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module ListSet where
type Set a = [a]
empty :: Set a
empty = []
insert :: Ord a => a -> Set a -> Set a
insert a [] = [a]
insert a (x:xs)
| a < x = a:x:xs
| a > x = x:insert a xs
| a == x = x:xs
set :: Ord a => [a] -> Set a
set = foldr insert empty
ordered [] = True
ordered [x] = True
ordered (x:y:zs) = x <= y && ordered (y:zs)
allDiff [] = True
allDiff (x:xs) = x `notElem` xs && allDiff xs
isSet s = ordered s && allDiff s
-- Properties
infixr 0 -->
False --> _ = True
True --> x = x
prop_insertSet :: (Char, Set Char) -> Bool
prop_insertSet (c, s) = ordered s --> ordered (insert c s)
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