request: a Nat ordering constraint that is not an equality constraint

Simon Peyton Jones simonpj at
Wed May 22 11:14:22 UTC 2019

I’m afraid I’ve lost sync with this conversation. At some point it’d be good to write up conclusions on a wiki page and/or a ticket, so we don’t lose them.  Email threads are hard to parse later.


From: Libraries <libraries-bounces at> On Behalf Of Nicolas Frisby
Sent: 21 May 2019 18:19
To: Richard Eisenberg <rae at>
Cc: Haskell Libraries <libraries at>
Subject: Re: request: a Nat ordering constraint that is not an equality constraint

Yep, agreed. Totally practical.

FYI: my code base at work seems quite happy with the IsTrue approach; a very happy simplification! Like so:

class IsTrue
  (msg :: ErrorMessage) (b :: Bool) where
  isTrue :: Proxy# msg -> b :~: 'True

instance IsTrue msg 'True where
  isTrue = \_ -> Refl

instance TypeError msg => IsTrue msg 'False
  isTrue = \_ -> error "impossible"

type (<=) n m = IsTrue (Msg n m) (n GHC.TypeLits.<=? m)

type Msg (n :: Nat) (m :: Nat) =
    'ShowType n
    'Text " is not <= "
    'ShowType m

(An `absurd` for TypeError might be nice?

Via isTrue, I can explicitly convert between the two <= constraints wherever I need to. Which I could use to explicitly manage those hypothetical hypotheticals we were discussing.

Just to spell it out: the SMT solver would indeed supercede TcTypeNats, but TcTypeNats in turn already supercedes the IsTrue-based encoding. So I've already lost some <= deductions. But my work code doesn't use any of them in this middle ground -- they're all simple enough that IsTrue works as-is or complicated enough that TcTypeNats didn't work already (I'm grumpily writing and invoking Trusted Code Base "axia" in those cases -- I'm avoiding plugin dependencies at work for now.)
On Tue, May 21, 2019, 09:14 Richard Eisenberg <rae at<mailto:rae at>> wrote:
You're right that, without special support, the IsTrue approach won't work with any deductions from Givens. But -- short of strapping on an SMT solver -- we're always going to fall short there, so we should analyze a particular on-the-ground use case before taking any drastic action. (It sounds like you agree with this.)


On May 21, 2019, at 5:52 PM, Nicolas Frisby <nicolas.frisby at<mailto:nicolas.frisby at>> wrote:

Yes, it seems possible that a user space declaration of <= via IsTrue as in my first email could get much of the desired behavior. I plan on trying it with the work code base soon, maybe even today -- it'll probably do better than my current workaround.

If, however, we want the Nat solver to do anything at all with a Given `IsTrue (n <=? m)`, then I think it will need changes. I don't know that machinery well, but it seems very likely it would ignore such Givens.

For example, I would naively expect the Nat solver should discharge a Wanted `IsTrue (n <=? m)` from two Givens `(IsTrue (n <=? x),IsTrue (x <=? m))`.

Simon's exploration of IsTrue/IsEqual might shed more light on what exactly the Nat solver should and should not do with such a Given. If it's in fact nothing at all, then yes, maybe a user space solution fully supplants the proposed Passive.<=. But I currently anticipate that it should do something with such Givens.

Thanks. -Nick

On Tue, May 21, 2019, 00:29 Richard Eisenberg <rae at<mailto:rae at>> wrote:
This is an interesting proposal. When I started reading it, I wondered why anyone would want to avoid the current definition. But you motivate that part well. I would want a larger test of the IsTrue approach to make sure it does what you want before supporting this. But wait: couldn't you write your GHC.TypeLits.Passive today, in a library, with no ill effect? If so, there isn't a strict reason GHC needs to adopt this. (Of course, if the new definition proves useful, then it might make sense to do so in time.)

> On May 21, 2019, at 3:48 AM, Nicolas Frisby <nicolas.frisby at<mailto:nicolas.frisby at>> wrote:
> P.P.S. - Is there a standard place to find something like `IsTrue`? More generally: a test for type equality that does not drive unification? Thanks again.

If something like this ends up in GHC, Data.Type.Bool seems like the right place.


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