Lab 4

[BertLisser]

Exercise 5

type Rel a = [(a,a)]

symClos :: Ord a => Rel a -> Rel a
symClos []         = []

symClos :: Ord a => Rel a -> Rel a
symClos ((x,y):xs) = (nub $ sort $ (y,x):(x,y):(symClos xs))

Better and cheaper

symClos' []         = []
symClos' ls@((x,y):xs) = (y,x):ls
symClos xs = nub $ sort symclos' xs

Exercise 7
Nice readable documentation by using mathematical symbols!
{-
trClos:
Test properties:
∀ x, y, z: xSy, ySz => xSz
R ⊆ S
∀ S' ⊂ S: S' is not a transitive closure of R
-}
The following function is not a good idea.

prop_TrClosSmallest r = isSmallest $ trClos r
        where
            isSmallest s = r ⊆ s && isTransitive s && (not . hasSmaller) s
            hasSmaller []       = False
            hasSmaller [x]      = isSmallest []
            hasSmaller (x:y:xs) = isSmallest (x:xs) && isSmallest (y:xs)
            types = (r :: Rel Int)

Why not

prop_TrClosSmallest r  = not (Any (\x -> not( isTransitive r\\ [x] )  r)