Type inference of singular matches on GADTs

[Richard Eisenberg]

As usual, I want to separate out the specification of a feature from the implementation. So let's just focus on specification for now -- with the caveat that there might be no possible implementation of these ideas.

The key innovation I see lurking here is the idea of an *exhaustive* function, where we know that any pattern-match on an argument is always exhaustive. I will write such a thing with @->, in both the type and in the arrow that appears after the lambda. The @-> type is a subtype of -> (and perhaps does not need to be written differently from ->).

EX1: \x @-> case x of HNil -> blah

This is easy: we can infer HList '[] @-> blah's type, because the pattern match is declared to be exhaustive, and no other type grants that property.

EX2: \x @-> (x, case x of HNit -> blah)

Same as EX1.

EX3: \x @-> case x of { HNil1 -> blah; HNil2 -> blah }

Same as EX1. There is still a unique type for which the patten-match is exhaustive.

EX4: Reject. There are multiple valid types, and we don't know which one to pick. This is like classic untouchable-variables territory.

EX5: This is hard. A declarative spec would probably choose HL2 [a] -> ... as you suggest, but there may be no implementation of such an idea.

EX6: Reject. No type leads to exhaustive matches.

I'm not saying this is a good idea for GHC or that it's implementable. But the idea of having type inference account for exhaustivity in this way does not seem, a priori, unspecified.

Richard



> On Mar 29, 2021, at 5:00 AM, Simon Peyton Jones <simonpj at microsoft.com> wrote:
> 
> I haven't thought about how to implement such a thing. At the least, it would probably require some annotation saying that we expect `\HNil -> ()` to be exhaustive (as GHC won't, in general, make that assumption). Even with that, could we get type inference to behave? Possibly.
>  
> As I wrote in another post on this thread, it’s a bit tricky. 
>  
> What would you expect of (EX1)
>  
>    \x -> case x of HNil -> blah
>  
> Here the lambda and the case are separated
>  
> Now (EX2)
>  
> \x -> (x, case x of HNil -> blah)
>  
> Here the lambda and the case are separated more, and x is used twice.
> What if there are more data constructors that share a common return type? (EX3)
>  
> data HL2 a where
> HNil1 :: HL2 []
> HNil2 :: HL2 []
> HCons :: …blah…
>  
> \x -> case x of { HNil1 -> blah; HNil 2 -> blah  }
>  
> Here HNil1 and HNil2 both return HL2 [].   Is that “singular”?   
>  
> What if one was a bit more general than the other?   Do we seek the least common generalisation of the alternatives given? (EX4)
>  
> data HL3 a where
> HNil1 :: HL2 [Int]
> HNil2 :: HL2 [a]
> HCons :: …blah…
>  
> \x -> case x of { HNil1 -> blah; HNil 2 -> blah  }
>  
> What if the cases were incompatible?  (EX5)
>  
> data HL4 a where
> HNil1 :: HL2 [Int]
> HNil2 :: HL2 [Bool]
> HCons :: …blah…
>  
> \x -> case x of { HNil1 -> blah; HNil 2 -> blah  }
>  
> Would you expect that to somehow generalise to `HL4 [a] -> blah`?
> 
> What if x matched multiple times, perhaps on different constructors (EX6)
> 
> \x -> (case s of HNil1 -> blah1;  case x of HNil2 -> blah)
>  
>  
> The water gets deep quickly here.  I don’t (yet) see an obviously-satisfying design point that isn’t massively ad-hoc.
>  
> Simon
>  
>  
> From: ghc-devs <ghc-devs-bounces at haskell.org> On Behalf Of Richard Eisenberg
> Sent: 29 March 2021 03:18
> To: Alexis King <lexi.lambda at gmail.com>
> Cc: ghc-devs at haskell.org
> Subject: Re: Type inference of singular matches on GADTs
>  
>  
> 
> 
> On Mar 26, 2021, at 8:41 PM, Alexis King <lexi.lambda at gmail.com <mailto:lexi.lambda at gmail.com>> wrote:
>  
> If there’s a single principal type that makes my function well-typed and exhaustive, I’d really like GHC to pick it.
>  
> I think this is the key part of Alexis's plea: that the type checker take into account exhaustivity in choosing how to proceed.
>  
> Another way to think about this:
>  
> f1 :: HList '[] -> ()
> f1 HNil = ()
>  
> f2 :: HList as -> ()
> f2 HNil = ()
>  
> Both f1 and f2 are well typed definitions. In any usage site where both are well-typed, they will behave the same. Yet f1 is exhaustive while f2 is not. This isn't really about an open-world assumption or the possibility of extra cases -- it has to do with what the runtime behaviors of the two functions are. f1 never fails, while f2 must check a constructor tag and perhaps throw an exception.
>  
> If we just see \HNil -> (), Alexis seems to be suggesting we prefer the f1 interpretation over the f2 interpretation. Why? Because f1 is exhaustive, and when we can choose an exhaustive interpretation, that's probably a good idea to pursue.
>  
> I haven't thought about how to implement such a thing. At the least, it would probably require some annotation saying that we expect `\HNil -> ()` to be exhaustive (as GHC won't, in general, make that assumption). Even with that, could we get type inference to behave? Possibly.
>  
> But first: does this match your understanding?
>  
> Richard

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