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

OAlg.Structure.Fibred.Root

Contents

Root
Applicative
R

Description

Definition of Root.

Synopsis

type family Root x
class Show (Root x) => ShowRoot x
class Eq (Root x) => EqRoot x
class Ord (Root x) => OrdRoot x
class Singleton (Root f) => SingletonRoot f
type TotalRoot = SingletonRoot
class Validable (Root x) => ValidableRoot x
class Typeable (Root x) => TypeableRoot x
type EntityRoot x = (ShowRoot x, EqRoot x, ValidableRoot x, TypeableRoot x)
class XStandard (Root f) => XStandardRoot f
newtype Rt x = Rt (Root x)
fromRtG :: (Rt x -> Rt y) -> Root x -> Root y
idRt :: Root x ~ Root y => Rt x -> Rt y
type ApplicativeRoot (h :: Type -> Type -> Type) = ApplicativeG Rt h (->)
rmap :: ApplicativeRoot h => h x y -> Root x -> Root y
amapRt :: (Root x -> Root y) -> Rt x -> Rt y
type FunctorialRoot (h :: Type -> Type -> Type) = FunctorialG Rt h (->)
newtype R x = R x

Root

type family Root x Source #

the type of roots.

Instances

Instances details

type Root N Source #
Instance details Defined in OAlg.Structure.Fibred.Root type Root N = Orientation ()
type Root Q Source #
Instance details Defined in OAlg.Structure.Fibred.Root type Root Q = Orientation ()
type Root Z Source #
Instance details Defined in OAlg.Structure.Fibred.Root type Root Z = Orientation ()
type Root N' Source #
Instance details Defined in OAlg.Entity.Natural type Root N' = Orientation ()
type Root W' Source #
Instance details Defined in OAlg.Entity.Natural type Root W' = Orientation ()
type Root Integer Source #
Instance details Defined in OAlg.Structure.Fibred.Root type Root Integer = Orientation ()
type Root () Source #
Instance details Defined in OAlg.Structure.Fibred.Root type Root () = Orientation ()
type Root Int Source #
Instance details Defined in OAlg.Structure.Fibred.Root type Root Int = Orientation ()
type Root (Id x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root type Root (Id x) = Root x
type Root (Matrix x) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition type Root (Matrix x) = Orientation (Dim' x)
type Root (Vector r) Source #
Instance details Defined in OAlg.Entity.Matrix.Vector type Root (Vector r) = ()
type Root (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition type Root (Sheaf f) = Root f
type Root (R x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root type Root (R x) = ()
type Root (Rt x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root type Root (Rt x) = Root x
type Root (Op x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root type Root (Op x) = Root x
type Root (Orientation p) Source #
Instance details Defined in OAlg.Structure.Fibred.Root type Root (Orientation p) = Orientation p
type Root (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition type Root (Sum r a) = Root a
type Root (SumForm r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition type Root (SumForm r a) = Root a
type Root (SumSymbol r a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol type Root (SumSymbol r a) = ()
type Root (FSequenceForm d i x) Source #
Instance details Defined in OAlg.Entity.Sequence.FSequence type Root (FSequenceForm d i x) = d
type Root (Slice s i x) Source #
Instance details Defined in OAlg.Entity.Slice.Definition type Root (Slice s i x) = Point x
type Root (ConsecutiveZeroHom t n x) Source #
Instance details Defined in OAlg.Limes.Exact.ConsecutiveZero type Root (ConsecutiveZeroHom t n x) = Orientation (ConsecutiveZero t n x)
type Root (DiagramTrafo t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation type Root (DiagramTrafo t n m a) = Orientation (Diagram t n m a)
type Root (FSequence s d i x) Source #
Instance details Defined in OAlg.Entity.Sequence.FSequence type Root (FSequence s d i x) = d

class Show (Root x) => ShowRoot x Source #

helper class to avoid undecidable instances.

Instances

Instances details

ShowRoot N Source #
Instance details Defined in OAlg.Structure.Fibred.Root
ShowRoot Q Source #
Instance details Defined in OAlg.Structure.Fibred.Root
ShowRoot Z Source #
Instance details Defined in OAlg.Structure.Fibred.Root
ShowRoot N' Source #
Instance details Defined in OAlg.Entity.Natural
ShowRoot W' Source #
Instance details Defined in OAlg.Entity.Natural
ShowRoot Integer Source #
Instance details Defined in OAlg.Structure.Fibred.Root
ShowRoot () Source #
Instance details Defined in OAlg.Structure.Fibred.Root
ShowRoot Int Source #
Instance details Defined in OAlg.Structure.Fibred.Root
ShowRoot x => ShowRoot (Id x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
Oriented x => ShowRoot (Matrix x) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition
ShowRoot (Vector r) Source #
Instance details Defined in OAlg.Entity.Matrix.Vector
ShowRoot f => ShowRoot (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
ShowRoot (R x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
ShowRoot x => ShowRoot (Op x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
Show p => ShowRoot (Orientation p) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
ShowRoot a => ShowRoot (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition
ShowRoot a => ShowRoot (SumForm r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition
ShowRoot (SumSymbol r a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol
Show d => ShowRoot (FSequenceForm d i x) Source #
Instance details Defined in OAlg.Entity.Sequence.FSequence
ShowPoint x => ShowRoot (Slice s i x) Source #
Instance details Defined in OAlg.Entity.Slice.Definition
(Show x, ShowPoint x) => ShowRoot (ConsecutiveZeroHom t n x) Source #
Instance details Defined in OAlg.Limes.Exact.ConsecutiveZero
(Show a, ShowPoint a) => ShowRoot (DiagramTrafo t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation
Show d => ShowRoot (FSequence s d i x) Source #
Instance details Defined in OAlg.Entity.Sequence.FSequence

class Eq (Root x) => EqRoot x Source #

helper class to avoid undecidable instances.

Instances

Instances details

EqRoot N Source #
Instance details Defined in OAlg.Structure.Fibred.Root
EqRoot Q Source #
Instance details Defined in OAlg.Structure.Fibred.Root
EqRoot Z Source #
Instance details Defined in OAlg.Structure.Fibred.Root
EqRoot N' Source #
Instance details Defined in OAlg.Entity.Natural
EqRoot W' Source #
Instance details Defined in OAlg.Entity.Natural
EqRoot Integer Source #
Instance details Defined in OAlg.Structure.Fibred.Root
EqRoot () Source #
Instance details Defined in OAlg.Structure.Fibred.Root
EqRoot Int Source #
Instance details Defined in OAlg.Structure.Fibred.Root
EqRoot x => EqRoot (Id x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
Oriented x => EqRoot (Matrix x) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition
EqRoot (Vector r) Source #
Instance details Defined in OAlg.Entity.Matrix.Vector
EqRoot f => EqRoot (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
EqRoot (R x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
EqRoot x => EqRoot (Op x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
Eq p => EqRoot (Orientation p) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
EqRoot a => EqRoot (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition
EqRoot a => EqRoot (SumForm r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition
EqRoot (SumSymbol r a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol
Eq d => EqRoot (FSequenceForm d i x) Source #
Instance details Defined in OAlg.Entity.Sequence.FSequence
EqPoint x => EqRoot (Slice s i x) Source #
Instance details Defined in OAlg.Entity.Slice.Definition
(Eq x, EqPoint x) => EqRoot (ConsecutiveZeroHom t n x) Source #
Instance details Defined in OAlg.Limes.Exact.ConsecutiveZero
(Eq a, EqPoint a) => EqRoot (DiagramTrafo t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation
Eq d => EqRoot (FSequence s d i x) Source #
Instance details Defined in OAlg.Entity.Sequence.FSequence

class Ord (Root x) => OrdRoot x Source #

helper class to avoid undecidable instances.

Instances

Instances details

OrdRoot (R x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root

class Singleton (Root f) => SingletonRoot f Source #

helper class to avoid undecidable instances.

Instances

Instances details

SingletonRoot N Source #
Instance details Defined in OAlg.Structure.Fibred.Root
SingletonRoot Q Source #
Instance details Defined in OAlg.Structure.Fibred.Root
SingletonRoot Z Source #
Instance details Defined in OAlg.Structure.Fibred.Root
SingletonRoot N' Source #
Instance details Defined in OAlg.Entity.Natural
SingletonRoot W' Source #
Instance details Defined in OAlg.Entity.Natural
SingletonRoot Integer Source #
Instance details Defined in OAlg.Structure.Fibred.Root
SingletonRoot () Source #
Instance details Defined in OAlg.Structure.Fibred.Root
SingletonRoot Int Source #
Instance details Defined in OAlg.Structure.Fibred.Root
SingletonRoot x => SingletonRoot (Id x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
SingletonRoot (R x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
SingletonRoot x => SingletonRoot (Op x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
Singleton p => SingletonRoot (Orientation p) Source #
Instance details Defined in OAlg.Structure.Fibred.Root

type TotalRoot = SingletonRoot Source #

Roots admitting total operations.

class Validable (Root x) => ValidableRoot x Source #

helper class to avoid undecidable instances.

Instances

Instances details

ValidableRoot N Source #
Instance details Defined in OAlg.Structure.Fibred.Root
ValidableRoot Q Source #
Instance details Defined in OAlg.Structure.Fibred.Root
ValidableRoot Z Source #
Instance details Defined in OAlg.Structure.Fibred.Root
ValidableRoot N' Source #
Instance details Defined in OAlg.Entity.Natural
ValidableRoot W' Source #
Instance details Defined in OAlg.Entity.Natural
ValidableRoot Integer Source #
Instance details Defined in OAlg.Structure.Fibred.Root
ValidableRoot () Source #
Instance details Defined in OAlg.Structure.Fibred.Root
ValidableRoot Int Source #
Instance details Defined in OAlg.Structure.Fibred.Root
ValidableRoot x => ValidableRoot (Id x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
Oriented x => ValidableRoot (Matrix x) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition
ValidableRoot (Vector r) Source #
Instance details Defined in OAlg.Entity.Matrix.Vector
ValidableRoot f => ValidableRoot (Sheaf f) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition
ValidableRoot (R x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
ValidableRoot x => ValidableRoot (Op x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
Validable p => ValidableRoot (Orientation p) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
ValidableRoot a => ValidableRoot (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition
ValidableRoot a => ValidableRoot (SumForm r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition
ValidableRoot (SumSymbol r a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol
Validable d => ValidableRoot (FSequenceForm d i x) Source #
Instance details Defined in OAlg.Entity.Sequence.FSequence
ValidablePoint x => ValidableRoot (Slice s i x) Source #
Instance details Defined in OAlg.Entity.Slice.Definition
Distributive x => ValidableRoot (ConsecutiveZeroHom t n x) Source #
Instance details Defined in OAlg.Limes.Exact.ConsecutiveZero
Oriented a => ValidableRoot (DiagramTrafo t n m a) Source #
Instance details Defined in OAlg.Entity.Diagram.Transformation
Validable d => ValidableRoot (FSequence s d i x) Source #
Instance details Defined in OAlg.Entity.Sequence.FSequence

class Typeable (Root x) => TypeableRoot x Source #

helper class to avoid undecidable instances.

Instances

Instances details

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

type EntityRoot x = (ShowRoot x, EqRoot x, ValidableRoot x, TypeableRoot x) Source #

Roots as Entitys.

class XStandard (Root f) => XStandardRoot f Source #

helper class to avoid undecidable instances.

Instances

Instances details

XStandard p => XStandardRoot (Orientation p) Source #
Instance details Defined in OAlg.Structure.Fibred.Root

newtype Rt x Source #

wraper for Roots.

Constructors

Rt (Root x)

Instances

Instances details

(TransformableGRefl o s, DualisableDistributive s o, TransformableGRefl Matrix s) => ApplicativeMorCo Rt Matrix s o (->) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition
(TransformableGRefl o s, DualisableDistributive s o, TransformableGRefl Matrix s) => FunctorialHomCo Rt Matrix s o (->) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition
ApplicativeG Rt h c => ApplicativeG Rt (Inv2 h) c Source #
Instance details Defined in OAlg.Structure.Fibred.Root Methods amapG :: Inv2 h x y -> c (Rt x) (Rt y) Source #
ApplicativeG Rt (HomEmpty s) c Source #
Instance details Defined in OAlg.Hom.Definition Methods amapG :: HomEmpty s x y -> c (Rt x) (Rt y) Source #
Transformable s FbrOrt => ApplicativeG Rt (Ornt s) (->) Source #
Instance details Defined in OAlg.Data.Ornt Methods amapG :: Ornt s x y -> Rt x -> Rt y Source #
ApplicativeG Rt (HomSymbol r) (->) Source #
Instance details Defined in OAlg.Entity.Matrix.Vector Methods amapG :: HomSymbol r x y -> Rt x -> Rt y Source #
(Transformable r Type, TransformableOp r, Transformable r FbrOrt) => DualisableG r (->) Op Rt Source #
Instance details Defined in OAlg.Category.Dualisable Methods toDualG :: Struct r x -> Rt x -> Rt (Op x) Source # fromDualG :: Struct r x -> Rt (Op x) -> Rt x Source #
Transformable r Type => ReflexiveG r (->) Op Rt Source #
Instance details Defined in OAlg.Category.Dualisable Methods reflG :: Struct r x -> Inv2 (->) (Rt x) (Rt (Op (Op x))) Source #
(TransformableGRefl o s, DualisableDistributive s o, TransformableGRefl Matrix s) => ApplicativeG Rt (MorCo Matrix s o) (->) Source #
Instance details Defined in OAlg.Entity.Matrix.Definition Methods amapG :: MorCo Matrix s o x y -> Rt x -> Rt y Source #
(Morphism h, ApplicativeRoot h, DualisableG s (->) o Rt) => ApplicativeG Rt (HomDisj s o h) (->) Source #
Instance details Defined in OAlg.Hom.Definition Methods amapG :: HomDisj s o h x y -> Rt x -> Rt y Source #
(Morphism h, ApplicativeRoot h, DualisableG s (->) o Rt) => FunctorialG Rt (HomDisj s o h) (->) Source #
Instance details Defined in OAlg.Hom.Definition
ApplicativeG Rt h c => ApplicativeG Rt (Variant2 v h) c Source #
Instance details Defined in OAlg.Data.Variant Methods amapG :: Variant2 v h x y -> c (Rt x) (Rt y) Source #
ShowRoot x => Show (Rt x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root Methods showsPrec :: Int -> Rt x -> ShowS # show :: Rt x -> String # showList :: [Rt x] -> ShowS #
EqRoot x => Eq (Rt x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root Methods (==) :: Rt x -> Rt x -> Bool # (/=) :: Rt x -> Rt x -> Bool #
ValidableRoot x => Validable (Rt x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root Methods valid :: Rt x -> Statement Source #
type Root (Rt x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root type Root (Rt x) = Root x

fromRtG :: (Rt x -> Rt y) -> Root x -> Root y Source #

from Rt.

idRt :: Root x ~ Root y => Rt x -> Rt y Source #

casting a Root.

Applicative

type ApplicativeRoot (h :: Type -> Type -> Type) = ApplicativeG Rt h (->) Source #

application on Roots.

rmap :: ApplicativeRoot h => h x y -> Root x -> Root y Source #

the induced mapping of Roots.

amapRt :: (Root x -> Root y) -> Rt x -> Rt y Source #

the induced mapping.

type FunctorialRoot (h :: Type -> Type -> Type) = FunctorialG Rt h (->) Source #

functorials for Root.

R

newtype R x Source #

adjoining the total root () to x.

Constructors

R x

Instances

Instances details

Show x => Show (R x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root Methods showsPrec :: Int -> R x -> ShowS # show :: R x -> String # showList :: [R x] -> ShowS #
Eq x => Eq (R x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root Methods (==) :: R x -> R x -> Bool # (/=) :: R x -> R x -> Bool #
Ord x => Ord (R x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root Methods compare :: R x -> R x -> Ordering # (<) :: R x -> R x -> Bool # (<=) :: R x -> R x -> Bool # (>) :: R x -> R x -> Bool # (>=) :: R x -> R x -> Bool # max :: R x -> R x -> R x # min :: R x -> R x -> R x #
Validable x => Validable (R x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root Methods valid :: R x -> Statement Source #
Entity a => Fibred (R a) Source #
Instance details Defined in OAlg.Structure.Fibred.Definition Methods root :: R a -> Root (R a) Source #
EqRoot (R x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
OrdRoot (R x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
ShowRoot (R x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
SingletonRoot (R x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
TypeableRoot (R x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
ValidableRoot (R x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root
type Root (R x) Source #
Instance details Defined in OAlg.Structure.Fibred.Root type Root (R x) = ()