<html><head><style>body{font-family:Helvetica,Arial;font-size:13px}</style></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;"><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;">Hmm… I think Pudlák's Store as given in the stackoveflow post is a genuine example of where the two laws differ. That’s unfortunate.</div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;"><br></div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;">The quantification law allows the reasonable instance given in the post. Even with clever use of GADTs I don’t see how to produce a type to fulfill the injectivity law, though I’m not ruling out the possibility altogether.</div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;"><br></div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;">We can cook up something even simpler with the same issue, unfortunately.</div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;"><br></div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;">data Foo a = Foo [Int] (Int -> a)</div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;"><br></div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;">Again, there doesn’t seem to be a way to produce a GADT with an injection that also has traversable. But there is an obvious foldable instance, and it again passes the quantification law.</div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;"><br></div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;">The problem is that injectivity is too strong, but we need to get “almost” there for the law to work. We hit the same problem in fact if we have an `a` in any nontraversable position or structure, even of we have some other ones lying around. So also failing is:</div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;"><br></div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;">data Foo a = Foo [a] (a -> Int).</div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;"><br></div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;">I guess not only is the invectivity law genuinely stronger, it really is _too_ strong.</div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;"><br></div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;">What we want is the “closest thing” to an injection. I sort of know how to say this, but it results in something with the same complicated universal quantification statement (sans GenericSet) that you already dislike in the quantification law.</div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;"><br></div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;">So given “a GADT `u a` and function `toTrav :: forall a. f a -> u a`” we no longer require `toTrav` to be injective and instead require:</div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;"><br></div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;">`forall <span style="font-family: "helvetica Neue", helvetica; font-size: 14px;">(g :: forall a. f a -> Maybe a), exists (h :: forall a. u a -> Maybe a) such that g === h . toTrav`.</span></div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;"><span style="font-family: "helvetica Neue", helvetica; font-size: 14px;"><br></span></div><div id="bloop_customfont" style="margin: 0px;"><font face="helvetica Neue, helvetica"><span style="font-size: 14px;">In a sense, rather than requiring a global retract, we instead require that each individual “way of getting an `a`” induces a local retract.</span></font></div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;"><span style="font-family: "helvetica Neue", helvetica; font-size: 14px;"><br></span></div><div id="bloop_customfont" style="margin: 0px;"><font face="helvetica Neue, helvetica"><span style="font-size: 14px;">This is certainly a more complicated condition than “injective”. On the other hand it still avoids the ad-hoc feeling of `GenericSet` that Edward has been concerned about.</span></font></div><div id="bloop_customfont" style="margin: 0px;"><br></div><div id="bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px; color: rgba(0,0,0,1.0); margin: 0px; line-height: auto;">—Gershom</div> <br> <div id="bloop_sign_1525584726508983040" class="bloop_sign"></div> <br><p class="airmail_on">On May 6, 2018 at 12:41:11 AM, David Feuer (<a href="mailto:david.feuer@gmail.com">david.feuer@gmail.com</a>) wrote:</p> <blockquote type="cite" class="clean_bq"><span><div><div></div><div>
<title></title>
<div dir="auto">Two more points:
<div dir="auto"><br></div>
<div dir="auto">People have previously considered unusual Foldable
instances that this law would prohibit. See for example Petr
Pudlák's example instance for Store f a [*]. I don't have a very
strong opinion about whether such things should be allowed, but I
think it's only fair to mention them.</div>
<div dir="auto"><br></div>
<div dir="auto">If the Committee chooses to accept the proposal, I
suspect it would be reasonable to add that if the type is also a
Functor, then it should be possible to write a Traversable instance
compatible with the Functor and Foldable instances. This would
subsume the current foldMap f = fold . fmap f law.</div>
<div dir="auto"><br></div>
<div dir="auto">[*] <a href="https://stackoverflow.com/a/12896512/1477667">https://stackoverflow.com/a/12896512/1477667</a><br>
</div>
</div>
<br>
<div class="gmail_quote">
<div dir="ltr">On Sat, May 5, 2018, 10:37 PM Edward Kmett
<<a href="mailto:ekmett@gmail.com">ekmett@gmail.com</a>>
wrote:<br></div>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div dir="ltr">I actually don't have any real objection to
something like David's version of the law.
<div><br></div>
<div>Unlike the GenericSet version, it at first glance feels like
it handles the GADT-based cases without tripping on the
cases where the law doesn't apply because it doesn't just
doesn't type check. That had been my major objection to Gershom's
law.
<div>
<div><br></div>
<div>-Edward</div>
</div>
</div>
</div>
<div class="gmail_extra"><br>
<div class="gmail_quote">On Sat, May 5, 2018 at 5:09 PM, David
Feuer <span dir="ltr"><<a href="mailto:david.feuer@gmail.com" target="_blank" rel="noreferrer">david.feuer@gmail.com</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div dir="auto">I have another idea that might be worth
considering. I think it's a lot simpler than yours.
<div dir="auto"><br></div>
<div dir="auto">Law: If t is a Foldable instance, then there must
exist:</div>
<div dir="auto"><br></div>
<div dir="auto">1. A Traversable instance u and</div>
<div dir="auto">2. An injective function</div>
<div dir="auto"> toTrav :: t a -> u
a</div>
<div dir="auto"><br></div>
<div dir="auto">Such that</div>
<div dir="auto"><br></div>
<div dir="auto"> foldMap @t = foldMapDefault .
toTrav</div>
<div dir="auto"><br></div>
<div dir="auto">I'm pretty sure this gets at the point you're
trying to make.</div>
<div>
<div class="m_-4446501889732117611h5"><br>
<div class="gmail_extra" dir="auto"><br>
<div class="gmail_quote">On May 3, 2018 11:58 AM, "Gershom B"
<<a href="mailto:gershomb@gmail.com" target="_blank" rel="noreferrer">gershomb@gmail.com</a>> wrote:<br type="attribution">
<blockquote class="m_-4446501889732117611m_7487089496557678815quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
This came up before (see the prior thread):<br>
<a href="https://mail.haskell.org/pipermail/libraries/2015-February/024943.html" rel="noreferrer noreferrer noreferrer" target="_blank">https://mail.haskell.org/pipermail/libraries/2015-February/024943.html</a><br>
<br>
The thread at that time grew rather large, and only at the end did
I<br>
come up with what I continue to think is a satisfactory formulation
of<br>
the law.<br>
<br>
However, at that point nobody really acted to do anything about
it.<br>
<br>
I would like to _formally request that the core libraries
committee<br>
review_ the final version of the law as proposed, for addition
to<br>
Foldable documentation:<br>
<br>
==<br>
Given a fresh newtype GenericSet = GenericSet Integer deriving
(Eq,<br>
Ord), where GenericSet is otherwise fully abstract:<br>
<br>
forall (g :: forall a. f a -> Maybe a), (x :: f
GenericSet).<br>
maybe True (`Foldable.elem` x) (g x) =/= False<br>
==<br>
<br>
The intuition is: "there is no general way to get an `a` out of `f
a`<br>
which cannot be seen by the `Foldable` instance". The use of<br>
`GenericSet` is to handle the case of GADTs, since even
parametric<br>
polymorphic functions on them may at given _already known_ types
have<br>
specific behaviors.<br>
<br>
This law also works over infinite structures.<br>
<br>
It rules out "obviously wrong" instances and accepts all the
instances<br>
we want to that I am aware of.<br>
<br>
My specific motivation for raising this again is that I am
rather<br>
tired of people saying "well, Foldable has no laws, and it is in
base,<br>
so things without laws are just fine." Foldable does a have a law
we<br>
all know to obey. It just has been rather tricky to state. The
above<br>
provides a decent way to state it. So we should state it.<br>
<br>
Cheers,<br>
Gershom<br>
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</blockquote>
</div>
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</blockquote>
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