úÎ~õlJÄ      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~€‚ƒ„…†‡ˆ‰Š‹ŒŽ‘’“”•–—˜™š›œžŸ ¡¢£¤¥¦§¨©ª«¬­®¯°±²³´µ¶·¸¹º»¼½¾¿ÀÁÂÃNone% ÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖרÙÚÛÜÝÞ   $ ÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖרÙÚÛÜÝÞ Safe-Inferred ßàáâãä   NoneHType-level constraint to check whether it forms polynomial ring or not. 3n-ary polynomial ring over some noetherian ring R. EMonomial order which can be use to calculate n-th elimination ideal. F This should judge it as bigger that contains variables to eliminate. Class for Monomial orders. %Data.Proxy provides kind-polymorphic ä) data-type, but due to bug of GHC 7.4.1, F It canot be used as kind-polymorphic. So I define another type here. *Graded order from another monomial order. "&Graded lexicographical order. Same as  Graded Lex. $/Graded reversed lexicographical order. Same as  Graded Revlex. &Reversed lexicographical order (Lexicographical order *5Class to lookup ordering from its (type-level) name. ,9A wrapper for monomials with a certain (monomial) order. /BMonomial order (of degree n). This should satisfy following laws:  (1) Totality: forall a, b (a < b || a == b || b < a)  (2) Additivity: a  =b == a + c <= b + c  (3) Non-negative: forall a, 0 <= a 0)Monomorphic representation for monomial. 36N-ary Monomial. IntMap contains degrees for each x_i. 4!convert NAry list into Monomial. å2apply monomial ordering to monomorphic monomials. 73Lexicographical order. This *is* a monomial order. 8@Reversed lexicographical order. This is *not* a monomial order. 9"Convert ordering into graded one. ::Graded lexicographical order. This *is* a monomial order. ;CGraded reversed lexicographical order. This *is* a monomial order. Acoefficient for a degree. æ;We provide Num instance to use trivial injection R into R[X].  Do not use signum or abs. ç By Hilbert'\s finite basis theorem, a polynomial ring over a noetherian ring is also a noetherian ring. èbFor simplicity, we choose grevlex for the default monomial ordering (for the sake of efficiency). é(Special ordering for ordered-monomials. êëìíîï !"#$%&'()*+,-./01234å56789:;<=>?@ABCDEFGHIJðKLMNñOPQRSTUVWXYZ[\]^æòóôõö÷øùúûüýþÿçèé     T  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^T3/?@789:;<=Z>56AUVNBDMWXCEHJIFGKLY\[ST4  OPQR]^,-.012$%&'()"# !*+v  êëìíîï !"#$%&'()*+,-./01234å56789:;<=>?@ABCDEFGHIJðKLMNñOPQRSTUVWXYZ[\]^æòóôõö÷øùúûüýþÿçèé     None2_`abcdefghijklmnopqrstuvwxyz{ !"#$%&'()*+,-./012_`abcdefghijklmnopqrstuvwxyz{ghijfcdeklmnopq_`abrstuvwxyz{*_`abcdefghijklmnopqrstuvwxyz{ !"#$%&'()*+,-./012None'3456789:;<=>?@ABCDEFG|H}~IJKLMNO€‚PƒQ|}~€‚ƒ|}~€‚ƒ3456789:;<=>?@ABCDEFG|H}~IJKLMNO€‚PƒQNone„„Sugar strategy. This chooses the pair with the least phantom homogenized degree and then break the tie with the given strategy (say s). ˆ8Choose the pair with the least LCM(LT(f), LT(g)) w.r.t. $ order. Š Buchberger'8s normal selection strategy. This selects the pair with ? the least LCM(LT(f), LT(g)) w.r.t. current monomial ordering. Œ1Type-class for selection strategies in Buchberger' s algorithm. FCalculate a polynomial quotient and remainder w.r.t. second argument. :Remainder of given polynomial w.r.t. the second argument. ‘;A Quotient of given polynomial w.r.t. the second argument. ’The Naive buchberger'=s algorithm to calculate Groebner basis for the given ideal. “ Buchberger'Cs algorithm slightly improved by discarding relatively prime pair. ”7Calculate Groebner basis applying (modified) Buchberger' s algorithm.  This function is same as •. • Buchberger'Os 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'Rt have strong reason to avoid this function, this function is recommended to use. –apply buchberger',s algorithm using given selection strategy. —ECalculate the weight of given polynomials w.r.t. the given strategy.  Buchberger'Bs algorithm proccesses the pair with the most least weight first.  This function requires the Ord: instance for the weight; this constraint is not required  in the Ž> because of the ease of implementation. So use this function. ™;Reduce minimum Groebner basis into reduced Groebner basis. š\Caliculating reduced Groebner basis of the given ideal w.r.t. the specified monomial order. ›\Caliculating reduced Groebner basis of the given ideal w.r.t. the specified monomial order. œ8Caliculating reduced Groebner basis of the given ideal. 9Test if the given polynomial is the member of the ideal. RFTest if the given polynomial can be divided by the given polynomials. ž'Calculate n-th elimination ideal using  ordering. ŸRCalculate n-th elimination ideal using the specified n-th elimination type order.  RCalculate 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. ¡-An intersection ideal of given ideals (using ). ¢&Ideal quotient by a principal ideals. £#Ideal quotient by the given ideal. ¤!Saturation by a principal ideal. ¥Saturation ideal ¦6Calculate resultant for given two unary polynomimals. SQChoose the pair with the least LCM(LT(f), LT(g)) w.r.t. graded current ordering. 1„…†‡ˆ‰Š‹ŒŽ‘’“TUVW”•X–—˜™Yš›œRžŸ ¡¢£¤¥¦§ZS[\]^$„…†‡ˆ‰Š‹ŒŽ‘’“”•–—˜™š›œžŸ ¡¢£¤¥¦§$‘œš›”•’“™˜–ŒŽ—ˆ‰Š‹„…†‡¡žŸ £¢¥¤¦§+„…†‡ˆ‰Š‹ŒŽ‘’“TUVW”•X–—˜™Yš›œRžŸ ¡¢£¤¥¦§ZS[\]^None ¨Synonym ©*Calculate a intersection of given ideals. ªTCalculate saturation ideal by the principal ideal generated by the second argument. «Calculate saturation ideal. ¬1Calculate ideal quotient of I by principal ideal ­(Calculate the ideal quotient of I of J. ¾8Computes the ideal with specified variables eliminated. À Computes nth elimination ideal. Â7Calculates resultants for given two unary-polynomials. Ã<Determin if given two unary polynomials have common factor. ¨©_ª«¬­®¯°±²³´µ¶·¸¹º»¼½¾¿ÀÁÂÃ`3"#$%&'()*+„…†‡ˆ‰Š‹ŒŽ—¨©ª«¬­®¯°±²³´µ¶·¸¹º»¼½¾¿ÀÁÂÃ3¨®°±¯²³´µº¼»¸¹¶·½©À¿Á¾­¬«ªÂÃ()&'"#$%*+ŒŽŠ‹„…ˆ‰†‡—¨©_ª«¬­®¯°±²³´µ¶·¸¹º»¼½¾¿ÀÁÂÃ` Safe-Inferredabcdefghijklmno       !!"#$%%&&''(())**+,--./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abc3ddefFghijklmnopqrstuGvwxyz{|}~~€€‚ƒ„…†‡ˆ‰Š‹ŒŽ‘’“”•–—˜™š›œž—š›˜™…Ÿ‡† ¡’ˆ¢‰£‹Œ¤“¥¦”•œ§¨©ª«¬­®¯°±²³´µ¶·¸¹º»¼½¾¿ÀÁÂÃÄÂÃÅÂÃÆÂÃÇÂÃÈÂÃÈÉÊËÌÍÎÏÐÑÑÒÓÔÕÖרÙÚÛÜÝÞßàáâãäåæçèéêëìíîïðñòóôõö÷øùúûüýþÿ      !"#$%&'()*+,-./0123456789:;<=>?@ABCDBCEBCFBCGBCHBCIBCJBCKBCBCBCLBCMBCNBCOPcomputational-algebra-0.3.0.0Algebra.Ring.NoetherianAlgebra.InternalAlgebra.Ring.Polynomial#Algebra.Ring.Polynomial.MonomorphicAlgebra.Ring.Polynomial.ParserAlgebra.Algorithms.Groebner'Algebra.Algorithms.Groebner.Monomorphic MonomorphicIdealNoetherianRing addToIdealtoIdeal appendIdeal generators filterIdealprincipalIdealmapIdealtoProxy CoefficientEpsPositiveNegativeZero IsPolynomial 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$fIsOrderLex$fIsOrderRevlex$fIsOrderGrevlex$fIsOrderProductOrder$fIsOrderWeightOrder$fToWeightVector:$fToWeightVector[]$fIsMonomialOrderGraded$fIsOrderGraded3$fWrappedVectorVectorOrderedMonomialOrderedMonomial$fMonomorphicableNatVector$fOrdOrderedMonomial'#$fMonomorphicableNatOrderedMonomial$fMonomorphicableNat:.:%$fMonomorphicableNatOrderedPolynomial$fShowPolynomial$fShowPolynomial0$fAdditivePolynomial$fAbelianPolynomial$fSemiringPolynomial$fRightModuleIntegerPolynomial$fLeftModuleIntegerPolynomial$fRightModuleNaturalPolynomial$fLeftModuleNaturalPolynomial$fMonoidalPolynomial$fUnitalPolynomial$fRigPolynomial$fGroupPolynomial$fRingPolynomial$fMultiplicativePolynomial$fCommutativePolynomial$fNoetherianRingPolynomial$fShowVariable$fNumPolynomial MemoTabletbl_skip tbl_delimitertbl_expression tbl_letter tbl_variabletbl_variableWithPowertbl_expr tbl_expr_tailtbl_term tbl_monomstbl_monoms_tailtbl_fact tbl_fact_tail tbl_monomial tbl_number tbl_integer tbl_naturalskip delimiterletterexpr expr_tailtermmonoms monoms_tailfact fact_tailtoPolyn$fMemoTableMemoTable groebnerTest!$fSelectionStrategyGradedStrategy.=%=padVec combinations=@=monoize $fSelectionStrategySugarStrategy"$fSelectionStrategyGrevlexStrategy!$fSelectionStrategyNormalStrategy$fOrdMonomorphic$fEqMonomorphicfreshVar$fGroebnerablermonomorphic-0.0.3.0Data.Type.MonomorphicviaPolyliftPolywithPolymorhicmonomorphicComposedemoteComposeddemote'Comp:.:demotepromoteMonomorphicRepMonomorphicable