úÎBv>*/      !"#$%&'()*+,-.portable provisionalEdward Kmett <ekmett@gmail.com>A / sans 0. Minimal definition: Either  or  OIf defining both, then the following laws (the default definitions) must hold:   join = (>>- id)  m >>- f = join (fmap f m) Laws:  4 induced definition of <.>: f <.> x = f >>- (<$> x) 1Finally, there are two associativity conditions:  G associativity of (>>-): (m >>- f) >>- g == m >>- (\x -> f x >>- g) ; associativity of join: join . join = join . fmap join ?These can both be seen as special cases of the constraint that ; associativity of (->-): (f ->- g) ->- h = f ->- (g ->- h) 8Transform a Apply into an Applicative by adding a unit.  Wrap an 1 to be used as a member of   )A strong lax semi-monoidal endofunctor.  This is equivalent to an 1 without 2. Laws: B associative composition: (.) <$> u <.> v <.> w = u <.> (v <.> w) a .> b = const id  $ a  . b a  . b = const <$ a  . b TODO: move into Data.Functor  A variant of  with the arguments reversed. 3Lift a binary function into a comonad with zipping 4Lift a ternary function into a comonad with zipping 3An 4 is a 1, but it is an instance of  5A 6 is not a /, but it is an instance of  7An IntMap is not 1, but it is an instance of   8 A Map is not 1, but it is an instance of          portable provisionalEdward Kmett <ekmett@gmail.com>1A subset of monad transformers can transform any  as well. portable provisionalEdward Kmett <ekmett@gmail.com>    portable provisionalEdward Kmett <ekmett@gmail.com> 9:; !"EUsable default for foldMap, but only if you define foldMap1 yourself  !" !" !"portable provisionalEdward Kmett <ekmett@gmail.com>#$%&#$%&#$%&#$%$%&portable provisionalEdward Kmett <ekmett@gmail.com>'GA strong lax symmetric semi-monoidal comonad. As such, an instance of ' is required to satisfy:  + extract (a <.> b) = extract a $ extract b 'This class is based on ComonadZip from "#The Essence of Dataflow Programming" Mby Tarmo Uustalu and Varmo Vene, but adapted to fit the programming style of Control.Applicative. 1. can be seen as a similar law over and above  Apply that: " pure (a $ b) = pure a <.> pure b (3Lift a binary function into a comonad with zipping )4Lift a ternary function into a comonad with zipping <A sad little orphan =>Both required because Semigroup is not a superclass of Monoid #>?@ABCDEFGHIJK '()'()'()portable provisionalEdward Kmett <ekmett@gmail.com>*Laws:  C <!> is associative: (a <!> b) <!> c = a <!> (b <!> c) K <$> left-distributes over <!>: f <$> (a <!> b) = (f <$> a) <!> (f <$> b) If extended to an L then + should equal M. Ideally, an instance of * also satisfies the "left distributon" law of  MonadPlus:  K <.> right-distributes over <!>: (a <!> b) <.> c = (a <.> c) <!> (b <.> c) But N, O, P a, QR e m, and STM satisfy the alternative  " left catch" law instead:   pure a <!> b = pure a PHowever, this variation cannot be stated purely in terms of the dependencies of *. JWhen and if MonadPlus is successfully refactored, this class should also * be refactored to remove these instances. BThe right distributive law should extend in the cases where the a  or / is = provided to yield variations of the right distributive law: + (m <!> n) >>- f = (m >>- f) <!> (m >>- f) + (m <!> n) >>= f = (m >>= f) <!> (m >>= f) +( |) without a required empty S-This instance does not actually satisfy the ( .) right distributive law  It instead satisfies the  Left-Catch law  *+*+*++portable provisionalEdward Kmett <ekmett@gmail.com>,-. *+,-.,-.,-.-.T    !"#$%&'()*+,-./0123456789 : ; <= <>?@ABC@DEFGHHIJKLMNLMOLMPLMQLMRLMSLMSLTULTVLTWLTXLTYLTZLT[ <\ <] ^_`ab cdefgefghifunctor-apply-0.9.3Data.Functor.BindData.Functor.Bind.TransData.Semigroup.FoldableData.Semigroup.TraversableControl.Comonad.ApplyData.Functor.AltControl.Applicative.AltData.Functor.ApplybaseGHC.BaseFunctor Data.Functor<$><$fmapBind>>-join MaybeApply runMaybeApplyWrappedApplicativeWrapApplicativeunwrapApplicativeApply<.>.><.$><..>liftF2liftF3 returning-<<->--<- apDefault BindTransliftB Foldable1fold1foldMap1 traverse1_for1_ sequenceA1_foldMapDefault1 Traversable1 traverse1 sequence1foldMap1Default ComonadApplyliftW2liftW3AltApplicativeAltsomemanyMonadreturnControl.Applicative Applicativepure $fBindIntMapcontainers-0.4.0.0 Data.IntMapIntMap $fBindMapData.MapMap $fApplyIntMap $fApplyMapActgetAct$fArrowLoopCokleisli$fComonadApply(,)comonad-0.9.0.1Control.ComonadwfixliftWextractComonad runCokleisli CokleisliData.Functor.Extend=>==<=<<==>>extend duplicateExtend Alternative<|> Data.MaybeMaybeghc-prim GHC.TypesIO Data.EitherEithertransformers-0.2.2.0Control.Monad.Trans.ErrorErrorT$fAltIO