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

OAlg.Entity.Sum.SumSymbol

Contents

SumSymbol
R

Description

free sums with symbols in a.

Synopsis

newtype SumSymbol r a = SumSymbol (Sum r (R a))
ssypsq :: Semiring r => SumSymbol r a -> PSequence a r
ssylc :: Semiring r => SumSymbol r a -> LinearCombination r a
sumSymbol :: (Semiring r, Commutative r, Entity a, Ord a) => [(r, a)] -> SumSymbol r a
sy :: (Semiring r, Commutative r, Entity a, Ord a) => a -> SumSymbol r a
ssyMap :: (Semiring r, Commutative r, Entity y, Ord y) => (x -> y) -> SumSymbol r x -> SumSymbol r y
ssySum :: (Semiring r, Commutative r, Entity y, Ord y) => (x -> LinearCombination r y) -> SumSymbol r x -> SumSymbol r y
ssyJoin :: (Semiring r, Commutative r, Entity x, Ord x) => SumSymbol r (SumSymbol r x) -> SumSymbol r x
ssyprj :: (Semiring r, Commutative r, Ord a, Entity a) => Set a -> SumSymbol r a -> SumSymbol r a
newtype R a = R a

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 #
(Entity a, Entity 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 #
(Entity a, Entity r, Ord r, Ord a) => 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, Entity a) => Validable (SumSymbol r a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol Methods valid :: SumSymbol r a -> Statement Source #
(Semiring r, Commutative r, Entity a) => Entity (SumSymbol r a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol
(Distributive r, Total r, Commutative r, Ord a, Abelian r, Entity 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, Entity 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 Associated Types type Root (SumSymbol r a) Source # Methods root :: SumSymbol r a -> Root (SumSymbol r a) Source #
(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) Source # Methods (!) :: Scalar (SumSymbol r a) -> SumSymbol 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 a Source #

adjoining the root ().

Constructors

R a

Instances

Instances details

Show a => Show (R a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol Methods showsPrec :: Int -> R a -> ShowS # show :: R a -> String # showList :: [R a] -> ShowS #
Eq a => Eq (R a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol Methods (==) :: R a -> R a -> Bool # (/=) :: R a -> R a -> Bool #
Ord a => Ord (R a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol Methods compare :: R a -> R a -> Ordering # (<) :: R a -> R a -> Bool # (<=) :: R a -> R a -> Bool # (>) :: R a -> R a -> Bool # (>=) :: R a -> R a -> Bool # max :: R a -> R a -> R a # min :: R a -> R a -> R a #
Validable a => Validable (R a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol Methods valid :: R a -> Statement Source #
Entity a => Entity (R a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol
Entity a => Fibred (R a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol Associated Types type Root (R a) Source # Methods root :: R a -> Root (R a) Source #
OrdRoot (R a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol
type Root (R a) Source #
Instance details Defined in OAlg.Entity.Sum.SumSymbol type Root (R a) = ()