[Haskell-cafe] Injective type family on a sub domain

Guillaume Bouchard guillaum.bouchard+haskell at gmail.com
Wed Dec 7 07:43:09 UTC 2016


I have the following GADT :

type family Or a b where
  Or 'False 'False = 'False
  Or _ _ = 'True

data Expr t where
  Add :: Expr t -> Expr t' -> Expr (Or t t')
  Lit :: Int -> Expr 'False
  Var :: Expr 'True

The idea is that if the `Expr` contains a sub `Var`, its type is `Expr
'True`, else it is `Expr 'False`.

I now want to evaluate my expression, something like that :

eval :: Expr t -> Int
eval (Lit i) = i
eval (Add a b) = eval a + eval b
eval Var = error "Cannot evaluate expression with variable"

Using the GADT I previously defined, I'm tempted to remove the impossible
"Var" case with :

eval' :: Expr 'False -> Int
eval' (Lit i) = i
eval' (Add a b) = eval' a + eval' b

However this does not typecheck because GHC cannot deduce that `a` and `b`
are `~ Expr 'False`. Because the type family `Or` is not injective.

The wiki https://ghc.haskell.org/trac/ghc/wiki/InjectiveTypeFamilies classifies
injectives types families in three categories, but I don't think my `Or`
appears in any of them.

Actually `Or` is injective only if `Or t t' ~ 'False` and in this case, we
can deduce that `t ~ 'False` and `t' ~ 'False`. I qualify it as a "partialy
injective type family".

The type checker does not know that, hence my code does not compile.

Is there a solution, other than writing a custom type checker plugin? Is
there a way to provide the inverse type family function, something such as:

type family InverseOr a where
     InverseOr 'False = ('False, 'False)

Thank you.

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