# Natural number comparisons with evidence

Simon Peyton Jones simonpj at microsoft.com
Thu May 24 21:30:47 UTC 2018

```I see this in GHC.TypeNats

sameNat :: (KnownNat a, KnownNat b) =>

Proxy a -> Proxy b -> Maybe (a :~: b)

sameNat x y

| natVal x == natVal y = Just (unsafeCoerce Refl)

| otherwise            = Nothing

The unsafeCoerce says that sameNat is part of the trusted code base.  And indeed, it’s only because SNat is a private newtype (i.e its data constructor is private to GHC.TypeNats) that you can’t bogusly say    (SNat 7 :: SNat 8)

You want exactly the same thing, but for a comparison oriented data CompareEv, rather than its equality counterpart :~:.   So the same approach seems legitimate.

I always want code with unsafeCoerce to be clear about (a) why it’s necessary and (b) why it’s sound.

Simon

From: Glasgow-haskell-users <glasgow-haskell-users-bounces at haskell.org> On Behalf Of Conal Elliott
Sent: 24 May 2018 00:39
To: glasgow-haskell-users at haskell.org
Subject: Natural number comparisons with evidence

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`):

> 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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