úÎE@:@      !"#$%&'()*+,-./012345678 9 : ; < = > ? portable provisionalEdward Kmett <ekmett@gmail.com>A @ sans A. Minimal definition: Either  or  OIf defining both, then the following laws (the default definitions) must hold:   join = (>>- id)  m >>- f = join (fmap f m) Laws:  4 induced definition of <.>: f <.> x = f >>- (<$> 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 B to be used as a member of   )A strong lax semi-monoidal endofunctor.  This is equivalent to an B without C. 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 DAn E is a B, but it is an instance of  FA G is not a @, but it is an instance of  HAn IntMap is not B, but it is an instance of   I A Map is not B, 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> JKL M!"GUsable default for foldMap, but only if you define bifoldMap1 yourself  !" !" !"portable provisionalEdward Kmett <ekmett@gmail.com>#$%&#$%&#$%&#$%$%&portable provisionalEdward Kmett <ekmett@gmail.com> NOP'()*+,-EUsable default for foldMap, but only if you define foldMap1 yourself '()*+,-'()*+,-'()()*+,-portable provisionalEdward Kmett <ekmett@gmail.com>./01./01./01./0/01portable provisionalEdward Kmett <ekmett@gmail.com>2Laws:  C <!> is associative: (a <!> b) <!> c = a <!> (b <!> c) K <$> left-distributes over <!>: f <$> (a <!> b) = (f <$> a) <!> (f <$> b) If extended to an Q then 3 should equal R. Ideally, an instance of 2 also satisfies the "left distributon" law of  MonadPlus with respect to  .:  K <.> right-distributes over <!>: (a <!> b) <.> c = (a <.> c) <!> (b <.> c) But S, T, U a, VW e m, and STM satisfy the alternative  " left catch" law instead:   pure a <!> b = pure a PHowever, this variation cannot be stated purely in terms of the dependencies of 2. JWhen and if MonadPlus is successfully refactored, this class should also * be refactored to remove these instances. BThe right distributive law should extend in the cases where the a  or @ is = provided to yield variations of the right distributive law: + (m <!> n) >>- f = (m >>- f) <!> (m >>- f) + (m <!> n) >>= f = (m >>= f) <!> (m >>= f) 3( |) without a required empty 45X-This instance does not actually satisfy the ( .) right distributive law  It instead satisfies the  Left-Catch law  234523452345345portable provisionalEdward Kmett <ekmett@gmail.com>6Laws:   zero <!> m = m  m <!> zero = m If extended to an Q then 7 should equal Y. 7 23456767677 portable provisionalEdward Kmett <ekmett@gmail.com>8)A strong lax semi-monoidal endofunctor.  This is equivalent to an  Applicative without pure. Laws: B associative composition: (.) <$> u <.> v <.> w = u <.> (v <.> w) 9:a .> b = const id  $ a  . b ;a  . b = const <$ a  . b <=>3Lift a binary function into a comonad with zipping ?4Lift a ternary function into a comonad with zipping Z[\]89:;<=>?89:;<=>?89:;9:;<=>?^     !"#$%&'()*+,-./0123456789:;<=>?@ABCD E F G H I J K L M N OP OQRSTUVSWXYZ[[\][[\ O^ O_ `abcd efghighij Oklmnlmolmplmqrfunctor-apply-0.10.1Data.Functor.BindData.Functor.Bind.TransData.Semigroup.BifoldableData.Semigroup.BitraversableData.Semigroup.FoldableData.Semigroup.TraversableData.Functor.AltData.Functor.PlusData.Bifunctor.ApplyData.Functor.ApplybaseGHC.BaseFunctor Data.Functor<$><$fmapBind>>-join MaybeApply runMaybeApplyWrappedApplicativeWrapApplicativeunwrapApplicativeApply<.>.><.$><..>liftF2liftF3 returning-<<->--<- apDefault BindTransliftB Bifoldable1bifold1 bifoldMap1 bitraverse1_bifor1_ bisequenceA1_bifoldMapDefault1Bitraversable1 bitraverse1 bisequence1bifoldMap1Default Foldable1fold1foldMap1 traverse1_for1_ sequenceA1_foldMapDefault1 Traversable1 traverse1 sequence1foldMap1DefaultAltsomemanyPluszeroBiapply<<.>>.>><<.<<$>><<..>>bilift2bilift3MonadreturnControl.Applicative Applicativepure $fBindIntMapcontainers-0.4.0.0 Data.IntMapIntMap $fBindMapData.MapMap $fApplyIntMap $fApplyMapActgetActignore Alternative<|> Data.MaybeMaybeghc-prim GHC.TypesIO Data.EitherEithertransformers-0.2.2.0Control.Monad.Trans.ErrorErrorT$fAltIOemptybifunctors-0.1Data.Bifunctorsecondfirstbimap Bifunctor