l      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUV W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~                                                                       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~    !!!"## $$$$$$$$$$$$$$$ $ $ $ $ $$$$$$$$$$$$$$$$$%% %!&"&#&$'%'&''((()(*(+),)-).)/*0*1*2*3+4+5,6,7,8,9,:,;,<,=,>,?,@,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/u0v0w0x0y0z0{0|0}0~0000000000000011111111111222222222223333333333333333333334444444444444444444444444444444444444444444444444444444444444444445555555555555555 5 5 5 5 5555555555555555555 5!5"5#5$5%5&5'5(5)5*5+5,5-5.5/505152535455565758696:6;6<6=6>6?6@6A6B6C6D6E6F6G6H6I6J6K6L6M6N6O6P6Q6R6S6T6U6V6W6X6Y6Z6[6\6]6^6_6`6a6b6c6d6e6f6g6h6i6j6k6l6m6n6o6p6q6r6s6t7u7v7w7x7y7z7{7|7}7~7777777777777777777777777777777777777777777777777777777777788888888888888888888888888888888888888888888888888888888888888899999999999999999 9 9 9 9 9999999999999999999 9!9"9#9$9%9&9'9(9)9*9+9,9-9.9/909192939495969798999:9;9<9=9>9?9@9A9B9C9D9E9F9G9H:I:J:K:L:M:N:O:P:Q:R:S:T:U:V:W:X:Y:Z:[:\:]:^:_:`:a:b:c:d:e:f:g:h:i:j:k:l:m:n:o:p:q:r:s:t:u:v:w:x:y:z:{:|:}:~::::;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<== = = = = =============>>>>>> >!>">#?$?%?&?'?(?)?*?+?,?-?.?/?0@1@2@3@4@5@6@7@8@9@:@;@<@=@>@?@@@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@ZA[A\A]A^A_A`AaAbAcAdAeAfAgAhAiAjAkAlAmAnAoBpBqBrBsBtBuBvBwBxByBzB{B|B}B~BBBBBBBBBBBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDDDDDDDDDHSafe  Safe !"#$%&'()*+ !"#$%&'()*+Safe,-./0123456789:;<=>?@,-./,-./,-./0123456789:;<=>?@SafeABCDEFGHIJKLMNOPQRSTUVWXYZ[\] AFECBHDGIJ ABCDEFGHIJABCDEFGHIJKLMNOPQRSTUVWXYZ[\]SafeQR^/An additive semigroup with idempotent addition.  a + a = a_an additive abelian semigroup a + b = b + aaTpartitionWith f c returns a list containing f a b for each a b such that a + b = c, b (a + b) + c = a + (b + c) sinnum 1 a = a sinnum (2 * n) a = sinnum n a + sinnum n a sinnum (2 * n + 1) a = sinnum n a + sinnum n a + adsinnum1p n r = sinnum (1 + n) r?^_`abcdefghijklmnopqrstuvwxyz{|}~ ^_`abcdefg bcdef_^g`a;^_`abcdefghijklmnopqrstuvwxyz{|}~c6Safe9:;<=QR An additive monoid zero + a = a = a + zero?An associative algebra built with a free module over a semiringdA pair of an additive abelian semigroup, and a multiplicative semigroup, with the distributive laws: a(b + c) = ab + ac -- left distribution (we are a LeftNearSemiring) (a + b)c = ac + bc -- right distribution (we are a [Right]NearSemiring)WCommon notation includes the laws for additive and multiplicative identity in semiring.If you want that, look at Rig instead.'Ideally we'd use the cyclic definition: ^class (LeftModule r r, RightModule r r, Additive r, Abelian r, Multiplicative r) => Semiring r[to enforce that every semiring r is an r-module over itself, but Haskell doesn't like that.A multiplicative semigroup1the free commutative coalgebra over a set and Int?the free commutative coalgebra over a set and a given semigroup,the free commutative band coalgebra over Int#the free commutative band coalgebraThe tensor Hopf algebra The tensor Hopf algebra0Every coalgebra gives rise to an algebra by vector space duality classically. Sadly, it requires vector space duality, which we cannot use constructively. The dual argument only relies in the fact that any constructive coalgebra can only inspect a finite number of coefficients, which we CAN exploit.The tensor algebraThe tensor algebra      !"#$%&'()*+,-./0123456789:;<=>?      !"#$%&'()*+,-./0123456789:;<=>?7778Safe9:;<=QR@ABCDEFGHIJKLMNOPQRSTU@ABCD@ABCD@ABCDEFGHIJKLMNOPQRSTUA6D7 SafeVb`factorWith f c` returns a non-empty list containing `f a b` for all `a, b` such that `a * b = c`.^Results of factorWith f 0 are undefined and may result in either an error or an infinite list. VWXYZ[\]VWVWVWXYZ[\] Safe9;<=^HA bialgebra is both a unital algebra and counital coalgebra where the  and b) are compatible in some sense with the  and `. That is to say that  and b8 are a coalgebra homomorphisms or (equivalently) that  and ` are an algebra homomorphisms.aHAn associative unital algebra over a semiring, built using a free module3^_`abcdefghijklmnopqrstuvwxyz{|}~ ^_`abcedfg cdefgab_`^.^_`abcdefghijklmnopqrstuvwxyz{|}~e8 Safe9;<= 778 Safe9;<=>A HopfAlgebra algebra on a semiring, where the module is free.When antipode . antipode = idM and antipode is an antihomomorphism then we are an InvolutiveBialgebra with inv = antipode as well Safe9;<=;An multiplicative semigroup with idempotent multiplication.  a * a = aESafeSafeAb is an associate of a if there exists a unit u such that b = a*uTThis relationship is symmetric because if u is a unit, u^-1 exists and is a unit, so b*u^-1 = a*u*u^-1 = aSafea rectangular band is a nowhere commutative semigroup. That is to say, if ab = ba then a = b. From this it follows classically that aa = a and that such a band is isomorphic to the following structureSafe<Safe  FSafeSafe A Ring without (n)egation            Safe3:NB: we're using the boolean semiring, not the boolean ring !"#$%&'()*+,-./0123 ! ! !"#$%&'()*+,-./0123Safe9;4A Ring without an identity.454445Safe6789:;<=>?@ABCDEFGH6786786789:;<=>?@ABCDEFGHSafe9;IJIIIJSafeKKKKSafeL,A zero-product semiring has no zero divisors "a * b = 0 implies a == 0 || b == 0LMNOLLLMNOSafe6PQRSTPQRSPQQRSPQQRSTSafe9;<=QRX&A commutative multiplicative semigroup)UVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}UVWXXWVU)UVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}Safe9;<=QR  adjoint = id'adjoint (x + y) = adjoint x + adjoint yAn semigroup with involution 'adjoint a * adjoint b = adjoint (b * a)Z~ ~ ~W~GSafeNone09;<=IMHSafe69;Euclidean (degree) function on r.Division algorithm. a  b) calculates quotient and remainder of a divided by b.>let (q, r) = divide a p in p*q + r == a && degree r < degree q8An integral domain is a commutative domain in which 1"`0.+(Integral) domain is the integral semiring.Extended euclidean algorithm.>euclid f g == xs ==> all (\(r, s, t) -> r == f * s + g * t) xs 77777"Safe9;SafeList of  (m_i, v_i)f with f = v_i (mod v_i) #Safe9; Safe!Safe9;SafeSafe$Safe9:;<=ILinear functionals from elements of an (infinite) free module to a scalarSconvolveM antipodeM return = convolveM return antipodeM = comultM >=> uncurry joinM$       "     0%Safe  &Safe!"#!"!"!"#'Safe$%&$%$%$%&(Safe'()*'()'()'()*)Safe+,-.+-,+,-+,-.*Safe/012/10/01/012+Safe9;343334,Safe$)<=T5Convenient synonym for 6.6Fraction field k(D) of  domain D.#56789:;<=>?@ABCDEFGHIJKLMNOPQRSTUV5678968957"56789:;<=>?@ABCDEFGHIJKLMNOPQRSTUV77-None): W!`Additive.(+)` default definitionXIJ default definitionYKL default definitionZKM default definition[NO default definition\`Group.(-)` default definition]NP default definition^NQ default definition_'`Multiplicative.(*)` default definition`RS default definitionaTU default definitionbVW default definition WXYZ[\]^_`ab WXYZ[\]^_`ab WXYZ[\]^_`ab WXYZ[\]^_`ab.Safec(z + x <= z + y = x <= y = x + z <= y + z cdefghijkcc cdefghijk/Safe lmnopqrstll lmnopqrst0Safeuvwxyz{|}~uvwxuvwxuvwxyz{|}~1Safe09;<=2Safe09;<= the dual incidence algebra basis3Safe9;<=NoneAFECBHDGI^_`abcdefg@ABCDVW^_`abcedfg   !467IKUVWX~3WXYZ[\]^_`abclubcdef_^g`a@ABCDXcdefgVW34  67KIab_`^~WUV !ABCDEFGHIlcuWXYZ[\]^_`ab4None 09;<=DQR2half of the Cayley-Dickson quaternion isomorphism 6 !" !"35None 09;<=DR#dual number basis, D^2 = 0. D /= 0.3      !"#$%&'()*+ $% $%0      !"#$%&'()*+6None 09;<=DR089:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefg+-,89:;<+,-:;<89-89:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefg7None 09;<=DQR{/Cayley-Dickson quaternion isomorphism (one way)the trivial diagonal coalgebrathe quaternion algebra9tuvwxyz{|}~!"'()tuvwxyz{|}!"'()vwxyztu{}|4tuvwxyz{|}~8None 09;<=DR#dual number basis, D^2 = 0. D /= 0.3 $% $%09None09;<=DIMQRT                     :None 09;<=DR\&the hyperbolic trigonometric coalgebra^the trivial diagonal algebra.HIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstu+-,HIJKL+,-JKLHI+HIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstu;None 09;<=DQR/Cayley-Dickson quaternion isomorphism (one way) dual quaternion comultiplicationthe trivial diagonal algebra9!"'()!"'()4<None 09;<=DQR4/10/011=None An element x is nilpotent if there exists n s.t.  pow1p x n is zero.          >None  !" !"?None<= #$%&'()*+,-./#$%#$% #$%&'()*+,-./@None9;<=DR0Hlinear maps from elements of a free module to another free module over r >f $# x + y = (f $# x) + (f $# y) f $# (r .* x) = r .* (f $# x) Map r b a; represents a linear mapping from a free module with basis a over r to a free module with basis b over r.\Note well the reversed direction of the arrow, due to the contravariance of change of basis!]This way enables we can employ arbitrary pure functions as linear maps by lifting them using {, or build them by using the monad instance for Map r b. As a consequence Map is an instance of, well, almost everything.2-extract a linear functional from a linear map3(inefficiently) combine a linear combination of basis vectors to make a map. arrMap :: (Monoidal r, Semiring r) => (b -> [(r, a)]) -> Map r b a arrMap f = Map $ k b -> sum [ r * k a | (r, a) <- f b ]7Dconvolution given an associative algebra and coassociative coalgebra+0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXY 0123456789: 0124635:987*0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXY020ANone9;<=Z[The endomorphism ring of an abelian group or the endomorphism semiring of an abelian monoid .http://en.wikipedia.org/wiki/Endomorphism_ringZ[\]^_`abcdefghijklmnZ[\]^_Z[\]^_Z[\]^_`abcdefghijklmnBNone9;<=o *http://en.wikipedia.org/wiki/Opposite_ringopqrstuvwxyz{|}~opqopqopqrstuvwxyz{|}~CNone9;<=<The free Ring given a Rng obtained by adjoining Z, such that RngRing r = n*1 + r!This ring is commonly denoted r^.(The rng homomorphism from r to RngRing rigiven a rng homomorphism from a rng r into a ring s, liftRngHom yields a ring homomorphism from the ring `r^` into s.DNone9;<=XYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~IJKLM      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~NOPQ              R S                                                                                            !"#$$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWTUXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~VW      !""#$%&'()*+,H-H.H/H0H1H2H3H4H5H6H7H8H9H:H;H<H=>?#@#A B$C$C$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*x*y*z+{+|,},~,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,------------........./////////00000000000000000000000111111111112222222222233333333333333333333344444444444444 4 4 4 4 4444444444444444444 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<!<"<#<$<%<&<'<(<)<*<+<,<-<.</<0=1=2=3=4=5=6=7=8=9=:=;=<===>=?=@=A=B=C>D>D>E>F>G>H>I>J>K?L?L?M?N?O?P?Q?R?S?T?U?V?W@X@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@{@|@}@~@@AAAAAAAAAAAAAAAAAAAAABBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBCCCCCCCCCCCCCCCCCCCCCCCDDDDDDDDDDDDDDDDDDDD HHHHHH,~3999==X@A"algebra-4.3-D0DMsPAxIWsLDh5SSB4VxMNumeric.AlgebraNumeric.Partial.SemigroupNumeric.Partial.MonoidNumeric.Partial.GroupNumeric.Order.ClassNumeric.Additive.ClassNumeric.Algebra.ClassNumeric.Additive.GroupNumeric.Algebra.FactorableNumeric.Algebra.UnitalNumeric.Algebra.DivisionNumeric.Algebra.HopfNumeric.Algebra.IdempotentNumeric.Decidable.AssociatesNumeric.Band.RectangularNumeric.Decidable.UnitsNumeric.Decidable.ZeroNumeric.Rig.ClassNumeric.Rig.CharacteristicNumeric.Rng.ClassNumeric.Ring.ClassNumeric.Ring.DivisionNumeric.Ring.LocalNumeric.Semiring.ZeroProduct%Numeric.Algebra.Unital.UnitNormalFormNumeric.Algebra.CommutativeNumeric.Algebra.InvolutiveNumeric.Coalgebra.CategoricalNumeric.Domain.EuclideanNumeric.Domain.PIDNumeric.Domain.UFDNumeric.Domain.GCDNumeric.Domain.IntegralNumeric.Domain.ClassNumeric.Field.ClassNumeric.Covector#Numeric.Algebra.Distinguished.ClassNumeric.Algebra.Complex.ClassNumeric.Algebra.Dual.Class Numeric.Algebra.Quaternion.Class"Numeric.Coalgebra.Hyperbolic.Class%Numeric.Coalgebra.Trigonometric.ClassNumeric.Dioid.ClassNumeric.Field.FractionNumeric.Module.RepresentableNumeric.Order.AdditiveNumeric.Rig.OrderedNumeric.Order.LocallyFiniteNumeric.Algebra.IncidenceNumeric.Coalgebra.IncidenceNumeric.Quadrance.ClassNumeric.Algebra.ComplexNumeric.Algebra.DualNumeric.Algebra.HyperbolicNumeric.Algebra.QuaternionNumeric.Coalgebra.DualNumeric.Coalgebra.GeometricNumeric.Coalgebra.HyperbolicNumeric.Coalgebra.QuaternionNumeric.Coalgebra.TrigonometricNumeric.Decidable.Nilpotent Numeric.Exp Numeric.Log Numeric.MapNumeric.Ring.EndomorphismNumeric.Ring.OppositeNumeric.Ring.RngNumeric.Rng.ZeroNumeric.Band.ClassNumeric.Module.ClassNumeric.Semiring.InvolutiveNumeric.Domain.InternalAdditivesinnum1pMonoidalzerosinnumGroupnegatesubtracttimesUnitaloneRig fromNaturalRing fromIntegerbase GHC.NaturalNaturalPartialSemigrouppadd$fPartialSemigroupEither$fPartialSemigroup(,,,,)$fPartialSemigroup(,,,)$fPartialSemigroup(,,)$fPartialSemigroup(,)$fPartialSemigroup()$fPartialSemigroupBool$fPartialSemigroupMaybe$fPartialSemigroupWord64$fPartialSemigroupWord32$fPartialSemigroupWord16$fPartialSemigroupWord8$fPartialSemigroupWord$fPartialSemigroupInt64$fPartialSemigroupInt32$fPartialSemigroupInt16$fPartialSemigroupInt8$fPartialSemigroupNatural$fPartialSemigroupInteger$fPartialSemigroupInt PartialMonoidpzero$fPartialMonoid(,,,,)$fPartialMonoid(,,,)$fPartialMonoid(,,)$fPartialMonoid(,)$fPartialMonoidMaybe$fPartialMonoid()$fPartialMonoidWord64$fPartialMonoidWord32$fPartialMonoidWord16$fPartialMonoidWord8$fPartialMonoidWord$fPartialMonoidInt64$fPartialMonoidInt32$fPartialMonoidInt16$fPartialMonoidInt8$fPartialMonoidNatural$fPartialMonoidInteger$fPartialMonoidInt$fPartialMonoidBool PartialGrouppnegatepminus psubtract$fPartialGroup(,,,,)$fPartialGroup(,,,)$fPartialGroup(,,)$fPartialGroup(,)$fPartialGroup()$fPartialGroupNatural$fPartialGroupWord64$fPartialGroupWord32$fPartialGroupWord16$fPartialGroupWord8$fPartialGroupWord$fPartialGroupInt64$fPartialGroupInt32$fPartialGroupInt16$fPartialGroupInt8$fPartialGroupInteger$fPartialGroupIntOrder<~<>~>~~/~order comparableorderOrd $fOrder(,,,,) $fOrder(,,,) $fOrder(,,) $fOrder(,) $fOrder() $fOrderSet $fOrderWord64 $fOrderWord32 $fOrderWord16 $fOrderWord8 $fOrderWord$fOrderNatural $fOrderInt64 $fOrderInt32 $fOrderInt16 $fOrderInt8 $fOrderInt$fOrderInteger $fOrderBool IdempotentAbelian Partitionable partitionWith+sumWith1sum1sinnum1pIdempotent$fIdempotent(,,,,)$fIdempotent(,,,)$fIdempotent(,,)$fIdempotent(,)$fIdempotent(->)$fIdempotentBool$fIdempotent()$fAbelian(,,,,)$fAbelian(,,,) $fAbelian(,,) $fAbelian(,)$fAbelianWord64$fAbelianWord32$fAbelianWord16$fAbelianWord8 $fAbelianWord$fAbelianInt64$fAbelianInt32$fAbelianInt16 $fAbelianInt8 $fAbelianInt$fAbelianNatural$fAbelianInteger $fAbelianBool $fAbelian() $fAbelian(->)$fPartitionable(,,,,)$fPartitionable(,,,)$fPartitionable(,,)$fPartitionable(,)$fPartitionable()$fPartitionableNatural$fPartitionableBool$fAdditive(,,,,)$fAdditive(,,,)$fAdditive(,,) $fAdditive(,) $fAdditive()$fAdditiveWord64$fAdditiveWord32$fAdditiveWord16$fAdditiveWord8$fAdditiveWord$fAdditiveInt64$fAdditiveInt32$fAdditiveInt16$fAdditiveInt8 $fAdditiveInt$fAdditiveInteger$fAdditiveNatural$fAdditiveBool$fAdditive(->)sumWithModule RightModule*. LeftModule.* CoalgebracomultAlgebramultSemiringMultiplicative*pow1p productWith1product1 pow1pIntegralsumsinnumIdempotent$fMonoidal(,,,,)$fMonoidal(,,,)$fMonoidal(,,) $fMonoidal(,) $fMonoidal()$fMonoidal(->)$fMonoidalWord64$fMonoidalWord32$fMonoidalWord16$fMonoidalWord8$fMonoidalWord$fMonoidalInt64$fMonoidalInt32$fMonoidalInt16$fMonoidalInt8 $fMonoidalInt$fMonoidalInteger$fMonoidalNatural$fMonoidalBool $fModulerm$fRightModuler(,,,,)$fRightModuler(,,,)$fRightModuler(,,)$fRightModuler(,)$fRightModule()m$fRightModuler(->)$fRightModuler()$fRightModuleIntegerWord64$fRightModuleNaturalWord64$fRightModuleIntegerWord32$fRightModuleNaturalWord32$fRightModuleIntegerWord16$fRightModuleNaturalWord16$fRightModuleIntegerWord8$fRightModuleNaturalWord8$fRightModuleIntegerWord$fRightModuleNaturalWord$fRightModuleIntegerInt64$fRightModuleNaturalInt64$fRightModuleIntegerInt32$fRightModuleNaturalInt32$fRightModuleIntegerInt16$fRightModuleNaturalInt16$fRightModuleIntegerInt8$fRightModuleNaturalInt8$fRightModuleIntegerInt$fRightModuleNaturalInt$fRightModuleIntegerInteger$fRightModuleNaturalInteger$fRightModuleNaturalNatural$fRightModuleNaturalBool$fLeftModuler(,,,,)$fLeftModuler(,,,)$fLeftModuler(,,)$fLeftModuler(,)$fLeftModule()m$fLeftModuler(->)$fLeftModuler()$fLeftModuleIntegerWord64$fLeftModuleNaturalWord64$fLeftModuleIntegerWord32$fLeftModuleNaturalWord32$fLeftModuleIntegerWord16$fLeftModuleNaturalWord16$fLeftModuleIntegerWord8$fLeftModuleNaturalWord8$fLeftModuleIntegerWord$fLeftModuleNaturalWord$fLeftModuleIntegerInt64$fLeftModuleNaturalInt64$fLeftModuleIntegerInt32$fLeftModuleNaturalInt32$fLeftModuleIntegerInt16$fLeftModuleNaturalInt16$fLeftModuleIntegerInt8$fLeftModuleNaturalInt8$fLeftModuleIntegerInt$fLeftModuleNaturalInt$fLeftModuleIntegerInteger$fLeftModuleNaturalInteger$fLeftModuleNaturalNatural$fLeftModuleNaturalBool$fCoalgebrarIntMap$fCoalgebrarMap$fCoalgebrarIntSet$fCoalgebrarSet$fCoalgebrarSeq$fCoalgebrar[]$fCoalgebrar(,,,,)$fCoalgebrar(,,,)$fCoalgebrar(,,)$fCoalgebrar(,)$fCoalgebrar()$fCoalgebrar(->)$fAlgebrar(,,,,)$fAlgebrar(,,,)$fAlgebrar(,,) $fAlgebrar(,)$fAlgebrarIntSet $fAlgebrarSet $fAlgebrar() $fAlgebrarSeq $fAlgebrar[] $fAlgebra()a$fSemiring(->)$fSemiring(,,,,)$fSemiring(,,,)$fSemiring(,,) $fSemiring(,) $fSemiring()$fSemiringWord64$fSemiringWord32$fSemiringWord16$fSemiringWord8$fSemiringWord$fSemiringInt64$fSemiringInt32$fSemiringInt16$fSemiringInt8 $fSemiringInt$fSemiringBool$fSemiringNatural$fSemiringInteger$fMultiplicative(->)$fMultiplicative(,,,,)$fMultiplicative(,,,)$fMultiplicative(,,)$fMultiplicative(,)$fMultiplicative()$fMultiplicativeWord64$fMultiplicativeWord32$fMultiplicativeWord16$fMultiplicativeWord8$fMultiplicativeWord$fMultiplicativeInt64$fMultiplicativeInt32$fMultiplicativeInt16$fMultiplicativeInt8$fMultiplicativeInt$fMultiplicativeInteger$fMultiplicativeNatural$fMultiplicativeBool- $fGroup(,,,,) $fGroup(,,,) $fGroup(,,) $fGroup(,) $fGroup() $fGroupWord64 $fGroupWord32 $fGroupWord16 $fGroupWord8 $fGroupWord $fGroupInt64 $fGroupInt32 $fGroupInt16 $fGroupInt8 $fGroupInt$fGroupInteger $fGroup(->) Factorable factorWith$fFactorable(,,,,)$fFactorable(,,,)$fFactorable(,,)$fFactorable(,)$fFactorable()$fFactorableBool BialgebraCounitalCoalgebracounit UnitalAlgebraunitpow productWithproduct$fBialgebrarSeq$fBialgebrar[]$fBialgebrar(,,,,)$fBialgebrar(,,,)$fBialgebrar(,,)$fBialgebrar(,)$fBialgebrar()$fCounitalCoalgebrarSeq$fCounitalCoalgebrar[]$fCounitalCoalgebrar(,,,,)$fCounitalCoalgebrar(,,,)$fCounitalCoalgebrar(,,)$fCounitalCoalgebrar(,)$fCounitalCoalgebrar()$fCounitalCoalgebrar(->)$fUnitalAlgebrarSeq$fUnitalAlgebrar[]$fUnitalAlgebrar(,,,,)$fUnitalAlgebrar(,,,)$fUnitalAlgebrar(,,)$fUnitalAlgebrar(,)$fUnitalAlgebrar() $fUnital(->)$fUnital(,,,,) $fUnital(,,,) $fUnital(,,) $fUnital(,) $fUnital()$fUnitalWord64$fUnitalWord32$fUnitalWord16 $fUnitalWord8 $fUnitalWord$fUnitalNatural $fUnitalInt64 $fUnitalInt32 $fUnitalInt16 $fUnitalInt8 $fUnitalInt$fUnitalInteger $fUnitalBoolDivisionAlgebra recipriocalDivisionrecip/\\^$fDivision(->)$fDivision(,,,,)$fDivision(,,,)$fDivision(,,) $fDivision(,) $fDivision() HopfAlgebraantipode$fHopfAlgebrar(,,,,)$fHopfAlgebrar(,,,)$fHopfAlgebrar(,,)$fHopfAlgebrar(,)IdempotentBialgebraIdempotentCoalgebraIdempotentAlgebraBand pow1pBandpowBand$fIdempotentBialgebrarh$fIdempotentCoalgebrar(,,,,)$fIdempotentCoalgebrar(,,,)$fIdempotentCoalgebrar(,,)$fIdempotentCoalgebrar(,)$fIdempotentCoalgebrar()$fIdempotentCoalgebrarIntSet$fIdempotentCoalgebrarSet$fIdempotentAlgebrar(,,,,)$fIdempotentAlgebrar(,,,)$fIdempotentAlgebrar(,,)$fIdempotentAlgebrar(,)$fIdempotentAlgebrar()$fIdempotentAlgebrarIntSet$fIdempotentAlgebrarSet $fBand(,,,,) $fBand(,,,) $fBand(,,) $fBand(,) $fBandBool$fBand()DecidableAssociates isAssociateisAssociateIntegralisAssociateWhole$fDecidableAssociates(,,,,)$fDecidableAssociates(,,,)$fDecidableAssociates(,,)$fDecidableAssociates(,)$fDecidableAssociates()$fDecidableAssociatesWord64$fDecidableAssociatesWord32$fDecidableAssociatesWord16$fDecidableAssociatesWord8$fDecidableAssociatesWord$fDecidableAssociatesNatural$fDecidableAssociatesInt64$fDecidableAssociatesInt32$fDecidableAssociatesInt16$fDecidableAssociatesInt8$fDecidableAssociatesInt$fDecidableAssociatesInteger$fDecidableAssociatesBoolRect $fBandRect$fMultiplicativeRect$fSemigroupoidTYPERect$fEqRect $fOrdRect $fShowRect $fReadRectDecidableUnits recipUnitisUnit^?recipUnitIntegralrecipUnitWhole$fDecidableUnits(,,,,)$fDecidableUnits(,,,)$fDecidableUnits(,,)$fDecidableUnits(,)$fDecidableUnits()$fDecidableUnitsWord64$fDecidableUnitsWord32$fDecidableUnitsWord16$fDecidableUnitsWord8$fDecidableUnitsWord$fDecidableUnitsNatural$fDecidableUnitsInt64$fDecidableUnitsInt32$fDecidableUnitsInt16$fDecidableUnitsInt8$fDecidableUnitsInt$fDecidableUnitsInteger$fDecidableUnitsBool DecidableZeroisZero$fDecidableZero(,,,,)$fDecidableZero(,,,)$fDecidableZero(,,)$fDecidableZero(,)$fDecidableZero()$fDecidableZeroWord64$fDecidableZeroWord32$fDecidableZeroWord16$fDecidableZeroWord8$fDecidableZeroWord$fDecidableZeroNatural$fDecidableZeroInt64$fDecidableZeroInt32$fDecidableZeroInt16$fDecidableZeroInt8$fDecidableZeroInt$fDecidableZeroInteger$fDecidableZeroBool $fRig(,,,,) $fRig(,,,) $fRig(,,)$fRig(,)$fRig() $fRigWord64 $fRigWord32 $fRigWord16 $fRigWord8 $fRigWord $fRigInt64 $fRigInt32 $fRigInt16 $fRigInt8$fRigInt $fRigBool $fRigNatural $fRigIntegerCharacteristiccharcharIntcharWord$fCharacteristic(,,,,)$fCharacteristic(,,,)$fCharacteristic(,,)$fCharacteristic(,)$fCharacteristic()$fCharacteristicWord64$fCharacteristicWord32$fCharacteristicWord16$fCharacteristicWord8$fCharacteristicWord$fCharacteristicInt64$fCharacteristicInt32$fCharacteristicInt16$fCharacteristicInt8$fCharacteristicInt$fCharacteristicNatural$fCharacteristicInteger$fCharacteristicBoolRng$fRngr fromIntegral $fRing(,,,,) $fRing(,,,) $fRing(,,) $fRing(,)$fRing() $fRingWord64 $fRingWord32 $fRingWord16 $fRingWord8 $fRingWord $fRingInt64 $fRingInt32 $fRingInt16 $fRingInt8 $fRingInt $fRingInteger DivisionRing$fDivisionRingr LocalRingZeroProductSemiring$fZeroProductSemiringBool$fZeroProductSemiringNatural$fZeroProductSemiringIntegerUnitNormalForm splitUnit normalize leadingUnit$fUnitNormalFormIntegerCommutativeBialgebraCocommutativeCoalgebraCommutativeAlgebra Commutative$fCommutativeBialgebrarh$fCocommutativeCoalgebrarIntMap$fCocommutativeCoalgebrarMap$fCocommutativeCoalgebrarIntSet$fCocommutativeCoalgebrarSet$fCocommutativeCoalgebrar(,,,,)$fCocommutativeCoalgebrar(,,,)$fCocommutativeCoalgebrar(,,)$fCocommutativeCoalgebrar(,)$fCocommutativeCoalgebrar()$fCocommutativeCoalgebrar(->)$fCommutativeAlgebrarIntSet$fCommutativeAlgebrarSet$fCommutativeAlgebrar(,,,,)$fCommutativeAlgebrar(,,,)$fCommutativeAlgebrar(,,)$fCommutativeAlgebrar(,)$fCommutativeAlgebrar()$fCommutative(->)$fCommutative(,,,,)$fCommutative(,,,)$fCommutative(,,)$fCommutative(,)$fCommutativeWord64$fCommutativeWord32$fCommutativeWord16$fCommutativeWord8$fCommutativeWord$fCommutativeNatural$fCommutativeInt64$fCommutativeInt32$fCommutativeInt16$fCommutativeInt8$fCommutativeInt$fCommutativeInteger$fCommutativeBool$fCommutative()TriviallyInvolutiveBialgebraInvolutiveBialgebraTriviallyInvolutiveCoalgebraInvolutiveCoalgebracoinvTriviallyInvolutiveAlgebraInvolutiveAlgebrainvTriviallyInvolutiveInvolutiveSemiringInvolutiveMultiplicationadjoint $fTriviallyInvolutiveBialgebrarh$fInvolutiveBialgebrarh%$fTriviallyInvolutiveCoalgebrar(,,,,)$$fTriviallyInvolutiveCoalgebrar(,,,)#$fTriviallyInvolutiveCoalgebrar(,,)"$fTriviallyInvolutiveCoalgebrar(,)!$fTriviallyInvolutiveCoalgebrar()$fInvolutiveCoalgebrar(,,,,)$fInvolutiveCoalgebrar(,,,)$fInvolutiveCoalgebrar(,,)$fInvolutiveCoalgebrar(,)$fInvolutiveCoalgebrar()#$fTriviallyInvolutiveAlgebrar(,,,,)"$fTriviallyInvolutiveAlgebrar(,,,)!$fTriviallyInvolutiveAlgebrar(,,) $fTriviallyInvolutiveAlgebrar(,)$fTriviallyInvolutiveAlgebrar()$fInvolutiveAlgebrar(,,,,)$fInvolutiveAlgebrar(,,,)$fInvolutiveAlgebrar(,,)$fInvolutiveAlgebrar(,)$fInvolutiveAlgebrar()$fTriviallyInvolutive(->)$fTriviallyInvolutive(,,,,)$fTriviallyInvolutive(,,,)$fTriviallyInvolutive(,,)$fTriviallyInvolutive(,)$fTriviallyInvolutive()$fTriviallyInvolutiveWord64$fTriviallyInvolutiveWord32$fTriviallyInvolutiveWord16$fTriviallyInvolutiveWord8$fTriviallyInvolutiveNatural$fTriviallyInvolutiveWord$fTriviallyInvolutiveInt64$fTriviallyInvolutiveInt32$fTriviallyInvolutiveInt16$fTriviallyInvolutiveInt8$fTriviallyInvolutiveInteger$fTriviallyInvolutiveInt$fTriviallyInvolutiveBool$fInvolutiveSemiring(,,,,)$fInvolutiveSemiring(,,,)$fInvolutiveSemiring(,,)$fInvolutiveSemiring(,)$fInvolutiveSemiringWord64$fInvolutiveSemiringWord32$fInvolutiveSemiringWord16$fInvolutiveSemiringWord8$fInvolutiveSemiringWord$fInvolutiveSemiringNatural$fInvolutiveSemiringInt64$fInvolutiveSemiringInt32$fInvolutiveSemiringInt16$fInvolutiveSemiringInt8$fInvolutiveSemiringInt$fInvolutiveSemiringInteger$fInvolutiveSemiringBool$fInvolutiveSemiring()$fInvolutiveMultiplication(->) $fInvolutiveMultiplication(,,,,)$fInvolutiveMultiplication(,,,)$fInvolutiveMultiplication(,,)$fInvolutiveMultiplication(,)$fInvolutiveMultiplication() $fInvolutiveMultiplicationWord64 $fInvolutiveMultiplicationWord32 $fInvolutiveMultiplicationWord16$fInvolutiveMultiplicationWord8!$fInvolutiveMultiplicationNatural$fInvolutiveMultiplicationWord$fInvolutiveMultiplicationBool$fInvolutiveMultiplicationInt64$fInvolutiveMultiplicationInt32$fInvolutiveMultiplicationInt16$fInvolutiveMultiplicationInt8!$fInvolutiveMultiplicationInteger$fInvolutiveMultiplicationIntMorphism$fCounitalCoalgebrarMorphism$fCoalgebrarMorphism $fEqMorphism $fOrdMorphism$fShowMorphism$fReadMorphism$fPartialSemigroupMorphism$fPartialMonoidMorphism$fPartialGroupMorphism$fDataMorphism EuclideandegreedividequotremPIDegcdUFD GCDDomaingcdreduceFractionlcmIntegralDomaindivides maybeQuotDomaineuclidprschineseRemainderField$fFielddgcd'Covector$*multMunitMcomultMcounitM convolveMinvMcoinvM antipodeM$fRightModulerCovector$fRightModuleCovectorCovector$fLeftModulerCovector$fLeftModuleCovectorCovector$fGroupCovector$fAbelianCovector$fMonoidalCovector$fBandCovector$fIdempotentCovector$fRingCovector $fRigCovector$fUnitalCovector$fSemiringCovector$fCommutativeCovector$fMultiplicativeCovector$fAdditiveCovector$fMonadPlusCovector$fAlternativeCovector$fPlusCovector $fAltCovector$fMonadCovector$fBindCovector$fApplicativeCovector$fApplyCovector$fFunctorCovector Distinguishede$fDistinguishedCovector Complicatedi$fComplicatedCovector Infinitesimald$fInfinitesimalCovector Hamiltonianjk$fHamiltonianCovector Hyperboliccoshsinh$fHyperbolicCovector Trigonometriccossin$fTrigonometricCovectorDioid$fDioidrRatioFraction% numerator denominator$fEuclideanFraction $fPIDFraction $fUFDFraction$fGCDDomainFraction$fIntegralDomainFraction$fUnitNormalFormFraction$fCharacteristicFraction $fRigFraction$fMultiplicativeFraction$fUnitalFraction$fAdditiveFraction$fRightModuleNaturalFraction$fLeftModuleNaturalFraction$fRightModuleIntegerFraction$fLeftModuleIntegerFraction$fMonoidalFraction$fGroupFraction$fSemiringFraction$fAbelianFraction$fRingFraction$fDecidableAssociatesFraction$fDecidableUnitsFraction$fDecidableZeroFraction$fCommutativeFraction$fDivisionFraction $fOrdFraction $fEqFraction$fZeroProductSemiringFraction$fShowFractionaddRep sinnum1pRepzeroRep sinnumRep negateRepminusRep subtractReptimesRepmulReponeRepfromNaturalRepfromIntegerRep AdditiveOrder$fAdditiveOrder(,,,,)$fAdditiveOrder(,,,)$fAdditiveOrder(,,)$fAdditiveOrder(,)$fAdditiveOrder()$fAdditiveOrderBool$fAdditiveOrderNatural$fAdditiveOrderInteger OrderedRig$fOrderedRig(,,,,)$fOrderedRig(,,,)$fOrderedRig(,,)$fOrderedRig(,)$fOrderedRig()$fOrderedRigBool$fOrderedRigNatural$fOrderedRigIntegerLocallyFiniteOrderrange rangeSizemoebiusInversion$fLocallyFiniteOrder(,,,,)$fLocallyFiniteOrder(,,,)$fLocallyFiniteOrder(,,)$fLocallyFiniteOrder(,)$fLocallyFiniteOrder()$fLocallyFiniteOrderWord64$fLocallyFiniteOrderWord32$fLocallyFiniteOrderWord16$fLocallyFiniteOrderWord8$fLocallyFiniteOrderWord$fLocallyFiniteOrderInt64$fLocallyFiniteOrderInt32$fLocallyFiniteOrderInt16$fLocallyFiniteOrderInt8$fLocallyFiniteOrderInt$fLocallyFiniteOrderBool$fLocallyFiniteOrderSet$fLocallyFiniteOrderInteger$fLocallyFiniteOrderNaturalIntervalzetamoebius$fUnitalAlgebrarInterval$fAlgebrarInterval $fEqInterval $fOrdInterval$fShowInterval$fReadInterval$fDataInterval Interval'zeta'moebius'$fCounitalCoalgebrarInterval'$fCoalgebrarInterval' $fEqInterval'$fOrdInterval'$fShowInterval'$fReadInterval'$fDataInterval' Quadrance quadrance$fQuadrancerWord64$fQuadrancerWord32$fQuadrancerWord16$fQuadrancerWord8$fQuadrancerInt64$fQuadrancerInt32$fQuadrancerInt16$fQuadrancerInt8$fQuadrancerInteger$fQuadrancerNatural$fQuadrancerWord$fQuadrancerInt$fQuadrancerBool$fQuadrancer(,,,,)$fQuadrancer(,,,)$fQuadrancer(,,)$fQuadrancer(,)$fQuadrancer()$fQuadrance()aComplex ComplexBasisEIrealPartimagPart uncomplicate$fDivisionComplex$fQuadrancerComplex$fInvolutiveSemiringComplex!$fInvolutiveMultiplicationComplex$fRightModuleComplexComplex$fLeftModuleComplexComplex $fRingComplex $fRigComplex$fUnitalComplex$fSemiringComplex$fCommutativeComplex$fMultiplicativeComplex$fHopfAlgebrakComplexBasis"$fInvolutiveCoalgebrakComplexBasis $fInvolutiveAlgebrakComplexBasis$fBialgebrakComplexBasis $fCounitalCoalgebrakComplexBasis$fCoalgebrakComplexBasis$fUnitalAlgebrakComplexBasis$fAlgebrakComplexBasis$fPartitionableComplex$fIdempotentComplex$fAbelianComplex$fGroupComplex$fMonoidalComplex$fRightModulerComplex$fLeftModulerComplex$fAdditiveComplex$fTraversable1Complex$fFoldable1Complex$fTraversableComplex$fFoldableComplex $fMonadReaderComplexBasisComplex$fMonadComplex $fBindComplex$fApplicativeComplex$fApplyComplex$fFunctorComplex$fDistributiveComplex$fRepresentableComplex$fComplicated(->)$fDistinguished(->)$fComplicatedComplex$fDistinguishedComplex$fComplicatedComplexBasis$fDistinguishedComplexBasis$fEqComplexBasis$fOrdComplexBasis$fShowComplexBasis$fReadComplexBasis$fEnumComplexBasis$fIxComplexBasis$fBoundedComplexBasis$fDataComplexBasis $fEqComplex $fShowComplex $fReadComplex $fDataComplexDual DualBasisD$fDivisionDual$fQuadrancerDual$fInvolutiveSemiringDual$fInvolutiveMultiplicationDual$fRightModuleDualDual$fLeftModuleDualDual $fRingDual $fRigDual $fUnitalDual$fSemiringDual$fCommutativeDual$fMultiplicativeDual$fHopfAlgebrakDualBasis$fInvolutiveCoalgebrakDualBasis$fInvolutiveAlgebrakDualBasis$fBialgebrakDualBasis$fCounitalCoalgebrakDualBasis$fCoalgebrakDualBasis$fUnitalAlgebrakDualBasis$fAlgebrakDualBasis$fPartitionableDual$fIdempotentDual $fAbelianDual $fGroupDual$fMonoidalDual$fRightModulerDual$fLeftModulerDual$fAdditiveDual$fTraversable1Dual$fFoldable1Dual$fTraversableDual$fFoldableDual$fMonadReaderDualBasisDual $fMonadDual $fBindDual$fApplicativeDual $fApplyDual $fFunctorDual$fDistributiveDual$fRepresentableDual$fInfinitesimal(->)$fInfinitesimalDual$fDistinguishedDual$fInfinitesimalDualBasis$fDistinguishedDualBasis $fEqDualBasis$fOrdDualBasis$fShowDualBasis$fReadDualBasis$fEnumDualBasis $fIxDualBasis$fBoundedDualBasis$fDataDualBasis$fEqDual $fShowDual $fReadDual $fDataDualHyper' HyperBasis'Cosh'Sinh'$fDivisionHyper'$fQuadrancerHyper'$fInvolutiveSemiringHyper' $fInvolutiveMultiplicationHyper'$fRightModuleHyper'Hyper'$fLeftModuleHyper'Hyper' $fRingHyper' $fRigHyper'$fUnitalHyper'$fSemiringHyper'$fCommutativeHyper'$fMultiplicativeHyper'$fHopfAlgebrakHyperBasis'!$fInvolutiveCoalgebrakHyperBasis'$fInvolutiveAlgebrakHyperBasis'$fBialgebrakHyperBasis'$fCounitalCoalgebrakHyperBasis'$fCoalgebrakHyperBasis'$fUnitalAlgebrakHyperBasis'$fAlgebrakHyperBasis'$fPartitionableHyper'$fIdempotentHyper'$fAbelianHyper' $fGroupHyper'$fMonoidalHyper'$fRightModulerHyper'$fLeftModulerHyper'$fAdditiveHyper'$fTraversable1Hyper'$fFoldable1Hyper'$fTraversableHyper'$fFoldableHyper'$fMonadReaderHyperBasis'Hyper' $fMonadHyper' $fBindHyper'$fApplicativeHyper' $fApplyHyper'$fFunctorHyper'$fDistributiveHyper'$fRepresentableHyper'$fHyperbolic(->)$fHyperbolicHyper'$fHyperbolicHyperBasis'$fEqHyperBasis'$fOrdHyperBasis'$fShowHyperBasis'$fReadHyperBasis'$fEnumHyperBasis'$fIxHyperBasis'$fBoundedHyperBasis'$fDataHyperBasis' $fEqHyper' $fShowHyper' $fReadHyper' $fDataHyper' QuaternionQuaternionBasisJK complicate scalarPart vectorPart$fDivisionQuaternion$fQuadrancerQuaternion$$fInvolutiveMultiplicationQuaternion!$fRightModuleQuaternionQuaternion $fLeftModuleQuaternionQuaternion$fRingQuaternion$fRigQuaternion$fUnitalQuaternion$fSemiringQuaternion$fMultiplicativeQuaternion$fHopfAlgebrarQuaternionBasis%$fInvolutiveCoalgebrarQuaternionBasis#$fInvolutiveAlgebrarQuaternionBasis$fBialgebrarQuaternionBasis#$fCounitalCoalgebrarQuaternionBasis$fCoalgebrarQuaternionBasis$fUnitalAlgebrarQuaternionBasis$fAlgebrarQuaternionBasis$fPartitionableQuaternion$fIdempotentQuaternion$fAbelianQuaternion$fGroupQuaternion$fMonoidalQuaternion$fRightModulerQuaternion$fLeftModulerQuaternion$fAdditiveQuaternion$fTraversable1Quaternion$fFoldable1Quaternion$fTraversableQuaternion$fFoldableQuaternion&$fMonadReaderQuaternionBasisQuaternion$fMonadQuaternion$fBindQuaternion$fApplicativeQuaternion$fApplyQuaternion$fFunctorQuaternion$fDistributiveQuaternion$fRepresentableQuaternion$fHamiltonian(->)$fHamiltonianQuaternion$fComplicatedQuaternion$fDistinguishedQuaternion$fHamiltonianQuaternionBasis$fComplicatedQuaternionBasis$fDistinguishedQuaternionBasis$fEqQuaternionBasis$fOrdQuaternionBasis$fEnumQuaternionBasis$fReadQuaternionBasis$fShowQuaternionBasis$fBoundedQuaternionBasis$fIxQuaternionBasis$fDataQuaternionBasis$fEqQuaternion$fShowQuaternion$fReadQuaternion$fDataQuaternionDual' DualBasis'$fDivisionDual'$fQuadrancerDual'$fInvolutiveSemiringDual'$fInvolutiveMultiplicationDual'$fRightModuleDual'Dual'$fLeftModuleDual'Dual' $fRingDual' $fRigDual' $fUnitalDual'$fSemiringDual'$fCommutativeDual'$fMultiplicativeDual'$fHopfAlgebrakDualBasis' $fInvolutiveCoalgebrakDualBasis'$fInvolutiveAlgebrakDualBasis'$fBialgebrakDualBasis'$fCounitalCoalgebrakDualBasis'$fCoalgebrakDualBasis'$fUnitalAlgebrakDualBasis'$fAlgebrakDualBasis'$fPartitionableDual'$fIdempotentDual'$fAbelianDual' $fGroupDual'$fMonoidalDual'$fRightModulerDual'$fLeftModulerDual'$fAdditiveDual'$fTraversable1Dual'$fFoldable1Dual'$fTraversableDual'$fFoldableDual'$fMonadReaderDualBasis'Dual' $fMonadDual' $fBindDual'$fApplicativeDual' $fApplyDual'$fFunctorDual'$fDistributiveDual'$fRepresentableDual'$fInfinitesimalDual'$fDistinguishedDual'$fInfinitesimalDualBasis'$fDistinguishedDualBasis'$fEqDualBasis'$fOrdDualBasis'$fShowDualBasis'$fReadDualBasis'$fEnumDualBasis'$fIxDualBasis'$fBoundedDualBasis'$fDataDualBasis' $fEqDual' $fShowDual' $fReadDual' $fDataDual' Comultivector Eigenmetricmetric Eigenbasis euclidean antiEuclideanv BasisCobladerunBasisCobladegrade filterGradereversecliffordConjugategradeInversion geometricouter contractL contractRdothestenes liftProduct $fCounitalCoalgebrarBasisCoblade$fCoalgebrarBasisCoblade$fEigenmetricrEuclidean$fEigenbasisEuclidean$fEqBasisCoblade$fOrdBasisCoblade$fNumBasisCoblade$fBitsBasisCoblade$fEnumBasisCoblade$fIxBasisCoblade$fBoundedBasisCoblade$fShowBasisCoblade$fReadBasisCoblade$fRealBasisCoblade$fIntegralBasisCoblade$fAdditiveBasisCoblade$fAbelianBasisCoblade$fLeftModuleBasisCoblade$fRightModuleBasisCoblade$fMonoidalBasisCoblade$fMultiplicativeBasisCoblade$fUnitalBasisCoblade$fCommutativeBasisCoblade$fSemiringBasisCoblade$fRigBasisCoblade$fDecidableZeroBasisCoblade!$fDecidableAssociatesBasisCoblade$fDecidableUnitsBasisCoblade $fEqEuclidean$fOrdEuclidean$fShowEuclidean$fReadEuclidean$fNumEuclidean $fIxEuclidean$fEnumEuclidean$fRealEuclidean$fIntegralEuclidean$fDataEuclidean$fAdditiveEuclidean$fLeftModuleEuclidean$fRightModuleEuclidean$fMonoidalEuclidean$fAbelianEuclidean$fLeftModuleEuclidean0$fRightModuleEuclidean0$fGroupEuclidean$fMultiplicativeEuclidean$fTriviallyInvolutiveEuclidean#$fInvolutiveMultiplicationEuclidean$fInvolutiveSemiringEuclidean$fUnitalEuclidean$fCommutativeEuclidean$fSemiringEuclidean$fRigEuclidean$fRingEuclideanHyper HyperBasisCoshSinh$fInvolutiveSemiringHyper$fInvolutiveMultiplicationHyper$fRightModuleHyperHyper$fLeftModuleHyperHyper $fRingHyper $fRigHyper $fUnitalHyper$fSemiringHyper$fCommutativeHyper$fMultiplicativeHyper$fHopfAlgebrakHyperBasis $fInvolutiveCoalgebrakHyperBasis$fInvolutiveAlgebrakHyperBasis$fBialgebrakHyperBasis$fCounitalCoalgebrakHyperBasis$fCoalgebrakHyperBasis$fUnitalAlgebrakHyperBasis$fAlgebrakHyperBasis$fPartitionableHyper$fIdempotentHyper$fAbelianHyper $fGroupHyper$fMonoidalHyper$fRightModulerHyper$fLeftModulerHyper$fAdditiveHyper$fTraversable1Hyper$fFoldable1Hyper$fTraversableHyper$fFoldableHyper$fMonadReaderHyperBasisHyper $fMonadHyper $fBindHyper$fApplicativeHyper $fApplyHyper$fFunctorHyper$fDistributiveHyper$fRepresentableHyper$fHyperbolicHyper$fHyperbolicHyperBasis$fEqHyperBasis$fOrdHyperBasis$fShowHyperBasis$fReadHyperBasis$fEnumHyperBasis$fIxHyperBasis$fBoundedHyperBasis$fDataHyperBasis $fEqHyper $fShowHyper $fReadHyper $fDataHyper Quaternion'QuaternionBasis'E'I'J'K' complicate' scalarPart' vectorPart'$fDivisionQuaternion'$fQuadrancerQuaternion'%$fInvolutiveMultiplicationQuaternion'#$fRightModuleQuaternion'Quaternion'"$fLeftModuleQuaternion'Quaternion'$fRingQuaternion'$fRigQuaternion'$fUnitalQuaternion'$fSemiringQuaternion'$fMultiplicativeQuaternion'$fHopfAlgebrarQuaternionBasis'&$fInvolutiveCoalgebrarQuaternionBasis'$$fInvolutiveAlgebrarQuaternionBasis'$fBialgebrarQuaternionBasis'$$fCounitalCoalgebrarQuaternionBasis'$fCoalgebrarQuaternionBasis' $fUnitalAlgebrarQuaternionBasis'$fAlgebrarQuaternionBasis'$fPartitionableQuaternion'$fIdempotentQuaternion'$fAbelianQuaternion'$fGroupQuaternion'$fMonoidalQuaternion'$fRightModulerQuaternion'$fLeftModulerQuaternion'$fAdditiveQuaternion'$fTraversable1Quaternion'$fFoldable1Quaternion'$fTraversableQuaternion'$fFoldableQuaternion'($fMonadReaderQuaternionBasis'Quaternion'$fMonadQuaternion'$fBindQuaternion'$fApplicativeQuaternion'$fApplyQuaternion'$fFunctorQuaternion'$fDistributiveQuaternion'$fRepresentableQuaternion'$fHamiltonianQuaternion'$fComplicatedQuaternion'$fDistinguishedQuaternion'$fHamiltonianQuaternionBasis'$fComplicatedQuaternionBasis'$fDistinguishedQuaternionBasis'$fEqQuaternionBasis'$fOrdQuaternionBasis'$fEnumQuaternionBasis'$fReadQuaternionBasis'$fShowQuaternionBasis'$fBoundedQuaternionBasis'$fIxQuaternionBasis'$fDataQuaternionBasis'$fEqQuaternion'$fShowQuaternion'$fReadQuaternion'$fDataQuaternion'Trig TrigBasisCosSin$fInvolutiveSemiringTrig$fInvolutiveMultiplicationTrig$fRightModuleTrigTrig$fLeftModuleTrigTrig $fRingTrig $fRigTrig $fUnitalTrig$fSemiringTrig$fCommutativeTrig$fMultiplicativeTrig$fCounitalCoalgebrakTrigBasis$fHopfAlgebrakTrigBasis$fInvolutiveCoalgebrakTrigBasis$fInvolutiveAlgebrakTrigBasis$fBialgebrakTrigBasis$fCoalgebrakTrigBasis$fUnitalAlgebrakTrigBasis$fAlgebrakTrigBasis$fPartitionableTrig$fIdempotentTrig $fAbelianTrig $fGroupTrig$fMonoidalTrig$fRightModulerTrig$fLeftModulerTrig$fAdditiveTrig$fTraversable1Trig$fFoldable1Trig$fTraversableTrig$fFoldableTrig$fMonadReaderTrigBasisTrig $fMonadTrig $fBindTrig$fApplicativeTrig $fApplyTrig $fFunctorTrig$fDistributiveTrig$fRepresentableTrig$fTrigonometric(->)$fTrigonometricTrig$fComplicatedTrig$fDistinguishedTrig$fTrigonometricTrigBasis$fComplicatedTrigBasis$fDistinguishedTrigBasis $fEqTrigBasis$fOrdTrigBasis$fShowTrigBasis$fReadTrigBasis$fEnumTrigBasis $fIxTrigBasis$fBoundedTrigBasis$fDataTrigBasis$fEqTrig $fShowTrig $fReadTrig $fDataTrigDecidableNilpotent isNilpotent$fDecidableNilpotent(,,,,)$fDecidableNilpotent(,,,)$fDecidableNilpotent(,,)$fDecidableNilpotent(,)$fDecidableNilpotentWord64$fDecidableNilpotentWord32$fDecidableNilpotentWord16$fDecidableNilpotentWord8$fDecidableNilpotentInt64$fDecidableNilpotentInt32$fDecidableNilpotentInt16$fDecidableNilpotentInt8$fDecidableNilpotentInt$fDecidableNilpotentInteger$fDecidableNilpotentNatural$fDecidableNilpotentBool$fDecidableNilpotent()ExprunExp$fFactorableExp $fBandExp$fCommutativeExp $fDivisionExp $fUnitalExp$fMultiplicativeExpLogrunLog$fPartitionableLog$fIdempotentLog $fAbelianLog $fGroupLog$fRightModuleIntegerLog$fLeftModuleIntegerLog $fMonoidalLog$fRightModuleNaturalLog$fLeftModuleNaturalLog $fAdditiveLogMap$@ comultMapmultMap counitMapunitMap convolveMap antipodeMapcoinvMapinvMap $fRingMap$fRigMap$fCommutativeMap $fGroupMap $fAbelianMap $fMonoidalMap$fMonadPlusMap$fAlternativeMap $fPlusMap$fAltMap$fRightModulerMap$fRightModuleMapMap$fLeftModulerMap$fLeftModuleMapMap $fSemiringMap $fUnitalMap$fMultiplicativeMap $fAdditiveMap$fArrowChoiceMap$fArrowPlusMap$fArrowZeroMap$fMonadReaderbMap$fArrowApplyMap $fArrowMap $fMonadMap $fBindMap$fApplicativeMap $fApplyMap $fFunctorMap$fSemigroupoidTYPEMap$fCategoryTYPEMapEndappEndtoEndfromEnd frobenius$fRightModulerEnd$fLeftModulerEnd$fRightModuleEndEnd$fLeftModuleEndEnd $fRingEnd$fRigEnd $fSemiringEnd$fCommutativeEnd $fUnitalEnd$fMultiplicativeEnd $fGroupEnd $fMonoidalEnd $fAbelianEnd $fAdditiveEnd $fMonoidEndOpposite runOpposite$fRingOpposite $fRigOpposite$fSemiringOpposite$fDivisionOpposite$fUnitalOpposite$fBandOpposite$fIdempotentOpposite$fCommutativeOpposite$fMultiplicativeOpposite$fDecidableAssociatesOpposite$fDecidableUnitsOpposite$fDecidableZeroOpposite$fAbelianOpposite$fGroupOpposite$fRightModuleOppositeOpposite$fRightModulerOpposite$fLeftModulerOpposite$fLeftModuleOppositeOpposite$fMonoidalOpposite$fAdditiveOpposite$fTraversable1Opposite$fFoldable1Opposite$fTraversableOpposite$fFoldableOpposite$fFunctorOpposite $fOrdOpposite $fEqOpposite$fShowOpposite$fReadOppositeRngRing rngRingHom liftRngHom $fRingRngRing $fRigRngRing$fSemiringRngRing$fDivisionRngRing$fUnitalRngRing$fRightModuleRngRingRngRing$fLeftModuleRngRingRngRing$fCommutativeRngRing$fMultiplicativeRngRing$fGroupRngRing$fRightModuleIntegerRngRing$fLeftModuleIntegerRngRing$fMonoidalRngRing$fRightModuleNaturalRngRing$fLeftModuleNaturalRngRing$fAbelianRngRing$fAdditiveRngRing $fShowRngRing $fReadRngRingZeroRng runZeroRng$fRightModuleIntegerZeroRng$fLeftModuleIntegerZeroRng$fRightModuleNaturalZeroRng$fLeftModuleNaturalZeroRng $fRngZeroRng$fCommutativeZeroRng$fSemiringZeroRng$fMultiplicativeZeroRng$fGroupZeroRng$fMonoidalZeroRng$fAbelianZeroRng$fIdempotentZeroRng$fAdditiveZeroRng $fEqZeroRng $fOrdZeroRng $fShowZeroRng $fReadZeroRngpaddNumconcatProxy asProxyTypeOf$fEuclideanInteger $fPIDInteger $fUFDInteger$fGCDDomainInteger$fIntegralDomainInteger $fDomaindsqlsb m1powTimesreorderunsignedBitsNilpotentsignedBitsNilpotent Control.Arrowarr$#ofRing