Safe Haskell	None
Language	Haskell2010

Proarrow.Object.Dual

Documentation

class Ob (Dual a) => ObDual (a :: k) Source Comments #

Instances

Instances details

Ob (Dual a) => ObDual (a :: k) Source Comments #
Instance details Defined in Proarrow.Object.Dual

class (SymMonoidal k, Closed k, forall (a :: k). Ob a => ObDual a, Ob (Unit :: k)) => StarAutonomous k where Source Comments #

Associated Types

type Dual (a :: k) :: k Source Comments #

Methods

dual :: forall (a :: k) (b :: k). (a ~> b) -> Dual b ~> Dual a Source Comments #

dualInv :: forall (a :: k) (b :: k). (Ob a, Ob b) => (Dual a ~> Dual b) -> b ~> a Source Comments #

linDist :: forall (a :: k) (b :: k) (c :: k). (Ob a, Ob b, Ob c) => ((a ** b) ~> Dual c) -> a ~> Dual (b ** c) Source Comments #

linDistInv :: forall (a :: k) (b :: k) (c :: k). (Ob a, Ob b, Ob c) => (a ~> Dual (b ** c)) -> (a ** b) ~> Dual c Source Comments #

Instances

Instances details

StarAutonomous BOOL Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Bool

Associated Types

type Dual 'TRU
Instance details Defined in Proarrow.Category.Instance.Bool type Dual 'TRU = 'FLS
type Dual 'FLS
Instance details Defined in Proarrow.Category.Instance.Bool type Dual 'FLS = 'TRU

Methods

dual :: forall (a :: BOOL) (b :: BOOL). (a ~> b) -> Dual b ~> Dual a Source Comments #

dualInv :: forall (a :: BOOL) (b :: BOOL). (Ob a, Ob b) => (Dual a ~> Dual b) -> b ~> a Source Comments #

linDist :: forall (a :: BOOL) (b :: BOOL) (c :: BOOL). (Ob a, Ob b, Ob c) => ((a ** b) ~> Dual c) -> a ~> Dual (b ** c) Source Comments #

linDistInv :: forall (a :: BOOL) (b :: BOOL) (c :: BOOL). (Ob a, Ob b, Ob c) => (a ~> Dual (b ** c)) -> (a ** b) ~> Dual c Source Comments #

StarAutonomous KIND Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Cat

Associated Types

type Dual ('K a :: KIND)
Instance details Defined in Proarrow.Category.Instance.Cat type Dual ('K a :: KIND) = 'K (OPPOSITE a)

Methods

dual :: forall (a :: KIND) (b :: KIND). (a ~> b) -> Dual b ~> Dual a Source Comments #

dualInv :: forall (a :: KIND) (b :: KIND). (Ob a, Ob b) => (Dual a ~> Dual b) -> b ~> a Source Comments #

linDist :: forall (a :: KIND) (b :: KIND) (c :: KIND). (Ob a, Ob b, Ob c) => ((a ** b) ~> Dual c) -> a ~> Dual (b ** c) Source Comments #

linDistInv :: forall (a :: KIND) (b :: KIND) (c :: KIND). (Ob a, Ob b, Ob c) => (a ~> Dual (b ** c)) -> (a ** b) ~> Dual c Source Comments #

StarAutonomous LINEAR Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Linear

Associated Types

type Dual ('L a :: LINEAR)
Instance details Defined in Proarrow.Category.Instance.Linear type Dual ('L a :: LINEAR) = 'L (Not a)

Methods

dual :: forall (a :: LINEAR) (b :: LINEAR). (a ~> b) -> Dual b ~> Dual a Source Comments #

dualInv :: forall (a :: LINEAR) (b :: LINEAR). (Ob a, Ob b) => (Dual a ~> Dual b) -> b ~> a Source Comments #

linDist :: forall (a :: LINEAR) (b :: LINEAR) (c :: LINEAR). (Ob a, Ob b, Ob c) => ((a ** b) ~> Dual c) -> a ~> Dual (b ** c) Source Comments #

linDistInv :: forall (a :: LINEAR) (b :: LINEAR) (c :: LINEAR). (Ob a, Ob b, Ob c) => (a ~> Dual (b ** c)) -> (a ** b) ~> Dual c Source Comments #

StarAutonomous () Source Comments #

Instance details

Defined in Proarrow.Object.Dual

Associated Types

type Dual '()
Instance details Defined in Proarrow.Object.Dual type Dual '() = '()

Methods

dual :: forall (a :: ()) (b :: ()). (a ~> b) -> Dual b ~> Dual a Source Comments #

dualInv :: forall (a :: ()) (b :: ()). (Ob a, Ob b) => (Dual a ~> Dual b) -> b ~> a Source Comments #

linDist :: forall (a :: ()) (b :: ()) (c :: ()). (Ob a, Ob b, Ob c) => ((a ** b) ~> Dual c) -> a ~> Dual (b ** c) Source Comments #

linDistInv :: forall (a :: ()) (b :: ()) (c :: ()). (Ob a, Ob b, Ob c) => (a ~> Dual (b ** c)) -> (a ** b) ~> Dual c Source Comments #

(TracedMonoidal k, Ob (Unit :: k)) => StarAutonomous (INT k) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.IntConstruction

Methods

dual :: forall (a :: INT k) (b :: INT k). (a ~> b) -> Dual b ~> Dual a Source Comments #

dualInv :: forall (a :: INT k) (b :: INT k). (Ob a, Ob b) => (Dual a ~> Dual b) -> b ~> a Source Comments #

linDist :: forall (a :: INT k) (b :: INT k) (c :: INT k). (Ob a, Ob b, Ob c) => ((a ** b) ~> Dual c) -> a ~> Dual (b ** c) Source Comments #

linDistInv :: forall (a :: INT k) (b :: INT k) (c :: INT k). (Ob a, Ob b, Ob c) => (a ~> Dual (b ** c)) -> (a ** b) ~> Dual c Source Comments #

Num a => StarAutonomous (MatK a) Source Comments #

Instance details

Defined in Proarrow.Category.Instance.Mat

Methods

dual :: forall (a0 :: MatK a) (b :: MatK a). (a0 ~> b) -> Dual b ~> Dual a0 Source Comments #

dualInv :: forall (a0 :: MatK a) (b :: MatK a). (Ob a0, Ob b) => (Dual a0 ~> Dual b) -> b ~> a0 Source Comments #

linDist :: forall (a0 :: MatK a) (b :: MatK a) (c :: MatK a). (Ob a0, Ob b, Ob c) => ((a0 ** b) ~> Dual c) -> a0 ~> Dual (b ** c) Source Comments #

linDistInv :: forall (a0 :: MatK a) (b :: MatK a) (c :: MatK a). (Ob a0, Ob b, Ob c) => (a0 ~> Dual (b ** c)) -> (a0 ** b) ~> Dual c Source Comments #

StarAutonomous (KLEISLI (Cont r)) Source Comments #

Instance details

Defined in Proarrow.Promonad.Cont

Methods

dual :: forall (a :: KLEISLI (Cont r)) (b :: KLEISLI (Cont r)). (a ~> b) -> Dual b ~> Dual a Source Comments #

dualInv :: forall (a :: KLEISLI (Cont r)) (b :: KLEISLI (Cont r)). (Ob a, Ob b) => (Dual a ~> Dual b) -> b ~> a Source Comments #

linDist :: forall (a :: KLEISLI (Cont r)) (b :: KLEISLI (Cont r)) (c :: KLEISLI (Cont r)). (Ob a, Ob b, Ob c) => ((a ** b) ~> Dual c) -> a ~> Dual (b ** c) Source Comments #

linDistInv :: forall (a :: KLEISLI (Cont r)) (b :: KLEISLI (Cont r)) (c :: KLEISLI (Cont r)). (Ob a, Ob b, Ob c) => (a ~> Dual (b ** c)) -> (a ** b) ~> Dual c Source Comments #

(StarAutonomous j, StarAutonomous k) => StarAutonomous (j, k) Source Comments #

Instance details

Defined in Proarrow.Object.Dual

Methods

dual :: forall (a :: (j, k)) (b :: (j, k)). (a ~> b) -> Dual b ~> Dual a Source Comments #

dualInv :: forall (a :: (j, k)) (b :: (j, k)). (Ob a, Ob b) => (Dual a ~> Dual b) -> b ~> a Source Comments #

linDist :: forall (a :: (j, k)) (b :: (j, k)) (c :: (j, k)). (Ob a, Ob b, Ob c) => ((a ** b) ~> Dual c) -> a ~> Dual (b ** c) Source Comments #

linDistInv :: forall (a :: (j, k)) (b :: (j, k)) (c :: (j, k)). (Ob a, Ob b, Ob c) => (a ~> Dual (b ** c)) -> (a ** b) ~> Dual c Source Comments #

dualObj :: forall {k} (a :: k). (StarAutonomous k, Ob a) => Obj (Dual a) Source Comments #

doubleNeg :: forall {k} (a :: k). (StarAutonomous k, Ob a) => Dual (Dual a) ~> a Source Comments #

doubleNegInv :: forall {k} (a :: k). (StarAutonomous k, Ob a) => a ~> Dual (Dual a) Source Comments #

type ExpSA (a :: k) (b :: k) = Dual (a ** Dual b) Source Comments #

currySA :: forall {k} (a :: k) (b :: k) (c :: k). (StarAutonomous k, Ob a, Ob b) => ((a ** b) ~> c) -> a ~> ExpSA b c Source Comments #

uncurrySA :: forall {k} (a :: k) (b :: k) (c :: k). (StarAutonomous k, Ob b, Ob c) => (a ~> ExpSA b c) -> (a ** b) ~> c Source Comments #

expSA :: forall {k} (a :: k) (b :: k) (x :: k) (y :: k). StarAutonomous k => (b ~> y) -> (x ~> a) -> ExpSA a b ~> ExpSA x y Source Comments #

dualityUnitSA :: forall {k} (a :: k). (StarAutonomous k, Ob a) => (Unit :: k) ~> Dual (Dual a ** a) Source Comments #

dualityCounitSA :: forall {k} (a :: k). (StarAutonomous k, Ob a) => (Dual a ** a) ~> Dual (Unit :: k) Source Comments #

class (StarAutonomous k, SymMonoidal k) => CompactClosed k where Source Comments #

Methods

distribDual :: forall (a :: k) (b :: k). (Ob a, Ob b) => Dual (a ** b) ~> (Dual a ** Dual b) Source Comments #

dualUnit :: Dual (Unit :: k) ~> (Unit :: k) Source Comments #

Instances

Instances details

CompactClosed KIND Source Comments #
Instance details Defined in Proarrow.Category.Instance.Cat Methods distribDual :: forall (a :: KIND) (b :: KIND). (Ob a, Ob b) => Dual (a b) ~> (Dual a Dual b) Source Comments # dualUnit :: Dual (Unit :: KIND) ~> (Unit :: KIND) Source Comments #
CompactClosed () Source Comments #
Instance details Defined in Proarrow.Object.Dual Methods distribDual :: forall (a :: ()) (b :: ()). (Ob a, Ob b) => Dual (a b) ~> (Dual a Dual b) Source Comments # dualUnit :: Dual (Unit :: ()) ~> (Unit :: ()) Source Comments #
(TracedMonoidal k, Ob (Unit :: k)) => CompactClosed (INT k) Source Comments #
Instance details Defined in Proarrow.Category.Instance.IntConstruction Methods distribDual :: forall (a :: INT k) (b :: INT k). (Ob a, Ob b) => Dual (a b) ~> (Dual a Dual b) Source Comments # dualUnit :: Dual (Unit :: INT k) ~> (Unit :: INT k) Source Comments #
Num a => CompactClosed (MatK a) Source Comments #
Instance details Defined in Proarrow.Category.Instance.Mat Methods distribDual :: forall (a0 :: MatK a) (b :: MatK a). (Ob a0, Ob b) => Dual (a0 b) ~> (Dual a0 Dual b) Source Comments # dualUnit :: Dual (Unit :: MatK a) ~> (Unit :: MatK a) Source Comments #
(CompactClosed j, CompactClosed k) => CompactClosed (j, k) Source Comments #
Instance details Defined in Proarrow.Object.Dual Methods distribDual :: forall (a :: (j, k)) (b :: (j, k)). (Ob a, Ob b) => Dual (a b) ~> (Dual a Dual b) Source Comments # dualUnit :: Dual (Unit :: (j, k)) ~> (Unit :: (j, k)) Source Comments #

dualityUnit :: forall {k} (a :: k). (CompactClosed k, Ob a) => (Unit :: k) ~> (a ** Dual a) Source Comments #

dualityCounit :: forall {k} (a :: k). (CompactClosed k, Ob a) => (Dual a ** a) ~> (Unit :: k) Source Comments #

dualUnitInv :: CompactClosed k => (Unit :: k) ~> Dual (Unit :: k) Source Comments #

combineDual :: forall {k} (a :: k) (b :: k). (CompactClosed k, Ob a, Ob b) => (Dual a ** Dual b) ~> Dual (a ** b) Source Comments #

compactClosedTrace :: forall {k} (u :: k) (x :: k) (y :: k). (CompactClosed k, Ob x, Ob y, Ob u) => ((x ** u) ~> (y ** u)) -> x ~> y Source Comments #