positive type-level naturals

[Carter Schonwald]

(aside, pardon my earlier tone, been a bit overloaded the past few weeks,
that crossed over to the list)

OOO
that works?
I guess that gives a decent way of using TypeLits as a concrete input
syntax for Peano numbers. Thanks for pointing that out

I think i'm gonna go drop 7.6 support on some code i'm working on if this
works :)





On Mon, Mar 17, 2014 at 2:05 PM, Richard Eisenberg <eir at cis.upenn.edu>wrote:

> So much to respond to!
>
> First, a little relevant context: Iavor Diatchki is the primary
> implementor of the type-lits stuff; I am not. But, he and I are playing in
> the same playground, so to speak, so we confer a fair amount and I may have
> some helpful perspective on all of this.
>
> Henning asks:
> > How can I convince GHC that n+1 is always at least 1?
>
> You can ramrod facts like this down GHC's throat when necessary. For
> example, the following works:
>
> > foo :: (1 <= n + 1) => Proxy n -> ()
> > foo _ = ()
> >
> > lt :: Proxy n -> (1 <=? n + 1) :~: True
> > lt _ = unsafeCoerce Refl
> >
> > bar :: forall (n :: Nat). Proxy n -> ()
> > bar p = gcastWith (lt p) (foo p)
>
>
> In case you're unfamiliar with them, here are some helpful definitions
> from Data.Type.Equality, used above:
>
> > data a :~: b where
> >   Refl :: a :~: a
> > gcastWith :: (a :~: b) -> ((a ~ b) => r) -> r
>
> Also helpful when doing things like this is this definition, from Edward
> Kmett's `constraints` package:
>
> > data Dict c where
> >   Dict :: c => Dict c
>
> An `unsafeCoerce Dict` can be used to satisfy any arbitrary constraint.
>
> Of course, it is often possible to use an inductive proof (er, recursive
> function) to produce Refl or Dict without resorting to unsafeCoerce. But,
> as the TypeLits Nat isn't inductive, we're forced to use drastic measures
> here.
>
> Carter says:
> > I've been toying with the idea that the type lits syntax should be just
> that, a type level analogue of from integer that you can give to user land
> types, but I'm not going to suggest that till 7.8 is fully released.
>
>
> I like this idea.
>
> Christiaan says:
> > Iavor is working on a branch that allows the constraint solver to call
> an external solver: https://github.com/ghc/ghc/tree/decision-procedure
>
>
> Yes, though I don't know how active this branch is currently. There are
> whispers afoot of going in the direction of strapping an SMT solver into
> GHC, though much work remains to be done before this happens. My sense is
> that an SMT solver will be necessary before TypeLits really becomes
> fluently useful. I'm confident this will happen eventually, but it may be
> over a year out, still. It's even possible that I will be the one to do it,
> but it's not on my short-to-do-list.
>
> Christiaan says:
> > I myself worked on a patch that can only work with equalities:
> https://gist.github.com/christiaanb/8396614
>
>
> Cool! Have you conferred with Iavor about this?
>
> Carter says:
> > The current typeLits story for nats is kinda a fuster cluck to put it
> politely . We have type lits but we cant use them (well, we can't compute
> on them, which is the same thing).
>
> I disagree on both counts here. TypeLits is a work in progress, as are
> many parts of GHC. That's one of the beautiful things about Haskell/GHC! Is
> there more progress to be made? Absolutely. But, without the work that's
> already been done, I'm not sure we would be as convinced as we are (still
> not 100%, to be sure, but getting there) that we need an SMT solver. We
> have to build on previous work, and to do that, we have to write
> potentially incomplete features. And, I've been able to use TypeLits most
> of the way toward implementing my `units` library (a way of type-checking
> with respect to units-of-measure). The only feature that they couldn't
> support was automatic unit conversions.
>
> Carter says:
> > I'm still waiting (2 years later) for a solver we can actually include
> in ghc or even a user land solver!
>
>
> I've done some thinking about user-land solvers and am quite interested in
> seeing it done. My chief thrust right now is about dependent types in
> Haskell, not this, but Iavor's "decision-procedure" branch lays a lot of
> the groundwork down for integrating perhaps multiple solvers in with GHC.
>
> Henning says:
> > A minimal invasive solution would be to provide a kind for unary type
> level numbers and type functions that convert between Unary and Nat.
>
>
> This definition is quite straightforward and works beautifully:
>
> > data Nat1 = Zero | Succ Nat1
> > type family U n where
> >   U 0 = Zero
> >   U n = Succ (U (n-1))
>
> Iavor made sure that subtraction worked specifically in this case because
> it was so useful.
>
> I hope this is helpful!
> Richard
>
> On Mar 16, 2014, at 4:52 PM, Henning Thielemann <
> lemming at henning-thielemann.de> wrote:
>
> > Am 16.03.2014 20:02, schrieb Carter Schonwald:
> >> respectfully,
> >> The current typeLits story for nats is kinda a fuster cluck to put it
> >> politely . We have type lits but we cant use them (well, we can't
> >> compute on them, which is the same thing).
> >>
> >> For the past 2 years, every ghc release cycle, I first discover, then
> >> have to communicate to everyone else "you can't compute on type lits".
> >
> > A minimal invasive solution would be to provide a kind for unary type
> level numbers and type functions that convert between Unary and Nat.
> >
> > _______________________________________________
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>
>
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