Is `data PoE a = Empty | Pair a a` a monad?

Question

This question comes from this answer in example of a functor that is Applicative but not a Monad: It is claimed that the

data PoE a = Empty | Pair a a deriving (         


        

Answer 1

Since you are interested in how to do it systematically, here's how I found a counterexample with quickcheck:

{-# LANGUAGE DeriveFunctor #-}

import Control.Monad ((>=>))
import Test.QuickCheck

-- 

Defining an arbitrary instance to generate random PoEs.

instance (Arbitrary a) => Arbitrary (PoE a) where
    arbitrary = do
      emptyq <- arbitrary
      if emptyq
        then return Empty
        else Pair <$> arbitrary <*> arbitrary

And tests for the monad laws:

prop_right_id m = (m >>= return) == m
    where
    _types = (m :: PoE Int)

prop_left_id fun x = (return x >>= f) == f x
    where
    _types = fun :: Fun Int (PoE Int)
    f = applyFun fun

prop_assoc fun gun hun x = (f >=> (g >=> h)) x == ((f >=> g) >=> h) x
    where
    _types = (fun :: Fun Int (PoE Int),
              gun :: Fun Int (PoE Int),
              hun :: Fun Int (PoE Int),
              x :: Int)
    f = applyFun fun
    g = applyFun gun
    h = applyFun hun

I don't get any failures for the identity laws, but prop_assoc does generate a counterexample:

ghci> quickCheck prop_assoc
*** Failed! Falsifiable (after 7 tests and 36 shrinks):
{6->Pair 1 (-1), _->Empty}
{-1->Pair (-3) (-4), 1->Pair (-1) (-2), _->Empty}
{-3->Empty, _->Pair (-2) (-4)}
6

Not that it's terribly helpful for understanding why the failure occurs, it does give you a place to start. If we look carefully, we see that we are passing (-3) and (-2) to the third function; (-3) maps to Empty and (-2) maps to a Pair, so we can't defer to the laws of either of the two monads PoE is composed of.