"<      !"#$%&'()*+,-./012345678 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R STUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ !!"""##$$$%%%&''''''''''''()****++++,,,,--........//////////000001111122222333334444444444444444444444444555 5 5 5 5 5556666677788899999 9!9"9#9$9%9&:':(:):*:+:,;-;.;/<0<1<2<3=4=5=6>7>8>9>:>;>A Safe-Inferred<=>?@ABCDEFGHIJKLMNOP<=>?@ABCDEFGHIJKLMNOP Safe-InferredQRSTUVWXYZ[\]^_`abcQRSTUVWXYZ[\]^_`abc Safe-Inferreddefghijklmnopqrstdefghijklmnopqrst Safe-Inferred uvwxyz{|}~    uvwxyz{|}~ Safe-InferredJK/An additive semigroup with idempotent addition.  a + a = aan additive abelian semigroup a + b = b + aTpartitionWith f c returns a list containing f a b for each a b such that a + b = c,  (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 + asinnum1p n r = sinnum (1 + n) r?  ; Safe-Inferred2346JK An additive monoid zero + a = a = a + zero(?An associative algebra built with a free module over a semiring*dA 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 algebraThe 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 !"#$%&'()*+,-./012      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKL !"#$%&'()*+,-./012+,-.0/*$%"#! 12()&' !"#$%&'()*+,-./012      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKL#%,- Safe-Inferred2346JK34567MNOPQRSTUVWXYZ[\]345673456734567MNOPQRSTUVWXYZ[\]47  Safe-Inferred8b`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. 89^_`abcd898989^_`abcd  Safe-Inferred246:HA bialgebra is both a unital algebra and counital coalgebra where the ) and >) are compatible in some sense with the ' and <. That is to say that ) and >8 are a coalgebra homomorphisms or (equivalently) that ' and < are an algebra homomorphisms.=HAn associative unital algebra over a semiring, built using a free module3:;<=>?@ABCefghijklmnopqrstuvwxyz{|}~ :;<=>?@ABC ?@ABC=>;<:.:;<=>?@ABCefghijklmnopqrstuvwxyz{|}~A  Safe-Inferred246G;An multiplicative semigroup with idempotent multiplication.  a * a = aDEFGHIDEFGHIGHIFEDDEFGHI? Safe-InferredGHIGHI  Safe-Inferred246 JKLMNOPJKLMNOPLMNOPJKJKLMNOPNOP  Safe-Inferred246Q>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 wellQRQRQRQR Safe-InferredTAb 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 = aSTUVSTUVSTUVSTUV Safe-InferredWXYZ[\WXYZ[\WXYZ[\WXYZ[\ Safe-Inferred]^]^]^]^ Safe-Inferred_A Ring without (n)egation_`_`_`_` Safe-Inferred:NB: we're using the boolean semiring, not the boolean ringabcd abcdabcdabcd  Safe-Inferred24eA Ring without an identity.e eee  Safe-Inferredfgh   fghfghfgh    Safe-Inferrediiii Safe-Inferred24jjjj Safe-Inferredka 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 structureklklklkl Safe-Inferredm)An integral semiring has no zero divisors "a * b = 0 implies a == 0 || b == 0m !mmm ! Safe-Inferred24n+(Integral) domain is the integral semiring.nonnonoNone %&2346BHMq splitUnit r- calculates its leading unit and normal form.Nlet (u, n) = splitUnit r in r == u * n && fst (splitUnit n) == one && isUnit urEuclidean (degree) function on r.sDivision algorithm. a s b( calculates quotient and reminder of a divided by b.>let (q, r) = divide a p in p*q + r == a && degree r < degree qvv a b' calculates greatest common divisor of a and b.wExtended euclidean algorithm.>euclid f g == xs ==> all (\(r, s, t) -> r == f * s + g * t) xspqrselements divided bydivisorquotient and remintuvwxyz{|List of  (m_i, v_i)f with f = v_i (mod v_i)" pqrstuvwxyz{| pqrstuvwxzy{|pqrstuvwxyz{|"stu@ Safe-Inferred!"#$%$%"#! Safe-Inferred246JK&A commutative multiplicative semigroup+}~#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHI}~~}+}~#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHI Safe-Inferred246JK  adjoint = id'adjoint (x + y) = adjoint x + adjoint yAn semigroup with involution 'adjoint a * adjoint b = adjoint (b * a)ZJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ WJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~A Safe-Inferred Safe-Inferred24None+246BF Safe-Inferred2346ILinear functionals from elements of an (infinite) free module to a scalarSconvolveM antipodeM return = convolveM return antipodeM = comultM >=> uncurry joinM$ "  Safe-Inferred! Safe-Inferred" Safe-Inferred# Safe-Inferred$ Safe-Inferred% Safe-Inferred& Safe-Inferred24'None%3 !`Additive.(+)` default definitionBC default definitionDE default definitionDF default definitionGH default definition`Group.(-)` default definitionGI default definitionGJ default definition'`Multiplicative.(*)` default definitionKL default definitionMN default definitionOP default definition ( Safe-Inferred(z + x <= z + y = x <= y = x + z <= y + z  ) Safe-Inferred  * Safe-Inferred+ Safe-Inferred+246, Safe-Inferred+246 the dual incidence algebra basis- Safe-Inferred246None  !"#$%&'()*+,-./123456789:;<=>?@ABCDFGHIJKLMNOPQRSW]_`abcdefgij}~Ƅ2 134567+,-./?@ABCGHILMNOP89*e_`fgij$%"#!()&'=>;<:FD}~JKQRabcd ]WS.None +246=JK2half of the Cayley-Dickson quaternion isomorphism 6      !"# 3      !"#/None +246=JK/Cayley-Dickson quaternion isomorphism (one way)$the trivial diagonal coalgebra%the quaternion algebra9&'()*+,-./01234$5%6789:;<=>?@ABCDEFGHIJKLMNOPQR4&'()*+,-./01234$5%6789:;<=>?@ABCDEFGHIJKLMNOPQR0None +246=K#dual number basis, D^2 = 0. D /= 0.3STUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ 0STUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~1None +246=K0-2None +246=K&the hyperbolic trigonometric coalgebrathe trivial diagonal algebra.+3None +246=K#dual number basis, D^2 = 0. D /= 0.3 04None+246=BFJKM   5None +246=JK /Cayley-Dickson quaternion isomorphism (one way)  dual quaternion comultiplication the trivial diagonal algebra9          !"#$%&'()*+,-./012345678          4          !"#$%&'()*+,-./0123456786None +246=JK49:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefg19:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefg7None hijklmhijklm8None6 nopqrstuvw nopqrstuvw9None246=KHlinear 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 x{, or build them by using the monad instance for Map r b. As a consequence Map is an instance of, well, almost everything.-extract a linear functional from a linear map(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 ]"Dconvolution given an associative algebra and coassociative coalgebra+y !"#$%z{|}~  !"#$% ! %$#"*y !"#$%z{|}~y:None246&[The endomorphism ring of an abelian group or the endomorphism semiring of an abelian monoid .http://en.wikipedia.org/wiki/Endomorphism_ring&'()*+&'()*+&'()*+&'()*+;None246, *http://en.wikipedia.org/wiki/Opposite_ring,-.,-.,-.,-.<None246/<The free Ring given a Rng obtained by adjoining Z, such that RngRing r = n*1 + r!This ring is commonly denoted r^.1(The rng homomorphism from r to RngRing r2igiven a rng homomorphism from a rng r into a ring s, liftRngHom yields a ring homomorphism from the ring `r^` into s./012/012/012/012=None246345345345345>None %6M6Convenient synonym for 7.7Fraction field k(D) of p domain D.6789:;6789:;7:;6986789:;9QRSTUVWXYZ[\]^_`abcdefghiBjCklmDEFnopqrstuvwxyz{|}~GHIJ     K L        MNOP  !!"""##$$$%%%&''''''''''''()****++++,,,,--..... . . . / / ///// ///00000 1111122222333 33 444!4"4#4$4%4&4'44(4(4)4*4+4,4-4.4/40414243444556565758595:5;5<5=5>6?6?6@6A6B7C7C7D8E8E8F9G9G9H9I9J9K9L9M9N9O9P:Q:Q:R:S:T:U;V;V;W<X<X<Y<Z=[=[=\>]>^>_>`>a>bcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~                                                                                     !"#$%&'()*+,-./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/C/D/p/q/r/s/t/u0v0w0x0y0z0{0|0}0~000000000000000000000000000000000D000011111111111111111111111111111111111111111112222222222222222222222222222222222222222233333333333333333333 3 3 3 3 333333333333333333D333 3!4"4#4$4%4&4'4(5)5*5+5,5-5.5/505152535455565758595:5;5<5=5>5?5@5A5B5C5D5E5F5G5H5I5J5K5L5M5N5o5C5D5O5P5Q5R5S5T6U6V6W6X6Y6Z6[6\6]6^6_6`6a6b6c6d6e6f6g6h6i6j6k6l6m6n6o6p6q6r6s6t6u6v6w6x6y6z6{6C6D6|6}6~666777777888888888899999999999999999999999999999999::::::::::::::::;;;;;;;;;;;;;;;;;;;;;;;;;;;<<<<<<<<<<<<<<<<<=============>^>>>>>>>>>>>> > > > > >>>>>> algebra-4.2Numeric.AlgebraNumeric.Partial.SemigroupNumeric.Partial.MonoidNumeric.Partial.GroupNumeric.Order.ClassNumeric.Additive.ClassNumeric.Algebra.ClassNumeric.Additive.GroupNumeric.Algebra.FactorableNumeric.Algebra.UnitalNumeric.Algebra.IdempotentNumeric.Algebra.DivisionNumeric.Algebra.HopfNumeric.Decidable.AssociatesNumeric.Decidable.UnitsNumeric.Decidable.ZeroNumeric.Rig.ClassNumeric.Rig.CharacteristicNumeric.Rng.ClassNumeric.Ring.ClassNumeric.Ring.LocalNumeric.Ring.DivisionNumeric.Band.RectangularNumeric.Semiring.IntegralNumeric.Domain.ClassNumeric.Domain.EuclideanNumeric.Algebra.CommutativeNumeric.Algebra.InvolutiveNumeric.Field.ClassNumeric.Coalgebra.CategoricalNumeric.Covector#Numeric.Algebra.Distinguished.ClassNumeric.Algebra.Complex.Class Numeric.Algebra.Quaternion.ClassNumeric.Algebra.Dual.Class"Numeric.Coalgebra.Hyperbolic.Class%Numeric.Coalgebra.Trigonometric.ClassNumeric.Dioid.ClassNumeric.Module.RepresentableNumeric.Order.AdditiveNumeric.Rig.OrderedNumeric.Order.LocallyFiniteNumeric.Algebra.IncidenceNumeric.Coalgebra.IncidenceNumeric.Quadrance.ClassNumeric.Algebra.ComplexNumeric.Algebra.QuaternionNumeric.Algebra.DualNumeric.Algebra.HyperbolicNumeric.Coalgebra.HyperbolicNumeric.Coalgebra.DualNumeric.Coalgebra.GeometricNumeric.Coalgebra.QuaternionNumeric.Coalgebra.Trigonometric Numeric.Exp Numeric.Log Numeric.MapNumeric.Ring.EndomorphismNumeric.Ring.OppositeNumeric.Ring.RngNumeric.Rng.ZeroNumeric.Field.FractionNumeric.Band.ClassNumeric.Module.ClassNumeric.Semiring.InvolutiveAdditivesinnum1pMonoidalzerosinnumGroupnegatesubtracttimesUnitaloneRig fromNaturalRing fromIntegernats-1Numeric.NaturalNaturalPartialSemigrouppadd PartialMonoidpzero PartialGrouppnegatepminus psubtractOrder<~<>~>~~/~order comparableorderOrd IdempotentAbelian Partitionable partitionWith+sumWith1sum1sinnum1pIdempotentsumWithModule RightModule*. LeftModule.* CoalgebracomultAlgebramultSemiringMultiplicative*pow1p productWith1product1 pow1pIntegralsumsinnumIdempotent- Factorable factorWith BialgebraCounitalCoalgebracounit UnitalAlgebraunitpow productWithproductIdempotentBialgebraIdempotentCoalgebraIdempotentAlgebraBand pow1pBandpowBandDivisionAlgebra recipriocalDivisionrecip/\\^ HopfAlgebraantipodeDecidableAssociates isAssociateisAssociateIntegralisAssociateWholeDecidableUnits recipUnitisUnit^?recipUnitIntegralrecipUnitWhole DecidableZeroisZeroCharacteristiccharcharIntcharWordRng fromIntegral LocalRing DivisionRingRectIntegralSemiringDomain $fDomaind Euclidean splitUnitdegreedividequotremgcdeuclidprsgcd' normalize leadingUnitchineseRemainderCommutativeBialgebraCocommutativeCoalgebraCommutativeAlgebra CommutativeTriviallyInvolutiveBialgebraInvolutiveBialgebraTriviallyInvolutiveCoalgebraInvolutiveCoalgebracoinvTriviallyInvolutiveAlgebraInvolutiveAlgebrainvTriviallyInvolutiveInvolutiveSemiringInvolutiveMultiplicationadjointFieldMorphismCovector$*multMunitMcomultMcounitM convolveMinvMcoinvM antipodeM Distinguishede Complicatedi Hamiltonianjk Infinitesimald Hyperboliccoshsinh TrigonometriccossinDioidaddRep sinnum1pRepzeroRep sinnumRep negateRepminusRep subtractReptimesRepmulReponeRepfromNaturalRepfromIntegerRep AdditiveOrder OrderedRigLocallyFiniteOrderrange rangeSizemoebiusInversionIntervalzetamoebius Interval'zeta'moebius' Quadrance quadranceComplex ComplexBasisIErealPartimagPart uncomplicate QuaternionQuaternionBasisKJ complicate scalarPart vectorPartDual DualBasisDHyper' HyperBasis'Sinh'Cosh'Hyper HyperBasisSinhCoshDual' DualBasis' Comultivector Eigenmetricmetric Eigenbasis euclidean antiEuclideanv BasisCobladerunBasisCobladegrade filterGradereversecliffordConjugategradeInversion 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denominatorpaddNum$fPartialSemigroupEither$fPartialSemigroup(,,,,)$fPartialSemigroup(,,,)$fPartialSemigroup(,,)$fPartialSemigroup(,)$fPartialSemigroup()$fPartialSemigroupBool$fPartialSemigroupMaybe$fPartialSemigroupWord64$fPartialSemigroupWord32$fPartialSemigroupWord16$fPartialSemigroupWord8$fPartialSemigroupWord$fPartialSemigroupInt64$fPartialSemigroupInt32$fPartialSemigroupInt16$fPartialSemigroupInt8$fPartialSemigroupNatural$fPartialSemigroupInteger$fPartialSemigroupInt$fPartialMonoid(,,,,)$fPartialMonoid(,,,)$fPartialMonoid(,,)$fPartialMonoid(,)$fPartialMonoidMaybe$fPartialMonoid()$fPartialMonoidWord64$fPartialMonoidWord32$fPartialMonoidWord16$fPartialMonoidWord8$fPartialMonoidWord$fPartialMonoidInt64$fPartialMonoidInt32$fPartialMonoidInt16$fPartialMonoidInt8$fPartialMonoidNatural$fPartialMonoidInteger$fPartialMonoidInt$fPartialMonoidBool$fPartialGroup(,,,,)$fPartialGroup(,,,)$fPartialGroup(,,)$fPartialGroup(,)$fPartialGroup()$fPartialGroupNatural$fPartialGroupWord64$fPartialGroupWord32$fPartialGroupWord16$fPartialGroupWord8$fPartialGroupWord$fPartialGroupInt64$fPartialGroupInt32$fPartialGroupInt16$fPartialGroupInt8$fPartialGroupInteger$fPartialGroupInt $fOrder(,,,,) $fOrder(,,,) $fOrder(,,) $fOrder(,) $fOrder() $fOrderSet $fOrderWord64 $fOrderWord32 $fOrderWord16 $fOrderWord8 $fOrderWord$fOrderNatural $fOrderInt64 $fOrderInt32 $fOrderInt16 $fOrderInt8 $fOrderInt$fOrderInteger $fOrderBoolconcat$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$fAdditive(->)$fCoalgebrarIntMap$fCoalgebrarMap$fCoalgebrarIntSet$fCoalgebrarSet$fCoalgebrarSeq$fCoalgebrar[]$fCoalgebrar(->) $fAlgebrarSeq $fAlgebrar[]$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$fLeftModuleNaturalBool$fCoalgebrar(,,,,)$fCoalgebrar(,,,)$fCoalgebrar(,,)$fCoalgebrar(,)$fCoalgebrar()$fAlgebrar(,,,,)$fAlgebrar(,,,)$fAlgebrar(,,) $fAlgebrar(,)$fAlgebrarIntMap $fAlgebrarMap$fAlgebrarIntSet $fAlgebrarSet $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 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$fUnitalBool$fIdempotentBialgebrarh$fIdempotentCoalgebrar(,,,,)$fIdempotentCoalgebrar(,,,)$fIdempotentCoalgebrar(,,)$fIdempotentCoalgebrar(,)$fIdempotentCoalgebrar()$fIdempotentCoalgebrarIntSet$fIdempotentCoalgebrarSet$fIdempotentAlgebrar(,,,,)$fIdempotentAlgebrar(,,,)$fIdempotentAlgebrar(,,)$fIdempotentAlgebrar(,)$fIdempotentAlgebrar()$fIdempotentAlgebrarIntSet$fIdempotentAlgebrarSet $fBand(,,,,) $fBand(,,,) $fBand(,,) $fBand(,) $fBandBool$fBand()$fDivision(->)$fDivision(,,,,)$fDivision(,,,)$fDivision(,,) $fDivision(,) $fDivision()$fHopfAlgebrar(,,,,)$fHopfAlgebrar(,,,)$fHopfAlgebrar(,,)$fHopfAlgebrar(,)$fDecidableAssociates(,,,,)$fDecidableAssociates(,,,)$fDecidableAssociates(,,)$fDecidableAssociates(,)$fDecidableAssociates()$fDecidableAssociatesWord64$fDecidableAssociatesWord32$fDecidableAssociatesWord16$fDecidableAssociatesWord8$fDecidableAssociatesWord$fDecidableAssociatesNatural$fDecidableAssociatesInt64$fDecidableAssociatesInt32$fDecidableAssociatesInt16$fDecidableAssociatesInt8$fDecidableAssociatesInt$fDecidableAssociatesInteger$fDecidableAssociatesBool$fDecidableUnits(,,,,)$fDecidableUnits(,,,)$fDecidableUnits(,,)$fDecidableUnits(,)$fDecidableUnits()$fDecidableUnitsWord64$fDecidableUnitsWord32$fDecidableUnitsWord16$fDecidableUnitsWord8$fDecidableUnitsWord$fDecidableUnitsNatural$fDecidableUnitsInt64$fDecidableUnitsInt32$fDecidableUnitsInt16$fDecidableUnitsInt8$fDecidableUnitsInt$fDecidableUnitsInteger$fDecidableUnitsBool$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 $fRigInteger$fCharacteristicBoolProxy asProxyTypeOf$fCharacteristic(,,,,)$fCharacteristic(,,,)$fCharacteristic(,,)$fCharacteristic(,)$fCharacteristic()$fCharacteristicWord64$fCharacteristicWord32$fCharacteristicWord16$fCharacteristicWord8$fCharacteristicWord$fCharacteristicInt64$fCharacteristicInt32$fCharacteristicInt16$fCharacteristicInt8$fCharacteristicInt$fCharacteristicNatural$fCharacteristicInteger$fRngr $fRing(,,,,) $fRing(,,,) $fRing(,,) $fRing(,)$fRing() $fRingWord64 $fRingWord32 $fRingWord16 $fRingWord8 $fRingWord $fRingInt64 $fRingInt32 $fRingInt16 $fRingInt8 $fRingInt $fRingInteger$fDivisionRingr $fBandRect$fMultiplicativeRect$fSemigroupoidRect$fIntegralSemiringBool$fIntegralSemiringNatural$fIntegralSemiringInteger$fEuclideanInteger$fCommutativeBialgebrarh$fCocommutativeCoalgebrarIntMap$fCocommutativeCoalgebrarMap$fCocommutativeCoalgebrarIntSet$fCocommutativeCoalgebrarSet$fCocommutativeCoalgebrar(,,,,)$fCocommutativeCoalgebrar(,,,)$fCocommutativeCoalgebrar(,,)$fCocommutativeCoalgebrar(,)$fCocommutativeCoalgebrar()$fCocommutativeCoalgebrar(->)$fCommutativeAlgebrarIntMap$fCommutativeAlgebrarMap$fCommutativeAlgebrarIntSet$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() $fTriviallyInvolutiveBialgebrarh$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$fInvolutiveMultiplicationInt$fFieldr$fCounitalCoalgebrarMorphism$fCoalgebrarMorphism$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$fDistinguishedCovector$fComplicatedCovector$fHamiltonianCovector$fInfinitesimalCovector$fHyperbolicCovector$fTrigonometricCovector$fDioidr$fAdditiveOrder(,,,,)$fAdditiveOrder(,,,)$fAdditiveOrder(,,)$fAdditiveOrder(,)$fAdditiveOrder()$fAdditiveOrderBool$fAdditiveOrderNatural$fAdditiveOrderInteger$fOrderedRig(,,,,)$fOrderedRig(,,,)$fOrderedRig(,,)$fOrderedRig(,)$fOrderedRig()$fOrderedRigBool$fOrderedRigNatural$fOrderedRigInteger$fLocallyFiniteOrder(,,,,)$fLocallyFiniteOrder(,,,)$fLocallyFiniteOrder(,,)$fLocallyFiniteOrder(,)$fLocallyFiniteOrder()$fLocallyFiniteOrderWord64$fLocallyFiniteOrderWord32$fLocallyFiniteOrderWord16$fLocallyFiniteOrderWord8$fLocallyFiniteOrderWord$fLocallyFiniteOrderInt64$fLocallyFiniteOrderInt32$fLocallyFiniteOrderInt16$fLocallyFiniteOrderInt8$fLocallyFiniteOrderInt$fLocallyFiniteOrderBool$fLocallyFiniteOrderSet$fLocallyFiniteOrderInteger$fLocallyFiniteOrderNatural$fUnitalAlgebrarInterval$fAlgebrarInterval$fCounitalCoalgebrarInterval'$fCoalgebrarInterval'sq$fQuadrancerWord64$fQuadrancerWord32$fQuadrancerWord16$fQuadrancerWord8$fQuadrancerInt64$fQuadrancerInt32$fQuadrancerInt16$fQuadrancerInt8$fQuadrancerInteger$fQuadrancerNatural$fQuadrancerWord$fQuadrancerInt$fQuadrancerBool$fQuadrancer(,,,,)$fQuadrancer(,,,)$fQuadrancer(,,)$fQuadrancer(,)$fQuadrancer()$fQuadrance()a$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$fCoalgebrarQuaternionBasis$fAlgebrarQuaternionBasis$fDivisionQuaternion$fQuadrancerQuaternion$$fInvolutiveMultiplicationQuaternion!$fRightModuleQuaternionQuaternion $fLeftModuleQuaternionQuaternion$fRingQuaternion$fRigQuaternion$fUnitalQuaternion$fSemiringQuaternion$fMultiplicativeQuaternion$fHopfAlgebrarQuaternionBasis%$fInvolutiveCoalgebrarQuaternionBasis#$fInvolutiveAlgebrarQuaternionBasis$fBialgebrarQuaternionBasis#$fCounitalCoalgebrarQuaternionBasis$fUnitalAlgebrarQuaternionBasis$fPartitionableQuaternion$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$fDistinguishedQuaternionBasis$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$fDivisionHyper'$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'$fHyperbolicHyperBasis'$fCoalgebrakHyperBasis$fAlgebrakHyperBasis$fInvolutiveSemiringHyper$fInvolutiveMultiplicationHyper$fRightModuleHyperHyper$fLeftModuleHyperHyper $fRingHyper $fRigHyper $fUnitalHyper$fSemiringHyper$fCommutativeHyper$fMultiplicativeHyper$fHopfAlgebrakHyperBasis $fInvolutiveCoalgebrakHyperBasis$fInvolutiveAlgebrakHyperBasis$fBialgebrakHyperBasis$fCounitalCoalgebrakHyperBasis$fUnitalAlgebrakHyperBasis$fPartitionableHyper$fIdempotentHyper$fAbelianHyper $fGroupHyper$fMonoidalHyper$fRightModulerHyper$fLeftModulerHyper$fAdditiveHyper$fTraversable1Hyper$fFoldable1Hyper$fTraversableHyper$fFoldableHyper$fMonadReaderHyperBasisHyper $fMonadHyper $fBindHyper$fApplicativeHyper $fApplyHyper$fFunctorHyper$fDistributiveHyper$fRepresentableHyper$fHyperbolicHyper$fHyperbolicHyperBasis$fDivisionDual'$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'lsb m1powTimesreorder $fCounitalCoalgebrarBasisCoblade$fCoalgebrarBasisCoblade$fEigenmetricrEuclidean$fEigenbasisEuclidean$fCoalgebrarQuaternionBasis'$fAlgebrarQuaternionBasis'$fDivisionQuaternion'$fQuadrancerQuaternion'%$fInvolutiveMultiplicationQuaternion'#$fRightModuleQuaternion'Quaternion'"$fLeftModuleQuaternion'Quaternion'$fRingQuaternion'$fRigQuaternion'$fUnitalQuaternion'$fSemiringQuaternion'$fMultiplicativeQuaternion'$fHopfAlgebrarQuaternionBasis'&$fInvolutiveCoalgebrarQuaternionBasis'$$fInvolutiveAlgebrarQuaternionBasis'$fBialgebrarQuaternionBasis'$$fCounitalCoalgebrarQuaternionBasis' $fUnitalAlgebrarQuaternionBasis'$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'$fDistinguishedQuaternionBasis'$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$fFactorableExp $fBandExp$fCommutativeExp $fDivisionExp $fUnitalExp$fMultiplicativeExp$fPartitionableLog$fIdempotentLog $fAbelianLog $fGroupLog$fRightModuleIntegerLog$fLeftModuleIntegerLog $fMonoidalLog$fRightModuleNaturalLog$fLeftModuleNaturalLog $fAdditiveLogbase Control.Arrowarr$# $fRingMap$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 $fFunctorMap$fSemigroupoidMap$fCategory*MapofRing$fRightModulerEnd$fLeftModulerEnd$fRightModuleEndEnd$fLeftModuleEndEnd $fRingEnd$fRigEnd $fSemiringEnd$fCommutativeEnd $fUnitalEnd$fMultiplicativeEnd $fGroupEnd $fMonoidalEnd $fAbelianEnd $fAdditiveEnd $fMonoidEnd$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 $fRingRngRing $fRigRngRing$fSemiringRngRing$fDivisionRngRing$fUnitalRngRing$fRightModuleRngRingRngRing$fLeftModuleRngRingRngRing$fCommutativeRngRing$fMultiplicativeRngRing$fGroupRngRing$fRightModuleIntegerRngRing$fLeftModuleIntegerRngRing$fMonoidalRngRing$fRightModuleNaturalRngRing$fLeftModuleNaturalRngRing$fAbelianRngRing$fAdditiveRngRing$fRightModuleIntegerZeroRng$fLeftModuleIntegerZeroRng$fRightModuleNaturalZeroRng$fLeftModuleNaturalZeroRng $fRngZeroRng$fCommutativeZeroRng$fSemiringZeroRng$fMultiplicativeZeroRng$fGroupZeroRng$fMonoidalZeroRng$fAbelianZeroRng$fIdempotentZeroRng$fAdditiveZeroRng$fCharacteristicFraction $fRigFraction$fMultiplicativeFraction$fUnitalFraction$fAdditiveFraction$fRightModuleNaturalFraction$fLeftModuleNaturalFraction$fRightModuleIntegerFraction$fLeftModuleIntegerFraction$fMonoidalFraction$fGroupFraction$fSemiringFraction$fAbelianFraction$fRingFraction$fDecidableUnitsFraction$fDecidableZeroFraction$fCommutativeFraction$fDivisionFraction $fOrdFraction $fEqFraction$fIntegralSemiringFraction$fShowFraction