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

OAlg.Entity.Product.ProductSymbol

Contents

ProductSymbol
U
X

Description

free products of symbols in x with index type N.

Synopsis

newtype ProductSymbol x = ProductSymbol (Product N (U x))
sy :: Entity x => x -> ProductSymbol x
psyShow :: Entity x => ProductSymbol x -> String
psyxs :: ProductSymbol x -> [(x, N)]
psywrd :: Entity x => ProductSymbol x -> Word N x
wrdpsy :: Entity x => Word N x -> ProductSymbol x
nProxy :: Proxy N
psyJoin :: Entity x => ProductSymbol (ProductSymbol x) -> ProductSymbol x
productSymbol :: Entity x => [x] -> ProductSymbol x
psyLength :: ProductSymbol x -> N
psyFactor :: ProductSymbol x -> N -> x
psyMap :: Entity y => (x -> y) -> ProductSymbol x -> ProductSymbol y
newtype U x = U x
fromU :: U x -> x
xProductSymbol :: Entity x => N -> X x -> X (ProductSymbol x)

ProductSymbol

newtype ProductSymbol x Source #

free product of symbols in x with index type N.

Example

The expression sy 'a' constructs a free product of exactly one symbol in Char consisting just of the character 'a'.

>>> sy 'a'
ProductSymbol['a']

they are Total Multiplicative

>>> sy 'a' * sy 'b' * sy 'c'
ProductSymbol['a'*'b'*'c']

and admit a listing

>>> list (Proxy :: Proxy N) (sy 'a' * sy 'b' * sy 'c')
[('a',0),('b',1),('c',2)]

they have a compact representation for repetitions

>>> sy 'a' * sy 'b' * sy 'b' * sy 'a' * sy 'c'
ProductSymbol['a'*'b'^2*'a'*'c']

>>> sy 'a' * sy 'b' * sy 'b' * sy 'a' * sy 'c' == sy 'a' * sy 'b' ^ 2 * sy 'a' * sy 'c'
True

but they are not Commutative

>>> sy 'a' * sy 'b' ^ 2 * sy 'a' * sy 'c' == sy 'a' ^ 2 * sy 'b' ^ 2 * sy 'c'
False

and they admit a total right operation <* of Permutation N

>>> (sy 'a' * sy 'b' ^ 2 * sy 'a' * sy 'c') <* (pmtSwap 1 3 :: Permutation N)
ProductSymbol['a'^2*'b'^2*'c']

Note

Free products of symbols are finite complete sequences and allow a compact representation for repetitions and serve merely as dimensions for matrices (see OAlg.Entity.Matrix.Dim).
Possibly infinite complete sequences are represented by [x].

Constructors

ProductSymbol (Product N (U x))

Instances

Instances details

Foldable ProductSymbol Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Methods fold :: Monoid m => ProductSymbol m -> m # foldMap :: Monoid m => (a -> m) -> ProductSymbol a -> m # foldMap' :: Monoid m => (a -> m) -> ProductSymbol a -> m # foldr :: (a -> b -> b) -> b -> ProductSymbol a -> b # foldr' :: (a -> b -> b) -> b -> ProductSymbol a -> b # foldl :: (b -> a -> b) -> b -> ProductSymbol a -> b # foldl' :: (b -> a -> b) -> b -> ProductSymbol a -> b # foldr1 :: (a -> a -> a) -> ProductSymbol a -> a # foldl1 :: (a -> a -> a) -> ProductSymbol a -> a # toList :: ProductSymbol a -> [a] # null :: ProductSymbol a -> Bool # length :: ProductSymbol a -> Int # elem :: Eq a => a -> ProductSymbol a -> Bool # maximum :: Ord a => ProductSymbol a -> a # minimum :: Ord a => ProductSymbol a -> a # sum :: Num a => ProductSymbol a -> a # product :: Num a => ProductSymbol a -> a #
Entity x => ConstructableSequence ProductSymbol N x Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Methods sequence :: (N -> Maybe x) -> Set N -> ProductSymbol x Source # (<&) :: ProductSymbol x -> Set N -> ProductSymbol x Source #
Sequence ProductSymbol N x Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Methods graph :: p N -> ProductSymbol x -> Graph N x Source # list :: p N -> ProductSymbol x -> [(x, N)] Source # (??) :: ProductSymbol x -> N -> Maybe x Source #
Entity x => PermutableSequence CSequence N x Source #
Instance details Defined in OAlg.Entity.Sequence.Permutation Methods permuteBy :: p N -> (w -> w -> Ordering) -> (x -> w) -> CSequence x -> (CSequence x, Permutation N) Source #
Entity x => Show (ProductSymbol x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Methods showsPrec :: Int -> ProductSymbol x -> ShowS # show :: ProductSymbol x -> String # showList :: [ProductSymbol x] -> ShowS #
Entity x => Eq (ProductSymbol x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Methods (==) :: ProductSymbol x -> ProductSymbol x -> Bool # (/=) :: ProductSymbol x -> ProductSymbol x -> Bool #
(Entity x, Ord x) => Ord (ProductSymbol x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Methods compare :: ProductSymbol x -> ProductSymbol x -> Ordering # (<) :: ProductSymbol x -> ProductSymbol x -> Bool # (<=) :: ProductSymbol x -> ProductSymbol x -> Bool # (>) :: ProductSymbol x -> ProductSymbol x -> Bool # (>=) :: ProductSymbol x -> ProductSymbol x -> Bool # max :: ProductSymbol x -> ProductSymbol x -> ProductSymbol x # min :: ProductSymbol x -> ProductSymbol x -> ProductSymbol x #
Entity x => Constructable (ProductSymbol x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Methods make :: Form (ProductSymbol x) -> ProductSymbol x Source #
Exposable (ProductSymbol x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Associated Types type Form (ProductSymbol x) Source # Methods form :: ProductSymbol x -> Form (ProductSymbol x) Source #
LengthN (ProductSymbol x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Methods lengthN :: ProductSymbol x -> N Source #
Entity x => Validable (ProductSymbol x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Methods valid :: ProductSymbol x -> Statement Source #
Entity x => Entity (ProductSymbol x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol
Entity x => Exponential (ProductSymbol x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Associated Types type Exponent (ProductSymbol x) Source # Methods (^) :: ProductSymbol x -> Exponent (ProductSymbol x) -> ProductSymbol x Source #
Entity x => Multiplicative (ProductSymbol x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Methods one :: Point (ProductSymbol x) -> ProductSymbol x Source # (*) :: ProductSymbol x -> ProductSymbol x -> ProductSymbol x Source # npower :: ProductSymbol x -> N -> ProductSymbol x Source #
Entity x => Oriented (ProductSymbol x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Associated Types type Point (ProductSymbol x) Source # Methods orientation :: ProductSymbol x -> Orientation (Point (ProductSymbol x)) Source # start :: ProductSymbol x -> Point (ProductSymbol x) Source # end :: ProductSymbol x -> Point (ProductSymbol x) Source #
Entity x => Opr (Permutation N) (CSequence x) Source #
Instance details Defined in OAlg.Entity.Sequence.Permutation Methods (<*) :: CSequence x -> Permutation N -> CSequence x Source #
Entity x => TotalOpr (Permutation N) (CSequence x) Source #
Instance details Defined in OAlg.Entity.Sequence.Permutation
type Form (ProductSymbol x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol type Form (ProductSymbol x) = ProductForm N (U x)
type Exponent (ProductSymbol x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol type Exponent (ProductSymbol x) = N
type Point (ProductSymbol x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol type Point (ProductSymbol x) = ()

sy :: Entity x => x -> ProductSymbol x Source #

symbol of an entity, i.e. the complete sequence of psyLength one consisting just of it.

Example

>>> sy 'a'
ProductSymbol['a']

>>> sy 'a' * sy 'b' * sy 'b' ^ 5 * sy 'c'
ProductSymbol['a'*'b'^6*'c']

psyShow :: Entity x => ProductSymbol x -> String Source #

showing as a product of symbols.

psyxs :: ProductSymbol x -> [(x, N)] Source #

the indexed listing of the symbols.

psywrd :: Entity x => ProductSymbol x -> Word N x Source #

the underlying word.

wrdpsy :: Entity x => Word N x -> ProductSymbol x Source #

from word.

nProxy :: Proxy N Source #

proxy for N.

psyJoin :: Entity x => ProductSymbol (ProductSymbol x) -> ProductSymbol x Source #

joining complete sequences.

productSymbol :: Entity x => [x] -> ProductSymbol x Source #

the induced product of symbols.

psyLength :: ProductSymbol x -> N Source #

the length of a complete sequence.

psyFactor :: ProductSymbol x -> N -> x Source #

the symbol for the given index.

psyMap :: Entity y => (x -> y) -> ProductSymbol x -> ProductSymbol y Source #

mapping free products of symbols.

U

newtype U x Source #

adjoins the point () to an entity.

Note Serves to build sums or products over symbols in x.

Constructors

U x

Instances

Instances details

Foldable U Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Methods fold :: Monoid m => U m -> m # foldMap :: Monoid m => (a -> m) -> U a -> m # foldMap' :: Monoid m => (a -> m) -> U a -> m # foldr :: (a -> b -> b) -> b -> U a -> b # foldr' :: (a -> b -> b) -> b -> U a -> b # foldl :: (b -> a -> b) -> b -> U a -> b # foldl' :: (b -> a -> b) -> b -> U a -> b # foldr1 :: (a -> a -> a) -> U a -> a # foldl1 :: (a -> a -> a) -> U a -> a # toList :: U a -> [a] # null :: U a -> Bool # length :: U a -> Int # elem :: Eq a => a -> U a -> Bool # maximum :: Ord a => U a -> a # minimum :: Ord a => U a -> a # sum :: Num a => U a -> a # product :: Num a => U a -> a #
Functor U Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Methods fmap :: (a -> b) -> U a -> U b # (<$) :: a -> U b -> U a #
Show x => Show (U x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Methods showsPrec :: Int -> U x -> ShowS # show :: U x -> String # showList :: [U x] -> ShowS #
Eq x => Eq (U x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Methods (==) :: U x -> U x -> Bool # (/=) :: U x -> U x -> Bool #
Ord x => Ord (U x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Methods compare :: U x -> U x -> Ordering # (<) :: U x -> U x -> Bool # (<=) :: U x -> U x -> Bool # (>) :: U x -> U x -> Bool # (>=) :: U x -> U x -> Bool # max :: U x -> U x -> U x # min :: U x -> U x -> U x #
Validable x => Validable (U x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Methods valid :: U x -> Statement Source #
Entity x => Entity (U x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol
OrdPoint (U x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol
Entity x => Oriented (U x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol Associated Types type Point (U x) Source # Methods orientation :: U x -> Orientation (Point (U x)) Source # start :: U x -> Point (U x) Source # end :: U x -> Point (U x) Source #
type Point (U x) Source #
Instance details Defined in OAlg.Entity.Product.ProductSymbol type Point (U x) = ()

fromU :: U x -> x Source #

deconstructor.

X

xProductSymbol :: Entity x => N -> X x -> X (ProductSymbol x) Source #

random variable of complete sequences with the given maximal length.