<      !"#$%&'()*+,-./01234567 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 fghijklmno p q r s t u v w x y z { | } ~  portable provisionalEdward Kmett <ekmett@gmail.com> Trustworthy portable provisionalEdward Kmett <ekmett@gmail.com> Trustworthy9A vector is an additive group with additional structure. The zero vector Compute the sum of two vectors V2 1 2 ^+^ V2 3 4V2 4 6+Compute the difference between two vectors V2 4 5 - V2 3 1V2 1 4*Linearly interpolate between two vectors. Apply a function to merge the  'non-zero'= components of two vectors, unioning the rest of the values. * For a dense vector this is equivalent to . + For a sparse vector this is equivalent to . 3Apply a function to the components of two vectors. * For a dense vector this is equivalent to . + For a sparse vector this is equivalent to . Basis element !Compute the negation of a vector negated (V2 2 4) V2 (-2) (-4) Sum over multiple vectors sumV [V2 1 1, V2 3 4]V2 4 5  Compute the left scalar product  2 *^ V2 3 4V2 6 8!Compute the right scalar product  V2 3 4 ^* 2V2 6 8+Compute division by a scalar on the right. BProduce a default basis for a vector space. If the dimensionality 2 of the vector space is not statically known, see . :Produce a default basis for a vector space from which the  argument is drawn. )Produce a diagonal matrix from a vector. Create a unit vector. unit _x :: V2 IntV2 1 0&Outer (tensor) product of two vectors .       portable provisionalEdward Kmett <ekmett@gmail.com> Safe-InferredEProvides a fairly subjective test to see if a quantity is near zero. nearZero (1e-11 :: Double)FalsenearZero (1e-17 :: Double)TruenearZero (1e-5 :: Float)FalsenearZero (1e-7 :: Float)True&Determine if a quantity is near zero.  a  1e-12  a  1e-6  a  1e-12  a  1e-6 non-portable experimentalEdward Kmett <ekmett@gmail.com> Trustworthy Free and sparse inner product/metric spaces. ;Compute the inner product of two vectors or (equivalently)  convert a vector f a into a covector f a -> a. V2 1 2 `dot` V2 3 4119Compute the squared norm. The name quadrance arises from  Norman J. Wildberger's rational trigonometry. (Compute the quadrance of the difference ;Compute the distance between two vectors in a metric space /Compute the norm of a vector in a metric space *Convert a non-zero vector to unit vector.  Normalize a  functor to have unit . This function $ does not change the functor if its  is 0 or 1.  project u v computes the projection of v onto u.    non-portable experimentalEdward Kmett <ekmett@gmail.com> Trustworthy A 0-dimensional vector pure 1 :: V0 IntV0V0 + V0V0 !      ! ! !      non-portable experimentalEdward Kmett <ekmett@gmail.com> Trustworthy")A space that has at least 1 basis vector #. # V1 2 ^._x2V1 2 & _x .~ 3V1 3$A 1-dimensional vector pure 1 :: V1 IntV1 1 V1 2 + V1 3V1 5 V1 2 * V1 3V1 6 sum (V1 2)2"#$%& !"#$%&'()*"#$%&$%"#&"#$%& !"#$%&'()* non-portable experimentalEdward Kmett <ekmett@gmail.com> Trustworthy'6A space that distinguishes 2 orthogonal basis vectors # and (, but may have more. ( V2 1 2 ^._y2V2 1 2 & _y .~ 3V2 1 3   ( :: + (t a) a )   ) :: + (t a) (* a) *A 2-dimensional vector pure 1 :: V2 IntV2 1 1V2 1 2 + V2 3 4V2 4 6V2 1 2 * V2 3 4V2 3 8 sum (V2 1 2)3-+the counter-clockwise perpendicular vector perp $ V2 10 20 V2 (-20) 10''()*+,-.,-./0123456789:;<=>?@ABCDEFGHIJ "#&'()*+,-. *+"#'()&,-.$'()*+,-.,-./0123456789:;<=>?@ABCDEFGHIJ non-portable experimentalEdward Kmett <ekmett@gmail.com> Trustworthy/7A space that distinguishes 3 orthogonal basis vectors: #, (, and 0. (It may have more) 0   0 :: + (t a) a 1   1 :: + (t a) (2 a) 2A 3-dimensional vector 5cross product 6scalar triple product (/0123456KLMNOPQRSTUVWXYZ[\]^_`abcdefghij"#&'(),/01234562356"#'()/01&,4%/0123456KLMNOPQRSTUVWXYZ[\]^_`abcdefghij  non-portable experimentalEdward Kmett <ekmett@gmail.com> Trustworthy74A space that distinguishes orthogonal basis vectors #, (, 0, 8. (It may have more.) 8   8 :: + (t a) a 9   9 :: + (t a) (: a) :A 4-dimensional vector. =OConvert a 3-dimensional affine vector into a 4-dimensional homogeneous vector. >NConvert a 3-dimensional affine point into a 4-dimensional homogeneous vector. ?@Convert 4-dimensional projective coordinates to a 3-dimensional ' point. This operation may be denoted, euclidean [x:y:z:w] = (x/w,  y/w, z/w). where the projective, homogenous, coordinate  [x:y:z:w]/ is one of many associated with a single point (x/w,  y/w, z/w). *789:;<=>?klmnopqrstuvwxyz{|}~"#&'(),/014789:;<=>?:;=>?"#'()/01789&,4<'789:;<=>?klmnopqrstuvwxyz{|}~  non-portable experimentalEdward Kmett <ekmett@gmail.com> Trustworthy @ When lines are represented as PlC(Describe how two lines pass each other. D+The lines pass each other counterclockwise  (left-handed screw). E2The lines pass each other clockwise (right-handed  screw) F3The lines are coplanar (parallel or intersecting). GPlI>Given a pair of points represented by homogeneous coordinates  generate PlJ&Given a pair of 3D points, generate PlK&These elements form a basis for the PlL&These elements form a basis for the PlM&These elements form a basis for the PlN&These elements form a basis for the PlO&These elements form a basis for the PlP&These elements form a basis for the PlQ1These elements form an alternate basis for the PlR1These elements form an alternate basis for the PlS1These elements form an alternate basis for the PlT1These elements form an alternate basis for the PlU1These elements form an alternate basis for the PlV1These elements form an alternate basis for the Pl]Valid Pl^This isn'Yt th actual metric because this bilinear form gives rise to an isotropic quadratic space _@Checks if the line is near-isotropic (isotropic vectors in this 4 quadratic space represent lines in real 3d space). `5Checks if two lines intersect (or nearly intersect). a%Check how two lines pass each other.  passes l1 l2 describes  l2 when looking down l1. b"Checks if two lines are parallel. Represent a PlLChecks if two lines coincide in space. In other words, undirected equality. 9Checks if two lines coincide in space, and have the same  orientation. c8The minimum squared distance of a line from the origin. d1The point where a line is closest to the origin. e>Not all 6-dimensional points correspond to a line in 3D. This  predicate tests that a PlH@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcde&@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcde&GH]_^IJb`CFEDacde@BAKLMQPURVNSOTWXY\[ZB@BACFEDGHIJKLMNOPQRSTUVWXYZ[\]^_`abcde non-portable experimentalEdward Kmett <ekmett@gmail.com> Trustworthy fghijklmn fghijklmn fghkijlmnfghijklmn  non-portable experimentalEdward Kmett <ekmett@gmail.com> Safe-Inferredo5Requires and provides a default definition such that   q =  pAn involutive ring q<Conjugate a value. This defaults to the trivial involution. conjugate (1 :+ 2) 1.0 :+ (-2.0) conjugate 11"opqopqpqo!opq  non-portable experimentalEdward Kmett <ekmett@gmail.com> Trustworthyr0A vector space that includes the basis elements w, x, s and t v0A vector space that includes the basis elements w and x y Quaternions %quadrance of the imaginary component  norm of the imaginary component raise a y to a scalar power 1Helper for calculating with specific branch cuts 1Helper for calculating with specific branch cuts  with a specified branch cut.  with a specified branch cut.  with a specified branch cut.  with a specified branch cut.  with a specified branch cut.  with a specified branch cut. 8Spherical linear interpolation between two quaternions. Apply a rotation to a vector.  axis theta builds a y representing a  rotation of theta radians about axis. =rstuvwxyz{|}~     rstuvwxyz{|}~yzvwxrstu{|}~7rstuvwxyz{|}~       non-portable experimentalEdward Kmett <ekmett@gmail.com> TrustworthyCompute the trace of a matrix trace (V2 (V2 a b) (V2 c d))a + d!Compute the diagonal of a matrix diagonal (V2 (V2 a b) (V2 c d))V2 a d non-portable experimentalEdward Kmett <ekmett@gmail.com> Trustworthy+A 4x3 matrix with row-major representation +A 4x4 matrix with row-major representation +A 3x3 matrix with row-major representation +A 2x2 matrix with row-major representation This is a generalization of " to work over any corepresentable .     ::  f =>   s t a b ->   (f s) (f t) (f a) (f b) 7In practice it is used to access a column of a matrix. V2 (V3 1 2 3) (V3 4 5 6) ^._xV3 1 2 3$V2 (V3 1 2 3) (V3 4 5 6) ^.column _xV2 1 4UMatrix product. This can compute any combination of sparse and dense multiplication. :V2 (V3 1 2 3) (V3 4 5 6) !*! V3 (V2 1 2) (V2 3 4) (V2 4 5)V2 (V2 19 25) (V2 43 58)SV2 (fromList [(1,2)]) (fromList [(2,3)]) !*! fromList [(1,V3 0 0 1), (2, V3 0 0 5)]V2 (V3 0 0 2) (V3 0 0 15)Entry-wise matrix addition. 5V2 (V3 1 2 3) (V3 4 5 6) !+! V2 (V3 7 8 9) (V3 1 2 3)V2 (V3 8 10 12) (V3 5 7 9)Entry-wise matrix subtraction. 5V2 (V3 1 2 3) (V3 4 5 6) !-! V2 (V3 7 8 9) (V3 1 2 3)!V2 (V3 (-6) (-6) (-6)) (V3 3 3 3)Matrix * column vector $V2 (V3 1 2 3) (V3 4 5 6) !* V3 7 8 9 V2 50 122Row vector * matrix "V2 1 2 *! V2 (V3 3 4 5) (V3 6 7 8) V3 15 18 21Scalar-matrix product 5 *!! V2 (V2 1 2) (V2 3 4)V2 (V2 5 10) (V2 15 20)Matrix-scalar product V2 (V2 1 2) (V2 3 4) !!* 5V2 (V2 5 10) (V2 15 20)+Hermitian conjugate or conjugate transpose :adjoint (V2 (V2 (1 :+ 2) (3 :+ 4)) (V2 (5 :+ 6) (7 :+ 8)))LV2 (V2 (1.0 :+ (-2.0)) (5.0 :+ (-6.0))) (V2 (3.0 :+ (-4.0)) (7.0 :+ (-8.0)))$Build a rotation matrix from a unit y. ;Build a transformation matrix from a rotation matrix and a  translation vector. =Build a transformation matrix from a rotation expressed as a  y and a translation vector. AConvert from a 4x3 matrix to a 4x4 matrix, extending it with the  [ 0 0 0 1 ] column vector 8Convert a 3x3 matrix to a 4x4 matrix extending it with 0's in the new row and column. 2x2 identity matrix. eye2V2 (V2 1 0) (V2 0 1)3x3 identity matrix. eye3#V3 (V3 1 0 0) (V3 0 1 0) (V3 0 0 1)4x4 identity matrix. eye46V4 (V4 1 0 0 0) (V4 0 1 0 0) (V4 0 0 1 0) (V4 0 0 0 1)@Extract the translation vector (first three entries of the last # column) from a 3x4 or 4x4 matrix. 2x2 matrix determinant. det22 (V2 (V2 a b) (V2 c d)) a * d - b * c3x3 matrix determinant. +det33 (V3 (V3 a b c) (V3 d e f) (V3 g h i))?a * (e * i - f * h) - d * (b * i - c * h) + g * (b * f - c * e)2x2 matrix inverse. inv22 $ V2 (V2 1 2) (V2 3 4))Just (V2 (V2 (-2.0) 1.0) (V2 1.5 (-0.5)))3x3 matrix inverse. +inv33 $ V3 (V3 1 2 4) (V3 4 2 2) (V3 1 1 1)JJust (V3 (V3 0.0 0.5 (-1.0)) (V3 (-0.5) (-0.75) 3.5) (V3 0.5 0.25 (-1.5)))portable provisionalEdward Kmett <ekmett@gmail.com> Trustworthy?A handy wrapper to help distinguish points from vectors at the  type level ;An affine space is roughly a vector space in which we have = forgotten or at least pretend to have forgotten the origin.  a .+^ (b .-. a) = b@ & (a .+^ u) .+^ v = a .+^ (u ^+^ v)@ & (a .-. b) ^+^ v = (a .+^ v) .-. q@ :Get the difference between two points as a vector offset.  Add a vector offset to a point. 'Subtract a vector offset from a point. ECompute the quadrance of the difference (the square of the distance) /Distance between two points in an affine space Vector spaces have origins. %!"#$%&'()*+,-./0123456789: !"#$%&'()*+,-./0123456789: non-portable experimentalEdward Kmett <ekmett@gmail.com> Safe-InferredSerialize a linear type. Deserialize a linear type. None/A coassociative counital coalgebra over a ring *An associative unital algebra over a ring ;<=>?@ABCDEFGHIJK ;<=>?@ABCDEFGHIJKNoneJLinear functionals from elements of an (infinite) free module to a scalar LMNOPQRSTU LMNOPQRSTU non-portable experimentalEdward Kmett <ekmett@gmail.com>None  !"#$%&'()*+,-./0123456789:;<=>?opqrstuvwxyz{|}~V !"#$%&'()*+,-./012345667899:;<=>>?@ABCDEEFGH 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 uvvwxyz{|} ~             !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~                                                                                                                                  ! " # $ % & ' ( ) * + , - . / 0 1 2 3>456789>?:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnop linear-1.11Linear.V Linear.VectorLinear.Epsilon Linear.Metric Linear.V0 Linear.V1 Linear.V2 Linear.V3 Linear.V4Linear.PluckerLinear.ConjugateLinear.Quaternion Linear.Trace Linear.Matrix Linear.Affine Linear.BinaryLinear.AlgebraLinear.CovectorLinear.InstancesLinearreflection-1.5.1Data.ReflectionintAdditivezero^+^^-^lerpliftU2liftI2EelnegatedsumV*^^*^/basisbasisFor kroneckerunitouterEpsilonnearZeroMetricdot quadranceqddistancenormsignorm normalizeprojectV0R1_xV1exR2_y_xyV2eyperpangleR3_z_xyzV3ezcrosstripleR4_w_xyzwV4ewvectorpointnormalizePoint CoincidesRayLineLinePassCounterclockwise ClockwiseCoplanarPluckerplucker plucker3Dp01p02p03p23p31p12p10p20p30p32p13p21e01e02e03e23e31e12 squaredError>< isotropic intersectspassesparallelquadranceToOriginclosestToOriginisLineVtoVectorDim reflectDimdimreifyDim reifyVector fromVectorTrivialConjugate Conjugate conjugate Hamiltonian_j_k_ijk Complicated_e_i Quaternioneeeiejekabsipowasinqacosqatanqasinhqacoshqatanhqslerprotate axisAngleTracetracediagonalM43M44M33M22column!*!!+!!-!!**!*!!!!*adjointfromQuaternionmkTransformationMatmkTransformation m43_to_m44 m33_to_m44eye2eye3eye4 translationdet22det33inv22inv33PointPAffineDiff.-..+^.-^qdA distanceAlensPorigin putLinear getLinear CoalgebracomultcounitalAlgebramultunitalmultRep unitalRep comultRep counitalRepCovector runCovector$*$fTraversable1Complex$fFoldable1Complex$fTraversableComplex$fFoldableComplex$fMonadFixComplex$fMonadZipComplex$fMonadComplex $fBindComplex$fApplicativeComplex$fApplyComplex$fFunctorComplex $fBindHashMap$fApplyHashMapbaseControl.ApplicativeliftA2containers-0.5.0.0 Data.Map.Base unionWithintersectionWithSetOne_fillerchoices GAdditivegzerogliftU2gliftI2 fillFromList$fApplicativeSetOne$fFunctorSetOne$fAdditiveIdentity$fAdditiveComplex$fAdditive(->)$fAdditiveHashMap $fAdditiveMap$fAdditiveIntMap $fAdditive[]$fAdditiveMaybe$fAdditiveVector$fAdditiveZipList$fGAdditivePar1 $fGAdditiveM1$fGAdditiveRec1$fGAdditive:*: $fGAdditiveU1$fEpsilonCDoubleGHC.Numabsghc-prim GHC.Classes<=$fEpsilonCFloat$fEpsilonDouble$fEpsilonFloat$fMetricVector$fMetricHashMap $fMetricMap$fMetricIntMap$fMetricIdentity $fMonadFixV0 $fMonadZipV0$fVectorVectorV0$fMVectorMVectorV0 $fUnboxV0 $fEachV0V0ab$fIxedV0$fRepresentableV0$fTraversableWithIndexEV0$fFoldableWithIndexEV0$fFunctorWithIndexEV0 $fStorableV0 $fEpsilonV0 $fHashableV0$fDistributiveV0 $fMetricV0$fFractionalV0$fNumV0 $fMonadV0$fBindV0 $fAdditiveV0$fApplicativeV0 $fApplyV0$fTraversableV0 $fFoldableV0 $fFunctorV0 $fMonadFixV1 $fMonadZipV1$fVectorVectorV1$fMVectorMVectorV1 $fUnboxV1 $fEachV1V1ab$fIxedV1$fTraversableWithIndexEV1$fFoldableWithIndexEV1$fFunctorWithIndexEV1$fRepresentableV1$fIxV1$fDistributiveV1 $fR1Identity$fR1V1 $fMetricV1 $fHashableV1$fFractionalV1$fNumV1 $fMonadV1$fBindV1 $fAdditiveV1$fApplicativeV1 $fApplyV1$fTraversable1V1 $fFoldable1V1lens-4.5Control.Lens.TypeLens' $fMonadFixV2 $fMonadZipV2$fVectorVectorV2$fMVectorMVectorV2 $fUnboxV2 $fEachV2V2ab$fIxedV2$fTraversableWithIndexEV2$fFoldableWithIndexEV2$fFunctorWithIndexEV2$fRepresentableV2$fIxV2 $fStorableV2 $fEpsilonV2$fDistributiveV2$fR2V2$fR1V2 $fMetricV2$fFractionalV2$fNumV2 $fMonadV2$fBindV2 $fAdditiveV2 $fHashableV2$fApplicativeV2 $fApplyV2$fTraversable1V2 $fFoldable1V2$fTraversableV2 $fFoldableV2 $fFunctorV2 $fMonadFixV3 $fMonadZipV3$fVectorVectorV3$fMVectorMVectorV3 $fUnboxV3 $fEachV3V3ab$fIxedV3$fTraversableWithIndexEV3$fFoldableWithIndexEV3$fFunctorWithIndexEV3$fRepresentableV3$fIxV3 $fEpsilonV3 $fStorableV3$fR3V3$fR2V3$fR1V3$fDistributiveV3 $fMetricV3 $fHashableV3$fFractionalV3$fNumV3 $fMonadV3$fBindV3 $fAdditiveV3$fApplicativeV3 $fApplyV3$fTraversable1V3 $fFoldable1V3$fTraversableV3 $fFoldableV3 $fFunctorV3 $fMonadFixV4 $fMonadZipV4$fVectorVectorV4$fMVectorMVectorV4 $fUnboxV4 $fEachV4V4ab$fIxedV4$fTraversableWithIndexEV4$fFoldableWithIndexEV4$fFunctorWithIndexEV4$fRepresentableV4$fIxV4 $fEpsilonV4 $fStorableV4$fR4V4$fR3V4$fR2V4$fR1V4 $fHashableV4$fDistributiveV4 $fMetricV4$fFractionalV4$fNumV4 $fMonadV4$fBindV4 $fAdditiveV4 $fApplyV4$fApplicativeV4$fTraversable1V4 $fFoldable1V4$fTraversableV4 $fFoldableV4 $fFunctorV4toUV coincides coincides'anti$fMonadFixPlucker$fMonadZipPlucker$fVectorVectorPlucker$fMVectorMVectorPlucker$fUnboxPlucker $fEqCoincides$fEachPluckerPluckerab $fIxedPlucker$fTraversableWithIndexEPlucker$fFoldableWithIndexEPlucker$fFunctorWithIndexEPlucker$fEpsilonPlucker$fMetricPlucker$fStorablePlucker$fHashablePlucker$fFractionalPlucker $fNumPlucker $fIxPlucker$fTraversable1Plucker$fFoldable1Plucker$fTraversablePlucker$fFoldablePlucker$fRepresentablePlucker$fDistributivePlucker$fMonadPlucker $fBindPlucker$fAdditivePlucker$fApplicativePlucker$fApplyPlucker$fFunctorPlucker ReifiedDimretagDim $fEachVVab $fMonadFixV $fMonadZipV$fIxedV$fRepresentableV $fMetricV $fEpsilonV $fStorableV$fDistributiveV $fFractionalV$fNumV $fAdditiveV$fMonadV$fBindV$fApplicativeV$fApplyV$fTraversableV $fFoldableV $fFunctorV$fDimV$fDimReifiedDimGHC.Baseid$fTrivialConjugateCDouble$fTrivialConjugateCFloat$fTrivialConjugateFloat$fTrivialConjugateDouble$fTrivialConjugateWord8$fTrivialConjugateWord16$fTrivialConjugateWord32$fTrivialConjugateWord64$fTrivialConjugateWord$fTrivialConjugateInt8$fTrivialConjugateInt16$fTrivialConjugateInt32$fTrivialConjugateInt64$fTrivialConjugateInt$fTrivialConjugateInteger$fConjugateComplex$fConjugateCDouble$fConjugateCFloat$fConjugateFloat$fConjugateDouble$fConjugateWord8$fConjugateWord16$fConjugateWord32$fConjugateWord64$fConjugateWord$fConjugateInt8$fConjugateInt16$fConjugateInt32$fConjugateInt64$fConjugateInt$fConjugateIntegerqicutcutWith GHC.FloatasinacosatanasinhacoshatanhqNaN reimagine$fMonadFixQuaternion$fMonadZipQuaternion$fVectorVectorQuaternion$fMVectorMVectorQuaternion$fUnboxQuaternion$fEpsilonQuaternion$fFloatingQuaternion$fConjugateQuaternion$fDistributiveQuaternion$fHamiltonianQuaternion$fComplicatedQuaternion$fComplicatedComplex$fMetricQuaternion$fFractionalQuaternion$fHashableQuaternion$fNumQuaternion$fStorableQuaternion$fTraversableQuaternion$fFoldableQuaternion$fEachQuaternionQuaternionab$fIxedQuaternion!$fTraversableWithIndexEQuaternion$fFoldableWithIndexEQuaternion$fFunctorWithIndexEQuaternion$fRepresentableQuaternion$fIxQuaternion$fMonadQuaternion$fBindQuaternion$fAdditiveQuaternion$fApplicativeQuaternion$fApplyQuaternion$fFunctorQuaternion$fTraceCompose$fTraceProduct$fTraceComplex$fTraceQuaternion$fTracePlucker $fTraceV4 $fTraceV3 $fTraceV2 $fTraceV0$fTraceV$fTraceHashMap $fTraceMap $fTraceIntMapControl.Lens.LensinsideFunctoradjunctions-4.2Data.Functor.Rep RepresentableLens $fAffinePoint $fR4Point $fR3Point $fR2Point $fR1Point$fRepresentablePoint$fDistributivePoint $fBindPoint $fAffineV$fAffineHashMap $fAffineMap $fAffine(->)$fAffineQuaternion$fAffinePlucker $fAffineV4 $fAffineV3 $fAffineV2 $fAffineV1 $fAffineV0$fAffineVector$fAffineIdentity$fAffineIntMap $fAffineMaybe$fAffineZipList$fAffineComplex $fAffine[]$fCoalgebrar(,) $fCoalgebrarE$fCoalgebrarE0$fCoalgebrarE1$fCoalgebrarE2$fCoalgebrarE3$fCoalgebrarE4$fCoalgebrarE5$fCoalgebrar()$fCoalgebrarVoid $fAlgebrarE $fAlgebrarE0 $fAlgebrar(,) $fAlgebrar() $fAlgebrarE1 $fAlgebrarE2$fAlgebrarVoid $fNumCovector$fMonadPlusCovector$fAlternativeCovector$fPlusCovector $fAltCovector$fMonadCovector$fBindCovector$fApplicativeCovector$fApplyCovector$fFunctorCovector