Data.Functor.Invariant

Haskell98 invariant functors (also known as exponential functors).

For more information, see Edward Kmett's article "Rotten Bananas":

http://comonad.com/reader/2008/rotten-bananas/

Invariant

class Invariant f where

Any * -> * type parametric in the argument permits an instance of Invariant.

Instances should satisfy the following laws:

invmap id id = id
invmap f2 f2' . invmap f1 f1' = invmap (f2 . f1) (f1' . f2')

Minimal complete definition

Nothing

Methods

invmap :: (a -> b) -> (b -> a) -> f a -> f b

Instances

Invariant []

Invariant IO

Invariant V1

from GHC.Generics

Invariant U1

from GHC.Generics

Invariant Par1

from GHC.Generics

Invariant StateVar

from the StateVar package

Invariant SettableStateVar

from the StateVar package

Invariant ArgOrder

from System.Console.GetOpt

Invariant OptDescr

from System.Console.GetOpt

Invariant ArgDescr

from System.Console.GetOpt

Invariant Identity

from Data.Functor.Identity

Invariant ZipList

from Control.Applicative

Invariant Handler

from Control.Exception

Invariant STM

from the stm package

Invariant Dual

from Data.Monoid

Invariant Endo

from Data.Monoid

Invariant First

from Data.Monoid

Invariant Last

from Data.Monoid

Invariant ReadPrec

Invariant ReadP

Invariant Maybe

Invariant IntMap

from the containers package

Invariant Tree

from the containers package

Invariant Seq

from the containers package

Invariant ViewL

from the containers package

Invariant ViewR

from the containers package

Invariant Predicate

from the contravariant package

Invariant Comparison

from the contravariant package

Invariant Equivalence

from the contravariant package

Invariant Min

from the semigroups package

Invariant Max

from the semigroups package

Invariant First

from the semigroups package

Invariant Last

from the semigroups package

Invariant Option

from the semigroups package

Invariant NonEmpty

from the semigroups package

Invariant ((->) a)

Invariant (Either a)

Invariant f => Invariant (Rec1 f)

from GHC.Generics

Invariant ((,) a)

Invariant (ST s)

Ix i => Invariant (Array i)

from the array package

Invariant (ST s)

Invariant (Const a)

from Control.Applicative

Monad m => Invariant (WrappedMonad m)

from Control.Applicative

Arrow a => Invariant (ArrowMonad a)

from Control.Arrow

Invariant (Proxy *)

from Data.Proxy

Invariant m => Invariant (IdentityT m)

from the transformers package

Invariant (Map k)

from the containers package

Invariant (Op a)

from the contravariant package

Invariant2 p => Invariant (Prep p)

from the profunctors package

Invariant2 p => Invariant (Coprep p)

from the profunctors package

Invariant (Arg a)

from the semigroups package

Invariant f => Invariant (Reverse f)

from the transformers package

Invariant f => Invariant (Backwards f)

from the transformers package

Invariant m => Invariant (MaybeT m)

from the transformers package

Invariant m => Invariant (ListT m)

from the transformers package

Invariant f => Invariant (Lift f)

from the transformers package

Invariant (Constant a)

from the transformers package

Invariant (HashMap k)

from the unordered-containers package

Invariant (K1 i c)

from GHC.Generics

(Invariant l, Invariant r) => Invariant ((:+:) l r)

from GHC.Generics

(Invariant l, Invariant r) => Invariant ((:*:) l r)

from GHC.Generics

(Invariant f, Invariant g) => Invariant ((:.:) f g)

from GHC.Generics; genuinely relying on this instance likely requires writing your Generic1 instance by hand

Invariant ((,,) a b)

Arrow arr => Invariant (WrappedArrow arr a)

from Control.Applicative

Invariant f => Invariant (Alt * f)

from Data.Monoid

Invariant2 p => Invariant (Join * p)

from the bifunctors package

Invariant2 p => Invariant (Fix * p)

from the bifunctors package

(Invariant f, Invariant g) => Invariant (Compose f g)

from the contravariant package

(Invariant f, Invariant g) => Invariant (ComposeFC f g)

from the contravariant package

(Invariant f, Invariant g) => Invariant (ComposeCF f g)

from the contravariant package

(Invariant f, Invariant g) => Invariant (Compose f g)

from the transformers package

Invariant2 p => Invariant (Codensity p a)

from the profunctors package

Invariant2 p => Invariant (TambaraSum p a)

from the profunctors package

Invariant (CotambaraSum p a)

from the profunctors package

Invariant2 p => Invariant (Closure p a)

from the profunctors package

Invariant2 p => Invariant (Tambara p a)

from the profunctors package

Invariant (Cotambara p a)

from the profunctors package

Invariant f => Invariant (Star f a)

from the profunctors package

Invariant (Costar f a)

from the profunctors package

Arrow arr => Invariant (WrappedArrow arr a)

from the profunctors package

Invariant (Forget r a)

from the profunctors package

Invariant (Tagged k s)

from the tagged package

(Invariant f, Invariant g) => Invariant (Sum f g)

from the transformers package

(Invariant f, Invariant g) => Invariant (Product f g)

from the transformers package

Invariant m => Invariant (WriterT w m)

from the transformers package

Invariant m => Invariant (WriterT w m)

from the transformers package

Invariant m => Invariant (ErrorT e m)

Invariant m => Invariant (ExceptT e m)

from the transformers package

Invariant m => Invariant (StateT s m)

from the transformers package

Invariant m => Invariant (StateT s m)

from the transformers package

Invariant m => Invariant (ReaderT r m)

from the transformers package

Invariant (ContT r m)

from the transformers package

Contravariant f => Invariant (WrappedContravariant * f)

Functor f => Invariant (WrappedFunctor * f)

Invariant f => Invariant (M1 i t f)

from GHC.Generics

Invariant ((,,,) a b c)

Invariant2 q => Invariant (Ran p q a)

from the profunctors package

Invariant2 p => Invariant (Procompose p q a)

from the profunctors package

Invariant2 p => Invariant (Rift p q a)

from the profunctors package

Invariant ((,,,,) a b c d)

Bifunctor p => Invariant (WrappedBifunctor * * p a)

from the bifunctors package

Invariant g => Invariant (Joker k * g a)

from the bifunctors package

Invariant2 p => Invariant (Flip * * p a)

from the bifunctors package

Invariant (Clown * k f a)

from the bifunctors package

Invariant m => Invariant (RWST r w s m)

from the transformers package

Invariant m => Invariant (RWST r w s m)

from the transformers package

Profunctor p => Invariant (WrappedProfunctor * * p a)

(Invariant f, Invariant2 p) => Invariant (Tannen * * * f p a)

from the bifunctors package

(Invariant2 p, Invariant g) => Invariant (Biff * * k * p f g a)

from the bifunctors package

invmapFunctor :: Functor f => (a -> b) -> (b -> a) -> f a -> f b

Every Functor is also an Invariant functor.

GHC.Generics

With GHC 7.2 or later, Invariant instances can be defined easily using GHC generics like so:

{-# LANGUAGE DeriveGeneric, FlexibleContexts #-}

import Data.Functor.Invariant
import GHC.Generics

data T f a = T (f a) deriving Generic1

instance Invariant f => Invariant (T f)

Be aware that generic Invariant instances cannot be derived for data types that have function arguments in which the last type parameter appears in a position other than the result type (e.g., data Fun a = Fun (a -> a)). For these, you can derive them using the Data.Functor.Invariant.TH module.

genericInvmap :: (Generic1 f, Invariant (Rep1 f)) => (a -> b) -> (b -> a) -> f a -> f b

A generic implementation of invmap.

newtype WrappedFunctor f a

Wrap a Functor to be used as a member of Invariant.

Constructors

WrapFunctor

Fields

unwrapFunctor :: f a

Instances

Monad m => Monad (WrappedFunctor * m)

Functor f => Functor (WrappedFunctor * f)

Applicative f => Applicative (WrappedFunctor * f)

Foldable f => Foldable (WrappedFunctor * f)

Traversable f => Traversable (WrappedFunctor * f)

Alternative f => Alternative (WrappedFunctor * f)

MonadPlus m => MonadPlus (WrappedFunctor * m)

Functor f => Invariant (WrappedFunctor * f)

Eq (f a) => Eq (WrappedFunctor k f a)

Ord (f a) => Ord (WrappedFunctor k f a)

Read (f a) => Read (WrappedFunctor k f a)

Show (f a) => Show (WrappedFunctor k f a)

invmapContravariant :: Contravariant f => (a -> b) -> (b -> a) -> f a -> f b

Every Contravariant functor is also an Invariant functor.

newtype WrappedContravariant f a

Wrap a Contravariant functor to be used as a member of Invariant.

Constructors

WrapContravariant

Fields

unwrapContravariant :: f a

Instances

Divisible f => Divisible (WrappedContravariant * f)

Decidable f => Decidable (WrappedContravariant * f)

Contravariant f => Contravariant (WrappedContravariant * f)

Contravariant f => Invariant (WrappedContravariant * f)

Eq (f a) => Eq (WrappedContravariant k f a)

Ord (f a) => Ord (WrappedContravariant k f a)

Read (f a) => Read (WrappedContravariant k f a)

Show (f a) => Show (WrappedContravariant k f a)

Invariant2

class Invariant2 f where

Any * -> * -> * type parametric in both arguments permits an instance of Invariant2.

Instances should satisfy the following laws:

invmap2 id id id id = id
invmap2 f2 f2' g2 g2' . invmap2 f1 f1' g1 g1' =
  invmap2 (f2 . f1) (f1' . f2') (g2 . g1) (g1' . g2')

Methods

invmap2 :: (a -> c) -> (c -> a) -> (b -> d) -> (d -> b) -> f a b -> f c d

Instances

Invariant2 (->)

Invariant2 Either

Invariant2 (,)

Invariant2 Const

from Control.Applicative

Invariant2 Op

from the contravariant package

Invariant2 Arg

from the semigroups package

Invariant2 Constant

from the transformers package

Invariant2 (K1 i)

from GHC.Generics

Invariant2 ((,,) a)

Arrow arr => Invariant2 (WrappedArrow arr)

from Control.Applicative

Invariant2 p => Invariant2 (Codensity p)

from the profunctors package

Invariant2 p => Invariant2 (CofreeMapping p)

from the profunctors package

Invariant2 (FreeMapping p)

from the profunctors package

Invariant2 p => Invariant2 (CofreeTraversing p)

from the profunctors package

Invariant2 (FreeTraversing p)

from the profunctors package

Invariant2 p => Invariant2 (TambaraSum p)

from the profunctors package

Invariant2 (PastroSum p)

from the profunctors package

Invariant2 (CotambaraSum p)

from the profunctors package

Invariant2 (CopastroSum p)

from the profunctors package

Invariant2 p => Invariant2 (Closure p)

from the profunctors package

Invariant2 (Environment p)

from the profunctors package

Invariant2 p => Invariant2 (Tambara p)

from the profunctors package

Invariant2 (Pastro p)

from the profunctors package

Invariant2 (Cotambara p)

from the profunctors package

Invariant f => Invariant2 (Star f)

from the profunctors package

Invariant f => Invariant2 (Costar f)

from the profunctors package

Arrow arr => Invariant2 (WrappedArrow arr)

from the profunctors package

Invariant2 (Forget r)

from the profunctors package

Invariant2 (Tagged *)

from the tagged package

Invariant2 ((,,,) a b)

(Invariant2 p, Invariant2 q) => Invariant2 (Ran p q)

from the profunctors package

(Invariant2 p, Invariant2 q) => Invariant2 (Procompose p q)

from the profunctors package

(Invariant2 p, Invariant2 q) => Invariant2 (Rift p q)

from the profunctors package

(Invariant f, Invariant2 p) => Invariant2 (Cayley f p)

from the profunctors package

Invariant2 ((,,,,) a b c)

Bifunctor p => Invariant2 (WrappedBifunctor * * p)

from the bifunctors package

Invariant g => Invariant2 (Joker * * g)

from the bifunctors package

Invariant2 p => Invariant2 (Flip * * p)

from the bifunctors package

Invariant f => Invariant2 (Clown * * f)

from the bifunctors package

Profunctor p => Invariant2 (WrappedProfunctor * * p)

(Invariant2 f, Invariant2 g) => Invariant2 (Product * * f g)

from the bifunctors package

(Invariant2 p, Invariant2 q) => Invariant2 (Sum * * p q)

from the bifunctors package

(Invariant f, Invariant2 p) => Invariant2 (Tannen * * * f p)

from the bifunctors package

(Invariant2 p, Invariant f, Invariant g) => Invariant2 (Biff * * * * p f g)

from the bifunctors package

invmap2Bifunctor :: Bifunctor f => (a -> c) -> (c -> a) -> (b -> d) -> (d -> b) -> f a b -> f c d

Every Bifunctor is also an Invariant2 functor.

newtype WrappedBifunctor p a b :: (k -> k1 -> *) -> k -> k1 -> *

Make a Functor over the second argument of a Bifunctor.

Constructors

WrapBifunctor

Fields

unwrapBifunctor :: p a b

Instances

Bifunctor p => Bifunctor (WrappedBifunctor * * p)

Biapplicative p => Biapplicative (WrappedBifunctor * * p)

Bitraversable p => Bitraversable (WrappedBifunctor * * p)

Bifoldable p => Bifoldable (WrappedBifunctor * * p)

Bifunctor p => Invariant2 (WrappedBifunctor * * p)

from the bifunctors package

Bifunctor p => Functor (WrappedBifunctor * * p a)

Bifoldable p => Foldable (WrappedBifunctor * * p a)

Bitraversable p => Traversable (WrappedBifunctor * * p a)

Generic1 (WrappedBifunctor k * p a)

Bifunctor p => Invariant (WrappedBifunctor * * p a)

from the bifunctors package

Eq (p a b) => Eq (WrappedBifunctor k k1 p a b)

Ord (p a b) => Ord (WrappedBifunctor k k1 p a b)

Read (p a b) => Read (WrappedBifunctor k k1 p a b)

Show (p a b) => Show (WrappedBifunctor k k1 p a b)

Generic (WrappedBifunctor k k1 p a b)

type Rep1 (WrappedBifunctor k k1 p a) = D1 D1WrappedBifunctor (C1 C1_0WrappedBifunctor (S1 S1_0_0WrappedBifunctor (Rec1 (p a))))

type Rep (WrappedBifunctor k k1 p a b) = D1 D1WrappedBifunctor (C1 C1_0WrappedBifunctor (S1 S1_0_0WrappedBifunctor (Rec0 (p a b))))

invmap2Profunctor :: Profunctor f => (a -> c) -> (c -> a) -> (b -> d) -> (d -> b) -> f a b -> f c d

Every Profunctor is also an Invariant2 functor.

newtype WrappedProfunctor p a b

Wrap a Profunctor to be used as a member of Invariant2.

Constructors

WrapProfunctor

Fields

unwrapProfunctor :: p a b

Instances

ProfunctorFunctor (WrappedProfunctor * *)

ProfunctorMonad (WrappedProfunctor * *)

ProfunctorComonad (WrappedProfunctor * *)

Mapping p => Mapping (WrappedProfunctor * * p)

Traversing p => Traversing (WrappedProfunctor * * p)

Choice p => Choice (WrappedProfunctor * * p)

Cochoice p => Cochoice (WrappedProfunctor * * p)

Closed p => Closed (WrappedProfunctor * * p)

Strong p => Strong (WrappedProfunctor * * p)

Costrong p => Costrong (WrappedProfunctor * * p)

Profunctor p => Profunctor (WrappedProfunctor * * p)

Profunctor p => Invariant2 (WrappedProfunctor * * p)

Profunctor p => Invariant (WrappedProfunctor * * p a)

Eq (p a b) => Eq (WrappedProfunctor k k p a b)

Ord (p a b) => Ord (WrappedProfunctor k k p a b)

Read (p a b) => Read (WrappedProfunctor k k p a b)

Show (p a b) => Show (WrappedProfunctor k k p a b)