e      !"#$%&'()*+,-./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 NoneB`3 Minus () is reserved for where both the left and right cancellative laws hold. This then implies that the AdditiveGroup is also Abelian.Syntactic unary negation - substituting "negate a" for "-a" in code - is hard-coded in the language to assume a Num instance. So, for example, using ''-a = zero - a' for the second rule below doesn't work. Ha - a = zero negate a = zero - a negate a + a = zero a + negate a = zeroNon-commutative right minus a `plus` negate a = zeroNon-commutative left minus negate a `plus` a = zero>Additive is commutative, unital and associative under addition Ezero + a == a a + zero == a (a + b) + c == a + (b + c) a + b == b + a Idempotent magma for addition. a `plus` a == a Invertible magma for addition. " a " A: negate a " A&law is true by construction in Haskell Commutative magma for addition. a `plus` b == b `plus` aAssociative magma for addition. .(a `plus` b) `plus` c == a `plus` (b `plus` c)Unital magma for addition. %zero `plus` a == a a `plus` zero == aD is used as the operator for the additive magma to distinguish from  , which, by convention, implies commutativity " a,b " A: a `plus` b " A&law is true by construction in Haskell^sum definition avoiding a clash with the Sum monoid in base fixme: fit in with the Sum monoid     666 6SafeB`D CAn Abelian Group is associative, unital, invertible and commutative-A Group is associative, unital and invertibleA Loop is unital and invertible9A CMonoidal Magma is commutative, associative and unital.+A Monoidal Magma is associative and unital.An Idempotent Magma  a " a = aAn Invertible Magma " a " T: inv a " T&law is true by construction in HaskellA Commutative Magma  a " b = b " aAn Associative Magma (a " b) " c = a " (b " c)A Unital Magma unit " a = a a " unit = aA  -https://en.wikipedia.org/wiki/Magma_(algebra)Magma is a tuple (T,") consisting of a type a, anda function (") :: T -> T -> T&The mathematical laws for a magma are:2" is defined for all possible pairs of type T, and7" is closed in the set of all possible values of type Tor, more tersly, " a, b " T: a " b " TFThese laws are true by construction in haskell: the type signature of magma. and the above mathematical laws are synonyms.see 8http://chris-taylor.github.io/blog/2013/02/25/xor-trick/NoneB`W> Divide () is reserved for where both the left and right cancellative laws hold. This then implies that the MultiplicativeGroup is also Abelian. Aa / a = one recip a = one / a recip a * a = one a * recip a = oneNon-commutative right divide a `times` recip a = oneNon-commutative left divide recip a `times` a = oneJMultiplicative is commutative, associative and unital under multiplication Cone * a == a a * one == a (a * b) * c == a * (b * c) a * b == b * a$Idempotent magma for multiplication. a `times` a == a$Invertible magma for multiplication. " a " A: recip a " A&law is true by construction in Haskell%Commutative magma for multiplication. a `times` b == b `times` a%Associative magma for multiplication. 2(a `times` b) `times` c == a `times` (b `times` c) Unital magma for multiplication. %one `times` a == a a `times` one == aJ is used as the operator for the multiplicative magam to distinguish from , which, by convention, implies commutativity " a,b " A: a `times` b " A&law is true by construction in Haskellgproduct definition avoiding a clash with the Product monoid in base fixme: fit in with Product in base7777NoneB`Z$Distribution (and annihilation) laws [a * (b + c) == a * b + a * c (a + b) * c == a * c + b * c a * zero == zero zero * a == zeroNone;=>?B`` element by element division a ./. a == singleton one!element by element multiplication e(a .*. b) .*. c == a .*. (b .*. c) singleton one .*. a = a a .*. singelton one = a a .*. b == b .*. aelement by element subtraction a .-. a = singleton zeroelement by element addition S(a .+. b) .+. c == a .+. (b .+. c) zero .+. a = a a .+. zero = a a .+. b == b .+. a       7767None;=B`tInvolutive Ring ]adj (a + b) ==> adj a + adj b adj (a * b) ==> adj a * adj b adj one ==> one adj (adj a) ==> aNote: elements for which  adj a == a are called "self-adjoint". KleeneAlgebra aa `times` x + x = a ==> star a `times` x + x = x x `times` a + x = a ==> x `times` star a + x = x StarSemiring star a = one + a `times` star aICRing is a Ring with Multiplicative Commutation. It arises often due to 9 being defined as a multiplicative commutative operation.Ring A Ring consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication; it is an abelian group with a second binary operation that is associative, is distributive over the abelian group operation, and has an identity element.:Summary of the laws inherited from the ring super-classes: szero + a == a a + zero == a (a + b) + c == a + (b + c) a + b == b + a a - a = zero negate a = zero - a negate a + a = zero a + negate a = zero one `times` a == a a `times` one == a (a `times` b) `times` c == a `times` (b `times` c) a `times` (b + c) == a `times` b + a `times` c (a + b) `times` c == a `times` c + b `times` c a `times` zero == zero zero `times` a == zeroSemiring  NoneB`|X+fromInteger is the most problematic of the Num class operators. Particularly heinous, it is assumed that any number type can be constructed from an Integer, so that the broad classes of objects that are composed of multiple elements is avoided in haskell.ZKtoInteger is kept separate from Integral to help with compatability issues.\ Integral laws /b == zero || b * (a `div` b) + (a `mod` b) == ac coercion of \s fromIntegral a == af/raise a number to a non-negative integral powerXYZ[\`a]^b_cdefg\]^_`abZ[XYcdefgXYZ[\]^_`ab]7^7 None *;=>?B`LTrigonometric Field`A bounded field includes the concepts of infinity and NaN, thus moving away from error throwing. Mone / zero + infinity == infinity infinity + a == infinity zero / zero != nan}Note the tricky law that, although nan is assigned to zero/zero, they are never-the-less not equal. A committee decided this.Pquotient fields explode constraints if they allow for polymorphic integral types Ra - one < floor a <= a <= ceiling a < a + one round a == floor (a + one/(one+one))}fixme: had to redefine Signed operators here because of the Field import in Metric, itself due to Complex being defined thereA hyperbolic field class Ysqrt . (**2) == identity log . exp == identity for +ive b, a != 0,1: a ** logBase a b "H bTA Field is a Ring plus additive invertible and multiplicative invertible operations.<A summary of the rules inherited from super-classes of Field pzero + a == a a + zero == a (a + b) + c == a + (b + c) a + b == b + a a - a = zero negate a = zero - a negate a + a = zero a + negate a = zero one * a == a a * one == a (a * b) * c == a * (b * c) a * (b + c) == a * b + a * c (a + b) * c == a * c + b * c a * zero == zero zero * a == zero a * b == b * a a / a = one recip a = one / a recip a * a = one a * recip a = oneDA Semifield is chosen here to be a Field without an Additive InverseOtodo: work out boundings for complex as it stands now, complex is different eg $one / (zero :: Complex Float) == nan!todo: bottom is here somewhere???!!  None;=>?B`ftodo: This should probably be split off into some sort of alternative Equality logic, but to what end?1distance between numbers using L1, L2 or Lp-norms distanceL2 a b >= zero distanceL2 a a == zero \a b c -> distanceL2 a c + distanceL2 b c - distanceL2 a b >= zero && distanceL2 a b + distanceL2 b c - distanceL2 a c >= zero && distanceL2 a b + distanceL2 a c - distanceL2 b c >= zero &&WL1 and L2 norms are provided for potential speedups, as well as the generalized p-norm. for p >= 1 (normLp p a >= zero normLp p zero == zero=Note that the Normed codomain can be different to the domain.ez from base is not an operator replicated in numhask, being such a very silly name, and preferred is the much more obvious . Compare with Norm and Banach$ where there is a change in codomain abs a * sign a == aGeneralising this class tends towards size and direction (abs is the size on the one-dim number line of a vector with its tail at zero, and sign is the direction, right?).-todo: is utf perfectly acceptable these days?4 None;=>?B` Fractional6 in base splits into fromRatio and MultiplicativeGrouptoRatio is equivalent to Real in base. coercion of s fromRational a == affixme: use coerce is a subsidiary function used only in this module. It normalises a ratio by dividing both numerator and denominator by their greatest common divisor. x y$ is the non-negative factor of both x and y" of which every common factor of x and y is also a factor; for example  4 2 = 2,  (-4) 6 = 2,  0 4 = 4.  0 0 = 0U. (That is, the common divisor that is "greatest" in the divisibility preordering.)2Note: Since for signed fixed-width integer types,  minBound < 09, the result may be negative if one of the arguments is minBound& (and necessarily is if the other is 0 or minBound) for such types.   None,;=>?BFST`[ Kgeneralised outer product 3a><b + c><b == (a+c) >< b a><b + a><c == a >< (b+c)6todo: work out why these laws down't apply > a *. (b> c)== (a<b) .* c > (a> b).* c == a *. (b<c)grepresentation synthesisPtensorial typeQ4the inner product of a representable over a semiring ba <.> b == b <.> a a <.> (b +c) == a <.> b + a <.> c a <.> (s *. b + c) == s * (a <.> b) + a <.> c (s0 *. a)  . (s1 *. b) == s0 * s1 * (a  . b)SzBanach (with Norm) laws form rules around size and direction of a number, with a potential crossing into another codomain. 5a == singleton zero || normalizeL2 a *. normL2 a == aWDivision Module Laws ?nearZero a || a ./ one == a b == zero || a ./ b == recip b *. aZMultiplicative Module Laws xa .* one == a (a + b) .* c == (a .* c) + (b .* c) c *. (a + b) == (c *. a) + (c *. b) a .* zero == zero a .* b == b *. a]Subtraction Module Laws `(a + b) .- c == a + (b .- c) (a + b) .- c == (a .- c) + b a .- zero == a a .- b == negate b +. a`Additive Module Laws Y(a + b) .+ c == a + (b .+ c) (a + b) .+ c == (a .+ c) + b a .+ zero == a a .+ b == b +. aKMLNOPQRSTUVWXYZ[\]^_`ab`ab]^_Z[\WXYSTUVQRPKLMNOKLMNOQRSTUVWXYZ[\]^_`ab L8P8R8X7Y7[7\7^6_6a6b6NoneB`ɟ   XYZ[\`a]^b_cdefgKMLNOPQRSTUVWXYZ[\]^_`ab Safe;=>?B`GczThis class could also be called replicate. Looking forward, however, it may be useful to consider a Representable such as *VectorThing a = Vector a | Single a | Zeroand then ,singleton a = Single a singleton zero = Zeroshort-circuiting an expensive computation. As the class action then doesn't actually involve replication, it would be mis-named.cdcdcdSafeB`Ҹhijklmnop !"#$%&'()*+,-./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 p q r s tuv w xyz{|}~&numhask-0.2.2.0-E3E7C0joxeXCcSuuMGRqvgNumHask.AlgebraNumHask.Algebra.AdditiveNumHask.Algebra.MagmaNumHask.Algebra.MultiplicativeNumHask.Algebra.DistributionNumHask.Algebra.BasisNumHask.Algebra.RingNumHask.Algebra.IntegralNumHask.Algebra.FieldNumHask.Algebra.MetricNumHask.Algebra.RationalNumHask.Algebra.ModuleNumHask.Algebra.Singleton Paths_numhaskbase Data.Complex:+Complex AdditiveGroup-AdditiveRightCancellative-~AdditiveLeftCancellative~-Additive+AdditiveIdempotentAdditiveInvertiblenegateAdditiveCommutativeAdditiveAssociativeAdditiveUnitalzero 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$fAdditiveInt$fAdditiveFloat$fAdditiveDouble$fAdditiveGroupWord64$fAdditiveGroupWord32$fAdditiveGroupWord16$fAdditiveGroupWord8$fAdditiveGroupWord$fAdditiveGroupInt64$fAdditiveGroupInt32$fAdditiveGroupInt16$fAdditiveGroupInt8$fAdditiveGroupComplex$fAdditiveGroupInteger$fAdditiveGroupInt$fAdditiveGroupFloat$fAdditiveGroupDoubleAbelianGroupLoop CMonoidalMonoidal Idempotent Invertibleinv Commutative AssociativeUnitalunitMagma⊕ groupSwapMultiplicativeGroup/MultiplicativeRightCancellative/~MultiplicativeLeftCancellative~/Multiplicative*MultiplicativeIdempotentMultiplicativeInvertiblerecipMultiplicativeCommutativeMultiplicativeAssociativeMultiplicativeUnitaloneMultiplicativeMagmatimesproduct$fMultiplicativeMagmaWord64$fMultiplicativeMagmaWord32$fMultiplicativeMagmaWord16$fMultiplicativeMagmaWord8$fMultiplicativeMagmaWord$fMultiplicativeMagmaInt64$fMultiplicativeMagmaInt32$fMultiplicativeMagmaInt16$fMultiplicativeMagmaInt8$fMultiplicativeMagmaNatural$fMultiplicativeMagmaComplex$fMultiplicativeMagmaBool$fMultiplicativeMagmaInteger$fMultiplicativeMagmaInt$fMultiplicativeMagmaFloat$fMultiplicativeMagmaDouble$fMultiplicativeUnitalWord64$fMultiplicativeUnitalWord32$fMultiplicativeUnitalWord16$fMultiplicativeUnitalWord8$fMultiplicativeUnitalWord$fMultiplicativeUnitalInt64$fMultiplicativeUnitalInt32$fMultiplicativeUnitalInt16$fMultiplicativeUnitalInt8$fMultiplicativeUnitalNatural$fMultiplicativeUnitalComplex$fMultiplicativeUnitalBool$fMultiplicativeUnitalInteger$fMultiplicativeUnitalInt$fMultiplicativeUnitalFloat$fMultiplicativeUnitalDouble!$fMultiplicativeAssociativeWord64!$fMultiplicativeAssociativeWord32!$fMultiplicativeAssociativeWord16 $fMultiplicativeAssociativeWord8$fMultiplicativeAssociativeWord $fMultiplicativeAssociativeInt64 $fMultiplicativeAssociativeInt32 $fMultiplicativeAssociativeInt16$fMultiplicativeAssociativeInt8"$fMultiplicativeAssociativeNatural"$fMultiplicativeAssociativeComplex$fMultiplicativeAssociativeBool"$fMultiplicativeAssociativeInteger$fMultiplicativeAssociativeInt $fMultiplicativeAssociativeFloat!$fMultiplicativeAssociativeDouble!$fMultiplicativeCommutativeWord64!$fMultiplicativeCommutativeWord32!$fMultiplicativeCommutativeWord16 $fMultiplicativeCommutativeWord8$fMultiplicativeCommutativeWord $fMultiplicativeCommutativeInt64 $fMultiplicativeCommutativeInt32 $fMultiplicativeCommutativeInt16$fMultiplicativeCommutativeInt8"$fMultiplicativeCommutativeNatural"$fMultiplicativeCommutativeComplex$fMultiplicativeCommutativeBool"$fMultiplicativeCommutativeInteger$fMultiplicativeCommutativeInt 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AdditiveBasis.+.InvolutiveRingadj KleeneAlgebra StarSemiringstarplus'CRingRingSemiring$fSemiringWord64$fSemiringWord32$fSemiringWord16$fSemiringWord8$fSemiringWord$fSemiringInt64$fSemiringInt32$fSemiringInt16$fSemiringInt8$fSemiringNatural$fSemiringComplex$fSemiringBool$fSemiringInteger $fSemiringInt$fSemiringFloat$fSemiringDouble $fRingWord64 $fRingWord32 $fRingWord16 $fRingWord8 $fRingWord $fRingInt64 $fRingInt32 $fRingInt16 $fRingInt8 $fRingComplex $fRingInteger $fRingInt $fRingFloat $fRingDouble $fCRingWord64 $fCRingWord32 $fCRingWord16 $fCRingWord8 $fCRingWord $fCRingInt64 $fCRingInt32 $fCRingInt16 $fCRingInt8$fCRingComplex$fCRingInteger $fCRingInt $fCRingFloat $fCRingDouble$fInvolutiveRingWord64$fInvolutiveRingWord32$fInvolutiveRingWord16$fInvolutiveRingWord8$fInvolutiveRingWord$fInvolutiveRingInt64$fInvolutiveRingInt32$fInvolutiveRingInt16$fInvolutiveRingInt8$fInvolutiveRingNatural$fInvolutiveRingComplex$fInvolutiveRingInt$fInvolutiveRingInteger$fInvolutiveRingFloat$fInvolutiveRingDouble FromInteger fromInteger ToInteger toIntegerIntegraldivmoddivModquotremquotRem fromIntegralevenodd^^^$fIntegralWord64$fIntegralWord32$fIntegralWord16$fIntegralWord8$fIntegralWord$fIntegralInt64$fIntegralInt32$fIntegralInt16$fIntegralInt8$fIntegralNatural$fIntegralInteger $fIntegralInt$fToIntegerWord64$fToIntegerWord32$fToIntegerWord16$fToIntegerWord8$fToIntegerWord$fToIntegerInt64$fToIntegerInt32$fToIntegerInt16$fToIntegerInt8$fToIntegerNatural$fToIntegerInteger$fToIntegerInt$fFromIntegerWord64$fFromIntegerWord32$fFromIntegerWord16$fFromIntegerWord8$fFromIntegerWord$fFromIntegerInt64$fFromIntegerInt32$fFromIntegerInt16$fFromIntegerInt8$fFromIntegerNatural$fFromIntegerInteger$fFromIntegerInt$fFromIntegerFloat$fFromIntegerDouble TrigFieldpisincostanasinacosatansinhcoshtanhasinhacoshatanh BoundedFieldLowerBoundedField negInfinityUpperBoundedFieldinfinitynan QuotientFieldproperFractionroundceilingfloorExpFieldexploglogBase**sqrtField Semifield$fSemifieldComplex$fSemifieldFloat$fSemifieldDouble$fFieldComplex $fFieldFloat $fFieldDouble$fExpFieldFloat$fExpFieldDouble$fQuotientFieldDoubleInteger$fQuotientFieldFloatInteger$fUpperBoundedFieldComplex$fUpperBoundedFieldDouble$fUpperBoundedFieldFloat$fLowerBoundedFieldDouble$fLowerBoundedFieldFloat$fBoundedFielda$fTrigFieldFloat$fTrigFieldDouble$fExpFieldComplexEpsilonnearZero aboutEqualpositive veryPositive veryNegativeMetric distanceL1 distanceL2 distanceLpNormednormL1normL2normLpSignedsignabs≈$fSignedWord64$fSignedWord32$fSignedWord16 $fSignedWord8 $fSignedWord $fSignedInt64 $fSignedInt32 $fSignedInt16 $fSignedInt8$fSignedNatural$fSignedInteger $fSignedInt $fSignedFloat$fSignedDouble$fNormedWord64Word64$fNormedWord32Word32$fNormedWord16Word16$fNormedWord8Word8$fNormedWordWord$fNormedInt64Int64$fNormedInt32Int32$fNormedInt16Int16$fNormedInt8Int8$fNormedNaturalNatural$fNormedComplexa$fNormedIntegerInteger$fNormedIntInt$fNormedFloatFloat$fNormedDoubleDouble$fMetricWord64Word64$fMetricWord32Word32$fMetricWord16Word16$fMetricWord8Word8$fMetricWordWord$fMetricInt64Int64$fMetricInt32Int32$fMetricInt16Int16$fMetricInt8Int8$fMetricNaturalNatural$fMetricComplexa$fMetricIntegerInteger$fMetricIntInt$fMetricFloatFloat$fMetricDoubleDouble$fEpsilonWord64$fEpsilonWord32$fEpsilonWord16$fEpsilonWord8 $fEpsilonWord$fEpsilonInt64$fEpsilonInt32$fEpsilonInt16 $fEpsilonInt8$fEpsilonComplex$fEpsilonInteger $fEpsilonInt$fEpsilonFloat$fEpsilonDouble FromRatio fromRatioToRatiotoRatioRationalRatio:% fromRationalreducegcd$fFromIntegerRatio$fEpsilonRatio$fMetricRatioRatio$fNormedRatioRatio $fSignedRatio$fLowerBoundedFieldRatio$fUpperBoundedFieldRatio$fQuotientFieldRatiob $fFieldRatio$fSemifieldRatio$fInvolutiveRingRatio $fCRingRatio $fRingRatio$fSemiringRatio$fDistributionRatio$fMultiplicativeGroupRatio$fMultiplicativeRatio$fMultiplicativeInvertibleRatio $fMultiplicativeCommutativeRatio $fMultiplicativeAssociativeRatio$fMultiplicativeUnitalRatio$fMultiplicativeMagmaRatio$fAdditiveGroupRatio$fAdditiveRatio$fAdditiveInvertibleRatio$fAdditiveCommutativeRatio$fAdditiveAssociativeRatio$fAdditiveUnitalRatio$fAdditiveMagmaRatio $fOrdRatio$fToRatioWord64$fToRatioWord32$fToRatioWord16$fToRatioWord8 $fToRatioWord$fToRatioInt64$fToRatioInt32$fToRatioInt16 $fToRatioInt8$fToRatioRatio$fToRatioNatural$fToRatioInteger $fToRatioInt$fToRatioFloat$fToRatioDouble$fToRatioRatio0$fFromRatioFloat$fFromRatioDouble$fFromRatioRatio $fEqRatio $fShowRatio TensorProduct><outer timesleft timesrightHilbert<.>Banach normalizeL1 normalizeL2 normalizeLpMultiplicativeGroupModule.//.MultiplicativeModule.**.AdditiveGroupModule.--.AdditiveModule.++. Singleton singletonGHC.NumsignumfromBaseRational TensorRepversion getBinDir getLibDir getDynLibDir getDataDir getLibexecDir getSysconfDirgetDataFileName