Testing 'DAG' aka acyclic graphs properties in Haskell QuickCheck

Question

module Graph where

import Control.Monad.State
import Data.Maybe
import Data.Set as Set

-- | 'Edge' represents an edge entre two nodes.
data Edge v = Edge {source :: v, target :: v}
              deriving (Show,Eq,Ord)

data Graph v = Graph {nodes :: Set v, edges :: Set (Edge v)}
               deriving Show

-- | 'Eq' for graphs.
instance Ord v => Eq (Graph v) where
    g == g' = nodes g == nodes g' && edges g == edges g'

isValid :: Ord v => Graph v -> Bool
isValid g = (Set.map source (edges g) `Set.union` Set.map target (edges g)) `isSubsetOf` nodes g

-- | 'isDAG' tests if a graph is acyclic
isDAG :: Ord v => Graph v -> Bool
isDAG g = isValid g && all nocycle (nodes g)
    where nocycle v = all (\a -> v `notMember` reachable g a) $ Set.map target (adj g v)

I'm trying to use QuickCheck to test 'isDAG', could you give me an property that 'isDAG' abides?

prop_isDAG :: Graph Int -> Property
prop_isDAG g = 

I believe that's the accurate function declaration, if not, feel free to change it!

---

Here is another example that I've made for another function, to guide possible helpers on what i'm looking for:

-- | The function 'isSubgraphOf' tests if the first graph is subgraph of the second.
isSubgraphOf :: Ord v => Graph v -> Graph v -> Bool
isSubgraphOf g g' = isSubsetOf (nodes g) (nodes g') && isSubsetOf (edges g) (edges g')

And this is the QuickCheck property that it abides:

-- QuickCheck proprety to test 'isSubgraphOf'
prop_isSubgraphOf :: Graph Int -> Property
prop_isSubgraphOf g = forAll (elements $ elems $ nodes g) $ \v -> adj g v `isSubsetOf` edges g

Answer 1

prop_dag :: Property
prop_dag = forAll (dag :: Gen (DAG Int)) $ \g -> collect (length (edges g)) $ isDAG g

This was what I was looking for.