{-# 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@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 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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