September 2014 - Haskell-Cafe

optional ghc-options based on GHC version?
by Mateusz Kowalczyk 18 Sep '14

18 Sep '14

Hi, I have a project here where GHC's option -flate-dmd-anal gets out some extra speed but the flag is not available on 7.6.3. How can I specify in the cabal file that I only want to pass this through when building with 7.8.3 or newer? It'd be a shame to not take advantage of the extra speed on 7.8.3 just because we want to support 7.6.3 too for a bit longer. I was unable to find the info online. Thanks. -- Mateusz K.

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Re: [Haskell-cafe] Hello and type-level SKI
by oleg＠okmij.org 18 Sep '14

18 Sep '14

Yet another demonstration of Turing completeness of the type checker (given the unrestricted context reduction stack) was lambda-calculator in types, posted back in 2006. http://okmij.org/ftp/Computation/lambda-calc.html#haskell-type-level The examples included the Fibonacci function (encoded using the fix-point combinator) and the SKI calculator (expressing combinators as lambda-terms).

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Improving ghc error messages
by Toby Goodwin 18 Sep '14

18 Sep '14

Interesting article: https://izbicki.me/blog/error-messages-in-ghc-vs-g++.html I do hope some of these changes make it into a future ghc. I'd particularly like to highlight this one: No instance for (Show ([a0] -> Int)) arising from a use of `show' Possible fix: add an instance declaration for (Show ([a0] -> Int)) The second line looks like it was added by someone who'd read an article about making error messages helpful: "suggest a possible way to fix the error". Like creating a Show instance for a function type!?! I must have seen that "possible fix" 1000 times, and 999 of them it was nothing like the right fix. Now I know just to ignore it. Simply removing the second line would be an improvement: it doesn't add anything to the first line. Toby.

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cryptography and homomorphic library support?
by Vasili I. Galchin 18 Sep '14

18 Sep '14

Hi, I just perused the crypto section of Hackage and I didn't see any support for homomorphc functions. I didn't see anything .. maybe I missed. If no support, I wouldn't mind giving a shot at working on this as long as no one has already implemented. If someone is working on this, please inform me of contact info so perhaps I can join the team. Kind regards, Vasya

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Haskell Weekly News: Issue 306
by Daniel Santa Cruz 18 Sep '14

18 Sep '14

Welcome to issue 306 of the HWN, an issue covering crowd-sourced bits of information about Haskell from around the web. This issue covers from September 07 to 13, 2014 Quotes of the Week * prophile: at this point ekmett is more part of the language than a practitioner of it * Qfwfq: Simon is a title, not a forename. Top Reddit Stories * Haskell for all: Morte: an intermediate language for super-optimizing functional programs Domain: haskellforall.com, Score: 110, Comments: 83 Original: [1] http://goo.gl/MjPyyr On Reddit: [2] http://goo.gl/v2btl4 * AMP has landed in GHC Main! Domain: haskell.org, Score: 88, Comments: 71 Original: [3] http://goo.gl/giu1Xh On Reddit: [4] http://goo.gl/ZGJ76S * [Meta] What's with all these downvotes on beginner questions? Domain: self.haskell, Score: 69, Comments: 67 Original: [5] http://goo.gl/ik2uu2 On Reddit: [6] http://goo.gl/ik2uu2 * Haskell Implementors Workshop 2014 videos [youtube] Domain: youtube.com, Score: 59, Comments: 11 Original: [7] http://goo.gl/ijM6hs On Reddit: [8] http://goo.gl/48GOSr * FP Complete is hiring: Haskell web UI developer Domain: fpcomplete.com, Score: 59, Comments: 14 Original: [9] http://goo.gl/0kPlFl On Reddit: [10] http://goo.gl/WtivSo * How to get (approx) stack traces with profiled builds Domain: self.haskell, Score: 46, Comments: 15 Original: [11] http://goo.gl/ZAurK6 On Reddit: [12] http://goo.gl/ZAurK6 * Hakaru: An embedded probabilistic programming language Domain: indiana.edu, Score: 40, Comments: 10 Original: [13] http://goo.gl/PLcCGW On Reddit: [14] http://goo.gl/aP7giY * Why does David Turner say type classes were a bad idea? Domain: self.haskell, Score: 37, Comments: 96 Original: [15] http://goo.gl/WqRLZQ On Reddit: [16] http://goo.gl/WqRLZQ * Let's Build a Browser Engine in Haskell: Part 2 Domain: hrothen.github.io, Score: 33, Comments: 14 Original: [17] http://goo.gl/PyxMOy On Reddit: [18] http://goo.gl/45L7hs * Proposal for copatterns in Idris Domain: github.com, Score: 30, Comments: 38 Original: [19] http://goo.gl/7Zv7U1 On Reddit: [20] http://goo.gl/pPIq8p * GHC Blog: Static pointers and serialisation (by SPJ) Domain: ghc.haskell.org, Score: 28, Comments: 2 Original: [21] http://goo.gl/HJrm4t On Reddit: [22] http://goo.gl/kPxJj5 * introduction to the basic lens operators Domain: intolerable.me, Score: 26, Comments: 12 Original: [23] http://goo.gl/VXlKnm On Reddit: [24] http://goo.gl/b0bLfE * Prisms lead to typeclasses for subtractive types Domain: gelisam.blogspot.ca, Score: 22, Comments: 6 Original: [25] http://goo.gl/l9QIIm On Reddit: [26] http://goo.gl/GMD5Cj * Preventing memoization in (AI) search problems Domain: okmij.org, Score: 22, Comments: 16 Original: [27] http://goo.gl/oEb6WD On Reddit: [28] http://goo.gl/jqVKPS * A shallow survey of formal methods for C code Domain: imperialviolet.org, Score: 21, Comments: 7 Original: [29] http://goo.gl/rJgycy On Reddit: [30] http://goo.gl/RrXwns * Haskell Lectures Domain: shuklan.com, Score: 21, Comments: 1 Original: [31] http://goo.gl/Ef0E4 On Reddit: [32] http://goo.gl/VASdCX * A Netwire 5 Tutorial Domain: self.haskell, Score: 20, Comments: 5 Original: [33] http://goo.gl/BmcKWr On Reddit: [34] http://goo.gl/BmcKWr * Haskell Symposium talk on inline Objective-C in Haskell Domain: speakerdeck.com, Score: 20, Comments: 0 Original: [35] http://goo.gl/VDHOiP On Reddit: [36] http://goo.gl/TyL90G * HaLVM / Unikernel talk at the Xen User Summit on Monday, September 15th Domain: sched.co, Score: 19, Comments: 8 Original: [37] http://goo.gl/HoVwWT On Reddit: [38] http://goo.gl/9lqCE0 * Fun with (Extended Kalman) Filters Domain: idontgetoutmuch.wordpress.com, Score: 18, Comments: 2 Original: [39] http://goo.gl/8EGSQR On Reddit: [40] http://goo.gl/aTASRU Top StackOverflow Questions * Why does OCaml sometimes require eta expansion? votes: 12, answers: 1 Read on SO: [41] http://goo.gl/TsBuz5 * Can Haskell evaluate and not garbage collect random indexes in a list? votes: 11, answers: 2 Read on SO: [42] http://goo.gl/zXsBT1 * How do I install dependencies when cross compiling haskell code? votes: 10, answers: 1 Read on SO: [43] http://goo.gl/3qH6zK * Haskell lenses: how to make view play nicely with traverse? votes: 10, answers: 3 Read on SO: [44] http://goo.gl/EWrR5j * Is there an instance of Monad but not of MonadFix? votes: 10, answers: 2 Read on SO: [45] http://goo.gl/EAh0qc * How can the continuation monad be expressed using the free monad? votes: 10, answers: 3 Read on SO: [46] http://goo.gl/ymMoFw Until next time, [47]+Daniel Santa Cruz References 1. http://www.haskellforall.com/2014/09/morte-intermediate-language-for-super.… 2. http://www.reddit.com/r/haskell/comments/2g6wsx/haskell_for_all_morte_an_in… 3. http://www.haskell.org/pipermail/ghc-devs/2014-September/006265.html 4. http://www.reddit.com/r/haskell/comments/2fxqts/amp_has_landed_in_ghc_main/ 5. http://www.reddit.com/r/haskell/comments/2g3em9/meta_whats_with_all_these_d… 6. http://www.reddit.com/r/haskell/comments/2g3em9/meta_whats_with_all_these_d… 7. https://www.youtube.com/playlist?list=PL4UWOFngo5DW6nKDjK0UB5Oy9zmdWdo7K 8. http://www.reddit.com/r/haskell/comments/2frhfn/haskell_implementors_worksh… 9. https://www.fpcomplete.com/blog/2014/09/hiring-haskell-web-ui-dev 10. http://www.reddit.com/r/haskell/comments/2g4mdu/fp_complete_is_hiring_haske… 11. http://www.reddit.com/r/haskell/comments/2fwwx3/how_to_get_approx_stack_tra… 12. http://www.reddit.com/r/haskell/comments/2fwwx3/how_to_get_approx_stack_tra… 13. http://indiana.edu/~ppaml/ 14. http://www.reddit.com/r/haskell/comments/2g4q81/hakaru_an_embedded_probabil… 15. http://www.reddit.com/r/haskell/comments/2foggq/why_does_david_turner_say_t… 16. http://www.reddit.com/r/haskell/comments/2foggq/why_does_david_turner_say_t… 17. http://hrothen.github.io/2014/09/08/lets-build-a-browser-engine-in-haskell-… 18. http://www.reddit.com/r/haskell/comments/2fu2l5/lets_build_a_browser_engine… 19. https://github.com/idris-lang/Idris-dev/wiki/Copatterns 20. http://www.reddit.com/r/haskell/comments/2fy1hj/proposal_for_copatterns_in_… 21. https://ghc.haskell.org/trac/ghc/blog/simonpj/StaticPointers 22. http://www.reddit.com/r/haskell/comments/2g4vvf/ghc_blog_static_pointers_an… 23. http://intolerable.me/lens-operators-intro/ 24. http://www.reddit.com/r/haskell/comments/2fud8k/introduction_to_the_basic_l… 25. http://gelisam.blogspot.ca/2014/09/prisms-lead-to-typeclasses-for.html 26. http://www.reddit.com/r/haskell/comments/2foz4a/prisms_lead_to_typeclasses_… 27. http://okmij.org/ftp/Haskell/#memo-off 28. http://www.reddit.com/r/haskell/comments/2g9akh/preventing_memoization_in_a… 29. https://www.imperialviolet.org/2014/09/07/provers.html 30. http://www.reddit.com/r/haskell/comments/2fsvns/a_shallow_survey_of_formal_… 31. http://shuklan.com/haskell/ 32. http://www.reddit.com/r/haskell/comments/2g672e/haskell_lectures/ 33. http://www.reddit.com/r/haskell/comments/2fs4mv/a_netwire_5_tutorial/ 34. http://www.reddit.com/r/haskell/comments/2fs4mv/a_netwire_5_tutorial/ 35. https://speakerdeck.com/mchakravarty/foreign-inline-code-in-haskell-haskell… 36. http://www.reddit.com/r/haskell/comments/2fve6k/haskell_symposium_talk_on_i… 37. http://sched.co/1rFBMed 38. http://www.reddit.com/r/haskell/comments/2g2sa0/halvm_unikernel_talk_at_the… 39. http://idontgetoutmuch.wordpress.com/2014/09/09/fun-with-extended-kalman-fi… 40. http://www.reddit.com/r/haskell/comments/2fwbhd/fun_with_extended_kalman_fi… 41. http://stackoverflow.com/questions/25763412/why-does-ocaml-sometimes-requir… 42. http://stackoverflow.com/questions/25751194/can-haskell-evaluate-and-not-ga… 43. http://stackoverflow.com/questions/25765893/how-do-i-install-dependencies-w… 44. http://stackoverflow.com/questions/25773444/haskell-lenses-how-to-make-view… 45. http://stackoverflow.com/questions/25814489/is-there-an-instance-of-monad-b… 46. http://stackoverflow.com/questions/25827271/how-can-the-continuation-monad-… 47. https://plus.google.com/105107667630152149014/about

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Boolean formula: part2
by Dmitry Bogatov 17 Sep '14

17 Sep '14

Hello! I have following code: buildFormulas :: [Variable] -> Int -> [Formula] buildFormulas vars 0 = map Var vars buildFormulas vars n = join $ forM [0 .. n - 1 ] $ \i -> do x <- buildFormulas vars i y <- buildFormulas vars (n - i - 1) z <- buildFormulas vars (n - 1) let t1 = case z of (Negation _) -> [] _ -> [Negation z] t1 ++ [conj x y, impl x y, disj x y] It is correct, typechecks, but it is insanely slow (calculating take 35 (buildFormulas [P, Q, R] 5) hangs my computer). My guess is that it builds too much before returning first results. Can you suggest more efficent way enumerate formulas of set of variables? -- Best regards, Dmitry Bogatov <KAction(a)gnu.org>, Free Software supporter, esperantisto and netiquette guardian. GPG: 54B7F00D

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Re: [Haskell-cafe] Hackage package "synopsis" sections
by Mike Izbicki 17 Sep '14

17 Sep '14

One problem with using the README file as the synopsis is that a lot of the standard information in a README file would be out of place when displayed on hackage. For example, many READMEs have: * installation instructions * a list of contributors * a list of known bugs * how to contribute bug reports * licensing information Some of this information is very out of place on hackage, some of it is duplication of the standard fields in the cabal file, and some of it is just not things that a new user needs to see on their first visit to the package documentation. Maybe this could be solved by having the name of the synopsis file as a variable the author can specify. It can default to README(.md) but can be overridden to something else for authors who choose to do so. On Tue, Sep 16, 2014 at 11:20 PM, Mike Izbicki <mike.izbicki(a)gmail.com> wrote: > One problem with using the README file as the synopsis is that a lot > of the standard information in a README file would be out of place > when displayed on hackage. For example, many READMEs have: > > * installation instructions > * a list of contributors > * a list of known bugs > * how to contribute bug reports > * licensing information > > Some of this information is very out of place on hackage, some of it > is duplication of the standard fields in the cabal file, and some of > it is just not things that a new user needs to see on their first > visit to the package documentation. > > Maybe this could be solved by having the name of the synopsis file as > a variable the author can specify. It can default to README(.md) but > can be overridden to something else for authors who choose to do so. > > On Tue, Sep 16, 2014 at 8:58 PM, Michael Snoyman <michael(a)snoyman.com> wrote: >> >> >> On Wed, Sep 17, 2014 at 12:10 AM, Richard Lewis <richard(a)rjlewis.me.uk> >> wrote: >>> >>> So there's clearly an important technical hurdle to be overcome for >>> this: what markup language is to be used?; and where should the >>> synposis be stored? >>> >>> But I can't help thinking that the real work is in getting the >>> synopses written. Encouraging package authors/maintainers to add them >>> to their own packages is one possible way forward. >>> >> >> Actually, if we get the README support in place, I think it should be much >> easier to get package authors to write synopses. Most projects have a README >> file already, and getting it included in Hackage would now be a one-line >> pull request (adding `extra-source-files: README.extension`). It will >> hopefully be easier to get authors to elaborate on their READMEs then >> because: >> >> 1. They're serving double duty, both in the project repository and on >> Hackage. >> 2. It's trivial for outside developers to send pull requests against a >> simple text file. I've received quite a few such contributions on projects I >> maintain, even from people not ready to work on the code itself. >> >>> >>> At Mon, 15 Sep 2014 11:28:38 -0700, >>> Tikhon Jelvis wrote: >>> >>> > Maybe we could have a guerrilla campaign of pull requests adding >>> > examples and a bit of explanation to every package you like that >>> > doesn't have them... That could also be a good way for beginners who >>> > want to contribute to start. >>> >>> Another, as Tikhon suggests, would be for others to write them and >>> send pull requests (or whatever) to the maintainers. >>> >>> At Mon, 15 Sep 2014 20:10:18 -0400, >>> Dominick Samperi wrote: >>> >>> > I think this is a great idea, but it probably needs a complementary >>> > "nudge" if it is going to have a significant impact. This could be >>> > incorporated into the package submission process where the submitter >>> > runs a final check and is warned when examples are not provided for >>> > exported functions (unless an "opt out" flag is turned on for >>> > functions that have "obvious" semantics). >>> >>> And yet another, as Dominick suggests, is effectively to require a >>> synposis when a package is submitted. In CPAN, it's just become part >>> of the culture, but it's also not required and you do find packages >>> without a synposis. >>> >>> If there's interest, I'd like to solicit some discussion on this part >>> of the proposal on this thread... >>> >>> There may also be a potentially significant difference between >>> Hackage/Haskell and CPAN/Perl: most CPAN packages tend to be quite >>> small and specific in their purpose and consequently have just a few, >>> simple common use cases which suit a synposis very well. Hackage >>> packages, on the other hand, are quite often more broad in their >>> scope, often comprising many modules. Also, Perl has only one >>> semantics for organising code at the finest level: sequential, >>> imperative statements. In Haskell, some packages actually define whole >>> coding styles. As a result, it's always pretty obvious how to write a >>> few isolated lines of Perl code, but not necessarily so with Haskell >>> code. Anyway, I'm sure all this can be overcome, and/or argued >>> against. >>> >>> Richard >>> -- >>> -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- >>> Richard Lewis >>> j: ironchicken(a)jabber.earth.li >>> @: lewisrichard >>> http://www.richardlewis.me.uk/ >>> -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- >>> _______________________________________________ >>> Haskell-Cafe mailing list >>> Haskell-Cafe(a)haskell.org >>> http://www.haskell.org/mailman/listinfo/haskell-cafe >> >> >> >> _______________________________________________ >> Haskell-Cafe mailing list >> Haskell-Cafe(a)haskell.org >> http://www.haskell.org/mailman/listinfo/haskell-cafe >>

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Mapping over Type Level Literals
by Dominic Steinitz 17 Sep '14

17 Sep '14

I was playing around with Haar functions and realised I could encode the constraints in the types > > {-# LANGUAGE DataKinds #-} > > {-# LANGUAGE TypeOperators #-} > > {-# LANGUAGE KindSignatures #-} > > {-# LANGUAGE TypeFamilies #-} > > {-# LANGUAGE TypeOperators #-} > > {-# LANGUAGE Rank2Types #-} > > {-# LANGUAGE ScopedTypeVariables #-} > > > import GHC.TypeLits > > import Data.Proxy > > > data Haar (a :: Nat) (b :: Nat) = Haar { unHaar :: Double -> Double } > > > haar :: forall n k . > > (KnownNat n, KnownNat k, (2 * k - 1 <=? 2^n - 1) ~ 'True) => > > Haar n (2 * k - 1) > > haar = Haar g where > > g t | (k' - 1) * 2 ** (-n') < t && t <= k' * 2 ** (-n') = 2 ** ((n' - 1) / 2) > > | k' * 2 ** (-n') < t && t <= (k' + 1) * 2 ** (-n') = -2 ** ((n' - 1) / 2) > > | otherwise = 0 > > where > > n' = fromIntegral (natVal (Proxy :: Proxy n)) > > k' = 2 * (fromIntegral (natVal (Proxy :: Proxy k))) - 1 So now I can do e.g. > *Main> unHaar (haar :: Haar 3 7) 0.05 > 0.0 > *Main> unHaar (haar :: Haar 3 6) 0.05 > > <interactive>:359:9: > Couldn't match type ‘2 * k0’ with ‘7’ > The type variable ‘k0’ is ambiguous > Expected type: 'True > Actual type: ((2 * k0) - 1) <=? ((2 ^ 3) - 1) > In the first argument of ‘unHaar’, namely ‘(haar :: Haar 3 6)’ > In the expression: unHaar (haar :: Haar 3 6) 0.05 > In an equation for ‘it’: it = unHaar (haar :: Haar 3 6) 0.05 > *Main> unHaar (haar :: Haar 3 9) 0.05 > > <interactive>:360:9: > Couldn't match type ‘'False’ with ‘'True’ > Expected type: 'True > Actual type: ((2 * 5) - 1) <=? ((2 ^ 3) - 1) > In the first argument of ‘unHaar’, namely ‘(haar :: Haar 3 9)’ > In the expression: unHaar (haar :: Haar 3 9) 0.05 > In an equation for ‘it’: it = unHaar (haar :: Haar 3 9) 0.05 > *Main> So errors get rejected at compilation time as one would like and with moderately informative error messages. But now of course I would like to map over n and k but these are at the type level. Can this be done? I imagine unsafeCoerce would have to come into it somewhere. Dominic Steinitz dominic(a)steinitz.org http://idontgetoutmuch.wordpress.com

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Re: [Haskell-cafe] Type union
by oleg＠okmij.org 17 Sep '14

17 Sep '14

Type unions are also necessary for extensible effects. In fact, they are the foundation of extensible effects. One implementation http://okmij.org/ftp/Haskell/extensible/TList.hs uses only common and uncontroversial extensions (No Typeable). It could be converted to closed type families. The other, more optimal implementation, relies on Typeable, so that injection and projection become constant time, regardless of the size of the union. The Typeable constraint can be dispensed with; see the HList implementation of type-indexed coproducts (TIC), which are essentially open unions.

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Re: [Haskell-cafe] [QuickCheck] Fwd: an idea for improving 'shrink'
by Petr Pudlák 17 Sep '14

17 Sep '14

Hi Nick, you're right, this example is indeed problematic. I'll try to think about it how to get around that, keeping backward compatibility. Petr Dne 11. 9. 2014 21:37 "Nick Smallbone" <nicsma(a)chalmers.se> napsal(a): > Hi Petr, > > My attitude has always been "just use the shrink instance for tuples". > I don't think there's all that much difference between: > shrink (C x y) = [ C x' y' | (x', y') <- shrink (x, y) ] > and: > shrink (C x y) = C <$> shrink x <*> shrink y > The second is certainly nicer, but neither way is really very painful. > > However, your shrinkOne example shows a case where shrinking tuples > doesn't work and you really need a shrinking Applicative. > So this is a good reason to add a shrinking Applicative to QuickCheck > and I would probably like to do it at some point. > > There is a problem with your current patch, though. Namely the default > implementation of shrinkA: > shrinkA = pure > If someone defines shrink but not shrinkA, then shrinkA will not > shrink! In fact, one of your examples exhibits this bug: > shrinkA (Branch x l r) = > Branch <$> shrinkA x <*> shrinkA l <*> shrinkA r > If the Arbitrary instance for whatever x is defines shrink but not > shrinkA, then this will do the wrong thing. > > So we can't provide a default implementation for shrink in terms of > shrinkA, or vice versa, because we want the user to be able to define > either. We can't even define both in terms of the other because then > the default implementation will loop instead of returning []. > > Probably we would just have to move shrinkA outside of the Arbitrary > class. This does make using the Applicative more cumbersome, e.g.: > shrink (C x y) = fromShrink (C <$> shrinkA x <*> shrinkA y) > shrinkOne shr = fromShrink . traverse (Shrink . shr) > but maybe this would still be useful for defining trickier shrink > functions. > > Anyway, this is the only worry I have - how to integrate the > traditional and applicative-based shrinking without too much syntactic > overhead. If we can come up with a good solution for that then I'd be > very eager to add this to QuickCheck. Or maybe calling fromShrink > isn't so painful and we should add it anyway. I will think it over a > bit... > > Nick > > On Saturday 06 September, 2014 at 11:17 am, Petr Pudlák wrote: > > 2014-09-05 18:22 GMT+02:00 Bryan O’Sullivan <bos(a)serpentine.com>: > > > > > > > On Thu, Sep 4, 2014 at 1:06 PM, Petr Pudlák <petr.mvd(a)gmail.com> > wrote: > > > > > >> since I got no response from the QuickCheck mail address, so I'm > > >> resending it here, if someone wants to comment on the idea. > > >> > > > > > > I don't think it pays its way: it makes shrink a bit tidier, at the > cost > > > of breaking backwards compatibility in a widely used library. > > > > > That’s a problem. But I believe it the patch addresses it. QuickCheck > > always uses ‘shrink’, not ‘shrinkA’, and as the default implementation of > > ‘shrink’ is defined in terms of ‘shrinkA’, then it’s possible to define > > instances both using ‘shrink’ and using ‘shrinkA’. So both old code and > new > > code that wants to take advantage of the Applicative interface work. > > > > I believe that in the long term this change would pay off: Compare the > old > > code > > > > shrinkOne [] = [] > > shrinkOne (x:xs) = [ x':xs | x' <- shr x ] > > ++ [ x:xs' | xs' <- shrinkOne xs ] > > > > shrink (x, y) = > > [ (x', y) | x' <- shrink x ] > > ++ [ (x, y') | y' <- shrink y ] > > > > shrink (x, y, z) = > > [ (x', y', z') > > | (x', (y', z')) <- shrink (x, (y, z)) ] > > > > against the improved piece > > > > traverse shr xs -- replaces the whole shrinkOne > > shrinkA (x, y) = liftA2 (,) (shrinkA x) (shrinkA y) > > shrinkA (x, y, z) = liftA3 (,,) (shrinkA x) (shrinkA y) (shrinkA z) >

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