úÎpkÅI      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHNone569:;QR) kA class of types that have additional invariants defined upon them that aren't enforced by the type systemPurpose } checks whether a given value is a valid value and reports all reasons why the given value is not valid if that is the case. @ only checks whether a given value is a valid value of its type.Instantiating  To instantiate  , one has to implement both   and  . Start by implementing  §. Use the helper functions below to define all the reasons why a given value would be a valid value of its type. Then define `isValid = isValidbyValidating' for now.Example: ™newtype Even = Even Int instance Validity Even validate (Event i) even i <?@> "The contained 'Int' is even." isValid = isValidByValidating%If it turns out that, at this point,  D is too slow for your taste, you can replace the implementation of  A by a custom implementation. However, it is important that this  3 implementation has exactly the same semantics as isValidbyValidating.Example: –newtype Even = Even Int instance Validity Even validate (Event i) even i <?@> "The contained 'Int' is even." isValid (Event i) = even i Semantics 4 should be an underapproximation of actual validity.This means that if  a is not a perfect representation of actual validity, for safety reasons, it should never return I( for invalid values, but it may return J for valid values. For example: isValid = const FalseAis a valid implementation for any type, because it never returns I for invalid values. isValid (Even i) = i == 2is a valid implementation for newtype Even = Even Int, but #isValid (Even i) = even i || i == 1is not because it returns I for an invalid value: '1'.Automatic instances with KSAn instance of this class can be made automatically if the type in question has a K) instance. This instance will try to use  Q to on all structural sub-parts of the value that is being checked for validity.Example: £{-# LANGUAGE DeriveGeneric #-} data MyType = MyType { myDouble :: Double { myString :: String } deriving (Show, Eq, Generic) instance Validity MyTypegenerates something like: ªinstance Validity MyType where isValid (MyType d s) = isValid d && isValid s validate (MyType d s) = d <?!> "myDouble" <> s <?!> "myString" Declare any value to be valid. triviallyValid a = seq a True Implement   by using  C and checking that there are no reasons that the value is invalid.+Declare any value to be valid in validation "trivialValidation a = seq a mempty Implement   by using  C and using the given string as the reason if the value is invalid. Implement   by using  H and using the given name to define the reason if the value is invalid. RvalidateByCheckingName name = validateByChecking $ unwords ["The", name, "valid."] Implement   by using  5 and using a default reason if the value is invalid. DvalidateByCheckingDefault = validateByChecking "The value is valid."#Check that a given invariant holds.BThe given string should describe the invariant, not the violation.Example: ,check (x < 5) "x is strictly smaller than 5" instead of #check (x < 5) "x is greater than 5"Infix operator for Example: )x < 5 <?@> "x is strictly smaller than 5"SDeclare a sub-part as a necessary part for validation, and annotate it with a name.Example: ”validate (a, b) = mconcat [ annotate a "The first element of the tuple" , annotate b "The second element of the tuple" ]Infix operator for Example: Œvalidate (a, b) = mconcat [ a <?!> "The first element of the tuple" , b <?!> "The second element of the tuple" ]Check whether  is not valid. isInvalid = not . isValid3Construct a valid element from an unchecked element>Construct a valid element from an unchecked element, throwing L on invalid elements.validate a given value.gThis function returns either all the reasons why the given value is invalid, in the form of a list of s, or it returns M5 with the input value, as evidence that it is valid.Note: You map want to use ( instead, if you want to display these  s to a user.Hvalidate a given value, and return a nice error if the value is invalid.$!Valid according to the contained N.%Valid if the contained N6s are valid and the denominator is strictly positive.&Valid according to OOnly available with  base >= 4.8.'Trivially validBInteger is not trivially valid under the hood, but instantiating  N correctly would force validity to depend on a specific (big integer library  integer-gmp versus integer-simpleY). This is rather impractical so for the time being we have opted for assuming that an NX is always valid. Even though this is not technically sound, it is good enough for now.(NOT trivially valid:NaN is not valid.Infinite values are not valid.)NOT trivially valid:NaN is not valid.Infinite values are not valid.*Trivially valid+Trivially valid,Trivially valid-Trivially valid.Trivially valid/Trivially valid0Trivially valid1Trivially valid2Trivially valid3Trivially valid4cA Maybe thing is valid if the thing inside is valid or it's nothing It makes sense to assume that PZ is valid. If Nothing wasn't valid, you wouldn't have used a Maybe in the datastructure.57A nonempty list is valid if all the elements are valid.9See the instance for 'Validity [a]' for more information.69A list of things is valid if all of the things are valid.¥This means that the empty list is considered valid. If the empty list should not be considered valid as part of your custom data type, make sure to write a custom Validity instance7DAny sextuple of things is valid if all six of its elements are valid8FAny quintuple of things is valid if all five of its elements are valid9FAny quadruple of things is valid if all four of its elements are valid:DAny triple of things is valid if all three of its elements are valid;OAny Either of things is valid if the contents are valid in either of the cases.<>Any tuple of things is valid if both of its elements are valid@QRS T !"#$%&'()*+,-./0123456789:;<=>?    8QRS  T !"#$%&'()*+,-./0123456789:;<=>?00Safe9;<=FkA class of types that have additional invariants defined upon them that aren't enforced by the type systemIf there is a  Validity a instance as well, then a G b should imply  isValid a for any b.If there is a  Validity b instance as well, then a G b should imply  isValid b for any a.FGHFGHFGHFGHU       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOMNPQRSTUVWXYZ[\]^_`a'validity-0.4.0.1-F00UQ9bb5AvBAkzc3JlmhO Data.ValidityData.RelativeValiditybaseGHC.BasememptymappendmconcatMonoid Validation unValidationValidationChainViolatedLocationValidityvalidateisValidtriviallyValidisValidByValidatingtrivialValidationvalidateByCheckingvalidateByCheckingNamevalidateByCheckingDefaultcheckannotate isInvalidconstructValidconstructValidUnsafe checkValidityprettyValidation $fGValidityK1 $fGValidityM1$fGValidityM10$fGValidityM11$fGValidity:+:$fGValidity:*: $fGValidityV1 $fGValidityU1$fValidityFixed$fValidityRatio$fValidityNatural$fValidityInteger$fValidityDouble$fValidityFloat$fValidityWord64$fValidityWord32$fValidityWord16$fValidityWord8$fValidityWord $fValidityInt$fValidityChar$fValidityOrdering$fValidityBool $fValidity()$fValidityMaybe$fValidityNonEmpty $fValidity[]$fValidity(,,,,,)$fValidity(,,,,)$fValidity(,,,)$fValidity(,,)$fValidityEither $fValidity(,)$fMonoidValidation$fValidityValidation$fValidityValidationChain$fShowValidationChain$fEqValidationChain$fGenericValidationChain$fShowValidation$fEqValidation$fGenericValidationRelativeValidity isValidFor isInvalidForghc-prim GHC.TypesTrueFalse GHC.GenericsGenericGHC.Errerror Data.EitherRight integer-gmpGHC.Integer.TypeInteger GHC.NaturalisValidNaturalNothing GValiditygIsValid gValidateannotateValidation