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

[David Feuer]

Another option, I believe, would be to include singletons (or constraints
providing them) to the Add constructor.

On Dec 8, 2016 3:13 AM, "Oleg Grenrus" <oleg.grenrus at iki.fi> wrote:

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@
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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