Copyright	(c) Conal Elliott 2007-2013
License	BSD3
Maintainer	[email protected]
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 Source

Unary functions

type Binop a = a -> a -> a Source

Binary functions

Specialized semantic editor combinators

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

Add post-processing

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

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 Source

Add pre- and post processing

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

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

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

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

Contravariant functors

class ContraFunctor h where Source

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

Methods

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

Instances

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

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

Bijections on contravariant functors

Unary/unary composition

newtype (g :. f) a infixl 9 Source

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

type O = (:.) Source

Compatibility synonym

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

Unwrap a '(:.)'.

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

newtype bijection

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

Compose a bijection, Functor style

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

Compose a bijection, ContraFunctor style

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

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'' Source

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''' Source

Apply a ternary function within the O constructor.

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

Handy combination of O and pure.

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

Handy combination of inO and fmap.

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

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''' Source

Handy combination of inO3 and liftA3.

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

Used for the Functor :. Functor instance of Functor

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

Used for the ContraFunctor :. ContraFunctor instance of Functor

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

Used for the Functor :. ContraFunctor instance of Functor

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

Used for the ContraFunctor :. Functor instance of Functor

class DistribM m n where Source

Monad distributivity.

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

Methods

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

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

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 Source

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

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

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

join-like function for implicitly composed monads

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

join-like function for explicitly composed monads

Type composition

Unary/binary

newtype OO f j a b Source

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) Source
(Applicative f, Arrow arr) => Arrow (OO f arr) Source

(->)/unary

newtype FunA h a b Source

Common pattern for Arrows.

Constructors

FunA
Fields unFunA :: h a -> h b

Instances

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

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

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'' Source

Apply binary function in side a FunA representation.

class FunAble h where Source

Support needed for a FunA to be an Arrow.

Minimal complete definition

arrFun, firstFun, secondFun

Methods

arrFun Source

Arguments

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

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

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

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

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

Monoid constructors

class Monoid_f m where Source

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 Source

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

Instances

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

Flip a binary constructor's type arguments

newtype Flip j b a Source

Flip type arguments

Constructors

Flip
Fields unFlip :: a `j` b

Instances

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

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

newtype bijection

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

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

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''' Source

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

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

class ToOI sink where Source

Convert to an OI.

Methods

toOI :: sink b -> OI b Source

Instances

ToOI OI Source

Type application

newtype f :$ a infixr 0 Source

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) Source

type App = (:$) Source

Compatibility synonym for (:$).

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

newtype bijection

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

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

Identity

newtype Id a Source

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

unId :: Id a -> a Source

biId :: a :<->: Id a Source

newtype bijection

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

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

Constructor pairing

Unary

newtype (f :*: g) a infixl 7 Source

Pairing of unary type constructors

Constructors

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

Instances

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

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

Handy infix & curried Prod

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

newtype bijection

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

Compose a bijection

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

Combine two binary functions into a binary function on pairs

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

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

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

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'' Source

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''' Source

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

Binary

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

Pairing of binary type constructors

Constructors

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

Instances

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

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

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' Source

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'' Source

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

Arrow between two constructor applications

newtype Arrw j f g a Source

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)) Source
(Arrow j, ContraFunctor f, Functor g) => Functor (Arrw j f g) Source
(Arrow j, Functor f, ContraFunctor g) => ContraFunctor (Arrw j f g) Source
(Arrow j, Unpair f, Pair g) => Pair (Arrw j f g) Source
(Arrow j, Unzip f, Zip g) => Zip (Arrw j f g) Source
(Arrow j, Unlambda f f', Lambda g g') => Lambda (Arrw j f g) (Arrw j f' g') Source
Monoid (j (f a) (g a)) => Monoid (Arrw j f g a) Source

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

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

newtype bijection

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

Compose a bijection

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

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'' Source

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''' Source

Apply ternary function inside of Arrw j f g representation.

Augment other modules

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

newtype bijection

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

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

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

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

newtype bijection

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

Convenience for partial-manipulating functions