Algebra.Prelude.Core

Documentation

(%) :: (IsPolynomial poly, Division (Coefficient poly)) => Coefficient poly -> Coefficient poly -> poly infixl 7

newtype Scalar r

Scalar r provides almost the same type-instances as r, but it can also behave as a Module over r itself.

Constructors

Scalar

Fields

• runScalar :: r

Instances

LeftModule Integer r => LeftModule Integer (Scalar r)

Methods

(.*) :: Integer -> Scalar r -> Scalar r

Semiring r => LeftModule r (Scalar r)

Methods

(.*) :: r -> Scalar r -> Scalar r

LeftModule Natural r => LeftModule Natural (Scalar r)

Methods

(.*) :: Natural -> Scalar r -> Scalar r

RightModule Integer r => RightModule Integer (Scalar r)

Methods

(*.) :: Scalar r -> Integer -> Scalar r

Semiring r => RightModule r (Scalar r)

Methods

(*.) :: Scalar r -> r -> Scalar r

RightModule Natural r => RightModule Natural (Scalar r)

Methods

(*.) :: Scalar r -> Natural -> Scalar r

Enum r => Enum (Scalar r)

Methods

succ :: Scalar r -> Scalar r

pred :: Scalar r -> Scalar r

toEnum :: Int -> Scalar r

fromEnum :: Scalar r -> Int

enumFrom :: Scalar r -> [Scalar r]

enumFromThen :: Scalar r -> Scalar r -> [Scalar r]

enumFromTo :: Scalar r -> Scalar r -> [Scalar r]

enumFromThenTo :: Scalar r -> Scalar r -> Scalar r -> [Scalar r]

Eq r => Eq (Scalar r)

Methods

(==) :: Scalar r -> Scalar r -> Bool

(/=) :: Scalar r -> Scalar r -> Bool

Fractional r => Fractional (Scalar r)

Methods

(/) :: Scalar r -> Scalar r -> Scalar r

recip :: Scalar r -> Scalar r

fromRational :: Rational -> Scalar r

Integral r => Integral (Scalar r)

Methods

quot :: Scalar r -> Scalar r -> Scalar r

rem :: Scalar r -> Scalar r -> Scalar r

div :: Scalar r -> Scalar r -> Scalar r

mod :: Scalar r -> Scalar r -> Scalar r

quotRem :: Scalar r -> Scalar r -> (Scalar r, Scalar r)

divMod :: Scalar r -> Scalar r -> (Scalar r, Scalar r)

toInteger :: Scalar r -> Integer

Num r => Num (Scalar r)

Methods

(+) :: Scalar r -> Scalar r -> Scalar r

(-) :: Scalar r -> Scalar r -> Scalar r

(*) :: Scalar r -> Scalar r -> Scalar r

negate :: Scalar r -> Scalar r

abs :: Scalar r -> Scalar r

signum :: Scalar r -> Scalar r

fromInteger :: Integer -> Scalar r

Ord r => Ord (Scalar r)

Methods

compare :: Scalar r -> Scalar r -> Ordering

(<) :: Scalar r -> Scalar r -> Bool

(<=) :: Scalar r -> Scalar r -> Bool

(>) :: Scalar r -> Scalar r -> Bool

(>=) :: Scalar r -> Scalar r -> Bool

max :: Scalar r -> Scalar r -> Scalar r

min :: Scalar r -> Scalar r -> Scalar r

Read r => Read (Scalar r)

Methods

readsPrec :: Int -> ReadS (Scalar r)

readList :: ReadS [Scalar r]

readPrec :: ReadPrec (Scalar r)

readListPrec :: ReadPrec [Scalar r]

Real r => Real (Scalar r)

Methods

toRational :: Scalar r -> Rational

Show r => Show (Scalar r)

Methods

showsPrec :: Int -> Scalar r -> ShowS

show :: Scalar r -> String

showList :: [Scalar r] -> ShowS

Commutative r => Commutative (Scalar r)

UnitNormalForm r => UnitNormalForm (Scalar r)

Methods

splitUnit :: Scalar r -> (Scalar r, Scalar r)

Ring r => Ring (Scalar r)

Methods

fromInteger :: Integer -> Scalar r

Rig r => Rig (Scalar r)

Methods

fromNatural :: Natural -> Scalar r

DecidableUnits r => DecidableUnits (Scalar r)

Methods

recipUnit :: Scalar r -> Maybe (Scalar r)

isUnit :: Scalar r -> Bool

(^?) :: Integral n => Scalar r -> n -> Maybe (Scalar r)

DecidableAssociates r => DecidableAssociates (Scalar r)

Methods

isAssociate :: Scalar r -> Scalar r -> Bool

Division r => Division (Scalar r)

Methods

recip :: Scalar r -> Scalar r

(/) :: Scalar r -> Scalar r -> Scalar r

(\\) :: Scalar r -> Scalar r -> Scalar r

(^) :: Integral n => Scalar r -> n -> Scalar r

Unital r => Unital (Scalar r)

Methods

one :: Scalar r

pow :: Scalar r -> Natural -> Scalar r

productWith :: Foldable f => (a -> Scalar r) -> f a -> Scalar r

Group r => Group (Scalar r)

Methods

(-) :: Scalar r -> Scalar r -> Scalar r

negate :: Scalar r -> Scalar r

subtract :: Scalar r -> Scalar r -> Scalar r

times :: Integral n => n -> Scalar r -> Scalar r

Multiplicative r => Multiplicative (Scalar r)

Methods

(*) :: Scalar r -> Scalar r -> Scalar r

pow1p :: Scalar r -> Natural -> Scalar r

productWith1 :: Foldable1 f => (a -> Scalar r) -> f a -> Scalar r

Semiring r => Semiring (Scalar r)

Monoidal r => Monoidal (Scalar r)

Methods

zero :: Scalar r

sinnum :: Natural -> Scalar r -> Scalar r

sumWith :: Foldable f => (a -> Scalar r) -> f a -> Scalar r

Additive r => Additive (Scalar r)

Methods

(+) :: Scalar r -> Scalar r -> Scalar r

sinnum1p :: Natural -> Scalar r -> Scalar r

sumWith1 :: Foldable1 f => (a -> Scalar r) -> f a -> Scalar r

Abelian r => Abelian (Scalar r)

Normed r => Normed (Scalar r)

Associated Types

type Norm (Scalar r) :: *

Methods

norm :: Scalar r -> Norm (Scalar r)

liftNorm :: Norm (Scalar r) -> Scalar r

LeftModule (Scalar (Fraction Integer)) Algebraic

Methods

(.*) :: Scalar (Fraction Integer) -> Algebraic -> Algebraic

RightModule (Scalar (Fraction Integer)) Algebraic

Methods

(*.) :: Algebraic -> Scalar (Fraction Integer) -> Algebraic

Semiring r => LeftModule (Scalar r) (Scalar r)

Methods

(.*) :: Scalar r -> Scalar r -> Scalar r

(DecidableZero r, Semiring r) => LeftModule (Scalar r) (Unipol r)

Methods

(.*) :: Scalar r -> Unipol r -> Unipol r

(DecidableZero r, Semiring r, Multiplicative r) => LeftModule (Scalar r) (Matrix r)

Methods

(.*) :: Scalar r -> Matrix r -> Matrix r

Semiring r => RightModule (Scalar r) (Scalar r)

Methods

(*.) :: Scalar r -> Scalar r -> Scalar r

(DecidableZero r, Semiring r) => RightModule (Scalar r) (Unipol r)

Methods

(*.) :: Unipol r -> Scalar r -> Unipol r

(DecidableZero r, Semiring r, Multiplicative r) => RightModule (Scalar r) (Matrix r)

Methods

(*.) :: Matrix r -> Scalar r -> Matrix r

(Wraps vars poly, LeftModule (Scalar r) poly) => LeftModule (Scalar r) (LabPolynomial poly vars)

Methods

(.*) :: Scalar r -> LabPolynomial poly vars -> LabPolynomial poly vars

(Wraps vars poly, RightModule (Scalar r) poly) => RightModule (Scalar r) (LabPolynomial poly vars)

Methods

(*.) :: LabPolynomial poly vars -> Scalar r -> LabPolynomial poly vars

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly, LeftModule (Scalar r) poly) => LeftModule (Scalar r) (PadPolyL n ord poly)

Methods

(.*) :: Scalar r -> PadPolyL n ord poly -> PadPolyL n ord poly

((~) * r (Coefficient poly), Field (Coefficient poly), IsOrderedPolynomial poly) => LeftModule (Scalar r) (Quotient k poly ideal)

Methods

(.*) :: Scalar r -> Quotient k poly ideal -> Quotient k poly ideal

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly, RightModule (Scalar r) poly) => RightModule (Scalar r) (PadPolyL n ord poly)

Methods

(*.) :: PadPolyL n ord poly -> Scalar r -> PadPolyL n ord poly

((~) * r (Coefficient poly), IsOrderedPolynomial poly) => RightModule (Scalar r) (Quotient k poly ideal)

Methods

(*.) :: Quotient k poly ideal -> Scalar r -> Quotient k poly ideal

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => LeftModule (Scalar r) (OrderedPolynomial * r order n)

Methods

(.*) :: Scalar r -> OrderedPolynomial * r order n -> OrderedPolynomial * r order n

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => RightModule (Scalar r) (OrderedPolynomial * r order n)

Methods

(*.) :: OrderedPolynomial * r order n -> Scalar r -> OrderedPolynomial * r order n

type Norm (Scalar r)

type Norm (Scalar r) = Norm r

(.*.) :: Module (Scalar r) m => r -> m -> m infixr 8

od :: QuasiQuoter

Quasiquoter for ordinal indexed by GHC's built-n Nat.

Since 0.7.0.0

type Ordinal n = Ordinal Nat n

Set-theoretic (finite) ordinals:

n = {0, 1, ..., n-1}

So, Ordinal n has exactly n inhabitants. So especially Ordinal 0 is isomorphic to Void. This module exports a variant of polymorphic Ordinal specialised to GHC's builtin numeral Nat.

Since 0.7.0.0

enumOrdinal :: Sing Nat n -> [Ordinal n]

Enumerate all Ordinals less than n.

Since 0.7.0.0

logBase2 :: Int -> Int

ceilingLogBase2 :: Int -> Int

module AlgebraicPrelude

module Algebra.Ring.Polynomial.Monomial

module Algebra.Ring.Polynomial.Class

type Polynomial r = OrderedPolynomial r Grevlex

transformMonomial :: (IsMonomialOrder m o, CoeffRing k, KnownNat m) => (Monomial n -> Monomial m) -> OrderedPolynomial k o n -> OrderedPolynomial k o m

castPolynomial :: (CoeffRing r, KnownNat n, KnownNat m, IsMonomialOrder n o, IsMonomialOrder m o') => OrderedPolynomial r o n -> OrderedPolynomial r o' m

changeOrder :: (CoeffRing k, Eq (Monomial n), IsMonomialOrder n o, IsMonomialOrder n o', KnownNat n) => o' -> OrderedPolynomial k o n -> OrderedPolynomial k o' n

changeOrderProxy :: (CoeffRing k, Eq (Monomial n), IsMonomialOrder n o, IsMonomialOrder n o', KnownNat n) => Proxy o' -> OrderedPolynomial k o n -> OrderedPolynomial k o' n

scastPolynomial :: (IsMonomialOrder n o, IsMonomialOrder m o', KnownNat m, CoeffRing r, KnownNat n) => SNat m -> OrderedPolynomial r o n -> OrderedPolynomial r o' m

newtype OrderedPolynomial r order n

n-ary polynomial ring over some noetherian ring R.

Constructors

Polynomial

Fields

• _terms :: Map (OrderedMonomial order n) r

Instances

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => LeftModule Integer (OrderedPolynomial * r order n)

Methods

(.*) :: Integer -> OrderedPolynomial * r order n -> OrderedPolynomial * r order n

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => LeftModule Natural (OrderedPolynomial * r order n)

Methods

(.*) :: Natural -> OrderedPolynomial * r order n -> OrderedPolynomial * r order n

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => RightModule Integer (OrderedPolynomial * r order n)

Methods

(*.) :: OrderedPolynomial * r order n -> Integer -> OrderedPolynomial * r order n

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => RightModule Natural (OrderedPolynomial * r order n)

Methods

(*.) :: OrderedPolynomial * r order n -> Natural -> OrderedPolynomial * r order n

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => LeftModule (Scalar r) (OrderedPolynomial * r order n)

Methods

(.*) :: Scalar r -> OrderedPolynomial * r order n -> OrderedPolynomial * r order n

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => RightModule (Scalar r) (OrderedPolynomial * r order n)

Methods

(*.) :: OrderedPolynomial * r order n -> Scalar r -> OrderedPolynomial * r order n

Eq r => Eq (OrderedPolynomial k r order n)

Methods

(==) :: OrderedPolynomial k r order n -> OrderedPolynomial k r order n -> Bool

(/=) :: OrderedPolynomial k r order n -> OrderedPolynomial k r order n -> Bool

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Num (OrderedPolynomial * r order n)

We provide Num instance to use trivial injection R into R[X]. Do not use signum or abs.

Methods

(+) :: OrderedPolynomial * r order n -> OrderedPolynomial * r order n -> OrderedPolynomial * r order n

(-) :: OrderedPolynomial * r order n -> OrderedPolynomial * r order n -> OrderedPolynomial * r order n

(*) :: OrderedPolynomial * r order n -> OrderedPolynomial * r order n -> OrderedPolynomial * r order n

negate :: OrderedPolynomial * r order n -> OrderedPolynomial * r order n

abs :: OrderedPolynomial * r order n -> OrderedPolynomial * r order n

signum :: OrderedPolynomial * r order n -> OrderedPolynomial * r order n

fromInteger :: Integer -> OrderedPolynomial * r order n

(CoeffRing r, IsOrder n ord, Ord r) => Ord (OrderedPolynomial * r ord n)

Methods

compare :: OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n -> Ordering

(<) :: OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n -> Bool

(<=) :: OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n -> Bool

(>) :: OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n -> Bool

(>=) :: OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n -> Bool

max :: OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n

min :: OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n

(KnownNat n, CoeffRing r, IsMonomialOrder n order, PrettyCoeff r) => Show (OrderedPolynomial * r order n)

Methods

showsPrec :: Int -> OrderedPolynomial * r order n -> ShowS

show :: OrderedPolynomial * r order n -> String

showList :: [OrderedPolynomial * r order n] -> ShowS

(Eq r, DecidableUnits r, DecidableZero r, KnownNat n, Field r, IsMonomialOrder n ord, ZeroProductSemiring r) => IntegralDomain (OrderedPolynomial * r ord n)

Methods

divides :: OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n -> Bool

maybeQuot :: OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n -> Maybe (OrderedPolynomial * r ord n)

(Eq r, DecidableUnits r, DecidableZero r, Field r, IsMonomialOrder 1 ord, ZeroProductSemiring r) => IntegralDomain (OrderedPolynomial * r ord 1)

Methods

divides :: OrderedPolynomial * r ord 1 -> OrderedPolynomial * r ord 1 -> Bool

maybeQuot :: OrderedPolynomial * r ord 1 -> OrderedPolynomial * r ord 1 -> Maybe (OrderedPolynomial * r ord 1)

(Eq r, DecidableUnits r, DecidableZero r, Field r, IsMonomialOrder 1 ord, ZeroProductSemiring r) => GCDDomain (OrderedPolynomial * r ord 1)

Methods

gcd :: OrderedPolynomial * r ord 1 -> OrderedPolynomial * r ord 1 -> OrderedPolynomial * r ord 1

reduceFraction :: OrderedPolynomial * r ord 1 -> OrderedPolynomial * r ord 1 -> (OrderedPolynomial * r ord 1, OrderedPolynomial * r ord 1)

lcm :: OrderedPolynomial * r ord 1 -> OrderedPolynomial * r ord 1 -> OrderedPolynomial * r ord 1

(Eq r, DecidableUnits r, DecidableZero r, Field r, IsMonomialOrder 1 ord, ZeroProductSemiring r) => UFD (OrderedPolynomial * r ord 1)

(Eq r, DecidableUnits r, DecidableZero r, Field r, IsMonomialOrder 1 ord, ZeroProductSemiring r) => PID (OrderedPolynomial * r ord 1)

Methods

egcd :: OrderedPolynomial * r ord 1 -> OrderedPolynomial * r ord 1 -> (OrderedPolynomial * r ord 1, OrderedPolynomial * r ord 1, OrderedPolynomial * r ord 1)

(Eq r, DecidableUnits r, DecidableZero r, Field r, IsMonomialOrder 1 ord, ZeroProductSemiring r) => Euclidean (OrderedPolynomial * r ord 1)

Methods

degree :: OrderedPolynomial * r ord 1 -> Maybe Natural

divide :: OrderedPolynomial * r ord 1 -> OrderedPolynomial * r ord 1 -> (OrderedPolynomial * r ord 1, OrderedPolynomial * r ord 1)

quot :: OrderedPolynomial * r ord 1 -> OrderedPolynomial * r ord 1 -> OrderedPolynomial * r ord 1

rem :: OrderedPolynomial * r ord 1 -> OrderedPolynomial * r ord 1 -> OrderedPolynomial * r ord 1

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Commutative (OrderedPolynomial * r order n)

(Eq r, Euclidean r, KnownNat n, IsMonomialOrder n ord) => UnitNormalForm (OrderedPolynomial * r ord n)

Methods

splitUnit :: OrderedPolynomial * r ord n -> (OrderedPolynomial * r ord n, OrderedPolynomial * r ord n)

(Eq r, DecidableUnits r, DecidableZero r, KnownNat n, Field r, IsMonomialOrder n ord, ZeroProductSemiring r) => ZeroProductSemiring (OrderedPolynomial * r ord n)

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Ring (OrderedPolynomial * r order n)

Methods

fromInteger :: Integer -> OrderedPolynomial * r order n

(IsMonomialOrder n ord, Characteristic r, KnownNat n, CoeffRing r) => Characteristic (OrderedPolynomial * r ord n)

Methods

char :: proxy (OrderedPolynomial * r ord n) -> Natural

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Rig (OrderedPolynomial * r order n)

Methods

fromNatural :: Natural -> OrderedPolynomial * r order n

(CoeffRing r, KnownNat n, IsMonomialOrder n ord) => DecidableZero (OrderedPolynomial * r ord n)

Methods

isZero :: OrderedPolynomial * r ord n -> Bool

(CoeffRing r, IsMonomialOrder n ord, DecidableUnits r, KnownNat n) => DecidableUnits (OrderedPolynomial * r ord n)

Methods

recipUnit :: OrderedPolynomial * r ord n -> Maybe (OrderedPolynomial * r ord n)

isUnit :: OrderedPolynomial * r ord n -> Bool

(^?) :: Integral n => OrderedPolynomial * r ord n -> n -> Maybe (OrderedPolynomial * r ord n)

(Eq r, KnownNat n, Euclidean r, IsMonomialOrder n ord) => DecidableAssociates (OrderedPolynomial * r ord n)

Methods

isAssociate :: OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n -> Bool

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Unital (OrderedPolynomial * r order n)

Methods

one :: OrderedPolynomial * r order n

pow :: OrderedPolynomial * r order n -> Natural -> OrderedPolynomial * r order n

productWith :: Foldable f => (a -> OrderedPolynomial * r order n) -> f a -> OrderedPolynomial * r order n

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Group (OrderedPolynomial * r order n)

Methods

(-) :: OrderedPolynomial * r order n -> OrderedPolynomial * r order n -> OrderedPolynomial * r order n

negate :: OrderedPolynomial * r order n -> OrderedPolynomial * r order n

subtract :: OrderedPolynomial * r order n -> OrderedPolynomial * r order n -> OrderedPolynomial * r order n

times :: Integral n => n -> OrderedPolynomial * r order n -> OrderedPolynomial * r order n

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Multiplicative (OrderedPolynomial * r order n)

Methods

(*) :: OrderedPolynomial * r order n -> OrderedPolynomial * r order n -> OrderedPolynomial * r order n

pow1p :: OrderedPolynomial * r order n -> Natural -> OrderedPolynomial * r order n

productWith1 :: Foldable1 f => (a -> OrderedPolynomial * r order n) -> f a -> OrderedPolynomial * r order n

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Semiring (OrderedPolynomial * r order n)

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Monoidal (OrderedPolynomial * r order n)

Methods

zero :: OrderedPolynomial * r order n

sinnum :: Natural -> OrderedPolynomial * r order n -> OrderedPolynomial * r order n

sumWith :: Foldable f => (a -> OrderedPolynomial * r order n) -> f a -> OrderedPolynomial * r order n

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Additive (OrderedPolynomial * r order n)

Methods

(+) :: OrderedPolynomial * r order n -> OrderedPolynomial * r order n -> OrderedPolynomial * r order n

sinnum1p :: Natural -> OrderedPolynomial * r order n -> OrderedPolynomial * r order n

sumWith1 :: Foldable1 f => (a -> OrderedPolynomial * r order n) -> f a -> OrderedPolynomial * r order n

(IsMonomialOrder n order, CoeffRing r, KnownNat n) => Abelian (OrderedPolynomial * r order n)

Hashable r => Hashable (OrderedPolynomial k r ord n)

Methods

hashWithSalt :: Int -> OrderedPolynomial k r ord n -> Int

hash :: OrderedPolynomial k r ord n -> Int

NFData r => NFData (OrderedPolynomial k r order n)

Methods

rnf :: OrderedPolynomial k r order n -> ()

(KnownNat n, CoeffRing r, IsMonomialOrder n order) => Wrapped (OrderedPolynomial * r order n)

Associated Types

type Unwrapped (OrderedPolynomial * r order n) :: *

Methods

_Wrapped' :: Iso' (OrderedPolynomial * r order n) (Unwrapped (OrderedPolynomial * r order n))

(KnownNat n, CoeffRing r, IsMonomialOrder n ord) => IsOrderedPolynomial (OrderedPolynomial * r ord n)

Associated Types

type MOrder (OrderedPolynomial * r ord n) :: *

Methods

coeff :: OrderedMonomial * (MOrder (OrderedPolynomial * r ord n)) (Arity (OrderedPolynomial * r ord n)) -> OrderedPolynomial * r ord n -> Coefficient (OrderedPolynomial * r ord n)

terms :: OrderedPolynomial * r ord n -> Map (OrderedMonomial * (MOrder (OrderedPolynomial * r ord n)) (Arity (OrderedPolynomial * r ord n))) (Coefficient (OrderedPolynomial * r ord n))

leadingTerm :: OrderedPolynomial * r ord n -> (Coefficient (OrderedPolynomial * r ord n), OrderedMonomial * (MOrder (OrderedPolynomial * r ord n)) (Arity (OrderedPolynomial * r ord n)))

leadingMonomial :: OrderedPolynomial * r ord n -> OrderedMonomial * (MOrder (OrderedPolynomial * r ord n)) (Arity (OrderedPolynomial * r ord n))

leadingCoeff :: OrderedPolynomial * r ord n -> Coefficient (OrderedPolynomial * r ord n)

orderedMonomials :: OrderedPolynomial * r ord n -> Set (OrderedMonomial * (MOrder (OrderedPolynomial * r ord n)) (Arity (OrderedPolynomial * r ord n)))

fromOrderedMonomial :: OrderedMonomial * (MOrder (OrderedPolynomial * r ord n)) (Arity (OrderedPolynomial * r ord n)) -> OrderedPolynomial * r ord n

toPolynomial :: (Coefficient (OrderedPolynomial * r ord n), OrderedMonomial * (MOrder (OrderedPolynomial * r ord n)) (Arity (OrderedPolynomial * r ord n))) -> OrderedPolynomial * r ord n

polynomial :: Map (OrderedMonomial * (MOrder (OrderedPolynomial * r ord n)) (Arity (OrderedPolynomial * r ord n))) (Coefficient (OrderedPolynomial * r ord n)) -> OrderedPolynomial * r ord n

(>*) :: OrderedMonomial * (MOrder (OrderedPolynomial * r ord n)) (Arity (OrderedPolynomial * r ord n)) -> OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n

(*<) :: OrderedPolynomial * r ord n -> OrderedMonomial * (MOrder (OrderedPolynomial * r ord n)) (Arity (OrderedPolynomial * r ord n)) -> OrderedPolynomial * r ord n

_Terms :: Iso' (OrderedPolynomial * r ord n) (Map (OrderedMonomial * (MOrder (OrderedPolynomial * r ord n)) (Arity (OrderedPolynomial * r ord n))) (Coefficient (OrderedPolynomial * r ord n)))

diff :: Ordinal Nat (Arity (OrderedPolynomial * r ord n)) -> OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n

mapMonomialMonotonic :: (OrderedMonomial * (MOrder (OrderedPolynomial * r ord n)) (Arity (OrderedPolynomial * r ord n)) -> OrderedMonomial * (MOrder (OrderedPolynomial * r ord n)) (Arity (OrderedPolynomial * r ord n))) -> OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n

(KnownNat n, IsMonomialOrder n ord, CoeffRing r) => IsPolynomial (OrderedPolynomial * r ord n)

Associated Types

type Coefficient (OrderedPolynomial * r ord n) :: *

type Arity (OrderedPolynomial * r ord n) :: Nat

Methods

liftMap :: (Module (Scalar (Coefficient (OrderedPolynomial * r ord n))) alg, Ring alg, Commutative alg) => (Ordinal Nat (Arity (OrderedPolynomial * r ord n)) -> alg) -> OrderedPolynomial * r ord n -> alg

subst :: (Ring alg, Commutative alg, Module (Scalar (Coefficient (OrderedPolynomial * r ord n))) alg) => Sized (Arity (OrderedPolynomial * r ord n)) alg -> OrderedPolynomial * r ord n -> alg

substWith :: (Unital r, Monoidal m) => (Coefficient (OrderedPolynomial * r ord n) -> r -> m) -> Sized (Arity (OrderedPolynomial * r ord n)) r -> OrderedPolynomial * r ord n -> m

sArity' :: OrderedPolynomial * r ord n -> SNat (Arity (OrderedPolynomial * r ord n))

sArity :: proxy (OrderedPolynomial * r ord n) -> SNat (Arity (OrderedPolynomial * r ord n))

arity :: proxy (OrderedPolynomial * r ord n) -> Integer

injectCoeff :: Coefficient (OrderedPolynomial * r ord n) -> OrderedPolynomial * r ord n

injectCoeff' :: proxy (OrderedPolynomial * r ord n) -> Coefficient (OrderedPolynomial * r ord n) -> OrderedPolynomial * r ord n

monomials :: OrderedPolynomial * r ord n -> HashSet (Monomial (Arity (OrderedPolynomial * r ord n)))

terms' :: OrderedPolynomial * r ord n -> Map (Monomial (Arity (OrderedPolynomial * r ord n))) (Coefficient (OrderedPolynomial * r ord n))

coeff' :: Monomial (Arity (OrderedPolynomial * r ord n)) -> OrderedPolynomial * r ord n -> Coefficient (OrderedPolynomial * r ord n)

constantTerm :: OrderedPolynomial * r ord n -> Coefficient (OrderedPolynomial * r ord n)

fromMonomial :: Monomial (Arity (OrderedPolynomial * r ord n)) -> OrderedPolynomial * r ord n

toPolynomial' :: (Coefficient (OrderedPolynomial * r ord n), Monomial (Arity (OrderedPolynomial * r ord n))) -> OrderedPolynomial * r ord n

polynomial' :: Map (Monomial (Arity (OrderedPolynomial * r ord n))) (Coefficient (OrderedPolynomial * r ord n)) -> OrderedPolynomial * r ord n

totalDegree' :: OrderedPolynomial * r ord n -> Natural

var :: Ordinal Nat (Arity (OrderedPolynomial * r ord n)) -> OrderedPolynomial * r ord n

mapCoeff' :: (Coefficient (OrderedPolynomial * r ord n) -> Coefficient (OrderedPolynomial * r ord n)) -> OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n

(>|*) :: Monomial (Arity (OrderedPolynomial * r ord n)) -> OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n

(*|<) :: OrderedPolynomial * r ord n -> Monomial (Arity (OrderedPolynomial * r ord n)) -> OrderedPolynomial * r ord n

(!*) :: Coefficient (OrderedPolynomial * r ord n) -> OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n

_Terms' :: Iso' (OrderedPolynomial * r ord n) (Map (Monomial (Arity (OrderedPolynomial * r ord n))) (Coefficient (OrderedPolynomial * r ord n)))

mapMonomial :: (Monomial (Arity (OrderedPolynomial * r ord n)) -> Monomial (Arity (OrderedPolynomial * r ord n))) -> OrderedPolynomial * r ord n -> OrderedPolynomial * r ord n

(KnownNat n, CoeffRing r, IsMonomialOrder n ord, (~) * t (OrderedPolynomial k q ord' m)) => Rewrapped (OrderedPolynomial * r ord n) t

type Unwrapped (OrderedPolynomial * r order n)

type Unwrapped (OrderedPolynomial * r order n) = Map (OrderedMonomial * order n) r

type MOrder (OrderedPolynomial * r ord n)

type MOrder (OrderedPolynomial * r ord n) = ord

type Coefficient (OrderedPolynomial * r ord n)

type Coefficient (OrderedPolynomial * r ord n) = r

type Arity (OrderedPolynomial * r ord n)

type Arity (OrderedPolynomial * r ord n) = n

allVars :: forall k ord n. (IsMonomialOrder n ord, CoeffRing k, KnownNat n) => Sized n (OrderedPolynomial k ord n)

substVar :: (CoeffRing r, KnownNat n, IsMonomialOrder n ord, (1 :<= n) ~ True) => Ordinal n -> OrderedPolynomial r ord n -> OrderedPolynomial r ord n -> OrderedPolynomial r ord n

substVar n f substitutes n-th variable with polynomial f, without changing arity.

homogenize :: forall k ord n. (CoeffRing k, KnownNat n, IsMonomialOrder (n + 1) ord, IsMonomialOrder n ord) => OrderedPolynomial k ord n -> OrderedPolynomial k ord (n + 1)

Calculate the homogenized polynomial of given one, with additional variable is the last variable.

unhomogenize :: forall k ord n. (CoeffRing k, KnownNat n, IsMonomialOrder n ord, IsMonomialOrder (n + 1) ord) => OrderedPolynomial k ord (Succ n) -> OrderedPolynomial k ord n

normalize :: DecidableZero r => OrderedPolynomial r order n -> OrderedPolynomial r order n

varX :: forall r n order. (CoeffRing r, KnownNat n, IsMonomialOrder n order, (0 :< n) ~ True) => OrderedPolynomial r order n

getTerms :: OrderedPolynomial k order n -> [(k, OrderedMonomial order n)]

shiftR :: forall k r n ord. (CoeffRing r, KnownNat n, IsMonomialOrder n ord, IsMonomialOrder (k + n) ord) => SNat k -> OrderedPolynomial r ord n -> OrderedPolynomial r ord (k :+ n)

orderedBy :: OrderedPolynomial k o n -> o -> OrderedPolynomial k o n

mapCoeff :: (KnownNat n, CoeffRing b, IsMonomialOrder n ord) => (a -> b) -> OrderedPolynomial a ord n -> OrderedPolynomial b ord n

reversal :: (CoeffRing k, IsMonomialOrder 1 o) => Int -> OrderedPolynomial k o 1 -> OrderedPolynomial k o 1

padeApprox :: (Field r, DecidableUnits r, CoeffRing r, ZeroProductSemiring r, IsMonomialOrder 1 order) => Natural -> Natural -> OrderedPolynomial r order 1 -> (OrderedPolynomial r order 1, OrderedPolynomial r order 1)

eval :: (CoeffRing r, IsMonomialOrder n order, KnownNat n) => Sized n r -> OrderedPolynomial r order n -> r

Evaluate polynomial at some point.

evalUnivariate :: (CoeffRing b, IsMonomialOrder 1 order) => b -> OrderedPolynomial b order 1 -> b

substUnivariate :: (Module (Scalar r) b, Unital b, CoeffRing r, IsMonomialOrder 1 order) => b -> OrderedPolynomial r order 1 -> b

Substitute univariate polynomial using Horner's rule

minpolRecurrent :: forall k. (Eq k, ZeroProductSemiring k, DecidableUnits k, DecidableZero k, Field k) => Natural -> [k] -> Polynomial k 1

class IsOrder n ordering where

Class to lookup ordering from its (type-level) name.

Minimal complete definition

cmpMonomial

Methods

cmpMonomial :: Proxy ordering -> MonomialOrder n

Instances

IsOrder n Grlex

Methods

cmpMonomial :: Proxy * Grlex -> MonomialOrder n

IsOrder n Lex

Methods

cmpMonomial :: Proxy * Lex -> MonomialOrder n

IsOrder n Revlex

Methods

cmpMonomial :: Proxy * Revlex -> MonomialOrder n

IsOrder n Grevlex

Methods

cmpMonomial :: Proxy * Grevlex -> MonomialOrder n

IsOrder n ord => IsOrder n (Graded ord)

Methods

cmpMonomial :: Proxy * (Graded ord) -> MonomialOrder n

(KnownNat n, IsOrder n ord, SingI [Nat] ws) => IsOrder n (WeightOrder ws ord)

Methods

cmpMonomial :: Proxy * (WeightOrder ws ord) -> MonomialOrder n

(IsOrder n ord, IsOrder m ord', KnownNat m, KnownNat n, (~) Nat k ((+) n m)) => IsOrder k (ProductOrder n m ord ord')

Methods

cmpMonomial :: Proxy * (ProductOrder n m ord ord') -> MonomialOrder k

newtype PadPolyL n ord poly

Constructors

PadPolyL

Fields

• runPadPolyL :: OrderedPolynomial poly (Graded ord) n

Instances

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => LeftModule Integer (PadPolyL n ord poly)

Methods

(.*) :: Integer -> PadPolyL n ord poly -> PadPolyL n ord poly

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => LeftModule Natural (PadPolyL n ord poly)

Methods

(.*) :: Natural -> PadPolyL n ord poly -> PadPolyL n ord poly

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => RightModule Integer (PadPolyL n ord poly)

Methods

(*.) :: PadPolyL n ord poly -> Integer -> PadPolyL n ord poly

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => RightModule Natural (PadPolyL n ord poly)

Methods

(*.) :: PadPolyL n ord poly -> Natural -> PadPolyL n ord poly

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly, LeftModule (Scalar r) poly) => LeftModule (Scalar r) (PadPolyL n ord poly)

Methods

(.*) :: Scalar r -> PadPolyL n ord poly -> PadPolyL n ord poly

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly, RightModule (Scalar r) poly) => RightModule (Scalar r) (PadPolyL n ord poly)

Methods

(*.) :: PadPolyL n ord poly -> Scalar r -> PadPolyL n ord poly

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Eq (PadPolyL n ord poly)

Methods

(==) :: PadPolyL n ord poly -> PadPolyL n ord poly -> Bool

(/=) :: PadPolyL n ord poly -> PadPolyL n ord poly -> Bool

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Commutative (PadPolyL n ord poly)

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Ring (PadPolyL n ord poly)

Methods

fromInteger :: Integer -> PadPolyL n ord poly

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Rig (PadPolyL n ord poly)

Methods

fromNatural :: Natural -> PadPolyL n ord poly

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => DecidableZero (PadPolyL n ord poly)

Methods

isZero :: PadPolyL n ord poly -> Bool

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Unital (PadPolyL n ord poly)

Methods

one :: PadPolyL n ord poly

pow :: PadPolyL n ord poly -> Natural -> PadPolyL n ord poly

productWith :: Foldable f => (a -> PadPolyL n ord poly) -> f a -> PadPolyL n ord poly

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Group (PadPolyL n ord poly)

Methods

(-) :: PadPolyL n ord poly -> PadPolyL n ord poly -> PadPolyL n ord poly

negate :: PadPolyL n ord poly -> PadPolyL n ord poly

subtract :: PadPolyL n ord poly -> PadPolyL n ord poly -> PadPolyL n ord poly

times :: Integral n => n -> PadPolyL n ord poly -> PadPolyL n ord poly

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Multiplicative (PadPolyL n ord poly)

Methods

(*) :: PadPolyL n ord poly -> PadPolyL n ord poly -> PadPolyL n ord poly

pow1p :: PadPolyL n ord poly -> Natural -> PadPolyL n ord poly

productWith1 :: Foldable1 f => (a -> PadPolyL n ord poly) -> f a -> PadPolyL n ord poly

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Semiring (PadPolyL n ord poly)

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Monoidal (PadPolyL n ord poly)

Methods

zero :: PadPolyL n ord poly

sinnum :: Natural -> PadPolyL n ord poly -> PadPolyL n ord poly

sumWith :: Foldable f => (a -> PadPolyL n ord poly) -> f a -> PadPolyL n ord poly

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Additive (PadPolyL n ord poly)

Methods

(+) :: PadPolyL n ord poly -> PadPolyL n ord poly -> PadPolyL n ord poly

sinnum1p :: Natural -> PadPolyL n ord poly -> PadPolyL n ord poly

sumWith1 :: Foldable1 f => (a -> PadPolyL n ord poly) -> f a -> PadPolyL n ord poly

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => Abelian (PadPolyL n ord poly)

(SingI [Nat] (Replicate Nat n 1), KnownNat n, IsMonomialOrder n ord, IsOrderedPolynomial poly) => IsOrderedPolynomial (PadPolyL n ord poly)

Associated Types

type MOrder (PadPolyL n ord poly) :: *

Methods

coeff :: OrderedMonomial * (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly -> Coefficient (PadPolyL n ord poly)

terms :: PadPolyL n ord poly -> Map (OrderedMonomial * (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly))) (Coefficient (PadPolyL n ord poly))

leadingTerm :: PadPolyL n ord poly -> (Coefficient (PadPolyL n ord poly), OrderedMonomial * (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly)))

leadingMonomial :: PadPolyL n ord poly -> OrderedMonomial * (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly))

leadingCoeff :: PadPolyL n ord poly -> Coefficient (PadPolyL n ord poly)

orderedMonomials :: PadPolyL n ord poly -> Set (OrderedMonomial * (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly)))

fromOrderedMonomial :: OrderedMonomial * (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly

toPolynomial :: (Coefficient (PadPolyL n ord poly), OrderedMonomial * (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly))) -> PadPolyL n ord poly

polynomial :: Map (OrderedMonomial * (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly))) (Coefficient (PadPolyL n ord poly)) -> PadPolyL n ord poly

(>*) :: OrderedMonomial * (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly -> PadPolyL n ord poly

(*<) :: PadPolyL n ord poly -> OrderedMonomial * (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly

_Terms :: Iso' (PadPolyL n ord poly) (Map (OrderedMonomial * (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly))) (Coefficient (PadPolyL n ord poly)))

diff :: Ordinal Nat (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly -> PadPolyL n ord poly

mapMonomialMonotonic :: (OrderedMonomial * (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly)) -> OrderedMonomial * (MOrder (PadPolyL n ord poly)) (Arity (PadPolyL n ord poly))) -> PadPolyL n ord poly -> PadPolyL n ord poly

(KnownNat n, IsMonomialOrder n ord, IsPolynomial poly) => IsPolynomial (PadPolyL n ord poly)

Associated Types

type Coefficient (PadPolyL n ord poly) :: *

type Arity (PadPolyL n ord poly) :: Nat

Methods

liftMap :: (Module (Scalar (Coefficient (PadPolyL n ord poly))) alg, Ring alg, Commutative alg) => (Ordinal Nat (Arity (PadPolyL n ord poly)) -> alg) -> PadPolyL n ord poly -> alg

subst :: (Ring alg, Commutative alg, Module (Scalar (Coefficient (PadPolyL n ord poly))) alg) => Sized (Arity (PadPolyL n ord poly)) alg -> PadPolyL n ord poly -> alg

substWith :: (Unital r, Monoidal m) => (Coefficient (PadPolyL n ord poly) -> r -> m) -> Sized (Arity (PadPolyL n ord poly)) r -> PadPolyL n ord poly -> m

sArity' :: PadPolyL n ord poly -> SNat (Arity (PadPolyL n ord poly))

sArity :: proxy (PadPolyL n ord poly) -> SNat (Arity (PadPolyL n ord poly))

arity :: proxy (PadPolyL n ord poly) -> Integer

injectCoeff :: Coefficient (PadPolyL n ord poly) -> PadPolyL n ord poly

injectCoeff' :: proxy (PadPolyL n ord poly) -> Coefficient (PadPolyL n ord poly) -> PadPolyL n ord poly

monomials :: PadPolyL n ord poly -> HashSet (Monomial (Arity (PadPolyL n ord poly)))

terms' :: PadPolyL n ord poly -> Map (Monomial (Arity (PadPolyL n ord poly))) (Coefficient (PadPolyL n ord poly))

coeff' :: Monomial (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly -> Coefficient (PadPolyL n ord poly)

constantTerm :: PadPolyL n ord poly -> Coefficient (PadPolyL n ord poly)

fromMonomial :: Monomial (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly

toPolynomial' :: (Coefficient (PadPolyL n ord poly), Monomial (Arity (PadPolyL n ord poly))) -> PadPolyL n ord poly

polynomial' :: Map (Monomial (Arity (PadPolyL n ord poly))) (Coefficient (PadPolyL n ord poly)) -> PadPolyL n ord poly

totalDegree' :: PadPolyL n ord poly -> Natural

var :: Ordinal Nat (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly

mapCoeff' :: (Coefficient (PadPolyL n ord poly) -> Coefficient (PadPolyL n ord poly)) -> PadPolyL n ord poly -> PadPolyL n ord poly

(>|*) :: Monomial (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly -> PadPolyL n ord poly

(*|<) :: PadPolyL n ord poly -> Monomial (Arity (PadPolyL n ord poly)) -> PadPolyL n ord poly

(!*) :: Coefficient (PadPolyL n ord poly) -> PadPolyL n ord poly -> PadPolyL n ord poly

_Terms' :: Iso' (PadPolyL n ord poly) (Map (Monomial (Arity (PadPolyL n ord poly))) (Coefficient (PadPolyL n ord poly)))

mapMonomial :: (Monomial (Arity (PadPolyL n ord poly)) -> Monomial (Arity (PadPolyL n ord poly))) -> PadPolyL n ord poly -> PadPolyL n ord poly

type MOrder (PadPolyL n ord poly)

type MOrder (PadPolyL n ord poly) = ProductOrder n (Arity poly) (Graded ord) (MOrder poly)

type Coefficient (PadPolyL n ord poly)

type Coefficient (PadPolyL n ord poly) = Coefficient poly

type Arity (PadPolyL n ord poly)

type Arity (PadPolyL n ord poly) = (+) n (Arity poly)

padLeftPoly :: (IsMonomialOrder n ord, IsPolynomial poly) => Sing n -> ord -> poly -> PadPolyL n ord poly

module Algebra.Ring.Ideal

module Algebra.Normed

module Algebra.Internal