r]      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\None ]^_`abcdef ]^_`abcdef ]^_`abcdefHaskell semiringsMITmail@doisinkidney.com experimentalNone /2345ITThe " .https://ncatlab.org/nlab/show/max-plus+algebraArctic0" or max-plus semiring. It is a semiring where:  = g  = -"  =   = Note that we can't use  from   % because annihilation needs to hold: -"  x = x  -" = -"Taking -" to be h# would break the above law. Using ! to represent it follows the law.The " /https://ncatlab.org/nlab/show/tropical+semiringTropical0" or min-plus semiring. It is a semiring where:  = i  = "  =   = Note that we can't use   from   % because annihilation needs to hold: "  x = x  " = "Taking " to be j" would break the above law. Using " to represent it follows the law.KA suitable definition of a square matrix for certain types which are both k and l%. For instance, given a type like so::{ (data Quad a = Quad a a a a deriving Showinstance Functor Quad where8 fmap f (Quad w x y z) = Quad (f w) (f x) (f y) (f z)instance Applicative Quad where pure x = Quad x x x x+ Quad fw fx fy fz <*> Quad xw xx xy xz =, Quad (fw xw) (fx xx) (fy xy) (fz xz)instance Foldable Quad where6 foldr f b (Quad w x y z) = f w (f x (f y (f z b)))instance Traversable Quad whereD traverse f (Quad w x y z) = Quad <$> f w <*> f x <*> f y <*> f z:})The newtype performs as you would expect:$getMatrix one :: Quad (Quad Integer)@Quad (Quad 1 0 0 0) (Quad 0 1 0 0) (Quad 0 0 1 0) (Quad 0 0 0 1)m1s are another type which works with this newtype::{7let xs = (Matrix . ZipList . map ZipList) [[1,2],[3,4]]7 ys = (Matrix . ZipList . map ZipList) [[5,6],[7,8]]8in (map getZipList . getZipList . getMatrix) (xs <.> ys):}[[19,22],[43,50]]  Monoid under . Analogous to   , but uses the  constraint, rather than n.  Monoid under . Analogous to  , but uses the  constraint, rather than n.A class for semirings with a concept of "negative infinity". It's important that this isn't regarded as the same as "bounded": x   should probably equal .A negative infinite value%Test if a value is negative infinity.xA class for semirings with a concept of "infinity". It's important that this isn't regarded as the same as "bounded": x   should probably equal .A positive infinite value%Test if a value is positive infinity.iUseful for operations where zeroes may need to be discarded: for instance in sparse matrix calculations.o if x is .A  5https://en.wikipedia.org/wiki/Semiring#Star_semirings Star semiring adds one operation,  to a  , such that it follows the law:  x =   x   x =    x  xNFor the semiring of types, this is equivalent to a list. When looking at the k and p; classes as (near-) semirings, this is equivalent to the q operation.Another operation,  , can be defined in relation to :  x = x   xBThis should be recognizable as a non-empty list on types, or the r operation in p.A  &https://en.wikipedia.org/wiki/SemiringSemiring% is like the the combination of two  s. The first is called ; it has the identity element /, and it is commutative. The second is called ; it has identity element , and it must distribute over .LawsNormal s laws (a  b)  c = a  (b  c)   a = a   = a (a  b)  c = a  (b  c)   a = a   = aCommutativity of  a  b = b  aDistribution of  over  a  (b  c) = (a  b)  (a  c) (a  b)  c = (a  c)  (b  c) Annihilation   a = a   = %An ordered semiring follows the laws: x t y => x  z t y  z x t y => x  z t y  z  t z u x t y => x  z t y  z u z  x t z  yThe identity of .The identity of .8An associative binary operation, which distributes over .-An associative, commutative binary operation.#Takes the sum of the elements of a v. Analogous to w on numbers, or x on ys. add [1..5]15add [False, False]Falseadd [False, True]Trueadd [True, undefined]True 'Takes the product of the elements of a v. Analogous to z on numbers, or { on ys. mul [1..5]120mul [True, True]Truemul [True, False]Falsemul [False, undefined]False!!The product of the contents of a v."The sum of the contents of a v.This is not a true semiring. In particular, it requires the underlying monoid to be commutative, and even then, it is only a near semiring. It is, however, extremely useful. For instance, this type:  forall a. | (| a)hIs a valid encoding of church numerals, with addition and multiplication being their semiring variants.The (->)& instance is analogous to the one for s.(getMin . foldMap Min) [1..10]1.0(getMax . foldMap Max) [1..10]10.0A polynomial in x: can be defined as a list of its coefficients, where the i!th element is the coefficient of x^i7. This is the semiring for such a list. Adapted from  Nhttps://pdfs.semanticscholar.org/702d/348c32133997e992db362a19697d5607ab32.pdfhere.}~  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~2@N#  !"# !"  }~    !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~2@N769 NoneI\The free semiring_Run a \.`Run a \-, interpreting it in the underlying semiring. \]^_`abc\]^_`a\]^a`_ \]^_`abc9 9 None2345T[ihAdds positive and negative infinity to a type. Useful for expressing detectable infinity in types like , etc.m[Adds positive infinity to a type. Useful for expressing detectable infinity in types like , etc.p[Adds negative infinity to a type. Useful for expressing detectable infinity in types like , etc.Not distributive.,Only lawful when used with positive numbers.Doesn't follow  or .,ijklmnopqrstuvwxyz{|}~ijklmnopqrpqrmnoijkl%ijklmnopqrstuvwxyz{|}~"Some interesting numeric semiringsMITmail@doisinkidney.com experimentalNone%&2345I(Adds a star operation to integral types. () = () () = ()  =   =   0 = 1  _ = *Adds a star operation to fractional types. () = () () = ()  =   =  $ x = if x < 1 then 1 / (1 - x) else  ;https://en.wikipedia.org/wiki/Semiring#cite_ref-droste_14-0 Wikipedia& has some information on this. Also  Shttp://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.304.6152&rep=rep1&type=pdfthis3 paper. Apparently used for probabilistic parsing. () = g () = ()  =   =  ;https://en.wikipedia.org/wiki/Semiring#cite_ref-droste_14-0 Wikipedia& has some information on this. Also  Shttp://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.304.6152&rep=rep1&type=pdfthis paper. () = g x  y = g 0 (x  y  1)  =   = Positive numbers only. () =  () =   =   = $Useful for some constraint problems. () = g () = i  = h  = jOnly expects positive numbers;/(Some functions for generating tests for s.MITmail@doisinkidney.com experimentalNoneTC0Typealias for ternary laws. Can be used like so: -smallCheck 6 (ternaryLaws :: TernaryLaws Int)D/Typealias for binary laws. Can be used like so: +smallCheck 8 (binaryLaws :: BinaryLaws Int)E.Typealias for unary laws. Can be used like so: *smallCheck 10 (unaryLaws :: UnaryLaws Int)FPlus is associative. (x  y)  z = x  (y  z)GMultiplication is associative. (x  y)  z = x  (y  z)HPlus is commutative. x  y = y  xI Multiplication distributes left. x  (y  z) = x  y  x  zJ!Multiplication distributes right. (x  y)  z = x  z  y  zKAdditive identity. x   =   x = xLMultiplicative identity. x   =   x = xMRight annihilation of  by .   x = NLeft annihilation of  by . x   = O$A test for all three unary laws for s (K, L, N, and M).P#A test for the unary laws for near-s (K, L, and M).Q,A test for all of the ternary laws for near-s (F, G, J).R&A test for all of the binary laws for s (just H).S'A test for all of the ternary laws for s (F, G, I, J).TThe star law for s.  x =   x   x =    x  xUThe plus law for s.  x = x   xV The laws for s (T, U).WAddition law for ordered s. x t y => x  z t y  z u z  x t z  yXMultiplication law for ordered s. x t y => x  z t y  z u z  x t z  yYLaws for ordered s (X, W).ZLaw for result of  operation. x   =   x =  x[Zero is zero law.   = o\ The laws for  s (Z, [).CDEFGHIJKLMNOPQRSTUVWXYZ[\CDEFGHIJKLMNOPQRSTUVWXYZ[\EDCKLNMOHRFGIJSPQTUVZ[\XWYCDEFGHIJKLMNOPQRSTUVWXYZ[\   !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefgghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnijolmplqrlstluvlwxiyzlq{lq|lq}lqij~ijllliylll lwlwllij+semiring-num-1.2.0.0-74YaMIouughCwA3JyIWLsy Data.SemiringData.Semiring.FreeData.Semiring.InfiniteData.Semiring.Numeric Test.SemiringData.Semiring.THData.SemigroupMaxData SemigroupMin Data.MonoidProductSumMonoid annihilateL mulDistribRgetMaxgetMinMatrix getMatrixMulgetMulAddgetAddHasNegativeInfinitynegativeInfinityisNegativeInfinityHasPositiveInfinitypositiveInfinityisPositiveInfinityDetectableZeroisZero StarSemiringstarplusSemiringzeroone<.><+>addmul mulFoldable addFoldable$fDetectableZeroFixed$fDetectableZeroComplex$fDetectableZeroRatio$fDetectableZeroNatural$fDetectableZeroCDev$fDetectableZeroCIno$fDetectableZeroCMode$fDetectableZeroCOff$fDetectableZeroCPid$fDetectableZeroCSsize$fDetectableZeroCGid$fDetectableZeroCNlink$fDetectableZeroCUid$fDetectableZeroCCc$fDetectableZeroCSpeed$fDetectableZeroCTcflag$fDetectableZeroCRLim$fDetectableZeroFd$fDetectableZeroWordPtr$fDetectableZeroIntPtr$fDetectableZeroCChar$fDetectableZeroCSChar$fDetectableZeroCUChar$fDetectableZeroCShort$fDetectableZeroCUShort$fDetectableZeroCInt$fDetectableZeroCUInt$fDetectableZeroCLong$fDetectableZeroCULong$fDetectableZeroCLLong$fDetectableZeroCULLong$fDetectableZeroCFloat$fDetectableZeroCDouble$fDetectableZeroCPtrdiff$fDetectableZeroCSize$fDetectableZeroCWchar$fDetectableZeroCSigAtomic$fDetectableZeroCClock$fDetectableZeroCTime$fDetectableZeroCUSeconds$fDetectableZeroCSUSeconds$fDetectableZeroCIntPtr$fDetectableZeroCUIntPtr$fDetectableZeroCIntMax$fDetectableZeroCUIntMax$fDetectableZeroDouble$fDetectableZeroFloat$fDetectableZeroWord64$fDetectableZeroWord32$fDetectableZeroWord16$fDetectableZeroWord8$fDetectableZeroWord$fDetectableZeroInteger$fDetectableZeroInt64$fDetectableZeroInt32$fDetectableZeroInt16$fDetectableZeroInt8$fDetectableZeroInt$fSemiringIdentity$fSemiringFixed$fSemiringComplex $fSemiringSum$fSemiringProduct$fSemiringRatio$fSemiringNatural$fSemiringCDev$fSemiringCIno$fSemiringCMode$fSemiringCOff$fSemiringCPid$fSemiringCSsize$fSemiringCGid$fSemiringCNlink$fSemiringCUid $fSemiringCCc$fSemiringCSpeed$fSemiringCTcflag$fSemiringCRLim $fSemiringFd$fSemiringWordPtr$fSemiringIntPtr$fSemiringCChar$fSemiringCSChar$fSemiringCUChar$fSemiringCShort$fSemiringCUShort$fSemiringCInt$fSemiringCUInt$fSemiringCLong$fSemiringCULong$fSemiringCLLong$fSemiringCULLong$fSemiringCFloat$fSemiringCDouble$fSemiringCPtrdiff$fSemiringCSize$fSemiringCWchar$fSemiringCSigAtomic$fSemiringCClock$fSemiringCTime$fSemiringCUSeconds$fSemiringCSUSeconds$fSemiringCIntPtr$fSemiringCUIntPtr$fSemiringCIntMax$fSemiringCUIntMax$fSemiringDouble$fSemiringFloat$fSemiringWord64$fSemiringWord32$fSemiringWord16$fSemiringWord8$fSemiringWord$fSemiringInteger$fSemiringInt64$fSemiringInt32$fSemiringInt16$fSemiringInt8 $fSemiringInt$fDetectableZeroAll$fDetectableZeroAny$fStarSemiringAll $fSemiringAll$fStarSemiringAny $fSemiringAny$fDetectableZeroEndo$fStarSemiringEndo$fSemiringEndo$fStarSemiring(->)$fSemiring(->)$fDetectableZeroMax$fDetectableZeroMin$fStarSemiringMin$fStarSemiringMax $fSemiringMin $fSemiringMax $fMonoidMin $fMonoidMax$fSemigroupMin$fSemigroupMax $fRead1Min $fShow1Min $fOrd1Min$fEq1Min $fRead1Max $fShow1Max $fOrd1Max$fEq1Max $fOrdMatrix $fEqMatrix $fReadMatrix $fShowMatrix $fOrd1Matrix $fEq1Matrix $fRead1Matrix $fShow1Matrix$fApplicativeState$fSemiringMatrix$fApplicativeMatrix $fMonoidMul $fMonoidAdd$fSemigroupMul$fSemigroupAdd $fRead1Mul $fShow1Mul $fOrd1Mul$fEq1Mul $fRead1Add $fShow1Add $fOrd1Add$fEq1Add$fDetectableZeroLog $fSemiringLog$fDetectableZeroSet $fSemiringMap $fSemiringSet$fDetectableZero[] $fSemiring[]$fStarSemiring()$fDetectableZero() $fSemiring()$fDetectableZeroBool$fStarSemiringBool$fSemiringBool$fHasNegativeInfinityCFloat$fHasPositiveInfinityCFloat$fHasNegativeInfinityCDouble$fHasPositiveInfinityCDouble$fHasNegativeInfinityFloat$fHasPositiveInfinityFloat$fHasNegativeInfinityDouble$fHasPositiveInfinityDouble$fEqMul$fOrdMul $fReadMul $fShowMul $fBoundedMul $fGenericMul $fGeneric1Mul$fNumMul $fEnumMul $fStorableMul$fFractionalMul $fRealMul $fRealFracMul $fFunctorMul $fFoldableMul$fTraversableMul $fSemiringMul$fDetectableZeroMul$fStarSemiringMul$fEqAdd$fOrdAdd $fReadAdd $fShowAdd $fBoundedAdd $fGenericAdd $fGeneric1Add$fNumAdd $fEnumAdd $fStorableAdd$fFractionalAdd $fRealAdd $fRealFracAdd $fFunctorAdd $fFoldableAdd$fTraversableAdd $fSemiringAdd$fDetectableZeroAdd$fStarSemiringAdd$fGenericMatrix$fGeneric1Matrix$fFunctorMatrix$fFoldableMatrix$fTraversableMatrix$fFunctorState$fEqMin$fOrdMin $fReadMin $fShowMin $fBoundedMin $fGenericMin $fGeneric1Min$fNumMin $fEnumMin $fStorableMin$fFractionalMin $fRealMin $fRealFracMin $fFunctorMin $fFoldableMin$fTraversableMin$fEqMax$fOrdMax $fReadMax $fShowMax $fBoundedMax $fGenericMax $fGeneric1Max$fNumMax $fEnumMax $fStorableMax$fFractionalMax $fRealMax $fRealFracMax $fFunctorMax $fFoldableMax$fTraversableMax$fDetectableZeroIdentity$fDetectableZeroSum$fDetectableZeroProduct$fSemiring(,,,,,,,,,,,,,,)$fSemiring(,,,,,,,,,,,,,)$fSemiring(,,,,,,,,,,,,)$fSemiring(,,,,,,,,,,,)$fSemiring(,,,,,,,,,,)$fSemiring(,,,,,,,,,)$fSemiring(,,,,,,,,)$fSemiring(,,,,,,,)$fSemiring(,,,,,,)$fSemiring(,,,,,)$fSemiring(,,,,)$fSemiring(,,,)$fSemiring(,,) $fSemiring(,)$fStarSemiring(,,,,,,,,,,,,,,)$fStarSemiring(,,,,,,,,,,,,,)$fStarSemiring(,,,,,,,,,,,,)$fStarSemiring(,,,,,,,,,,,)$fStarSemiring(,,,,,,,,,,)$fStarSemiring(,,,,,,,,,)$fStarSemiring(,,,,,,,,)$fStarSemiring(,,,,,,,)$fStarSemiring(,,,,,,)$fStarSemiring(,,,,,)$fStarSemiring(,,,,)$fStarSemiring(,,,)$fStarSemiring(,,)$fStarSemiring(,) $fDetectableZero(,,,,,,,,,,,,,,)$fDetectableZero(,,,,,,,,,,,,,)$fDetectableZero(,,,,,,,,,,,,)$fDetectableZero(,,,,,,,,,,,)$fDetectableZero(,,,,,,,,,,)$fDetectableZero(,,,,,,,,,)$fDetectableZero(,,,,,,,,)$fDetectableZero(,,,,,,,)$fDetectableZero(,,,,,,)$fDetectableZero(,,,,,)$fDetectableZero(,,,,)$fDetectableZero(,,,)$fDetectableZero(,,)$fDetectableZero(,)FreegetFreerunFree lowerFreeliftFree$fFoldableFree $fNumFree $fShowFree $fReadFree$fEqFree $fOrdFree$fSemiringFreeInfiniteNegativeFinitePositivePositiveInfinite PosFinitePositiveInfinityNegativeInfiniteNegativeInfinity NegFinite$fStorableInfinite$fStorablePositiveInfinite$fStorableNegativeInfinite $fNumInfinite$fNumPositiveInfinite$fNumNegativeInfinite$fMonoidInfinite$fMonoidPositiveInfinite$fMonoidNegativeInfinite$fEnumInfinite$fEnumPositiveInfinite$fEnumNegativeInfinite$fHasPositiveInfinityInfinite$fHasNegativeInfinityInfinite%$fHasPositiveInfinityPositiveInfinite%$fHasNegativeInfinityNegativeInfinite$fBoundedInfinite$fBoundedPositiveInfinite$fBoundedNegativeInfinite$fApplicativeInfinite$fDetectableZeroInfinite $fDetectableZeroPositiveInfinite $fDetectableZeroNegativeInfinite$fStarSemiringPositiveInfinite$fSemiringInfinite$fSemiringPositiveInfinite$fSemiringNegativeInfinite$fApplicativePositiveInfinite$fApplicativeNegativeInfinite$fEqNegativeInfinite$fOrdNegativeInfinite$fReadNegativeInfinite$fShowNegativeInfinite$fGenericNegativeInfinite$fGeneric1NegativeInfinite$fFunctorNegativeInfinite$fFoldableNegativeInfinite$fTraversableNegativeInfinite$fEqPositiveInfinite$fOrdPositiveInfinite$fReadPositiveInfinite$fShowPositiveInfinite$fGenericPositiveInfinite$fGeneric1PositiveInfinite$fFunctorPositiveInfinite$fFoldablePositiveInfinite$fTraversablePositiveInfinite $fEqInfinite $fOrdInfinite$fReadInfinite$fShowInfinite$fGenericInfinite$fGeneric1Infinite$fFunctorInfinite$fFoldableInfinite$fTraversableInfinitePosInt getPosIntPosFrac getPosFracViterbi getViterbi ŁukasiewiczgetŁukasiewiczDivision getDivision Bottleneck getBottleneck $fRead1PosInt $fShow1PosInt $fOrd1PosInt $fEq1PosInt$fStarSemiringPosInt$fDetectableZeroPosInt$fSemiringPosInt$fBoundedPosInt$fRead1PosFrac$fShow1PosFrac $fOrd1PosFrac $fEq1PosFrac$fStarSemiringPosFrac$fDetectableZeroPosFrac$fSemiringPosFrac$fBoundedPosFrac$fRead1Viterbi$fShow1Viterbi $fOrd1Viterbi $fEq1Viterbi$fSemiringViterbi$fRead1Łukasiewicz$fShow1Łukasiewicz$fOrd1Łukasiewicz$fEq1Łukasiewicz$fDetectableZeroŁukasiewicz$fSemiringŁukasiewicz$fRead1Division$fShow1Division$fOrd1Division $fEq1Division$fSemiringDivision$fRead1Bottleneck$fShow1Bottleneck$fOrd1Bottleneck$fEq1Bottleneck$fDetectableZeroBottleneck$fSemiringBottleneck$fEqBottleneck$fOrdBottleneck$fReadBottleneck$fShowBottleneck$fBoundedBottleneck$fGenericBottleneck$fGeneric1Bottleneck$fNumBottleneck$fEnumBottleneck$fStorableBottleneck$fFractionalBottleneck$fRealBottleneck$fRealFracBottleneck$fFunctorBottleneck$fFoldableBottleneck$fTraversableBottleneck $fEqDivision $fOrdDivision$fReadDivision$fShowDivision$fBoundedDivision$fGenericDivision$fGeneric1Division $fNumDivision$fEnumDivision$fStorableDivision$fFractionalDivision$fRealDivision$fRealFracDivision$fFunctorDivision$fFoldableDivision$fTraversableDivision$fDetectableZeroDivision$fEqŁukasiewicz$fOrdŁukasiewicz$fReadŁukasiewicz$fShowŁukasiewicz$fBoundedŁukasiewicz$fGenericŁukasiewicz$fGeneric1Łukasiewicz$fNumŁukasiewicz$fEnumŁukasiewicz$fStorableŁukasiewicz$fFractionalŁukasiewicz$fRealŁukasiewicz$fRealFracŁukasiewicz$fFunctorŁukasiewicz$fFoldableŁukasiewicz$fTraversableŁukasiewicz $fEqViterbi $fOrdViterbi $fReadViterbi $fShowViterbi$fBoundedViterbi$fGenericViterbi$fGeneric1Viterbi $fNumViterbi $fEnumViterbi$fStorableViterbi$fFractionalViterbi $fRealViterbi$fRealFracViterbi$fFunctorViterbi$fFoldableViterbi$fTraversableViterbi$fDetectableZeroViterbi $fEqPosFrac $fOrdPosFrac $fReadPosFrac $fShowPosFrac$fGenericPosFrac$fGeneric1PosFrac $fNumPosFrac $fEnumPosFrac$fStorablePosFrac$fFractionalPosFrac $fRealPosFrac$fRealFracPosFrac$fFunctorPosFrac$fFoldablePosFrac$fTraversablePosFrac $fEqPosInt $fOrdPosInt $fReadPosInt $fShowPosInt$fGenericPosInt$fGeneric1PosInt $fNumPosInt $fEnumPosInt$fStorablePosInt$fFractionalPosInt $fRealPosInt$fRealFracPosInt$fFunctorPosInt$fFoldablePosInt$fTraversablePosInt TernaryLaws BinaryLaws UnaryLaws plusAssocmulAssocplusComm mulDistribLplusIdmulId annihilateR unaryLaws nearUnaryLawsnearTernaryLaws binaryLaws ternaryLawsstarLawplusLawstarLaws ordAddLaw ordMulLawordLawszeroLaw zeroIsZerozeroLaws typeNamesvarNamesrepNappNcmbNstarInsinline semiringInszeroInsandAllghc-prim GHC.ClassesmaxbaseGHC.EnumminBoundminmaxBoundGHC.Base ApplicativeData.Traversable TraversableControl.ApplicativeZipListGHC.NumNum GHC.TypesTrue Alternativemanysome<=&& Data.FoldableFoldablesumorBoolproductandEndoState WrapBinaryisZeroEqdefaultPositiveInfinitydefaultIsPositiveInfinitydefaultIsNegativeInfinitydefaultNegativeInfinity showsNewtype readsNewtype evalStateimap#..#rep integer-gmpGHC.Integer.TypeInteger CoerceBinary maxBoundOfstrip stripFPtrstripPtr+-GHC.Realgcdlcm==