin p1 m1 l1 r1) (Bin p2 m2 l2 r2)
613 = (m1 /= m2) || (p1 /= p2) || (nequal l1 l2) || (nequal r1 r2)
614 nequal (Tip x) (Tip y)
616 nequal Nil Nil = False
619 {--------------------------------------------------------------------
621 --------------------------------------------------------------------}
623 instance Ord IntSet where
624 compare s1 s2 = compare (toAscList s1) (toAscList s2)
625 -- tentative implementation. See if more efficient exists.
627 {--------------------------------------------------------------------
629 --------------------------------------------------------------------}
630 instance Show IntSet where
631 showsPrec p xs = showParen (p > 10) $
632 showString "fromList " . shows (toList xs)
634 showSet :: [Int] -> ShowS
638 = showChar '{' . shows x . showTail xs
640 showTail [] = showChar '}'
641 showTail (x:xs) = showChar ',' . shows x . showTail xs
643 {--------------------------------------------------------------------
645 --------------------------------------------------------------------}
646 instance Read IntSet where
647 #ifdef __GLASGOW_HASKELL__
648 readPrec = parens $ prec 10 $ do
649 Ident "fromList" <- lexP
653 readListPrec = readListPrecDefault
655 readsPrec p = readParen (p > 10) $ \ r -> do
656 ("fromList",s) <- lex r
658 return (fromList xs,t)
661 {--------------------------------------------------------------------
663 --------------------------------------------------------------------}
665 #include "Typeable.h"
666 INSTANCE_TYPEABLE0(IntSet,intSetTc,"IntSet")
668 {--------------------------------------------------------------------
670 --------------------------------------------------------------------}
671 -- | /O(n)/. Show the tree that implements the set. The tree is shown
672 -- in a compressed, hanging format.
673 showTree :: IntSet -> String
675 = showTreeWith True False s
678 {- | /O(n)/. The expression (@'showTreeWith' hang wide map@) shows
679 the tree that implements the set. If @hang@ is
680 'True', a /hanging/ tree is shown otherwise a rotated tree is shown. If
681 @wide@ is 'True', an extra wide version is shown.
683 showTreeWith :: Bool -> Bool -> IntSet -> String
684 showTreeWith hang wide t
685 | hang = (showsTreeHang wide [] t) ""
686 | otherwise = (showsTree wide [] [] t) ""
688 showsTree :: Bool -> [String] -> [String] -> IntSet -> ShowS
689 showsTree wide lbars rbars t
692 -> showsTree wide (withBar rbars) (withEmpty rbars) r .
693 showWide wide rbars .
694 showsBars lbars . showString (showBin p m) . showString "\n" .
695 showWide wide lbars .
696 showsTree wide (withEmpty lbars) (withBar lbars) l
698 -> showsBars lbars . showString " " . shows x . showString "\n"
699 Nil -> showsBars lbars . showString "|\n"
701 showsTreeHang :: Bool -> [String] -> IntSet -> ShowS
702 showsTreeHang wide bars t
705 -> showsBars bars . showString (showBin p m) . showString "\n" .
707 showsTreeHang wide (withBar bars) l .
709 showsTreeHang wide (withEmpty bars) r
711 -> showsBars bars . showString " " . shows x . showString "\n"
712 Nil -> showsBars bars . showString "|\n"
715 = "*" -- ++ show (p,m)
718 | wide = showString (concat (reverse bars)) . showString "|\n"
721 showsBars :: [String] -> ShowS
725 _ -> showString (concat (reverse (tail bars))) . showString node
728 withBar bars = "| ":bars
729 withEmpty bars = " ":bars
732 {--------------------------------------------------------------------
734 --------------------------------------------------------------------}
735 {--------------------------------------------------------------------
737 --------------------------------------------------------------------}
738 join :: Prefix -> IntSet -> Prefix -> IntSet -> IntSet
740 | zero p1 m = Bin p m t1 t2
741 | otherwise = Bin p m t2 t1
746 {--------------------------------------------------------------------
747 @bin@ assures that we never have empty trees within a tree.
748 --------------------------------------------------------------------}
749 bin :: Prefix -> Mask -> IntSet -> IntSet -> IntSet
752 bin p m l r = Bin p m l r
755 {--------------------------------------------------------------------
756 Endian independent bit twiddling
757 --------------------------------------------------------------------}
758 zero :: Int -> Mask -> Bool
760 = (natFromInt i) .&. (natFromInt m) == 0
762 nomatch,match :: Int -> Prefix -> Mask -> Bool
769 mask :: Int -> Mask -> Prefix
771 = maskW (natFromInt i) (natFromInt m)
773 zeroN :: Nat -> Nat -> Bool
774 zeroN i m = (i .&. m) == 0
776 {--------------------------------------------------------------------
777 Big endian operations
778 --------------------------------------------------------------------}
779 maskW :: Nat -> Nat -> Prefix
781 = intFromNat (i .&. (complement (m-1) `xor` m))
783 shorter :: Mask -> Mask -> Bool
785 = (natFromInt m1) > (natFromInt m2)
787 branchMask :: Prefix -> Prefix -> Mask
789 = intFromNat (highestBitMask (natFromInt p1 `xor` natFromInt p2))
791 {----------------------------------------------------------------------
792 Finding the highest bit (mask) in a word [x] can be done efficiently in
794 * convert to a floating point value and the mantissa tells us the
795 [log2(x)] that corresponds with the highest bit position. The mantissa
796 is retrieved either via the standard C function [frexp] or by some bit
797 twiddling on IEEE compatible numbers (float). Note that one needs to
798 use at least [double] precision for an accurate mantissa of 32 bit
800 * use bit twiddling, a logarithmic sequence of bitwise or's and shifts (bit).
801 * use processor specific assembler instruction (asm).
803 The most portable way would be [bit], but is it efficient enough?
804 I have measured the cycle counts of the different methods on an AMD
805 Athlon-XP 1800 (~ Pentium III 1.8Ghz) using the RDTSC instruction:
807 highestBitMask: method cycles
814 highestBit: method cycles
821 Wow, the bit twiddling is on today's RISC like machines even faster
822 than a single CISC instruction (BSR)!
823 ----------------------------------------------------------------------}
825 {----------------------------------------------------------------------
826 [highestBitMask] returns a word where only the highest bit is set.
827 It is found by first setting all bits in lower positions than the
828 highest bit and than taking an exclusive or with the original value.
829 Allthough the function may look expensive, GHC compiles this into
830 excellent C code that subsequently compiled into highly efficient
831 machine code. The algorithm is derived from Jorg Arndt's FXT library.
832 ----------------------------------------------------------------------}
833 highestBitMask :: Nat -> Nat
835 = case (x .|. shiftRL x 1) of
836 x -> case (x .|. shiftRL x 2) of
837 x -> case (x .|. shiftRL x 4) of
838 x -> case (x .|. shiftRL x 8) of
839 x -> case (x .|. shiftRL x 16) of
840 x -> case (x .|. shiftRL x 32) of -- for 64 bit platforms
841 x -> (x `xor` (shiftRL x 1))
844 {--------------------------------------------------------------------
846 --------------------------------------------------------------------}
850 (x:xx) -> let z' = f z x in seq z' (foldlStrict f z' xx)
854 {--------------------------------------------------------------------
856 --------------------------------------------------------------------}
857 testTree :: [Int] -> IntSet
858 testTree xs = fromList xs
859 test1 = testTree [1..20]
860 test2 = testTree [30,29..10]
861 test3 = testTree [1,4,6,89,2323,53,43,234,5,79,12,9,24,9,8,423,8,42,4,8,9,3]
863 {--------------------------------------------------------------------
865 --------------------------------------------------------------------}
870 { configMaxTest = 500
871 , configMaxFail = 5000
872 , configSize = \n -> (div n 2 + 3)
873 , configEvery = \n args -> let s = show n in s ++ [ '\b' | _ <- s ]
877 {--------------------------------------------------------------------
878 Arbitrary, reasonably balanced trees
879 --------------------------------------------------------------------}
880 instance Arbitrary IntSet where
881 arbitrary = do{ xs <- arbitrary
882 ; return (fromList xs)
886 {--------------------------------------------------------------------
887 Single, Insert, Delete
888 --------------------------------------------------------------------}
889 prop_Single :: Int -> Bool
891 = (insert x empty == singleton x)
893 prop_InsertDelete :: Int -> IntSet -> Property
894 prop_InsertDelete k t
895 = not (member k t) ==> delete k (insert k t) == t
898 {--------------------------------------------------------------------
900 --------------------------------------------------------------------}
901 prop_UnionInsert :: Int -> IntSet -> Bool
903 = union t (singleton x) == insert x t
905 prop_UnionAssoc :: IntSet -> IntSet -> IntSet -> Bool
906 prop_UnionAssoc t1 t2 t3
907 = union t1 (union t2 t3) == union (union t1 t2) t3
909 prop_UnionComm :: IntSet -> IntSet -> Bool
911 = (union t1 t2 == union t2 t1)
913 prop_Diff :: [Int] -> [Int] -> Bool
915 = toAscList (difference (fromList xs) (fromList ys))
916 == List.sort ((List.\\) (nub xs) (nub ys))
918 prop_Int :: [Int] -> [Int] -> Bool
920 = toAscList (intersection (fromList xs) (fromList ys))
921 == List.sort (nub ((List.intersect) (xs) (ys)))
923 {--------------------------------------------------------------------
925 --------------------------------------------------------------------}
927 = forAll (choose (5,100)) $ \n ->
929 in fromAscList xs == fromList xs
931 prop_List :: [Int] -> Bool
933 = (sort (nub xs) == toAscList (fromList xs))