oalg-base-3.0.0.0: Algebraic structures on oriented entities and limits as a tool kit to solve algebraic problems.
Copyright(c) Erich Gut
LicenseBSD3
Maintainerzerich.gut@gmail.com
Safe HaskellNone
LanguageHaskell2010

OAlg.Entity.Sum.SumSymbol

Contents

Description

free sums with symbols in a.

Synopsis

SumSymbol

newtype SumSymbol r a Source #

free sum with symbols in a.

Constructors

SumSymbol (Sum r (R a)) 

Instances

Instances details
(Semiring r, Show a) => Show (SumSymbol r a) Source # 
Instance details

Defined in OAlg.Entity.Sum.SumSymbol

Methods

showsPrec :: Int -> SumSymbol r a -> ShowS #

show :: SumSymbol r a -> String #

showList :: [SumSymbol r a] -> ShowS #

(Show a, Show r, Eq a, Eq r, Validable a, Validable r, Typeable a, Typeable r) => Eq (SumSymbol r a) Source # 
Instance details

Defined in OAlg.Entity.Sum.SumSymbol

Methods

(==) :: SumSymbol r a -> SumSymbol r a -> Bool #

(/=) :: SumSymbol r a -> SumSymbol r a -> Bool #

(Ord r, Ord a, Show a, Show r, Validable a, Validable r, Typeable a, Typeable r) => Ord (SumSymbol r a) Source # 
Instance details

Defined in OAlg.Entity.Sum.SumSymbol

Methods

compare :: SumSymbol r a -> SumSymbol r a -> Ordering #

(<) :: SumSymbol r a -> SumSymbol r a -> Bool #

(<=) :: SumSymbol r a -> SumSymbol r a -> Bool #

(>) :: SumSymbol r a -> SumSymbol r a -> Bool #

(>=) :: SumSymbol r a -> SumSymbol r a -> Bool #

max :: SumSymbol r a -> SumSymbol r a -> SumSymbol r a #

min :: SumSymbol r a -> SumSymbol r a -> SumSymbol r a #

(Distributive r, Total r, Commutative r, Show a, Eq a, Validable a, Typeable a) => Validable (SumSymbol r a) Source # 
Instance details

Defined in OAlg.Entity.Sum.SumSymbol

Methods

valid :: SumSymbol r a -> Statement Source #

(Ord a, Distributive r, Total r, Abelian r, Commutative r, Show a, Validable a, Typeable a) => Abelian (SumSymbol r a) Source # 
Instance details

Defined in OAlg.Entity.Sum.SumSymbol

Methods

negate :: SumSymbol r a -> SumSymbol r a Source #

(-) :: SumSymbol r a -> SumSymbol r a -> SumSymbol r a Source #

ztimes :: Z -> SumSymbol r a -> SumSymbol r a Source #

(Distributive r, Total r, Commutative r, Ord a, Show a, Validable a, Typeable a) => Additive (SumSymbol r a) Source # 
Instance details

Defined in OAlg.Entity.Sum.SumSymbol

Methods

zero :: Root (SumSymbol r a) -> SumSymbol r a Source #

(+) :: SumSymbol r a -> SumSymbol r a -> SumSymbol r a Source #

ntimes :: N -> SumSymbol r a -> SumSymbol 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 #

EqRoot (SumSymbol r a) Source # 
Instance details

Defined in OAlg.Entity.Sum.SumSymbol

ShowRoot (SumSymbol r a) Source # 
Instance details

Defined in OAlg.Entity.Sum.SumSymbol

TypeableRoot (SumSymbol r a) Source # 
Instance details

Defined in OAlg.Entity.Sum.SumSymbol

ValidableRoot (SumSymbol r a) Source # 
Instance details

Defined in OAlg.Entity.Sum.SumSymbol

(Semiring r, Commutative r, Entity a, Ord a) => Euclidean (SumSymbol r a) Source # 
Instance details

Defined in OAlg.Entity.Sum.SumSymbol

Methods

(<!>) :: SumSymbol r a -> SumSymbol r a -> Scalar (SumSymbol r a) Source #

(Semiring r, Commutative r, Entity a, Ord a) => Vectorial (SumSymbol r a) Source # 
Instance details

Defined in OAlg.Entity.Sum.SumSymbol

Associated Types

type Scalar (SumSymbol r a) 
Instance details

Defined in OAlg.Entity.Sum.SumSymbol

type Scalar (SumSymbol r a) = r

Methods

(!) :: Scalar (SumSymbol r a) -> SumSymbol r a -> SumSymbol r a Source #

(Entity a, Ord a, Semiring r, Commutative r) => Projectible (SumSymbol r a) [a] Source # 
Instance details

Defined in OAlg.Entity.Sum.SumSymbol

Methods

prj :: [a] -> SumSymbol r a Source #

(Entity a, Ord a, Semiring r, Commutative r) => Projectible (SumSymbol r a) (LinearCombination r (SumSymbol r a)) Source # 
Instance details

Defined in OAlg.Entity.Sum.SumSymbol

Methods

prj :: LinearCombination r (SumSymbol r a) -> SumSymbol r a Source #

(Entity a, Ord a, Semiring r, Commutative r) => Projectible (SumSymbol r a) (LinearCombination r a) Source # 
Instance details

Defined in OAlg.Entity.Sum.SumSymbol

Methods

prj :: LinearCombination r a -> SumSymbol r a Source #

type Root (SumSymbol r a) Source # 
Instance details

Defined in OAlg.Entity.Sum.SumSymbol

type Root (SumSymbol r a) = ()
type Scalar (SumSymbol r a) Source # 
Instance details

Defined in OAlg.Entity.Sum.SumSymbol

type Scalar (SumSymbol r a) = r

ssypsq :: Semiring r => SumSymbol r a -> PSequence a r Source #

the underlying partial sequence.

ssylc :: Semiring r => SumSymbol r a -> LinearCombination r a Source #

the underlying linear combination.

sumSymbol :: (Semiring r, Commutative r, Entity a, Ord a) => [(r, a)] -> SumSymbol r a Source #

the induced free sum given by a list of scalars and symbols.

sy :: (Semiring r, Commutative r, Entity a, Ord a) => a -> SumSymbol r a Source #

the induced free sum given by the symbol.

ssyMap :: (Semiring r, Commutative r, Entity y, Ord y) => (x -> y) -> SumSymbol r x -> SumSymbol r y Source #

mapping of free sums

ssySum :: (Semiring r, Commutative r, Entity y, Ord y) => (x -> LinearCombination r y) -> SumSymbol r x -> SumSymbol r y Source #

additive homomorphism given by a mapping of a symbol in x to a linear combination of y.

ssyJoin :: (Semiring r, Commutative r, Entity x, Ord x) => SumSymbol r (SumSymbol r x) -> SumSymbol r x Source #

joining a free sum of free sums to a free sum.

ssyprj :: (Semiring r, Commutative r, Ord a, Entity a) => Set a -> SumSymbol r a -> SumSymbol r a Source #

the projectin of a free sum according to the given set of symbols.

Definition Let x be in SumSymbol r a and s a Set of symbols in a, then x is called representable according to s iff all symbols of ssylc x are elements of s.

Property Let s be a set of symbols in a and x be representable in SumSymbol r a according to s, then ssyprj x == x.

Examples 

>>> ssyprj (Set [A,D,E]) (3!sy D) :: SumSymbol Z Symbol SumSymbol[3!D] 
>>> ssyprj (Set [A,D,E]) (2!sy B) :: SumSymbol Z Symbol SumSymbol[] 
>>> ssyprj (Set [A,D,E]) (3!sy D + sy A - 5!sy E) :: SumSymbol Z Symbol SumSymbol[A+3!D+-5!E] 
>>> ssyprj (Set [A,D,E]) (2!sy D + 7!sy B - sy E + sy F) :: SumSymbol Z Symbol SumSymbol[2!D+-1!E]

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) = ()