[Haskell-cafe] Relationship between ((a -> Void) -> Void) and (forall r. (a -> r) -> r)

[Chris Smith]

This was indeed a fun puzzle to play with.  I think this becomes easier to
interpret if you factor out De Morgan's Law from the form you posted at the
beginning of your email.

https://code.world/haskell#PYGDwpaMZ_2iSs74NnwCUrg


On Fri, May 15, 2020 at 5:23 AM Ruben Astudillo <ruben.astud at gmail.com>
wrote:

> On 13-05-20 09:15, Olaf Klinke wrote:
> > Excersise: Prove that intuitionistically, it is absurd to deny the law
> > of excluded middle:
> >
> > Not (Not (Either a (Not a)))
>
> It took me a while but it was good effort. I will try to explain how I
> derived it. We need a term for
>
>    proof :: Not (Not (Either a (Not a)))
>    proof :: (Either a (Not a) -> Void) -> Void
>
> A first approximation is
>
>    -- Use the (cont :: Either a (Not a) -> Void) to construct the Void
>    -- We need to pass it an Either a (Not a)
>    proof :: (Either a (Not a) -> Void) -> Void
>    proof cont = cont $ Left <no a to fill in>
>
> Damn, we can't use the `Left` constructor as we are missing an `a` value
> to fill with. Let's try with `Right`
>
>    proof :: (Either a (Not a) -> Void) -> Void
>    proof cont = cont $ Right (\a -> cont (Left a))
>
> Mind bending. But it does make sense, on the `Right` constructor we
> assume we are have an `a` but we have to return a `Void`. Luckily we can
> construct a `Void` retaking the path we were gonna follow before filling
> with a `Left a`.
>
> Along the way I had other questions related to the original mail and
> given you seem knowledgeable I want to corroborate with you. I've seen
> claimed on the web that the CPS transform *is* the double negation [1]
> [2]. I don't think that true, it is almost true in my view. I'll
> explain, these are the types at hand:
>
>     type DoubleNeg a = (a -> Void) -> Void
>     type CPS a = forall r. (a -> r) -> r
>
> We want to see there is an equivalence/isomorphism between the two
> types. One implication is trivial
>
>     proof_CPS_DoubleNeg :: forall a. CPS a -> DoubleNeg a
>     proof_CPS_DoubleNeg cont = cont
>
> We only specialized `r ~ Void`, which mean we can transform a `CPS a`
> into a `DoubleNeg a`. So far so good, we are missing the other
> implication
>
>     -- bind type variables: a, r
>     -- cont   :: (a -> Void) -> Void
>     -- absurd :: forall b. Void -> b
>     -- cc     :: a -> r
>     proof_DoubleNeg_CPS :: forall a. DoubleNeg a -> CPS a
>     proof_DoubleNeg_CPS cont = \cc -> absurd $ cont (_missing . cc)
>
> Trouble, we can't fill `_missing :: r -> Void` as such function only
> exists when `r ~ Void` as it must be the empty function. This is why I
> don't think `CPS a` is the double negation.
>
> But I can see how people can get confused. Given a value `x :: a` we can
> embed it onto `CPS a` via `return x`. As we saw before we can pass from
> `CPS a` to `DoubleNeg a`. So we have *two* ways for passing from `a` to
> `DoubleNeg a`, the first one is directly as in the previous mail. The
> second one is using `proof_CPS_DoubleNeg`
>
>     embed_onto_DoubleNeg :: a -> DoubleNeg
>     embed_onto_DoubleNeg = proof_CPS_DoubleNeg . return
>       where
>         return :: a -> CPS a
>         return a = ($ a)
>
> So CPS is /almost/ the double negation. It is still interesting because
> it's enough to embed a classical fragment of logic onto the constructive
> fragment (LEM, pierce etc). But calling it a double negation really
> tripped me off.
>
> Am I correct? Or is there other reason why CPS is called the double
> negation transformation?
>
> Thank for your time reading this, I know it was long.
>
> [1]: http://jelv.is/talks/curry-howard.html#slide30
> [2]:
>
> https://www.quora.com/What-is-continuation-passing-style-in-functional-programming
>
> --
> -- Rubén
> -- pgp: 4EE9 28F7 932E F4AD
>
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