如何为该函数编写类型声明？

Question

I'm still rather new to Haskell and I've encountered a problem that probably shouldn't be too hard to solve but totally stumped me.

I've written a function:

maxFor l n = n * m * (m + 1) / 2 where m = l `div` n

The function belongs to a small module which loads without problems, but throws two errors that are embarrassingly cryptic to me, whenever I try to use the function after loading the module:

<interactive>:182:1:
    No instance for (Integral a0) arising from a use of `maxFor'
    The type variable `a0' is ambiguous
    Possible fix: add a type signature that fixes these type variable(s)
    Note: there are several potential instances:
      instance Integral Int -- Defined in `GHC.Real'
      instance Integral Integer -- Defined in `GHC.Real'
      instance Integral GHC.Types.Word -- Defined in `GHC.Real'
    In the expression: maxFor 999 5
    In an equation for `it': it = maxFor 999 5

<interactive>:182:8:
    No instance for (Num a0) arising from the literal `999'
    The type variable `a0' is ambiguous
    Possible fix: add a type signature that fixes these type variable(s)
    Note: there are several potential instances:
      instance Num Double -- Defined in `GHC.Float'
      instance Num Float -- Defined in `GHC.Float'
      instance Integral a => Num (GHC.Real.Ratio a)
        -- Defined in `GHC.Real'
      ...plus three others
    In the first argument of `maxFor', namely `999'
    In the expression: maxFor 999 5
    In an equation for `it': it = maxFor 999 5

I understand it's telling me my function is missing a proper type declaration, but I don't understand how to write it without the compiler nagging again. I've tried countless variants and that I don't understand the actual problem doesn't help me to solve the problem.

Part of the problem might be that I'm using (/) and div in one function, so the needed type declaration will probably contain Fractional and/or Integral , since:

(/) :: Fractional a => a -> a -> a

and:

div :: Integral a => a -> a -> a

But that's as far as I've got and I'm stuck. How would I write the type declaration for maxFor ? What am I doing wrong? Am I missing or overlooking something that should be obvious?

Any help and constructive feedback is greatly appreciated.

EDIT: I've found this answer on stack overflow, which already helps a little. I'd still appreciate help, but I will of course delete the question, if it turns out I'm being an idiot and end up being downvoted.

Answer 1

_{Note: This post is written in literate Haskell . You can save it as Max.lhs and try it in your GHCi.}

---

A quick solution

First of all, lets reason your use of (/2) . Is the number n * m * (m + 1) odd or even? Well, either m is odd, and m + 1 is even, and therefore the whole product is even, or m is even and the same arguments hold.

So instead of n * m * (m + 1) / 2 , we can use n * m * (m + 1) `div` 2 :

> maxFor l n = n * m * (m + 1) `div` 2 
>    where m = l `div` n

Now, all we have to to is to check the types of the functions we use:

(+)  :: Num a => a -> a -> a
(*)  :: Num a => a -> a -> a
div  :: Integral n => n -> n -> n

And that leads directly to your type:

> maxFor :: Integral n => n -> n -> n

Alternatively, you can use some specific type to get errors out:

maxFor :: Integer -> Integer -> Integer
-- Now GHC yells at you since Integer doesn't have a Fractional instance
-- and you try to use (/).

Background information

You've already recognized the problem. div indicates that l and n are some integral types, but (/) indicates that they're fractional. In your usual Prelude , there is no data type that's an instance of both typeclasses. You can check this in GHCi:

Prelude> :info Fractional
class Num a => Fractional a where
  (/) :: a -> a -> a
  recip :: a -> a
  fromRational :: Rational -> a
    -- Defined in ‘GHC.Real’

instance Fractional Float -- Defined in ‘GHC.Float’
instance Fractional Double -- Defined in ‘GHC.Float’

Prelude> :info Integral
class (Real a, Enum a) => Integral a where
  quot :: a -> a -> a
  rem :: a -> a -> a
  div :: a -> a -> a
  mod :: a -> a -> a
  quotRem :: a -> a -> (a, a)
  divMod :: a -> a -> (a, a)
  toInteger :: a -> Integer
    -- Defined in ‘GHC.Real’
instance Integral Word -- Defined in ‘GHC.Real’
instance Integral Integer -- Defined in ‘GHC.Real’
instance Integral Int -- Defined in ‘GHC.Real’

This is the first part.

The second part stems from literals. 999 or 5 has type Num a => a . Now GHC is between a rock and a hard place: it has to find a Num instance, that's both Integral and Fractional . Since there's no such type (see above), GHC gives up due to ambiguity. You can check this too in GHCi:

Prelude> 5 :: Num a => a
5

Prelude> 5 :: (Fractional a, Integral a) => a
<interactive>:7:1:
    No instance for (Fractional a0) arising from a use of ‘it’
    The type variable ‘a0’ is ambiguous
    Note: there are several potential instances:
      instance Integral a => Fractional (GHC.Real.Ratio a)
        -- Defined in ‘GHC.Real’
      instance Fractional Double -- Defined in ‘GHC.Float’
      instance Fractional Float -- Defined in ‘GHC.Float’
    In the first argument of ‘print’, namely ‘it’
    In a stmt of an interactive GHCi command: print it

So this is mainly the reason why you end up with the error.