}p      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w xyz{ | } ~                                                                                                                                      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnoNone%&+,9:;<=DRAdditional types for normed types.    None%&+,/9:;<=DIR r, provides almost the same type-instances as r!, but it can also behave as a p over r itself. 8None%&+,9:;<=DRfMThese Instances are not algebraically right, but for the sake of convenience.623456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefg623456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefgNone%&+,/9:;<=DOQRTb hijklmnopqrsqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~hijklmnopqrnkjoipqhmlrs hijklmnopqrsNone%&+,9:;<=ADRTtuvwxyz{|}~ tuvwxyz{|} tuwxyz{}|v tuvwxyz{|}~w None%&*+,/9:;<=ADFINOQRTbf8A wrapper for monomials with a certain (monomial) order.\N-ary Monomial. IntMap contains degrees for each x_i- type Monomial (n :: Nat) = Sized n IntGMonomial ordering which can do with monomials of arbitrary large arity.Monomial order which can be use to calculate n-th elimination ideal of m-ary polynomial. This should judge monomial to be bigger if it contains variables to eliminate.Class for Monomial orders.)Graded order from another monomial order.&Graded lexicographical order. Same as  Graded Lex./Graded reversed lexicographical order. Same as  Graded Revlex.Reversed lexicographical orderLexicographical order4Class to lookup ordering from its (type-level) name.Monomial order (of degree n). This should satisfy following laws: (1) Totality: forall a, b (a < b || a == b || b < a) (2) Additivity: a  =b ==3 a + c <= b + c (3) Non-negative: forall a, 0 <= a convert NAry list into Monomial.2Lexicographical order. This *is* a monomial order.?Reversed lexicographical order. This is *not* a monomial order.!Convert ordering into graded one.9Graded lexicographical order. This *is* a monomial order.BGraded reversed lexicographical order. This *is* a monomial order.:Comparing monomials with different arity, padding with 0? at bottom of the shorter monomial to make the length equal.aFor simplicity, we choose grevlex for the default monomial ordering (for the sake of efficiency).'Special ordering for ordered-monomials.S66INone%&*+,69:;<=DFNOQRT4@Pretty-printing conditional for coefficients. Each returning  must not have any sign.)Coefficients which admits pretty-printing4Class to lookup ordering from its (type-level) name. A variant of   which takes  instead of sThe default implementation is not enough efficient. So it is strongly recomended to give explicit definition to .3Leading term with respect to its monomial ordering.7Leading monomial with respect to its monomial ordering.:Leading coefficient with respect to its monomial ordering.\The collection of all monomials in the given polynomial, with metadata of their ordering. A variant of   which takes  as argument. A variant of   which takes  as argument. A variant of   which takes  as argument.$The default implementation combines  and  a, hence is not enough efficient. So it is strongly recomended to give explicit definition to . A variant of '(>|*)' which takes  as argument.Flipped version of (>*)diff n f partially diffrenciates n%-th variable in the given polynomial f%. The default implementation uses  and D and is really naive; please consider overrideing for efficiency.Same as C, but maping function is assumed to be strictly monotonic (i.e. a < b implies  f a < f b).Polynomial in terms of free associative commutative algebra generated by n-elements. To effectively compute all terms, we need / in addition to universality of free object.$Coefficient ring of polynomial type.Arity of polynomial type.`Universal mapping for free algebra. This corresponds to the algebraic substitution operation. A variant of , each value is given by k.Another variant of . This function relies on Y; if you have more efficient implementation, it is encouraged to override this method.Arity of given polynomial.,Arity of given polynomial, using type proxy.Non-dependent version of arity.#Inject coefficient into polynomial.EInject coefficient into polynomial with result-type explicitly given. f6 returns the finite set of all monomials appearing in f. f2 returns the finite set of all terms appearing in fA; Term is a finite map from monomials to non-zero coefficient.  'coeff m f'% returns the coefficient of monomial m in polynomial f.  Calculates constant coefficient. Inject monic monomial. !Inject coefficient with monomial. RConstruct polynomial from the given finite mapping from monomials to coefficients.Returns total degree. n1 returns a polynomial representing n-th variable.$Adjusting coefficients of each term.m  f multiplies polynomial f by monomial m.Flipped version of ()HConstraint synonym for rings that can be used as polynomial coefficient.*1-norm of given polynomial, taking sum of s of each coefficients.8Maximum norm of given polynomial, taking maximum of the s of each coefficients.YMake the given polynomial monic. If the given polynomial is zero, it returns as it is.6 calculates the S-Polynomial of given two polynomials.pDivModPoly f g0 calculates the pseudo quotient and reminder of f by g.%The content of a polynomial f is the  of all its coefficients. f3 calculates the primitive part of given polynomial f , namely f / content(f)."ShowS coefficients as term. showsCoeffAsTerm  "" = "" showsCoeffAsTerm (, (shows "12")) "" = "-12" showsCoeffAsTerm ( (shows "12")) "" = "12" #0ShowS coefficients prefixed with infix operator. (shows 12 . showsCoeffWithOp *) "" = "12" (shows 12 . showsCoeffWithOp (; (shows 34))) "" = "12 - 34" (shows 12 . showsCoeffWithOp ( (shows 34))) "" = "12 + 34" &ECalculate a polynomial quotient and remainder w.r.t. second argument.'9Remainder of given polynomial w.r.t. the second argument.(:A Quotient of given polynomial w.r.t. the second argument.U      !"#$%&'()*+,-./012345678D      !"#$%&'(D      !"#%$&('(      !"#$%&'()*+,-./01234567877777&7'7(7 None$%&*+,/9:;<=DINOQRT:2n-ary polynomial ring over some noetherian ring R.B4Substitute univariate polynomial using Horner's ruleD"Evaluate polynomial at some point.E substVar n f substitutes n-th variable with polynomial f, without changing arity.MaCalculate the homogenized polynomial of given one, with additional variable is the last variable.]ZWe provide Num instance to use trivial injection R into R[X]. Do not use signum or abs.A9:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstu      !"#$%&'(9:;<=>?@ABCDEFGHIJKLMNOPQ9J=GH>:;<FEMN@AILK?OPDCBQ?9:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuNone%&+,9:;<=DRxyzDqr      !"#$%stuvwxyz{|&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~p      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ hijklmnopqrtuvwxyz{|}      !"#$%&'(9:;<=>?@ABCDEFGHIJKLMNOPQxyz xyzxyzx7 None%&*+,/9:;<=DIRTf{"Univariate polynomial. It uses S as its internal representation; so if you want to treat the power greater than maxBound :: Int/, please consider using other represntation.~)Polynomial multiplication, naive version.3Polynomial multiplication using Karatsuba's method.By this instance, you can use #x for the unique variable of { r.4{|}     ~      !"#$%&'({|}~{~}|2{|}     ~ None%&*+,/9:;<=DOQRT`aConvenient type-synonym for  LabPlynomial wrapping univariate polynomial {.Convenient type-synonym for  LabPlynomial wrapping : and {.So unsafe! Don't expose it!$This instance allows something like J#x :: LabPolynomial (OrderedPolynomial Integer Grevlex 3) '["x", "y", "z"].)  & None%&+,9:;<=DR Calculates n4-th power efficiently, using repeated square method.!Fermat-test for pseudo-primeness. x m p efficiently calculates x ^ p `mod' m. n tests if the given integer n is pseudo prime. It returns $ p if p < n divides n, %  if n is pseudo-prime, % 4 if it is not pseudo-prime but no clue can be found.  None%&+,9:;<=DIORTPrime field of characteristic p. p- should be prime, and not statically checked.2 !  . !None%&+,9:;<=DRT2Empty tag to reify Conway polynomial to type-level;Type-class to provide the dictionary for Conway polynomials"#$%&"%"#$%&None%&+,9:;<=DNRT+Macro to add Conway polynomials dictionary.bParse conway polynomial file and define instances for them. File-format must be the same as  Ohttp://www.math.rwth-aachen.de/~Frank.Luebeck/data/ConwayPol/index.html?LANG=enLueback's file.''None%&+,/9:;<=DFOQRT2Type-constraint synonym to work with Galois field.Galois Field of order p^nr. This uses conway polynomials as canonical minimal polynomial and it should be known at compile-time (i.e. #Reifies (Conway p n) (Unipol (F n))9 instances should be defined to use field operations).Galois field of order p^n. f, stands for the irreducible polynomial over F_p of degree n.generateIrreducible p n) generates irreducible polynomial over F_p of degree n.+Conway polynomial (if definition is known).4()*+,     - !"#$%&'()*+,-./0          2()*+,     - !"#$%&'()*+,-./0None%&+,9:;<=DRqqr      !"#$%stuvwxyz{|&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~p      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ hijklmnopqrtuvwxyz{|}      !"#$%&'(9:;<=>?@ABCDEFGHIJKLMNOPQxyz{|}~     None$%&*+,9:;<=DFOQRTa2Sugar strategy. This chooses the pair with the least phantom homogenized degree and then break the tie with the given strategy (say s).68Choose the pair with the least LCM(LT(f), LT(g)) w.r.t.  order.8Buchberger's normal selection strategy. This selects the pair with the least LCM(LT(f), LT(g)) w.r.t. current monomial ordering.:>Type-class for selection strategies in Buchberger's algorithm.<SCalculates the weight for the given pair of polynomial used for selection strategy.=^Test if the given ideal is Groebner basis, using Buchberger criteria and relatively primeness.>QThe Naive buchberger's algorithm to calculate Groebner basis for the given ideal.?MBuchberger's algorithm slightly improved by discarding relatively prime pair.@_Calculate Groebner basis applying (modified) Buchberger's algorithm. This function is same as A.ABuchberger's algorithm greately improved using the syzygy theory with the sugar strategy. Utilizing priority queues, this function reduces division complexity and comparison time. If you don't have strong reason to avoid this function, this function is recommended to use.B<apply buchberger's algorithm using given selection strategy.CCalculate the weight of given polynomials w.r.t. the given strategy. Buchberger's algorithm proccesses the pair with the most least weight first. This function requires the OrdD instance for the weight; this constraint is not required in the <= because of the ease of implementation. So use this function.E:Reduce minimum Groebner basis into reduced Groebner basis.F[Caliculating reduced Groebner basis of the given ideal w.r.t. the specified monomial order.G[Caliculating reduced Groebner basis of the given ideal w.r.t. the specified monomial order.H7Caliculating reduced Groebner basis of the given ideal.I8Test if the given polynomial is the member of the ideal..ETest if the given polynomial can be divided by the given polynomials.J'Calculate n-th elimination ideal using  ordering.KQCalculate n-th elimination ideal using the specified n-th elimination type order.LCalculate n-th elimination ideal using the specified n-th elimination type order. This function should be used carefully because it does not check whether the given ordering is n-th elimintion type or not.M-An intersection ideal of given ideals (using ).N%Ideal quotient by a principal ideals.O"Ideal quotient by the given ideal.P Saturation by a principal ideal.QSaturation idealR5Calculate resultant for given two unary polynomimals.S\Determine whether two polynomials have a common factor with positive degree using resultant.TFCalculates the Least Common Multiply of the given pair of polynomials.UHCalculates the Greatest Common Divisor of the given pair of polynomials.WPChoose the pair with the least LCM(LT(f), LT(g)) w.r.t. graded current ordering.2/0123456789:;<=>?234@ABCDEFGHI.567JKLMNOPQRSTUVWXY$23456789:;<=>?@ABCDEFGHIJKLMNOPQRSTU$=HFG@A>?EDB:;<C67892345IMJKLONQPRSTU*/0123456789:;<=>?234@ABCDEFGHI.567JKLMNOPQRSTUVWXYNone%&*+,/9:;<=BDIOQRTkCThe polynomial modulo the ideal indexed at the last type-parameter.l7Representative polynomial of given quotient polynomial.rFind the standard monomials of the quotient ring for the zero-dimensional ideal, which are form the basis of it as k-vector space.vPolynomial modulo ideal.x!Polynomial modulo ideal given by Proxy.yEReifies the ideal at the type-level. The ideal can be recovered with 8.z!Computes polynomial modulo ideal.{Reduce polynomial modulo ideal..9j:;<=>k?@lmnopqABCrstuvwxDyzE{|}~jklmnopqrstuvwxyz{|kjyvxlzmutwosrn{qp|'9j:;<=><k?@lmnopqABCrstuvwxDyzE{|}~None%&+,9:;<=DFLRTbdistinctDegFactor fe computes the distinct-degree decomposition of the given square-free polynomial over finite field f.JFactorise a polynomial over finite field using Cantor-Zassenhaus algorithmUFactorise the given integer-coefficient polynomial, choosing a large enough prime.FFactorise the given interger-coefficient polynomial by Hensel lifting. Given that f = gh (mod m) with sg + th = 1 (mod m) and leadingCoeff f isn't zero divisor mod m, henselStep m f g h s t0 calculates the unique (g', h', s', t') s.t. Sf = g' h' (mod m^2), g' = g (mod m), h' = h (mod m), s' = s (mod m), t' = t (mod m), h' monic.F-Repeatedly applies hensel lifting for monics.G&Monic hensel lifting for many factors.)Square-free polynomial over finite field.Distinct-degree decomposition.HIJKLMNOPQRSmodulusFGprime piteration count k.original polynomialcoprime factorisation mod pcoprime factorisation mod p^(2^k).TUVWX  HIJKLMNOPQRSFGTUVWXQ5None%&+,9:;<=DR`UAlgebraic real numbers, which can be expressed as a root of a rational polynomial.$Takes intersection of two intervals.0Test if the former interval includes the latter.Smart constructor.  f i6 represents the unique root of rational polynomial f in the interval iZ. If no root is found, or more than one root belongs to the given interval, returns .Unsafe version of . nthRoot n r< tries to computes n-th root of the given algebraic real r. It returns  if it's undefined. See also . rD returns the same algebraic number, but with more tighter bounds.(Pseudo resultant. should we expose this? f1 finds all real roots of the rational polynomial f.YSame as D, but assumes that the given polynomial is monic and irreducible. f4 finds all complex roots of the rational polynomial f.ZCAUTION: This function currently comes with really naive implementation. Easy to explode.4Choose representative element of the given interval. eps r returns rational number r' close to r , with abs (r - r') < eps.Same as , but returns  value instead.hZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Y`Z[\]^_`abcdefghijklmnopqrstuvwxyz{|}~YNone%&*+,9:;<=DR 44None$%&+,9:;<=DR#Recovers rational number from Z/pZ.&Chinese Remainder for raional numbers.Bound for numeratormodulus+integer corresponds to the rational number.recovered rational numberNone#$%&+,/9:;<=DILORT[PSolving linear equation using linearly recurrent sequence (Wiedemann algorithm).      (m', bs, as) with m is full-rank submatrix, bs are independent and as are dependent. prime number poriginal matrix Minverse matrix of M mod pcoefficient vector vvector x with  Mx = b mod p !"#$%&'()>     >           !"#$%&'()9 None%&*+,9:;<=DRZgaussReduction a = (a', p) where a' is row echelon form and p is pivoting matrix.gaussReduction a = (a', p) where a' is row echelon form and p is pivoting matrix..89:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`ab(89;RHSPFTC<>@:DQBAJILKENOMGVWU?=XYZ[\]^_(89:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ\][^_89:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abNone%&*+,/69:;<=DFIQRTb c`Finds complex approximate roots of given zero-dimensional ideal, using randomized altorithm. See also e and g.dsolveWith f is? finds complex approximate roots of the given zero-dimensional n-variate polynomial system is1, using the given relatively prime polynomial f.ee err isS finds numeric approximate root of the given zero-dimensional polynomial system is, with error <err. See also g and c.gg err isS finds numeric approximate root of the given zero-dimensional polynomial system is, with error <err. See also e and c.i Calculates n-th reduction of f: f  f, "_{x_n} f.j2Calculate the monic generator of k[X_0, ..., X_n] [ k[X_i].kFSolves linear system. If the given matrix is degenerate, this returns Nothing.l:Calculate the radical of the given zero-dimensional ideal.m;Test if the given zero-dimensional ideal is radical or not.nCalculate the Groebner basis w.r.t. lex ordering of the zero-dimensional ideal using FGLM algorithm. If the given ideal is not zero-dimensional this function may diverge.oVCompute the kernel and image of the given linear map using generalized FGLM algorithm.cdefghijklmno Linear map from polynomial ring. The tuple of:9lex-Groebner basis of the kernel of the given linear map.0The vector basis of the image of the linear map.mcdefghijklmnocegklmnodjihfmcdefghijklmno !"#$%&$%'$%()*+,-./01123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOP Q R Q S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~                                                                                                                                                    !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIIJJKKLLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJK    L MN"#OPQR2STUVWXYZ[\]^_`abcdefghijiklmnlmoipiqirisitiuiviwixiyizi{i|i}i~iiiiiiiiiii!"!"!"!"!"!!!$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ $ $ $ $$$$$$$$$$$$$$$$$$$ $!$"$#$$$%$&$'$($)$*$+$,$-$.$/$0$1$2$3$4$5$6$7$8$9$:$;$<$=$>$?$@$A$B$B$CD$CE$CF$CG$CH$CI$CJ$CK$CL$CM$CN$CO$CP$C$CP$CQ$CR$CS$CT$CU$CV$CW$CX$CY$CZ$C[$C\$C]$C^$C_$C`$Ca$Cb$Cc$Cd$Ce$Cf$Cg$Ch$Ci$Cj$Ck$Cl$Cm$Cn$Co$Cp$Cq$Cr$Cs$Ct$Cu$Cv$Cw$Cx$Cy$Cz$C{$C|$C}$C~f    llllllllllll     lmlmlmlm lm!lm"lm#$%$&'()*+,lm-lm.lm/lm0lm1lm234353678797:;l<=>7?f@Af@BfCDfEFfGHfIJfIKfILfIMfINfIOfIPfIQfIRfISfITfIUfVWfVXfVYfVZf[\f]^f]_f]`f]af]bf]cf]df]ef]ff]ff]gfhifjkflmflnfofpfqfrfsftfufvfwfxfyfzf{f|f}f~ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffftffffgfgfgfgfgqfgfgfgfgfgfgfgfgfgfgfgfgfgLfgfgffffpffffff          ! "#$%&$'()*+),-./0./1./2./3./4./5./6./7.89.8.8:.8;.8<.8=.8>.8?.8@.8A.8B.8C.8D.8E.8FlGlHIJKLKMNOPQRSTUVWXYZ[\]^_`abacadaeafagahaijklmnopqrstuvwxyz{|}~777IIII      !"#$%&'()*+,-./0123456789:;:<=>?=>@=>r=>A=>B=>C=>D=>=>E=>E=>F=>G=>G=>H=>I=>I=>J=>K=>K=>L=>M=>M=>N=>O=>O=>PQRQSQTQUVWXY  Z [ \ ] ^ _ ` a b  c d e f g h i j k l m n o  p q rstuvwxyXz{|}~~}~ W    J  !"#$%&'()*+,-./0123456789:;3computational-algebra-0.4.0.0-2CAoodiRVH5fVaWUJl6prAlgebra.InternalAlgebra.Prelude.CoreAlgebra.NormedAlgebra.ScalarAlgebra.InstancesAlgebra.Ring.Ideal Algebra.Ring.Polynomial.MonomialAlgebra.Ring.Polynomial.ClassAlgebra.Ring.Polynomial"Algebra.Ring.Polynomial.UnivariateAlgebra.Ring.Polynomial.LabeledAlgebra.Algorithms.PrimeTestAlgebra.Field.FiniteAlgebra.Field.GaloisAlgebra.Algorithms.Groebner Algebra.Ring.Polynomial.Quotient!Algebra.Ring.Polynomial.FactoriseAlgebra.Field.AlgebraicReal#Algebra.Algorithms.ChineseRemainderAlgebra.LinkedMatrixAlgebra.MatrixAlgebra.Algorithms.ZeroDimData.MapmapKeysAlgebra.Field.Galois.InternalAlgebra.Field.Galois.ConwayAlgebra.PreludeAlgebra.Algorithms.FGLMbaseData.Type.EqualityRefl:~:3equational-reasoning-0.4.1.1-GRjaU1j1xXfBpgNWit1P3ZProof.EquationalwithRefl+type-natural-0.7.1.2-9oKjIGd17va5CLPNxEwIvmData.Type.Ordinal.Builtin enumOrdinalodOrdinalNormedNormnormliftNorm$fNormedFraction$fNormedInteger $fNormedInt$fNormedDoubleScalar runScalar.*.$fRightModuleScalarScalar$fLeftModuleScalarScalar$fLeftModulerScalar$fRightModulerScalar$fNormedScalar $fReadScalar $fShowScalar $fEqScalar $fOrdScalar$fAdditiveScalar$fIntegralScalar $fRealScalar $fEnumScalar$fMultiplicativeScalar$fUnitalScalar$fRightModuleScalar$fLeftModuleScalar$fRightModuleScalar0$fLeftModuleScalar0$fDivisionScalar$fCommutativeScalar $fRigScalar$fAbelianScalar $fRingScalar$fSemiringScalar $fGroupScalar$fMonoidalScalar$fFractionalScalar $fNumScalar$fUnitNormalFormScalar$fDecidableUnitsScalar$fDecidableAssociatesScalar$fHashableVector$fRandomFraction$fConvertibleFractionComplex$fConvertibleFractionDouble$fDecidableUnitsRatio$fDecidableZeroRatio $fRingRatio $fRigRatio$fSemiringRatio$fMonoidalRatio$fDivisionRatio $fUnitalRatio$fMultiplicativeRatio$fRightModuleScalarRatio$fLeftModuleScalarRatio$fCommutativeRatio $fGroupRatio$fRightModuleIntegerRatio$fLeftModuleIntegerRatio$fRightModuleNaturalRatio$fLeftModuleNaturalRatio$fAbelianRatio$fAdditiveRatio$fDivisionDouble$fDecidableZeroDouble $fRingDouble$fAbelianDouble$fSemiringDouble $fRigDouble$fRightModuleIntegerDouble$fLeftModuleIntegerDouble $fGroupDouble$fCommutativeDouble$fMultiplicativeDouble$fUnitalDouble$fMonoidalDouble$fRightModuleNaturalDouble$fLeftModuleNaturalDouble$fMultiplicativeComplex$fAdditiveDouble$fUnitalComplex$fMonoidalComplex$fRightModuleaComplex$fLeftModuleaComplex$fGroupComplex $fRingComplex$fCommutativeComplex $fRigComplex$fSemiringComplex$fAbelianComplex$fAdditiveComplex$fNFDataFraction$fDecidableZeroComplex$fAdditiveVectorFlippedSNatSized'SizedOLttoProxy coerceLength sizedLengthpadVecs sNatToInt $fHashableSeqIdeal isEmptyIdeal addToIdealtoIdeal appendIdeal generators filterIdealprincipalIdealmapIdeal $fNFDataIdeal $fShowIdeal $fOrdIdeal $fEqIdealOrderedMonomial getMonomialMonomial$fNFDataOrderedMonomialIsStrongMonomialOrderWeightedEliminationOrderEliminationOrderEliminationTypeIsMonomialOrder WeightOrder WeightProxy ProductOrderGradedGrlexGrevlexRevlexLexIsOrder cmpMonomial MonomialOrderfromListisRelativelyPrime totalDegreelexrevlexgradedgrlexgrevlex productOrder productOrder' weightOrder lcmMonomial gcdMonomialdivs isPowerOftryDivvarMonomeliminationOrdersOnesweightedEliminationOrder castMonomial scastMonomialchangeMonomialOrderchangeMonomialOrderProxywithStrongMonomialOrdercmpAnyMonomial orderMonomial$$f:=>IsMonomialOrder'IsMonomialOrder$fIsMonomialOrder'ordn $fOrdSized$fOrdOrderedMonomial!$fEliminationTypeNatknWeightOrder"$fEliminationTypeNatknProductOrder$fEliminationTypeknmLex$fIsMonomialOrderkWeightOrder$fIsMonomialOrderkProductOrder$fIsMonomialOrdernLex$fIsMonomialOrdernGrevlex$fIsMonomialOrdernGrlex$fIsOrdernGrlex $fIsOrdernLex$fIsOrdernRevlex$fIsOrdernGrevlex$fIsOrderkProductOrder$fIsOrdernWeightOrder$fIsMonomialOrdernGraded$fIsOrdernGraded$fUnitalOrderedMonomial$fDivisionOrderedMonomial$fMultiplicativeOrderedMonomial$fShowOrderedMonomial$fWrappedOrderedMonomial$fRewrappedOrderedMonomialt $fShowLex$fEqLex$fOrdLex $fShowRevlex $fEqRevlex $fOrdRevlex $fShowGrevlex $fEqGrevlex $fOrdGrevlex $fShowGrlex $fEqGrlex $fOrdGrlex $fReadGraded $fShowGraded $fEqGraded $fOrdGraded$fEqOrderedMonomial$fHashableOrderedMonomial ShowSCoeffNegativeVanishedOneCoeffPositive PrettyCoeff showsCoeffIsOrderedPolynomialMOrdercoeffterms leadingTermleadingMonomial leadingCoefforderedMonomialsfromOrderedMonomial toPolynomial polynomial>**<_TermsdiffmapMonomialMonotonic IsPolynomial CoefficientArityliftMapsubst substWithsArity'sArityarity injectCoeff injectCoeff' monomialsterms'coeff' constantTerm fromMonomial toPolynomial' polynomial' totalDegree'var mapCoeff'>|**|<!*_Terms' mapMonomial CoeffRing liftMapCoeff substCoeffoneNormmaxNormmonoize sPolynomial pDivModPolycontentpp injectVarsvarsshowsCoeffAsTermshowsCoeffWithOpshowPolynomialWithshowsPolynomialWithdivModPolynomial modPolynomial divPolynomial$fPrettyCoeffFraction$fPrettyCoeffFraction0$fPrettyCoeffComplex$fPrettyCoeffRatio$fPrettyCoeffWord8$fPrettyCoeffWord32$fPrettyCoeffWord16$fPrettyCoeffWord64$fPrettyCoeffInt8$fPrettyCoeffInt32$fPrettyCoeffInt16$fPrettyCoeffInt64$fPrettyCoeffInt$fPrettyCoeffNatural$fPrettyCoeffInteger $fCoeffRingr PolynomialOrderedPolynomial_termscastPolynomialscastPolynomialmapCoeff normalizevarXsubstUnivariateevalUnivariateevalsubstVarallVars changeOrderchangeOrderProxygetTermstransformMonomial orderedByshiftR homogenize unhomogenizereversal padeApproxminpolRecurrent!$fDecidableUnitsOrderedPolynomial!$fIntegralDomainOrderedPolynomial&$fZeroProductSemiringOrderedPolynomial$fEuclideanOrderedPolynomial$fPIDOrderedPolynomial$fUFDOrderedPolynomial$fGCDDomainOrderedPolynomial!$fUnitNormalFormOrderedPolynomial&$fDecidableAssociatesOrderedPolynomial'$fZeroProductSemiringOrderedPolynomial0 $fDecidableZeroOrderedPolynomial$fNumOrderedPolynomial$fShowOrderedPolynomial!$fCharacteristicOrderedPolynomial$$fRightModuleScalarOrderedPolynomial#$fLeftModuleScalarOrderedPolynomial$fAbelianOrderedPolynomial$fCommutativeOrderedPolynomial$fSemiringOrderedPolynomial!$fMultiplicativeOrderedPolynomial$fUnitalOrderedPolynomial%$fRightModuleNaturalOrderedPolynomial$$fLeftModuleNaturalOrderedPolynomial$fMonoidalOrderedPolynomial$fAdditiveOrderedPolynomial%$fRightModuleIntegerOrderedPolynomial$$fLeftModuleIntegerOrderedPolynomial$fGroupOrderedPolynomial$fRigOrderedPolynomial$fRingOrderedPolynomial$fEqOrderedPolynomial$fRewrappedOrderedPolynomialt$fWrappedOrderedPolynomial&$fIsOrderedPolynomialOrderedPolynomial$fIsPolynomialOrderedPolynomial$fHashableOrderedPolynomial$fNFDataOrderedPolynomial$fOrdOrderedPolynomial%logBase2ceilingLogBase2Unipol divModUnipoldivModUnipolByMult naiveMult karatsubamapCoeffUnipol $fShowUnipol$fIsOrderedPolynomialUnipol$fIsPolynomialUnipol$fDecidableZeroUnipol$fMonoidalUnipol$fRightModuleScalarUnipol $fRingUnipol $fRigUnipol$fSemiringUnipol$fLeftModuleScalarUnipol$fCommutativeUnipol$fUnitalUnipol $fGroupUnipol$fMultiplicativeUnipol$fLeftModuleIntegerUnipol$fRightModuleIntegerUnipol$fLeftModuleNaturalUnipol$fRightModuleNaturalUnipol$fAbelianUnipol$fAdditiveUnipol $fOrdUnipol $fEqUnipol $fNumUnipol$fEuclideanUnipol $fPIDUnipol $fUFDUnipol$fIntegralDomainUnipol$fZeroProductSemiringUnipol$fGCDDomainUnipol$fUnitNormalFormUnipol$fDecidableAssociatesUnipol$fDecidableUnitsUnipol$fIsLabel"x"Unipol$fHashableUnipol$fNFDataUnipol IsSubsetOf LabUnipolLabPolynomial' LabPolynomialLabelPolynomialunLabelPolynomial IsUniqueList canonicalMap canonicalMap'$fIsSubsetOfaxsys"$fIsOrderedPolynomialLabPolynomial$fIsPolynomialLabPolynomial$fOrdLabPolynomial$fEqLabPolynomial$fDecidableZeroLabPolynomial $fRightModuleScalarLabPolynomial$fLeftModuleScalarLabPolynomial$fRingLabPolynomial$fRigLabPolynomial$fSemiringLabPolynomial$fMonoidalLabPolynomial $fLeftModuleIntegerLabPolynomial!$fRightModuleIntegerLabPolynomial $fLeftModuleNaturalLabPolynomial!$fRightModuleNaturalLabPolynomial$fGroupLabPolynomial$fUnitalLabPolynomial$fCommutativeLabPolynomial$fAbelianLabPolynomial$fMultiplicativeLabPolynomial$fAdditiveLabPolynomial$fNumLabPolynomial$fShowLabPolynomial$fIsLabelsymbLabPolynomial$fIsUniqueListxsrepeatedSquare fermatTestmodPow isPseudoPrime$fReadPrimeResult$fShowPrimeResult$fEqPrimeResult$fOrdPrimeResult FiniteFieldpowerelementsF naturalReprmodNatmodNat'reifyPrimeFieldwithPrimeFieldordermodRatmodRat' $fRandomF$fFiniteFieldF$fCharacteristicF$fCommutativeF $fFractionalF $fDivisionF $fEuclideanF$fZeroProductSemiringF$fPIDF$fUFDF $fGCDDomainF$fIntegralDomainF$fUnitNormalFormF$fDecidableAssociatesF$fDecidableUnitsF $fUnitalF$fDecidableZeroF$fRingF$fRigF $fSemiringF $fAbelianF$fGroupF$fRightModuleIntegerF$fLeftModuleIntegerF$fRightModuleNaturalF$fLeftModuleNaturalF $fMonoidalF$fMultiplicativeF $fAdditiveF$fNumF $fNormedF$fEqF$fPrettyCoeffF$fShowF $fNFDataFConwayConwayPolynomialconwayPolynomialaddConwayPolynomials conwayFileIsGF'GFGF'modPolymodVecgenerateIrreduciblewithIrreduciblereifyGF' linearRepGF linearRepGF'withGF' primitiveconway$fFiniteFieldGF'$fZeroProductSemiringGF' $fIsGF'kpnf$fFractionalGF'$fNumGF'$fEuclideanGF'$fPIDGF'$fUFDGF'$fGCDDomainGF'$fIntegralDomainGF'$fUnitNormalFormGF'$fZeroProductSemiringGF'0$fDecidableAssociatesGF' $fDivisionGF'$fCharacteristicGF'$fDecidableUnitsGF'$fDecidableZeroGF' $fRingGF'$fCommutativeGF'$fRigGF' $fSemiringGF' $fUnitalGF'$fMultiplicativeGF' $fAbelianGF' $fGroupGF'$fRightModuleIntegerGF'$fLeftModuleIntegerGF'$fRightModuleNaturalGF'$fLeftModuleNaturalGF' $fMonoidalGF' $fAdditiveGF'$fPrettyCoeffGF' $fShowGF'$fEqGF' SugarStrategyGradedStrategyGrevlexStrategyNormalStrategySelectionStrategyWeight calcWeightisGroebnerBasissimpleBuchbergerprimeTestBuchberger buchbergersyzygyBuchbergersyzygyBuchbergerWithStrategy calcWeight'minimizeGroebnerBasisreduceMinimalGroebnerBasiscalcGroebnerBasisWithcalcGroebnerBasisWithStrategycalcGroebnerBasis isIdealMemberthEliminationIdealthEliminationIdealWithunsafeThEliminationIdealWith intersectionquotByPrincipalIdeal quotIdealsaturationByPrincipalIdealsaturationIdeal resultanthasCommonFactor lcmPolynomial gcdPolynomial%$fSelectionStrategyTYPEnSugarStrategy&$fSelectionStrategyTYPEnGradedStrategy'$fSelectionStrategyTYPEnGrevlexStrategy&$fSelectionStrategyTYPEnNormalStrategy$fReadNormalStrategy$fShowNormalStrategy$fEqNormalStrategy$fOrdNormalStrategy$fReadGrevlexStrategy$fShowGrevlexStrategy$fEqGrevlexStrategy$fOrdGrevlexStrategy$fReadGradedStrategy$fShowGradedStrategy$fEqGradedStrategy$fOrdGradedStrategy$fReadSugarStrategy$fShowSugarStrategy$fEqSugarStrategy$fOrdSugarStrategyQIdealQuotientquotRepr vectorRepmatRepr'matRep0 multUnamb multWithTablestandardMonomials'standardMonomials genQuotVars' genQuotVarsmodIdealgBasis' modIdeal' reifyQuotient withQuotientreduceisZeroDimensional $fNumQuotient$fRightModuleScalarQuotient$fLeftModuleScalarQuotient$fRingQuotient $fRigQuotient$fUnitalQuotient$fSemiringQuotient$fMultiplicativeQuotient$fRightModuleIntegerQuotient$fLeftModuleIntegerQuotient$fRightModuleNaturalQuotient$fLeftModuleNaturalQuotient$fShowQuotient$fNFDataQuotient $fEqQuotient$fAbelianQuotient$fGroupQuotient$fMonoidalQuotient$fAdditiveQuotientdistinctDegFactorequalDegreeSplitMequalDegreeFactorMsquareFreePartsquareFreeDecomp factorise clearDenomfactorQBigPrime factorHensel henselStepIntervallowerupper Algebraicfactors intersectincludes sqFreePart algebraicnthRoot'nthRootstrumimprove presultant realRoots complexRoots realPartPoly imagPartPolyrepresentative approximateapproxFractional$fTriviallyInvolutiveAlgebraic#$fInvolutiveMultiplicationAlgebraic$fShowInterval$fMultiplicativeInterval$fAdditiveInterval$fFractionalAlgebraic$fNumAlgebraic$fUnitNormalFormAlgebraic$fDecidableAssociatesAlgebraic$fDecidableUnitsAlgebraic$fDecidableZeroAlgebraic$fDivisionAlgebraic$fZeroProductSemiringAlgebraic$fRingAlgebraic$fRigAlgebraic$fAbelianAlgebraic$fSemiringAlgebraic$fUnitalAlgebraic$fCommutativeAlgebraic$fOrdAlgebraic $fEqAlgebraic$fRightModuleScalarAlgebraic$fLeftModuleScalarAlgebraic$fMultiplicativeAlgebraic$fRightModuleFractionAlgebraic$fRightModuleIntegerAlgebraic$fLeftModuleFractionAlgebraic$fLeftModuleIntegerAlgebraic$fRightModuleNaturalAlgebraic$fLeftModuleNaturalAlgebraic$fMonoidalAlgebraic$fGroupAlgebraic$fAdditiveAlgebraic$fShowAlgebraic $fEqInterval $fOrdInterval recoverRatrationalChineseRemainderEntry $fReadEntry $fShowEntry $fEqEntry $fOrdEntryMatrixidxvalue $fReadMatrix $fShowMatrixheightwidthempty fromListsgetDiagdiagProdtracetoListsswapRowsswapColsclearRowclearColscaleRowscaleColgetRowgetColaddRowaddColinBoundindex! combineRows combineColsncolsnrowsidentitydiag rowVector colVectortoRowstoCols<--><||>catRowcatCol switchRows switchColscmap transposezeroMatrowCountcolCount traverseRow traverseColstructuredGaussstructuredGauss'nonZeroEntriesmultWithVector nonZeroRows nonZeroCols substMatrixsolveWiedemannrankLMsplitIndependentDirstriangulateModular henselLift solveHensel$fSemiringMatrix$fAbelianMatrix $fGroupMatrix$fRightModuleScalarMatrix$fLeftModuleScalarMatrix$fAdditiveMatrix$fMonoidalMatrix$fLeftModuleIntegerMatrix$fRightModuleIntegerMatrix$fLeftModuleNaturalMatrix$fRightModuleNaturalMatrix$fMultiplicativeMatrix$fUnitalSquare$fMonoidMaxEntry$fSemigroupMaxEntry $fEqMatrix$fReadMaxEntry$fShowMaxEntry $fEqMaxEntry $fOrdMaxEntry$fReadDirection$fShowDirection $fEqDirection$fOrdDirection $fShowSquare $fEqSquare$fAdditiveSquare$fMultiplicativeSquare$fRightModuleSquare$fLeftModuleSquareElemfromColsfromRowszerotrans buildMatrixdelta companiongaussReductiondetrankWithinverse inverseWith$fMatrixMatrix$fMatrixMatrix0$fMatrixMatrix1solveM solveWithsolve' subspMatrixsolveViaCompanion matrixRep reductionunivPoly solveLinearradical isRadicalfglmfglmMap"algebra-4.3-D0DMsPAxIWsLDh5SSB4VxMNumeric.Algebra.ClassModule GHC.TypeLitsKnownNat KnownSymbolghc-prim GHC.TypesNatSymbol+*^<=?- CmpSymbolCmpNat TypeError sameSymbolsameNat someSymbolVal someNatVal symbolVal'natVal' symbolValnatValSomeNat SomeSymbol<= ErrorMessageText:<>::$$:ShowType Data.Proxy asProxyTypeOfProxyKProxy%singletons-2.2-6SkjYmmXU9eBuVim0IqZfqData.SingletonsSomeSingSingKind DemoteRepfromSingtoSingSingIsingSing!Data.Singletons.Prelude.InstancesSTrueSFalse withSingIcoercestart=~====becauseProof.Propositional withWitnessIsTrueWitnessData.Singletons.Prelude.EnumPEnumPredSuccToEnumFromEnum EnumFromToEnumFromThenToSEnumsSuccsPredsToEnum sFromEnum sEnumFromTosEnumFromThenToData.Singletons.Prelude.NumPNum:+:-AbsNegate:*Signum FromIntegerSNum%:+%:-%:*sNegatesAbssSignum sFromInteger!Data.Singletons.TypeLits.Internal withKnownNatData.Singletons.Prelude.OrdPOrdMaxMin:>:<Compare:<=:>=SOrdsCompare%:<%:<=%:>%:>=sMaxsMin$sized-0.2.1.0-85JdzGbJwJx7NlODSUfIQfData.Sized.BuiltinNilLNilR Data.SizedunsafeFromList'unsafeFromList zipWithSamegenerate singletonsIndexData.Type.Natural.ClassmkSNatQQData.Type.Natural.Class.Order truncMinusLeqminPlusTruncMinus%:-. sLeqCongR sLeqCongLsLeqCong sFlipOrdering FlipOrdering PeanoOrderleqToCmpeqlCmpEQeqToRefl flipCompareltToNeq leqNeqToLT succLeqToLTltToLeqgtToLeq ltToSuccLeqcmpZeroleqToGTcmpZero'zeroNoLTltRightPredSucccmpSuccltSucc cmpSuccStepR ltSuccLToLTleqToLTleqZeroleqSucc fromLeqView leqViewReflviewLeq leqWitnessleqStepleqNeqToSuccLeqleqRefl leqSuccStepR leqSuccStepL leqReflexiveleqTrans leqAntisymm plusMonotone leqZeroElim plusMonotoneL plusMonotoneRplusLeqLplusLeqRplusCancelLeqRplusCancelLeqLsuccLeqZeroAbsurdsuccLeqZeroAbsurd' succLeqAbsurdsuccLeqAbsurd' notLeqToLeqleqSucc'leqToMingeqToMinminCommminLeqLminLeqR minLargestleqToMaxgeqToMaxmaxCommmaxLeqRmaxLeqLmaxLeast leqReversed lneqSuccLeq lneqReversedlneqToLTltToLneqlneqZerolneqSucc succLneqSucclneqRightPredSucc lneqSuccStepL lneqSuccStepRplusStrictMonotonemaxZeroLmaxZeroRminZeroLminZeroR minusSucclneqZeroAbsurd minusPlus:-. coerceLeqR coerceLeqLLeqViewLeqZeroLeqSuccDiffNat"Data.Type.Natural.Class.Arithmetic minusCongR minusCongL minusCong multCongR multCongLmultCongsuccCong plusCongR plusCongLplusCongsOnesZeroZeroOneS ZeroOrSuccIsZeroIsSuccIsPeano succOneCongsuccInjsuccInj' succNonCyclic induction plusMinus plusMinus' plusZeroL plusSuccL plusZeroR plusSuccRplusComm plusAssoc multZeroL multSuccL multZeroR multSuccRmultCommmultOneRmultOneLplusMultDistribmultPlusDistribminusNilpotent multAssoc plusEqCancelL plusEqCancelRsuccAndPlusOneLsuccAndPlusOneRpredSucc zeroOrSucc plusEqZeroL plusEqZeroR predUniquemultEqSuccElimLmultEqSuccElimR minusZero multEqCancelRsuccPred multEqCancelLsPred'IsMonomialOrder'head'calcOrderWeightcalcOrderWeight'D:R:UnwrappedOrderedMonomialGHC.ShowShowSNumeric.Domain.InternalgcddefaultShowsOrdCoeffordVecdecZeroshowPolynomialWithVarsisConstantMonomialRationalGHC.PrimseqGHC.Listfilterzip Data.TuplefstsndGHC.Base otherwise$GHC.Real fromIntegral realToFrac Control.MonadguardjoinGHC.EnumBoundedminBoundmaxBoundEnumenumFrom enumFromThenenumFromThenTo enumFromTofromEnumtoEnumsuccpred GHC.ClassesEq==/= GHC.FloatFloatingcoshsinhcossinatanhacoshasinhtanhatanacosasintanlogBase**sqrtlogexppi FractionalIntegral toIntegermoddivMonad>>=>>returnfailFunctorfmap<$GHC.NumNumsignumabsOrd>=<>minmaxcompareGHC.ReadRead readsPrecreadListReal toRational RealFloatatan2isIEEEisNegativeZeroisDenormalized isInfiniteisNaN scaleFloat significandexponent encodeFloat decodeFloat floatRange floatDigits floatRadixRealFracfloorceilingroundtruncateproperFractionShow showsPrecshowshowListData.Typeable.InternalTypeable Data.StringIsString fromString Applicativepure<*>*><* Data.FoldableFoldablefoldlfoldl'foldl1foldrfoldr'foldr1foldMapnulllengthmaximumminimumelemData.Traversable TraversablesequencemapMtraverse sequenceAMonoidmemptymappendmconcatBoolFalseTrueCharDoubleFloatIntGHC.IntInt32Int64 integer-gmpGHC.Integer.TypeIntegerMaybeNothingJustOrderingLTEQGTIOWordGHC.WordWord8Word32Word64 Data.EitherEitherLeftRightStringnotidliftMeitherNumeric.Quadrance.Class Quadrance quadranceNumeric.Order.LocallyFiniteLocallyFiniteOrderNumeric.Rig.Ordered OrderedRigNumeric.Order.Additive AdditiveOrderNumeric.Module.RepresentablefromIntegerRepfromNaturalReponeRepmulReptimesRep subtractRepminusRep negateRep sinnumRepzeroRep sinnum1pRepaddRepNumeric.Field.Fraction denominator numeratorFractionRatioNumeric.Dioid.ClassDioidNumeric.Covector antipodeMcoinvMinvM convolveMcounitMcomultMunitMmultMCovector$*Numeric.Domain.GCDgcd'Numeric.Field.ClassFieldNumeric.Domain.EuclideanchineseRemainderprseuclidDomainIntegralDomaindivides maybeQuot GCDDomainreduceFractionlcmUFDPIDegcd EuclideandegreedividequotremNumeric.Algebra.InvolutiveInvolutiveMultiplicationadjointInvolutiveSemiringTriviallyInvolutiveInvolutiveAlgebrainvTriviallyInvolutiveAlgebraInvolutiveCoalgebracoinvTriviallyInvolutiveCoalgebraInvolutiveBialgebraTriviallyInvolutiveBialgebraNumeric.Algebra.Commutative CommutativeCommutativeAlgebraCocommutativeCoalgebraCommutativeBialgebra%Numeric.Algebra.Unital.UnitNormalForm leadingUnitUnitNormalForm splitUnitNumeric.Semiring.ZeroProductZeroProductSemiringNumeric.Ring.Local LocalRingNumeric.Ring.Division DivisionRingNumeric.Ring.ClassRingNumeric.Rng.ClassRngNumeric.Rig.CharacteristiccharWordcharIntCharacteristiccharNumeric.Rig.ClassRig fromNaturalNumeric.Decidable.Zero DecidableZeroisZeroNumeric.Decidable.UnitsrecipUnitWholerecipUnitIntegralDecidableUnits recipUnitisUnit^?Numeric.Decidable.AssociatesisAssociateWholeisAssociateIntegralDecidableAssociates isAssociateNumeric.Algebra.IdempotentpowBand pow1pBandBandIdempotentAlgebraIdempotentBialgebraNumeric.Algebra.Hopf HopfAlgebraantipodeNumeric.Algebra.DivisionDivisionrecip/\\DivisionAlgebra recipriocalNumeric.Algebra.UnitalproductUnitalonepow productWith UnitalAlgebraunitCounitalCoalgebracounit BialgebraNumeric.Algebra.Factorable Factorable factorWithNumeric.Additive.GroupGroupnegatesubtracttimessinnumIdempotentsumproduct1Multiplicativepow1p productWith1SemiringAlgebramult Coalgebracomult LeftModule.* RightModule*.MonoidalsinnumsumWithNumeric.Additive.Classsinnum1pIdempotentsum1Additivesinnum1psumWith1 Partitionable partitionWithAbelian Idempotent GHC.NaturalNaturalbytestring-0.10.8.1Data.ByteString.Internal ByteStringcontainers-0.5.7.1Data.IntMap.BaseIntMapData.IntSet.BaseIntSet Data.Map.BaseMap Data.SequenceSeq Data.Set.BaseSetfilepath-1.4.1.0System.FilePath.Posix<.>'hashable-1.2.4.0-Ctl752zbguF6QanxurLOm2Data.Hashable.ClassHashablehash hashWithSalt*lifted-base-0.2.3.8-LSXKdE75JIl3uzD4Y2GaXOControl.Exception.Lifted onExceptionfinallybracketOnErrorbracket_bracketuninterruptibleMask_uninterruptibleMaskmask_masktryJusttry handleJusthandle catchJustcatchthrowIO#text-1.2.2.1-9Yh8rJoh8fO2JMLWffT3QsData.Text.InternalData.Text.Encoding encodeUtf83unordered-containers-0.2.7.2-5FvILdEAx092lIiPKjCugu Data.HashSetHashSetData.HashMap.BaseHashMap&vector-0.11.0.0-6uB77qGCxR6GPLxI2sqsX3 Data.VectorVectorData.Vector.Unboxed.BaseUnbox*basic-prelude-0.6.1-IfkhRuZU18I1i3XwoymVgG CorePreludeSVectorUVector LByteStringLTextequatingprintreadArgsterror BasicPreludemap++concat intercalatetshowreadIO appendFile textToString ltextToStringfpToText fpFromText fpToString decodeUtf8readMay||&& GHC.Exception SomeExceptionGHC.Errerror undefined MonadPlusmzeromplus=<<whenliftM2liftM3liftM4liftM5apconst.flip$!untilasTypeOfcurryuncurryswap Data.MaybemaybeisJust isNothing fromMaybe maybeToList listToMaybe catMaybesmapMaybeheadunconstaillastinitfoldl1'scanlscanl1scanl'scanrscanr1iteraterepeat replicatecycle takeWhile dropWhiletakedropsplitAtspanbreakreverselookup!!zip3zipWithzipWith3unzipunzip3showsshowChar showString showParenevenodd Data.Functor<$>void Data.Functionon Data.BoolboolText.ParserCombinators.ReadPReadS readParenData.OrdDown comparing Data.OldList dropWhileEnd stripPrefix elemIndex elemIndices findIndex findIndices isPrefixOf isSuffixOf isInfixOfnubnubBydeletedeleteByunionunionBy intersectBy intersperse partitioninsertinsertBy genericLength genericTake genericDropgenericSplitAt genericIndexgenericReplicatezip4zip5zip6zip7zipWith4zipWith5zipWith6zipWith7unzip4unzip5unzip6unzip7deleteFirstsBygroupgroupByinitstails subsequences permutationssortsortBysortOnunfoldrlinesunlineswordsunwordsleftsrightspartitionEithers Exception toException fromExceptiondisplayExceptionGHC.IO.ExceptionIOError IOException userErrorGHC.IOFilePath Data.Monoid<>mapM_forM_ sequence_asummsum concatMapandoranyall maximumBy minimumBynotElemfindForeign.StorableStorable IOErrorTypeioError Text.ReadreadsSystem.IO.Error tryIOError mkIOErrorisAlreadyExistsErrorisDoesNotExistErrorisAlreadyInUseError isFullError isEOFErrorisIllegalOperationisPermissionError isUserErroralreadyExistsErrorTypedoesNotExistErrorTypealreadyInUseErrorType fullErrorType eofErrorTypeillegalOperationErrorTypepermissionErrorType userErrorTypeisAlreadyExistsErrorTypeisDoesNotExistErrorTypeisAlreadyInUseErrorTypeisFullErrorTypeisEOFErrorTypeisIllegalOperationErrorTypeisPermissionErrorTypeisUserErrorTypeioeGetErrorTypeioeGetErrorStringioeGetLocation ioeGetHandleioeGetFileNameioeSetErrorTypeioeSetErrorStringioeSetLocation ioeSetHandleioeSetFileName modifyIOErrorannotateIOError catchIOError System.IOputCharputStrputStrLngetChargetLine getContentsinteractreadFile writeFilereadLnforM mapAccumL mapAccumRfilterM>=><=<forever mapAndUnzipMzipWithM zipWithM_foldMfoldM_ replicateM replicateM_unless<$!>mfilter Data.ListisSubsequenceOfControl.Monad.IO.ClassMonadIOliftIO0algebraic-prelude-0.1.0.1-3sNF7QAsKNSILQsGTrsRJ9AlgebraicPrelude ifThenElse^^ normaliseUnit fromRational fromInteger' fromIntegerWrapNum unwrapNumWrapFractionalunwrapFractional WrapIntegralunwrapIntegral WrapAlgebra unwrapAlgebraAddrunAddMultrunMult Control.Arrowfirstsecond***&&&<|>transformers-0.5.2.0Control.Monad.Trans.Classlift runUnipol normaliseUP recipBinPow modVarPower reversalIMreversalIMWith leadingTermIM%!! varUnipoldiffIMappermute0 _suppressWraps UniqueList UniqueList'permute PrimeResult Composite ProbablyPrimePrimebinRep splitFactorrunFproxyFpows parseLineplusOptoPoly buildInstance$fReifiesTYPEConwayUnipol$fConwayPolynomial1110runGF' vecToPoly polyToVecproxyGF' groebnerTestLengthReplicaterunLengthReplicate.=%= combinationslengthReplicate lengthCong replicateCong'reflection-2.1.2-Lwt0A3NRHka1hAeW9AOpLqData.ReflectionreflectTable ZeroDimIdeal_gBasis_vBasis multTable quotRepr_buildMultTable stdMonoms buildQIdeal asProxyOf repeatHensel multiHensel traceCharTwofactorSquareFreeyun charUnipol powerUnipolpthRoot wrapSQFFactorsecondM<@>factorSqFreeQBPfactorHenselSqFreerecipMod modFractioncomb normalizeMod isSquareFreerealRootsIrreducibleeqn _strumseq _intervalstdGenFromEntropy viewNthRoot showsNthRootnegateAminusAdisjointscaleplusInt rootSumPolyplusAisIn algebraic'catchershiftP stabilize nthRootRatnthRootRatCeilnthRootRatFloornthRootInterval rootMultPolymultIntmultAdefEqnimproveNonzerorecipIntrecipA rootRecipPolysizebisect signChange countRootsIn countChangeIn signChangeAt rootBound isolateRoots improveWithpowAiterateImproveiterateImproveStrum fromFraction complexRoots'FGLMEnv_lMap_gLex_bLex_proced _monomialMachinebLexgLexlMapmonomialprocedlook.==%==image_value_idx_rowNext_colNext _coefficients _rowStart _colStart_height_widthcolNextrowNextnewEntry BuildState_colMap_rowMap_curIdx coefficientscolStartrowStart GaussianState_input_output_prevCol _heavyCols_curRow_detAcccolMapcurIdxrowMapSquare runSquare DirectionRowColumnMaxEntry_weightentrycurRowdetAcc heavyColsinputoutputprevColswapperscaleDirclearAtclearDirmapDir traverseDir traverseDirMgetDirigetDirlenLcountLcoordLnthLstartLnextLaddDirperp combineDircatDir dirVectortoDirs appendDir clearZerodirCountnewGaussianState getHeaviest matListView prettyMat prettyEntry nonZeroDirstestCasetoSquare krylovMinpolintDet shiftHalf.!lcm'gaussReduction'swapIJtoIntLA matToLists toComplexmainLoop toContinue nextMonomialbeta