Copyright	(c) Erich Gut
License	BSD3
Maintainer	zerich.gut@gmail.com
Safe Haskell	Safe-Inferred
Language	Haskell2010

OAlg.Structure.Oriented.Definition

Contents

Oriented
Transposable
Orientation
Path
X

Description

definition of Oriented structures.

Synopsis

class (Entity q, Entity (Point q)) => Oriented q where
- type Point q
- orientation :: q -> Orientation (Point q)
- start :: q -> Point q
- end :: q -> Point q
class Singleton (Point x) => Total x
class Entity (Point x) => EntityPoint x
class Ord (Point x) => OrdPoint x
isEndo :: Oriented q => q -> Bool
isEndoAt :: Oriented a => Point a -> a -> Bool
type OS = Orientation Symbol
data Ort
structOrtOp :: Struct Ort x -> Struct Ort (Op x)
class Transformable s Ort => ForgetfulOrt s
class (Transposable q, Oriented q) => TransposableOriented q
data Orientation p = p :> p
opposite :: Orientation p -> Orientation p
data Path q = Path (Point q) [q]
pthLength :: Path q -> N
pthOne :: Point q -> Path q
pthMlt :: Oriented q => Path q -> Path q -> Path q
data XOrtSite s q where
- XStart :: X (Point q) -> (Point q -> X q) -> XOrtSite From q
- XEnd :: X (Point q) -> (Point q -> X q) -> XOrtSite To q
class XStandardOrtSite s a where
- xStandardOrtSite :: XOrtSite s a
class XStandardOrtSite To a => XStandardOrtSiteTo a
class XStandardOrtSite From a => XStandardOrtSiteFrom a
coXOrtSite :: XOrtSite s q -> Dual (XOrtSite s q)
coXOrtSiteInv :: (Dual (Dual s) :~: s) -> Dual (XOrtSite s q) -> XOrtSite s q
xosFromOpOp :: XOrtSite s (Op (Op q)) -> XOrtSite s q
xosStart :: XOrtSite From q -> Point q -> X q
xosEnd :: XOrtSite To q -> Point q -> X q
xosPathMaxAt :: Oriented q => XOrtSite s q -> N -> Point q -> X (Path q)
xosPathMax :: Oriented q => XOrtSite s q -> N -> X (Path q)
data XOrtOrientation q = XOrtOrientation (X (Orientation (Point q))) (Orientation (Point q) -> X q)
xoOrientation :: XOrtOrientation q -> X (Orientation (Point q))
xoArrow :: XOrtOrientation q -> Orientation (Point q) -> X q
xoPoint :: Oriented q => XOrtOrientation q -> X (Point q)
coXOrtOrientation :: XOrtOrientation q -> Dual (XOrtOrientation q)
xoTo :: Oriented q => XOrtOrientation q -> XOrtSite To q
xoFrom :: Oriented q => XOrtOrientation q -> XOrtSite From q
xoTtl :: Total q => X q -> XOrtOrientation q
xoOrnt :: X p -> XOrtOrientation (Orientation p)
class XStandardOrtOrientation q where
- xStandardOrtOrientation :: XOrtOrientation q
class XStandard (Point a) => XStandardPoint a
xStartOrnt :: X p -> XOrtSite From (Orientation p)
xEndOrnt :: X p -> XOrtSite To (Orientation p)

Oriented

class (Entity q, Entity (Point q)) => Oriented q where Source #

types with a Oriented structure. The values of an Oriented structure will be called arrows and the values of the associated Point type points. To each arrow there is a start and a end point assigned.

Property Let q be a type instance of the class Oriented, then holds:

For all a in q holds: orientation a == start a :> end a.

Note

If the types q and Point q are interpreted as sets A and P and start, end as functions from A to P then this structure forms a quiver with arrows in A and points in P.
Morphisms can be interpreted as Oriented structures via SomeMorphism. The bad thing about this is that we lose the check for composability of two Morphisms given by the type checker, but we gain all the functionality of Oriented structures, i.e we can define homomorphisms, limits etc on Morphisms.

Minimal complete definition

orientation | start, end

Associated Types

type Point q Source #

the associated type of points.

Methods

orientation :: q -> Orientation (Point q) Source #

the orientation of an arrow.

start :: q -> Point q Source #

the start point of an arrow.

end :: q -> Point q Source #

the end point of an arrow.

Instances

Instances details

Oriented N Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Associated Types type Point N Source # Methods orientation :: N -> Orientation (Point N) Source # start :: N -> Point N Source # end :: N -> Point N Source #
Oriented Q Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Associated Types type Point Q Source # Methods orientation :: Q -> Orientation (Point Q) Source # start :: Q -> Point Q Source # end :: Q -> Point Q Source #
Oriented Z Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Associated Types type Point Z Source # Methods orientation :: Z -> Orientation (Point Z) Source # start :: Z -> Point Z Source # end :: Z -> Point Z Source #
Oriented N' Source #
Instance details Defined in OAlg.Entity.Natural Associated Types type Point N' Source # Methods orientation :: N' -> Orientation (Point N') Source # start :: N' -> Point N' Source # end :: N' -> Point N' Source #
Oriented W' Source #
Instance details Defined in OAlg.Entity.Natural Associated Types type Point W' Source # Methods orientation :: W' -> Orientation (Point W') Source # start :: W' -> Point W' Source # end :: W' -> Point W' Source #
Oriented Integer Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Associated Types type Point Integer Source # Methods orientation :: Integer -> Orientation (Point Integer) Source # start :: Integer -> Point Integer Source # end :: Integer -> Point Integer Source #
Oriented () Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Associated Types type Point () Source # Methods orientation :: () -> Orientation (Point ()) Source # start :: () -> Point () Source # end :: () -> Point () Source #
Oriented Int Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Associated Types type Point Int Source # Methods orientation :: Int -> Orientation (Point Int) Source # start :: Int -> Point Int Source # end :: Int -> Point Int Source #
(EmbeddableMorphismTyp m, Entity2 m) => Oriented (SomeMorphism m) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Associated Types type Point (SomeMorphism m) Source # Methods orientation :: SomeMorphism m -> Orientation (Point (SomeMorphism m)) Source # start :: SomeMorphism m -> Point (SomeMorphism m) Source # end :: SomeMorphism m -> Point (SomeMorphism m) Source #
Oriented q => Oriented (Op q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Associated Types type Point (Op q) Source # Methods orientation :: Op q -> Orientation (Point (Op q)) Source # start :: Op q -> Point (Op q) Source # end :: Op q -> Point (Op q) Source #
(Additive x, FibredOriented x) => Oriented (Matrix x) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Associated Types type Point (Matrix x) Source # Methods orientation :: Matrix x -> Orientation (Point (Matrix x)) Source # start :: Matrix x -> Point (Matrix x) Source # end :: Matrix x -> Point (Matrix x) Source #
Galoisian x => Oriented (GL2 x) Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup Associated Types type Point (GL2 x) Source # Methods orientation :: GL2 x -> Orientation (Point (GL2 x)) Source # start :: GL2 x -> Point (GL2 x) Source # end :: GL2 x -> Point (GL2 x) Source #
Oriented x => Oriented (GLT x) Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup Associated Types type Point (GLT x) Source # Methods orientation :: GLT x -> Orientation (Point (GLT x)) Source # start :: GLT x -> Point (GLT x) Source # end :: GLT x -> Point (GLT x) Source #
Oriented x => Oriented (Transformation x) Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup Associated Types type Point (Transformation x) Source # Methods orientation :: Transformation x -> Orientation (Point (Transformation x)) Source # start :: Transformation x -> Point (Transformation x) Source # end :: Transformation x -> Point (Transformation x) Source #
Oriented x => Oriented (ColTrafo x) Source #
Instance details Defined in OAlg.Entity.Matrix.Transformation Associated Types type Point (ColTrafo x) Source # Methods orientation :: ColTrafo x -> Orientation (Point (ColTrafo x)) Source # start :: ColTrafo x -> Point (ColTrafo x) Source # end :: ColTrafo x -> Point (ColTrafo x) Source #
Oriented a => Oriented (RowTrafo a) Source #
Instance details Defined in OAlg.Entity.Matrix.Transformation Associated Types type Point (RowTrafo a) Source # Methods orientation :: RowTrafo a -> Orientation (Point (RowTrafo a)) Source # start :: RowTrafo a -> Point (RowTrafo a) Source # end :: RowTrafo a -> Point (RowTrafo a) Source #
Entity x => Oriented (ProductSymbol x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Associated Types type Point (ProductSymbol x) Source # Methods orientation :: ProductSymbol x -> Orientation (Point (ProductSymbol x)) Source # start :: ProductSymbol x -> Point (ProductSymbol x) Source # end :: ProductSymbol x -> Point (ProductSymbol x) Source #
Entity x => Oriented (U x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Associated Types type Point (U x) Source # Methods orientation :: U x -> Orientation (Point (U x)) Source # start :: U x -> Point (U x) Source # end :: U x -> Point (U x) Source #
(Entity i, Ord i) => Oriented (Permutation i) Source #
Instance details Defined in OAlg.Entity.Sequence.Permutation Associated Types type Point (Permutation i) Source # Methods orientation :: Permutation i -> Orientation (Point (Permutation i)) Source # start :: Permutation i -> Point (Permutation i) Source # end :: Permutation i -> Point (Permutation i) Source #
(Entity i, Ord i) => Oriented (PermutationForm i) Source #
Instance details Defined in OAlg.Entity.Sequence.Permutation Associated Types type Point (PermutationForm i) Source # Methods orientation :: PermutationForm i -> Orientation (Point (PermutationForm i)) Source # start :: PermutationForm i -> Point (PermutationForm i) Source # end :: PermutationForm i -> Point (PermutationForm i) Source #
Fibred f => Oriented (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Associated Types type Point (Sheaf f) Source # Methods orientation :: Sheaf f -> Orientation (Point (Sheaf f)) Source # start :: Sheaf f -> Point (Sheaf f) Source # end :: Sheaf f -> Point (Sheaf f) Source #
Multiplicative c => Oriented (Inv c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Associated Types type Point (Inv c) Source # Methods orientation :: Inv c -> Orientation (Point (Inv c)) Source # start :: Inv c -> Point (Inv c) Source # end :: Inv c -> Point (Inv c) Source #
Entity p => Oriented (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Associated Types type Point (Orientation p) Source # Methods orientation :: Orientation p -> Orientation (Point (Orientation p)) Source # start :: Orientation p -> Point (Orientation p) Source # end :: Orientation p -> Point (Orientation p) Source #
Oriented q => Oriented (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Associated Types type Point (Path q) Source # Methods orientation :: Path q -> Orientation (Point (Path q)) Source # start :: Path q -> Point (Path q) Source # end :: Path q -> Point (Path q) Source #
(Oriented x, Typeable p) => Oriented (Dim x p) Source #
Instance details Defined in OAlg.Entity.Matrix.Dim Associated Types type Point (Dim x p) Source # Methods orientation :: Dim x p -> Orientation (Point (Dim x p)) Source # start :: Dim x p -> Point (Dim x p) Source # end :: Dim x p -> Point (Dim x p) Source #
(Oriented a, Integral r) => Oriented (Product r a) Source #
Instance details Defined in OAlg.Entity.Product.Definition Associated Types type Point (Product r a) Source # Methods orientation :: Product r a -> Orientation (Point (Product r a)) Source # start :: Product r a -> Point (Product r a) Source # end :: Product r a -> Point (Product r a) Source #
(Oriented a, Number r) => Oriented (ProductForm r a) Source #
Instance details Defined in OAlg.Entity.Product.Definition Associated Types type Point (ProductForm r a) Source # Methods orientation :: ProductForm r a -> Orientation (Point (ProductForm r a)) Source # start :: ProductForm r a -> Point (ProductForm r a) Source # end :: ProductForm r a -> Point (ProductForm r a) Source #
(Multiplicative c, Sliced i c, Typeable s) => Oriented (SliceFactor s i c) Source #
Instance details Defined in OAlg.Entity.Slice.Definition Associated Types type Point (SliceFactor s i c) Source # Methods orientation :: SliceFactor s i c -> Orientation (Point (SliceFactor s i c)) Source # start :: SliceFactor s i c -> Point (SliceFactor s i c) Source # end :: SliceFactor s i c -> Point (SliceFactor s i c) Source #
(Oriented a, Typeable t, Typeable n, Typeable m) => Oriented (Diagram ('Chain t) n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Definition Associated Types type Point (Diagram ('Chain t) n m a) Source # Methods orientation :: Diagram ('Chain t) n m a -> Orientation (Point (Diagram ('Chain t) n m a)) Source # start :: Diagram ('Chain t) n m a -> Point (Diagram ('Chain t) n m a) Source # end :: Diagram ('Chain t) n m a -> Point (Diagram ('Chain t) n m a) Source #
(Oriented a, Typeable d, Typeable n, Typeable m) => Oriented (Diagram ('Parallel d) n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Definition Associated Types type Point (Diagram ('Parallel d) n m a) Source # Methods orientation :: Diagram ('Parallel d) n m a -> Orientation (Point (Diagram ('Parallel d) n m a)) Source # start :: Diagram ('Parallel d) n m a -> Point (Diagram ('Parallel d) n m a) Source # end :: Diagram ('Parallel d) n m a -> Point (Diagram ('Parallel d) n m a) Source #
(Multiplicative a, Typeable t, Typeable n, Typeable m) => Oriented (Transformation t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation Associated Types type Point (Transformation t n m a) Source # Methods orientation :: Transformation t n m a -> Orientation (Point (Transformation t n m a)) Source # start :: Transformation t n m a -> Point (Transformation t n m a) Source # end :: Transformation t n m a -> Point (Transformation t n m a) Source #
(Oriented a, Typeable s, Typeable p, Typeable d, Typeable m) => Oriented (Cone s p ('Parallel d) N2 m a) Source #
Instance details Defined in OAlg.Limes.Cone.Definition Associated Types type Point (Cone s p ('Parallel d) N2 m a) Source # Methods orientation :: Cone s p ('Parallel d) N2 m a -> Orientation (Point (Cone s p ('Parallel d) N2 m a)) Source # start :: Cone s p ('Parallel d) N2 m a -> Point (Cone s p ('Parallel d) N2 m a) Source # end :: Cone s p ('Parallel d) N2 m a -> Point (Cone s p ('Parallel d) N2 m a) Source #

class Singleton (Point x) => Total x Source #

structures where its associated Point type is singleton. They yield globally defiend algebraic operations.

Instances

Instances details

Total N Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
Total Q Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
Total Z Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
Total N' Source #
Instance details Defined in OAlg.Entity.Natural
Total W' Source #
Instance details Defined in OAlg.Entity.Natural
Total Integer Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
Total () Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
Total Int Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
Total x => Total (Op x) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
Galoisian x => Total (GL2 x) Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup
Total (Permutation i) Source #
Instance details Defined in OAlg.Entity.Sequence.Permutation
Total q => Total (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
Total (Dim x p) Source #
Instance details Defined in OAlg.Entity.Matrix.Dim

class Entity (Point x) => EntityPoint x Source #

helper class to avoid undecidable instances.

Instances

Instances details

EntityPoint N Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
EntityPoint Q Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
EntityPoint Z Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
EntityPoint Integer Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
EntityPoint () Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
EntityPoint Int Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
EntityPoint x => EntityPoint (Op x) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
EntityPoint q => EntityPoint (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition

class Ord (Point x) => OrdPoint x Source #

helper class to circumvent undecidable instances.

Instances

Instances details

OrdPoint N Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
OrdPoint Q Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
OrdPoint Z Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
OrdPoint Integer Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
OrdPoint () Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
OrdPoint Int Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
OrdPoint (U x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol
OrdPoint q => OrdPoint (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition

isEndo :: Oriented q => q -> Bool Source #

check for being an endo.

Definition Let q be a Oriented structure, then an arrow a in q is called endo if and only if start a == end a.

isEndoAt :: Oriented a => Point a -> a -> Bool Source #

check for being an endo at the given point.

type OS = Orientation Symbol Source #

as Orientation p is an instance of almost every structured class it serves as a standard type for validating.

data Ort Source #

type representing the class of Oriented structures.

Instances

Instances details

ForgetfulTyp Ort Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
TransformableOp Ort Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
ForgetfulOrt Ort Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
EmbeddableMorphism GLApp Ort Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup
EmbeddableMorphism TrApp Ort Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup
Transformable Dst Ort Source #
Instance details Defined in OAlg.Structure.Distributive.Definition Methods tau :: Struct Dst x -> Struct Ort x Source #
Transformable FbrOrt Ort Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods tau :: Struct FbrOrt x -> Struct Ort x Source #
Transformable Mlt Ort Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods tau :: Struct Mlt x -> Struct Ort x Source #
Transformable Ort Ent Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods tau :: Struct Ort x -> Struct Ent x Source #
Transformable Ort Typ Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods tau :: Struct Ort x -> Struct Typ x Source #
Transformable1 Op Ort Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods tau1 :: Struct Ort x -> Struct Ort (Op x) Source #
EmbeddableMorphism (SliceFactorDrop s) Ort Source #
Instance details Defined in OAlg.Entity.Slice.Definition
EmbeddableMorphism h Ort => EmbeddableMorphism (OpHom h) Ort Source #
Instance details Defined in OAlg.Hom.Oriented.Definition
Transformable (Alg k) Ort Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition Methods tau :: Struct (Alg k) x -> Struct Ort x Source #
(Multiplicative c, Sliced i c) => EmbeddableMorphism (SliceCokernelKernel i c) Ort Source #
Instance details Defined in OAlg.Entity.Slice.Adjunction
type Hom Ort h Source #
Instance details Defined in OAlg.Hom.Oriented.Definition type Hom Ort h = HomOriented h
type Structure Ort x Source #
Instance details Defined in OAlg.Structure.Oriented.Definition type Structure Ort x = Oriented x

structOrtOp :: Struct Ort x -> Struct Ort (Op x) Source #

attest that if x is Oriented then also Op x is Oriented.

class Transformable s Ort => ForgetfulOrt s Source #

transformable to Oriented structure.

Instances

Instances details

ForgetfulOrt Dst Source #
Instance details Defined in OAlg.Structure.Distributive.Definition
ForgetfulOrt FbrOrt Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
ForgetfulOrt Mlt Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
ForgetfulOrt Ort Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
ForgetfulOrt (Alg k) Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition

Transposable

class (Transposable q, Oriented q) => TransposableOriented q Source #

transposable oriented structures.

Property Let q be a TransposableOriented structure, then holds: For all a in q holds: orientation (transpose a) == opposite (orientation a).

Instances

Instances details

TransposableOriented N Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
TransposableOriented Q Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
TransposableOriented Z Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
(Distributive x, TransposableDistributive x) => TransposableOriented (Matrix x) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition
(Galoisian x, TransposableDistributive x) => TransposableOriented (GL2 x) Source #
Instance details Defined in OAlg.Entity.Matrix.GeneralLinearGroup
TransposableMultiplicative c => TransposableOriented (Inv c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
Entity p => TransposableOriented (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition

Orientation

data Orientation p Source #

orientation given by the start point as its first component and the end point as its second.

Property For all o in Orientation p holds: o == start o :> end o.

Note As Orientations are instances of almost all algebraic structures defined here, they serve as a proof that this structures are instanceable.

Constructors

p :> p infix 5

Instances

Instances details

Functor Orientation Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods fmap :: (a -> b) -> Orientation a -> Orientation b # (<$) :: a -> Orientation b -> Orientation a #
XStandard p => XStandardOrtSite 'From (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods xStandardOrtSite :: XOrtSite 'From (Orientation p) Source #
XStandard p => XStandardOrtSite 'To (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods xStandardOrtSite :: XOrtSite 'To (Orientation p) Source #
XStandardOrtSite 'From (SliceFactor 'To (Proxy :: Type -> Type) OS) Source #
Instance details Defined in OAlg.Entity.Slice.Definition Methods xStandardOrtSite :: XOrtSite 'From (SliceFactor 'To Proxy OS) Source #
Show p => Show (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods showsPrec :: Int -> Orientation p -> ShowS # show :: Orientation p -> String # showList :: [Orientation p] -> ShowS #
Eq p => Eq (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods (==) :: Orientation p -> Orientation p -> Bool # (/=) :: Orientation p -> Orientation p -> Bool #
Ord p => Ord (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods compare :: Orientation p -> Orientation p -> Ordering # (<) :: Orientation p -> Orientation p -> Bool # (<=) :: Orientation p -> Orientation p -> Bool # (>) :: Orientation p -> Orientation p -> Bool # (>=) :: Orientation p -> Orientation p -> Bool # max :: Orientation p -> Orientation p -> Orientation p # min :: Orientation p -> Orientation p -> Orientation p #
Transposable (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods transpose :: Orientation p -> Orientation p Source #
Singleton u => Singleton (Orientation u) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods unit :: Orientation u Source #
Validable p => Validable (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods valid :: Orientation p -> Statement Source #
XStandard p => XStandard (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods xStandard :: X (Orientation p) Source #
Entity p => Entity (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
Entity p => Abelian (Orientation p) Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods negate :: Orientation p -> Orientation p Source # (-) :: Orientation p -> Orientation p -> Orientation p Source # ztimes :: Z -> Orientation p -> Orientation p Source #
Entity p => Additive (Orientation p) Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods zero :: Root (Orientation p) -> Orientation p Source # (+) :: Orientation p -> Orientation p -> Orientation p Source # ntimes :: N -> Orientation p -> Orientation p Source #
(Entity p, XStandard p) => XStandardAdd (Orientation p) Source #
Instance details Defined in OAlg.Structure.Additive.Proposition Methods xStandardAdd :: XAdd (Orientation p) Source #
Entity p => Algebraic (Orientation p) Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
Entity p => Distributive (Orientation p) Source #
Instance details Defined in OAlg.Structure.Distributive.Definition
Entity p => TransposableDistributive (Orientation p) Source #
Instance details Defined in OAlg.Structure.Distributive.Definition
(Entity p, XStandard p) => XStandardDst (Orientation p) Source #
Instance details Defined in OAlg.Structure.Distributive.Proposition Methods xStandardDst :: XDst (Orientation p) Source #
Entity p => Real (Orientation p) Source #
Instance details Defined in OAlg.Structure.Exponential Methods power :: Number r => Orientation p -> r -> Orientation p Source #
Entity p => Fibred (Orientation p) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Associated Types type Root (Orientation p) Source # Methods root :: Orientation p -> Root (Orientation p) Source #
Entity p => FibredOriented (Orientation p) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
Entity p => Cayleyan (Orientation p) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
Entity p => Invertible (Orientation p) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods tryToInvert :: Orientation p -> Solver (Orientation p) Source # invert :: Orientation p -> Orientation p Source # isInvertible :: Orientation p -> Bool Source # zpower :: Orientation p -> Z -> Orientation p Source #
Entity p => Multiplicative (Orientation p) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods one :: Point (Orientation p) -> Orientation p Source # (*) :: Orientation p -> Orientation p -> Orientation p Source # npower :: Orientation p -> N -> Orientation p Source #
Entity p => TransposableMultiplicative (Orientation p) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition
(Entity p, XStandard p) => XStandardMlt (Orientation p) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Proposition Methods xStandardMlt :: XMlt (Orientation p) Source #
Entity p => Oriented (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Associated Types type Point (Orientation p) Source # Methods orientation :: Orientation p -> Orientation (Point (Orientation p)) Source # start :: Orientation p -> Point (Orientation p) Source # end :: Orientation p -> Point (Orientation p) Source #
Entity p => TransposableOriented (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
XStandard p => XStandardOrtOrientation (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods xStandardOrtOrientation :: XOrtOrientation (Orientation p) Source #
XStandard p => XStandardOrtSiteFrom (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
XStandard p => XStandardOrtSiteTo (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
XStandard p => XStandardPoint (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
Entity p => Euclidean (Orientation p) Source #
Instance details Defined in OAlg.Structure.Vectorial.Definition Methods (<!>) :: Orientation p -> Orientation p -> Scalar (Orientation p) Source #
Entity p => Vectorial (Orientation p) Source #
Instance details Defined in OAlg.Structure.Vectorial.Definition Associated Types type Scalar (Orientation p) Source # Methods (!) :: Scalar (Orientation p) -> Orientation p -> Orientation p Source #
Sliced (Proxy :: Type -> Type) OS Source #
Instance details Defined in OAlg.Entity.Slice.Definition Methods slicePoint :: Proxy OS -> Point OS Source #
XStandardOrtSiteFrom (SliceFactor 'To (Proxy :: Type -> Type) OS) Source #
Instance details Defined in OAlg.Entity.Slice.Definition
(Entity p, t ~ 'Parallel 'RightToLeft, n ~ N2, XStandard p, XStandard (Diagram t n m (Orientation p))) => XStandard (Cone Dst 'Injective t n m (Orientation p)) Source #
Instance details Defined in OAlg.Limes.Cone.Definition Methods xStandard :: X (Cone Dst 'Injective t n m (Orientation p)) Source #
(Entity p, t ~ 'Parallel 'LeftToRight, n ~ N2, XStandard p, XStandard (Diagram t n m (Orientation p))) => XStandard (Cone Dst 'Projective t n m (Orientation p)) Source #
Instance details Defined in OAlg.Limes.Cone.Definition Methods xStandard :: X (Cone Dst 'Projective t n m (Orientation p)) Source #
(Entity p, XStandard p, XStandard (Diagram t n m (Orientation p))) => XStandard (Cone Mlt 'Injective t n m (Orientation p)) Source #
Instance details Defined in OAlg.Limes.Cone.Definition Methods xStandard :: X (Cone Mlt 'Injective t n m (Orientation p)) Source #
(Entity p, XStandard p, XStandard (Diagram t n m (Orientation p))) => XStandard (Cone Mlt 'Projective t n m (Orientation p)) Source #
Instance details Defined in OAlg.Limes.Cone.Definition Methods xStandard :: X (Cone Mlt 'Projective t n m (Orientation p)) Source #
type Root (Orientation p) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition type Root (Orientation p) = Orientation p
type Point (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition type Point (Orientation p) = p
type Scalar (Orientation p) Source #
Instance details Defined in OAlg.Structure.Vectorial.Definition type Scalar (Orientation p) = Q

opposite :: Orientation p -> Orientation p Source #

the opposite orientation.

Path

data Path q Source #

a path in a Oriented structure q starting at a given point.

Definition Let q be a Oriented structure and p = Path s [a 0..a (n-1)] a path in q, then p is valid if and only if

s is valid and a i are valid for all i = 0..n-1.
start (a (n-1)) == s and start (a i) == end (a (n+1)) for all i = 0..n-2.

furthermore n is called the length of p.

Note Paths admit a canonical embedding in to Product.

Constructors

Path (Point q) [q]

Instances

Instances details

Foldable Path Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods fold :: Monoid m => Path m -> m # foldMap :: Monoid m => (a -> m) -> Path a -> m # foldMap' :: Monoid m => (a -> m) -> Path a -> m # foldr :: (a -> b -> b) -> b -> Path a -> b # foldr' :: (a -> b -> b) -> b -> Path a -> b # foldl :: (b -> a -> b) -> b -> Path a -> b # foldl' :: (b -> a -> b) -> b -> Path a -> b # foldr1 :: (a -> a -> a) -> Path a -> a # foldl1 :: (a -> a -> a) -> Path a -> a # toList :: Path a -> [a] # null :: Path a -> Bool # length :: Path a -> Int # elem :: Eq a => a -> Path a -> Bool # maximum :: Ord a => Path a -> a # minimum :: Ord a => Path a -> a # sum :: Num a => Path a -> a # product :: Num a => Path a -> a #
Oriented q => Embeddable q (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods inj :: q -> Path q Source #
Multiplicative c => Projectible c (Path c) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods prj :: Path c -> c Source #
Oriented q => Show (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods showsPrec :: Int -> Path q -> ShowS # show :: Path q -> String # showList :: [Path q] -> ShowS #
Oriented q => Eq (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods (==) :: Path q -> Path q -> Bool # (/=) :: Path q -> Path q -> Bool #
Oriented q => Dualisable (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods toDual :: Path q -> Dual (Path q) Source # fromDual :: Dual (Path q) -> Path q Source #
Reflexive (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods toBidual :: Path q -> Dual (Dual (Path q)) Source # fromBidual :: Dual (Dual (Path q)) -> Path q Source #
LengthN (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods lengthN :: Path q -> N Source #
Oriented q => Validable (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods valid :: Path q -> Statement Source #
Oriented q => Entity (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
Oriented q => Multiplicative (Path q) Source #
Instance details Defined in OAlg.Structure.Multiplicative.Definition Methods one :: Point (Path q) -> Path q Source # (*) :: Path q -> Path q -> Path q Source # npower :: Path q -> N -> Path q Source #
EntityPoint q => EntityPoint (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
OrdPoint q => OrdPoint (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
Oriented q => Oriented (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Associated Types type Point (Path q) Source # Methods orientation :: Path q -> Orientation (Point (Path q)) Source # start :: Path q -> Point (Path q) Source # end :: Path q -> Point (Path q) Source #
Total q => Total (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
(Oriented a, Integral r) => Embeddable (Path a) (Product r a) Source #
Instance details Defined in OAlg.Entity.Product.Definition Methods inj :: Path a -> Product r a Source #
Embeddable (Path a) (ProductForm r a) Source #
Instance details Defined in OAlg.Entity.Product.Definition Methods inj :: Path a -> ProductForm r a Source #
Oriented a => Projectible (Path a) (Product N a) Source #
Instance details Defined in OAlg.Entity.Product.Definition Methods prj :: Product N a -> Path a Source #
Oriented a => Projectible (Path a) (ProductForm N a) Source #
Instance details Defined in OAlg.Entity.Product.Definition Methods prj :: ProductForm N a -> Path a Source #
ForgetfulOrt s => Applicative (IsoOpMap Path s) Source #
Instance details Defined in OAlg.Hom.Oriented.Definition Methods amap :: IsoOpMap Path s a b -> a -> b Source #
ForgetfulOrt s => Applicative (OpMap Path s) Source #
Instance details Defined in OAlg.Hom.Oriented.Definition Methods amap :: OpMap Path s a b -> a -> b Source #
ForgetfulOrt s => Functorial (IsoOpMap Path s) Source #
Instance details Defined in OAlg.Hom.Oriented.Definition
(TransformableOp s, ForgetfulOrt s, ForgetfulTyp s, Typeable s) => FunctorialHomOriented (IsoOpMap Path s) Source #
Instance details Defined in OAlg.Hom.Oriented.Definition
(TransformableOp s, ForgetfulOrt s, ForgetfulTyp s, Typeable s) => HomOriented (IsoOpMap Path s) Source #
Instance details Defined in OAlg.Hom.Oriented.Definition Methods pmap :: IsoOpMap Path s a b -> Point a -> Point b Source #
(TransformableOp s, ForgetfulOrt s, ForgetfulTyp s, Typeable s) => HomOriented (OpMap Path s) Source #
Instance details Defined in OAlg.Hom.Oriented.Definition Methods pmap :: OpMap Path s a b -> Point a -> Point b Source #
type Dual (Path q :: Type) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition type Dual (Path q :: Type) = Path (Op q)
type Point (Path q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition type Point (Path q) = Point q

pthLength :: Path q -> N Source #

the length of a path.

pthOne :: Point q -> Path q Source #

path of length 0 at the given point.

pthMlt :: Oriented q => Path q -> Path q -> Path q Source #

composition of two paths.

X

Site

data XOrtSite s q where Source #

random variables X q and X (Point q) for Oriented structure q.

Properties Let q be an instance of the class Oriented, then holds:

Let XStart xp xStart be in XOrtSite From q, then holds: For all p in Point q and x in the range of xStart p holds: start x == p.
Let XEnd xp xEnd be in XOrtSite To q, then holds: For all p in Point q and x in the range of xEnd p holds: end x == p.

Note The random variables xp should have a bias to non trivial random variables xp >>= xStart or xp >>= xEnd.

Constructors

XStart :: X (Point q) -> (Point q -> X q) -> XOrtSite From q
XEnd :: X (Point q) -> (Point q -> X q) -> XOrtSite To q

Instances

Instances details

Dualisable (XOrtSite 'To q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods toDual :: XOrtSite 'To q -> Dual (XOrtSite 'To q) Source # fromDual :: Dual (XOrtSite 'To q) -> XOrtSite 'To q Source #
Oriented q => Validable (XOrtSite s q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods valid :: XOrtSite s q -> Statement Source #
type Dual (XOrtSite s q :: Type) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition type Dual (XOrtSite s q :: Type) = XOrtSite (Dual s) (Op q)

class XStandardOrtSite s a where Source #

standard random variable for XOrtSite.

Methods

xStandardOrtSite :: XOrtSite s a Source #

Instances

Instances details

XStandardOrtSite 'From (Matrix Z) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Methods xStandardOrtSite :: XOrtSite 'From (Matrix Z) Source #
XStandard p => XStandardOrtSite 'From (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods xStandardOrtSite :: XOrtSite 'From (Orientation p) Source #
XStandardOrtSite 'From a => XStandardOrtSite 'To (Op a) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods xStandardOrtSite :: XOrtSite 'To (Op a) Source #
XStandardOrtSite 'To (Matrix Z) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Methods xStandardOrtSite :: XOrtSite 'To (Matrix Z) Source #
XStandard p => XStandardOrtSite 'To (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods xStandardOrtSite :: XOrtSite 'To (Orientation p) Source #
(Multiplicative c, Sliced i c, XStandardOrtSite 'From c) => XStandardOrtSite 'From (SliceFactor 'From i c) Source #
Instance details Defined in OAlg.Entity.Slice.Definition Methods xStandardOrtSite :: XOrtSite 'From (SliceFactor 'From i c) Source #
XStandardOrtSite 'From (SliceFactor 'To (Proxy :: Type -> Type) OS) Source #
Instance details Defined in OAlg.Entity.Slice.Definition Methods xStandardOrtSite :: XOrtSite 'From (SliceFactor 'To Proxy OS) Source #
(Multiplicative c, Sliced i c, XStandardOrtSite 'To c) => XStandardOrtSite 'To (SliceFactor 'To i c) Source #
Instance details Defined in OAlg.Entity.Slice.Definition Methods xStandardOrtSite :: XOrtSite 'To (SliceFactor 'To i c) Source #

class XStandardOrtSite To a => XStandardOrtSiteTo a Source #

standard random variable for XOrtSite To, helper class to avoid undecidable instances.

Instances

Instances details

XStandard p => XStandardOrtSiteTo (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
(Multiplicative c, Sliced i c, XStandardOrtSite 'To c) => XStandardOrtSiteTo (SliceFactor 'To i c) Source #
Instance details Defined in OAlg.Entity.Slice.Definition

class XStandardOrtSite From a => XStandardOrtSiteFrom a Source #

standard random variable for XOrtSite From, helper class to avoid undecidable instances.

Instances

Instances details

XStandardOrtSiteFrom (Matrix Z) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition
XStandard p => XStandardOrtSiteFrom (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
(Multiplicative c, Sliced i c, XStandardOrtSite 'From c) => XStandardOrtSiteFrom (SliceFactor 'From i c) Source #
Instance details Defined in OAlg.Entity.Slice.Definition
XStandardOrtSiteFrom (SliceFactor 'To (Proxy :: Type -> Type) OS) Source #
Instance details Defined in OAlg.Entity.Slice.Definition

coXOrtSite :: XOrtSite s q -> Dual (XOrtSite s q) Source #

to the dual of a XOrtSite s q, with inverse coXOrtSiteInv.

coXOrtSiteInv :: (Dual (Dual s) :~: s) -> Dual (XOrtSite s q) -> XOrtSite s q Source #

from the dual of a Dual (XOrtSite s q), with inverse coXOrtSite.

xosFromOpOp :: XOrtSite s (Op (Op q)) -> XOrtSite s q Source #

from the bidual.

xosStart :: XOrtSite From q -> Point q -> X q Source #

the random variable of arrows in q having all as start the given point.

xosEnd :: XOrtSite To q -> Point q -> X q Source #

the random variable of arrows in q having all as end the given point.

xosPathMaxAt :: Oriented q => XOrtSite s q -> N -> Point q -> X (Path q) Source #

tries to make a path at the given point with maximal length of the given length.

Properties Let xPath = xosPathMaxAt xos n x, then holds:

If xos matches XStart _ xq then for all p in the range of xPath holds:
1. start p == x.
2. If pthLength p < n then xq (end p) matches XEmpty.
If xos matches XEnd _ xq then for all p in the range of xPath holds:
1. end p == x.
2. If pthLength p < n then xq (start p) matches XEmpty.

xosPathMax :: Oriented q => XOrtSite s q -> N -> X (Path q) Source #

random variable of paths with maximal length of the given length.

Orientation

data XOrtOrientation q Source #

random variable of arrows given by an orientation.

Properties Let XOrtOrientation xo xArrow be in XOrtOrientation q for a Oriented structure q, then holds: For all o in Orientation q and x in the range of xArrow o holds: orientation x == o.

Note The random variable xo should have a bias to non trivial random variables xo >>= xArrow and as such the range of xo should be included in one connection component of q.

Constructors

XOrtOrientation (X (Orientation (Point q))) (Orientation (Point q) -> X q)

Instances

Instances details

Oriented q => Validable (XOrtOrientation q) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods valid :: XOrtOrientation q -> Statement Source #
type Dual (XOrtOrientation q :: Type) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition type Dual (XOrtOrientation q :: Type) = XOrtOrientation (Op q)

xoOrientation :: XOrtOrientation q -> X (Orientation (Point q)) Source #

the underlying random variable of orientations.

xoArrow :: XOrtOrientation q -> Orientation (Point q) -> X q Source #

the underlying random variable of arrow given by the orientation.

xoPoint :: Oriented q => XOrtOrientation q -> X (Point q) Source #

the underlying random variable of points, i.e. the union of the induced start and end random variable of xoOrientation.

coXOrtOrientation :: XOrtOrientation q -> Dual (XOrtOrientation q) Source #

to the dual.

xoTo :: Oriented q => XOrtOrientation q -> XOrtSite To q Source #

the induced XOrtSite To

xoFrom :: Oriented q => XOrtOrientation q -> XOrtSite From q Source #

the induced XOrtSite From.

xoTtl :: Total q => X q -> XOrtOrientation q Source #

random variable of XOrtOrientation q for a total q.

xoOrnt :: X p -> XOrtOrientation (Orientation p) Source #

the induced random variable of Orientation q.

class XStandardOrtOrientation q where Source #

standard random variable for XOrtOrientation.

Methods

xStandardOrtOrientation :: XOrtOrientation q Source #

Instances

Instances details

XStandardOrtOrientation Z Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods xStandardOrtOrientation :: XOrtOrientation Z Source #
(Additive x, FibredOriented x, XStandardOrtOrientation x, XStandardOrientationMatrix x) => XStandardOrtOrientation (Matrix x) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Methods xStandardOrtOrientation :: XOrtOrientation (Matrix x) Source #
XStandard p => XStandardOrtOrientation (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition Methods xStandardOrtOrientation :: XOrtOrientation (Orientation p) Source #

Orientation

class XStandard (Point a) => XStandardPoint a Source #

standard random variable of Points of a.

Instances

Instances details

XStandardPoint N Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
XStandardPoint Q Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
XStandardPoint Z Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
XStandardPoint (Matrix Z) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition
XStandard p => XStandardPoint (Orientation p) Source #
Instance details Defined in OAlg.Structure.Oriented.Definition
(Multiplicative c, Sliced i c, XStandardOrtSite 'From c) => XStandardPoint (SliceFactor 'From i c) Source #
Instance details Defined in OAlg.Entity.Slice.Definition
(Multiplicative c, Sliced i c, XStandardOrtSite 'To c) => XStandardPoint (SliceFactor 'To i c) Source #
Instance details Defined in OAlg.Entity.Slice.Definition

xStartOrnt :: X p -> XOrtSite From (Orientation p) Source #

the XOrtSite From for Orientation p of the given random variable.

xEndOrnt :: X p -> XOrtSite To (Orientation p) Source #

the XOrtSite To of Orientation p of the given random variable.