Copyright	(c) Erich Gut
License	BSD3
Maintainer	zerich.gut@gmail.com
Safe Haskell	None
Language	Haskell2010

OAlg.Structure.Fibred.Definition

Contents

Description

fibred structures, i.e. type f with an associated root type Root f such that every value in f has a root.

Synopsis

class (Entity f, EntityRoot f) => Fibred f where
- root :: f -> Root f
data Fbr
class Transformable s Fbr => TransformableFbr s
tauFbr :: Transformable s Fbr => Struct s x -> Struct Fbr x
module OAlg.Structure.Fibred.Root
data Sheaf f = Sheaf (Root f) [f]
type XFbr = X
xoFbr :: XOrtOrientation f -> XFbr f
xFbrOrnt :: X p -> XFbr (Orientation p)
data XStalk x = XStalk (X (Root x)) (Root x -> X x)
xRoot :: XStalk x -> X (Root x)
xSheafRootMax :: Fibred f => XStalk f -> N -> Root f -> X (Sheaf f)
xSheafMax :: Fibred f => XStalk f -> N -> X (Sheaf f)
xStalkOrnt :: X p -> XStalk (Orientation p)

Fibred

class (Entity f, EntityRoot f) => Fibred f where Source #

types with a Fibred structure. An entity of a Fibred structure will be called a stalk.

Note

On should accept the default for root only if one intens to implement a FibredOriented structure!
For Distributive structures the only thing to be implemented is the Root type and should be defined as Root d = Orientation p where-- p = Point d (see the default implementation of root).

Minimal complete definition

Nothing

Methods

root :: f -> Root f Source #

the root of a stalk in f.

default root :: (Root f ~ Orientation (Point f), Oriented f) => f -> Root f Source #

Instances

Instances details

Fibred N Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods root :: N -> Root N Source #
Fibred Q Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods root :: Q -> Root Q Source #
Fibred Z Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods root :: Z -> Root Z Source #
Fibred N' Source #
Instance details Defined in OAlg.Entity.Natural Methods root :: N' -> Root N' Source #
Fibred W' Source #
Instance details Defined in OAlg.Entity.Natural Methods root :: W' -> Root W' Source #
Fibred Integer Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods root :: Integer -> Root Integer Source #
Fibred () Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods root :: () -> Root () Source #
Fibred Int Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods root :: Int -> Root Int Source #
Fibred x => Fibred (Id x) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods root :: Id x -> Root (Id x) Source #
(Additive x, FibredOriented x) => Fibred (Matrix x) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Methods root :: Matrix x -> Root (Matrix x) Source #
Semiring r => Fibred (Vector r) Source #
Instance details Defined in OAlg.Entity.Matrix.Vector Methods root :: Vector r -> Root (Vector r) Source #
Fibred f => Fibred (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods root :: Sheaf f -> Root (Sheaf f) Source #
Entity a => Fibred (R a) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods root :: R a -> Root (R a) Source #
FibredOriented x => Fibred (Op x) Source #
Instance details Defined in OAlg.Structure.FibredOriented Methods root :: Op x -> Root (Op x) Source #
Entity p => Fibred (Orientation p) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods root :: Orientation p -> Root (Orientation p) Source #
(Fibred a, Semiring r, Commutative r) => Fibred (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods root :: Sum r a -> Root (Sum r a) Source #
(Fibred a, Semiring r, Commutative r) => Fibred (SumForm r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods root :: SumForm r a -> Root (SumForm r a) Source #
(Semiring r, Commutative r, Entity a) => Fibred (SumSymbol r a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol Methods root :: SumSymbol r a -> Root (SumSymbol r a) Source #
(Entity d, Entity i, Ord i, Entity x) => Fibred (FSequenceForm d i x) Source #
Instance details Defined in OAlg.Entity.Sequence.FSequence Methods root :: FSequenceForm d i x -> Root (FSequenceForm d i x) Source #
(Distributive x, Sliced i x, Typeable s) => Fibred (Slice s i x) Source #
Instance details Defined in OAlg.Entity.Slice.Definition Methods root :: Slice s i x -> Root (Slice s i x) Source #
(Distributive x, Typeable t, Typeable n) => Fibred (ConsecutiveZeroHom t n x) Source #
Instance details Defined in OAlg.Limes.Exact.ConsecutiveZero Methods root :: ConsecutiveZeroHom t n x -> Root (ConsecutiveZeroHom t n x) Source #
(Distributive a, Typeable t, Typeable n, Typeable m) => Fibred (DiagramTrafo t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation Methods root :: DiagramTrafo t n m a -> Root (DiagramTrafo t n m a) Source #
(DefaultValue d i x, Entity d, Entity i, Entity x, Ord i, Typeable s) => Fibred (FSequence s d i x) Source #
Instance details Defined in OAlg.Entity.Sequence.FSequence Methods root :: FSequence s d i x -> Root (FSequence s d i x) Source #

data Fbr Source #

type representing the class of Fibred structures.

Instances

Instances details

TransformableTyp Fbr Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
TransformableFbr Fbr Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
Transformable Abl Fbr Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods tau :: Struct Abl x -> Struct Fbr x Source #
Transformable Add Fbr Source #
Instance details Defined in OAlg.Structure.Additive.Definition Methods tau :: Struct Add x -> Struct Fbr x Source #
Transformable Dst Fbr Source #
Instance details Defined in OAlg.Structure.Distributive.Definition Methods tau :: Struct Dst x -> Struct Fbr x Source #
Transformable DstX Fbr Source #
Instance details Defined in OAlg.Structure.Distributive.Proposition Methods tau :: Struct DstX x -> Struct Fbr x Source #
Transformable Fbr Ent Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods tau :: Struct Fbr x -> Struct Ent x Source #
Transformable Fbr Typ Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods tau :: Struct Fbr x -> Struct Typ x Source #
Transformable FbrOrt Fbr Source #
Instance details Defined in OAlg.Structure.FibredOriented Methods tau :: Struct FbrOrt x -> Struct Fbr x Source #
Transformable FbrOrtX Fbr Source #
Instance details Defined in OAlg.Structure.FibredOriented Methods tau :: Struct FbrOrtX x -> Struct Fbr x Source #
Transformable (Alg k) Fbr Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition Methods tau :: Struct (Alg k) x -> Struct Fbr x Source #
Transformable (Vec k) Fbr Source #
Instance details Defined in OAlg.Structure.Vectorial.Definition Methods tau :: Struct (Vec k) x -> Struct Fbr x Source #
Transformable s Fbr => Transformable (s, SldFr) Fbr Source #
Instance details Defined in OAlg.Entity.Slice.Free Methods tau :: Struct (s, SldFr) x -> Struct Fbr x Source #
Transformable s Fbr => Transformable (s, Sld i) Fbr Source #
Instance details Defined in OAlg.Entity.Slice.Sliced Methods tau :: Struct (s, Sld i) x -> Struct Fbr x Source #
type Structure Fbr x Source #
Instance details Defined in OAlg.Structure.Fibred.Definition type Structure Fbr x = Fibred x

class Transformable s Fbr => TransformableFbr s Source #

helper class to avoid undecidable instances.

Instances

Instances details

TransformableFbr Abl Source #
Instance details Defined in OAlg.Structure.Additive.Definition
TransformableFbr Add Source #
Instance details Defined in OAlg.Structure.Additive.Definition
TransformableFbr Dst Source #
Instance details Defined in OAlg.Structure.Distributive.Definition
TransformableFbr DstX Source #
Instance details Defined in OAlg.Structure.Distributive.Proposition
TransformableFbr Fbr Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
TransformableFbr FbrOrt Source #
Instance details Defined in OAlg.Structure.FibredOriented
TransformableFbr FbrOrtX Source #
Instance details Defined in OAlg.Structure.FibredOriented
TransformableFbr (Alg k) Source #
Instance details Defined in OAlg.Structure.Algebraic.Definition
TransformableFbr (Vec k) Source #
Instance details Defined in OAlg.Structure.Vectorial.Definition
TransformableFbr s => TransformableFbr (s, SldFr) Source #
Instance details Defined in OAlg.Entity.Slice.Free
TransformableFbr s => TransformableFbr (s, Sld i) Source #
Instance details Defined in OAlg.Entity.Slice.Sliced

tauFbr :: Transformable s Fbr => Struct s x -> Struct Fbr x Source #

tau for Fbr.

module OAlg.Structure.Fibred.Root

Sheaf

data Sheaf f Source #

a list in a Fibred structure having all the same root.

Definition Let f be a Fibred structure and s = Sheaf r [t 0 .. t (n-1)] a sheaf in Sheaf f, then s is valid if and only if

r is valid and t i are valid for all i = 0..n-1.
root (t i) == r for all i = 0..n-1.

furthermore n is called the length of s.

If two sheafs have the same root then there stalks can be composed - via (++) - to a new sheaf having the same root. But as (++) is not commutative they are equipped with a Multiplicative structure.

Constructors

Sheaf (Root f) [f]

Instances

Instances details

Foldable Sheaf Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods fold :: Monoid m => Sheaf m -> m # foldMap :: Monoid m => (a -> m) -> Sheaf a -> m # foldMap' :: Monoid m => (a -> m) -> Sheaf a -> m # foldr :: (a -> b -> b) -> b -> Sheaf a -> b # foldr' :: (a -> b -> b) -> b -> Sheaf a -> b # foldl :: (b -> a -> b) -> b -> Sheaf a -> b # foldl' :: (b -> a -> b) -> b -> Sheaf a -> b # foldr1 :: (a -> a -> a) -> Sheaf a -> a # foldl1 :: (a -> a -> a) -> Sheaf a -> a # toList :: Sheaf a -> [a] # null :: Sheaf a -> Bool # length :: Sheaf a -> Int # elem :: Eq a => a -> Sheaf a -> Bool # maximum :: Ord a => Sheaf a -> a # minimum :: Ord a => Sheaf a -> a # sum :: Num a => Sheaf a -> a # product :: Num a => Sheaf a -> a #
Fibred f => Embeddable f (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods inj :: f -> Sheaf f Source #
(Show f, ShowRoot f) => Show (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods showsPrec :: Int -> Sheaf f -> ShowS # show :: Sheaf f -> String # showList :: [Sheaf f] -> ShowS #
(Eq f, EqRoot f) => Eq (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods (==) :: Sheaf f -> Sheaf f -> Bool # (/=) :: Sheaf f -> Sheaf f -> Bool #
FibredOriented f => Dualisable (Sheaf f) Source #
Instance details Defined in OAlg.Structure.FibredOriented Methods toDual :: Sheaf f -> Dual (Sheaf f) Source # fromDual :: Dual (Sheaf f) -> Sheaf f Source #
Fibred f => Validable (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods valid :: Sheaf f -> Statement Source #
Fibred f => Fibred (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods root :: Sheaf f -> Root (Sheaf f) Source #
EqRoot f => EqRoot (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
ShowRoot f => ShowRoot (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
TypeableRoot f => TypeableRoot (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
ValidableRoot f => ValidableRoot (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
Fibred f => Multiplicative (Sheaf f) Source #	`'Data.List.(++)'` is not commutative!
Instance details Defined in OAlg.Structure.Fibred.Definition Methods one :: Point (Sheaf f) -> Sheaf f Source # (*) :: Sheaf f -> Sheaf f -> Sheaf f Source # npower :: Sheaf f -> N -> Sheaf f Source #
Fibred f => Oriented (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods orientation :: Sheaf f -> Orientation (Point (Sheaf f)) Source # start :: Sheaf f -> Point (Sheaf f) Source # end :: Sheaf f -> Point (Sheaf f) Source #
EqRoot f => EqPoint (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
ShowRoot f => ShowPoint (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
TypeableRoot f => TypeablePoint (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
ValidableRoot f => ValidablePoint (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
(Fibred a, Ord a, Semiring r, Commutative r) => Projectible (Sum r a) (Sheaf a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods prj :: Sheaf a -> Sum r a Source #
type Dual (Sheaf f :: Type) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition type Dual (Sheaf f :: Type) = Sheaf (Op f)
type Root (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition type Root (Sheaf f) = Root f
type Point (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition type Point (Sheaf f) = Root f

X

type XFbr = X Source #

random variable for validating Fibred structures.

xoFbr :: XOrtOrientation f -> XFbr f Source #

the associated random variable for validating Fibred structures.

xFbrOrnt :: X p -> XFbr (Orientation p) Source #

random variable for the Fibred structure of Orientation p.

Stalk

data XStalk x Source #

random variable for stalks.

Constructors

XStalk (X (Root x)) (Root x -> X x)

xRoot :: XStalk x -> X (Root x) Source #

the underlying random variable of roots.

xSheafRootMax :: Fibred f => XStalk f -> N -> Root f -> X (Sheaf f) Source #

random variable of sheafs in a Fibred structure all rooted in the given root and with a length of either 0 - for empty roots - or with the given length.

xSheafMax :: Fibred f => XStalk f -> N -> X (Sheaf f) Source #

random variable of sheafs, based on the underlying random variable of roots, with a length of either 0 - for empty roots - or with the given length.

xStalkOrnt :: X p -> XStalk (Orientation p) Source #

random variable of XStalk on Orientation p.