[Haskell-beginners] type error in sub-function

[James Toll]

Hi,

I am trying to write a simple function to determine the divisors of an integer.  In its simplest form the type signature should be something like:

divisors :: Int -> [Int]
divisors x = 1 : lower ++ upper ++ x : []
    where lower = filter (\y -> mod x y == 0) [2..(ceiling . sqrt) x]
          upper = sort $ map (div x) lower

Although, I think my type signature isn’t complete as it ignores a lot of what’s going on in the function.  Regardless, the function throws an error when evaluated.  I don’t know if I just need a more accurate type signature, or if there is a bigger problem.  

From what I can tell, my problem is in the lower function, but when I deconstruct it into its parts, they work individually and as a whole, but not as a stand-alone function.  For example, using x = 36:

Prelude> [2..(ceiling . sqrt) 36]
[2,3,4,5,6]
Prelude> filter (\y -> mod 36 y == 0) it
[2,3,4,6]

Or as a whole:

Prelude> filter (\y -> mod 36 y == 0) [2..(ceiling . sqrt) 36]
[2,3,4,6]

But when I define this as a function, it throws an error when evaluated.

Prelude> let lower x = filter (\y -> mod x y == 0) [2..(ceiling . sqrt) x]
Prelude> lower 36

<interactive>:6:1:
    No instance for (RealFrac b0) arising from a use of `lower'
    The type variable `b0' is ambiguous
    Possible fix: add a type signature that fixes these type variable(s)
    Note: there are several potential instances:
      instance RealFrac Double -- Defined in `GHC.Float'
      instance RealFrac Float -- Defined in `GHC.Float'
      instance Integral a => RealFrac (GHC.Real.Ratio a)
        -- Defined in `GHC.Real'
    In the expression: lower 36
    In an equation for `it': it = lower 36

<interactive>:6:7:
    No instance for (Num b0) arising from the literal `36'
    The type variable `b0' 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 `lower', namely `36'
    In the expression: lower 36
    In an equation for `it': it = lower 36


At this point, I’m not sure what I need to do to get his working properly.  It appears to be a type error, but I’m apparently not understanding the error message enough to fix the problem.  Any suggestions would be appreciated.

 Thanks,

James