<div dir="ltr">Fair point. I was more thinking about the fact that you don't necessarily visit all of the a's because you can use the combinators you have there to generate an unbounded number of combination of them than the injectivity concern.</div><div class="gmail_extra"><br><div class="gmail_quote">On Sun, May 6, 2018 at 7:28 PM, David Feuer <span dir="ltr"><<a href="mailto:david.feuer@gmail.com" target="_blank">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 don't understand. Something like<div dir="auto"><br></div><div dir="auto">data Foo a where</div><div dir="auto"> Foo :: Num a => a -> Foo a<br><br>has a perfectly good "forgetful" injection:</div><div dir="auto"><br></div><div dir="auto">toTrav (Foo a) = Identity a</div><div dir="auto"><br></div><div dir="auto">This is injective because of class coherence.</div><div><div class="h5"><div dir="auto"><br><div class="gmail_quote" dir="auto"><div dir="ltr">On Sun, May 6, 2018, 6:04 PM Edward Kmett <<a href="mailto:ekmett@gmail.com" target="_blank">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">You can get stuck with contravariance in some fashion on the backswing, though, just from holding onto instance constraints about the type 'a'. e.g. If your Foldable data type captures a Num instance for 'a', it could build fresh 'a's out of the ones you have lying around.<div><br></div><div>There isn't a huge difference between that and just capturing the member of Num that you use.</div><div><br></div><div>-Edward</div></div><div class="gmail_extra"><br><div class="gmail_quote">On Sun, May 6, 2018 at 3:37 PM, David Feuer <span dir="ltr"><<a href="mailto:david.feuer@gmail.com" rel="noreferrer" target="_blank">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">The question, of course, is what we actually want to require. The strong injectivity law prohibits certain instances, yes. But it's not obvious, a priori, that those are "good" instances. Should a Foldable instance be allowed contravariance? Maybe that's just too weird. Nor is it remotely clear to me that the enumeration-based instance you give (that simply ignores the Foldable within) is something we want to accept. If we want Foldable to be as close to Traversable as possible while tolerating types that restrict their arguments in some fashion (i.e., things that look kind of like decorated lists) then I think the strong injectivity law is the way to go. Otherwise we need something else. I don't think your new law is entirely self-explanatory. Perhaps you can break it down a bit?</div><div class="m_-4690414243659487450m_5696484313029683073HOEnZb"><div class="m_-4690414243659487450m_5696484313029683073h5"><br><div class="gmail_quote"><div dir="ltr">On Sun, May 6, 2018, 1:56 PM Gershom B <<a href="mailto:gershomb@gmail.com" rel="noreferrer" target="_blank">gershomb@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 style="word-wrap:break-word;line-break:after-white-space"><div id="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px;color:rgba(0,0,0,1.0);margin:0px;line-height:auto">An amendment to the below, for clarity. There is still a problem, and the fix I suggest is still the fix I suggest.</div><div id="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px;color:rgba(0,0,0,1.0);margin:0px;line-height:auto">The contravarient example `data Foo a = Foo [a] (a -> Int)` is as I described, and passes quantification and almost-injectivity (as I suggested below), but not strong-injectivity (as proposed by David originally), and is the correct example to necessitate the fix.</div><div id="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px;color:rgba(0,0,0,1.0);margin:0px;line-height:auto">However, in the other two cases, while indeed they have instances that pass the quantification law (and the almost-injectivity law I suggest), these instances are more subtle than one would imagine. In other words, I wrote that there was an “obvious” foldable instance. But the instances, to pass the laws, are actually somewhat nonobvious. Furthermore, the technique to give these instances can _also_ be used to construct a type that allows them to pass strong-injectivity</div><div id="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px;color:rgba(0,0,0,1.0);margin:0px;line-height:auto">In particular, these instances are not the ones that come from only feeding the elements of the first component into the projection function of the second component. Rather, they arise from the projection function alone.</div><div id="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px;color:rgba(0,0,0,1.0);margin:0px;line-height:auto">So for `data Store f a b = Store (f a) (a -> b)`, then we have a Foldable instance for any enumerable type `a` that just foldMaps over every `b` produced by the function as mapped over every `a` in the enumeration, and the first component is discarded. I.e. we view the function as “an a-indexed container of b” and fold over it by knowledge of the index. Similarly for the `data Foo a = Foo [Int] (Int -> a)` case.</div><div id="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px;color:rgba(0,0,0,1.0);margin:0px;line-height:auto">So, while in general `r -> a` is not traversable, in the case when there is _any_ full enumeration on `r` (i.e., when `r` is known), then it _is_ able to be injected into something traversable, and hence these instances also pass the strong-injectivity law.</div><div id="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px;color:rgba(0,0,0,1.0);margin:0px;line-height:auto">Note that if there were universal quantification on `Store` then we’d have `Coyoneda` and the instance that _just_ used the `f a` in the first component (as described in Pudlák's SO post) would be the correct one, and furthermore that instance would pass all three versions of the law.</div><div id="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_customfont" style="font-family:Helvetica,Arial;font-size:13px;color:rgba(0,0,0,1.0);margin:0px;line-height:auto">Cheers,</div><div id="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_sign_1525627535162356736" class="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_sign"></div> <br><p class="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894airmail_on">On May 6, 2018 at 2:37:12 AM, Gershom B (<a href="mailto:gershomb@gmail.com" rel="noreferrer noreferrer" target="_blank">gershomb@gmail.com</a>) wrote:</p> <blockquote type="cite" class="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894clean_bq"><span><div style="word-wrap:break-word;line-break:after-white-space"><div></div><div>
<div id="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_customfont" style="margin:0px"><br></div>
<div id="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_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="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_sign_1525584726508983040" class="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894bloop_sign"></div>
<br>
<p class="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894airmail_on">On May 6, 2018 at 12:41:11 AM, David Feuer
(<a href="mailto:david.feuer@gmail.com" rel="noreferrer noreferrer" target="_blank">david.feuer@gmail.com</a>)
wrote:</p>
<blockquote type="cite" class="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894clean_bq">
<div>
<div>
<div dir="auto"><span>Two more points:</span>
<div dir="auto"><span><br></span></div>
<div dir="auto"><span>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.</span></div>
<div dir="auto"><span><br></span></div>
<div dir="auto"><span>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.</span></div>
<div dir="auto"><span><br></span></div>
<div dir="auto"><span>[*] <a href="https://stackoverflow.com/a/12896512/1477667" rel="noreferrer noreferrer" target="_blank">https://stackoverflow.com/a/<wbr>12896512/1477667</a><br>
</span></div>
</div>
<span><br></span>
<div class="gmail_quote">
<div dir="ltr"><span>On Sat, May 5, 2018, 10:37 PM Edward Kmett
<<a href="mailto:ekmett@gmail.com" rel="noreferrer noreferrer" target="_blank">ekmett@gmail.com</a>>
wrote:<br></span></div>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">
<div dir="ltr"><span>I actually don't have any real objection to
something like David's version of the law. </span>
<div><span><br></span></div>
<div><span>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.</span>
<div>
<div><span><br></span></div>
<div><span>-Edward</span></div>
</div>
</div>
</div>
<div class="gmail_extra"><span><br></span>
<div class="gmail_quote"><span>On Sat, May 5, 2018 at 5:09 PM,
David Feuer <span dir="ltr"><<a href="mailto:david.feuer@gmail.com" rel="noreferrer noreferrer noreferrer" target="_blank">david.feuer@gmail.com</a>></span>
wrote:<br></span>
<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>
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<div class="gmail_quote">On May 3, 2018 11:58 AM, "Gershom B"
<<a href="mailto:gershomb@gmail.com" rel="noreferrer noreferrer noreferrer" target="_blank">gershomb@gmail.com</a>> wrote:<br type="attribution">
<blockquote class="m_-4690414243659487450m_5696484313029683073m_-6140885288162992737m_-7017140726322644894m_-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 noreferrer noreferrer" target="_blank">https://mail.haskell.org/<wbr>pipermail/libraries/2015-<wbr>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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