[Haskell-cafe] Fixing undeduceable instance ==> overlapping instance

[Michael Orlitzky]

I'm trying to write a determinant function that works on matrices
parameterized by their dimensions (Peano naturals). If I declare the
following,

  -- Define a class so that we get a different determinant function
  -- on the base/recursive cases.
  class (Eq a, Ring.C a) => Determined m a where
    determinant :: (m a) -> a

  -- Base case, 1x1 matrices
  instance (Eq a, Ring.C a) => Determined (Mat (S Z) (S Z)) a where
    determinant m = m !!! (0,0)

  -- Recursive case, (n+2) x (n+2) matrices.
  instance (Eq a, Ring.C a, Arity n)
         => Determined (Mat (S (S n)) (S (S n))) a where
  determinant m =
    ...
    -- Recursive algorithm, the i,jth minor has dimension
    -- (n+1) x (n+1).
    foo bar (determinant (minor m i j))

I get an error stating that I'm missing an instance:

  Could not deduce (Determined (Mat (S n) (S n)) a)
  ...

Clearly, I *have* an instance for that case: if n == Z, then it's the
base case. If not, it's the recursive case. But GHC can't figure that
out. So maybe if I define a dummy instance to make it happy, it won't
notice that they overlap?

  instance (Eq a, Ring.C a) => Determined (Mat m m) a where
    determinant _ = undefined

No such luck:

  >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
  >>> determinant m

    Overlapping instances for Determined (Mat N2 N2) Int
      arising from a use of `determinant'
    Matching instances:
      instance (Eq a, Ring.C a) => Determined (Mat m m) a
        -- Defined at Linear/Matrix2.hs:353:10
      instance (Eq a, Ring.C a, Arity n) =>
               Determined (Mat (S (S n)) (S (S n))) a
    ...

I even tried generalizing the (Mat m m) instance definition so that
OverlappingInstances would pick the one I want, but I can't get that to
work either.

Is there some way to massage this?