SubHask.Algebra.Group

This module contains most of the math types not directly related to linear algebra

FIXME: there is probably a better name for this

Documentation

newtype NonNegative t

Constructors

NonNegative

Fields

unNonNegative :: t

Instances

Read t0 => Read (NonNegative t)

Show t0 => Show (NonNegative t)

Arbitrary t0 => Arbitrary (NonNegative t)

NFData t0 => NFData (NonNegative t)

IsMutable t0 => IsMutable (NonNegative t)

(Metric t0, HasScalar t0, Eq_ t0, Boolean (Logic t0), (~) * (Logic (Scalar t0)) (Logic t0)) => Metric (NonNegative t)

(Rig t0, Monoid t0, Rg t0) => Rig (NonNegative t)

(Rg t0, Abelian t0, Monoid t0) => Rg (NonNegative t)

(Abelian t0, Semigroup t0) => Abelian (NonNegative t)

(Ord t, Group t) => Cancellative (NonNegative t)

(Monoid t0, Semigroup t0) => Monoid (NonNegative t)

(Semigroup t0, IsMutable t0) => Semigroup (NonNegative t)

(Boolean t0, Complemented t0, Heyting t0) => Boolean (NonNegative t)

(Heyting t0, Bounded t0) => Heyting (NonNegative t)

(Complemented t0, Bounded t0) => Complemented (NonNegative t)

(Bounded t0, Lattice_ t0, MinBound_ t0) => Bounded (NonNegative t)

(Ord_ t0, Lattice_ t0) => Ord_ (NonNegative t)

(Graded t0, Lattice t0) => Graded (NonNegative t)

(Enum t0, Graded t0, Ord_ t0) => Enum (NonNegative t)

(Lattice_ t0, POrd_ t0) => Lattice_ (NonNegative t)

(MinBound_ t0, POrd_ t0) => MinBound_ (NonNegative t)

(POrd_ t0, Eq_ t0) => POrd_ (NonNegative t)

Eq_ t0 => Eq_ (NonNegative t)

data Mutable m (NonNegative t0) = Mutable_NonNegative (Mutable m t)

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Elem (NonNegative t0) = Elem t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Scalar (NonNegative t0) = Scalar t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Actor (NonNegative t0) = Actor t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

type Logic (NonNegative t0) = Logic t0

class Quotient a b where

Maps members of an equivalence class into the "canonical" element.

Methods

mkQuotient :: a -> a / b

Instances

KnownNat n => Quotient Nat Int n

KnownNat n => Quotient Nat Integer n

newtype a / b

The type of equivalence classes created by a mod b.

Constructors

Mod a

Instances

KnownNat n => FiniteType (Z n)

Eq a0 => Eq ((/) k a b)

(Ord a0, Eq a0) => Ord ((/) k a b)

Read a0 => Read ((/) k a b)

Show a0 => Show ((/) k a b)

(Quotient k a b, Arbitrary a) => Arbitrary ((/) k a b)

NFData a0 => NFData ((/) k a b)

IsMutable a0 => IsMutable ((/) k a b)

(Module a, Quotient k a b) => Module ((/) k a b)

(Ring a, Quotient k a b) => Ring ((/) k a b)

(Rig a, Quotient k a b) => Rig ((/) k a b)

(Rg a, Quotient k a b) => Rg ((/) k a b)

(Abelian a, Quotient k a b) => Abelian ((/) k a b)

(Group a, Quotient k a b) => Group ((/) k a b)

(Cancellative a, Quotient k a b) => Cancellative ((/) k a b)

(Monoid a, Quotient k a b) => Monoid ((/) k a b)

(Semigroup a, Quotient k a b) => Semigroup ((/) k a b)

Eq_ a0 => Eq_ ((/) k a b)

type (><) * k ((/) k1 a b) c = (/) k1 ((><) * k a c) b

data Mutable m ((/) k a0 b0) = Mutable_Mod (Mutable m a)

type Order (Z n) = n

type Elem ((/) k a0 b0) = Elem a0

type Elem ((/) k a0 b0) = Elem a0

type Elem ((/) k a0 b0) = Elem a0

type Elem ((/) k a0 b0) = Elem a0

type Elem ((/) k a0 b0) = Elem a0

type Elem ((/) k a0 b0) = Elem a0

type Scalar ((/) k a0 b0) = Scalar a0

type Scalar ((/) k a0 b0) = Scalar a0

type Scalar ((/) k a0 b0) = Scalar a0

type Scalar ((/) k a0 b0) = Scalar a0

type Scalar ((/) k a0 b0) = Scalar a0

type Scalar ((/) k a0 b0) = Scalar a0

type Actor ((/) k a0 b0) = Actor a0

type Actor ((/) k a0 b0) = Actor a0

type Actor ((/) k a0 b0) = Actor a0

type Actor ((/) k a0 b0) = Actor a0

type Actor ((/) k a0 b0) = Actor a0

type Actor ((/) k a0 b0) = Actor a0

type Logic ((/) k a0 b0) = Logic a0

type Logic ((/) k a0 b0) = Logic a0

type Logic ((/) k a0 b0) = Logic a0

type Logic ((/) k a0 b0) = Logic a0

type Logic ((/) k a0 b0) = Logic a0

type Logic ((/) k a0 b0) = Logic a0

type Z n = Integer / n

The type of integers modulo n

extendedEuclid :: (Eq t, Integral t) => t -> t -> (t, t, t, t, t, t)

Extended Euclid's algorithm is used to calculate inverses in modular arithmetic

newtype Galois p k

Galois p k is the type of integers modulo p^k, where p is prime. All finite fields have this form.

See wikipedia https://en.wikipedia.org/wiki/Finite_field for more details.

FIXME: Many arithmetic operations over Galois Fields can be implemented more efficiently than the standard operations. See http://en.wikipedia.org/wiki/Finite_field_arithmetic.

Constructors

Galois (Z (p ^ k))

Instances

Read (Z ((^) p0 k0)) => Read (Galois p k)

Show (Z ((^) p0 k0)) => Show (Galois p k)

Arbitrary (Z ((^) p0 k0)) => Arbitrary (Galois p k)

NFData (Z ((^) p0 k0)) => NFData (Galois p k)

IsMutable (Galois p k)

KnownNat ((^) p k) => Module (Galois p k)

(Prime p, KnownNat ((^) p k)) => Field (Galois p k)

(Ring (Z ((^) p0 k0)), Rng (Z ((^) p0 k0)), Rig (Z ((^) p0 k0))) => Ring (Galois p k)

(Rig (Z ((^) p0 k0)), Monoid (Z ((^) p0 k0)), Rg (Z ((^) p0 k0))) => Rig (Galois p k)

(Rg (Z ((^) p0 k0)), Abelian (Z ((^) p0 k0)), Monoid (Z ((^) p0 k0))) => Rg (Galois p k)

(Abelian (Z ((^) p0 k0)), Semigroup (Z ((^) p0 k0))) => Abelian (Galois p k)

(Group (Z ((^) p0 k0)), Cancellative (Z ((^) p0 k0)), Monoid (Z ((^) p0 k0))) => Group (Galois p k)

(Cancellative (Z ((^) p0 k0)), Semigroup (Z ((^) p0 k0))) => Cancellative (Galois p k)

(Monoid (Z ((^) p0 k0)), Semigroup (Z ((^) p0 k0))) => Monoid (Galois p k)

(Semigroup (Z ((^) p0 k0)), IsMutable (Z ((^) p0 k0))) => Semigroup (Galois p k)

Eq_ (Z ((^) p0 k0)) => Eq_ (Galois p k)

type (><) * * (Galois p k) Integer = Galois p k

data Mutable m (Galois p0 k0) = Mutable_Galois (Mutable m (Z ((^) p k)))

type Elem (Galois p0 k0) = Elem (Z ((^) p0 k0))

type Elem (Galois p0 k0) = Elem (Z ((^) p0 k0))

type Elem (Galois p0 k0) = Elem (Z ((^) p0 k0))

type Elem (Galois p0 k0) = Elem (Z ((^) p0 k0))

type Elem (Galois p0 k0) = Elem (Z ((^) p0 k0))

type Elem (Galois p0 k0) = Elem (Z ((^) p0 k0))

type Elem (Galois p0 k0) = Elem (Z ((^) p0 k0))

type Elem (Galois p0 k0) = Elem (Z ((^) p0 k0))

type Elem (Galois p0 k0) = Elem (Z ((^) p0 k0))

type Elem (Galois p0 k0) = Elem (Z ((^) p0 k0))

type Elem (Galois p0 k0) = Elem (Z ((^) p0 k0))

type Elem (Galois p0 k0) = Elem (Z ((^) p0 k0))

type Elem (Galois p0 k0) = Elem (Z ((^) p0 k0))

type Scalar (Galois p0 k0) = Scalar (Z ((^) p0 k0))

type Scalar (Galois p0 k0) = Scalar (Z ((^) p0 k0))

type Scalar (Galois p0 k0) = Scalar (Z ((^) p0 k0))

type Scalar (Galois p0 k0) = Scalar (Z ((^) p0 k0))

type Scalar (Galois p0 k0) = Scalar (Z ((^) p0 k0))

type Scalar (Galois p0 k0) = Scalar (Z ((^) p0 k0))

type Scalar (Galois p0 k0) = Scalar (Z ((^) p0 k0))

type Scalar (Galois p0 k0) = Scalar (Z ((^) p0 k0))

type Scalar (Galois p0 k0) = Scalar (Z ((^) p0 k0))

type Scalar (Galois p0 k0) = Scalar (Z ((^) p0 k0))

type Scalar (Galois p0 k0) = Scalar (Z ((^) p0 k0))

type Scalar (Galois p0 k0) = Scalar (Z ((^) p0 k0))

type Scalar (Galois p0 k0) = Scalar (Z ((^) p0 k0))

type Actor (Galois p0 k0) = Actor (Z ((^) p0 k0))

type Actor (Galois p0 k0) = Actor (Z ((^) p0 k0))

type Actor (Galois p0 k0) = Actor (Z ((^) p0 k0))

type Actor (Galois p0 k0) = Actor (Z ((^) p0 k0))

type Actor (Galois p0 k0) = Actor (Z ((^) p0 k0))

type Actor (Galois p0 k0) = Actor (Z ((^) p0 k0))

type Actor (Galois p0 k0) = Actor (Z ((^) p0 k0))

type Actor (Galois p0 k0) = Actor (Z ((^) p0 k0))

type Actor (Galois p0 k0) = Actor (Z ((^) p0 k0))

type Actor (Galois p0 k0) = Actor (Z ((^) p0 k0))

type Actor (Galois p0 k0) = Actor (Z ((^) p0 k0))

type Actor (Galois p0 k0) = Actor (Z ((^) p0 k0))

type Actor (Galois p0 k0) = Actor (Z ((^) p0 k0))

type Logic (Galois p0 k0) = Logic (Z ((^) p0 k0))

type Logic (Galois p0 k0) = Logic (Z ((^) p0 k0))

type Logic (Galois p0 k0) = Logic (Z ((^) p0 k0))

type Logic (Galois p0 k0) = Logic (Z ((^) p0 k0))

type Logic (Galois p0 k0) = Logic (Z ((^) p0 k0))

type Logic (Galois p0 k0) = Logic (Z ((^) p0 k0))

type Logic (Galois p0 k0) = Logic (Z ((^) p0 k0))

type Logic (Galois p0 k0) = Logic (Z ((^) p0 k0))

type Logic (Galois p0 k0) = Logic (Z ((^) p0 k0))

type Logic (Galois p0 k0) = Logic (Z ((^) p0 k0))

type Logic (Galois p0 k0) = Logic (Z ((^) p0 k0))

type Logic (Galois p0 k0) = Logic (Z ((^) p0 k0))

type Logic (Galois p0 k0) = Logic (Z ((^) p0 k0))

class Prime n

Instances

Prime 1

Prime 2

Prime 3

Prime 5

Prime 7

Prime 11

Prime 13

Prime 17

Prime 19

Prime 23

data GrothendieckGroup g where

The symmetric group is one of the simplest and best studied finite groups. It is efficiently implemented as a "BijectiveT SparseFunction (Z n) (Z n)". See https://en.wikipedia.org/wiki/Symmetric_group

The GrothendieckGroup is a general way to construct groups from cancellative semigroups.

FIXME: How should this be related to the Ratio type?

See wikipedia for more details.

Constructors

GrotheindieckGroup :: Cancellative g => g -> GrothendieckGroup g

newtype VedicSquare n

The Vedic Square always forms a monoid, and sometimes forms a group depending on the value of "n". (The type system isn't powerful enough to encode these special cases.)

See wikipedia for more detail.

Constructors

VedicSquare (Z n)

Instances

Read (Z n0) => Read (VedicSquare n)

Show (Z n0) => Show (VedicSquare n)

Arbitrary (Z n0) => Arbitrary (VedicSquare n)

NFData (Z n0) => NFData (VedicSquare n)

IsMutable (VedicSquare n)

KnownNat n => Monoid (VedicSquare n)

KnownNat n => Semigroup (VedicSquare n)

Eq_ (Z n0) => Eq_ (VedicSquare n)

data Mutable m (VedicSquare n0) = Mutable_VedicSquare (Mutable m (Z n))

type Elem (VedicSquare n0) = Elem (Z n0)

type Elem (VedicSquare n0) = Elem (Z n0)

type Elem (VedicSquare n0) = Elem (Z n0)

type Elem (VedicSquare n0) = Elem (Z n0)

type Elem (VedicSquare n0) = Elem (Z n0)

type Scalar (VedicSquare n0) = Scalar (Z n0)

type Scalar (VedicSquare n0) = Scalar (Z n0)

type Scalar (VedicSquare n0) = Scalar (Z n0)

type Scalar (VedicSquare n0) = Scalar (Z n0)

type Scalar (VedicSquare n0) = Scalar (Z n0)

type Actor (VedicSquare n0) = Actor (Z n0)

type Actor (VedicSquare n0) = Actor (Z n0)

type Actor (VedicSquare n0) = Actor (Z n0)

type Actor (VedicSquare n0) = Actor (Z n0)

type Actor (VedicSquare n0) = Actor (Z n0)

type Logic (VedicSquare n0) = Logic (Z n0)

type Logic (VedicSquare n0) = Logic (Z n0)

type Logic (VedicSquare n0) = Logic (Z n0)

type Logic (VedicSquare n0) = Logic (Z n0)

type Logic (VedicSquare n0) = Logic (Z n0)