Haskell could not unify type instance equations

Question

I am trying to tag canonical Nat datatype with (Even/Odd) Parity kind to see if we can get any free theorems. Here is the code:

{-# LANGUAGE GADTs #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE DataKinds #-}

-- Use DataKind promotion with type function for even-odd

module EvenOdd where

  data Parity = Even | Odd
  -- Parity is promoted to kind level Parity.
  -- Even & Odd to type level 'Even & 'Odd of kind Parity

  -- We define type-function opp to establish the relation that
  -- type 'Even is opposite of 'Odd, and vice-versa
  type family Opp (n :: Parity) :: Parity
  type instance Opp 'Even = 'Odd
  type instance Opp 'Odd = 'Even

 -- We tag natural number with the type of its parity
  data Nat :: Parity -> * where
     Zero :: Nat 'Even
     Succ :: Nat p -> Nat (Opp p)

  -- Now we (should) get free theorems.
  -- 1. Plus of two even numbers is even
  evenPlus :: Nat 'Even -> Nat 'Even -> Nat 'Even
  evenPlus Zero n2 = n2  -- Line 31
  evenPlus (Succ (Succ n1)) n2 = Succ (Succ (evenPlus n1 n2))

However, GHC throws type error:

Could not deduce (p1 ~ 'Even)
from the context ('Even ~ Opp p)
  bound by a pattern with constructor
             Succ :: forall (p :: Parity). Nat p -> Nat (Opp p),
           in an equation for `evenPlus'
  at even-odd.hs:31:13-26
or from (p ~ Opp p1)
  bound by a pattern with constructor
             Succ :: forall (p :: Parity). Nat p -> Nat (Opp p),
           in an equation for `evenPlus'
  at even-odd.hs:31:19-25
  `p1' is a rigid type variable bound by
       a pattern with constructor
         Succ :: forall (p :: Parity). Nat p -> Nat (Opp p),
       in an equation for `evenPlus'
       at even-odd.hs:31:19
Expected type: Nat 'Even
  Actual type: Nat p
In the first argument of `evenPlus', namely `n1'
In the first argument of `Succ', namely `(evenPlus n1 n2)'

As I understand it, the gist of the above error is that GHC was unable to deduce (p1 ~ 'Even), when the context has the equation: ((Opp (Opp p1)) ~ 'Even).

Why does this happen? Is there something wrong with my approach?

Answer 1

I don't think GADT pattern match refinement works this way round. You have Opp p as the result type of a constructor. So if you write something like

f :: Nat 'Even -> ...
f (Succ n) = ...

then the type checker knows that Nat (Opp t) ~ Nat 'Even and therefore Opp t ~ 'Even. But to solve this, the type checker has to invert the function Opp, which is asking a lot.

I suggest you change the definition of Nat to say:

data Nat :: Parity -> * where
  Zero :: Nat 'Even
  Succ :: Nat (Opp p) -> Nat p

This should just work.

Edit

Actually, let me expand slightly.

The suggestion above is not without a (minor) price. You lose a bit of type inference. For example, the type of Succ Zero is now Succ Zero :: Opp p ~ 'Even => Nat p and not Nat 'Odd. With an explicit type annotation, it resolves ok.

You can improve on this by adding a constraint to Succ that requires that Opp is self-inverse. The only two elements of Parity are Even and Odd, and for these the constraint holds, so it should never cause any problems:

data Nat :: Parity -> * where
  Zero :: Nat 'Even
  Succ :: (Opp (Opp p) ~ p) => Nat (Opp p) -> Nat p

Now Succ Zero is inferred to be of type Nat 'Odd, and the pattern match still works.