Re: Natural number comparisons with evidence

To throw out additional related work, I had a need for runtime values that capture type-level naturals. It’s used primarily for representing the widths in a type-safe bitvector formulas. The code is a few years old, but publicly available here: https://github.com/GaloisInc/parameterized-utils/blob/master/src/Data/Parame... I’d be interested in knowing if there are now any comprehensive libraries for doing this sort of runtime reflection of type-level naturals. Regards, Joe

On May 24, 2018, at 1:07 PM, Conal Elliott wrote:

I'm glad to know. Thanks for the endorsement, Richard.

On Thu, May 24, 2018 at 1:05 PM, Richard Eisenberg wrote: Just to add my 2 cents: I've played in this playground and used the same structures as David. I second his suggestions.

Richard

...

On May 24, 2018, at 3:54 PM, Conal Elliott wrote:

Great! Thanks for the suggestion to use type equality and coerced `Refl`. - Conal

On Thu, May 24, 2018 at 10:43 AM, David Feuer wrote: On Thu, May 24, 2018, 1:03 PM Conal Elliott wrote: Thanks for this suggestion, David. It seems to work out well, though I haven't tried running yet.

...

unsafeDict :: Dict c unsafeDict = unsafeCoerce (Dict @ ())

unsafeSatisfy :: forall c a. (c => a) -> a unsafeSatisfy z | Dict <- unsafeDict @ c = z

This doesn't really smell right to me, no. Dict @() is actually a rather different value than you seek. In general, these look like they do way more than they ever can. I would suggest you look through those comparison *constraints* to the underlying type equalities involving the primitive CmpNat type family.

-- Better, because there's only one Refl unsafeEqual :: forall a b. a :~: b unsafeEqual :: unsafeCoerce Refl

unsafeWithEqual :: forall a b r. (a ~ b => r) -> r unsafeWithEqual r | Refl <- unsafeEqual @a @b = r

compareEv = case .... of LT -> unsafeWithEqual @(CmpNat u v) @LT CompareLT ...

Now we can define `compareEv`:

...

compareEv :: forall u v. KnownNat2 u v => CompareEv u v compareEv = case natValAt @ u `compare` natValAt @ v of LT -> unsafeSatisfy @ (u < v) CompareLT EQ -> unsafeSatisfy @ (u ~ v) CompareEQ GT -> unsafeSatisfy @ (u > v) CompareGT

If anyone has other techniques to suggest, I'd love to hear.

-- Conal

On Wed, May 23, 2018 at 5:44 PM, David Feuer wrote: I think the usual approach for defining these sorts of primitive operations is to use unsafeCoerce.

On Wed, May 23, 2018, 7:39 PM Conal Elliott wrote: When programming with GHC's type-level natural numbers and `KnownNat` constraints, how can one construct *evidence* of the result of comparisons to be used in further computations? For instance, we might define a type for augmenting the results of `compare` with evidence:

...

data CompareEv u v = (u < v) => CompareLT | (u ~ v) => CompareEQ | (u > v) => CompareGT

Then I'd like to define a comparison operation (to be used with `AllowAmbiguousTypes` and `TypeApplications`, alternatively taking proxy arguments):

...

compareEv :: (KnownNat m, KnownNat n) => CompareEv u v

With `compareEv`, we can bring evidence into scope in `case` expressions.

I don't know how to implement `compareEv`. The following attempt fails to type-check, since `compare` doesn't produce evidence (which is the motivation for `compareEv` over `compare`):

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compareEv = case natVal (Proxy @ u) `compare` natVal (Proxy @ v) of LT -> CompareLT EQ -> CompareEQ GT -> CompareGT

Can `compareEv` be implemented in GHC Haskell? Is there already an implementation of something similar? Any other advice?

Thanks, -- Conal

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