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

[Oleg Grenrus]

The david’s approach is ingenious, but a more direct way, is to construct
the type equality proof yourself.

It’s more like what it would look like in say Agda:

{-# LANGUAGE DataKinds #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
{-# OPTIONS_GHC -Wall -Wno-redundant-constraints #-}
import Data.Type.Bool
import Data.Type.Equality

-- http://hackage.haskell.org/package/singleton-bool-0.1.2.0/docs/Data-Singletons-Bool.html
import Data.Singletons.Bool

import Unsafe.Coerce (unsafeCoerce)

-- Safe version
proof
    :: forall a b. (SBoolI a, SBoolI b, (a || b) ~ 'False)
    => (a :~: 'False, b :~: 'False)
proof = case (sbool :: SBool a, sbool :: SBool b) of
    (SFalse, SFalse) -> (Refl, Refl)

-- with unsafeCoerce we don't need the contexts. they can be satisfied for all a, b :: Bool
-- and we don't want runtime SBool dictionaries hanging around (would need to change Expr definition)
proof'
    :: forall a b. ((a || b) ~ 'False)
    => (a :~: 'False, b :~: 'False)
proof' = (unsafeCoerce trivialRefl, unsafeCoerce trivialRefl)

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

eval' :: Expr 'False -> Int
eval' (Lit i) = i
eval' (Add a b) = add a b
  where
    add :: forall t t'. ((t || t') ~ 'False) => Expr t -> Expr t' ->  Int
    add x y = case proof' :: (t :~: 'False, t' :~: 'False) of
        (Refl, Refl) -> eval' x + eval' y


> On 07 Dec 2016, at 09:43, Guillaume Bouchard <guillaum.bouchard+haskell at gmail.com> wrote:
> 
> Hi.
> 
> 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 <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.
> 
> --
> G.
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