Data.Group.Cyclic

Cyclic groups: integers modulo n (clock arithmetic).

Documentation

data Cyclic n where

Cyclic group of order n: integers with addition modulo n.

Bundled Patterns

pattern Cyclic :: forall n. KnownNat n => Int -> Cyclic n

Smart pattern and constructor for elements of cyclic groups.

Instances

Generic1 Cyclic

Defined in Data.Group.Cyclic

Associated Types

type Rep1 Cyclic :: k -> Type

Methods

from1 :: forall (a :: k). Cyclic a -> Rep1 Cyclic a

to1 :: forall (a :: k). Rep1 Cyclic a -> Cyclic a

KnownNat n => Bounded (Cyclic n)

Defined in Data.Group.Cyclic

Methods

minBound :: Cyclic n

maxBound :: Cyclic n

KnownNat n => Enum (Cyclic n)

Defined in Data.Group.Cyclic

Methods

succ :: Cyclic n -> Cyclic n

pred :: Cyclic n -> Cyclic n

toEnum :: Int -> Cyclic n

fromEnum :: Cyclic n -> Int

enumFrom :: Cyclic n -> [Cyclic n]

enumFromThen :: Cyclic n -> Cyclic n -> [Cyclic n]

enumFromTo :: Cyclic n -> Cyclic n -> [Cyclic n]

enumFromThenTo :: Cyclic n -> Cyclic n -> Cyclic n -> [Cyclic n]

Eq (Cyclic n)

Defined in Data.Group.Cyclic

Methods

(==) :: Cyclic n -> Cyclic n -> Bool

(/=) :: Cyclic n -> Cyclic n -> Bool

Ord (Cyclic n)

Defined in Data.Group.Cyclic

Methods

compare :: Cyclic n -> Cyclic n -> Ordering

(<) :: Cyclic n -> Cyclic n -> Bool

(<=) :: Cyclic n -> Cyclic n -> Bool

(>) :: Cyclic n -> Cyclic n -> Bool

(>=) :: Cyclic n -> Cyclic n -> Bool

max :: Cyclic n -> Cyclic n -> Cyclic n

min :: Cyclic n -> Cyclic n -> Cyclic n

Show (Cyclic n)

Defined in Data.Group.Cyclic

Methods

showsPrec :: Int -> Cyclic n -> ShowS

show :: Cyclic n -> String

showList :: [Cyclic n] -> ShowS

Generic (Cyclic n)

Defined in Data.Group.Cyclic

Associated Types

type Rep (Cyclic n) :: Type -> Type

Methods

from :: Cyclic n -> Rep (Cyclic n) x

to :: Rep (Cyclic n) x -> Cyclic n

KnownNat n => Semigroup (Cyclic n)

Defined in Data.Group.Cyclic

Methods

(<>) :: Cyclic n -> Cyclic n -> Cyclic n

sconcat :: NonEmpty (Cyclic n) -> Cyclic n

stimes :: Integral b => b -> Cyclic n -> Cyclic n

(KnownNat n, 1 <= n) => Monoid (Cyclic n)

Defined in Data.Group.Cyclic

Methods

mempty :: Cyclic n

mappend :: Cyclic n -> Cyclic n -> Cyclic n

mconcat :: [Cyclic n] -> Cyclic n

NFData (Cyclic n)

Defined in Data.Group.Cyclic

Methods

rnf :: Cyclic n -> ()

(KnownNat n, 1 <= n) => Group (Cyclic n)

Defined in Data.Group.Cyclic

Methods

inverse :: Cyclic n -> Cyclic n

gtimes :: Integral n0 => n0 -> Cyclic n -> Cyclic n

KnownNat n => Act (Cyclic n) Int

Defined in Data.Group.Cyclic

Methods

(•) :: Cyclic n -> Int -> Int

act :: Cyclic n -> Int -> Int

Act (Cyclic 2) Bool

Defined in Data.Group.Cyclic

Methods

(•) :: Cyclic 2 -> Bool -> Bool

act :: Cyclic 2 -> Bool -> Bool

(Enum a, Bounded a, KnownNat n, 1 <= n) => Torsor (Cyclic n) (CyclicEnum a)

Defined in Data.Group.Cyclic

Methods

(<--) :: CyclicEnum a -> CyclicEnum a -> Cyclic n

(-->) :: CyclicEnum a -> CyclicEnum a -> Cyclic n

(Enum a, Bounded a, KnownNat n) => Act (Cyclic n) (CyclicEnum a)

Defined in Data.Group.Cyclic

Methods

(•) :: Cyclic n -> CyclicEnum a -> CyclicEnum a

act :: Cyclic n -> CyclicEnum a -> CyclicEnum a

Num a => Act (Cyclic 2) (Complex a)

Defined in Data.Group.Cyclic

Methods

(•) :: Cyclic 2 -> Complex a -> Complex a

act :: Cyclic 2 -> Complex a -> Complex a

Num i => Act (Cyclic 2) (Sum i)

Defined in Data.Group.Cyclic

Methods

(•) :: Cyclic 2 -> Sum i -> Sum i

act :: Cyclic 2 -> Sum i -> Sum i

Fractional i => Act (Cyclic 2) (Product i)

Defined in Data.Group.Cyclic

Methods

(•) :: Cyclic 2 -> Product i -> Product i

act :: Cyclic 2 -> Product i -> Product i

type Rep1 Cyclic

Defined in Data.Group.Cyclic

type Rep1 Cyclic = D1 ('MetaData "Cyclic" "Data.Group.Cyclic" "acts-0.1.0.0-inplace" 'True) (C1 ('MetaCons "MkCyclic" 'PrefixI 'True) (S1 ('MetaSel ('Just "runCyclic") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec1 Finite)))

type Rep (Cyclic n)

Defined in Data.Group.Cyclic

type Rep (Cyclic n) = D1 ('MetaData "Cyclic" "Data.Group.Cyclic" "acts-0.1.0.0-inplace" 'True) (C1 ('MetaCons "MkCyclic" 'PrefixI 'True) (S1 ('MetaSel ('Just "runCyclic") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (Finite n))))

getCyclic :: forall n. KnownNat n => Cyclic n -> Int

Obtain a representative in the range \( [0, n[ \).

type C (n :: Nat) = Cyclic n

Synonym for finite cyclic group.

type Z = Sum Int

Synonym for infinite cyclic group.

newtype CyclicEnum a

Newtype for cycling through elements in a finite enumeration.

data ABCD = A | B | C | D 
  deriving stock ( Enum, Bounded )
  deriving ( Act ( Cyclic 4 ), Torsor ( Cyclic 4 ) )
    via CyclicEnum ABCD

> act ( Cyclic 2 ) C
A

> act ( Cyclic (-1) ) A
D

> ( C --> B :: Cyclic 4 )
Cyclic 3

Warning It is unfortunately not checked that the size of the group matches the size of the finite enumeration. Please manually ensure this condition.

Constructors

CyclicEnum

Fields

• getCyclicEnum :: a

Instances

Bounded a => Bounded (CyclicEnum a)

Defined in Data.Group.Cyclic

Methods

minBound :: CyclicEnum a

maxBound :: CyclicEnum a

Enum a => Enum (CyclicEnum a)

Defined in Data.Group.Cyclic

Methods

succ :: CyclicEnum a -> CyclicEnum a

pred :: CyclicEnum a -> CyclicEnum a

toEnum :: Int -> CyclicEnum a

fromEnum :: CyclicEnum a -> Int

enumFrom :: CyclicEnum a -> [CyclicEnum a]

enumFromThen :: CyclicEnum a -> CyclicEnum a -> [CyclicEnum a]

enumFromTo :: CyclicEnum a -> CyclicEnum a -> [CyclicEnum a]

enumFromThenTo :: CyclicEnum a -> CyclicEnum a -> CyclicEnum a -> [CyclicEnum a]

Eq a => Eq (CyclicEnum a)

Defined in Data.Group.Cyclic

Methods

(==) :: CyclicEnum a -> CyclicEnum a -> Bool

(/=) :: CyclicEnum a -> CyclicEnum a -> Bool

Data a => Data (CyclicEnum a)

Defined in Data.Group.Cyclic

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> CyclicEnum a -> c (CyclicEnum a)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (CyclicEnum a)

toConstr :: CyclicEnum a -> Constr

dataTypeOf :: CyclicEnum a -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (CyclicEnum a))

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (CyclicEnum a))

gmapT :: (forall b. Data b => b -> b) -> CyclicEnum a -> CyclicEnum a

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> CyclicEnum a -> r

gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> CyclicEnum a -> r

gmapQ :: (forall d. Data d => d -> u) -> CyclicEnum a -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> CyclicEnum a -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> CyclicEnum a -> m (CyclicEnum a)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> CyclicEnum a -> m (CyclicEnum a)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> CyclicEnum a -> m (CyclicEnum a)

Ord a => Ord (CyclicEnum a)

Defined in Data.Group.Cyclic

Methods

compare :: CyclicEnum a -> CyclicEnum a -> Ordering

(<) :: CyclicEnum a -> CyclicEnum a -> Bool

(<=) :: CyclicEnum a -> CyclicEnum a -> Bool

(>) :: CyclicEnum a -> CyclicEnum a -> Bool

(>=) :: CyclicEnum a -> CyclicEnum a -> Bool

max :: CyclicEnum a -> CyclicEnum a -> CyclicEnum a

min :: CyclicEnum a -> CyclicEnum a -> CyclicEnum a

Show a => Show (CyclicEnum a)

Defined in Data.Group.Cyclic

Methods

showsPrec :: Int -> CyclicEnum a -> ShowS

show :: CyclicEnum a -> String

showList :: [CyclicEnum a] -> ShowS

Generic (CyclicEnum a)

Defined in Data.Group.Cyclic

Associated Types

type Rep (CyclicEnum a) :: Type -> Type

Methods

from :: CyclicEnum a -> Rep (CyclicEnum a) x

to :: Rep (CyclicEnum a) x -> CyclicEnum a

NFData a => NFData (CyclicEnum a)

Defined in Data.Group.Cyclic

Methods

rnf :: CyclicEnum a -> ()

Generic1 CyclicEnum

Defined in Data.Group.Cyclic

Associated Types

type Rep1 CyclicEnum :: k -> Type

Methods

from1 :: forall (a :: k). CyclicEnum a -> Rep1 CyclicEnum a

to1 :: forall (a :: k). Rep1 CyclicEnum a -> CyclicEnum a

(Enum a, Bounded a, KnownNat n, 1 <= n) => Torsor (Cyclic n) (CyclicEnum a)

Defined in Data.Group.Cyclic

Methods

(<--) :: CyclicEnum a -> CyclicEnum a -> Cyclic n

(-->) :: CyclicEnum a -> CyclicEnum a -> Cyclic n

(Enum a, Bounded a, KnownNat n) => Act (Cyclic n) (CyclicEnum a)

Defined in Data.Group.Cyclic

Methods

(•) :: Cyclic n -> CyclicEnum a -> CyclicEnum a

act :: Cyclic n -> CyclicEnum a -> CyclicEnum a

type Rep (CyclicEnum a)

Defined in Data.Group.Cyclic

type Rep (CyclicEnum a) = D1 ('MetaData "CyclicEnum" "Data.Group.Cyclic" "acts-0.1.0.0-inplace" 'True) (C1 ('MetaCons "CyclicEnum" 'PrefixI 'True) (S1 ('MetaSel ('Just "getCyclicEnum") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a)))

type Rep1 CyclicEnum

Defined in Data.Group.Cyclic

type Rep1 CyclicEnum = D1 ('MetaData "CyclicEnum" "Data.Group.Cyclic" "acts-0.1.0.0-inplace" 'True) (C1 ('MetaCons "CyclicEnum" 'PrefixI 'True) (S1 ('MetaSel ('Just "getCyclicEnum") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) Par1))

pattern Involution :: Cyclic 2

Nontrivial element of cyclic group of order 2.

involution :: Act (Cyclic 2) x => x -> x

Act by an involution.

rootOfUnity :: forall a n. (KnownNat n, Floating a) => Cyclic n -> Complex a

Natural complex representations of finite cyclic groups as roots of unity.