Z      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~                                  ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T U V W X Y SafeB`%| 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 + aIdempotent 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` a Associative 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     6666SafeB`6d <CAn Abelian Group is associative, unital, invertible and commutative=-A Group is associative, unital and invertible>A Loop is unital and invertible?9A CMonoidal Magma is commutative, associative and unital.@+A Monoidal Magma is associative and unital.AAn Idempotent Magma  a " a = aBAn Invertible Magma " a " T: inv a " T&law is true by construction in HaskellDA Commutative Magma  a " b = b " aEAn Associative Magma (a " b) " c = a " (b " c)FA Unital Magma unit " a = a a " unit = aHA  -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.Jsee 8http://chris-taylor.github.io/blog/2013/02/25/xor-trick/<=>?@ABCDEFGHIJHIFGEDBCA@?>=J<BCFGHISafeB`H KDivide (L) 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 = oneMNon-commutative right divide a `times` recip a = oneONon-commutative left divide recip a `times` a = oneQJMultiplicative is commutative, associative and unital under multiplication Cone * a == a a * one == a (a * b) * c == a * (b * c) a * b == b * aZ$Idempotent magma for multiplication. a `times` a == aS$Invertible magma for multiplication. " a " A: recip a " A&law is true by construction in HaskellU%Commutative magma for multiplication. a `times` b == b `times` aV%Associative magma for multiplication. 2(a `times` b) `times` c == a `times` (b `times` c)W Unital magma for multiplication. %one `times` a == a a `times` one == aYZJ is used as the operator for the multiplicative magam to distinguish from R, which, by convention, implies commutativity " a,b " A: a `times` b " A&law is true by construction in Haskell[Cproduct definition avoiding a clash with the Product monoid in baseKLMNOPQRSTUVWXYZ[YZWXVUST[QRMNOPKLKLMNOPQRSTWXYZL7N7P7R7SafeB`K$Distribution (and annihilation) laws [a * (b + c) == a * b + a * c (a + b) * c == a * c + b * c a * zero == zero zero * a == zeroSafe;=>?B`Relement 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 .+. a7767SafeB`^ 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 R9 being defined as a multiplicative commutative operation.ARing a 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 == zeroSemiringSafeB`f+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.KtoInteger is kept separate from Integral to help with compatability issues. Integral laws /b == zero || b * (a `div` b) + (a `mod` b) == a coercion of s fromIntegral a == a 77SafeB`} Trigonometric Field`A bounded field includes the concepts of infinity and NaN, thus moving away from error throwing. one / zero + infinity == infinity infinity + a == infinity isNaN (infinity - infinity) isNaN (infinity / infinity) isNaN (nan + a) 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))A 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 = one:A Semifield is a Field without Commutative Multiplication. prints as Infinity prints as  `-Infinity`Otodo: work out boundings for complex as it stands now, complex is different eg $one / (zero :: Complex Float) == nan!todo: bottom is here somewhere???## Safe;=>?B`Oftodo: This should probably be split off into some sort of alternative Equality logic, but to what end?distance between numbers distance a b >= zero distance a a == zero \a b c -> distance a c + distance b c - distance a b >= zero && distance a b + distance b c - distance a c >= zero && distance a b + distance a c - distance b c >= zero &&@Like Signed, except the codomain can be different to the domain.signumz from base is not an operator replicated in numhask, being such a very silly name, and preferred is the much more obvious . Compare with Norm and Banach$ 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 Safe,;=>?BFST`T  generalised 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)[representation synthesistensorial type4the 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)zBanach (with Norm) laws form rules around size and direction of a number, with a potential crossing into another codomain. 1a == singleton zero || normalize a *. size a == aDivision Module Laws ?nearZero a || a ./ one == a b == zero || a ./ b == recip b *. aMultiplicative 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 *. aSubtraction Module Laws `(a + b) .- c == a + (b .- c) (a + b) .- c == (a .- c) + b a .- zero == a a .- b == negate b +. aAdditive Module Laws Y(a + b) .+ c == a + (b .+ c) (a + b) .+ c == (a .+ c) + b a .+ zero == a a .+ b == b +. a   ! !     !  888777766 6!6 SafeB` <=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[   ! Safe;=>?B`"zThis 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."#"#"#SafeB`\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ <=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[   !"# None<B` WsemiringXringY starsemiring6$%&'()*+,-./2013456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXY6/012345&'()*+,-.%$6789:;<>=?@ABCDEFGHISTUVLMNOKJRPQWXY&'()*+,-./012345Safe -B`[SafeB` !"#$%&'()*+,-./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 ij klmnopqlrslrtluvluwlmxlmylz{l|}lm~lllllllllooollllllzlzlmlmlmlmlmlmlmlloooooooollzlzllllllllllllllzlzlzllllmlmlmlmlmlmllllllllllllllllllllllllllllllllllllllllllllmlmlmlmoooooooooll l l   lmlmlmoooolzlzlzloolll l!l"#l"$l%&l%'l%(o)l*l+l+l,l,l-l.l.l/l0l1l2l3l3l4l5l6l7l8l9l:l;o<o=l>o?l@AlBClDElFGlHIlJKlJLlJMlNOlJPQRSQRTQRUQRVQRWQRXQRYQRZQR[QR\QR]QR^QR_QR`QRaQRbQRcQRdQReQRfQRgQRhQRiQRjQRkQRlQRmQRnQRoQRoQRplzqlzrlzslztlzulzvlzwlzxlzylzzlz{lz|lz}lz~lzlzlzlzlzlllmlmllllllllllllllllllllllllllllNlNlNlNlNlNlNlNlNlNllllllllllllllllllllllllllllllllllllllll|l|l|l|l|l|l|l|l|l|l|l|l|l|lllllllllllllllllllllllllll l  l  l l llllllllllllllllllll l l!l!l"l"l#l#lJ$lJ%lJ&lJ'lJ(lJ)lJ*lJ+lJ,lJ-lJ.lJ/lJ0lJ1lJ2l34l35l36l37l37l38l38l39l39l3:l3:l3;l3;l3<l3<l3=l3=l3>l3?l3@l3Al3Bl3Cl3Dl3El3Fl3Gl3HlIlJKlJLlJMlJNlJOlJPlJQlJRlJSlJTlJUlJVlJWl3XlYZlY[lY\lY]lY^lY_lY`lY`lYalYblYclYdlYelYflYglYhlijliklillimlnolnolnplqlrlsltlulvlwlxlylzl{l|l}l~lllllllFlFlFlFlFlFlFlFlFlFlFlFlFlFlFlFlFlFlFlFlFlFlFlFlFlFlFlllllllllllllllllllllllllllllllllllllllllll%l%l%l%l%l%lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lz lz lz lz lzlzlzllDlDlrlrlrlrlrlrlrlrlrlrlrlr lr!lr"lr#l$%l$&l$'l$(l$)l$*l$+l$,lu-lu.lu/l01l02l03l04l05l06l07l08l09l0:l;l<=lm>lm?lm@lmAlmBlmClmDlmElmFlmGlmHlmIlmJlmKlmLlmMlmNlmOlmPlmQlmRlmSlmTlmUlmVlYWlYWlBXlBYoZo[o\]^_`ab`cd`ef`gh`ijklmklnkloklpklqrstrsursvrswxyz{|}{|~{|{|{|{|{{{{{{{{xxxxxxxxxxxxxxxxxxxx&&&&&&      !"#$%&'()*+,-./0&numhask-0.1.4.0-Fbr4hCahrTL1ALEuc0R8G2NumHask.Algebra.AdditiveNumHask.Algebra.MagmaNumHask.Algebra.MultiplicativeNumHask.Algebra.DistributionNumHask.Algebra.BasisNumHask.Algebra.RingNumHask.Algebra.IntegralNumHask.Algebra.FieldNumHask.Algebra.MetricNumHask.Algebra.ModuleNumHask.Algebra.Singleton NumHask.LawsNumHask.AlgebraNumHask.PreludeNumHask.Examples Paths_numhask AdditiveGroup-AdditiveRightCancellative-~AdditiveLeftCancellative~-Additive+AdditiveIdempotentAdditiveInvertiblenegateAdditiveCommutativeAdditiveAssociativeAdditiveUnitalzero AdditiveMagmaplussum$fAdditiveMagmaComplex$fAdditiveMagmaBool$fAdditiveMagmaInteger$fAdditiveMagmaInt$fAdditiveMagmaFloat$fAdditiveMagmaDouble$fAdditiveUnitalComplex$fAdditiveUnitalBool$fAdditiveUnitalInteger$fAdditiveUnitalInt$fAdditiveUnitalFloat$fAdditiveUnitalDouble$fAdditiveAssociativeComplex$fAdditiveAssociativeBool$fAdditiveAssociativeInteger$fAdditiveAssociativeInt$fAdditiveAssociativeFloat$fAdditiveAssociativeDouble$fAdditiveCommutativeComplex$fAdditiveCommutativeBool$fAdditiveCommutativeInteger$fAdditiveCommutativeInt$fAdditiveCommutativeFloat$fAdditiveCommutativeDouble$fAdditiveInvertibleComplex$fAdditiveInvertibleBool$fAdditiveInvertibleInteger$fAdditiveInvertibleInt$fAdditiveInvertibleFloat$fAdditiveInvertibleDouble$fAdditiveIdempotentBool$fAdditiveComplex$fAdditiveBool$fAdditiveInteger $fAdditiveInt$fAdditiveFloat$fAdditiveDouble$fAdditiveGroupComplex$fAdditiveGroupInteger$fAdditiveGroupInt$fAdditiveGroupFloat$fAdditiveGroupDoubleAbelianGroupLoop CMonoidalMonoidal Idempotent Invertibleinv Commutative AssociativeUnitalunitMagma⊕ groupSwapMultiplicativeGroup/MultiplicativeRightCancellative/~MultiplicativeLeftCancellative~/Multiplicative*MultiplicativeInvertiblerecipMultiplicativeCommutativeMultiplicativeAssociativeMultiplicativeUnitaloneMultiplicativeMagmatimesproduct$fMultiplicativeMagmaComplex$fMultiplicativeMagmaBool$fMultiplicativeMagmaInteger$fMultiplicativeMagmaInt$fMultiplicativeMagmaFloat$fMultiplicativeMagmaDouble$fMultiplicativeUnitalComplex$fMultiplicativeUnitalBool$fMultiplicativeUnitalInteger$fMultiplicativeUnitalInt$fMultiplicativeUnitalFloat$fMultiplicativeUnitalDouble"$fMultiplicativeAssociativeComplex$fMultiplicativeAssociativeBool"$fMultiplicativeAssociativeInteger$fMultiplicativeAssociativeInt $fMultiplicativeAssociativeFloat!$fMultiplicativeAssociativeDouble"$fMultiplicativeCommutativeComplex$fMultiplicativeCommutativeBool"$fMultiplicativeCommutativeInteger$fMultiplicativeCommutativeInt $fMultiplicativeCommutativeFloat!$fMultiplicativeCommutativeDouble!$fMultiplicativeInvertibleComplex$fMultiplicativeInvertibleFloat $fMultiplicativeInvertibleDouble$fMultiplicativeIdempotentBool$fMultiplicativeComplex$fMultiplicativeBool$fMultiplicativeInteger$fMultiplicativeInt$fMultiplicativeFloat$fMultiplicativeDouble$fMultiplicativeGroupComplex$fMultiplicativeGroupFloat$fMultiplicativeGroupDouble Distribution$fDistributionComplex$fDistributionBool$fDistributionInteger$fDistributionInt$fDistributionFloat$fDistributionDoubleMultiplicativeGroupBasis./.MultiplicativeBasis.*.AdditiveGroupBasis.-. AdditiveBasis.+. KleeneAlgebra StarSemiringstarplus'CRingRingSemiring$fSemiringComplex$fSemiringBool$fSemiringInteger $fSemiringInt$fSemiringFloat$fSemiringDouble $fRingComplex $fRingInteger $fRingInt $fRingFloat $fRingDouble$fCRingComplex$fCRingInteger $fCRingInt $fCRingFloat $fCRingDouble FromInteger fromInteger ToInteger toIntegerIntegraldivmoddivMod fromIntegral$fIntegralInteger $fIntegralInt$fToIntegerInteger$fToIntegerInt$fFromIntegerInteger$fFromIntegerInt$fFromIntegerFloat$fFromIntegerDouble TrigFieldpisincostanasinacosatansinhcoshtanhasinhacoshatanhatan2 BoundedFieldmaxBoundminBoundnanisNaN QuotientFieldroundceilingfloor^^ExpFieldexploglogBase**sqrtField Semifieldinfinity neginfinity$fSemifieldComplex$fSemifieldFloat$fSemifieldDouble$fFieldComplex $fFieldFloat $fFieldDouble$fExpFieldFloat$fExpFieldDouble$fQuotientFieldDouble$fQuotientFieldFloat$fBoundedFieldComplex$fBoundedFieldDouble$fBoundedFieldFloat$fTrigFieldFloat$fTrigFieldDouble$fExpFieldComplexEpsilonnearZero aboutEqualpositive veryPositive veryNegativeMetricdistanceNormedsizeSignedsignabs≈$fSignedInteger $fSignedInt $fSignedFloat$fSignedDouble$fNormedComplexa$fNormedIntegerInteger$fNormedIntInt$fNormedFloatFloat$fNormedDoubleDouble$fMetricComplexa$fMetricIntegerInteger$fMetricIntInt$fMetricFloatFloat$fMetricDoubleDouble$fEpsilonComplex$fEpsilonInteger $fEpsilonInt$fEpsilonFloat$fEpsilonDouble TensorProduct><outer timesleft timesrightHilbert<.>Banach normalizeMultiplicativeGroupModule.//.MultiplicativeModule.**.AdditiveGroupModule.--.AdditiveModule.++. Singleton singletonLaw2Law LawArity2Unary2Binary2Ternary2 Ternary2' Ternary2''Quad31Quad22 Failiary2LawArityNonaryUnaryBinaryTernaryOrnaryFailiary testLawOf testLawOf2idempotentLaws additiveLawsadditiveLawsFailadditiveGroupLawsmultiplicativeLawsmultiplicativeMonoidalLawsmultiplicativeLawsFailmultiplicativeGroupLawsdistributionLawsdistributionLawsFail integralLaws signedLawsmetricFloatLawsmetricComplexFloatLawsboundedFieldFloatLawsquotientFieldLaws expFieldLawsexpFieldComplexLooseLawsmetricNaperianFloatLawsexpFieldNaperianLawsadditiveModuleLawsadditiveGroupModuleLawsmultiplicativeModuleLaws!multiplicativeGroupModuleLawsFail banachLaws hilbertLawstensorProductLawsadditiveBasisLawsadditiveGroupBasisLawsmultiplicativeBasisLawsmultiplicativeGroupBasisLaws semiringLawsringLawsstarSemiringLawsMultiplicativeIdempotent TensorRepbaseGHC.Base++ghc-primGHC.PrimseqGHC.Listfilterzip Data.Tuplefstsnd otherwise$GHC.Real realToFrac Control.MonadguardjoinGHC.EnumEnumenumFrom enumFromThenenumFromThenTo enumFromTofromEnumtoEnumsuccpred GHC.ClassesEq==/= GHC.FloatFloatinglog1pexpm1log1pexplog1mexp Fractional fromRationalMonad>>=>>returnFunctorfmap<$GHC.NumNumsignumOrd>=minmax><<=compareGHC.ReadReadReal toRational RealFloat floatRadix floatDigits floatRange decodeFloat encodeFloatexponent significand scaleFloat isInfiniteisDenormalizedisNegativeZeroisIEEERealFracproperFractiontruncateGHC.ShowShowData.Typeable.InternalTypeable Data.StringIsString Applicativepure<*>*>liftA2<* Data.FoldableFoldabletoListfoldrfoldfoldMapnulllengthfoldlfoldl'maximumminimumelemfoldr'Data.Traversable TraversabletraversemapM sequenceAsequence GHC.GenericsGenericfromtoGeneric1Datatype datatypeName moduleName packageName isNewtype ConstructorconName conFixity conIsRecordSelectorselNameselSourceUnpackednessselSourceStrictnessselDecidedStrictness GHC.TypeNatsKnownNat GHC.TypeLits KnownSymbolGHC.OverloadedLabelsIsLabel fromLabelData.Semigroup SemigroupsconcatstimesMonoidmemptymappendmconcat GHC.TypesBoolFalseTrueCharDoubleD#FloatF#IntGHC.IntInt8Int16Int32Int64 integer-gmpGHC.Integer.TypeIntegerMaybeJustNothingOrderingLTEQGTRatio:%Rational GHC.Stable StablePtrIOWordGHC.WordWord8Word16Word32Word64GHC.PtrPtrFunPtr Data.EitherEitherLeftRight ConstraintV1U1K1unK1M1unM1:+:L1R1:*::.:Comp1unComp1Rec0D1C1S1URecNatSymbolCmpNat Coercible GHC.StaticPtr StaticPtrGHC.Stack.Types CallStackGHC.STST Data.Monoid<>GHC.IO.Handle.TypesHandle GHC.Conc.SyncforkOnWithUnmaskforkIOWithUnmaskforkOnControl.ConcurrentforkOSThreadId$async-2.1.1.1-H3j65XcXMHtBvmNwGCQ80GControl.Concurrent.Async concurrentlyrace_racelink2linkwaitBothwaitEitherCancel waitEither_ waitEitherwaitEitherCatchCancelwaitEitherCatch waitAnyCancelwaitAnywaitAnyCatchCancel waitAnyCatch cancelWithcancelpoll waitCatchwait withAsyncOnwithAsyncBound withAsyncasyncOn asyncBoundasyncAsync asyncThreadId 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BifunctorfirstsecondbimapGHC.ExecutionStackshowStackTrace getStackTraceGHC.ExecutionStack.InternalSrcLoc sourceFile sourceLine sourceColumnLocation objectName functionNamesrcLocControl.Monad.IO.ClassMonadIOliftIOSystem.EnvironmentgetArgs System.Exit exitSuccess exitFailureexitWithmfilter<$!>unless replicateM_ replicateMfoldM_foldM zipWithM_zipWithM mapAndUnzipMforever<=<>=>filterMfoldMapDefault fmapDefault mapAccumR mapAccumLforMforControl.ApplicativeoptionalZipList getZipListData.Functor.IdentityIdentity runIdentity System.IOwithFileGHC.IO.Handle.FDopenFilestderrstdin GHC.Conc.IO threadDelayControl.Concurrent.MVar mkWeakMVaraddMVarFinalizermodifyMVarMaskedmodifyMVarMasked_ modifyMVar modifyMVar_withMVarMaskedwithMVarswapMVar GHC.StackwithFrozenCallStack callStackControl.ExceptionallowInterruptcatchesControl.Exception.BasebracketOnErrorbracket_finallybracket onExceptiontryJusttry mapException handleJusthandle catchJustPatternMatchFail RecSelError RecConError RecUpdError NoMethodError TypeErrorNonTerminationNestedAtomicallyalwaysalwaysSucceedscatchSTMthrowSTMorElseretry atomicallymkWeakThreadIdthreadCapability myThreadId killThreadsetNumCapabilitiesgetNumCapabilitiesforkIOSTMGHC.IO.ExceptionioErrorasyncExceptionFromExceptionasyncExceptionToExceptionBlockedIndefinitelyOnMVarBlockedIndefinitelyOnSTMDeadlockAllocationLimitExceededCompactionFailedAssertionFailedSomeAsyncExceptionAsyncException StackOverflow HeapOverflow ThreadKilled UserInterruptArrayExceptionIndexOutOfBoundsUndefinedElementExitCode ExitSuccess ExitFailurestdoutGHC.IOevaluateuninterruptibleMaskuninterruptibleMask_maskmask_getMaskingState interruptiblecatchFilePath MaskingStateUnmaskedMaskedInterruptibleMaskedUninterruptible IOException GHC.ExceptionprettyCallStack prettySrcLoc Exception toException fromExceptiondisplayException ErrorCallErrorCallWithLocationArithExceptionOverflow UnderflowLossOfPrecision DivideByZeroDenormalRatioZeroDenominator 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