8{      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGH I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l mnopqrstuvwxyz(C) 2011-2012 Edward Kmett BSD-style (see the file LICENSE)Edward Kmett <ekmett@gmail.com> provisionalGADTs Safe-Inferred!" Coend of  from  Hask -> Hask.(C) 2011-2013 Edward Kmett BSD-style (see the file LICENSE)Edward Kmett <ekmett@gmail.com> provisionalportable TrustworthyMFormally, the class  represents a profunctor from Hask -> Hask.nIntuitively it is a bifunctor where the first argument is contravariant and the second argument is covariant.You can define a  by either defining  or by defining both  and .If you supply , you should ensure that:  { { "a {If you supply  and  , ensure:  { "a {  { "a { +If you supply both, you should also ensure:  f g "a  f |  gThese ensure by parametricity:  (f | g) (h | i) "a  g h |  f i  (f | g) "a  g |  f  (f | g) "a  f |  g )Map over both arguments at the same time.  f g "a  f |  g'Map the first argument contravariantly.  f "a  f {$Map the second argument covariantly.  "a  {Strictly map the second argument argument covariantly with a function that is assumed operationally to be a cast, such as a newtype constructor.Note: This operation is explicitly unsafe- since an implementation may choose to use  unsafeCoerceh to implement this combinator and it has no way to validate that your function meets the requirements.'If you implement this combinator with  unsafeCoerceo, then you are taking upon yourself the obligation that you don't use GADT-like tricks to distinguish values.If you import Data.Profunctor.Unsafe you are taking upon yourself the obligation that you will only call this with a first argument that is operationally identity.\The semantics of this function with respect to bottoms should match the default definition: () "a \f -> \p -> p `seq`  f pStrictly map the first argument argument contravariantly with a function that is assumed operationally to be a cast, such as a newtype constructor.Note: This operation is explicitly unsafe- since an implementation may choose to use  unsafeCoerceh to implement this combinator and it has no way to validate that your function meets the requirements.'If you implement this combinator with  unsafeCoerceo, then you are taking upon yourself the obligation that you don't use GADT-like tricks to distinguish values.If you import Data.Profunctor.Unsafe you are taking upon yourself the obligation that you will only call this with a second argument that is operationally identity. () "a \p -> p `seq` \f ->  f p }~}~ (C) 2011-2013 Edward Kmett, BSD-style (see the file LICENSE)Edward Kmett <ekmett@gmail.com> provisionalportable TrustworthyHJKM  Analogous to ,  =  The generalization of  of ! that is strong with respect to .Note: This is also a notion of strength, except with regards to another monoidal structure that we can choose to equip Hask with: the cocartesian coproduct. Generalizing  of a strong Note: Every & in Haskell is strong with respect to (,).RThis describes profunctor strength with respect to the product structure of Hask. <http://www-kb.is.s.u-tokyo.ac.jp/~asada/papers/arrStrMnd.pdfWrap an arrow for use as a .Lift a  into a  (backwards).Lift a  into a  (forwards). approximates  costrength is  A       1     Safe-InferredHJKM!"#$%&'(!"#$%&'('($%&!"#!"#$%&'( Safe-Inferred68HJKM)*+)*+)*+)*+(C) 2014 Edward Kmett BSD-style (see the file LICENSE)Edward Kmett <ekmett@gmail.com> experimentalportable Trustworthy8Cayley transforms Comonads in Hask into comonads on Prof9Cayley transforms Monads in Hask into monads on Prof,-./0123456789:,-.,-.:9876543210/ ,-./0123456789: Trustworthy !"6HJKM== adjoins a @ structure to any . Analogous to  for .@BA strong profunctor allows the monoidal structure to pass through.@A closed profunctor allows the closed structure to pass through.B B | C "a { C | B "a { C B | C "a { C | B "a { #;<=>?@ABC ;<=>?@ABC @A=>?BC;<;<=>?@ABC(C) 2015 Edward Kmett BSD-style (see the file LICENSE)Edward Kmett <ekmett@gmail.com> provisionalMPTCs, fundeps Trustworthy23468DA  p is a F on f if it is a subprofunctor of  f.!That is to say it is a subset of  Hom(f-,=) closed under  and .*Alternately, you can view it as a cosieve in the comma category f/Hask.FA  p is a F on f if it is a subprofunctor of  f.!That is to say it is a subset of  Hom(-,f=) closed under  and .(Alternately, you can view it as a sieve in the comma category Hask/f. DEFGDEFGFGDE DEFG (C) 2011-2015 Edward Kmett, BSD-style (see the file LICENSE)Edward Kmett <ekmett@gmail.com> provisional Type-Families Trustworthy 234=HKMHA  p is H if there exists a  f such that p d c is isomorphic to f d -> c.KA  p is K if there exists a  f such that p d c is isomorphic to d -> f c.NDefault definition for  given that p is K.ODefault definition for  given that p is K.PM and G# form two halves of an isomorphism./This can be used with the combinators from the lens package. P :: K p => Iso' (d -> L p c) (p d c)QJ and E# form two halves of an isomorphism./This can be used with the combinators from the lens package. Q :: I f p => Iso' (f d -> c) (p d c)HIJKLMNOPQ HIJKLMNOPQ KLMPNOHIJQHIJKLMNOPQ (C) 2014 Edward Kmett BSD-style (see the file LICENSE)Edward Kmett <ekmett@gmail.com> experimentalGADTs, TFs, MPTCs, RankN Trustworthy !"6=HJKM R(This represents the right Kan lift of a  q along a  p in a limited version of the 2-category of Profunctors where the only object is the category Hask, 1-morphisms are profunctors composed and compose with Profunctor composition, and 2-morphisms are just natural transformations.UU p q is the  composition of the s p and q.For a good explanation of 2 composition in Haskell see Dan Piponi's article: :http://blog.sigfpe.com/2011/07/profunctors-in-haskell.htmlX(->)! functions as a lax identity for  composition.This provides an  for the lens1 package that witnesses the isomorphism between U (->) q d c and q d c", which is the left identity law. X ::  q => Iso' (U (->) q d c) (q d c) Y(->)! functions as a lax identity for  composition.This provides an  for the lens1 package that witnesses the isomorphism between U q (->) d c and q d c#, which is the right identity law. Y ::  q => Iso' (U q (->) d c) (q d c) ZThe associator for  composition.This provides an  for the lens1 package that witnesses the isomorphism between U p (U q r) a b and U (U p q) r a b, which arises because Prof7 is only a bicategory, rather than a strict 2-category.[ composition generalizes  composition in two ways.$This is the first, which shows that exists b. (a -> f b, b -> g c) is isomorphic to  a -> f (g c). [ ::  f => Iso' (U ( f) ( g) d c) ( ( f g) d c)\ composition generalizes  composition in two ways.%This is the second, which shows that exists b. (f a -> b, g b -> c) is isomorphic to  g (f a) -> c. \ ::  f => Iso' (U ( f) ( g) d c) ( ( g f) d c)]This is a variant on [ that uses  instead of . ] ::  f => Iso' (U ( f) ( g) d c) ( ( f g) d c)^This is a variant on \ that uses  instead of . ^ ::  f => Iso' (U ( f) ( g) d c) ( ( g f) d c)_,The 2-morphism that defines a left Kan lift. Note: When p is right adjoint to R p (->) then _ is the + of the adjunction.R p p forms a  in the . 2-category, which is isomorphic to a Haskell  instance.The composition of two K s is K. by the composition of their representations. RSTUVWXYZ[\]^_RSTUVWXYZ[\]^_UVWXYZ[]\^RST_RSTUVWXYZ[\]^_ (C) 2014 Edward Kmett BSD-style (see the file LICENSE)Edward Kmett <ekmett@gmail.com> provisionalRank2Types, TFs Trustworthy=HKM`-This represents the right Kan extension of a  p\ along itself. This provides a generalization of the "difference list" trick to profunctors.`abc`abc`abc`abc  Safe-InferredHJKMda  that is also a  is a Monoid in Profdededede '(C) 2013-2014 Edward Kmett and Dan Doel BSD-style (see the file LICENSE)Edward Kmett <ekmett@gmail.com> provisionalRank2Types, TFs Trustworthy=HJKMf-This represents the right Kan extension of a  q along a  p in a limited version of the 2-category of Profunctors where the only object is the category Hask, 1-morphisms are profunctors composed and compose with Profunctor composition, and 2-morphisms are just natural transformations.i2The 2-morphism that defines a right Kan extension. Note: When q is left adjoint to f q (->) then i is the counit of the adjunction.f p p forms a  in the . 2-category, which is isomorphic to a Haskell  instance. fghijklfghijklfghijkl fghijkl(C) 2014 Edward Kmett BSD-style (see the file LICENSE)Edward Kmett <ekmett@gmail.com> provisional Rank2Types Trustworthy !"6HJKMmCopastro -| CotambaraoMCotambara is freely adjoins respect for cocartesian structure to a profunctorrPastro -| Tambaraw w | x "a { x | w "a { x w | x "a { x | w "a { y y | z "a { z | y "a { z y | z "a { z | y "a { ,mnopqrstuvwxyz     mnopqrstuvwxyztuvwxrsopqyzmn&mnopqrstuvwxyz      !"#$$%&'())*++,-./01234567899:;<=>?@ABCDEFGGHHIJKLMNOPQ R S T U V W X Y Z [ \ \ ] ^ ^ _ ` a b c d e f g h h i j k l m m n o p q rssttuvvwxyz{|}~|}||||||}  ||        profunctors-5Data.Profunctor.TraceData.Profunctor.UnsafeData.ProfunctorData.Profunctor.MonadData.Profunctor.AdjunctionData.Profunctor.CayleyData.Profunctor.ClosedData.Profunctor.SieveData.Profunctor.RepData.Profunctor.CompositionData.Profunctor.CodensityData.Profunctor.MonoidData.Profunctor.RanData.Profunctor.Tambara ProfunctorProfuctor.Unsafe#.TambaraTracedimaplmaprmap.#CochoiceunleftunrightCostrongunfirstunsecondChoiceleft'right'Strongfirst'second'Forget runForget WrappedArrow WrapArrow unwrapArrowCostar runCostarStarrunStar:->ProfunctorComonad proextract produplicateProfunctorMonad proreturnprojoinProfunctorFunctorpromapProfunctorAdjunctionunitcounitCayley runCayley$fArrowPlusCayley$fArrowZeroCayley$fArrowLoopCayley$fArrowChoiceCayley $fArrowCayley$fCategory*Cayley$fChoiceCayley$fStrongCayley$fProfunctorCayley$fProfunctorComonadCayley$fProfunctorMonadCayley$fProfunctorFunctorCayley EnvironmentClosure runClosureClosedclosedcloseuncloseCosievecosieveSievesieveCorepresentableCorep cotabulate RepresentableReptabulatefirstRep secondRep tabulated cotabulatedRiftrunRift Procompose procomposedidlidrassocstarscostarskleislis cokleislis decomposeRift Codensity runCodensitydecomposeCodensityetamuRanrunRan decomposeRan precomposeRancurryRan uncurryRanCopastro Cotambara runCotambaraPastro runTambaratambara untambara cotambara uncotambarabaseControl.Categoryid.$fProfunctorCokleisli$fProfunctorKleisli$fProfunctorTagged$fProfunctor(->) Control.Arrow ArrowLooploopGHC.BaseFunctor Data.EitherEither$fChoiceCokleisli comonad-4.2.6Control.Comonadextract$fStrongWrappedArrowArrowCategory$fCostrongKleisli$fCostrongWrappedArrow$fCostrongTagged$fCostrong(->)$fChoiceForget$fChoiceWrappedArrow$fChoiceTagged$fChoiceCostar $fChoiceStar$fChoiceKleisli $fChoice(->)$fStrongForget $fStrongStar$fStrongKleisli 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$fClosed(->)$fClosedTagged$fCosieveCostarf$fCosieveTaggedProxy$fCosieveCokleisliw$fCosieve(->)Identity$fSieveForgetConst $fSieveStarf$fSieveKleislim$fSieve(->)IdentityIso$fCorepresentableCostar$fCorepresentableTagged$fCorepresentableCokleisli$fCorepresentable(->)$fRepresentableForget$fRepresentableStar$fRepresentableKleisli$fRepresentable(->)transformers-0.3.0.0Data.Functor.ComposeComposeKleisliMonad Cokleisli$fCategory*Rift$fRepresentableProcompose$$fProfunctorAdjunctionProcomposeRift $fFunctorRift$fProfunctorRift$fProfunctorComonadRift$fProfunctorFunctorRift$fClosedProcompose$fChoiceProcompose$fStrongProcompose$fCorepresentableProcompose$fCosieveProcomposeCompose$fSieveProcomposeCompose$fFunctorProcompose$fProfunctorProcompose$fProfunctorMonadProcompose$fProfunctorFunctorProcompose$fCategory*Codensity$fFunctorCodensity$fProfunctorCodensity$fCategory*Ran $fFunctorRan$fProfunctorRan$fProfunctorComonadRan$fProfunctorFunctorRan$fProfunctorMonadCopastro$fProfunctorFunctorCopastro'$fProfunctorAdjunctionCopastroCotambara$fProfunctorCopastro$fFunctorCotambara$fCategory*Cotambara$fChoiceCotambara$fProfunctorCotambara$fProfunctorComonadCotambara$fProfunctorFunctorCotambara#$fProfunctorAdjunctionPastroTambara$fProfunctorMonadPastro$fProfunctorFunctorPastro$fProfunctorPastro$fMonoidTambara$fAlternativeTambara$fApplicativeTambara$fFunctorTambara$fArrowPlusTambara$fArrowZeroTambara$fArrowLoopTambara$fArrowApplyTambara$fArrowChoiceTambara$fArrowTambara$fCategory*Tambara$fChoiceTambara$fStrongTambara$fProfunctorComonadTambara$fProfunctorFunctorTambara$fProfunctorTambara