ToySolver.Data.Polynomial

Polynomials

References:

• Monomial order http://en.wikipedia.org/wiki/Monomial_order
• Polynomial class for Ruby http://www.math.kobe-u.ac.jp/~kodama/tips-RubyPoly.html
• constructive-algebra package http://hackage.haskell.org/package/constructive-algebra

Polynomial type

data Polynomial r v

Polynomial over commutative ring r

Instances

SQFree (UPolynomial Integer)

Defined in ToySolver.Data.Polynomial.Factorization.SquareFree

Methods

sqfree :: UPolynomial Integer -> [(UPolynomial Integer, Integer)]

SQFree (UPolynomial Rational)

Defined in ToySolver.Data.Polynomial.Factorization.SquareFree

Methods

sqfree :: UPolynomial Rational -> [(UPolynomial Rational, Integer)]

KnownNat p => SQFree (UPolynomial (PrimeField p))

Defined in ToySolver.Data.Polynomial.Factorization.FiniteField

Methods

sqfree :: UPolynomial (PrimeField p) -> [(UPolynomial (PrimeField p), Integer)]

Factor (UPolynomial Integer)

Defined in ToySolver.Data.Polynomial.Factorization.Integer

Methods

factor :: UPolynomial Integer -> [(UPolynomial Integer, Integer)]

Factor (UPolynomial Rational)

Defined in ToySolver.Data.Polynomial.Factorization.Rational

Methods

factor :: UPolynomial Rational -> [(UPolynomial Rational, Integer)]

KnownNat p => Factor (UPolynomial (PrimeField p))

Defined in ToySolver.Data.Polynomial.Factorization.FiniteField

Methods

factor :: UPolynomial (PrimeField p) -> [(UPolynomial (PrimeField p), Integer)]

(Eq v, Eq r) => Eq (Polynomial r v)

Defined in ToySolver.Data.Polynomial.Base

Methods

(==) :: Polynomial r v -> Polynomial r v -> Bool

(/=) :: Polynomial r v -> Polynomial r v -> Bool

(Eq k, Num k, Ord v) => Num (Polynomial k v)

Defined in ToySolver.Data.Polynomial.Base

Methods

(+) :: Polynomial k v -> Polynomial k v -> Polynomial k v

(-) :: Polynomial k v -> Polynomial k v -> Polynomial k v

(*) :: Polynomial k v -> Polynomial k v -> Polynomial k v

negate :: Polynomial k v -> Polynomial k v

abs :: Polynomial k v -> Polynomial k v

signum :: Polynomial k v -> Polynomial k v

fromInteger :: Integer -> Polynomial k v

(Ord v, Ord r) => Ord (Polynomial r v)

Defined in ToySolver.Data.Polynomial.Base

Methods

compare :: Polynomial r v -> Polynomial r v -> Ordering

(<) :: Polynomial r v -> Polynomial r v -> Bool

(<=) :: Polynomial r v -> Polynomial r v -> Bool

(>) :: Polynomial r v -> Polynomial r v -> Bool

(>=) :: Polynomial r v -> Polynomial r v -> Bool

max :: Polynomial r v -> Polynomial r v -> Polynomial r v

min :: Polynomial r v -> Polynomial r v -> Polynomial r v

(Show v, Show k) => Show (Polynomial k v)

Defined in ToySolver.Data.Polynomial.Base

Methods

showsPrec :: Int -> Polynomial k v -> ShowS

show :: Polynomial k v -> String

showList :: [Polynomial k v] -> ShowS

(Eq k, Num k, Ord v, IsString v) => IsString (Polynomial k v)

Defined in ToySolver.Data.Polynomial.Base

Methods

fromString :: String -> Polynomial k v

(NFData k, NFData v) => NFData (Polynomial k v)

Defined in ToySolver.Data.Polynomial.Base

Methods

rnf :: Polynomial k v -> ()

(Hashable k, Hashable v) => Hashable (Polynomial k v)

Defined in ToySolver.Data.Polynomial.Base

Methods

hashWithSalt :: Int -> Polynomial k v -> Int

hash :: Polynomial k v -> Int

(Ord k, Num k, Ord v, PrettyCoeff k, PrettyVar v) => Pretty (Polynomial k v)

Defined in ToySolver.Data.Polynomial.Base

Methods

pPrintPrec :: PrettyLevel -> Rational -> Polynomial k v -> Doc

pPrint :: Polynomial k v -> Doc

pPrintList :: PrettyLevel -> [Polynomial k v] -> Doc

(Eq k, Num k, Ord v) => VectorSpace (Polynomial k v)

Defined in ToySolver.Data.Polynomial.Base

Associated Types

type Scalar (Polynomial k v) :: Type

Methods

(*^) :: Scalar (Polynomial k v) -> Polynomial k v -> Polynomial k v

(Eq k, Num k, Ord v) => AdditiveGroup (Polynomial k v)

Defined in ToySolver.Data.Polynomial.Base

Methods

zeroV :: Polynomial k v

(^+^) :: Polynomial k v -> Polynomial k v -> Polynomial k v

negateV :: Polynomial k v -> Polynomial k v

(^-^) :: Polynomial k v -> Polynomial k v -> Polynomial k v

(Num c, Ord c, PrettyCoeff c, Ord v, PrettyVar v) => PrettyCoeff (Polynomial c v)

Defined in ToySolver.Data.Polynomial.Base

Methods

pPrintCoeff :: PrettyLevel -> Rational -> Polynomial c v -> Doc

isNegativeCoeff :: Polynomial c v -> Bool

Degree (Polynomial k v)

Defined in ToySolver.Data.Polynomial.Base

Methods

deg :: Polynomial k v -> Integer

Ord v => Vars (Polynomial k v) v

Defined in ToySolver.Data.Polynomial.Base

Methods

vars :: Polynomial k v -> Set v

(Eq k, Num k, Ord v) => Var (Polynomial k v) v

Defined in ToySolver.Data.Polynomial.Base

Methods

var :: v -> Polynomial k v

type Scalar (Polynomial k v)

Defined in ToySolver.Data.Polynomial.Base

type Scalar (Polynomial k v) = k

Conversion

class Vars a v => Var a v | a -> v where

Methods

var :: v -> a

Instances

Ord v => Var (Monomial v) v

Defined in ToySolver.Data.Polynomial.Base

Methods

var :: v -> Monomial v

(Eq k, Num k, Ord v) => Var (Polynomial k v) v

Defined in ToySolver.Data.Polynomial.Base

Methods

var :: v -> Polynomial k v

constant :: (Eq k, Num k) => k -> Polynomial k v

construct a polynomial from a constant

terms :: Polynomial k v -> [Term k v]

list of terms

fromTerms :: (Eq k, Num k, Ord v) => [Term k v] -> Polynomial k v

construct a polynomial from a list of terms

coeffMap :: Polynomial r v -> Map (Monomial v) r

fromCoeffMap :: (Eq k, Num k) => Map (Monomial v) k -> Polynomial k v

fromTerm :: (Eq k, Num k) => Term k v -> Polynomial k v

construct a polynomial from a singlet term

Query

class Degree t where

total degree of a given polynomial

Methods

deg :: t -> Integer

Instances

Degree AReal

Degree of the algebraic number.

If the algebraic number's minimalPolynomial has degree n, then the algebraic number is said to be degree n.

Defined in ToySolver.Data.AlgebraicNumber.Real

Methods

deg :: AReal -> Integer

Degree (Monomial v)

Defined in ToySolver.Data.Polynomial.Base

Methods

deg :: Monomial v -> Integer

Degree (Polynomial k v)

Defined in ToySolver.Data.Polynomial.Base

Methods

deg :: Polynomial k v -> Integer

class Ord v => Vars a v | a -> v where

Methods

vars :: a -> Set v

Instances

Ord v => Vars (Monomial v) v

Defined in ToySolver.Data.Polynomial.Base

Methods

vars :: Monomial v -> Set v

Ord v => Vars (Polynomial k v) v

Defined in ToySolver.Data.Polynomial.Base

Methods

vars :: Polynomial k v -> Set v

lt :: Num k => MonomialOrder v -> Polynomial k v -> Term k v

leading term with respect to a given monomial order

lc :: Num k => MonomialOrder v -> Polynomial k v -> k

leading coefficient with respect to a given monomial order

lm :: Num k => MonomialOrder v -> Polynomial k v -> Monomial v

leading monomial with respect to a given monomial order

coeff :: (Num k, Ord v) => Monomial v -> Polynomial k v -> k

Look up a coefficient for a given monomial. If no such term exists, it returns 0.

lookupCoeff :: Ord v => Monomial v -> Polynomial k v -> Maybe k

Look up a coefficient for a given monomial. If no such term exists, it returns Nothing.

isPrimitive :: (Eq k, Num k, ContPP k, Ord v) => Polynomial k v -> Bool

a polynomial is called primitive if the greatest common divisor of its coefficients is 1.

Operations

class Factor a where

Prime factorization of polynomials

Methods

factor :: a -> [(a, Integer)]

factor a polynomial p into p1 ^ n1 * p2 ^ n2 * .. and return a list [(p1,n1), (p2,n2), ..].

Instances

Factor (UPolynomial Integer)

Defined in ToySolver.Data.Polynomial.Factorization.Integer

Methods

factor :: UPolynomial Integer -> [(UPolynomial Integer, Integer)]

Factor (UPolynomial Rational)

Defined in ToySolver.Data.Polynomial.Factorization.Rational

Methods

factor :: UPolynomial Rational -> [(UPolynomial Rational, Integer)]

KnownNat p => Factor (UPolynomial (PrimeField p))

Defined in ToySolver.Data.Polynomial.Factorization.FiniteField

Methods

factor :: UPolynomial (PrimeField p) -> [(UPolynomial (PrimeField p), Integer)]

class SQFree a where

Square-free factorization of polynomials

Methods

sqfree :: a -> [(a, Integer)]

factor a polynomial p into p1 ^ n1 * p2 ^ n2 * .. and return a list [(p1,n1), (p2,n2), ..].

Instances

SQFree (UPolynomial Integer)

Defined in ToySolver.Data.Polynomial.Factorization.SquareFree

Methods

sqfree :: UPolynomial Integer -> [(UPolynomial Integer, Integer)]

SQFree (UPolynomial Rational)

Defined in ToySolver.Data.Polynomial.Factorization.SquareFree

Methods

sqfree :: UPolynomial Rational -> [(UPolynomial Rational, Integer)]

KnownNat p => SQFree (UPolynomial (PrimeField p))

Defined in ToySolver.Data.Polynomial.Factorization.FiniteField

Methods

sqfree :: UPolynomial (PrimeField p) -> [(UPolynomial (PrimeField p), Integer)]

class ContPP k where

Associated Types

type PPCoeff k

Methods

cont :: Ord v => Polynomial k v -> k

Content of a polynomial

pp :: Ord v => Polynomial k v -> Polynomial (PPCoeff k) v

Primitive part of a polynomial

Instances

ContPP Integer

Defined in ToySolver.Data.Polynomial.Base

Associated Types

type PPCoeff Integer :: Type

Methods

cont :: Ord v => Polynomial Integer v -> Integer

pp :: Ord v => Polynomial Integer v -> Polynomial (PPCoeff Integer) v

Integral r => ContPP (Ratio r)

Defined in ToySolver.Data.Polynomial.Base

Associated Types

type PPCoeff (Ratio r) :: Type

Methods

cont :: Ord v => Polynomial (Ratio r) v -> Ratio r

pp :: Ord v => Polynomial (Ratio r) v -> Polynomial (PPCoeff (Ratio r)) v

deriv :: (Eq k, Num k, Ord v) => Polynomial k v -> v -> Polynomial k v

Formal derivative of polynomials

integral :: (Eq k, Fractional k, Ord v) => Polynomial k v -> v -> Polynomial k v

Formal integral of polynomials

eval :: Num k => (v -> k) -> Polynomial k v -> k

Evaluation

subst :: (Eq k, Num k, Ord v2) => Polynomial k v1 -> (v1 -> Polynomial k v2) -> Polynomial k v2

Substitution or bind

mapCoeff :: (Eq k1, Num k1) => (k -> k1) -> Polynomial k v -> Polynomial k1 v

Transform polynomial coefficients.

toMonic :: (Eq r, Fractional r) => MonomialOrder v -> Polynomial r v -> Polynomial r v

Transform a polynomial into a monic polynomial with respect to the given monomial order.

toUPolynomialOf :: (Ord k, Num k, Ord v) => Polynomial k v -> v -> UPolynomial (Polynomial k v)

Convert K[x,x1,x2,…] into K[x1,x2,…][x].

divModMP :: forall k v. (Eq k, Fractional k, Ord v) => MonomialOrder v -> Polynomial k v -> [Polynomial k v] -> ([Polynomial k v], Polynomial k v)

Multivariate division algorithm

divModMP cmp f [g1,g2,..] returns a pair ([q1,q2,…],r) such that

• f = g1*q1 + g2*q2 + .. + r and
• g1, g2, .. do not divide r.

reduce :: (Eq k, Fractional k, Ord v) => MonomialOrder v -> Polynomial k v -> [Polynomial k v] -> Polynomial k v

Multivariate division algorithm

reduce cmp f gs = snd (divModMP cmp f gs)

Univariate polynomials

type UPolynomial r = Polynomial r X

Univariate polynomials over commutative ring r

data X

Variable "x"

Constructors

X

Instances

Bounded X

Defined in ToySolver.Data.Polynomial.Base

Methods

minBound :: X

maxBound :: X

Enum X

Defined in ToySolver.Data.Polynomial.Base

Methods

succ :: X -> X

pred :: X -> X

toEnum :: Int -> X

fromEnum :: X -> Int

enumFrom :: X -> [X]

enumFromThen :: X -> X -> [X]

enumFromTo :: X -> X -> [X]

enumFromThenTo :: X -> X -> X -> [X]

Eq X

Defined in ToySolver.Data.Polynomial.Base

Methods

(==) :: X -> X -> Bool

(/=) :: X -> X -> Bool

Data X

Defined in ToySolver.Data.Polynomial.Base

Methods

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

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

toConstr :: X -> Constr

dataTypeOf :: X -> DataType

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

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

gmapT :: (forall b. Data b => b -> b) -> X -> X

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

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

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

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

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

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

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

Ord X

Defined in ToySolver.Data.Polynomial.Base

Methods

compare :: X -> X -> Ordering

(<) :: X -> X -> Bool

(<=) :: X -> X -> Bool

(>) :: X -> X -> Bool

(>=) :: X -> X -> Bool

max :: X -> X -> X

min :: X -> X -> X

Read X

Defined in ToySolver.Data.Polynomial.Base

Methods

readsPrec :: Int -> ReadS X

readList :: ReadS [X]

readPrec :: ReadPrec X

readListPrec :: ReadPrec [X]

Show X

Defined in ToySolver.Data.Polynomial.Base

Methods

showsPrec :: Int -> X -> ShowS

show :: X -> String

showList :: [X] -> ShowS

NFData X

Defined in ToySolver.Data.Polynomial.Base

Methods

rnf :: X -> ()

Hashable X

Defined in ToySolver.Data.Polynomial.Base

Methods

hashWithSalt :: Int -> X -> Int

hash :: X -> Int

PrettyVar X

Defined in ToySolver.Data.Polynomial.Base

Methods

pPrintVar :: PrettyLevel -> Rational -> X -> Doc

SQFree (UPolynomial Integer)

Defined in ToySolver.Data.Polynomial.Factorization.SquareFree

Methods

sqfree :: UPolynomial Integer -> [(UPolynomial Integer, Integer)]

SQFree (UPolynomial Rational)

Defined in ToySolver.Data.Polynomial.Factorization.SquareFree

Methods

sqfree :: UPolynomial Rational -> [(UPolynomial Rational, Integer)]

KnownNat p => SQFree (UPolynomial (PrimeField p))

Defined in ToySolver.Data.Polynomial.Factorization.FiniteField

Methods

sqfree :: UPolynomial (PrimeField p) -> [(UPolynomial (PrimeField p), Integer)]

Factor (UPolynomial Integer)

Defined in ToySolver.Data.Polynomial.Factorization.Integer

Methods

factor :: UPolynomial Integer -> [(UPolynomial Integer, Integer)]

Factor (UPolynomial Rational)

Defined in ToySolver.Data.Polynomial.Factorization.Rational

Methods

factor :: UPolynomial Rational -> [(UPolynomial Rational, Integer)]

KnownNat p => Factor (UPolynomial (PrimeField p))

Defined in ToySolver.Data.Polynomial.Factorization.FiniteField

Methods

factor :: UPolynomial (PrimeField p) -> [(UPolynomial (PrimeField p), Integer)]

type UTerm k = Term k X

type UMonomial = Monomial X

div :: (Eq k, Fractional k) => UPolynomial k -> UPolynomial k -> UPolynomial k infixl 7

division of univariate polynomials

mod :: (Eq k, Fractional k) => UPolynomial k -> UPolynomial k -> UPolynomial k infixl 7

division of univariate polynomials

divMod :: (Eq k, Fractional k) => UPolynomial k -> UPolynomial k -> (UPolynomial k, UPolynomial k)

division of univariate polynomials

divides :: (Eq k, Fractional k) => UPolynomial k -> UPolynomial k -> Bool

gcd :: (Eq k, Fractional k) => UPolynomial k -> UPolynomial k -> UPolynomial k

GCD of univariate polynomials

lcm :: (Eq k, Fractional k) => UPolynomial k -> UPolynomial k -> UPolynomial k

LCM of univariate polynomials

exgcd :: (Eq k, Fractional k) => UPolynomial k -> UPolynomial k -> (UPolynomial k, UPolynomial k, UPolynomial k)

Extended GCD algorithm

pdivMod :: (Eq r, Num r) => UPolynomial r -> UPolynomial r -> (r, UPolynomial r, UPolynomial r)

pseudo division

pdiv :: (Eq r, Num r) => UPolynomial r -> UPolynomial r -> UPolynomial r

pseudo quotient

pmod :: (Eq r, Num r) => UPolynomial r -> UPolynomial r -> UPolynomial r

pseudo reminder

gcd' :: Integral r => UPolynomial r -> UPolynomial r -> UPolynomial r

GCD of univariate polynomials

isRootOf :: (Eq k, Num k) => k -> UPolynomial k -> Bool

Is the number a root of the polynomial?

isSquareFree :: (Eq k, Fractional k) => UPolynomial k -> Bool

Is the polynomial square free?

nat :: MonomialOrder X

Natural ordering x^0 < x^1 < x^2 .. is the unique monomial order for univariate polynomials.

eisensteinsCriterion :: (ContPP k, PPCoeff k ~ Integer) => UPolynomial k -> Bool

Eisenstein's irreducibility criterion.

Given a polynomial with integer coefficients P[x] = an x^n + .. + a1*x + a0, it is irreducible over rational numbers if there exists a prime number p satisfying the following conditions:

• p divides ai for i /= n,
• p does not divides an, and
• p^2 does not divides a0.

Term

type Term k v = (k, Monomial v)

tdeg :: Term k v -> Integer

tscale :: Num k => k -> Term k v -> Term k v

tmult :: (Num k, Ord v) => Term k v -> Term k v -> Term k v

tdivides :: Ord v => Term k v -> Term k v -> Bool

tdiv :: (Fractional k, Ord v) => Term k v -> Term k v -> Term k v

tderiv :: (Num k, Ord v) => Term k v -> v -> Term k v

tintegral :: (Fractional k, Ord v) => Term k v -> v -> Term k v

Monic monomial

data Monomial v

Monic monomials

Instances

Eq v => Eq (Monomial v)

Defined in ToySolver.Data.Polynomial.Base

Methods

(==) :: Monomial v -> Monomial v -> Bool

(/=) :: Monomial v -> Monomial v -> Bool

Ord v => Ord (Monomial v)

Defined in ToySolver.Data.Polynomial.Base

Methods

compare :: Monomial v -> Monomial v -> Ordering

(<) :: Monomial v -> Monomial v -> Bool

(<=) :: Monomial v -> Monomial v -> Bool

(>) :: Monomial v -> Monomial v -> Bool

(>=) :: Monomial v -> Monomial v -> Bool

max :: Monomial v -> Monomial v -> Monomial v

min :: Monomial v -> Monomial v -> Monomial v

Show v => Show (Monomial v)

Defined in ToySolver.Data.Polynomial.Base

Methods

showsPrec :: Int -> Monomial v -> ShowS

show :: Monomial v -> String

showList :: [Monomial v] -> ShowS

(Ord v, IsString v) => IsString (Monomial v)

Defined in ToySolver.Data.Polynomial.Base

Methods

fromString :: String -> Monomial v

NFData v => NFData (Monomial v)

Defined in ToySolver.Data.Polynomial.Base

Methods

rnf :: Monomial v -> ()

Hashable v => Hashable (Monomial v)

Defined in ToySolver.Data.Polynomial.Base

Methods

hashWithSalt :: Int -> Monomial v -> Int

hash :: Monomial v -> Int

Degree (Monomial v)

Defined in ToySolver.Data.Polynomial.Base

Methods

deg :: Monomial v -> Integer

Ord v => Vars (Monomial v) v

Defined in ToySolver.Data.Polynomial.Base

Methods

vars :: Monomial v -> Set v

Ord v => Var (Monomial v) v

Defined in ToySolver.Data.Polynomial.Base

Methods

var :: v -> Monomial v

mone :: Monomial v

mfromIndices :: Ord v => [(v, Integer)] -> Monomial v

mfromIndicesMap :: Ord v => Map v Integer -> Monomial v

mindices :: Monomial v -> [(v, Integer)]

mindicesMap :: Monomial v -> Map v Integer

mmult :: Ord v => Monomial v -> Monomial v -> Monomial v

mpow :: Ord v => Monomial v -> Integer -> Monomial v

mdivides :: Ord v => Monomial v -> Monomial v -> Bool

mdiv :: Ord v => Monomial v -> Monomial v -> Monomial v

mderiv :: Ord v => Monomial v -> v -> (Integer, Monomial v)

mintegral :: Ord v => Monomial v -> v -> (Rational, Monomial v)

mlcm :: Ord v => Monomial v -> Monomial v -> Monomial v

mgcd :: Ord v => Monomial v -> Monomial v -> Monomial v

mcoprime :: Ord v => Monomial v -> Monomial v -> Bool

Monomial order

type MonomialOrder v = Monomial v -> Monomial v -> Ordering

lex :: Ord v => MonomialOrder v

Lexicographic order

revlex :: Ord v => Monomial v -> Monomial v -> Ordering

Reverse lexicographic order.

Note that revlex is NOT a monomial order.

grlex :: Ord v => MonomialOrder v

Graded lexicographic order

grevlex :: Ord v => MonomialOrder v

Graded reverse lexicographic order

Pretty Printing

data PrintOptions k v

Options for pretty printing polynomials

The default value can be obtained by def.

Constructors

PrintOptions

Fields

• pOptPrintVar :: PrettyLevel -> Rational -> v -> Doc
• pOptPrintCoeff :: PrettyLevel -> Rational -> k -> Doc
• pOptIsNegativeCoeff :: k -> Bool
• pOptMonomialOrder :: MonomialOrder v

Instances

(PrettyCoeff k, PrettyVar v, Ord v) => Default (PrintOptions k v)

Defined in ToySolver.Data.Polynomial.Base

Methods

def :: PrintOptions k v

prettyPrint :: (Ord k, Num k) => PrintOptions k v -> PrettyLevel -> Rational -> Polynomial k v -> Doc

class PrettyCoeff a where

Minimal complete definition

pPrintCoeff

Methods

pPrintCoeff :: PrettyLevel -> Rational -> a -> Doc

isNegativeCoeff :: a -> Bool

Instances

PrettyCoeff Integer

Defined in ToySolver.Data.Polynomial.Base

Methods

pPrintCoeff :: PrettyLevel -> Rational -> Integer -> Doc

isNegativeCoeff :: Integer -> Bool

PrettyCoeff AReal

Defined in ToySolver.Data.AlgebraicNumber.Real

Methods

pPrintCoeff :: PrettyLevel -> Rational -> AReal -> Doc

isNegativeCoeff :: AReal -> Bool

(PrettyCoeff a, Integral a) => PrettyCoeff (Ratio a)

Defined in ToySolver.Data.Polynomial.Base

Methods

pPrintCoeff :: PrettyLevel -> Rational -> Ratio a -> Doc

isNegativeCoeff :: Ratio a -> Bool

PrettyCoeff (PrimeField a)

Defined in ToySolver.Data.Polynomial.Base

Methods

pPrintCoeff :: PrettyLevel -> Rational -> PrimeField a -> Doc

isNegativeCoeff :: PrimeField a -> Bool

(Num c, Ord c, PrettyCoeff c, Ord v, PrettyVar v) => PrettyCoeff (Polynomial c v)

Defined in ToySolver.Data.Polynomial.Base

Methods

pPrintCoeff :: PrettyLevel -> Rational -> Polynomial c v -> Doc

isNegativeCoeff :: Polynomial c v -> Bool

class PrettyVar a where

Methods

pPrintVar :: PrettyLevel -> Rational -> a -> Doc

Instances

PrettyVar Int

Defined in ToySolver.Data.Polynomial.Base

Methods

pPrintVar :: PrettyLevel -> Rational -> Int -> Doc

PrettyVar X

Defined in ToySolver.Data.Polynomial.Base

Methods

pPrintVar :: PrettyLevel -> Rational -> X -> Doc