śĪ!• 36      !"#$%&'()*+,-./012345 (c) Levent ErkokBSD3erkokl@gmail.com experimentalSafe2 sbvPlugin,Importable type as an annotation alternative sbvPluginThe actual annotation. sbvPluginĻPlugin options. Note that we allow picking multiple solvers, which will all be run in parallel. You can pick and choose any number of them, or if you want to run all available solvers, then use the option v. The default behavior is to error-out on failure, using the default-SMT solver picked by SBV, which is currently Z3. sbvPluginContinue even if proof fails sbvPluginVSkip the proof. Can be handy for properties that we currently do not want to focus on. sbvPlugin*Produce verbose output, good for debugging sbvPluginDProduce really verbose output, use only when things go really wrong!  sbvPluginPerform quickCheck  sbvPlugin+Uninterpret this binding for proof purposes  sbvPlugin9Use these names for the arguments; need not be exhaustive  sbvPluginNIf a list-input is found, use this length. If not specified, we will complain!  sbvPluginUse Z3 sbvPlugin Use Yices sbvPlugin Use Boolector sbvPluginUse CVC4 sbvPlugin Use MathSAT sbvPluginUse ABC sbvPluginURun all installed solvers in parallel, and report the result from the first to finish sbvPlugin-A property annotation, using default options. sbvPlugin)Synonym for sbv really, just looks cooler    (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone$=?SX,«6 sbvPlugin+The values kept track of by the interpreter7 sbvPluginThe kinds used by the plugin8 sbvPlugin'Configuration info as we run the plugin9 sbvPluginThe interpreter monad: sbvPluginInterpreter environment; sbvPlugin<TyCon's are no longer Ord in GHC 8.2.1; so we make a newtype< sbvPlugin2Certain "very-polymorphic" things are just special= sbvPluginExtract the unique "key"> sbvPlugin8Given the user options, determine which solver(s) to use? sbvPlugin:Structural lifting of a boolean function (eq/neq) over Val@ sbvPluginSymbolic equality over valuesA sbvPlugin"Symbolic if-then-else over values.B sbvPluginMCompute the span given a Tick. Returns the old-span if the tick span useless.C sbvPluginCompute the span for a binding.D sbvPlugin‰Pick the first "good" span, hopefully corresponding to the closest location to where we are in the code when we issue an error message.E sbvPlugin%Show a GHC span in user-friendly formF sbvPlugin@This comes mighty handy! Wonder why GHC doesn't have it already:G sbvPluginOutputable instance for S.KindH sbvPlugin)Make a rudimentary Ord instance for TCKeyI sbvPlugin(Make a rudimentary Eq instance for TCKeyJ sbvPluginOutputable instance for SKindK sbvPluginOutputable instance for Val26LMNOP7QRST8UVWXY9:Z[\]^_`abcdefgh;i<jklm=>?@ABCDE(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone$:hn sbvPlugin#Dispatch the analyzer over bindingso sbvPluginProve an SBVTheoremp sbvPlugingSafely execute an action, catching the exceptions, printing and returning False if something goes wrongq sbvPluginReturns True if proof went thrur sbvPlugin’’Is this really a dictionary in disguise? This is a terrible hack, and the ice is thin here. But it seems to work. TODO: Figure out if there's a better way of doing this. Note that this function really does get applications, when those dictionaries are parameterized by others. Think of the case where "Eq [a]" dictionary depends on "Eq a", for instance. In these cases, GHC to produces applications.s sbvPluginUninterpret an expressiont sbvPluginBConvert a Core type to an SBV Type, retaining functions and tuplesn(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone$FQ7u sbvPluginTWhat tuple-sizes we support? We go upto 15, but would be easy to change if necessaryv sbvPlugin.Build the initial environment containing typesw sbvPlugin2Build the initial environment containing functionsx sbvPlugin#Basic conversions, only on one kindy sbvPlugin?Symbolic functions supported by the plugin; those from a class.z sbvPlugin Destructors{ sbvPluginlThese types show up during uninterpretation, but are not really "interesting" as they are singly inhabited.| sbvPluginTCertain things are just too special, as they uniformly apply to uninterpreted types.} sbvPlugin$Lift a binary type, with result bool~ sbvPluginLift a unary type sbvPluginLift a binary type€ sbvPlugin0Lift a binary type, where second argument is Int sbvPlugin(Construct the type for a split operation‚ sbvPlugin'Construct the type for a join operationƒ sbvPluginType of enumFromTo: [x .. y]„ sbvPlugin Type of enumFromThenTo: [x .. y]… sbvPlugin/Lift a unary SBV function that via kind/integer† sbvPlugin3Lift a unary SBV function to the plugin value space‡ sbvPlugin>Lift a two argument SBV function to our the plugin value spaceˆ sbvPluginLifting splits‰ sbvPluginDLifting an equality is special; since it acts uniformly over tuples.Š sbvPluginLifting enumFromTo: [x .. y]‹ sbvPlugin$Lifting sEnumFromThenTo: [x, y .. z]Œ sbvPluginCImplement [x .. y] or [x, y .. z]; provided the inputs are concretevwz{| (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone$Rą sbvPluginEntry point to the plugin(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneSß     (c) Nickolas FotopoulosBSD3nickolas.fotopoulos@gmail.com experimentalNoneX sbvPlugingA top-level binding with its type wrapped in Proved causes sbvPlugin to run a proof on the expression.Ž sbvPlugin)Simple booleans can be made theorems too.Ž(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonep±  sbvPluginGThe range detector must output if the range is larger than this amount.! sbvPluginNThe range detector must have sent an output before this many cycles have past." sbvPlugin+Given a last-signal-time calculator, named  calculate^, check that it satisfies the following three requirements: We must've just sent a signal if:minRate9: The last-time we sent is strictly less than the ! amountminRange=: We must've just sent a signal if the range is beyond  manualOverrideC: We must've just sent a signal if the manual-override is specified# sbvPluginmA "bad" implementation, see if you can spot the problem with it, before looking at the failed theorem below!$ sbvPluginUsing SBV, prove that the #K is indeed a bad implementation. Here's the output we get from the plugin: Į [SBV] MicroController.hs:85:1-8 Proving "checkBad", using Z3. [Z3] Falsifiable. Counter-example: range = 200 :: Int64 manual = False :: Bool timeSince = 9 :: Int64 šWe're being told that if the range is 200, and manual override is off, and time-since last is 9, then our "calculator" returns 10. But that violates the minRate) requirement, since we never want to go !! cycles without sending a signal!% sbvPluginPA "good" implementation, properly handling the off-by-one error of the original.& sbvPlugin=We now verify that the good variant is indeed good. We have: P [SBV] MicroController.hs:108:1-9 Proving "checkGood", using Z3. [Z3] Q.E.D.  !"#$%& !"#$%&(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone}Ō' sbvPlugin5Merging two given sorted lists, preserving the order.( sbvPlugin!Simple merge-sort implementation.) sbvPlugin1Check whether a given sequence is non-decreasing.* sbvPluginčCheck whether two given sequences are permutations. We simply check that each sequence is a subset of the other, when considered as a set. The check is slightly complicated for the need to account for possibly duplicated elements.+ sbvPluginÓAsserting correctness of merge-sort for a list of the given size. Note that we can only check correctness for fixed-size lists. Also, the proof will get more and more complicated for the backend SMT solver as n" increases. Here we try it with 4.We have: R [SBV] tests/T48.hs:100:1-16 Proving "mergeSortCorrect", using Z3. [Z3] Q.E.D. '()*+'()*+(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneŒß , sbvPluginGSBVPlugin can only see definitions in the current module. So we define , ourselves.- sbvPluginReturns 1 if bool is True. sbvPlugin Formalizes Dhttps://graphics.stanford.edu/~seander/bithacks.html#IntegerMinOrMax/ sbvPlugin Formalizes Dhttps://graphics.stanford.edu/~seander/bithacks.html#IntegerMinOrMax0 sbvPlugin Formalizes Hhttps://graphics.stanford.edu/~seander/bithacks.html#DetectOppositeSigns1 sbvPlugin Formalizes ^https://graphics.stanford.edu/~seander/bithacks.html#ConditionalSetOrClearBitsWithoutBranching2 sbvPlugin Formalizes Hhttps://graphics.stanford.edu/~seander/bithacks.html#DetermineIfPowerOf23 sbvPlugin Formalizes @https://graphics.stanford.edu/~seander/bithacks.html#MaskedMerge4 sbvPlugin Formalizes Dhttps://graphics.stanford.edu/~seander/bithacks.html#RoundUpPowerOf25 sbvPlugin Formalizes ?https://graphics.stanford.edu/~seander/bithacks.html#ZeroInWord ,-./012345 ,-./012345    !"#$%&'(&')&'*&'+ ,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`aEbcdeGfghijklmnopqrsHItuvwxyz{|}~€‚ƒ„…†‡ˆ‰Š‹ŒŽ‘’“”• – —˜%sbvPlugin-0.11-FI9gJtrJGKmEJjeFjB0HMpData.SBV.Plugin.DataData.SBV.Plugin(Data.SBV.Plugin.Examples.MicroController"Data.SBV.Plugin.Examples.MergeSort"Data.SBV.Plugin.Examples.BitTricksData.SBV.Plugin.CommonData.SBV.Plugin.AnalyzeData.SBV.Plugin.EnvData.SBV.Plugin.PluginData.SBV.Plugin.Examples.ProvedProved SBVAnnotationSBVoptions SBVOption IgnoreFailureSkipVerboseDebug QuickCheck UninterpretNamesListSizeZ3Yices BoolectorCVC4MathSATABC AnySolversbvtheorem$fShowSBVOption $fEqSBVOption$fDataSBVOption$fEqSBVAnnotation$fDataSBVAnnotation sbv-8.0.5-E6zRdapqbC0BTalXkjQTqEData.SBV.Core.Splittable Splittable#extendsplitpluginsafetyDistance maxTimeSince checkSpeccomputeLastBadcheckBadcomputeLastGood checkGoodmerge mergeSort nonDecreasingisPermutationOfmergeSortCorrectelemoneIffastMinCorrectfastMaxCorrectoppositeSignsCorrectconditionalSetClearCorrectpowerOfTwoCorrectmaskedMergeCorrectroundPowerOfTwoCorrect zeroInWordValSKindConfigEvalEnvTCKeySpecials tcKeyToUList pickSolvers liftEqValeqValiteValtickSpanvarSpanpickSpanshowSpan $fShowExpr$fOutputableKind $fOrdTCKey $fEqTCKey$fOutputableSKind$fOutputableValFuncLstTupTypBaseKFunKLstKTupKBasecfgEnv sbvAnnotationoptsisGHCibailOutcoreMapdestMapenvMaptcMapspecialsrUITypes rUsedNamesrUninterpreted uninteresting mbListSize machWordSizeflagscurLocisListisTuple isEquality analyzeBindprovesafelyproveItisReallyADictionary uninterpretgetTypesupportTupleSizes buildTCEnv buildFunEnv basicFuncssymFuncs buildDestsuninterestingTypes buildSpecials tlift2Booltlift1tlift2 tlift2ShRottSplittJoin tEnumFromTotEnumFromThenTolift1Intlift1lift2 liftSplitliftEq sEnumFromTosEnumFromThenToenumListintegerAssociativeisTrue