Copyright	(c) Conal Elliott 2007-2013
License	BSD3
Maintainer	conal@conal.net
Stability	experimental
Portability	see LANGUAGE pragma
Safe Haskell	None
Language	Haskell98

Control.Compose

Contents

Value transformers
Specialized semantic editor combinators
Contravariant functors
Unary/unary composition
Type composition
- Unary/binary
- (->)/unary
Monoid constructors
Flip a binary constructor's type arguments
Type application
Identity
Constructor pairing
- Unary
Binary
Arrow between two constructor applications
Augment other modules

Description

Various type constructor compositions and instances for them. Some come from "Applicative Programming with Effects" http://www.soi.city.ac.uk/~ross/papers/Applicative.html

Synopsis

type Unop a = a -> a
type Binop a = a -> a -> a
result :: Category cat => (b `cat` b') -> (a `cat` b) -> a `cat` b'
argument :: Category cat => (a' `cat` a) -> (a `cat` b) -> a' `cat` b
(~>) :: Category cat => (a' `cat` a) -> (b `cat` b') -> (a `cat` b) -> a' `cat` b'
(~>*) :: (Functor p, Functor q) => (a' -> a) -> (b -> b') -> (p a -> q b) -> p a' -> q b'
(<~) :: Category cat => (b `cat` b') -> (a' `cat` a) -> (a `cat` b) -> a' `cat` b'
(*<~) :: (Functor p, Functor q) => (b -> b') -> (a' -> a) -> (p a -> q b) -> p a' -> q b'
class ContraFunctor h where
- contraFmap :: (a -> b) -> h b -> h a
bicomap :: ContraFunctor f => (a :<->: b) -> f a :<->: f b
newtype (g :. f) a = O (g (f a))
type O = (:.)
unO :: (g :. f) a -> g (f a)
biO :: g (f a) :<->: (g :. f) a
convO :: Functor g => (b :<->: g c) -> (c :<->: f a) -> b :<->: (g :. f) a
coconvO :: ContraFunctor g => (b :<->: g c) -> (c :<->: f a) -> b :<->: (g :. f) a
inO :: (g (f a) -> g' (f' a')) -> (g :. f) a -> (g' :. f') a'
inO2 :: (g (f a) -> g' (f' a') -> g'' (f'' a'')) -> (g :. f) a -> (g' :. f') a' -> (g'' :. f'') a''
inO3 :: (g (f a) -> g' (f' a') -> g'' (f'' a'') -> g''' (f''' a''')) -> (g :. f) a -> (g' :. f') a' -> (g'' :. f'') a'' -> (g''' :. f''') a'''
oPure :: Applicative g => f a -> (g :. f) a
oFmap :: Functor g' => (f a -> f' a') -> (g' :. f) a -> (g' :. f') a'
oLiftA2 :: Applicative g'' => (f a -> f' a' -> f'' a'') -> (g'' :. f) a -> (g'' :. f') a' -> (g'' :. f'') a''
oLiftA3 :: Applicative g''' => (f a -> f' a' -> f'' a'' -> f''' a''') -> (g''' :. f) a -> (g''' :. f') a' -> (g''' :. f'') a'' -> (g''' :. f''') a'''
fmapFF :: (Functor g, Functor f) => (a -> b) -> (g :. f) a -> (g :. f) b
fmapCC :: (ContraFunctor g, ContraFunctor f) => (a -> b) -> (g :. f) a -> (g :. f) b
contraFmapFC :: (Functor g, ContraFunctor f) => (b -> a) -> (g :. f) a -> (g :. f) b
contraFmapCF :: (ContraFunctor g, Functor f) => (b -> a) -> (g :. f) a -> (g :. f) b
class DistribM m n where
- distribM :: n (m a) -> m (n a)
joinDistribM :: (Monad m, Monad n, DistribM m n) => (m :. n) ((m :. n) a) -> (m :. n) a
bindDistribM :: (Functor m, Functor n, Monad m, Monad n, DistribM m n) => (m :. n) a -> (a -> (m :. n) b) -> (m :. n) b
returnDistribM :: (Monad m, Monad n) => a -> (m :. n) a
joinMMT :: (Monad m, Monad n, Traversable n, Applicative m) => m (n (m (n a))) -> m (n a)
joinComposeT :: (Monad m, Monad n, Traversable n, Applicative m) => (m :. n) ((m :. n) a) -> (m :. n) a
newtype OO f j a b = OO {
- unOO :: f (a `j` b)
}
newtype FunA h a b = FunA {
- unFunA :: h a -> h b
}
inFunA :: ((h a -> h b) -> h' a' -> h' b') -> FunA h a b -> FunA h' a' b'
inFunA2 :: ((h a -> h b) -> (h' a' -> h' b') -> h'' a'' -> h'' b'') -> FunA h a b -> FunA h' a' b' -> FunA h'' a'' b''
class FunAble h where
- arrFun :: (a -> b) -> h a -> h b
- firstFun :: (h a -> h a') -> h (a, b) -> h (a', b)
- secondFun :: (h b -> h b') -> h (a, b) -> h (a, b')
- (***%) :: (h a -> h b) -> (h a' -> h b') -> h (a, a') -> h (b, b')
- (&&&%) :: (h a -> h b) -> (h a -> h b') -> h a -> h (b, b')
class Monoid_f m where
- mempty_f :: forall a. m a
- mappend_f :: forall a. m a -> m a -> m a
newtype Flip j b a = Flip {
- unFlip :: a `j` b
}
biFlip :: (a `j` b) :<->: Flip j b a
inFlip :: ((a `j` b) -> a' `k` b') -> Flip j b a -> Flip k b' a'
inFlip2 :: ((a `j` b) -> (a' `k` b') -> a'' `l` b'') -> Flip j b a -> Flip k b' a' -> Flip l b'' a''
inFlip3 :: ((a `j` b) -> (a' `k` b') -> (a'' `l` b'') -> a''' `m` b''') -> Flip j b a -> Flip k b' a' -> Flip l b'' a'' -> Flip m b''' a'''
type OI = Flip (->) (IO ())
class ToOI sink where
- toOI :: sink b -> OI b
newtype f :$ a = App {
- unApp :: f a
}
type App = (:$)
biApp :: f a :<->: App f a
inApp :: (f a -> f' a') -> App f a -> App f' a'
inApp2 :: (f a -> f' a' -> f'' a'') -> App f a -> App f' a' -> App f'' a''
newtype Id a = Id a
unId :: Id a -> a
biId :: a :<->: Id a
inId :: (a -> b) -> Id a -> Id b
inId2 :: (a -> b -> c) -> Id a -> Id b -> Id c
newtype (f :*: g) a = Prod {
- unProd :: (f a, g a)
}
(*:*) :: f a -> g a -> (f :*: g) a
biProd :: (f a, g a) :<->: (f :*: g) a
convProd :: (b :<->: f a) -> (c :<->: g a) -> (b, c) :<->: (f :*: g) a
(***#) :: (a -> b -> c) -> (a' -> b' -> c') -> (a, a') -> (b, b') -> (c, c')
($*) :: (a -> b, a' -> b') -> (a, a') -> (b, b')
inProd :: ((f a, g a) -> (f' a', g' a')) -> (f :*: g) a -> (f' :*: g') a'
inProd2 :: ((f a, g a) -> (f' a', g' a') -> (f'' a'', g'' a'')) -> (f :*: g) a -> (f' :*: g') a' -> (f'' :*: g'') a''
inProd3 :: ((f a, g a) -> (f' a', g' a') -> (f'' a'', g'' a'') -> (f''' a''', g''' a''')) -> (f :*: g) a -> (f' :*: g') a' -> (f'' :*: g'') a'' -> (f''' :*: g''') a'''
newtype (f ::*:: g) a b = Prodd {
- unProdd :: (f a b, g a b)
}
(*::*) :: f a b -> g a b -> (f ::*:: g) a b
inProdd :: ((f a b, g a b) -> (f' a' b', g' a' b')) -> (f ::*:: g) a b -> (f' ::*:: g') a' b'
inProdd2 :: ((f a b, g a b) -> (f' a' b', g' a' b') -> (f'' a'' b'', g'' a'' b'')) -> (f ::*:: g) a b -> (f' ::*:: g') a' b' -> (f'' ::*:: g'') a'' b''
newtype Arrw j f g a = Arrw {
- unArrw :: f a `j` g a
}
type (:->:) = Arrw (->)
biFun :: (f a -> g a) :<->: (f :->: g) a
convFun :: (b :<->: f a) -> (c :<->: g a) -> (b -> c) :<->: (f :->: g) a
inArrw :: ((f a `j` g a) -> f' a' `j` g' a') -> Arrw j f g a -> Arrw j f' g' a'
inArrw2 :: ((f a `j` g a) -> (f' a' `j` g' a') -> f'' a'' `j` g'' a'') -> Arrw j f g a -> Arrw j f' g' a' -> Arrw j f'' g'' a''
inArrw3 :: ((f a `j` g a) -> (f' a' `j` g' a') -> (f'' a'' `j` g'' a'') -> f''' a''' `j` g''' a''') -> Arrw j f g a -> Arrw j f' g' a' -> Arrw j f'' g'' a'' -> Arrw j f''' g''' a'''
biConst :: a :<->: Const a b
inConst :: (a -> b) -> Const a u -> Const b v
inConst2 :: (a -> b -> c) -> Const a u -> Const b v -> Const c w
inConst3 :: (a -> b -> c -> d) -> Const a u -> Const b v -> Const c w -> Const d x
biEndo :: (a -> a) :<->: Endo a
inEndo :: (Unop a -> Unop a') -> Endo a -> Endo a'

Value transformers

type Unop a = a -> a

Unary functions

type Binop a = a -> a -> a

Binary functions

Specialized semantic editor combinators

result :: Category cat => (b `cat` b') -> (a `cat` b) -> a `cat` b'

Add post-processing

argument :: Category cat => (a' `cat` a) -> (a `cat` b) -> a' `cat` b

Add pre-processing argument :: (a' -> a) -> ((a -> b) -> (a' -> b))

(~>) :: Category cat => (a' `cat` a) -> (b `cat` b') -> (a `cat` b) -> a' `cat` b' infixr 1

Add pre- and post processing

(~>*) :: (Functor p, Functor q) => (a' -> a) -> (b -> b') -> (p a -> q b) -> p a' -> q b' infixr 1

Like '(~>)' but specialized to functors and functions.

(<~) :: Category cat => (b `cat` b') -> (a' `cat` a) -> (a `cat` b) -> a' `cat` b' infixl 1

(*<~) :: (Functor p, Functor q) => (b -> b') -> (a' -> a) -> (p a -> q b) -> p a' -> q b' infixl 1

Contravariant functors

class ContraFunctor h where

Contravariant functors. often useful for acceptors (consumers, sinks) of values.

Methods

contraFmap :: (a -> b) -> h b -> h a

Instances

Arrow arr => ContraFunctor (Flip arr b)
(Arrow j, Functor f, ContraFunctor g) => ContraFunctor (Arrw j f g)

bicomap :: ContraFunctor f => (a :<->: b) -> f a :<->: f b

Bijections on contravariant functors

Unary/unary composition

newtype (g :. f) a infixl 9

Composition of unary type constructors

There are (at least) two useful Monoid instances, so you'll have to pick one and type-specialize it (filling in all or parts of g and/or f).

    -- standard Monoid instance for Applicative applied to Monoid
    instance (Applicative (g :. f), Monoid a) => Monoid ((g :. f) a) where
      { mempty = pure mempty; mappend = liftA2 mappend }
    -- Especially handy when g is a Monoid_f.
    instance Monoid (g (f a)) => Monoid ((g :. f) a) where
      { mempty = O mempty; mappend = inO2 mappend }

Corresponding to the first and second definitions above,

    instance (Applicative g, Monoid_f f) => Monoid_f (g :. f) where
      { mempty_f = O (pure mempty_f); mappend_f = inO2 (liftA2 mappend_f) }
    instance Monoid_f g => Monoid_f (g :. f) where
      { mempty_f = O mempty_f; mappend_f = inO2 mappend_f }

Similarly, there are two useful Functor instances and two useful ContraFunctor instances.

    instance (      Functor g,       Functor f) => Functor (g :. f) where fmap = fmapFF
    instance (ContraFunctor g, ContraFunctor f) => Functor (g :. f) where fmap = fmapCC

    instance (      Functor g, ContraFunctor f) => ContraFunctor (g :. f) where contraFmap = contraFmapFC
    instance (ContraFunctor g,       Functor f) => ContraFunctor (g :. f) where contraFmap = contraFmapCF

However, it's such a bother to define the Functor instances per composition type, I've left the fmapFF case in. If you want the fmapCC one, you're out of luck for now. I'd love to hear a good solution. Maybe someday Haskell will do Prolog-style search for instances, subgoaling the constraints, rather than just matching instance heads.

Constructors

O (g (f a))

Instances

Applicative f => Lambda f ((:.) (Flip (->) o) f)
Applicative f => Lambda f ((:.) f (Flip (->) o))
(Functor g, Functor f) => Functor ((:.) g f)
(Applicative g, Applicative f) => Applicative ((:.) g f)
(Foldable g, Foldable f, Functor g) => Foldable ((:.) g f)
(Traversable g, Traversable f) => Traversable ((:.) g f)
Title_f g => Title_f ((:.) g f)
(Functor h, Copair f) => Copair ((:.) h f)
(Functor h, Cozip f) => Cozip ((:.) h f)
Eq (g (f a)) => Eq ((:.) g f a)
Show (g (f a)) => Show ((:.) g f a)

type O = (:.)

Compatibility synonym

unO :: (g :. f) a -> g (f a)

Unwrap a '(:.)'.

biO :: g (f a) :<->: (g :. f) a

newtype bijection

convO :: Functor g => (b :<->: g c) -> (c :<->: f a) -> b :<->: (g :. f) a

Compose a bijection, Functor style

coconvO :: ContraFunctor g => (b :<->: g c) -> (c :<->: f a) -> b :<->: (g :. f) a

Compose a bijection, ContraFunctor style

inO :: (g (f a) -> g' (f' a')) -> (g :. f) a -> (g' :. f') a'

Apply a unary function within the O constructor.

inO2 :: (g (f a) -> g' (f' a') -> g'' (f'' a'')) -> (g :. f) a -> (g' :. f') a' -> (g'' :. f'') a''

Apply a binary function within the O constructor.

inO3 :: (g (f a) -> g' (f' a') -> g'' (f'' a'') -> g''' (f''' a''')) -> (g :. f) a -> (g' :. f') a' -> (g'' :. f'') a'' -> (g''' :. f''') a'''

Apply a ternary function within the O constructor.

oPure :: Applicative g => f a -> (g :. f) a

Handy combination of O and pure.

oFmap :: Functor g' => (f a -> f' a') -> (g' :. f) a -> (g' :. f') a'

Handy combination of inO and fmap.

oLiftA2 :: Applicative g'' => (f a -> f' a' -> f'' a'') -> (g'' :. f) a -> (g'' :. f') a' -> (g'' :. f'') a''

Handy combination of inO2 and liftA2.

oLiftA3 :: Applicative g''' => (f a -> f' a' -> f'' a'' -> f''' a''') -> (g''' :. f) a -> (g''' :. f') a' -> (g''' :. f'') a'' -> (g''' :. f''') a'''

Handy combination of inO3 and liftA3.

fmapFF :: (Functor g, Functor f) => (a -> b) -> (g :. f) a -> (g :. f) b

Used for the Functor :. Functor instance of Functor

fmapCC :: (ContraFunctor g, ContraFunctor f) => (a -> b) -> (g :. f) a -> (g :. f) b

Used for the ContraFunctor :. ContraFunctor instance of Functor

contraFmapFC :: (Functor g, ContraFunctor f) => (b -> a) -> (g :. f) a -> (g :. f) b

Used for the Functor :. ContraFunctor instance of Functor

contraFmapCF :: (ContraFunctor g, Functor f) => (b -> a) -> (g :. f) a -> (g :. f) b

Used for the ContraFunctor :. Functor instance of Functor

class DistribM m n where

Monad distributivity.

TODO: what conditions are required so that (m :. n) satisfies the monad laws?

Methods

distribM :: n (m a) -> m (n a)

joinDistribM :: (Monad m, Monad n, DistribM m n) => (m :. n) ((m :. n) a) -> (m :. n) a

A candidate join for (m :. n)

bindDistribM :: (Functor m, Functor n, Monad m, Monad n, DistribM m n) => (m :. n) a -> (a -> (m :. n) b) -> (m :. n) b

A candidate '(>>=)' for (m :. n)

returnDistribM :: (Monad m, Monad n) => a -> (m :. n) a

joinMMT :: (Monad m, Monad n, Traversable n, Applicative m) => m (n (m (n a))) -> m (n a)

join-like function for implicitly composed monads

joinComposeT :: (Monad m, Monad n, Traversable n, Applicative m) => (m :. n) ((m :. n) a) -> (m :. n) a

join-like function for explicitly composed monads

Type composition

Unary/binary

newtype OO f j a b

Composition of type constructors: unary with binary. Called StaticArrow in [1].

Constructors

OO
Fields unOO :: f (a `j` b)

Instances

(Applicative f, Category * cat) => Category * (OO f cat)
(Applicative f, Arrow arr) => Arrow (OO f arr)

(->)/unary

newtype FunA h a b

Common pattern for Arrows.

Constructors

FunA
Fields unFunA :: h a -> h b

Instances

FunAble h => Category * (FunA h)
FunAble h => Arrow (FunA h)

inFunA :: ((h a -> h b) -> h' a' -> h' b') -> FunA h a b -> FunA h' a' b'

Apply unary function in side a FunA representation.

inFunA2 :: ((h a -> h b) -> (h' a' -> h' b') -> h'' a'' -> h'' b'') -> FunA h a b -> FunA h' a' b' -> FunA h'' a'' b''

Apply binary function in side a FunA representation.

class FunAble h where

Support needed for a FunA to be an Arrow.

Minimal complete definition

arrFun, firstFun, secondFun

Methods

arrFun

Arguments

:: (a -> b)
-> h a -> h b	for `arr`

firstFun :: (h a -> h a') -> h (a, b) -> h (a', b)

secondFun :: (h b -> h b') -> h (a, b) -> h (a, b')

(***%) :: (h a -> h b) -> (h a' -> h b') -> h (a, a') -> h (b, b')

(&&&%) :: (h a -> h b) -> (h a -> h b') -> h a -> h (b, b')

Monoid constructors

class Monoid_f m where

Simulates universal constraint forall a. Monoid (f a).

See Simulating Quantified Class Constraints (http://flint.cs.yale.edu/trifonov/papers/sqcc.pdf) Instantiate this schema wherever necessary:

   instance Monoid_f f where { mempty_f = mempty ; mappend_f = mappend }

Methods

mempty_f :: forall a. m a

mappend_f :: forall a. m a -> m a -> m a

Instances

Monoid_f []
Monoid_f Endo
(Monoid_f f, Monoid_f g) => Monoid_f ((:*:) f g)
Monoid o => Monoid_f (Flip (->) o)

Flip a binary constructor's type arguments

newtype Flip j b a

Flip type arguments

Constructors

Flip
Fields unFlip :: a `j` b

Instances

ToOI OI
Lambda IO OI
Applicative f => Lambda f ((:.) (Flip (->) o) f)
Applicative f => Lambda f ((:.) f (Flip (->) o))
Lambda Id (Flip (->) o)
Monoid o => Monoid_f (Flip (->) o)
Arrow arr => ContraFunctor (Flip arr b)
Title o => Title_f (Flip (->) o)
Arrow j => Copair (Flip j o)
(Arrow j, Monoid_f (Flip j o)) => Pair (Flip j o)
Arrow j => Cozip (Flip j o)
(Arrow j, Monoid_f (Flip j o)) => Zip (Flip j o)
(Applicative (j a), Monoid o) => Monoid (Flip j o a)

biFlip :: (a `j` b) :<->: Flip j b a

newtype bijection

inFlip :: ((a `j` b) -> a' `k` b') -> Flip j b a -> Flip k b' a'

inFlip2 :: ((a `j` b) -> (a' `k` b') -> a'' `l` b'') -> Flip j b a -> Flip k b' a' -> Flip l b'' a''

inFlip3 :: ((a `j` b) -> (a' `k` b') -> (a'' `l` b'') -> a''' `m` b''') -> Flip j b a -> Flip k b' a' -> Flip l b'' a'' -> Flip m b''' a'''

type OI = Flip (->) (IO ())

(-> IO ()) as a Flip. A ContraFunctor.

class ToOI sink where

Convert to an OI.

Methods

toOI :: sink b -> OI b

Instances

ToOI OI

Type application

newtype f :$ a infixr 0

Type application We can also drop the App constructor, but then we overlap with many other instances, like [a]. Here's a template for App-free instances.

   instance (Applicative f, Monoid a) => Monoid (f a) where
     mempty  = pure mempty
     mappend = liftA2 mappend

Constructors

App
Fields unApp :: f a

Instances

(Applicative f, Monoid m) => Monoid (App f m)

type App = (:$)

Compatibility synonym for (:$).

biApp :: f a :<->: App f a

newtype bijection

inApp :: (f a -> f' a') -> App f a -> App f' a'

inApp2 :: (f a -> f' a' -> f'' a'') -> App f a -> App f' a' -> App f'' a''

Identity

newtype Id a

Identity type constructor. Until there's a better place to find it. I'd use Control.Monad.Identity, but I don't want to introduce a dependency on mtl just for Id.

Constructors

Id a

Instances

Monad Id
Functor Id
Applicative Id
Foldable Id
Traversable Id
Unpair Id
Pair Id
Unzip Id
Zip Id
Lambda Id (Flip (->) o)
Show a => Show (Id a)

unId :: Id a -> a

biId :: a :<->: Id a

newtype bijection

inId :: (a -> b) -> Id a -> Id b

inId2 :: (a -> b -> c) -> Id a -> Id b -> Id c

Constructor pairing

Unary

newtype (f :*: g) a infixl 7

Pairing of unary type constructors

Constructors

Prod
Fields unProd :: (f a, g a)

Instances

(Functor f, Functor g) => Functor ((:*:) f g)
(Applicative f, Applicative g) => Applicative ((:*:) f g)
(Monoid_f f, Monoid_f g) => Monoid_f ((:*:) f g)
(Copair f, Copair g) => Copair ((:*:) f g)
(Pair f, Pair g) => Pair ((:*:) f g)
(Cozip f, Cozip g) => Cozip ((:*:) f g)
(Zip f, Zip g) => Zip ((:*:) f g)
(Lambda src snk, Lambda dom' ran') => Lambda ((::) src dom') ((::) snk ran')
Eq (f a, g a) => Eq ((:*:) f g a)
Ord (f a, g a) => Ord ((:*:) f g a)
Show (f a, g a) => Show ((:*:) f g a)

(*:*) :: f a -> g a -> (f :*: g) a

Handy infix & curried Prod

biProd :: (f a, g a) :<->: (f :*: g) a

newtype bijection

convProd :: (b :<->: f a) -> (c :<->: g a) -> (b, c) :<->: (f :*: g) a

Compose a bijection

(***#) :: (a -> b -> c) -> (a' -> b' -> c') -> (a, a') -> (b, b') -> (c, c') infixr 3

Combine two binary functions into a binary function on pairs

($*) :: (a -> b, a' -> b') -> (a, a') -> (b, b') infixl 0

A handy combining form. See '(***#)' for an sample use.

inProd :: ((f a, g a) -> (f' a', g' a')) -> (f :*: g) a -> (f' :*: g') a'

Apply unary function inside of f :*: g representation.

inProd2 :: ((f a, g a) -> (f' a', g' a') -> (f'' a'', g'' a'')) -> (f :*: g) a -> (f' :*: g') a' -> (f'' :*: g'') a''

Apply binary function inside of f :*: g representation.

inProd3 :: ((f a, g a) -> (f' a', g' a') -> (f'' a'', g'' a'') -> (f''' a''', g''' a''')) -> (f :*: g) a -> (f' :*: g') a' -> (f'' :*: g'') a'' -> (f''' :*: g''') a'''

Apply ternary function inside of f :*: g representation.

Binary

newtype (f ::*:: g) a b

Pairing of binary type constructors

Constructors

Prodd
Fields unProdd :: (f a b, g a b)

Instances

(Category * f, Category * f') => Category * ((::*::) f f')
(Arrow f, Arrow f') => Arrow ((::*::) f f')
(Eq (f a b), Eq (g a b)) => Eq ((::*::) f g a b)
(Ord (f a b), Ord (g a b)) => Ord ((::*::) f g a b)
(Show (f a b), Show (g a b)) => Show ((::*::) f g a b)

(*::*) :: f a b -> g a b -> (f ::*:: g) a b

Handy infix & curried Prodd

inProdd :: ((f a b, g a b) -> (f' a' b', g' a' b')) -> (f ::*:: g) a b -> (f' ::*:: g') a' b'

Apply binary function inside of f :*: g representation.

inProdd2 :: ((f a b, g a b) -> (f' a' b', g' a' b') -> (f'' a'' b'', g'' a'' b'')) -> (f ::*:: g) a b -> (f' ::*:: g') a' b' -> (f'' ::*:: g'') a'' b''

Apply binary function inside of f :*: g representation.

Arrow between two constructor applications

newtype Arrw j f g a

Arrow-like type between type constructors (doesn't enforce Arrow (~>) here).

Constructors

Arrw
Fields unArrw :: f a `j` g a

Instances

Applicative f => Lambda f ((:->:) f (Const o))
(Arrow j, ContraFunctor f, Functor g) => Functor (Arrw j f g)
(Arrow j, Functor f, ContraFunctor g) => ContraFunctor (Arrw j f g)
(Arrow j, Unpair f, Pair g) => Pair (Arrw j f g)
(Arrow j, Unzip f, Zip g) => Zip (Arrw j f g)
(Arrow j, Unlambda f f', Lambda g g') => Lambda (Arrw j f g) (Arrw j f' g')
Monoid (j (f a) (g a)) => Monoid (Arrw j f g a)

type (:->:) = Arrw (->) infixr 1

biFun :: (f a -> g a) :<->: (f :->: g) a

newtype bijection

convFun :: (b :<->: f a) -> (c :<->: g a) -> (b -> c) :<->: (f :->: g) a

Compose a bijection

inArrw :: ((f a `j` g a) -> f' a' `j` g' a') -> Arrw j f g a -> Arrw j f' g' a'

Apply unary function inside of Arrw representation.

inArrw2 :: ((f a `j` g a) -> (f' a' `j` g' a') -> f'' a'' `j` g'' a'') -> Arrw j f g a -> Arrw j f' g' a' -> Arrw j f'' g'' a''

Apply binary function inside of Arrw j f g representation.

inArrw3 :: ((f a `j` g a) -> (f' a' `j` g' a') -> (f'' a'' `j` g'' a'') -> f''' a''' `j` g''' a''') -> Arrw j f g a -> Arrw j f' g' a' -> Arrw j f'' g'' a'' -> Arrw j f''' g''' a'''

Apply ternary function inside of Arrw j f g representation.

Augment other modules

biConst :: a :<->: Const a b

newtype bijection

inConst :: (a -> b) -> Const a u -> Const b v

inConst2 :: (a -> b -> c) -> Const a u -> Const b v -> Const c w

inConst3 :: (a -> b -> c -> d) -> Const a u -> Const b v -> Const c w -> Const d x

biEndo :: (a -> a) :<->: Endo a

newtype bijection

inEndo :: (Unop a -> Unop a') -> Endo a -> Endo a'

Convenience for partial-manipulating functions