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

OAlg.Entity.Sum.Definition

Contents

Sum
Form
Linear Combination

Description

definition of free Sums over Fibred symbols.

Synopsis

data Sum r a
smlc :: Semiring r => Sum r a -> LinearCombination r a
smJoin :: (Semiring r, Commutative r, Fibred a, Ord a) => Sum r (Sum r a) -> Sum r a
nSum :: (Hom Fbr h, Additive x) => h a x -> Sum N a -> x
zSum :: (Hom Fbr h, Abelian x) => h a x -> Sum Z a -> x
smMap :: (Singleton (Root y), Fibred y, Ord y, Semiring r, Commutative r) => (x -> y) -> Sum r x -> Sum r y
data SumForm r a
- = Zero (Root a)
- | S a
- | r :! (SumForm r a)
- | (SumForm r a) :+ (SumForm r a)
smfLength :: Number r => SumForm r a -> N
smflc :: Semiring r => SumForm r a -> LinearCombination r a
lcsmf :: Semiring r => Root a -> LinearCombination r a -> SumForm r a
smfMap :: Singleton (Root y) => (x -> y) -> SumForm r x -> SumForm r y
smfJoin :: SumForm r (SumForm r a) -> SumForm r a
smfReduce :: (Fibred a, Ord a, Semiring r, Commutative r) => SumForm r a -> SumForm r a
newtype LinearCombination r a = LinearCombination [(r, a)]
lcs :: LinearCombination r a -> [(r, a)]
lcAggr :: (Eq a, Semiring r) => LinearCombination r a -> LinearCombination r a
lcSort :: Ord a => LinearCombination r a -> LinearCombination r a
lcSclFilter :: (r -> Bool) -> LinearCombination r a -> LinearCombination r a

Sum

data Sum r a Source #

free sum over Fibred symbols in a with scalars in r.

Definition A Sum s is valid if and only if its underlying SumForm s' is valid and s' is reduced, i.e. s' == reduce s'.

Instances

Instances details

(Fibred a, Entity r) => Show (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods showsPrec :: Int -> Sum r a -> ShowS # show :: Sum r a -> String # showList :: [Sum r a] -> ShowS #
(Fibred a, Entity r) => Eq (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods (==) :: Sum r a -> Sum r a -> Bool # (/=) :: Sum r a -> Sum r a -> Bool #
(Fibred a, OrdRoot a, Ord r, Ord a, Entity r) => Ord (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods compare :: Sum r a -> Sum r a -> Ordering # (<) :: Sum r a -> Sum r a -> Bool # (<=) :: Sum r a -> Sum r a -> Bool # (>) :: Sum r a -> Sum r a -> Bool # (>=) :: Sum r a -> Sum r a -> Bool # max :: Sum r a -> Sum r a -> Sum r a # min :: Sum r a -> Sum r a -> Sum r a #
(Fibred a, Ord a, Semiring r, Commutative r) => Constructable (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods make :: Form (Sum r a) -> Sum r a Source #
Exposable (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Associated Types type Form (Sum r a) Source # Methods form :: Sum r a -> Form (Sum r a) Source #
(Distributive r, Total r, Commutative r, Fibred a) => Validable (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods valid :: Sum r a -> Statement Source #
(Fibred a, Semiring r, Commutative r) => Entity (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition
(Fibred a, Ord a, Ring r, Commutative r) => Abelian (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods negate :: Sum r a -> Sum r a Source # (-) :: Sum r a -> Sum r a -> Sum r a Source # ztimes :: Z -> Sum r a -> Sum r a Source #
(Fibred a, Ord a, Semiring r, Commutative r) => Additive (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods zero :: Root (Sum r a) -> Sum r a Source # (+) :: Sum r a -> Sum r a -> Sum r a Source # ntimes :: N -> Sum r a -> Sum r a Source #
(Fibred a, Semiring r, Commutative r) => Fibred (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Associated Types type Root (Sum r a) Source # Methods root :: Sum r a -> Root (Sum r a) Source #
(Fibred a, Ord a, Semiring r, Commutative r) => Vectorial (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Associated Types type Scalar (Sum r a) Source # Methods (!) :: Scalar (Sum r a) -> Sum r a -> Sum r a Source #
type Form (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition type Form (Sum r a) = SumForm r a
type Root (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition type Root (Sum r a) = Root a
type Scalar (Sum r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition type Scalar (Sum r a) = r

smlc :: Semiring r => Sum r a -> LinearCombination r a Source #

the associated linear combination.

Note The associated linear combination of a sum is sorted according to the second component!

smJoin :: (Semiring r, Commutative r, Fibred a, Ord a) => Sum r (Sum r a) -> Sum r a Source #

joining a sum of sums.

nSum :: (Hom Fbr h, Additive x) => h a x -> Sum N a -> x Source #

additive homomorphism for sums over N.

zSum :: (Hom Fbr h, Abelian x) => h a x -> Sum Z a -> x Source #

additive homomorphism for sums over Z.

smMap :: (Singleton (Root y), Fibred y, Ord y, Semiring r, Commutative r) => (x -> y) -> Sum r x -> Sum r y Source #

additive homomorphism to a totally defined sum.

Form

data SumForm r a Source #

form for a free sum over Fibred symbols in a with scalars in r.

Definition Let r be a Commutative Semiring and a a Fibred structure. A SumForm a is valid if and only if all scalars in a are valid and all symbols in a are valid and have the same root.

Constructors

Zero (Root a)
S a
r :! (SumForm r a) infixr 9
(SumForm r a) :+ (SumForm r a) infixr 6

Instances

Instances details

Foldable (SumForm N) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods fold :: Monoid m => SumForm N m -> m # foldMap :: Monoid m => (a -> m) -> SumForm N a -> m # foldMap' :: Monoid m => (a -> m) -> SumForm N a -> m # foldr :: (a -> b -> b) -> b -> SumForm N a -> b # foldr' :: (a -> b -> b) -> b -> SumForm N a -> b # foldl :: (b -> a -> b) -> b -> SumForm N a -> b # foldl' :: (b -> a -> b) -> b -> SumForm N a -> b # foldr1 :: (a -> a -> a) -> SumForm N a -> a # foldl1 :: (a -> a -> a) -> SumForm N a -> a # toList :: SumForm N a -> [a] # null :: SumForm N a -> Bool # length :: SumForm N a -> Int # elem :: Eq a => a -> SumForm N a -> Bool # maximum :: Ord a => SumForm N a -> a # minimum :: Ord a => SumForm N a -> a # sum :: Num a => SumForm N a -> a # product :: Num a => SumForm N a -> a #
(Fibred a, Entity r) => Show (SumForm r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods showsPrec :: Int -> SumForm r a -> ShowS # show :: SumForm r a -> String # showList :: [SumForm r a] -> ShowS #
(Fibred a, Entity r) => Eq (SumForm r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods (==) :: SumForm r a -> SumForm r a -> Bool # (/=) :: SumForm r a -> SumForm r a -> Bool #
(Fibred a, Entity r, OrdRoot a, Ord r, Ord a) => Ord (SumForm r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods compare :: SumForm r a -> SumForm r a -> Ordering # (<) :: SumForm r a -> SumForm r a -> Bool # (<=) :: SumForm r a -> SumForm r a -> Bool # (>) :: SumForm r a -> SumForm r a -> Bool # (>=) :: SumForm r a -> SumForm r a -> Bool # max :: SumForm r a -> SumForm r a -> SumForm r a # min :: SumForm r a -> SumForm r a -> SumForm r a #
LengthN (SumForm N a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods lengthN :: SumForm N a -> N Source #
(Fibred a, Ord a, Semiring r, Commutative r) => Reducible (SumForm r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods reduce :: SumForm r a -> SumForm r a Source #
(Fibred a, Semiring r, Commutative r) => Validable (SumForm r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods valid :: SumForm r a -> Statement Source #
(Fibred a, Semiring r, Commutative r) => Entity (SumForm r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition
(Fibred a, Semiring r, Commutative r) => Fibred (SumForm r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Associated Types type Root (SumForm r a) Source # Methods root :: SumForm r a -> Root (SumForm r a) Source #
type Root (SumForm r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition type Root (SumForm r a) = Root a

smfLength :: Number r => SumForm r a -> N Source #

the length of a sum form,

smflc :: Semiring r => SumForm r a -> LinearCombination r a Source #

transforming a sum form to its corresponding linear combination..

lcsmf :: Semiring r => Root a -> LinearCombination r a -> SumForm r a Source #

transforming a word to its corresponding sum form.

smfMap :: Singleton (Root y) => (x -> y) -> SumForm r x -> SumForm r y Source #

mapping of sum forms.

smfJoin :: SumForm r (SumForm r a) -> SumForm r a Source #

joining a sum form of sum forms.

smfReduce :: (Fibred a, Ord a, Semiring r, Commutative r) => SumForm r a -> SumForm r a Source #

reducing a sum form to its canonical form,

Linear Combination

newtype LinearCombination r a Source #

list of symbols in a together with a scalar in r.

Note valid linear combinations must not be sorted according to the second component!

Constructors

LinearCombination [(r, a)]

Instances

Instances details

(Show r, Show a) => Show (LinearCombination r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods showsPrec :: Int -> LinearCombination r a -> ShowS # show :: LinearCombination r a -> String # showList :: [LinearCombination r a] -> ShowS #
(Eq r, Eq a) => Eq (LinearCombination r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods (==) :: LinearCombination r a -> LinearCombination r a -> Bool # (/=) :: LinearCombination r a -> LinearCombination r a -> Bool #
(Validable r, Validable a) => Validable (LinearCombination r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition Methods valid :: LinearCombination r a -> Statement Source #
(Entity a, Entity r) => Entity (LinearCombination r a) Source #
Instance details Defined in OAlg.Entity.Sum.Definition

lcs :: LinearCombination r a -> [(r, a)] Source #

the underlying list of symbols with their scalar.

lcAggr :: (Eq a, Semiring r) => LinearCombination r a -> LinearCombination r a Source #

aggregating linear combinations with same symbols.

lcSort :: Ord a => LinearCombination r a -> LinearCombination r a Source #

sorting a linear combination according to its symbols.

lcSclFilter :: (r -> Bool) -> LinearCombination r a -> LinearCombination r a Source #

filtering a word according to the scalars.