-U      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvw x y z { | } ~   5Type for permutations, considered as group elements. Vx .^ g returns the image of a vertex or point x under the action of the permutation g .Construct a permutation from a list of cycles  |For example, p [[1,2,3],[4,5]]K returns the permutation that sends 1 to 2, 2 to 3, 3 to 1, 4 to 5, 5 to 4 'A trick: g^-1 returns the inverse of g 'g ~^ h returns the conjugate of g by h Eb -^ g returns the image of an edge or block b under the action of g <_C n returns generators for Cn, the cyclic group of order n <_S n returns generators for Sn, the symmetric group on [1..n] >_A n returns generators for An, the alternating group on [1..n] SGiven generators for a group, return a (sorted) list of all elements of the group. V |Implemented using a naive closure algorithm, so only suitable for small groups (|G| < 10000) VGiven generators for a group, return the order of the group (the number of elements). V |Implemented using a naive closure algorithm, so only suitable for small groups (|G| < 10000) I          !"#   vGiven generators for a group, determine whether a permutation is a member of the group, using Schreier-Sims algorithm qGiven generators for a group, return a (sorted) list of all elements of the group, using Schreier-Sims algorithm tGiven generators for a group, return the order of the group (the number of elements), using Schreier-Sims algorithm $%&'()*+,-./0123456789:KDatatype for graphs, represented as a list of vertices and a list of edges h |Both the list of vertices and the list of edges, and also the 2-element lists representing the edges, < |are required to be in ascending order, without duplicates 8combinationsOf k xs returns the subsets of xs of size k L |If xs is in ascending order, then the returned list is in ascending order =Safe constructor for graph from lists of vertices and edges. L |graph (vs,es) checks that vs and es are valid before returning the graph. The cyclic graph on n vertices !The complete graph on n vertices 1The complete bipartite graph on m and n vertices ) kb :: (Integral t) => t -> t -> Graph t <The complete bipartite graph on m left and n right vertices 4 kb :: (Integral t) => t -> t -> Graph (Either t t) >;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~  !"#$%&'()*+,-./01234567  !"#$%&'()*+,-./01234567 67453210/.-,+*)('&%$#"!   !"#$%&'()*+,-./01234556778.Phantom type for an elimination term ordering v |In the ordering, xis come before yjs come before zks, but within the xis, or yjs, or zks, grevlex ordering is used 90Phantom type representing grevlex term ordering :-Phantom type representing glex term ordering ;,Phantom type representing lex term ordering <=>?89:;<=>?<=;:98>?89:;<==>? @ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_ ^_\]YZ[XWVUTSRQPONMLKJIHGFEDCBA@ @ABCDEFGHIJKLMNOPQRSTUVWXYZ[Z[\]]^__`"Type for multivariate polynomials f |ord is a phantom type defining how terms are ordered, r is the type of the ring we are working over r |For example, a common choice will be MPoly Grevlex Q, meaning polynomials over Q with the grevlex term ordering ab*Create a variable with the supplied name. f |By convention, variable names should usually be a single letter followed by none, one or two digits cdefghijklmnopqrs*Convert a polynomial to lex term ordering t+Convert a polynomial to glex term ordering u.Convert a polynomial to grevlex term ordering v1`abcdefghijklmnopqrstuv`abdefghijklmncpqrostuv`aabcdefghijklmnopqrstuv wmGiven a list of polynomials over a field, return a Groebner basis for the ideal generated by the polynomials www xyz{|}~Axyz{|}~~}|{zyxxyz{|}~         " !"#$%&'()*+,-./01234 56789:;<=>?@ABCDEFGHIJ KLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~4'"      % !"#$%&'()*+,-./012345 6789:;<=>?@ABCDE%FGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghi !"#$%&'()'(*+,-./0123456789:;<=>?@ABCDEFGHIJKLMMNOPQRSTTUVWXYZ[\]^_`abcdefghijklmnopqrssttuvwxy.z{|}~                                     T M         +'%)      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOP(QRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~                                                        w           !                                      w |      z!"#$%&'x()*+,-./ED0123456)789:;<=>?@ABCDEFGHIJKLMNOPQ!R,STUVWXYZ[\]^_`abcdefghijklmno>pqrstuvwKxyz{|}~|>?y8{zy9     HaskellForMaths-0.1.6"Math.Algebra.Group.StringRewriting#Math.Algebra.Group.PermutationGroupMath.Algebra.Group.SchreierSimsMath.Combinatorics.GraphMath.Common.IntegerAsType!Math.Algebra.Commutative.MonomialMath.Algebra.Field.BaseMath.Algebra.Commutative.MPolyMath.Algebra.Commutative.GBasisMath.Algebra.Field.Extension"Math.Algebra.NonCommutative.NCPoly)Math.Algebra.NonCommutative.TensorAlgebra%Math.Projects.KnotTheory.LaurentMPolyMath.Projects.KnotTheory.Braid&Math.Projects.KnotTheory.TemperleyLieb%Math.Projects.KnotTheory.IwahoriHecke!Math.Combinatorics.FiniteGeometryMath.Combinatorics.Design'Math.Combinatorics.StronglyRegularGraphMath.Combinatorics.HypergraphMath.Projects.RootSystem(Math.Projects.ChevalleyGroup.ExceptionalMath.Common.ListSetMath.Combinatorics.GraphAuts#Math.Algebra.NonCommutative.GSBasisMath.Algebra.LinearAlgebra&Math.Projects.ChevalleyGroup.ClassicalS PermutationP.^p^-~^-^_C_S_AeltsorderisMemberGraphGcombinationsOf nullGraphck fromDigitsT97T89T83T79T73T71T67T61T59T53T47T43T41T37T31T29T23T19T17T13T11T7T5T3T2TOneTZeroTMinus1M IntegerAsTypevalueElimGrevlexGlexLexMonomialconvertMsupportMF97F89F83F79F73F71F67F61F59F53F47F43F41F37F31F29F23F19F17F13F11F7F5F3F2 FiniteFieldeltsFqbasisFqFpQMPolyMPvarabdstuvwxyzx0x1x2x3toLextoGlex toGrevlextoElimgb QSqrtMinus5 QSqrtMinus3 QSqrtMinus2 QSqrtMinus1QSqrt7QSqrt5QSqrt3QSqrt2SqrtF32 ConwayF32F27 ConwayF27F25 ConwayF25F16 ConwayF16F9ConwayF9F8ConwayF8F4ConwayF4ExtensionFieldExtPolynomialAsTypepvalueUPolyUP quotRemUP InvertibleinvVarZYXNPolyNPWeylGensDBasisE LaurentMPolyLPLaurentMonomialLM BraidGensLPQTemperleyLiebGensIwahoriHeckeGensT ZeroOneStarStarOneZeroDesign DesignVertexBC Incidence HypergraphHTypeFAbasisEltOctonionOrewrite rewrite''splitSubstring findOverlap knuthBendix1ordpairshortlex knuthBendix2merge knuthBendix3 knuthBendixnfss_s1s2s3tri_D toListSet isListSetunion intersect\\symDiffdisjointisSubsetrotateL fromPairs fromPairs'toPairsfromListsupp 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desargues2isVertexTransitiveisEdgeTransitive->^isArcTransitiveisArcTransitive'findArcsisnArcTransitiveis2ArcTransitiveis3ArcTransitiveisDistanceTransitiverefine isGraphAut graphAuts1 graphAuts2 graphAuts3 isSingleton graphAuts removeGens graphAutsNew graphIsosisIsodiffsdegMdividesMproperlyDividesMlcmMgcdMcoprimeM numeratorQ denominatorQextendedEuclid primitiveEltpowerscharf2f3f5f7f11f13f17f19f23f29f31f37f41f43f47f53f59f61f67f71f73f79f83f89f97cmpTerm mergeTermscollectx_ convertMPvarLexvarElimltlmdegmulTdivTdividesTproperlyDividesTlcmT.* quotRemMP%%divModMPdivMPmodMPinjecttoMonictoZsubstsupportsPolyisGBgb1pairWithreducegb2pairs!gb2bgb3bcmpFstmergeBygb4bsugar cmpNormalcmpSuggb3gb4toUPoly<+><*>convertmonomialmodUPextendedEuclidUPembedpolysf4x4f8x8f9x9f16x16f25x25f27x27f32x32sqrt2sqrt3sqrt5sqrt7i sqrtminus2 sqrtminus3 sqrtminus5divMlc quotRemNPremNPremNP2gb'gb2'mbasisQAe_e1e2e3e4dim tensorBasis extRelationsextnf exteriorBasis symRelationssymnfsymmetricBasis weylRelationsweylnf 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