h,(       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         !!!!!"""""""""##$$$$$$%%%%%&&&'''''((())))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))********************************+++++++++++++,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,-......../////////////////////000011222333333333333334444444444444444444455555555555555555555555555555555555555555666666777889:::::::::::::::::::::::::;;;<<<<<<<=====>?@@AAAAAAAAAAAAAAAAAAAAAAAAAAAAABCDEFFFFFFFFGGGGGGGGGGGGGHHHHHHHHHHHHHHIIIIIIIIIIIIIIIIIIJJJJJJJJJJJJJKKKKKKKKKKKKKKKKKKKKKKKKKKKLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLMMMMMMMMMMNNNNOPPPPPPPPPPQQQRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRSSSSSSSSSSSSSSSSSSSSSSTTTTTTTTTTTTTTTTTTTTTTTTUUUUUUUUUUUVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVWXXXXXXXXYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYZZZ[[\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\]]]]]]]]]]]]]^______________```````````````````````````````````````````````````````````````abbccddddddddeeeeeeeeeeeeeeeeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffgghhiijjjjkkkkkkkkkkkkkkkklllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllmmmnnnnnnnnnnnnooppqqqqqqqqqqqqrrrrrrrrrrrsssssssssstttttttttttttttttuuuuuuuuuuuuuuuuuuuvvvvvvvvvvvvvvvvvvvvvvvwwwwwwwwwwwwwwwwwwwwwwwxxxxxxxxxxxxxxxxxxxxxxxyyyyyyyyyyyyyyyyyyyyyyyzzzzzzzzzzzzzzzzzzzzz{{{{{{{{{{{{{{{{{{{{|10.3}(c) Levent ErkokBSD3erkokl@gmail.com experimental Safe-InferredX/sbvSMT-Lib logics. If left unspecified SBV will pick the logic based on what it determines is needed. However, the user can override this choice using a call to ~ This is especially handy if one is experimenting with custom logics that might be supported on new solvers. See  'http://smtlib.cs.uiowa.edu/logics.shtml for the official list.sbvFormulas over the theory of linear integer arithmetic and arrays extended with free sort and function symbols but restricted to arrays with integer indices and values.sbvLinear formulas with free sort and function symbols over one- and two-dimentional arrays of integer index and real value.sbvFormulas with free function and predicate symbols over a theory of arrays of arrays of integer index and real value.sbv*Linear formulas in linear real arithmetic.sbvQuantifier-free formulas over the theory of bitvectors and bitvector arrays.sbvQuantifier-free formulas over the theory of bitvectors and bitvector arrays extended with free sort and function symbols.sbvQuantifier-free linear formulas over the theory of integer arrays extended with free sort and function symbols.sbvQuantifier-free formulas over the theory of arrays with extensionality. sbvQuantifier-free formulas over the theory of fixed-size bitvectors. sbvDifference Logic over the integers. Boolean combinations of inequations of the form x - y < b where x and y are integer variables and b is an integer constant. sbvUnquantified linear integer arithmetic. In essence, Boolean combinations of inequations between linear polynomials over integer variables. sbvUnquantified linear real arithmetic. In essence, Boolean combinations of inequations between linear polynomials over real variables. sbv#Quantifier-free integer arithmetic.sbv Quantifier-free real arithmetic.sbvDifference Logic over the reals. In essence, Boolean combinations of inequations of the form x - y < b where x and y are real variables and b is a rational constant.sbvUnquantified formulas built over a signature of uninterpreted (i.e., free) sort and function symbols.sbvUnquantified formulas over bitvectors with uninterpreted sort function and symbols.sbvDifference Logic over the integers (in essence) but with uninterpreted sort and function symbols.sbvUnquantified linear integer arithmetic with uninterpreted sort and function symbols.sbvUnquantified linear real arithmetic with uninterpreted sort and function symbols.sbvUnquantified non-linear real arithmetic with uninterpreted sort and function symbols.sbvUnquantified non-linear real integer arithmetic with uninterpreted sort and function symbols.sbvLinear real arithmetic with uninterpreted sort and function symbols.sbvNon-linear integer arithmetic with uninterpreted sort and function symbols.sbvQuantifier-free formulas over the theory of floating point numbers, arrays, and bit-vectors.sbvQuantifier-free formulas over the theory of floating point numbers.sbvQuantifier-free finite domains.sbv4Quantifier-free formulas over the theory of strings.sbvThe catch-all value.sbv=Use this value when you want SBV to simply not set the logic.sbv(In case you need a really custom string! sbvOption values that can be set in the solver, following the SMTLib specification  )http://smtlib.cs.uiowa.edu/language.shtml.3Note that not all solvers may support all of these!Furthermore, SBV doesn't support the following options allowed by SMTLib.:interactive-mode+ (Deprecated in SMTLib, use " instead.):print-success (SBV critically needs this to be True in query mode.):produce-models (SBV always sets this option so it can extract models.):regular-output-channel (SBV always requires regular output to come on stdout for query purposes.):global-declarations (SBV always uses global declarations since definitions are accumulative.) Note that - and . are, strictly speaking, not SMTLib options. However, we treat it as such here uniformly, as it fits better with how options work./sbv(Collectable information from the solver.9sbvReason for reporting unknown.>sbv"Behavior of the solver for errors.Asbv(Collectable information from the solver.Jsbv Result of a  or  call.Ksbv=Satisfiable: A model is available, which can be queried with .LsbvDelta-satisfiable: A delta-sat model is available. String is the precision info, if available.MsbvUnsatisfiable: No model is available. Unsat cores might be obtained via .Nsbv Unknown: Use 5 to obtain an explanation why this might be the case.sbv7Can this command only be run at the very beginning? If  then we will reject setting these options in the query mode. Note that this classification follows the SMTLib document.sbvCan this option be set multiple times? I'm only making a guess here. If this returns True, then we'll only send the last instance we see. We might need to update as necessary.sbvTranslate an option setting to SMTLib. Note the SetLogic/SetInfo discrepancy.sbvShow instance for unknownsbvTrivial rnf instanceJLKNM  >@?ABCDEIFGH/123485607 !,("#%$&')*+.-9;:=<(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone7bIOsbv4Conversion from internal rationals to Haskell valuesPsbv'Root of a polynomial, cannot be reducedQsbvAn exact rationalRsbvAn approximated valueSsbv*Interval. Can be open/closed on both ends.TsbvA univariate polynomial, represented simply as a coefficient list. For instance, "5x^3 + 2x - 5" is represented as [(5, 3), (2, 1), (-5, 0)]VsbvAlgebraic reals. Note that the representation is left abstract. We represent rational results explicitly, while the roots-of-polynomials are represented implicitly by their defining equationWsbvbool says it's exact (i.e., SMT-solver did not return it with ? at the end.)Xsbvwhich root of this polynomial and an approximate decimal representation with given precision, if availableYsbv"interval, with low and high boundsZsbv)Is the endpoint included in the interval?[sbv%open: i.e., doesn't include the point\sbv closed: i.e., includes the point]sbv7Extract the point associated with the open-closed pointsbv3Check whether a given argument is an exact rationalsbvConstruct a poly-root real with a given approximate value (either as a decimal, or polynomial-root)sbvStructural equality for AlgReal; used when constants are Map keyssbvStructural comparisons for AlgReal; used when constants are Map keyssbv Render an V as an SMTLib2 value. Only supports rationals for the time being.sbv Render an V as a Haskell value. Only supports rationals, since there is no corresponding standard Haskell type that can represent root-of-polynomial variety.^sbv Convert an V to a  . If the V is exact, then you get a  value. Otherwise, you get a ( value which is simply an approximation.sbvMerge the representation of two algebraic reals, one assumed to be in polynomial form, the other in decimal. Arguments can be the same kind, so long as they are both rationals and equivalent; if not there must be one that is precise. It's an error to pass anything else to this function! (Used in reconstructing SMT counter-example values with reals).sbvNB: Following the other types we have, we require `a/0` to be `0` for all a.^]VYXWTUORQSPZ\[(c) Levent ErkokBSD3erkokl@gmail.com experimental Safe-InferredcH_sbvNames reserved by SMTLib. This list is current as of Dec 6 2015; but of course there's no guarantee it'll stay that way._(c) Brian SchroederBSD3erkokl@gmail.com experimental Safe-InferredfH`sbvMonads which support  operations and can extract all 2 behavior for interoperation with functions like , which takes an  action in negative position. This function can not be implemented for transformers like  ReaderT r or StateT s, whose resultant 6 actions are a function of some environment or state.asbv Law: the m a yielded by  is pure with respect to .sbvIO extraction for strict .sbvIO extraction for lazy .sbvIO extraction for .sbvIO extraction for .sbvTrivial IO extraction for .`a(c) Levent ErkokBSD3erkokl@gmail.com experimental Safe-Inferredk( sbvWe have a nasty issue with the usual String/List confusion in Haskell. However, we can do a simple dynamic trick to determine where we are. The ice is thin here, but it seems to work.sbvMonadic lift over 2-tuplessbvMonadic lift over 3-tuplessbvMonadic lift over 4-tuplessbvMonadic lift over 5-tuplessbvMonadic lift over 6-tuplessbvMonadic lift over 7-tuplessbvMonadic lift over 8-tuplessbvGiven a sequence of arguments, join them together in a manner that could be used on the command line, giving preference to the Windows cmd shell quoting conventions.For an alternative version, intended for actual running the result in a shell, see "System.Process.showCommandForUser"sbvGiven a string, split into the available arguments. The inverse of #. Courtesy of the cmdargs package.sbvGiven an SMTLib string (i.e., one that works in the string theory), convert it to a Haskell equivalentsbvGiven a Haskell string, convert it to SMTLib. if ord is 0x00020 to 0x0007E, then we print it as is to cover the printable ASCII range. (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone(17=ybsbvRounding mode to be used for the IEEE floating-point operations. Note that Haskell's default is c. If you use a different rounding mode, then the counter-examples you get may not match what you observe in Haskell.csbvRound to nearest representable floating point value. If precisely at half-way, pick the even number. (In this context, even% means the lowest-order bit is zero.)dsbvRound to nearest representable floating point value. If precisely at half-way, pick the number further away from 0. (That is, for positive values, pick the greater; for negative values, pick the smaller.)esbvRound towards positive infinity. (Also known as rounding-up or ceiling.)fsbvRound towards negative infinity. (Also known as rounding-down or floor.)gsbv/Round towards zero. (Also known as truncation.)hsbvA valid float has restrictions on eb/sb values. NB. In the below encoding, I found that CPP is very finicky about substitution of the machine-dependent macros. If you try to put the conditionals in the same line, it fails to substitute for some reason. Hence the awkward spacing. Filed this as a bug report for CPPHS at  1https://github.com/malcolmwallace/cpphs/issues/25.sbvCatch an invalid FP.isbv9Type family to create the appropriate non-zero constraintsbvCatch 0-width casesjsbvA class for capturing values that have a sign and a size (finite or infinite) minimal complete definition: kindOf, unless you can take advantage of the default signature: This class can be automatically derived for data-types that have a  instance; this is useful for creating uninterpreted sorts. So, in reality, end users should almost never need to define any methods.sbvKind of symbolic valuesbv;A version of show for kinds that says Bool instead of SBoolsbvPut parens if necessary. This test is rather crummy, but seems to work oksbvHow the type maps to SMT landsbv+Does this kind represent a signed quantity?sbvConstruct an uninterpreted/enumerated kind from a piece of data; we distinguish simple enumerations as those are mapped to proper SMT-Lib2 data-types; while others go completely uninterpretedsbvGrab the bit-size from the proxy. If the nat is too large to fit in an int, we throw an error. (This would mean too big of a bit-size, that we can't really deal with in any practical realm.) In fact, even the range allowed by this conversion (i.e., the entire range of a 64-bit int) is just impractical, but it's hard to come up with a better bound.sbvDo we have a completely uninterpreted sort lying around anywhere?sbvShould we ask the solver to flatten the output? This comes in handy so output is parseable Essentially, we're being conservative here and simply requesting flattening anything that has some structure to it.sbv;Convert a rounding mode to the format SMT-Lib2 understands.sbvThe interesting about the show instance is that it can tell apart two kinds nicely; since it conveniently ignores the enumeration constructors. Also, when we construct a <, we make sure we don't use any of the reserved names; see  for details.sbvThis instance allows us to use the `kindOf (Proxy @a)` idiom instead of the `kindOf (undefined :: a)`, which is safer and looks more idiomatic.sbvb kind5ijlmnowr}tqy|spzx{uvk~bdcfegh(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneM sbvThe SMT-Lib (in particular Z3) implementation for min/max for floats does not agree with Haskell's; and also it does not agree with what the hardware does. Sigh.. See:  ,http://ghc.haskell.org/trac/ghc/ticket/10378  'http://github.com/Z3Prover/z3/issues/68 So, we codify here what the Z3 (SMTLib) is implementing for fpMax. The discrepancy with Haskell is that the NaN propagation doesn't work in Haskell The discrepancy with x86 is that given +0/-0, x86 returns the second argument; SMTLib is non-deterministicsbv SMTLib compliant definition for . See the comments for .sbv.Convert double to float and back. Essentially fromRational . toRational, except careful on NaN, Infinities, and -0.sbvCompute the "floating-point" remainder function, the float/double value that remains from the division of x and y. There are strict rules around 0's, Infinities, and NaN's as coded below.sbvConvert a float to the nearest integral representable in that typesbvCheck that two floats are the exact same values, i.e., +0/-0 does not compare equal, and NaN's compare equal to themselves.sbv$Ordering for floats, avoiding the +0-0NaN issues. Note that this is essentially used for indexing into a map, so we need to be total. Thus, the order we pick is: NaN -oo -0 +0 +oo The placement of NaN here is questionable, but immaterial.sbvCheck if a number is "normal." Note that +0/-0 is not considered a normal-number and also this is not simply the negation of isDenormalized!sbvReinterpret-casts a  to a .sbvReinterpret-casts a  to a .sbvReinterpret-casts a  to a .sbvReinterpret-casts a  to a . (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone 179 sbv1Internal representation of a parameterized float.A note on cardinality: If we have eb exponent bits, and sb significand bits, then the total number of floats is 2^sb*(2^eb-1) + 3: All exponents except 11..11 is allowed. So we get, 2^eb-1, different combinations, each with a sign, giving us 2^sb*(2^eb-1) totals. Then we have two infinities, and one NaN, adding 3 more.sbvAbbreviation for IEEE quadruble precision float, bit width 128 = 15 + 113.sbvAbbreviation for IEEE double precision float, bit width 64 = 11 + 53.sbvAbbreviation for IEEE single precision float, bit width 32 = 8 + 24.sbvAbbreviation for brain-float precision float, bit width 16 = 8 + 8.sbvAbbreviation for IEEE half precision float, bit width 16 = 5 + 11.sbvA floating point value, indexed by its exponent and significand sizes.An IEEE SP is FloatingPoint 8 24 DP is FloatingPoint 11 53 etc.sbvRemove redundant p+0 etc.sbvShow a big float in the base given. NB. Do not be tempted to use BF.showFreeMin below; it produces arguably correct but very confusing results. See  0https://github.com/GaloisInc/cryptol/issues/1089! for a discussion of the issues.sbvDefault options for BF options.sbv,Construct a float, by appropriately roundingsbvConvert from an signexponentmantissa representation to a float. The values are the integers representing the bit-patterns of these values, i.e., the raw representation. We assume that these integers fit into the ranges given, i.e., no overflow checking is done here.sbvMake NaN. Exponent is all 1s. Significand is non-zero. The sign is irrelevant.sbv4Make Infinity. Exponent is all 1s. Significand is 0.sbvMake a signed zero.sbvMake from an integer value.sbv/Make a generalized floating-point value from a .sbv"Represent the FP in SMTLib2 formatsbv4Structural comparison only, for internal map indexessbv!Compute the signum of a big floatsbvEncode from exponent/mantissa form to a float representation. Corresponds to  in Haskell.sbvLift a unary operation, simple case of function with no status. Here, we call fpFromBigFloat since the big-float isn't size aware.sbvConvert from a IEEE float.sbvConvert from a IEEE double.sbvReal-frac instance for big-floats. Beware, not that well tested!sbv;Real instance for big-floats. Beware, not that well tested!sbvReal-float instance for big-floats. Beware! Some of these aren't really all that well tested.sbv Floating instance for big-floatssbv"Fractional instance for big-floatssbvNum instance for big-floatssbvNum instance for FloatingPointsbvShow instance for Floats. By default we print in base 10, with standard scientific notation.(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone7.sbvA simple expression type over extended values, covering infinity, epsilon and intervals.sbvA generalized CV allows for expressions involving infinite and epsilon values/intervals Used in optimization problems.sbv represents a concrete word of a fixed size: For signed words, the most significant digit is considered to be the sign.sbvA constant valuesbvAlgebraic realsbvBit-vector/unbounded integersbvFloatsbvDoublesbvArbitrary floatsbvRationalsbv CharactersbvStringsbvListsbv$Set. Can be regular or complemented.sbvValue of an uninterpreted/user kind. The Maybe Int shows index position for enumerationssbvTuplesbvMaybesbvDisjoint unionsbvA  is either a regular set or a set given by its complement from the corresponding universal set.sbvStructural equality for  . We need EqOrd instances for $ because we want to put them in mapstables. But we don't want to derive these, nor make it an instance! Why? Because the same set can have multiple representations if the underlying type is finite. For instance, {True} = U - {False} for boolean sets! Instead, we use the following two functions, which are equivalent to Eq/Ord instances and work for our purposes, but we do not export these to the user.sbv Comparing  values. See comments for  on why we don't define the  instance.sbvAssign a rank to constant values, this is structural and helps with orderingsbv*Show an extended CV, with kind if requiredsbvIs this a regular CV?sbvAre two CV's of the same type?sbvConvert a CV to a Haskell boolean (NB. Assumes input is well-kinded)sbvNormalize a CV. Essentially performs modular arithmetic to make sure the value can fit in the given bit-size. Note that this is rather tricky for negative values, due to asymmetry. (i.e., an 8-bit negative number represents values in the range -128 to 127; thus we have to be careful on the negative side.)sbvConstant False as a ,. We represent it using the integer value 0.sbvConstant True as a ,. We represent it using the integer value 1.sbv Lift a unary function through a .sbv!Lift a binary function through a .sbvMap a unary function through a .sbv Map a binary function through a .sbv)Show a CV, with kind info if bool is Truesbv(Create a constant word from an integral.sbv"Generate a random constant value () of the correct kind.sbv"Generate a random constant value () of the correct kind.sbvShow instance. Regular sets are shown as usual. Complements are shown "U -" notation.sbv-Ord instance for VWVal. Same comments as the % instance why this cannot be derived.sbvEq instance for CVVal. Note that we cannot simply derive Eq/Ord, since CVAlgReal doesn't have proper instances for these when values are infinitely precise reals. However, we do need a structural eq/ord for Map indexes; so define custom ones here:sbvShow instance for .sbv instance for CVsbv"Show instance, shows with the kindsbvKind instance for Extended CVsbvShow instance for Generalized sbv instance for generalized CV2(c) Levent ErkokBSD3erkokl@gmail.com experimental Safe-Inferredsbv2Specify how to save timing information, if at all.sbvShow  in human readable form.  is essentially picoseconds (10^-12 seconds). We show it so that it's represented at the day:hour:minute:second.XXX granularity.(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone"'()*679<= sbvQuery execution contextsbvTriggered from inside SBVsbvTriggered from user codesbv An SMT solversbv"Various capabilities of the solversbv=The solver engine, responsible for interpreting solver outputsbv Options to provide to the solversbvEach line sent to the solver will be passed through this function (typically id)sbvThe path to its executablesbvThe solver in usesbvSolvers that SBV is aware ofsbv An SMT enginesbv%A script, to be passed to the solver.sbv'Continuation script, to extract resultssbv Initial feedsbvThe result of an SMT solver call. Each constructor is tagged with the  that created it so that further tools can inspect it and build layers of results, if needed. For ordinary uses of the library, this type should not be needed, instead use the accessor functions on it. (Custom Show instances and model extractors.)sbvUnsatisfiable. If unsat-cores are enabled, they will be returned in the second parameter.sbvSatisfiable with modelsbv |x|. In this case, wrap-around can happen, so we reduce by the size of |x|. Case 2.2. Finite i, and it can't contain a value > |x|. In this case, no reduction is needed.sbvOverflow detection.sbvRead the array element at asbvUpdate the element at a to be bsbv?Merge two given arrays on the symbolic condition Intuitively: ,mergeArrays cond a b = if cond then a else b=. Merging pushes the if-then-else choice down on to elementssbvCreate a named new arraysbvCompare two arrays for equalitysbvCreate a symbolic two argument operation; with shortcut optimizationssbvCreate a symbolic two argument operation; no shortcut optimizationssbveqOpt says the references are to the same SV, thus we can optimize. Note that we explicitly disallow KFloatKDoubleKFloat here. Why? Because it's *NOT* true that NaN == NaN, NaN >= NaN, and so-forth. So, we have to make sure we don't optimize floats and doubles, in case the argument turns out to be NaN.sbv)Predicate to check if a value is concretesbvPredicate for optimizing word operations like (+) and (*). NB. We specifically do *not* match for Double/Float; because FP-arithmetic doesn't obey traditional rules. For instance, 0 * x = 0 fails if x happens to be NaN or +/- Infinity. So, we merely return False when given a floating-point value here.sbvPredicate for optimizing word operations like (+) and (*). NB. See comment on 6 for why we don't match for Float/Double values here.sbvPredicate for optimizing bitwise operations. The unbounded integer case of checking against -1 might look dubious, but that's how Haskell treats % as a member of the Bits class, try (-1 :: Integer)  i for any i and you'll get .sbv%Predicate for optimizing comparisons.sbv%Predicate for optimizing comparisons.sbvMost operations on concrete rationals require a compatibility check to avoid faulting on algebraic reals.sbv;Quot/Rem operations require a nonzero check on the divisor.sbv'Same as rationalCheck, except for SBV'ssbvGiven a composite structure, figure out how to compare for less thansbvStructural less-than for tuplessbvStructural less-than for maybessbvStructural less-than for eithersbv Convert an  to an , preserving the bit-correspondence. Note that since the representation for NaNs are not unique, this function will return a symbolic value when given a concrete NaN.Implementation note: Since there's no corresponding function in SMTLib for conversion to bit-representation due to partiality, we use a translation trick by allocating a new word variable, converting it to float, and requiring it to be equivalent to the input. In code-generation mode, we simply map it to a simple conversion.sbv Convert an  to an , preserving the bit-correspondence. Note that since the representation for NaNs are not unique, this function will return a symbolic value when given a concrete NaN.Implementation note: Since there's no corresponding function in SMTLib for conversion to bit-representation due to partiality, we use a translation trick by allocating a new word variable, converting it to float, and requiring it to be equivalent to the input. In code-generation mode, we simply map it to a simple conversion.sbvConvert a float to the word containing the corresponding bit pattern(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone13<=>t#sbv,Class of things that we can logically negatesbvNegation of a quantified formula. This operation essentially lifts  to quantified formulae. Note that you can achieve the same using  . , but that will hide the quantifiers, so prefer this version if you want to keep them around.sbvA class of values that can be skolemized. Note that we don't export this class. Use the  function instead.sbvSkolemization. For any formula, skolemization gives back an equisatisfiable formula that has no existential quantifiers in it. You have to provide enough names for all the existentials in the argument. (Extras OK, so you can pass an infinite list if you like.) The names should be distinct, and also different from any other uninterpreted name you might have elsewhere.sbvIf you use the same names for skolemized arguments in different functions, they will collide; which is undesirable. Unfortunately there's no easy way for SBV to detect this. In such cases, use 7 to add a scope to the skolem-function names generated.sbv5Not exported, used for implementing generic equality.sbv4Symbolic Equality. Note that we can't use Haskell's  class since Haskell insists on returning Bool Comparing symbolic values will necessarily return a symbolic value.>Minimal complete definition: None, if the type is instance of Generic . Otherwise .sbvSymbolic equality.sbvSymbolic inequality.sbvStrong equality. On floats (/0), strong equality is object equality; that is  NaN == NaN holds, but +0 == -0 doesn't. On other types, (.===) is simply (.==). Note that (.==) is the right notion of equality for floats per IEEE754 specs, since by definition +0 == -0 and NaN equals no other value including itself. But occasionally we want to be stronger and state NaN equals NaN and +0 and -0 are different from each other. In a context where your type is concrete, simply use . But in a polymorphic context, use the strong equality instead.NB. If you do not care about or work with floats, simply use (.==) and (./=).sbvNegation of strong equality. Equaivalent to negation of (.===) on all types.sbvReturns (symbolic) 5 if all the elements of the given list are different.sbvReturns (symbolic)  if all the elements of the given list are different. The second list contains exceptions, i.e., if an element belongs to that set, it will be considered distinct regardless of repetition.sbvReturns (symbolic) 4 if all the elements of the given list are the same.sbvSymbolic membership test.sbv!Symbolic negated membership test.sbv+Arrays implemented in terms of SMT-arrays: 2http://smtlib.cs.uiowa.edu/theories-ArraysEx.shtmlMaps directly to SMT-lib arraysReading from an uninitialized value is OK. If the default value is given in , it will be the result. Otherwise, the read yields an uninterpreted constant.&Can check for equality of these arrays:Cannot be used in code-generation (i.e., compilation to C)"Cannot quick-check theorems using SArray valuessbvArrays of symbolic values An  array a b! is an array indexed by the type  a, with elements of type  b.If a default value is supplied, then all the array elements will be initialized to this value. Otherwise, they will be left unspecified, i.e., a read from an unwritten location will produce an uninterpreted constant.The reason for this class is rather historic. In the past, SBV provided two different kinds of arrays: an  abstraction that mapped directly to SMTLib arrays (which is still available today), and a functional notion of arrays that used internal caching, called  SFunArray. The latter has been removed as the code turned out to be rather tricky and hard to maintain; so we only have one instance of this class. But end users can add their own instances, if needed.NB.  insists on a concrete initializer, because not having one would break referential transparency. See  -https://github.com/LeventErkok/sbv/issues/553 for details.sbvGeneralization of sbvGeneralization of sbvCreate a literal arraysbvRead the array element at asbvUpdate the element at a to be bsbv?Merge two given arrays on the symbolic condition Intuitively: ,mergeArrays cond a b = if cond then a else b=. Merging pushes the if-then-else choice down on to elementssbv+Internal function, not exported to the usersbvA  is a potential symbolic value that can be created instances of to be fed to a symbolic program.sbvGeneralization of sbv#Turn a literal constant to symbolicsbv+Extract a literal, from a CV representationsbv/Does it concretely satisfy the given predicate?sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbv+Extract a literal, if the value is concretesbvIs the symbolic word concrete?sbv%Is the symbolic word really symbolic?sbvA class representing what can be returned from a symbolic computation.sbvGeneralization of sbvActions we can do in a context: Either at problem description time or while we are dynamically querying.  and  are two instances of this class. Note that we use this mechanism internally and do not export it from SBV.sbvAdd a constraint, any satisfying instance must satisfy this condition.sbvAdd a soft constraint. The solver will try to satisfy this condition if possible, but won't if it cannot.sbvAdd a named constraint. The name is used in unsat-core extraction.sbv,Add a constraint, with arbitrary attributes.sbvSet info. Example: setInfo ":status" ["unsat"].sbvSet an option.sbvSet the logic.sbvSet a solver time-out value, in milli-seconds. This function essentially translates to the SMTLib call (set-info :timeout val), and your backend solver may or may not support it! The amount given is in milliseconds. Also see the function : for finer level control of time-outs, directly from SBV.sbv*Get the state associated with this contextsbv0A value that can be used as a quantified booleansbvTurn a quantified boolean into a regular boolean. That is, this function turns an exists/forall quantified formula to a simple boolean that can be used as a regular boolean value. An example is:  quantifiedBool $ \(Forall x) (Exists y) -> y .> (x :: SInteger) is equivalent to . You can think of this function as performing quantifier-elimination: It takes a quantified formula, and reduces it to a simple boolean that is equivalent to it, but has no quantifiers.sbv1Values that we can turn into a lambda abstractionsbvExactly n universal symbolic variables, used in in building quantified constraints. The name attached will be prefixed in front of _1, _2, ..., _n$ to form the names of the variables.sbvExactly n existential symbolic variables, used in building quantified constraints. The name attached will be prefixed in front of _1, _2, ..., _n$ to form the names of the variables.sbvA universal symbolic variable, used in building quantified constraints. The name attached via the symbol is used during skolemization. It names the corresponding argument to the skolem-functions within the scope of this quantifier.sbvAn existential unique symbolic variable, used in building quantified constraints. The name attached via the symbol is used during skolemization. It's split into two extra names, suffixed _eu1 and _eu25, to name the universals in the equivalent formula: \exists! x\,P(x)\Leftrightarrow \exists x\,P(x) \land \forall x_{eu1} \forall x_{eu2} (P(x_{eu1}) \land P(x_{eu2}) \Rightarrow x_{eu1} = x_{eu2}) sbvAn existential symbolic variable, used in building quantified constraints. The name attached via the symbol is used during skolemization to create a skolem-function name when this variable is eliminated.sbv)Values that we can turn into a constraintsbvThe symbolic variant of bsbv7Internal representation of a symbolic simulation resultsbv'SMTLib representation, given the configsbvSymbolic 8-tuple.sbvSymbolic 7-tuple.sbvSymbolic 6-tuple.sbvSymbolic 5-tuple.sbvSymbolic 4-tuple.sbvSymbolic 3-tuple.sbvSymbolic 2-tuple. NB.  and  are equivalent.sbvSymbolic 2-tuple. NB.  and  are equivalent.sbv Symbolic . Note that we use , which supports both regular sets and complements, i.e., those obtained from the universal set (of the right type) by removing elements.sbv Symbolic sbv Symbolic sbv7A symbolic list of items. Note that a symbolic list is not= a list of symbolic items, that is, it is not the case that  SList a = [a], unlike what one might expect following haskell lists/sequences. An  is a symbolic value of its own, of possibly arbitrary but finite length, and internally processed as one unit as opposed to a fixed-length list of items. Note that lists can be nested, i.e., we do allow lists of lists of ... items.sbvA symbolic rational value.sbv2A symbolic string. Note that a symbolic string is not a list of symbolic characters, that is, it is not the case that SString = [SChar]>, unlike what one might expect following Haskell strings. An  is a symbolic value of its own, of possibly arbitrary but finite length, and internally processed as one unit as opposed to a fixed-length list of characters.sbvA symbolic character. Note that this is the full unicode character set. see:  8http://smtlib.cs.uiowa.edu/theories-UnicodeStrings.shtml for details.sbvA symbolic quad-precision floatsbv!A symbolic double-precision floatsbv!A symbolic single-precision floatsbv&A symbolic brain-float precision floatsbvA symbolic half-precision floatsbv3A symbolic arbitrary precision floating point valuesbv0IEEE-754 double-precision floating point numberssbv0IEEE-754 single-precision floating point numberssbv0Infinite precision symbolic algebraic real valuesbv(Infinite precision signed symbolic valuesbv;64-bit signed symbolic value, 2's complement representationsbv;32-bit signed symbolic value, 2's complement representationsbv;16-bit signed symbolic value, 2's complement representationsbv:8-bit signed symbolic value, 2's complement representationsbv64-bit unsigned symbolic valuesbv32-bit unsigned symbolic valuesbv16-bit unsigned symbolic valuesbv8-bit unsigned symbolic valuesbvA symbolic boolean/bitsbvThe Symbolic value. The parameter a is phantom, but is extremely important in keeping the user interface strongly typed.sbvGet the current path conditionsbv4Extend the path condition with the given test value.sbvNot-A-Number for  and . Surprisingly, Haskell Prelude doesn't have this value defined, so we provide it here.sbv Infinity for  and . Surprisingly, Haskell Prelude doesn't have this value defined, so we provide it here.sbv;Symbolic variant of Not-A-Number. This value will inhabit ,  and . types.sbvSync-up the external solver with new context we have generatedsbvRetrieve the query contextsbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbv#Creating arrays, internal use only.sbvGeneralization of sbvGeneralization of sbv#Creating arrays, internal use only.sbvGeneralization of sbvGeneralization of sbvSend a string to the solver, and return the response. Except, if the response is one of the "ignore" ones, keep querying.sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvRegistering an uninterpreted SMT function. This is typically not necessary as uses of the UI function itself will register it automatically. But there are cases where doing this explicitly can come in handy.sbvPointwise function value extraction. If we get unlucky and can't parse z3's output (happens when we have all booleans and z3 decides to spit out an expression), just brute force our way out of it. Note that we only do this if we have a pure boolean type, as otherwise we'd blow up. And I think it'll only be necessary then, I haven't seen z3 try anything smarter in other scenarios.sbvFor saturation purposes, get a proper argument. The forall quantification is safe here since we only use in smtFunSaturate calls, which looks at the kind stored inside only.sbvGeneralization of sbvGeneralization of sbvGet the value of a term, but in CV form. Used internally. The model-index, in particular is extremely Z3 specific!sbv5"Make up" a CV for this type. Like zero, but smarter.sbv$Go from an SExpr directly to a valuesbvRecover a given solver-printed value with a possible interpretationsbvGeneralization of sbvRetrieve value from the solversbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvWhat are the top level inputs? Trackers are returned as top level existentialssbvGet observables, i.e., those explicitly labeled by the user with a call to .sbvGet UIs, both constants and functions. This call returns both the before and after query ones. Generalization of .sbvReturn all satisfying models.sbvGeneralization of sbvGeneralization of sbvBail out if a parse goes badsbvGeneralization of sbv(Convert a query result to an SMT ProblemsbvGeneralization of sbvGeneric  instance for things that are  and look like containers:sbvGeneric  instance for  valuessbv as a .sbvCurried functions of arity 8sbvCurried functions of arity 7sbvCurried functions of arity 6sbvCurried functions of arity 5sbvCurried functions of arity 4sbvCurried functions of arity 3sbvCurried functions of arity 2sbvFunctions of arity 8sbvFunctions of arity 7sbvFunctions of arity 6sbvFunctions of arity 5sbvFunctions of arity 4sbvFunctions of arity 3sbvFunctions of arity 2sbvFunctions of arity 1-(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone'(e$sbv An Assignment of a model bindingsbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvClassify a model based on whether it has unbound objectives or not.sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGet a model stored at an index. This is likely very Z3 specific!sbvJust after a check-sat is issued, collect objective values. Used internally only, not exposed to the user.sbvGeneralization of sbvGeneralization of sbvHelper for the two variants of checkSatAssuming we have. Internal only.sbvGeneralization of sbv;Upon a pop, we need to restore all arrays and tables. See: ,http://github.com/LeventErkok/sbv/issues/374sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvRetrieve the unsat core if it was asked for in the configurationsbvGeneralization of sbvGeneralization of  . Use this version with MathSAT.sbvGeneralization of .sbvGeneralization of .sbvGeneralization of . Use this version with Z3.sbvGeneralization of sbvGeneralization of sbvMake an assignment. The type 1 is abstract, the result is typically passed to :  mkSMTResult [ a |-> 332 , b |-> 2.3 , c |-> True ] End users should use  for automatically constructing models from the current solver state. However, an explicit  might be handy in complex scenarios where a model needs to be created manually.sbvGeneralization of  NB. This function does not allow users to create interpretations for UI-Funs. But that's probably not a good idea anyhow. Also, if you use the  or ? features, SBV will fail on models returned via this function.JLKNM  >@?ABCDEIFGH/123485607 !,("#%$&')*+.-9;:=<1(c) Brian Schroeder Levent ErkokBSD3erkokl@gmail.com experimentalNonesbvRun a custom query.JLKNM>@?ABCDEIFGH/123485607 !,("#%$&')*+.-`a`aJKLMNAHFBCDEGI>?@/012345678 !"#$%&'()*+,-.(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone '/[sbvSymbolically executable program fragments. This class is mainly used for  calls, and is sufficiently populated internally to cover most use cases. Users can extend it as they wish to allow  checks for SBV programs that return/take types that are user-defined.sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvA type a is provable if we can turn it into a predicate, i.e., it has to return a boolean. This class captures essentially prove calls.sbv"Reduce an arg, for proof purposes.sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvA type a is satisfiable if it has constraints, potentially returning a boolean. This class captures essentially sat and optimize calls.sbv Reduce an arg, for sat purposes.sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbv is specialization of  to the  monad. Unless you are using transformers explicitly, this is the type you should prefer.sbv is specialization of  to the  monad. Unless you are using transformers explicitly, this is the type you should prefer.sbvA constraint set is a symbolic program that returns no values. The idea is that the constraints/min-max goals will serve as the collection of constraints that will be used for sat/optimize calls.sbvA predicate is a symbolic program that returns a (symbolic) boolean value. For all intents and purposes, it can be treated as an n-ary function from symbolic-values to a boolean. The  monad captures the underlying representation, and can/should be ignored by the users of the library, unless you are building further utilities on top of SBV itself. Instead, simply use the  type when necessary.sbvIf supported, this makes all output go to stdout, which works better with SBV Alas, not all solvers support it..sbvDefault configuration for the ABC synthesis and verification tool.sbv2Default configuration for the Boolector SMT solversbv1Default configuration for the Bitwuzla SMT solversbv.Default configuration for the CVC4 SMT Solver.sbv.Default configuration for the CVC5 SMT Solver.sbv/Default configuration for the Yices SMT Solver.sbv0Default configuration for the MathSAT SMT solversbv/Default configuration for the Yices SMT Solver.sbv+Default configuration for the Z3 SMT solversbv0Default configuration for the OpenSMT SMT solversbv SInteger -> SInteger bar k = smtFunction "bar" (x -> x+k) -- Note the capture of k!  and you call bar 2 and bar 3, you *will* get the same SMTLib function. Obviously this is unsound. The reason is that the parameter value isn't captured by the name. In general, you should simply not do this, but if you must, have a concrete argument to make sure you can create a unique name. Something like:  bar :: String -> SInteger -> SInteger -> SInteger bar tag k = smtFunction ("bar_" ++ tag) (x -> x+k) -- Tag should make the name unique! Then, make sure you use  bar "two" 2 and  bar "three" 3 etc. to preserve the invariant.Note that this is a design choice, to keep function creation as easy to use as possible. SBV could've made  a monadic call and generated the name itself to avoid all these issues. But the ergonomics of that is worse, and doesn't fit with the general design philosophy. If you can think of a solution (perhaps using some nifty GHC tricks?) to avoid this issue without making . return a monadic result, please get in touch!sbvUninterpret a value, i.e., add this value as a completely undefined value/function that the solver is free to instantiate to satisfy other constraints.sbvUninterpret a value, with named arguments in case of functions. SBV will use these names when it shows the values for the arguments. If the given names are more than needed we ignore the excess. If not enough, we add from a stock set of variables.sbvUninterpret a value, only for the purposes of code-generation. For execution and verification the value is used as is. For code-generation, the alternate definition is used. This is useful when we want to take advantage of native libraries on the target languages.sbvMost generalized form of uninterpretation, this function should not be needed by end-user-code, but is rather useful for the library development.sbvA synonym for 8. Allows us to create variables without having to call 7 explicitly, i.e., without being in the symbolic monad.sbv4Render an uninterpeted value as an SMTLib definitionsbvNot exported. Used only in . Instances are provided for the generic representations of product types where each element is Mergeable.sbv)Symbolic conditionals are modeled by the  class, describing how to merge the results of an if-then-else call with a symbolic test. SBV provides all basic types as instances of this class, so users only need to declare instances for custom data-types of their programs as needed.A  instance may be automatically derived for a custom data-type with a single constructor where the type of each field is an instance of ?, such as a record of symbolic values. Users only need to add  and  to the deriving clause for the data-type. See ` for an example and an illustration of what the instance would look like if written by hand. The function  is a total-indexing function out of a list of choices with a default value, simulating array/list indexing. It's an n-way generalization of the   function.>Minimal complete definition: None, if the type is instance of Generic . Otherwise . Note that most types subject to merging are likely to be trivial instances of Generic.sbvMerge two values based on the condition. The first argument states whether we force the then-and-else branches before the merging, at the word level. This is an efficiency concern; one that we'd rather not make but unfortunately necessary for getting symbolic simulation working efficiently.sbvTotal indexing operation. select xs default index is intuitively the same as  xs !! index, except it evaluates to default if index underflows/overflows.sbvThe  class captures the essence of division. Unfortunately we cannot use Haskell's  class since the  and  superclasses are not implementable for symbolic bit-vectors. However,  and " both make perfect sense, and the  class captures this operation. One issue is how division by 0 behaves. The verification technology requires total functions, and there are several design choices here. We follow Isabelle/HOL approach of assigning the value 0 for division by 0. Therefore, we impose the following pair of laws:  x  0 = (0, x) x  0 = (0, x) 5Note that our instances implement this law even when x is 0 itself.NB.  truncates toward zero, while $ truncates toward negative infinity.(C code generation of division operationsIn the case of division or modulo of a minimal signed value (e.g. -128 for ) by -1, SMTLIB and Haskell agree on what the result should be. Unfortunately the result in C code depends on CPU architecture and compiler settings, as this is undefined behaviour in C. **SBV does not guarantee** what will happen in generated C code in this corner case.sbv;Finite bit-length symbolic values. Essentially the same as , but further leaves out . Loosely based on Haskell's  FiniteBits class, but with more methods defined and structured differently to fit into the symbolic world view. Minimal complete definition: .sbv Bit size.sbv:Least significant bit of a word, always stored at index 0.sbvMost significant bit of a word, always stored at the last position.sbv,Big-endian blasting of a word into its bits.sbv/Little-endian blasting of a word into its bits.sbv4Reconstruct from given bits, given in little-endian.sbv4Reconstruct from given bits, given in little-endian.sbvReplacement for  , returning  instead of .sbv Variant of 2, where we want to extract multiple bit positions.sbv Variant of , returning a symbolic value.sbv A combo of  and %, when the bit to be set is symbolic.sbv Variant of  when the index is symbolic. If the index it out-of-bounds, then the result is underspecified.sbvFull adder, returns carry-out from the addition. Only for unsigned quantities.sbvFull multiplier, returns both high and low-order bits. Only for unsigned quantities.sbv9Count leading zeros in a word, big-endian interpretation.sbv:Count trailing zeros in a word, big-endian interpretation.sbvSymbolic Numbers. This is a simple class that simply incorporates all number like base types together, simplifying writing polymorphic type-signatures that work for all symbolic numbers, such as ,  etc. For instance, we can write a generic list-minimum function as follows:  mm :: SIntegral a => [SBV a] -> SBV a mm = foldr1 (a b -> ite (a .<= b) a b) It is similar to the standard / class, except ranging over symbolic instances.sbv!Symbolic Comparisons. Similar to  , we cannot implement Haskell's + class since there is no way to return an 1 value from a symbolic comparison. Furthermore,  requires  to implement if-then-else, for the benefit of implementing symbolic versions of  and  functions.sbvSymbolic less than.sbvSymbolic less than or equal to.sbvSymbolic greater than.sbv"Symbolic greater than or equal to.sbvSymbolic minimum.sbvSymbolic maximum.sbv Is the value within the allowed  inclusive range?sbvIdentify tuple like things. Note that there are no methods, just instances to control type inferencesbvGenerate a variable, namedsbvGenerate an unnamed variablesbv$Generate a finite constant bitvectorsbv'Convert a constant to an integral valuesbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of sbvGeneralization of  sbvGeneralization of  sbvGeneralization of  sbvGeneralization of  sbvGeneralization of  sbvGeneralization of  sbvGeneralization of  sbvGeneralization of  sbvGeneralization of  sbvGeneralization of  sbvGeneralization of  sbvGeneralization of sbvGeneralization of sbvConvert an SReal to an SInteger. That is, it computes the largest integer n that satisfies sIntegerToSReal n <= r essentially giving us the floor.For instance, 1.3 will be 1, but -1.3 will be -2.sbvlabel: Label the result of an expression. This is essentially a no-op, but useful as it generates a comment in the generated C/SMT-Lib code. Note that if the argument is a constant, then the label is dropped completely, per the usual constant folding strategy. Compare this to . which is good for printing counter-examples. sbv$Check if an observable name is good.sbvObserve the value of an expression, if the given condition holds. Such values are useful in model construction, as they are printed part of a satisfying model, or a counter-example. The same works for quick-check as well. Useful when we want to see intermediate values, or expected/obtained pairs in a particular run. Note that an observed expression is always symbolic, i.e., it won't be constant folded. Compare this to  which is used for putting a label in the generated SMTLib-C code.sbv9Observe the value of an expression, unconditionally. See  for a generalized version.sbvA variant of observe that you can use at the top-level. This is useful with quick-check, for instance.sbvReturns 1 if the boolean is , otherwise 0. sbv+Lift a pseudo-boolean op, performing checkssbv if at most k of the input arguments are  sbv if at least k of the input arguments are  sbv if exactly k of the input arguments are  sbv if the sum of coefficients for  elements is at most k. Generalizes . sbv if the sum of coefficients for  elements is at least k. Generalizes  . sbv if the sum of coefficients for  elements is exactly least k. Useful for coding exactly K-of-N2 constraints, and in particular mutex constraints. sbv if there is at most one set bit sbv if there is exactly one set bit sbvConvert a concrete pseudo-boolean to given int; converting to integer sbv:Predicate for optimizing word operations like (+) and (*). sbv:Predicate for optimizing word operations like (+) and (*). sbvSymbolic exponentiation using bit blasting and repeated squaring.N.B. The exponent must be unsigned/bounded if symbolic. Signed exponents will be rejected. sbv&Lift a 1 arg FP-op, using sRNE default sbv7Lift a float/double unary function, only over constants sbv8Lift a float/double binary function, only over constants sbvLift an sreal unary function sbvLift an sreal binary function sbv?Conversion between integral-symbolic values, akin to Haskell's  sbvLift a binary operation thru it's dynamic counterpart. Note that we still want the actual functions here as differ in their type compared to their dynamic counterparts, but the implementations are the same. sbvGeneralization of  6, when the shift-amount is symbolic. Since Haskell's   only takes an  as the shift amount, it cannot be used when we have a symbolic amount to shift with. sbvGeneralization of  6, when the shift-amount is symbolic. Since Haskell's   only takes an  as the shift amount, it cannot be used when we have a symbolic amount to shift with.NB. If the shiftee is signed, then this is an arithmetic shift; otherwise it's logical, following the usual Haskell convention. See   for a variant that explicitly uses the msb as the sign bit, even for unsigned underlying types. sbvArithmetic shift-right with a symbolic unsigned shift amount. This is equivalent to   when the argument is signed. However, if the argument is unsigned, then it explicitly treats its msb as a sign-bit, and uses it as the bit that gets shifted in. Useful when using the underlying unsigned bit representation to implement custom signed operations. Note that there is no direct Haskell analogue of this function. sbvGeneralization of  6, when the shift-amount is symbolic. Since Haskell's   only takes an  as the shift amount, it cannot be used when we have a symbolic amount to shift with. The first argument should be a bounded quantity. sbvAn implementation of rotate-left, using a barrel shifter like design. Only works when both arguments are finite bitvectors, and furthermore when the second argument is unsigned. The first condition is enforced by the type, but the second is dynamically checked. We provide this implementation as an alternative to   since SMTLib logic does not support variable argument rotates (as opposed to shifts), and thus this implementation can produce better code for verification compared to  . sbvGeneralization of  6, when the shift-amount is symbolic. Since Haskell's   only takes an  as the shift amount, it cannot be used when we have a symbolic amount to shift with. The first argument should be a bounded quantity. sbvAn implementation of rotate-right, using a barrel shifter like design. See comments for   for details. sbv*Helper function for use in enum operations sbvLift 2 to symbolic words. Division by 0 is defined s.t. x/0 = 0; which holds even when x is 0 itself. sbvLift 2 to symbolic words. Division by 0 is defined s.t. x/0 = 0; which holds even when x is 0 itself. Essentially, this is conversion from quotRem (truncate to 0) to divMod (truncate towards negative infinity) sbvEuclidian division and modulus. sbvEuclidian division. sbvEuclidian modulus. sbv$If-then-else. This is by definition  with both branches forced. This is typically the desired behavior, but also see   should you need more laziness. sbvA Lazy version of ite, which does not force its arguments. This might cause issues for symbolic simulation with large thunks around, so use with care. sbvSymbolic assert. Check that the given boolean condition is always  in the given path. The optional first argument can be used to provide call-stack info via GHC's location facilities. sbv#Merge two symbolic values, at kind k , possibly force'ing the branches to make sure they do not evaluate to the same result. This should only be used for internal purposes; as default definitions provided should suffice in many cases. (i.e., End users should only need to define 2 when needed; which should be rare to start with.) sbv?Construct a useful error message if we hit an unmergeable case. sbv6Merge concrete values that can be checked for equalitysbvNot exported. Symbolic merge using the generic representation provided by . sbvGeneralization of sbvGeneralization of sbvGeneralization of  sbv1Quick check an SBV property. Note that a regular  quickCheck call will work just as well. Use this variant if you want to receive the boolean result. sbvExplicit sharing combinator. The SBV library has internal caching/hash-consing mechanisms built in, based on Andy Gill's type-safe observable sharing technique (see:  =http://ku-fpg.github.io/files/Gill-09-TypeSafeReification.pdf). However, there might be times where being explicit on the sharing can help, especially in experimental code. The   combinator ensures that its first argument is computed once and passed on to its continuation, explicitly indicating the intent of sharing. Most use cases of the SBV library should simply use Haskell's let construct for this purpose. sbvUsing a lambda as an array sbvSymbolic computations provide a context for writing symbolic programs. sbvUsing  or 6 on non-concrete values will result in an error. Use  or  instead. sbvSReal Floating instance, used in conjunction with the dReal solver for delta-satisfiability. Note that we do not constant fold these values (except for pi), as Haskell doesn't really have any means of computing them for arbitrary rationals. sbv We give a specific instance for , because the underlying floating-point type doesn't support fromRational directly. The overlap with the above instance is unfortunate. sbvDefine Floating instance on SBV's; only for base types that are already floating; i.e., , , and *. (See the separate definition below for +.) Note that unless you use delta-sat via  on , most of the fields are "undefined" for symbolic values. We will add methods as they are supported by SMTLib. Currently, the only symbolically available function in this class is   for ,  and . sbvRegular expressions can be compared for equality. Note that we diverge here from the equality in the concrete sense; i.e., the Eq instance does not match the symbolic case. This is a bit unfortunate, but unavoidable with the current design of how we "distinguish" operators. Hopefully shouldn't be a big deal, though one should be careful. sbvSymVal for 8-tuples sbvSymVal for 7-tuples sbvSymVal for 6-tuples sbvSymVal for 5-tuples sbvSymVal for 4-tuples sbvSymVal for 3-tuples sbvSymVal for 2-tuples sbvSymVal for 0-tuple (i.e., unit) sbvIf comparison is over something SMTLib can handle, just translate it. Otherwise desugar.*                                                        -(c) Joel Burget Levent ErkokBSD3erkokl@gmail.com experimentalNone 1tH sbv tuple @(Integer, Bool, (String, Char)) (untuple p) .== pQ.E.D. sbvConstructing a tuple from its parts. Forms an isomorphism pair with  :prove $ \p -> untuple @(Integer, Bool, (String, Char)) (tuple p) .== pQ.E.D. sbv The class  4 captures the notion that a type has a certain field sbv Field labels sbvField access, inspired by the lens library. This is merely reverse application, but allows us to write things like  (1, 2)^._1 which is likely to be familiar to most Haskell programmers out there. Note that this is precisely equivalent to  _1 (1, 2)', but perhaps it reads a little nicer. sbvSwap the elements of a 2-tuple sbvDynamic interface to exporting tuples, this function is not exported on purpose; use it only via the field functions  ,  , etc. sbvAccess the 1st element of an STupleN,  2 <= N <= 8 . Also see  . sbvAccess the 2nd element of an STupleN,  2 <= N <= 8 . Also see  . sbvAccess the 3rd element of an STupleN,  3 <= N <= 8 . Also see  . sbvAccess the 4th element of an STupleN,  4 <= N <= 8 . Also see  . sbvAccess the 5th element of an STupleN,  5 <= N <= 8 . Also see  . sbvAccess the 6th element of an STupleN,  6 <= N <= 8 . Also see  . sbvAccess the 7th element of an STupleN,  7 <= N <= 8 . Also see  . sbvAccess the 8th element of an STupleN,  8 <= N <= 8 . Also see  .   (c) Levent ErkokBSD3erkokl@gmail.com experimentalNonew sbvA symbolic tree containing values of type e, indexed by elements of type i. Note that these are full-trees, and their their shapes remain constant. There is no API provided that can change the shape of the tree. These structures are useful when dealing with data-structures that are indexed with symbolic values where access time is important.  7 structures provide logarithmic time reads and writes. sbvReading a value. We bit-blast the index and descend down the full tree according to bit-values. sbvWriting a value, similar to how reads are done. The important thing is that the tree representation keeps updates to a minimum. sbvConstruct the fully balanced initial tree using the given values.  -(c) Joel Burget Levent ErkokBSD3erkokl@gmail.com experimentalNone"# sbvLength of a string.sat $ \s -> length s .== 2Satisfiable. Model: s0 = "BA" :: Stringsat $ \s -> length s .< 0 Unsatisfiable=prove $ \s1 s2 -> length s1 + length s2 .== length (s1 ++ s2)Q.E.D. sbv  s is True iff the string is empty(prove $ \s -> null s .<=> length s .== 0Q.E.D."prove $ \s -> null s .<=> s .== ""Q.E.D. sbv  returns the head of a string. Unspecified if the string is empty.&prove $ \c -> head (singleton c) .== cQ.E.D. sbv  returns the tail of a string. Unspecified if the string is empty.-prove $ \h s -> tail (singleton h ++ s) .== sQ.E.D.prove $ \s -> length s .> 0 .=> length (tail s) .== length s - 1Q.E.D.prove $ \s -> sNot (null s) .=> singleton (head s) ++ tail s .== sQ.E.D. sbv@  returns the pair of the first character and tail. Unspecified if the string is empty. sbv  returns all but the last element of the list. Unspecified if the string is empty.-prove $ \c t -> init (t ++ singleton c) .== tQ.E.D. sbv  c is the string of length 1 that contains the only character whose value is the 8-bit value c.7prove $ \c -> c .== literal 'A' .=> singleton c .== "A"Q.E.D.(prove $ \c -> length (singleton c) .== 1Q.E.D. sbv  s offset. Substring of length 1 at offset in s*. Unspecified if offset is out of bounds.prove $ \s1 s2 -> strToStrAt (s1 ++ s2) (length s1) .== strToStrAt s2 0Q.E.D.sat $ \s -> length s .>= 2 .&& strToStrAt s 0 ./= strToStrAt s (length s - 1)Satisfiable. Model: s0 = "AB" :: String sbv  s i' is the 8-bit value stored at location i). Unspecified if index is out of bounds.prove $ \i -> i .>= 0 .&& i .<= 4 .=> "AAAAA" `strToCharAt` i .== literal 'A'Q.E.D. sbvShort cut for  sbv  cs is the string of length |cs| containing precisely those characters. Note that there is no corresponding function explode:, since we wouldn't know the length of a symbolic string.8prove $ \c1 c2 c3 -> length (implode [c1, c2, c3]) .== 3Q.E.D.prove $ \c1 c2 c3 -> map (strToCharAt (implode [c1, c2, c3])) (map literal [0 .. 2]) .== [c1, c2, c3]Q.E.D. sbv$Prepend an element, the traditional cons. sbvAppend an element sbvEmpty string. This value has the property that it's the only string with length 0:+prove $ \l -> length l .== 0 .<=> l .== nilQ.E.D. sbv"Concatenate two strings. See also  . sbvShort cut for  .sat $ \x y z -> length x .== 5 .&& length y .== 1 .&& x ++ y ++ z .== "Hello world!"Satisfiable. Model: s0 = "Hello" :: String s1 = " " :: String s2 = "world!" :: String sbv  sub s. Does s contain the substring sub?4prove $ \s1 s2 s3 -> s2 `isInfixOf` (s1 ++ s2 ++ s3)Q.E.D.prove $ \s1 s2 -> s1 `isInfixOf` s2 .&& s2 `isInfixOf` s1 .<=> s1 .== s2Q.E.D. sbv  pre s. Is pre a prefix of s?,prove $ \s1 s2 -> s1 `isPrefixOf` (s1 ++ s2)Q.E.D.prove $ \s1 s2 -> s1 `isPrefixOf` s2 .=> subStr s2 0 (length s1) .== s1Q.E.D. sbv  suf s. Is suf a suffix of s?,prove $ \s1 s2 -> s2 `isSuffixOf` (s1 ++ s2)Q.E.D.prove $ \s1 s2 -> s1 `isSuffixOf` s2 .=> subStr s2 (length s2 - length s1) (length s1) .== s1Q.E.D. sbv  len s. Corresponds to Haskell's   on symbolic-strings.3prove $ \s i -> i .>= 0 .=> length (take i s) .<= iQ.E.D. sbv  len s. Corresponds to Haskell's   on symbolic-strings..prove $ \s i -> length (drop i s) .<= length sQ.E.D.*prove $ \s i -> take i s ++ drop i s .== sQ.E.D. sbv  s offset len is the substring of s at offset offset with length len. This function is under-specified when the offset is outside the range of positions in s or len is negative or  offset+len exceeds the length of s.prove $ \s i -> i .>= 0 .&& i .< length s .=> subStr s 0 i ++ subStr s i (length s - i) .== sQ.E.D.+sat $ \i j -> subStr "hello" i j .== "ell"Satisfiable. Model: s0 = 1 :: Integer s1 = 3 :: Integer)sat $ \i j -> subStr "hell" i j .== "no" Unsatisfiable sbv  s src dst". Replace the first occurrence of src by dst in sprove $ \s -> replace "hello" s "world" .== "world" .=> s .== "hello"Q.E.D.prove $ \s1 s2 s3 -> length s2 .> length s1 .=> replace s1 s2 s3 .== s1Q.E.D. sbv  s sub. Retrieves first position of sub in s, -1- if there are no occurrences. Equivalent to   s sub 0.prove $ \s1 s2 -> length s2 .> length s1 .=> indexOf s1 s2 .== -1Q.E.D. sbv  s sub offset. Retrieves first position of sub at or after offset in s, -1 if there are no occurrences.9prove $ \s sub -> offsetIndexOf s sub 0 .== indexOf s subQ.E.D.prove $ \s sub i -> i .>= length s .&& length sub .> 0 .=> offsetIndexOf s sub i .== -1Q.E.D.prove $ \s sub i -> i .> length s .=> offsetIndexOf s sub i .== -1Q.E.D. sbv  s reverses the string. >>> sat $ s -> reverse s .== "abc" Satisfiable. Model: s0 = "cba" :: String >>> prove $ s -> reverse s .== "" . = null s Q.E.D. sbv  s%. Retrieve integer encoded by string s (ground rewriting only). Note that by definition this function only works when s only contains digits, that is, if it encodes a natural number. Otherwise, it returns '-1'.prove $ \s -> let n = strToNat s in length s .== 1 .=> (-1) .<= n .&& n .<= 9Q.E.D. sbv  i%. Retrieve string encoded by integer i (ground rewriting only). Again, only naturals are supported, any input that is not a natural number produces empty string, even though we take an integer as an argument.5prove $ \i -> length (natToStr i) .== 3 .=> i .<= 999Q.E.D. sbv#Lift a unary operator over strings. sbv$Lift a binary operator over strings. sbv%Lift a ternary operator over strings. sbv!Concrete evaluation for unary ops sbv"Concrete evaluation for binary ops sbv#Concrete evaluation for ternary ops sbv%Is the string concretely known empty?     (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone sbv Empty set.empty :: SSet Integer{} :: {SInteger} sbv Full set.full :: SSet IntegerU :: {SInteger}Note that the universal set over a type is represented by the letter U. sbv Synonym for  . sbvSingleton list.singleton 2 :: SSet Integer{2} :: {SInteger} sbvConversion from a list.fromList ([] :: [Integer]){} :: {SInteger}fromList [1,2,3]{1,2,3} :: {SInteger}fromList [5,5,5,12,12,3]{3,5,12} :: {SInteger} sbv Complement.+empty .== complement (full :: SSet Integer)True2Complementing twice gets us back the original set:?prove $ \(s :: SSet Integer) -> complement (complement s) .== sQ.E.D. sbvInsert an element into a set.Insertion is order independent:prove $ \x y (s :: SSet Integer) -> x `insert` (y `insert` s) .== y `insert` (x `insert` s)Q.E.D.5Deletion after insertion is not necessarily identity:prove $ \x (s :: SSet Integer) -> x `delete` (x `insert` s) .== sFalsifiable. Counter-example: s0 = 2 :: Integer s1 = U :: {Integer}But the above is true if the element isn't in the set to start with:prove $ \x (s :: SSet Integer) -> x `notMember` s .=> x `delete` (x `insert` s) .== sQ.E.D.'Insertion into a full set does nothing:6prove $ \x -> insert x full .== (full :: SSet Integer)Q.E.D. sbvDelete an element from a set.Deletion is order independent:prove $ \x y (s :: SSet Integer) -> x `delete` (y `delete` s) .== y `delete` (x `delete` s)Q.E.D.5Insertion after deletion is not necessarily identity:prove $ \x (s :: SSet Integer) -> x `insert` (x `delete` s) .== sFalsifiable. Counter-example: s0 = 2 :: Integer s1 = U - {2} :: {Integer}But the above is true if the element is in the set to start with:prove $ \x (s :: SSet Integer) -> x `member` s .=> x `insert` (x `delete` s) .== sQ.E.D.(Deletion from an empty set does nothing:8prove $ \x -> delete x empty .== (empty :: SSet Integer)Q.E.D. sbvTest for membership.2prove $ \x -> x `member` singleton (x :: SInteger)Q.E.D.;prove $ \x (s :: SSet Integer) -> x `member` (x `insert` s)Q.E.D./prove $ \x -> x `member` (full :: SSet Integer)Q.E.D. sbvTest for non-membership.prove $ \x -> x `notMember` observe "set" (singleton (x :: SInteger))Falsifiable. Counter-example: set = {0} :: {Integer} s0 = 0 :: Integer>prove $ \x (s :: SSet Integer) -> x `notMember` (x `delete` s)Q.E.D.3prove $ \x -> x `notMember` (empty :: SSet Integer)Q.E.D. sbvIs this the empty set?null (empty :: SSet Integer)True9prove $ \x -> null (x `delete` singleton (x :: SInteger))Q.E.D.#prove $ null (full :: SSet Integer) FalsifiableNote how we have to call  in the last case since dealing with infinite sets requires a call to the solver and cannot be constant folded. sbv Synonym for  . sbvIs this the full set?&prove $ isFull (empty :: SSet Integer) Falsifiableprove $ \x -> isFull (observe "set" (x `delete` (full :: SSet Integer)))Falsifiable. Counter-example: set = U - {2} :: {Integer} s0 = 2 :: IntegerisFull (full :: SSet Integer)TrueNote how we have to call  in the first case since dealing with infinite sets requires a call to the solver and cannot be constant folded. sbv Synonym for  . sbv Subset test.1prove $ empty `isSubsetOf` (full :: SSet Integer)Q.E.D.?prove $ \x (s :: SSet Integer) -> s `isSubsetOf` (x `insert` s)Q.E.D.?prove $ \x (s :: SSet Integer) -> (x `delete` s) `isSubsetOf` sQ.E.D. sbvProper subset test.7prove $ empty `isProperSubsetOf` (full :: SSet Integer)Q.E.D.prove $ \x (s :: SSet Integer) -> s `isProperSubsetOf` (x `insert` s)Falsifiable. Counter-example: s0 = 2 :: Integer s1 = U :: {Integer}prove $ \x (s :: SSet Integer) -> x `notMember` s .=> s `isProperSubsetOf` (x `insert` s)Q.E.D.prove $ \x (s :: SSet Integer) -> (x `delete` s) `isProperSubsetOf` sFalsifiable. Counter-example: s0 = 2 :: Integer s1 = U - {2,3} :: {Integer}prove $ \x (s :: SSet Integer) -> x `member` s .=> (x `delete` s) `isProperSubsetOf` sQ.E.D. sbvDisjoint test..disjoint (fromList [2,4,6]) (fromList [1,3])True2disjoint (fromList [2,4,6,8]) (fromList [2,3,5,7])False2disjoint (fromList [1,2]) (fromList [1,2,3,4])False9prove $ \(s :: SSet Integer) -> s `disjoint` complement sQ.E.D./allSat $ \(s :: SSet Integer) -> s `disjoint` s Solution #1: s0 = {} :: {Integer}This is the only solution.The last example is particularly interesting: The empty set is the only set where   is not reflexive!Note that disjointness of a set from its complement is guaranteed by the fact that all types are inhabited; an implicit assumption we have in classic logic which is also enjoyed by Haskell due to the presence of bottom! sbvUnion.union (fromList [1..10]) (fromList [5..15]) .== (fromList [1..15] :: SSet Integer)True=prove $ \(a :: SSet Integer) b -> a `union` b .== b `union` aQ.E.D.prove $ \(a :: SSet Integer) b c -> a `union` (b `union` c) .== (a `union` b) `union` cQ.E.D.7prove $ \(a :: SSet Integer) -> a `union` full .== fullQ.E.D.5prove $ \(a :: SSet Integer) -> a `union` empty .== aQ.E.D.?prove $ \(a :: SSet Integer) -> a `union` complement a .== fullQ.E.D. sbvUnions. Equivalent to      .-prove $ unions [] .== (empty :: SSet Integer)Q.E.D. sbv Intersection.intersection (fromList [1..10]) (fromList [5..15]) .== (fromList [5..10] :: SSet Integer)Trueprove $ \(a :: SSet Integer) b -> a `intersection` b .== b `intersection` aQ.E.D.prove $ \(a :: SSet Integer) b c -> a `intersection` (b `intersection` c) .== (a `intersection` b) `intersection` cQ.E.D.;prove $ \(a :: SSet Integer) -> a `intersection` full .== aQ.E.D.prove $ \(a :: SSet Integer) -> a `intersection` empty .== emptyQ.E.D.prove $ \(a :: SSet Integer) -> a `intersection` complement a .== emptyQ.E.D.prove $ \(a :: SSet Integer) b -> a `disjoint` b .=> a `intersection` b .== emptyQ.E.D. sbvIntersections. Equivalent to      . Note that Haskell's  does not support this operation as it does not have a way of representing universal sets.3prove $ intersections [] .== (full :: SSet Integer)Q.E.D. sbv Difference.>prove $ \(a :: SSet Integer) -> empty `difference` a .== emptyQ.E.D.:prove $ \(a :: SSet Integer) -> a `difference` empty .== aQ.E.D.prove $ \(a :: SSet Integer) -> full `difference` a .== complement aQ.E.D.:prove $ \(a :: SSet Integer) -> a `difference` a .== emptyQ.E.D. sbv Synonym for  .    (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone sbvConstruct a symbolic rational from a given numerator and denominator. Note that it is not possible to deconstruct a rational by taking numerator and denominator fields, since we do not represent them canonically. (This is due to the fact that SMTLib has no functions to compute the GCD. One can use the maximization engine to compute the GCD of numbers, but not as a function.)   -(c) Joel Burget Levent ErkokBSD3erkokl@gmail.com experimentalNone{ sbv The symbolic .sNothing :: SMaybe IntegerNothing :: SMaybe Integer sbv'Check if the symbolic value is nothing.&isNothing (sNothing :: SMaybe Integer)True"isNothing (sJust (literal "nope"))False sbv Construct an SMaybe a from an SBV a.sJust (3 :: SInteger)Just 3 :: SMaybe Integer sbv+Check if the symbolic value is not nothing.#isJust (sNothing :: SMaybe Integer)FalseisJust (sJust (literal "yep"))True,prove $ \x -> isJust (sJust (x :: SInteger))Q.E.D. sbvReturn the value of an optional value. The default is returned if Nothing. Compare to  .(fromMaybe 2 (sNothing :: SMaybe Integer) 2 :: SInteger'fromMaybe 2 (sJust 5 :: SMaybe Integer) 5 :: SInteger fromMaybe x (sNothing :: SMaybe Integer) .== xQ.E.D.?prove $ \x -> fromMaybe (x+1) (sJust x :: SMaybe Integer) .== xQ.E.D. sbvReturn the value of an optional value. The behavior is undefined if passed Nothing, i.e., it can return any value. Compare to  .fromJust (sJust (literal 'a')) 'a' :: SChar1prove $ \x -> fromJust (sJust x) .== (x :: SChar)Q.E.D..sat $ \x -> x .== (fromJust sNothing :: SChar)Satisfiable. Model: s0 = 'A' :: CharNote how we get a satisfying assignment in the last case: The behavior is unspecified, thus the SMT solver picks whatever satisfies the constraints, if there is one. sbv Construct an SMaybe a from a  Maybe (SBV a). liftMaybe (Just (3 :: SInteger))Just 3 :: SMaybe Integer%liftMaybe (Nothing :: Maybe SInteger)Nothing :: SMaybe Integer sbv Map over the   side of a .?prove $ \x -> fromJust (map (+1) (sJust x)) .== x+(1::SInteger)Q.E.D.,let f = uninterpret "f" :: SInteger -> SBool-prove $ \x -> map f (sJust x) .== sJust (f x)Q.E.D.map f sNothing .== sNothingTrue sbvMap over two maybe values. sbvCase analysis for symbolic s. If the value  #, return the default value; if it  , apply the function.*maybe 0 (`sMod` 2) (sJust (3 :: SInteger)) 1 :: SInteger/maybe 0 (`sMod` 2) (sNothing :: SMaybe Integer) 0 :: SInteger,let f = uninterpret "f" :: SInteger -> SBool+prove $ \x d -> maybe d f (sJust x) .== f xQ.E.D.&prove $ \d -> maybe d f sNothing .== dQ.E.D. sbvCustom  instance over  -(c) Joel Burget Levent ErkokBSD3erkokl@gmail.com experimentalNone#E. sbvLength of a list.,sat $ \(l :: SList Word16) -> length l .== 2Satisfiable. Model: s0 = [0,0] :: [Word16]+sat $ \(l :: SList Word16) -> length l .< 0 Unsatisfiableprove $ \(l1 :: SList Word16) (l2 :: SList Word16) -> length l1 + length l2 .== length (l1 ++ l2)Q.E.D. sbv  s is True iff the list is empty:prove $ \(l :: SList Word16) -> null l .<=> length l .== 0Q.E.D.4prove $ \(l :: SList Word16) -> null l .<=> l .== []Q.E.D. sbv  returns the first element of a list. Unspecified if the list is empty.4prove $ \c -> head (singleton c) .== (c :: SInteger)Q.E.D. sbv > returns the tail of a list. Unspecified if the list is empty.;prove $ \(h :: SInteger) t -> tail (singleton h ++ t) .== tQ.E.D.prove $ \(l :: SList Integer) -> length l .> 0 .=> length (tail l) .== length l - 1Q.E.D.prove $ \(l :: SList Integer) -> sNot (null l) .=> singleton (head l) ++ tail l .== lQ.E.D. sbv  returns the pair of the head and tail. Unspecified if the list is empty. sbv  returns all but the last element of the list. Unspecified if the list is empty.;prove $ \(h :: SInteger) t -> init (t ++ singleton h) .== tQ.E.D. sbv  x6 is the list of length 1 that contains the only value x.4prove $ \(x :: SInteger) -> head (singleton x) .== xQ.E.D.6prove $ \(x :: SInteger) -> length (singleton x) .== 1Q.E.D. sbv  l offset. List of length 1 at offset in l). Unspecified if index is out of bounds.prove $ \(l1 :: SList Integer) l2 -> listToListAt (l1 ++ l2) (length l1) .== listToListAt l2 0Q.E.D.sat $ \(l :: SList Word16) -> length l .>= 2 .&& listToListAt l 0 ./= listToListAt l (length l - 1)Satisfiable. Model: s0 = [0,64] :: [Word16] sbv  l i! is the value stored at location i8, starting at 0. Unspecified if index is out of bounds.prove $ \i -> i `inRange` (0, 4) .=> [1,1,1,1,1] `elemAt` i .== (1::SInteger)Q.E.D. sbvShort cut for  sbv  es is the list of length |es| containing precisely those elements. Note that there is no corresponding function explode8, since we wouldn't know the length of a symbolic list.prove $ \(e1 :: SInteger) e2 e3 -> length (implode [e1, e2, e3]) .== 3Q.E.D.prove $ \(e1 :: SInteger) e2 e3 -> P.map (elemAt (implode [e1, e2, e3])) (P.map literal [0 .. 2]) .== [e1, e2, e3]Q.E.D. sbv$Prepend an element, the traditional cons. sbvAppend an element sbvEmpty list. This value has the property that it's the only list with length 0:>prove $ \(l :: SList Integer) -> length l .== 0 .<=> l .== nilQ.E.D. sbvAppend two lists.sat $ \x y z -> length x .== 5 .&& length y .== 1 .&& x ++ y ++ z .== [1 .. 12]Satisfiable. Model:$ s0 = [1,2,3,4,5] :: [Integer]$ s1 = [6] :: [Integer]$ s2 = [7,8,9,10,11,12] :: [Integer] sbv  e l. Does l contain the element e? sbv  e l. Does l not contain the element e? sbv  sub l. Does l contain the subsequence sub?prove $ \(l1 :: SList Integer) l2 l3 -> l2 `isInfixOf` (l1 ++ l2 ++ l3)Q.E.D.prove $ \(l1 :: SList Integer) l2 -> l1 `isInfixOf` l2 .&& l2 `isInfixOf` l1 .<=> l1 .== l2Q.E.D. sbv  pre l. Is pre a prefix of l??prove $ \(l1 :: SList Integer) l2 -> l1 `isPrefixOf` (l1 ++ l2)Q.E.D.prove $ \(l1 :: SList Integer) l2 -> l1 `isPrefixOf` l2 .=> subList l2 0 (length l1) .== l1Q.E.D. sbv  suf l. Is suf a suffix of l?>prove $ \(l1 :: SList Word16) l2 -> l2 `isSuffixOf` (l1 ++ l2)Q.E.D.prove $ \(l1 :: SList Word16) l2 -> l1 `isSuffixOf` l2 .=> subList l2 (length l2 - length l1) (length l1) .== l1Q.E.D. sbv  len l. Corresponds to Haskell's   on symbolic lists.prove $ \(l :: SList Integer) i -> i .>= 0 .=> length (take i l) .<= iQ.E.D. sbv  len s. Corresponds to Haskell's   on symbolic-lists.prove $ \(l :: SList Word16) i -> length (drop i l) .<= length lQ.E.D. take i l ++ drop i l .== lQ.E.D. sbv  s offset len is the sublist of s at offset offset with length len. This function is under-specified when the offset is outside the range of positions in s or len is negative or  offset+len exceeds the length of s.prove $ \(l :: SList Integer) i -> i .>= 0 .&& i .< length l .=> subList l 0 i ++ subList l i (length l - i) .== lQ.E.D.?sat $ \i j -> subList [1..5] i j .== ([2..4] :: SList Integer)Satisfiable. Model: s0 = 1 :: Integer s1 = 3 :: Integer?sat $ \i j -> subList [1..5] i j .== ([6..7] :: SList Integer) Unsatisfiable sbv  l src dst". Replace the first occurrence of src by dst in sprove $ \l -> replace [1..5] l [6..10] .== [6..10] .=> l .== ([1..5] :: SList Word8)Q.E.D.prove $ \(l1 :: SList Integer) l2 l3 -> length l2 .> length l1 .=> replace l1 l2 l3 .== l1Q.E.D. sbv  l sub. Retrieves first position of sub in l, -1- if there are no occurrences. Equivalent to   l sub 0.prove $ \(l1 :: SList Word16) l2 -> length l2 .> length l1 .=> indexOf l1 l2 .== -1Q.E.D. sbv  l sub offset. Retrieves first position of sub at or after offset in l, -1 if there are no occurrences.prove $ \(l :: SList Int8) sub -> offsetIndexOf l sub 0 .== indexOf l subQ.E.D.prove $ \(l :: SList Int8) sub i -> i .>= length l .&& length sub .> 0 .=> offsetIndexOf l sub i .== -1Q.E.D.prove $ \(l :: SList Int8) sub i -> i .> length l .=> offsetIndexOf l sub i .== -1Q.E.D. sbv  s reverses the sequence.NB. We can define reverse in terms of foldl as: .foldl (soFar elt -> singleton elt ++ soFar) [] But in my experiments, I found that this definition performs worse instead of the recursive definition SBV generates for reverse calls. So we're keeping it intact.>sat $ \(l :: SList Integer) -> reverse l .== literal [3, 2, 1]Satisfiable. Model: s0 = [1,2,3] :: [Integer] reverse l .== [] .<=> null lQ.E.D. sbv  op s# maps the operation on to sequence.map (+1) [1 .. 5 :: Integer][2,3,4,5,6] :: [SInteger]map (+1) [1 .. 5 :: WordN 8][2,3,4,5,6] :: [SWord8]!map singleton [1 .. 3 :: Integer][[1],[2],[3]] :: [[SInteger]]import Data.SBV.Tupleimport GHC.Exts (fromList)map (\t -> t^._1 + t^._2) (fromList [(x, y) | x <- [1..3], y <- [4..6]] :: SList (Integer, Integer))![5,6,7,6,7,8,7,8,9] :: [SInteger]Of course, SBV's   can also be reused in reverse:-sat $ \l -> map (+1) l .== [1,2,3 :: Integer]Satisfiable. Model: s0 = [0,1,2] :: [Integer] sbv  op s maps the operation on to sequence, with the counter given at each element, starting at the given value. In Haskell terms, it is:  mapi :: (Integer -> a -> b) -> Integer -> [a] -> [b] mapi f i xs = zipWith f [i..] xs  Note that   is definable in terms of , with extra coding. The reason why SBV provides this function natively is because it maps to a native function in the underlying solver. So, hopefully it'll perform better in terms being decidable.mapi (+) 10 [1 .. 5 :: Integer][11,13,15,17,19] :: [SInteger] sbv  op base s folds the from the left.foldl (+) 0 [1 .. 5 :: Integer]15 :: SIntegerfoldl (*) 1 [1 .. 5 :: Integer]120 :: SIntegerfoldl (\soFar elt -> singleton elt ++ soFar) ([] :: SList Integer) [1 .. 5 :: Integer][5,4,3,2,1] :: [SInteger]Again, we can use  in the reverse too:sat $ \l -> foldl (\soFar elt -> singleton elt ++ soFar) ([] :: SList Integer) l .== [5, 4, 3, 2, 1 :: Integer]Satisfiable. Model: s0 = [1,2,3,4,5] :: [Integer] sbv  op i base s folds the sequence, with the counter given at each element, starting at the given value. In Haskell terms, it is:  foldli :: (Integer -> b -> a -> b) -> Integer -> b -> [a] -> b foldli f c e xs = foldl (b (i, a) -> f i b a) e (zip [c..] xs) While this function is rather odd looking, it maps directly to the implementation in the underlying solver, and proofs involving it might have better decidability.1foldli (\i b a -> i+b+a) 10 0 [1 .. 5 :: Integer]75 :: SInteger sbv  op base s# folds the sequence from the right.foldr (+) 0 [1 .. 5 :: Integer]15 :: SIntegerfoldr (*) 1 [1 .. 5 :: Integer]120 :: SIntegerfoldr (\elt soFar -> soFar ++ singleton elt) ([] :: SList Integer) [1 .. 5 :: Integer][5,4,3,2,1] :: [SInteger] sbv  op base i s folds the sequence from the right, with the counter given at each element, starting at the given value. This function is provided as a parallel to  . sbv  xs ys zips the lists to give a list of pairs. The length of the final list is the minumum of the lengths of the given lists.&zip [1..10::Integer] [11..20::Integer][(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20)] :: [(SInteger, SInteger)]import Data.SBV.Tuplefoldr (+) 0 (map (\t -> t^._1+t^._2::SInteger) (zip [1..10::Integer] [10, 9..1::Integer]))110 :: SInteger sbv  f xs ys zips the lists to give a list of pairs, applying the function to each pair of elements. The length of the final list is the minumum of the lengths of the given lists..zipWith (+) [1..10::Integer] [11..20::Integer]-[12,14,16,18,20,22,24,26,28,30] :: [SInteger]>foldr (+) 0 (zipWith (+) [1..10::Integer] [10, 9..1::Integer])110 :: SInteger sbvConcatenate list of lists.NB. Concat is typically defined in terms of foldr. Here we prefer foldl, since the underlying solver primitive is foldl: Otherwise, we'd induce an extra call to reverse.)concat [[1..3::Integer], [4..7], [8..10]]$[1,2,3,4,5,6,7,8,9,10] :: [SInteger] sbv)Check all elements satisfy the predicate.let isEven x = x `sMod` 2 .== 0&all isEven [2, 4, 6, 8, 10 :: Integer]True)all isEven [2, 4, 6, 1, 8, 10 :: Integer]False sbvCheck some element satisfies the predicate. -- >>> let isEven x = x  2 .== 0 >>> any (sNot . isEven) [2, 4, 6, 8, 10 :: Integer] False >>> any isEven [2, 4, 6, 1, 8, 10 :: Integer] True sbv filter f xs+ filters the list with the given predicate.4filter (\x -> x `sMod` 2 .== 0) [1 .. 10 :: Integer][2,4,6,8,10] :: [SInteger]4filter (\x -> x `sMod` 2 ./= 0) [1 .. 10 :: Integer][1,3,5,7,9] :: [SInteger] sbv!Lift a unary operator over lists. sbv"Lift a binary operator over lists. sbv#Lift a ternary operator over lists. sbv!Concrete evaluation for unary ops sbv"Concrete evaluation for binary ops sbv#Concrete evaluation for ternary ops sbv#Is the list concretely known empty?' '   -(c) Joel Burget Levent ErkokBSD3erkokl@gmail.com experimentalNoneC sbv Construct an  SEither a b from an SBV asLeft 3 :: SEither Integer BoolLeft 3 :: SEither Integer Bool sbvReturn  if the given symbolic value is ,  otherwise(isLeft (sLeft 3 :: SEither Integer Bool)True-isLeft (sRight sTrue :: SEither Integer Bool)False sbv Construct an  SEither a b from an SBV b%sRight sFalse :: SEither Integer Bool#Right False :: SEither Integer Bool sbvReturn  if the given symbolic value is ,  otherwise)isRight (sLeft 3 :: SEither Integer Bool)False.isRight (sRight sTrue :: SEither Integer Bool)True sbv Construct an  SEither a b from an Either (SBV a) (SBV b),liftEither (Left 3 :: Either SInteger SBool)Left 3 :: SEither Integer Bool1liftEither (Right sTrue :: Either SInteger SBool)"Right True :: SEither Integer Bool sbvCase analysis for symbolic s. If the value  #, apply the first function; if it  , apply the second function.either (*2) (*3) (sLeft 3) 6 :: SIntegereither (*2) (*3) (sRight 3) 9 :: SInteger/let f = uninterpret "f" :: SInteger -> SInteger/let g = uninterpret "g" :: SInteger -> SInteger*prove $ \x -> either f g (sLeft x) .== f xQ.E.D.+prove $ \x -> either f g (sRight x) .== g xQ.E.D. sbv"Map over both sides of a symbolic  at the same time/let f = uninterpret "f" :: SInteger -> SInteger/let g = uninterpret "g" :: SInteger -> SInteger4prove $ \x -> fromLeft (bimap f g (sLeft x)) .== f xQ.E.D.6prove $ \x -> fromRight (bimap f g (sRight x)) .== g xQ.E.D. sbvMap over the left side of an /let f = uninterpret "f" :: SInteger -> SIntegerprove $ \x -> first f (sLeft x :: SEither Integer Integer) .== sLeft (f x)Q.E.D.prove $ \x -> first f (sRight x :: SEither Integer Integer) .== sRight xQ.E.D. sbvMap over the right side of an /let f = uninterpret "f" :: SInteger -> SIntegerprove $ \x -> second f (sRight x :: SEither Integer Integer) .== sRight (f x)Q.E.D.prove $ \x -> second f (sLeft x :: SEither Integer Integer) .== sLeft xQ.E.D. sbvReturn the value from the left component. The behavior is undefined if passed a right value, i.e., it can return any value.6fromLeft (sLeft (literal 'a') :: SEither Char Integer) 'a' :: SCharprove $ \x -> fromLeft (sLeft x :: SEither Char Integer) .== (x :: SChar)Q.E.D.?sat $ \x -> x .== (fromLeft (sRight 4 :: SEither Char Integer))Satisfiable. Model: s0 = 'A' :: CharNote how we get a satisfying assignment in the last case: The behavior is unspecified, thus the SMT solver picks whatever satisfies the constraints, if there is one. sbvReturn the value from the right component. The behavior is undefined if passed a left value, i.e., it can return any value.8fromRight (sRight (literal 'a') :: SEither Integer Char) 'a' :: SCharprove $ \x -> fromRight (sRight x :: SEither Char Integer) .== (x :: SInteger)Q.E.D.sat $ \x -> x .== (fromRight (sLeft (literal 2) :: SEither Integer Char))Satisfiable. Model: s0 = 'A' :: CharNote how we get a satisfying assignment in the last case: The behavior is unspecified, thus the SMT solver picks whatever satisfies the constraints, if there is one. (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone 1". sbv3A symbolic signed bit-vector carrying its size info sbv*A signed bit-vector carrying its size info sbv5A symbolic unsigned bit-vector carrying its size info sbv-An unsigned bit-vector carrying its size info sbvGeneralization of  sbvGeneralization of  sbvGeneralization of  sbvGeneralization of  sbvGeneralization of  sbvGeneralization of  sbv7Extract a portion of bits to form a smaller bit-vector. sbvJoin two bitvectors. sbvZero extend a bit-vector. sbvSign extend a bit-vector. sbv'Drop bits from the top of a bit-vector. sbv'Take bits from the top of a bit-vector. sbvQuickcheck instance for WordN sbv Optimizing  sbvConstructing models for  sbv instance for  sbv instance for  sbv instance for  sbv instance for  sbv instance for  sbv instance for  sbv instance for  sbv  instance for  sbv instance for  sbv  has a kind sbvShow instance for  sbv instance for  sbvQuickcheck instance for IntN sbv Optimizing  sbvConstructing models for  sbv instance for  sbv instance for  sbv instance for  sbv instance for  sbv instance for  sbv instance for  sbv instance for  sbv  instance for  sbv instance for  sbv  has a kind sbvShow instance for  sbv instance for   sbvi : Start position, numbered from n-1 to 0sbvj: End position, numbered from n-1 to 0, j <= i must holdsbvInput bit vector of size nsbvOutput is of size  i - j + 1 sbvFirst input, of size n, becomes the left sidesbvSecond input, of size m, becomes the right sidesbvConcatenation, of size n+m sbvInput, of size nsbvOutput, of size m. n < m must hold sbvInput, of size nsbvOutput, of size m. n < m must hold sbvi: Number of bits to drop. i < n must hold.sbvInput, of size nsbvOutput, of size m.  m = n - i holds. sbvi: Number of bits to take.  0 < i <= n must hold.sbvInput, of size nsbvOutput, of size i  (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone 1q sbvImplements polynomial addition, multiplication, division, and modulus operations over GF(2^n). NB. Similar to , division by 0 is interpreted as follows: x   0 = (0, x)for all x (including 0)Minimal complete definition:  ,  ,  sbvGiven bit-positions to be set, create a polynomial For instance polynomial [0, 1, 3] :: SWord8will evaluate to 11, since it sets the bits 0, 1, and 31. Mathematicians would write this polynomial as  x^3 + x + 1. And in fact,   will show it like that. sbvAdd two polynomials in GF(2^n). sbvMultiply two polynomials in GF(2^n), and reduce it by the irreducible specified by the polynomial as specified by coefficients of the third argument. Note that the third argument is specifically left in this form as it is usually in GF(2^(n+1)), which is not available in our formalism. (That is, we would need SWord9 for SWord8 multiplication, etc.) Also note that we do not support symbolic irreducibles, which is a minor shortcoming. (Most GF's will come with fixed irreducibles, so this should not be a problem in practice.)Passing [] for the third argument will multiply the polynomials and then ignore the higher bits that won't fit into the resulting size. sbvDivide two polynomials in GF(2^n), see above note for division by 0. sbvCompute modulus of two polynomials in GF(2^n), see above note for modulus by 0. sbv%Division and modulus packed together. sbvDisplay a polynomial like a mathematician would (over the monomial x), with a type. sbvDisplay a polynomial like a mathematician would (over the monomial x), the first argument controls if the final type is shown as well. sbvPretty print as a polynomial sbvAdd two polynomials sbvRun down a boolean condition over two lists. Note that this is different than zipWith as shorter list is assumed to be filled with sFalse at the end (i.e., zero-bits); which nicely pads it when considered as an unsigned number in little-endian form. sbvMultiply two polynomials and reduce by the third (concrete) irreducible, given by its coefficients. See the remarks for the   function for this design choice sbv8Compute modulus/remainder of polynomials on bit-vectors. sbv(Compute CRCs over bit-vectors. The call  crcBV n m p" computes the CRC of the message m with respect to polynomial p. The inputs are assumed to be blasted big-endian. The number n5 specifies how many bits of CRC is needed. Note that n+ is actually the degree of the polynomial p, and thus it seems redundant to pass it in. However, in a typical proof context, the polynomial can be symbolic, so we cannot compute the degree easily. While this can be worked-around by generating code that accounts for all possible degrees, the resulting code would be unnecessarily big and complicated, and much harder to reason with. (Also note that a CRC is just the remainder from the polynomial division, but this routine is much faster in practice.)NB. The nth bit of the polynomial p must be set for the CRC to be computed correctly. Note that the polynomial argument p will not even have this bit present most of the time, as it will typically contain bits 0 through n-13 as usual in the CRC literature. The higher order nth bit is simply assumed to be set, as it does not make sense to use a polynomial of a lesser degree. This is usually not a problem since CRC polynomials are designed and expressed this way.NB. The literature on CRC's has many variants on how CRC's are computed. We follow the following simple procedure:Extend the message m by adding n 0 bits on the right+Divide the polynomial thus obtained by the pThe remainder is the CRC value.There are many variants on final XOR's, reversed polynomials etc., so it is essential to double check you use the correct  algorithm. sbvCompute CRC's over polynomials, i.e., symbolic words. The first 0 argument plays the same role as the one in the   function.       (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone  sbvA class of checked-arithmetic operations. These follow the usual arithmetic, except make calls to  to ensure no overflow/underflow can occur. Use them in conjunction with " to ensure no overflow can happen. sbv3Detecting overflow. Each function here will return  if the result will not fit in the target type, i.e., if it overflows or underflows. sbvBit-vector addition. Unsigned addition can only overflow. Signed addition can underflow and overflow.A tell tale sign of unsigned addition overflow is when the sum is less than minimum of the arguments.>prove $ \x y -> bvAddO x (y::SWord16) .<=> x + y .< x `smin` yQ.E.D. sbvBit-vector subtraction. Unsigned subtraction can only underflow. Signed subtraction can underflow and overflow. sbvBit-vector multiplication. Unsigned multiplication can only overflow. Signed multiplication can underflow and overflow. sbvBit-vector division. Unsigned division neither underflows nor overflows. Signed division can only overflow. In fact, for each signed bitvector type, there's precisely one pair that overflows, when x is minBound and y is -1:&allSat $ \x y -> x `bvDivO` (y::SInt8) Solution #1: s0 = -128 :: Int8 s1 = -1 :: Int8This is the only solution. sbvBit-vector negation. Unsigned negation neither underflows nor overflows. Signed negation can only overflow, when the argument is minBound:4prove $ \x -> x .== minBound .<=> bvNegO (x::SInt16)Q.E.D. sbvCheck all true sbv.Are all the bits between a b (inclusive) zero? sbv-Are all the bits between a b (inclusive) one? sbvDetecting underflow/overflow conditions for casting between bit-vectors. The first output is the result, the second component itself is a pair with the first boolean indicating underflow and the second indicating overflow.:sFromIntegralO (256 :: SInt16) :: (SWord8, (SBool, SBool))(0 :: SWord8,(False,True))9sFromIntegralO (-2 :: SInt16) :: (SWord8, (SBool, SBool))(254 :: SWord8,(True,False))8sFromIntegralO (2 :: SInt16) :: (SWord8, (SBool, SBool))(2 :: SWord8,(False,False))prove $ \x -> sFromIntegralO (x::SInt32) .== (sFromIntegral x :: SInteger, (sFalse, sFalse))Q.E.D.)As the last example shows, converting to - never underflows or overflows for any value. sbv Version of   that has calls to > for checking no overflow/underflow can happen. Use it with a  call.      (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone=9| sbvConversion to and from floats sbv.Convert from an IEEE74 single precision float. sbv.Convert to an IEEE-754 Single-precision float. sbv.Convert from an IEEE74 double precision float. sbv.Convert to an IEEE-754 Double-precision float. sbv)Convert from an arbitrary floating point. sbv'Convert to an arbitrary floating point. sbvA class of floating-point (IEEE754) operations, some of which behave differently based on rounding modes. Note that unless the rounding mode is concretely RoundNearestTiesToEven, we will not concretely evaluate these, but rather pass down to the SMT solver. sbv*Compute the floating point absolute value. sbv&Compute the unary negation. Note that 0 - x is not equivalent to -x for floating-point, since -0 and 0 are different. sbv c `elem` singleton cQ.E.D. prove $ \c -> sNot (c `elem` "")Q.E.D. sbv#Is the character not in the string?4prove $ \c s -> c `elem` s .<=> sNot (c `notElem` s)Q.E.D. sbvThe   of a character. sbv*Conversion from an integer to a character.8prove $ \x -> 0 .<= x .&& x .< 256 .=> ord (chr x) .== xQ.E.D.prove $ \x -> chr (ord x) .== xQ.E.D.!sbvLift a char function to a symbolic version. If the given char is not in the class recognized by predicate, the output is the same as the input. Only works for the Latin1 set, i.e., the first 256 characters. If the given character is outside this range, it's returned unchanged.!sbvLift a char predicate to a symbolic version. Only works for the Latin1 set, i.e., the first 256 characters. sbvConvert to lower-case. Only works for the Latin1 subset, otherwise returns its argument unchanged.5prove $ \c -> toLowerL1 (toLowerL1 c) .== toLowerL1 cQ.E.D.prove $ \c -> isLowerL1 c .&& c `notElem` "\181\255" .=> toLowerL1 (toUpperL1 c) .== cQ.E.D. sbvConvert to upper-case. Only works for the Latin1 subset, otherwise returns its argument unchanged.5prove $ \c -> toUpperL1 (toUpperL1 c) .== toUpperL1 cQ.E.D.;prove $ \c -> isUpperL1 c .=> toUpperL1 (toLowerL1 c) .== cQ.E.D. sbvConvert a digit to an integer. Works for hexadecimal digits too. If the input isn't a digit, then return -1.prove $ \c -> isDigit c .|| isHexDigit c .=> digitToInt c .>= 0 .&& digitToInt c .<= 15Q.E.D.prove $ \c -> sNot (isDigit c .|| isHexDigit c) .=> digitToInt c .== -1Q.E.D. sbv*Convert an integer to a digit, inverse of  . If the integer is out of bounds, we return the arbitrarily chosen space character. Note that for hexadecimal letters, we return the corresponding lowercase letter.prove $ \i -> i .>= 0 .&& i .<= 15 .=> digitToInt (intToDigit i) .== iQ.E.D.prove $ \i -> i .< 0 .|| i .> 15 .=> digitToInt (intToDigit i) .== -1Q.E.D.prove $ \c -> digitToInt c .== -1 .<=> intToDigit (digitToInt c) .== literal ' 'Q.E.D. sbvIs this a control character? Control characters are essentially the non-printing characters. Only works for the Latin1 subset, otherwise returns . sbvIs this white-space? Only works for the Latin1 subset, otherwise returns . sbvIs this a lower-case character? Only works for the Latin1 subset, otherwise returns .5prove $ \c -> isUpperL1 c .=> isLowerL1 (toLowerL1 c)Q.E.D. sbvIs this an upper-case character? Only works for the Latin1 subset, otherwise returns .0prove $ \c -> sNot (isLowerL1 c .&& isUpperL1 c)Q.E.D. sbvIs this an alphabet character? That is lower-case, upper-case and title-case letters, plus letters of caseless scripts and modifiers letters. Only works for the Latin1 subset, otherwise returns . sbvIs this an alphabetical character or a digit? Only works for the Latin1 subset, otherwise returns .>prove $ \c -> isAlphaNumL1 c .<=> isAlphaL1 c .|| isNumberL1 cQ.E.D. sbvIs this a printable character? Only works for the Latin1 subset, otherwise returns . sbv%Is this an ASCII digit, i.e., one of 0..9 . Note that this is a subset of   (prove $ \c -> isDigit c .=> isNumberL1 cQ.E.D. sbv%Is this an Octal digit, i.e., one of 0..7. sbv!Is this a Hex digit, i.e, one of 0..9, a..f, A..F.-prove $ \c -> isHexDigit c .=> isAlphaNumL1 cQ.E.D. sbvIs this an alphabet character. Only works for the Latin1 subset, otherwise returns .+prove $ \c -> isLetterL1 c .<=> isAlphaL1 cQ.E.D. sbvIs this a mark? Only works for the Latin1 subset, otherwise returns .Note that there are no marks in the Latin1 set, so this function always returns false!prove $ sNot . isMarkL1Q.E.D. sbvIs this a number character? Only works for the Latin1 subset, otherwise returns . sbvIs this a punctuation mark? Only works for the Latin1 subset, otherwise returns . sbvIs this a symbol? Only works for the Latin1 subset, otherwise returns . sbvIs this a separator? Only works for the Latin1 subset, otherwise returns .-prove $ \c -> isSeparatorL1 c .=> isSpaceL1 cQ.E.D. sbv;Is this an ASCII character, i.e., the first 128 characters. sbvIs this a Latin1 character? sbv*Is this an ASCII Upper-case letter? i.e., A thru Zprove $ \c -> isAsciiUpper c .<=> ord c .>= ord (literal 'A') .&& ord c .<= ord (literal 'Z')Q.E.D.;prove $ \c -> isAsciiUpper c .<=> isAscii c .&& isUpperL1 cQ.E.D. sbv*Is this an ASCII Lower-case letter? i.e., a thru zprove $ \c -> isAsciiLower c .<=> ord c .>= ord (literal 'a') .&& ord c .<= ord (literal 'z')Q.E.D.;prove $ \c -> isAsciiLower c .<=> isAscii c .&& isLowerL1 cQ.E.D.  (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone"c sbv/Matchable class. Things we can match against a .For instance, you can generate valid-looking phone numbers like this::set -XOverloadedStringslet dig09 = Range '0' '9'let dig19 = Range '1' '9'"let pre = dig19 * Loop 2 2 dig09"let post = dig19 * Loop 3 3 dig09let phone = pre * "-" * post(sat $ \s -> (s :: SString) `match` phoneSatisfiable. Model: s0 = "800-8000" :: String sbv  s r checks whether s! is in the language generated by r. sbv%Match everything, universal acceptor./prove $ \(s :: SString) -> s `match` everythingQ.E.D. sbv"Match nothing, universal rejector.3prove $ \(s :: SString) -> sNot (s `match` nothing)Q.E.D. sbv.Match any character, i.e., strings of length 1prove $ \(s :: SString) -> s `match` anyChar .<=> length s .== 1Q.E.D. sbvA literal regular-expression, matching the given string exactly. Note that with OverloadedStrings extension, you can simply use a Haskell string to mean the same thing, so this function is rarely needed.prove $ \(s :: SString) -> s `match` exactly "LITERAL" .<=> s .== "LITERAL"Q.E.D. sbv#Helper to define a character class.prove $ \(c :: SChar) -> c `match` oneOf "ABCD" .<=> sAny (c .==) (map literal "ABCD")Q.E.D. sbvRecognize a newline. Also includes carriage-return and form-feed.newline=(re.union (str.to.re "\n") (str.to.re "\r") (str.to.re "\f"))/prove $ \c -> c `match` newline .=> isSpaceL1 cQ.E.D. sbvRecognize a tab.tab(str.to.re "\t")2prove $ \c -> c `match` tab .=> c .== literal '\t'Q.E.D.!sbvLift a char function to a regular expression that recognizes it. sbv.Recognize white-space, but without a new line.prove $ \c -> c `match` whiteSpaceNoNewLine .=> c `match` whiteSpace .&& c ./= literal '\n'Q.E.D. sbvRecognize white space.2prove $ \c -> c `match` whiteSpace .=> isSpaceL1 cQ.E.D. sbv"Recognize a punctuation character.9prove $ \c -> c `match` punctuation .=> isPunctuationL1 cQ.E.D. sbv$Recognize an alphabet letter, i.e., A..Z, a..z. sbv$Recognize an ASCII lower case letter asciiLower(re.range "a" "z")prove $ \c -> (c :: SChar) `match` asciiLower .=> c `match` asciiLetterQ.E.D.prove $ \c -> c `match` asciiLower .=> toUpperL1 c `match` asciiUpperQ.E.D.prove $ \c -> c `match` asciiLetter .=> toLowerL1 c `match` asciiLowerQ.E.D. sbvRecognize an upper case letter asciiUpper(re.range "A" "Z")prove $ \c -> (c :: SChar) `match` asciiUpper .=> c `match` asciiLetterQ.E.D.prove $ \c -> c `match` asciiUpper .=> toLowerL1 c `match` asciiLowerQ.E.D.prove $ \c -> c `match` asciiLetter .=> toUpperL1 c `match` asciiUpperQ.E.D. sbvRecognize a digit. One of 0..9.digit(re.range "0" "9")prove $ \c -> c `match` digit .<=> let v = digitToInt c in 0 .<= v .&& v .< 10Q.E.D.7prove $ \c -> sNot ((c::SChar) `match` (digit - digit))Q.E.D. sbv!Recognize an octal digit. One of 0..7.octDigit(re.range "0" "7")prove $ \c -> c `match` octDigit .<=> let v = digitToInt c in 0 .<= v .&& v .< 8Q.E.D.?prove $ \(c :: SChar) -> c `match` octDigit .=> c `match` digitQ.E.D. sbv&Recognize a hexadecimal digit. One of 0..9, a..f, A..F.hexDigit(re.union (re.range "0" "9") (re.range "a" "f") (re.range "A" "F"))prove $ \c -> c `match` hexDigit .<=> let v = digitToInt c in 0 .<= v .&& v .< 16Q.E.D.?prove $ \(c :: SChar) -> c `match` digit .=> c `match` hexDigitQ.E.D. sbvRecognize a decimal number.decimal(re.+ (re.range "0" "9"))prove $ \s -> (s::SString) `match` decimal .=> sNot (s `match` KStar asciiLetter)Q.E.D. sbv:Recognize an octal number. Must have a prefix of the form 0o/0O.octal(re.++ (re.union (str.to.re "0o") (str.to.re "0O")) (re.+ (re.range "0" "7")))prove $ \s -> s `match` octal .=> sAny (.== take 2 s) ["0o", "0O"]Q.E.D. sbv?Recognize a hexadecimal number. Must have a prefix of the form 0x/0X. hexadecimal(re.++ (re.union (str.to.re "0x") (str.to.re "0X")) (re.+ (re.union (re.range "0" "9") (re.range "a" "f") (re.range "A" "F"))))prove $ \s -> s `match` hexadecimal .=> sAny (.== take 2 s) ["0x", "0X"]Q.E.D. sbvRecognize a floating point number. The exponent part is optional if a fraction is present. The exponent may or may not have a sign.3prove $ \s -> s `match` floating .=> length s .>= 3Q.E.D. sbvFor the purposes of this regular expression, an identifier consists of a letter followed by zero or more letters, digits, underscores, and single quotes. The first letter must be lowercase. s `match` identifier .=> isAsciiLower (head s)Q.E.D.5prove $ \s -> s `match` identifier .=> length s .>= 1Q.E.D.!sbv#Lift a unary operator over strings.!sbv!Concrete evaluation for unary ops!sbv$Quiet GHC about testing only imports sbvMatching symbolic strings. sbvMatching a character simply means the singleton string matches the regex.'  ' (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone6g sbvCheck whether the given solver is installed and is ready to go. This call does a simple call to the solver to ensure all is well. sbv:The default configs corresponding to supported SMT solvers sbvReturn the known available solver configs, installed on your machine.!sbvTurn a name into a symbolic type. If first argument is true, we'll also derive Eq and Ord instances. sbv$Make an enumeration a symbolic type. sbvMake an uninterpred sort.!sbv*Make sure the given type is an enumeration!sbv)Make sure the given type is an empty data 1(c) Brian Schroeder Levent ErkokBSD3erkokl@gmail.com experimentalNone7 sbvAsk solver for info.NB. For a version which generalizes over the underlying monad, see  sbvRetrieve the value of an 'SMTOption.' The curious function argument is on purpose here, simply pass the constructor name. Example: the call   will return either Nothing or Just (ProduceUnsatCores True) or Just (ProduceUnsatCores False).Result will be , if the solver does not support this option.NB. For a version which generalizes over the underlying monad, see  sbv-Get the reason unknown. Only internally used.NB. For a version which generalizes over the underlying monad, see  sbv0Get the observables recorded during a query run.NB. For a version which generalizes over the underlying monad, see !sbvGet the uninterpreted constants/functions recorded during a run.NB. For a version which generalizes over the underlying monad, see  sbv*Issue check-sat and get an SMT Result out.NB. For a version which generalizes over the underlying monad, see !sbvIssue check-sat and get results of a lexicographic optimization.NB. For a version which generalizes over the underlying monad, see !sbvIssue check-sat and get results of an independent (boxed) optimization.NB. For a version which generalizes over the underlying monad, see !sbv,Construct a pareto-front optimization resultNB. For a version which generalizes over the underlying monad, see  sbvCollect model values. It is implicitly assumed that we are in a check-sat context. See = for a variant that issues a check-sat first and returns an .NB. For a version which generalizes over the underlying monad, see  sbvCheck for satisfiability, under the given conditions. Similar to  except it allows making further assumptions as captured by the first argument of booleans. (Also see  for a variant that returns the subset of the given assumptions that led to the  conclusion.)NB. For a version which generalizes over the underlying monad, see  sbvCheck for satisfiability, under the given conditions. Returns the unsatisfiable set of assumptions. Similar to  except it allows making further assumptions as captured by the first argument of booleans. If the result is , the user will also receive a subset of the given assumptions that led to the  conclusion. Note that while this set will be a subset of the inputs, it is not necessarily guaranteed to be minimal.You must have arranged for the production of unsat assumptions first via   $   for this call to not error out! Usage note:  is usually easier to use than <, as it allows the use of named assertions, as obtained by . If 9 fills your needs, you should definitely prefer it over .NB. For a version which generalizes over the underlying monad, see  sbvThe current assertion stack depth, i.e., #push - #pops after start. Always non-negative.NB. For a version which generalizes over the underlying monad, see  sbvRun the query in a new assertion stack. That is, we push the context, run the query commands, and pop it back.NB. For a version which generalizes over the underlying monad, see  sbvPush the context, entering a new one. Pushes multiple levels if n > 1.NB. For a version which generalizes over the underlying monad, see  sbv 1. It's an error to pop levels that don't exist.NB. For a version which generalizes over the underlying monad, see  sbvSearch for a result via a sequence of case-splits, guided by the user. If one of the conditions lead to a satisfiable result, returns Just+ that result. If none of them do, returns Nothing. Note that we automatically generate a coverage case and search for it automatically as well. In that latter case, the string returned will be Coverage?. The first argument controls printing progress messages See ,Documentation.SBV.Examples.Queries.CaseSplit for an example use case.NB. For a version which generalizes over the underlying monad, see  sbvReset the solver, by forgetting all the assertions. However, bindings are kept as is, as opposed to a full reset of the solver. Use this variant to clean-up the solver state while leaving the bindings intact. Pops all assertion levels. Declarations and definitions resulting from the ~ command are unaffected. Note that SBV implicitly uses global-declarations, so bindings will remain intact.NB. For a version which generalizes over the underlying monad, see  sbvEcho a string. Note that the echoing is done by the solver, not by SBV.NB. For a version which generalizes over the underlying monad, see  sbvExit the solver. This action will cause the solver to terminate. Needless to say, trying to communicate with the solver after issuing "exit" will simply fail.NB. For a version which generalizes over the underlying monad, see  sbvRetrieve the unsat-core. Note you must have arranged for unsat cores to be produced first via   $   for this call to not error out!NB. There is no notion of a minimal unsat-core, in case unsatisfiability can be derived in multiple ways. Furthermore, Z3 does not guarantee that the generated unsat core does not have any redundant assertions either, as doing so can incur a performance penalty. (There might be assertions in the set that is not needed.) To ensure all the assertions in the core are relevant, use:   $  ":smt.core.minimize" ["true"] "Note that this only works with Z3.NB. For a version which generalizes over the underlying monad, see  sbvRetrieve the proof. Note you must have arranged for proofs to be produced first via   $   for this call to not error out!A proof is simply a , as returned by the solver. In the future, SBV might provide a better datatype, depending on the use cases. Please get in touch if you use this function and can suggest a better API.NB. For a version which generalizes over the underlying monad, see  sbv1Interpolant extraction for MathSAT. Compare with , which performs similar function (but with a different use model) in Z3.!Retrieve an interpolant after an  result is obtained. Note you must have arranged for interpolants to be produced first via   $   for this call to not error out!-To get an interpolant for a pair of formulas A and B, use a ( call to attach interplation groups to A and B . Then call  ["A"], assuming those are the names you gave to the formulas in the A group.An interpolant for A and B is a formula I such that: # A .=> I and B .=> sNot I That is, it's evidence that A and B cannot be true together since A implies I but B implies not I; establishing that A and B5 cannot be satisfied at the same time. Furthermore, I0 will have only the symbols that are common to A and B.NB. Interpolant extraction isn't standardized well in SMTLib. Currently both MathSAT and Z3 support them, but with slightly differing APIs. So, we support two APIs with slightly differing types to accommodate both. See /Documentation.SBV.Examples.Queries.Interpolants& for example usages in these solvers.NB. For a version which generalizes over the underlying monad, see  sbv,Interpolant extraction for z3. Compare with , which performs similar function (but with a different use model) in MathSAT.3Unlike the MathSAT variant, you should simply call  on symbolic booleans to retrieve the interpolant. Do not call  or create named constraints. This makes it harder to identify formulas, but the current state of affairs in interpolant API requires this kludge.An interpolant for A and B is a formula I such that: " A ==> I and B ==> not I That is, it's evidence that A and B cannot be true together since A implies I but B implies not I; establishing that A and B5 cannot be satisfied at the same time. Furthermore, I0 will have only the symbols that are common to A and B.In Z3, interpolants generalize to sequences: If you pass more than two formulas, then you will get a sequence of interpolants. In general, for N formulas that are not satisfiable together, you will be returned N-1 interpolants. If formulas are A1 .. An, then interpolants will be  I1 .. I(N-1) , such that  A1 ==> I1, A2 /\ I1 ==> I2, A3 /\ I2 ==> I3, ..., and finally AN ===> not I(N-1).Currently, SBV only returns simple and sequence interpolants, and does not support tree-interpolants. If you need these, please get in touch. Furthermore, the result will be a list of mere strings representing the interpolating formulas, as opposed to a more structured type. Please get in touch if you use this function and can suggest a better API.NB. Interpolant extraction isn't standardized well in SMTLib. Currently both MathSAT and Z3 support them, but with slightly differing APIs. So, we support two APIs with slightly differing types to accommodate both. See /Documentation.SBV.Examples.Queries.Interpolants& for example usages in these solvers.NB. For a version which generalizes over the underlying monad, see  sbvGet an abduct. The first argument is a conjecture. The return value will be an assertion such that in addition with the existing assertions you have, will imply this conjecture. The second argument is the grammar which guides the synthesis of this abduct, if given. Note that SBV doesn't do any checking on the grammar. See the relevant documentation on CVC5 for details.3NB. Before you use this function, make sure to call % setOption $ ProduceAbducts True to enable abduct generation. sbv@?ABCDEIFGH/123485607 !,("#%$&')*+.-`a`a JKLMN  AHFBCDEGI>?@/012345678   !"#$%&'()*+,-.(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone4 sbvDifferent kinds of "files" we can produce. Currently this is quite C specific. sbv5Representation of a collection of generated programs. sbvPossible mappings for the  type when translated to C. Used in conjunction with the function  . Note that the particular characteristics of the mapped types depend on the platform and the compiler used for compiling the generated C program. See  )http://en.wikipedia.org/wiki/C_data_types for details. sbv float sbv double sbv  long double sbvThe code-generation monad. Allows for precise layout of input values reference parameters (for returning composite values in languages such as C), and return values. sbvCode-generation state sbv%Abstraction of target language values sbvOptions for code-generation. sbvIf , then 8-bit unsigned values will be shown in hex as well, otherwise decimal. (Other types always shown in hex.) sbvIf 7, will overwrite the generated files without prompting. sbvIf , will ignore  calls sbvIf , will generate a makefile sbvIf  , will generate a driver program sbvValues to use for the driver program generated, useful for generating non-random drivers. sbv+Type to use for representing SReal (if any) sbv2Bit-size to use for representing SInteger (if any) sbvIf , perform run-time-checks for index-out-of-bounds or shifting-by-large values etc. sbv5Abstract over code generation for different languages sbvDefault options for code generation. The run-time checks are turned-off, and the driver values are completely random. sbv)Initial configuration for code-generation sbv.Reach into symbolic monad from code-generation sbvSets RTC (run-time-checks) for index-out-of-bounds, shift-with-large value etc. on/off. Default: . sbv4Sets number of bits to be used for representing the < type in the generated C code. The argument must be one of 8, 16, 32, or 64. Note that this is essentially unsafe as the semantics of unbounded Haskell integers becomes reduced to the corresponding bit size, as typical in most C implementations. sbv0Sets the C type to be used for representing the > type in the generated C code. The setting can be one of C's "float", "double", or  "long double", types, depending on the precision needed. Note that this is essentially unsafe as the semantics of infinite precision SReal values becomes reduced to the corresponding floating point type in C, and hence it is subject to rounding errors. sbv.Should we generate a driver program? Default: . When a library is generated, it will have a driver if any of the constituent functions has a driver. (See .) sbv(Should we generate a Makefile? Default: . sbvSets driver program run time values, useful for generating programs with fixed drivers for testing. Default: None, i.e., use random values. sbv&Ignore assertions (those generated by  calls) in the generated C code sbvAdds the given lines to the header file generated, useful for generating programs with uninterpreted functions. sbv If passed , then we will not ask the user if we're overwriting files as we generate the C code. Otherwise, we'll prompt. sbv If passed , then we will show 'SWord 8' type in hex. Otherwise we'll show it in decimal. All signed types are shown decimal, and all unsigned larger types are shown hexadecimal otherwise. sbvAdds the given lines to the program file generated, useful for generating programs with uninterpreted functions. sbvAdds the given words to the compiler options in the generated Makefile, useful for linking extra stuff in. sbv.Creates an atomic input in the generated code. sbv-Creates an array input in the generated code. sbv/Creates an atomic output in the generated code. sbv.Creates an array output in the generated code. sbv9Creates a returned (unnamed) value in the generated code. sbv?Creates a returned (unnamed) array value in the generated code. sbv.Creates an atomic input in the generated code. sbv-Creates an array input in the generated code. sbv/Creates an atomic output in the generated code. sbv.Creates an array output in the generated code. sbv9Creates a returned (unnamed) value in the generated code. sbv?Creates a returned (unnamed) array value in the generated code. sbvIs this a driver program? sbvIs this a make file? sbvGenerate code for a symbolic program, returning a Code-gen bundle, i.e., collection of makefiles, source code, headers, etc. sbv4Render a code-gen bundle to a directory or to stdout!sbvAn alternative to Pretty's render, which might have "leading" white-space in empty lines. This version eliminates such whitespace. (c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneѴ sbvGiven a symbolic computation, render it as an equivalent collection of files that make up a C program:The first argument is the directory name under which the files will be saved. To save files in the current directory pass   ".". Use  for printing to stdout.>The second argument is the name of the C function to generate.2The final argument is the function to be compiled.!Compilation will also generate a Makefile, a header file, and a driver (test) program, etc. As a result, we return whatever the code-gen function returns. Most uses should simply have () as the return type here, but the value can be useful if you want to chain the result of one compilation act to the next. sbvLower level version of  , producing a  sbvCreate code to generate a library archive (.a) from given symbolic functions. Useful when generating code from multiple functions that work together as a library.The first argument is the directory name under which the files will be saved. To save files in the current directory pass   ".". Use  for printing to stdout.;The second argument is the name of the archive to generate.The third argument is the list of functions to include, in the form of function-name/code pairs, similar to the second and third arguments of  , except in a list. sbvLower level version of  , producing a  !sbvPretty print a functions type. If there is only one output, we compile it as a function that returns that value. Otherwise, we compile it as a void function that takes return values as pointers to be updated.!sbvRenders as "const SWord8 s0", etc. the first parameter is the width of the typefield!sbvReturn the proper declaration and the result as a pair. No consts!sbv3Renders as "s0", etc, or the corresponding constant!sbv!Words as it would map to a C word!sbvAlmost a "show", but map SWord1 to SBool which is used for extracting one-bit words. This is OK since C's bool type handles arithmetic fine, and maps nicely to our `SWord 1`. (Same isn't true for `SInt 1`, which doesn't have an easy counter-part on the C side.!sbv!The printf specifier for the type!sbvMake a constant value of the given type. We don't check for out of bounds here, as it should not be needed. There are many options here, using binary, decimal, etc. We simply use decimal for values 8-bits or less, and hex otherwise.!sbvGenerate a makefile. The first argument is True if we have a driver.!sbvGenerate the header!sbv"Generate an example driver program!sbvGenerate the C program!sbvMerge a bunch of bundles to generate code for a library. For the final config, we simply return the first config we receive, or the default if none.!sbv!Create a Makefile for the library!sbvCreate a driver for a library (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone  (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone# sbvSend an arbitrary string to the solver in a query. Note that this is inherently dangerous as it can put the solver in an arbitrary state and confuse SBV. If you use this feature, you are on your own! sbvRetrieve multiple responses from the solver, until it responds with a user given tag that we shall arrange for internally. The optional timeout is in milliseconds. If the time-out is exceeded, then we will raise an error. Note that this is inherently dangerous as it can put the solver in an arbitrary state and confuse SBV. If you use this feature, you are on your own! sbvSend an arbitrary string to the solver in a query, and return a response. Note that this is inherently dangerous as it can put the solver in an arbitrary state and confuse SBV. sbvInverse transformation to  0. Note that this isn't a perfect inverse, since -0 maps to 0 and back to 0. Otherwise, it's faithful:prove $ \x -> let f = sComparableSWord32AsSFloat x in fpIsNaN f .|| fpIsNegativeZero f .|| sFloatAsComparableSWord32 f .== xQ.E.D.prove $ \x -> fpIsNegativeZero x .|| sComparableSWord32AsSFloat (sFloatAsComparableSWord32 x) `fpIsEqualObject` xQ.E.D. sbvInverse transformation to  0. Note that this isn't a perfect inverse, since -0 maps to 0 and back to 0. Otherwise, it's faithful:prove $ \x -> let d = sComparableSWord64AsSDouble x in fpIsNaN d .|| fpIsNegativeZero d .|| sDoubleAsComparableSWord64 d .== xQ.E.D.prove $ \x -> fpIsNegativeZero x .|| sComparableSWord64AsSDouble (sDoubleAsComparableSWord64 x) `fpIsEqualObject` xQ.E.D. sbvInverse transformation to  0. Note that this isn't a perfect inverse, since -0 maps to 0 and back to 0. Otherwise, it's faithful:prove $ \x -> let d :: SFPHalf = sComparableSWordAsSFloatingPoint x in fpIsNaN d .|| fpIsNegativeZero d .|| sFloatingPointAsComparableSWord d .== xQ.E.D.prove $ \x -> fpIsNegativeZero x .|| sComparableSWordAsSFloatingPoint (sFloatingPointAsComparableSWord x) `fpIsEqualObject` (x :: SFPHalf)Q.E.D.    _ VYXWTUjlmnowr}tqy|spzx{uvk~bdcfegVWXYbcdefgjklmnopqrstuvwxyz{|}~TU_    1(c) Brian Schroeder Levent ErkokBSD3erkokl@gmail.com experimentalNone '1=/!sbv7Conversion from a fixed-sized BV to a sized bit-vector. sbvConvert a fixed-sized bit-vector to the corresponding sized bit-vector, for instance  to 'SWord 16'. See also  .!sbv3Capture the correspondence in terms of a constraint sbv 5, y + z .< x] NB. For a version which generalizes over the underlying monad, see sbv4Introduce a soft assertion, with an optional penaltyNB. For a version which generalizes over the underlying monad, see sbvMinimize a named metricNB. For a version which generalizes over the underlying monad, see sbvMaximize a named metricNB. For a version which generalizes over the underlying monad, see !sbv toBytes ((fromBytes [a, b, c, d]) :: SWord 32) .== [a, b, c, d]Q.E.D.sbv Convert from a sequence of bytes7prove $ \r -> fromBytes (toBytes r) .== (r :: SWord 64)Q.E.D.sbvShow a value in detailed (cracked) form, if possible. This makes most sense with numbers, and especially floating-point types.sbvAn implementation of rotate-left, using a barrel shifter like design. Only works when both arguments are finite bitvectors, and furthermore when the second argument is unsigned. The first condition is enforced by the type, but the second is dynamically checked. We provide this implementation as an alternative to   since SMTLib logic does not support variable argument rotates (as opposed to shifts), and thus this implementation can produce better code for verification compared to  .prove $ \x y -> (x `sBarrelRotateLeft` y) `sBarrelRotateRight` (y :: SWord32) .== (x :: SWord64)Q.E.D.sbvAn implementation of rotate-right, using a barrel shifter like design. See comments for   for details.prove $ \x y -> (x `sBarrelRotateRight` y) `sBarrelRotateLeft` (y :: SWord32) .== (x :: SWord64)Q.E.D.sbv7Extract a portion of bits to form a smaller bit-vector.prove $ \x -> bvExtract (Proxy @7) (Proxy @3) (x :: SWord 12) .== bvDrop (Proxy @4) (bvTake (Proxy @9) x)Q.E.D.sbvJoin two bitvectors.prove $ \x y -> x .== bvExtract (Proxy @79) (Proxy @71) ((x :: SWord 9) # (y :: SWord 71))Q.E.D.sbvZero extend a bit-vector.prove $ \x -> bvExtract (Proxy @20) (Proxy @12) (zeroExtend (x :: SInt 12) :: SInt 21) .== 0Q.E.D.sbvSign extend a bit-vector.prove $ \x -> sNot (msb x) .=> bvExtract (Proxy @20) (Proxy @12) (signExtend (x :: SInt 12) :: SInt 21) .== 0Q.E.D.prove $ \x -> msb x .=> bvExtract (Proxy @20) (Proxy @12) (signExtend (x :: SInt 12) :: SInt 21) .== complement 0Q.E.D.sbv'Drop bits from the top of a bit-vector.5prove $ \x -> bvDrop (Proxy @0) (x :: SWord 43) .== xQ.E.D.prove $ \x -> bvDrop (Proxy @20) (x :: SWord 21) .== ite (lsb x) 1 (0 :: SWord 1)Q.E.D.sbv'Take bits from the top of a bit-vector.6prove $ \x -> bvTake (Proxy @13) (x :: SWord 13) .== xQ.E.D.prove $ \x -> bvTake (Proxy @1) (x :: SWord 13) .== ite (msb x) 1 0Q.E.D.prove $ \x -> bvTake (Proxy @4) x # bvDrop (Proxy @4) x .== (x :: SWord 23)Q.E.D.sbvCheck if a relation is a partial order. The string argument must uniquely identify this order.sbvCheck if a relation is a linear order. The string argument must uniquely identify this order.sbvCheck if a relation is a tree order. The string argument must uniquely identify this order.sbvCheck if a relation is a piece-wise linear order. The string argument must uniquely identify this order.!sbv$Make sure it's internally acceptablesbvCreate the transitive closure of a given relation. The string argument must uniquely identify the newly created relation.!sbv9Check if the given relation satisfies the required axiomssbv  1024 instance for sbv  512 instance for sbv  256 instance for sbv  128 instance for sbv  64 instance for sbv  32 instance for sbv  16 instance for sbv  8 instance for sbvi : Start position, numbered from n-1 to 0sbvj: End position, numbered from n-1 to 0, j <= i must holdsbvInput bit vector of size nsbvOutput is of size  i - j + 1sbvFirst input, of size n, becomes the left sidesbvSecond input, of size m, becomes the right sidesbvConcatenation, of size n+msbvInput, of size nsbvOutput, of size m. n < m must holdsbvInput, of size nsbvOutput, of size m. n < m must holdsbvi: Number of bits to drop. i < n must hold.sbvInput, of size nsbvOutput, of size m.  m = n - i holds.sbvi: Number of bits to take.  0 < i <= n must hold.sbvInput, of size nsbvOutput, of size i!!!!!!!!!!!!!!!!!!!!!        ^]     !!!!!!!!!!!!!!!!!!!!!! ! !!!!!!!!!   9;:= sFalse)[]"ranges (\(_ :: SInteger) -> sTrue) [(-oo,oo)]ranges (\(x :: SInteger) -> sAnd [x .<= 120, x .>= -12, x ./= 3])[[-12,3),(3,120]]ranges (\(x :: SInteger) -> sAnd [x .<= 75, x .>= 5, x ./= 6, x ./= 67])[[5,6),(6,67),(67,75]]?ranges (\(x :: SInteger) -> sAnd [x .<= 75, x ./= 3, x ./= 67])[(-oo,3),(3,67),(67,75]]6ranges (\(x :: SReal) -> sAnd [x .>= 3.2, x .<= 12.7]) [[3.2,12.7]]4ranges (\(x :: SReal) -> sAnd [x .<= 12.7, x ./= 8])[(-oo,8.0),(8.0,12.7]]5ranges (\(x :: SReal) -> sAnd [x .>= 12.7, x ./= 15])[[12.7,15.0),(15.0,oo)]1ranges (\(x :: SInt8) -> sAnd [x .<= 7, x ./= 6])[[-128,6),(6,7]]ranges $ \x -> x .>= (0::SReal) [[0.0,oo)]ranges $ \x -> x .<= (0::SReal) [(-oo,0.0]]$ranges $ \(x :: SWord8) -> 2*x .== 4[[2,3),(129,130]]sbv5Compute ranges, using the given solver configuration.sbvShow instance for (c) Levent ErkokBSD3erkokl@gmail.com experimentalNonezsbvPerform natural induction over the given function, which returns left and right hand-sides to be proven equal. Uses . That is, given f x = (lhs x, rhs x) , we inductively establish that lhs and rhs agree on 0, 1, ... n&, i.e., for all non-negative integers.import Data.SBV&import Data.SBV.Tools.NaturalInductionlet sumToN :: SInteger -> SInteger = smtFunction "sumToN" $ \x -> ite (x .<= 0) 0 (x + sumToN (x-1))let sumSquaresToN :: SInteger -> SInteger = smtFunction "sumSquaresToN" $ \x -> ite (x .<= 0) 0 (x*x + sumSquaresToN (x-1))1inductNat $ \n -> (sumToN n, (n*(n+1)) `sEDiv` 2)Q.E.D.inductNat $ \n -> (sumSquaresToN n, (n*(n+1)*(2*n+1)) `sEDiv` 6)Q.E.D.inductNat $ \n -> (sumSquaresToN n, ite (n .== 12) 0 ((n*(n+1)*(2*n+1)) `sEDiv` 6))Falsifiable. Counter-example:* P(0) = (0,0) :: (Integer, Integer)* P(k) = (506,506) :: (Integer, Integer)* P(k+1) = (650,0) :: (Integer, Integer) k = 11 :: Integersbv7Perform natural induction over, using the given solver.(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonesbvResult of an inductive proof, with a counter-example in case of failure.$If a proof is found (indicated by a  result), then the invariant holds and the goal is established once the termination condition holds. If it fails, then it can fail either in an initiation step or in a consecution step:A  result in an $ step means that the invariant does not hold for the initial state, and thus indicates a true failure.A  result in a  step will return a state s. This state is known as a CTI (counterexample to inductiveness): It will lead to a violation of the invariant in one step. However, this does not mean the property is invalid: It could be the case that it is simply not inductive. In this case, human intervention---or a smarter algorithm like IC3 for certain domains---is needed to see if one can strengthen the invariant so an inductive proof can be found. How this strengthening can be done remains an art, but the science is improving with algorithms like IC3.A  result in a  step means that the invariant holds, but assuming the termination condition the goal still does not follow. That is, the partial correctness does not hold.sbv;A step in an inductive proof. If the tag is present (i.e., Just nm), then the step belongs to the subproof that establishes the strengthening named nm.sbv0Induction engine, using the default solver. See 0Documentation.SBV.Examples.ProofTools.Strengthen and )Documentation.SBV.Examples.ProofTools.Sum for examples.sbv.Induction engine, configurable with the solversbvShow instance for , diagnostic purposes only.sbvShow instance for , diagnostic purposes only.sbv Verbose modesbv.Setup code, if necessary. (Typically used for  calls. Pass  return () if not needed.)sbvInitial conditionsbvTransition relationsbvStrengthenings, if any. The String is a simple tag.sbv0Invariant that ensures the goal upon terminationsbv/Termination condition and the goal to establishsbvEither proven, or a concrete state value that, if reachable, fails the invariant.  (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone#"sbv1Case analysis on a symbolic list. (Not exported.)sbvBounded fold from the right.sbv$Bounded monadic fold from the right.sbvBounded fold from the left.sbv#Bounded monadic fold from the left.sbv Bounded sum.sbvBounded product.sbv Bounded map.sbvBounded monadic map.sbvBounded filter.sbvBounded logical andsbvBounded logical orsbv Bounded anysbv Bounded allsbv,Bounded maximum. Undefined if list is empty.sbv,Bounded minimum. Undefined if list is empty.sbvBounded zipWithsbvBounded element checksbvBounded reverse"sbv$Bounded paramorphism (not exported)."sbv4Insert an element into a sorted list (not exported).sbvBounded insertion sort(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonelsbv(Bounded fixed-point operation. The call  bfix bnd nm f unrolls the recursion in f at most bnd5 times, and uninterprets the function (with the name nm) after the bound is reached.This combinator is handy for dealing with recursive definitions that are not symbolically terminating and when the property we are interested in does not require an infinite unrolling, or when we are happy with a bounded proof. In particular, this operator can be used as a basis of software-bounded model checking algorithms built on top of SBV. The bound can be successively refined in a CEGAR like loop as necessary, by analyzing the counter-examples and rejecting them if they are false-negatives.For instance, we can define the factorial function using the bounded fixed-point operator like this:  bfac :: SInteger -> SInteger bfac = bfix 10 "fac" fact where fact f n = ite (n .== 0) 1 (n * f (n-1)) This definition unrolls the recursion in factorial at most 10 times before uninterpreting the result. We can now prove:?prove $ \n -> n .>= 1 .&& n .<= 9 .=> bfac n .== n * bfac (n-1)Q.E.D.And we would get a bogus counter-example if the proof of our property needs a larger bound:-prove $ \n -> n .== 10 .=> bfac n .== 3628800Falsifiable. Counter-example: s0 = 10 :: Integer fac :: Integer -> Integer fac _ = 2The counter-example is telling us how it instantiated the function fac< when the recursion bottomed out: It simply made it return 2 for all arguments at that point, which provides the (unintended) counter-example.%By design, if a function defined via  is given a concrete argument, it will unroll the recursion as much as necessary to complete the call (which can of course diverge). The bound only applies if the given argument is symbolic. This fact can be used to observe concrete values to see where the bounded-model-checking approach fails:prove $ \n -> n .== 10 .=> observe "bfac_n" (bfac n) .== observe "bfac_10" (bfac 10)Falsifiable. Counter-example: bfac_10 = 3628800 :: Integer bfac_n = 7257600 :: Integer s0 = 10 :: Integer fac :: Integer -> Integer fac _ = 2Here, we see further evidence that the SMT solver must have decided to assign the value 2 in the final call just as it was reaching the base case, and thus got the final result incorrect. (Note that 7257600 = 2 * 3628800<.) A wrapper algorithm can then assert the actual value of bfac 10? here as an extra constraint and can search for "deeper bugs."(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonesbv6Bounded model checking, using the default solver. See )Documentation.SBV.Examples.ProofTools.BMC for an example use case.Note that the BMC engine does *not* guarantee that the solution is unique. However, if it does find a solution at depth i7, it is guaranteed that there are no shorter solutions.sbv4Bounded model checking, configurable with the solversbvOptional boundsbvVerbose: prints iteration countsbv.Setup code, if necessary. (Typically used for  calls. Pass  return () if not needed.)sbvInitial conditionsbvTransition relationsbvGoal to cover, i.e., we find a set of transitions that satisfy this predicate.sbvEither a result, or a satisfying path of given length and intermediate observations.(c) Brian HuffmanBSD3erkokl@gmail.com experimentalNone0sbvDynamic variant of quick-checksbv'Create SMT-Lib benchmark for a sat callsbv)Create SMT-Lib benchmark for a proof callsbv/Proves the predicate using the given SMT-solversbv7Find a satisfying assignment using the given SMT-solversbv'Check safety using the given SMT-solversbv:Find all satisfying assignments using the given SMT-solversbvProve a property with multiple solvers, running them in separate threads. The results will be returned in the order produced.sbvProve a property with multiple solvers, running them in separate threads. Only the result of the first one to finish will be returned, remaining threads will be killed.sbvProve a property with query mode using multiple threads. Each query computation will spawn a thread and a unique instance of your solver to run asynchronously. The   is duplicated for each thread. This function will block until all child threads return.sbvProve a property with query mode using multiple threads. Each query computation will spawn a thread and a unique instance of your solver to run asynchronously. The   is duplicated for each thread. This function will return the first query computation that completes, killing the others.sbvFind a satisfying assignment to a property with multiple solvers, running them in separate threads. The results will be returned in the order produced.sbvFind a satisfying assignment to a property with multiple solvers, running them in separate threads. Only the result of the first one to finish will be returned, remaining threads will be killed.sbvFind a satisfying assignment to a property with multiple threads in query mode. The   represents what is known to all child query threads. Each query thread will spawn a unique instance of the solver. Only the first one to finish will be returned and the other threads will be killed.sbvFind a satisfying assignment to a property with multiple threads in query mode. The   represents what is known to all child query threads. Each query thread will spawn a unique instance of the solver. This function will block until all child threads have completed.sbvExtract a model, the result is a tuple where the first argument (if True) indicates whether the model was "probable". (i.e., if the solver returned unknown.)sbvExtract a model dictionary. Extract a dictionary mapping the variables to their respective values as returned by the SMT solver. Also see .sbvCreate a named fresh existential variable in the current contextsbvCreate an unnamed fresh existential variable in the current context  jlmnowr}tqy|spzx{uvk~jklmnopqrstuvwxyz{|}~   (c) Levent ErkokBSD3erkokl@gmail.com experimentalNonesbv Formalizes http://graphics.stanford.edu/~seander/bithacks.html#IntegerMinOrMaxsbv Formalizes http://graphics.stanford.edu/~seander/bithacks.html#IntegerMinOrMaxsbv Formalizes http://graphics.stanford.edu/~seander/bithacks.html#DetectOppositeSignssbv Formalizes http://graphics.stanford.edu/~seander/bithacks.html#ConditionalSetOrClearBitsWithoutBranchingsbv Formalizes http://graphics.stanford.edu/~seander/bithacks.html#DetermineIfPowerOf2sbvCollection of queries(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonesbvModel the mid-point computation of the binary search, which is broken due to arithmetic overflow. Note how we use the overflow checking variants of the arithmetic operators. We have:!checkArithOverflow midPointBroken./Documentation/SBV/Examples/BitPrecise/BrokenSearch.hs:39:32:+!: SInt32 addition overflows: Violated. Model: low = 1073741832 :: Int32 high = 1107296257 :: Int32Indeed:*(1073741832 + 1107296257) `div` (2::Int32) -10569646041giving us quite a large negative mid-point value!sbvThe correct version of how to compute the mid-point. As expected, this version doesn't have any underflow or overflow issues: checkArithOverflow midPointFixedNo violations detected.1As expected, the value is computed correctly too:"checkCorrectMidValue midPointFixedQ.E.D.sbvShow that the variant suggested by the blog post is good as well: 4mid = ((unsigned int)low + (unsigned int)high) >> 1;In this case the overflow is eliminated by doing the computation at a wider range:&checkArithOverflow midPointAlternativeNo violations detected.)And the value computed is indeed correct:(checkCorrectMidValue midPointAlternativeQ.E.D.sbv=A helper predicate to check safety under the conditions that low is at least 0 and high is at least low.sbvAnother helper to show that the result is actually the correct value, if it was done over 64-bit integers, which is sufficiently large enough.(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone1<>|(sbv4Helper synonym for capturing relevant bits of MosteksbvAn instruction is modeled as a  transformer. We model mostek programs in direct continuation passing style.sbvPrograms are essentially state transformers (on the machine state)sbv0Given a machine state, compute a value out of itsbvAbstraction of the machine: The CPU consists of memory, registers, and flags. Unlike traditional hardware, we assume the program is stored in some other memory area that we need not model. (No self modifying programs!)+ is equipped with an automatically derived ! instance because each field is .sbv2Memory is simply an array from locations to valuessbv?We have three memory locations, sufficient to model our problemsbv multiplicandsbv multipliersbv'low byte of the result gets stored heresbv Flag banksbv Register banksbv-Convenient synonym for symbolic machine bits.sbvMostek was an 8-bit machine.sbvThe carry flag () and the zero flag ()sbvWe model only two registers of Mostek that is used in the above algorithm, can add more.sbv!Get the value of a given registersbv!Set the value of a given registersbvGet the value of a flagsbvSet the value of a flagsbv Read memorysbvWrite to memorysbv.Checking overflow. In Legato's multiplier the ADC instruction needs to see if the expression x + y + c overflowed, as checked by this function. Note that we verify the correctness of this check separately below in .sbvCorrectness theorem for our  implementation.We have:checkOverflowCorrectQ.E.D.sbvLDX: Set register X to value vsbvLDA: Set register A to value vsbvCLC: Clear the carry flagsbv9ROR, memory version: Rotate the value at memory location a to the right by 1 bit, using the carry flag as a transfer position. That is, the final bit of the memory location becomes the new carry and the carry moves over to the first bit. This very instruction is one of the reasons why Legato's multiplier is quite hard to understand and is typically presented as a verification challenge.sbvROR, register version: Same as , except through register r.sbvBCC: branch to label l if the carry flag is sFalsesbv%ADC: Increment the value of register A. by the value of memory contents at location a7, using the carry-bit as the carry-in for the addition.sbv%DEX: Decrement the value of register Xsbv&BNE: Branch if the zero-flag is sFalsesbvThe  combinator "stops" our program, providing the final continuation that does nothing.sbvMultiplies the contents of F1 and F2), storing the low byte of the result in LO% and the high byte of it in register A. The implementation is a direct transliteration of Legato's algorithm given at the top, using our notation.sbvGiven values for F1 and F2,  runLegato" takes an arbitrary machine state m; and returns the high and low bytes of the multiplication.sbvCreate an instance of the Mostek machine, initialized by the memory and the relevant values of the registers and the flagssbvThe correctness theorem. For all possible memory configurations, the factors (x and y below), the location of the low-byte result and the initial-values of registers and the flags, this function will return True only if running Legato's algorithm does indeed compute the product of x and y correctly.sbvThe correctness theorem.sbvGenerate a C program that implements Legato's algorithm automatically.00 (c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneÿsbvElement type of lists we'd like to sort. For simplicity, we'll just use ) here, but we can pick any symbolic type.sbv5Merging two given sorted lists, preserving the order.sbvSimple merge-sort implementation. We simply divide the input list in two halves so long as it has at least two elements, sort each half on its own, and then merge.sbv1Check whether a given sequence is non-decreasing.sbvCheck 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.sbvAsserting 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 the list size increases. A value around 5 or 6 should be fairly easy to prove. For instance, we have: correctness 5Q.E.D.sbv3Generate C code for merge-sorting an array of size n. Again, we're restricted to fixed size inputs. While the output is not how one would code merge sort in C by hand, it's a faithful rendering of all the operations merge-sort would do as described by its Haskell counterpart.!(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone15sbvParallel extraction: Given a source value and a mask, extract the bits in the source that are pointed to by the mask, and put it in the destination starting from the bottom.satWith z3{printBase = 16} $ \r -> r .== pext (0xAA :: SWord 8) 0xAASatisfiable. Model: s0 = 0x0f :: Word8prove $ \x -> pext @8 x 0 .== 0Q.E.D.,prove $ \x -> pext @8 x (complement 0) .== xQ.E.D.sbvParallel deposit: Given a source value and a mask, write into the destination that are allowed by the mask, grabbing the bits from the source starting from the bottom.satWith z3{printBase = 16} $ \r -> r .== pdep (0xFF :: SWord 8) 0xAASatisfiable. Model: s0 = 0xaa :: Word8prove $ \x -> pdep @8 x 0 .== 0Q.E.D.,prove $ \x -> pdep @8 x (complement 0) .== xQ.E.D.sbvProve that extraction and depositing with the same mask restore the source in all masked positions:extractThenDepositQ.E.D.sbvProve that depositing and extracting with the same mask will push preserve the bottom n-bits of the source, where n is the number of bits set in the mask.depositThenExtractQ.E.D.sbvWe can generate the code for these functions if they need to be used in SMTLib. Below is an example at 2-bits, which can be adjusted to produce any bit-size.putStrLn =<< sbv2smt pext_20; Automatically generated by SBV. Do not modify!); pext_2 :: SWord 2 -> SWord 2 -> SWord 2(define-fun pext_2 ((l1_s0 (_ BitVec 2)) (l1_s1 (_ BitVec 2))) (_ BitVec 2) (let ((l1_s3 #b0)) (let ((l1_s7 #b01)) (let ((l1_s8 #b00)) (let ((l1_s20 #b10))( (let ((l1_s2 ((_ extract 1 1) l1_s1)))' (let ((l1_s4 (distinct l1_s2 l1_s3)))( (let ((l1_s5 ((_ extract 0 0) l1_s1)))' (let ((l1_s6 (distinct l1_s3 l1_s5)))( (let ((l1_s9 (ite l1_s6 l1_s7 l1_s8)))! (let ((l1_s10 (= l1_s7 l1_s9)))& (let ((l1_s11 (bvlshr l1_s0 l1_s7)))* (let ((l1_s12 ((_ extract 0 0) l1_s11)))) (let ((l1_s13 (distinct l1_s3 l1_s12)))! (let ((l1_s14 (= l1_s8 l1_s9)))) (let ((l1_s15 ((_ extract 0 0) l1_s0)))) (let ((l1_s16 (distinct l1_s3 l1_s15)))* (let ((l1_s17 (ite l1_s16 l1_s7 l1_s8)))* (let ((l1_s18 (ite l1_s6 l1_s17 l1_s8)))% (let ((l1_s19 (bvor l1_s7 l1_s18)))' (let ((l1_s21 (bvand l1_s18 l1_s20))), (let ((l1_s22 (ite l1_s13 l1_s19 l1_s21))), (let ((l1_s23 (ite l1_s14 l1_s22 l1_s18)))& (let ((l1_s24 (bvor l1_s20 l1_s23)))& (let ((l1_s25 (bvand l1_s7 l1_s23))), (let ((l1_s26 (ite l1_s13 l1_s24 l1_s25))), (let ((l1_s27 (ite l1_s10 l1_s26 l1_s23)))+ (let ((l1_s28 (ite l1_s4 l1_s27 l1_s18)))$ l1_s28))))))))))))))))))))))))))))"(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone? sbvA poor man's representation of powerlists and basic operations on them:  (http://dl.acm.org/citation.cfm?id=1973564 We merely represent power-lists by ordinary lists.sbv The tie operator, concatenation.sbvThe zip operator, zips the power-lists of the same size, returns a powerlist of double the size.sbvInverse of zipping.sbvReference prefix sum (ps) is simply Haskell's scanl1 function.sbvThe Ladner-Fischer (lf$) implementation of prefix-sum. See  http://www.cs.utexas.edu/~plaxton/c/337/05f/slides/ParallelRecursion-4.pdf or pg. 16 of (http://dl.acm.org/citation.cfm?id=197356sbvCorrectness theorem, for a powerlist of given size, an associative operator, and its left-unit element.sbvProves Ladner-Fischer is equivalent to reference specification for addition. 0; is the left-unit element, and we use a power-list of size 8 . We have:thm1Q.E.D.sbvProves Ladner-Fischer is equivalent to reference specification for the function max. 0; is the left-unit element, and we use a power-list of size 16 . We have:thm2Q.E.D.  #(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneBsbv,Simple function that returns add/sum of argssbvGenerate C code for addSub. Here's the output showing the generated C code: genAddSub%== BEGIN: "Makefile" ================# Makefile for addSub. Automatically generated by SBV. Do not edit!=# include any user-defined .mk file in the current directory. -include *.mkCC?=gcc0CCFLAGS?=-Wall -O3 -DNDEBUG -fomit-frame-pointerall: addSub_driveraddSub.o: addSub.c addSub.h ${CC} ${CCFLAGS} -c $< -o $@ addSub_driver.o: addSub_driver.c ${CC} ${CCFLAGS} -c $< -o $@'addSub_driver: addSub.o addSub_driver.o ${CC} ${CCFLAGS} $^ -o $@clean: rm -f *.overyclean: clean rm -f addSub_driver%== END: "Makefile" ==================%== BEGIN: "addSub.h" ================/* Header file for addSub. Automatically generated by SBV. Do not edit! */##ifndef __addSub__HEADER_INCLUDED__##define __addSub__HEADER_INCLUDED__#include #include #include #include #include #include #include /* The boolean type */typedef bool SBool;/* The float type */typedef float SFloat;/* The double type */typedef double SDouble;/* Unsigned bit-vectors */typedef uint8_t SWord8;typedef uint16_t SWord16;typedef uint32_t SWord32;typedef uint64_t SWord64;/* Signed bit-vectors */typedef int8_t SInt8;typedef int16_t SInt16;typedef int32_t SInt32;typedef int64_t SInt64;/* Entry point prototype: */8void addSub(const SWord8 x, const SWord8 y, SWord8 *sum, SWord8 *dif);(#endif /* __addSub__HEADER_INCLUDED__ */%== END: "addSub.h" ==================,== BEGIN: "addSub_driver.c" ================(/* Example driver program for addSub. */:/* Automatically generated by SBV. Edit as you see fit! */#include #include "addSub.h"int main(void){ SWord8 sum; SWord8 dif; addSub(132, 241, &sum, &dif);. printf("addSub(132, 241, &sum, &dif) ->\n");$ printf(" sum = %"PRIu8"\n", sum);$ printf(" dif = %"PRIu8"\n", dif); return 0;},== END: "addSub_driver.c" ==================%== BEGIN: "addSub.c" ================/* File: "addSub.c". Automatically generated by SBV. Do not edit! */#include "addSub.h"8void addSub(const SWord8 x, const SWord8 y, SWord8 *sum, SWord8 *dif){ const SWord8 s0 = x; const SWord8 s1 = y; const SWord8 s2 = s0 + s1; const SWord8 s3 = s0 - s1; *sum = s2; *dif = s3;}%== END: "addSub.c" ==================$(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonesbvThe USB CRC polynomial:  x^5 + x^2 + 1. Although this polynomial needs just 6 bits to represent (5 if higher order bit is implicitly assumed to be set), we'll simply use a 16 bit number for its representation to keep things simple for code generation purposes.sbvGiven an 11 bit message, compute the CRC of it using the USB polynomial, which is 5 bits, and then append it to the msg to get a 16-bit word. Again, the incoming 11-bits is represented as a 16-bit word, with 5 highest bits essentially ignored for input purposes.sbv(Alternate method for computing the CRC, mathematically. We shift the number to the left by 5, and then compute the remainder from the polynomial division by the USB polynomial. The result is then appended to the end of the message.sbvProve that the custom   function is equivalent to the mathematical definition of CRC's for 11 bit messages. We have:crcGoodQ.E.D.sbvGenerate a C function to compute the USB CRC, using the internal CRC function.sbvGenerate a C function to compute the USB CRC, using the mathematical definition of the CRCs. While this version generates functionally equivalent C code, it's less efficient; it has about 30% more code. So, the above version is preferable for code generation purposes.%*(c) Lee Pike Levent ErkokBSD3erkokl@gmail.com experimentalNonesbvThis is a naive implementation of fibonacci, and will work fine (albeit slow) for concrete inputs:map fib0 [0..6][0 :: SWord64,1 :: SWord64,1 :: SWord64,2 :: SWord64,3 :: SWord64,5 :: SWord64,8 :: SWord64]However, it is not suitable for doing proofs or generating code, as it is not symbolically terminating when it is called with a symbolic value n. When we recursively call fib0 on n-1 (or n-2), the test against 0 will always explore both branches since the result will be symbolic, hence will not terminate. (An integrated theorem prover can establish termination after a certain number of unrollings, but this would be quite expensive to implement, and would be impractical.)sbvThe recursion-depth limited version of fibonacci. Limiting the maximum number to be 20, we can say:map (fib1 20) [0..6][0 :: SWord64,1 :: SWord64,1 :: SWord64,2 :: SWord64,3 :: SWord64,5 :: SWord64,8 :: SWord64]The function will work correctly, so long as the index we query is at most top*, and otherwise will return the value at top. Note that we also use accumulating parameters here for efficiency, although this is orthogonal to the termination concern.A note on modular arithmetic: The 64-bit word we use to represent the values will of course eventually overflow, beware! Fibonacci is a fast growing function..sbvWe can generate code for  using the  action. Note that the generated code will grow larger as we pick larger values of top, but only linearly, thanks to the accumulating parameter trick used by . The following is an excerpt from the code generated for the call  genFib1 10;, where the code will work correctly for indexes up to 10:  SWord64 fib1(const SWord64 x) { const SWord64 s0 = x; const SBool s2 = s0 == 0x0000000000000000ULL; const SBool s4 = s0 == 0x0000000000000001ULL; const SBool s6 = s0 == 0x0000000000000002ULL; const SBool s8 = s0 == 0x0000000000000003ULL; const SBool s10 = s0 == 0x0000000000000004ULL; const SBool s12 = s0 == 0x0000000000000005ULL; const SBool s14 = s0 == 0x0000000000000006ULL; const SBool s17 = s0 == 0x0000000000000007ULL; const SBool s19 = s0 == 0x0000000000000008ULL; const SBool s22 = s0 == 0x0000000000000009ULL; const SWord64 s25 = s22 ? 0x0000000000000022ULL : 0x0000000000000037ULL; const SWord64 s26 = s19 ? 0x0000000000000015ULL : s25; const SWord64 s27 = s17 ? 0x000000000000000dULL : s26; const SWord64 s28 = s14 ? 0x0000000000000008ULL : s27; const SWord64 s29 = s12 ? 0x0000000000000005ULL : s28; const SWord64 s30 = s10 ? 0x0000000000000003ULL : s29; const SWord64 s31 = s8 ? 0x0000000000000002ULL : s30; const SWord64 s32 = s6 ? 0x0000000000000001ULL : s31; const SWord64 s33 = s4 ? 0x0000000000000001ULL : s32; const SWord64 s34 = s2 ? 0x0000000000000000ULL : s33; return s34; }sbv,Compute the fibonacci numbers statically at code-generation0 time and put them in a table, accessed by the  call. sbv Once we have , we can generate the C code straightforwardly. Below is an excerpt from the code that SBV generates for the call  genFib2 64. Note that this code is a constant-time look-up table implementation of fibonacci, with no run-time overhead. The index can be made arbitrarily large, naturally. (Note that this function returns 0> if the index is larger than 64, as specified by the call to  with default 0.) SWord64 fibLookup(const SWord64 x) { const SWord64 s0 = x; static const SWord64 table0[] = { 0x0000000000000000ULL, 0x0000000000000001ULL, 0x0000000000000001ULL, 0x0000000000000002ULL, 0x0000000000000003ULL, 0x0000000000000005ULL, 0x0000000000000008ULL, 0x000000000000000dULL, 0x0000000000000015ULL, 0x0000000000000022ULL, 0x0000000000000037ULL, 0x0000000000000059ULL, 0x0000000000000090ULL, 0x00000000000000e9ULL, 0x0000000000000179ULL, 0x0000000000000262ULL, 0x00000000000003dbULL, 0x000000000000063dULL, 0x0000000000000a18ULL, 0x0000000000001055ULL, 0x0000000000001a6dULL, 0x0000000000002ac2ULL, 0x000000000000452fULL, 0x0000000000006ff1ULL, 0x000000000000b520ULL, 0x0000000000012511ULL, 0x000000000001da31ULL, 0x000000000002ff42ULL, 0x000000000004d973ULL, 0x000000000007d8b5ULL, 0x00000000000cb228ULL, 0x0000000000148addULL, 0x0000000000213d05ULL, 0x000000000035c7e2ULL, 0x00000000005704e7ULL, 0x00000000008cccc9ULL, 0x0000000000e3d1b0ULL, 0x0000000001709e79ULL, 0x0000000002547029ULL, 0x0000000003c50ea2ULL, 0x0000000006197ecbULL, 0x0000000009de8d6dULL, 0x000000000ff80c38ULL, 0x0000000019d699a5ULL, 0x0000000029cea5ddULL, 0x0000000043a53f82ULL, 0x000000006d73e55fULL, 0x00000000b11924e1ULL, 0x000000011e8d0a40ULL, 0x00000001cfa62f21ULL, 0x00000002ee333961ULL, 0x00000004bdd96882ULL, 0x00000007ac0ca1e3ULL, 0x0000000c69e60a65ULL, 0x0000001415f2ac48ULL, 0x000000207fd8b6adULL, 0x0000003495cb62f5ULL, 0x0000005515a419a2ULL, 0x00000089ab6f7c97ULL, 0x000000dec1139639ULL, 0x000001686c8312d0ULL, 0x000002472d96a909ULL, 0x000003af9a19bbd9ULL, 0x000005f6c7b064e2ULL, 0x000009a661ca20bbULL }; const SWord64 s65 = s0 >= 65 ? 0x0000000000000000ULL : table0[s0]; return s65; }&(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonesbv>The symbolic GCD algorithm, over two 8-bit numbers. We define sgcd a 0 to be a for all a, which implies  sgcd 0 0 = 0. Note that this is essentially Euclid's algorithm, except with a recursion depth counter. We need the depth counter since the algorithm is not symbolically terminating, as we don't have a means of determining that the second argument (b) will eventually reach 0 in a symbolic context. Hence we stop after 12 iterations. Why 12? We've empirically determined that this algorithm will recurse at most 12 times for arbitrary 8-bit numbers. Of course, this is a claim that we shall prove below.sbvWe have:prove sgcdIsCorrectQ.E.D.sbvThis call will generate the required C files. The following is the function body generated for ,. (We are not showing the generated header, Makefile, and the driver programs for brevity.) Note that the generated function is a constant time algorithm for GCD. It is not necessarily fastest, but it will take precisely the same amount of time for all values of x and y. /* File: "sgcd.c". Automatically generated by SBV. Do not edit! */ #include #include #include #include #include #include "sgcd.h" SWord8 sgcd(const SWord8 x, const SWord8 y) { const SWord8 s0 = x; const SWord8 s1 = y; const SBool s3 = s1 == 0; const SWord8 s4 = (s1 == 0) ? s0 : (s0 % s1); const SWord8 s5 = s3 ? s0 : s4; const SBool s6 = 0 == s5; const SWord8 s7 = (s5 == 0) ? s1 : (s1 % s5); const SWord8 s8 = s6 ? s1 : s7; const SBool s9 = 0 == s8; const SWord8 s10 = (s8 == 0) ? s5 : (s5 % s8); const SWord8 s11 = s9 ? s5 : s10; const SBool s12 = 0 == s11; const SWord8 s13 = (s11 == 0) ? s8 : (s8 % s11); const SWord8 s14 = s12 ? s8 : s13; const SBool s15 = 0 == s14; const SWord8 s16 = (s14 == 0) ? s11 : (s11 % s14); const SWord8 s17 = s15 ? s11 : s16; const SBool s18 = 0 == s17; const SWord8 s19 = (s17 == 0) ? s14 : (s14 % s17); const SWord8 s20 = s18 ? s14 : s19; const SBool s21 = 0 == s20; const SWord8 s22 = (s20 == 0) ? s17 : (s17 % s20); const SWord8 s23 = s21 ? s17 : s22; const SBool s24 = 0 == s23; const SWord8 s25 = (s23 == 0) ? s20 : (s20 % s23); const SWord8 s26 = s24 ? s20 : s25; const SBool s27 = 0 == s26; const SWord8 s28 = (s26 == 0) ? s23 : (s23 % s26); const SWord8 s29 = s27 ? s23 : s28; const SBool s30 = 0 == s29; const SWord8 s31 = (s29 == 0) ? s26 : (s26 % s29); const SWord8 s32 = s30 ? s26 : s31; const SBool s33 = 0 == s32; const SWord8 s34 = (s32 == 0) ? s29 : (s29 % s32); const SWord8 s35 = s33 ? s29 : s34; const SBool s36 = 0 == s35; const SWord8 s37 = s36 ? s32 : s35; const SWord8 s38 = s33 ? s29 : s37; const SWord8 s39 = s30 ? s26 : s38; const SWord8 s40 = s27 ? s23 : s39; const SWord8 s41 = s24 ? s20 : s40; const SWord8 s42 = s21 ? s17 : s41; const SWord8 s43 = s18 ? s14 : s42; const SWord8 s44 = s15 ? s11 : s43; const SWord8 s45 = s12 ? s8 : s44; const SWord8 s46 = s9 ? s5 : s45; const SWord8 s47 = s6 ? s1 : s46; const SWord8 s48 = s3 ? s0 : s47; return s48; }'(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonesbvGiven a 64-bit quantity, the simplest (and obvious) way to count the number of bits that are set in it is to simply walk through all the bits and add 1 to a running count. This is slow, as it requires 64 iterations, but is simple and easy to convince yourself that it is correct. For instance:popCountSlow 0x0123456789ABCDEF 32 :: SWord8sbvFaster version. This is essentially the same algorithm, except we go 8 bits at a time instead of one by one, by using a precomputed table of population-count values for each byte. This algorithm loops= only 8 times, and hence is at least 8 times more efficient.sbvLook-up table, containing population counts for all possible 8-bit value, from 0 to 255. Note that we do not "hard-code" the values, but merely use the slow version to compute them.sbvStates the correctness of faster population-count algorithm, with respect to the reference slow version. Turns out Z3's default solver is rather slow for this one, but there's a magic incantation to make it go fast. See  )http://github.com/Z3Prover/z3/issues/1150 for details.let cmd = "(check-sat-using (then (using-params ackermannize_bv :div0_ackermann_limit 1000000) simplify bit-blast sat))"0proveWith z3{satCmd = cmd} fastPopCountIsCorrectQ.E.D.sbvNot only we can prove that faster version is correct, but we can also automatically generate C code to compute population-counts for us. This action will generate all the C files that you will need, including a driver program for test purposes.'Below is the generated header file for :genPopCountInC%== BEGIN: "Makefile" ================# Makefile for popCount. Automatically generated by SBV. Do not edit!=# include any user-defined .mk file in the current directory. -include *.mkCC?=gcc0CCFLAGS?=-Wall -O3 -DNDEBUG -fomit-frame-pointerall: popCount_driver!popCount.o: popCount.c popCount.h ${CC} ${CCFLAGS} -c $< -o $@$popCount_driver.o: popCount_driver.c ${CC} ${CCFLAGS} -c $< -o $@-popCount_driver: popCount.o popCount_driver.o ${CC} ${CCFLAGS} $^ -o $@clean: rm -f *.overyclean: clean rm -f popCount_driver%== END: "Makefile" =================='== BEGIN: "popCount.h" ================/* Header file for popCount. Automatically generated by SBV. Do not edit! */%#ifndef __popCount__HEADER_INCLUDED__%#define __popCount__HEADER_INCLUDED__#include #include #include #include #include #include #include /* The boolean type */typedef bool SBool;/* The float type */typedef float SFloat;/* The double type */typedef double SDouble;/* Unsigned bit-vectors */typedef uint8_t SWord8;typedef uint16_t SWord16;typedef uint32_t SWord32;typedef uint64_t SWord64;/* Signed bit-vectors */typedef int8_t SInt8;typedef int16_t SInt16;typedef int32_t SInt32;typedef int64_t SInt64;/* Entry point prototype: */!SWord8 popCount(const SWord64 x);*#endif /* __popCount__HEADER_INCLUDED__ */'== END: "popCount.h" ==================.== BEGIN: "popCount_driver.c" ================*/* Example driver program for popCount. */:/* Automatically generated by SBV. Edit as you see fit! */#include #include "popCount.h"int main(void){: const SWord8 __result = popCount(0x1b02e143e4f0e0e5ULL); printf("popCount(0x1b02e143e4f0e0e5ULL) = %"PRIu8"\n", __result); return 0;}.== END: "popCount_driver.c" =================='== BEGIN: "popCount.c" ================/* File: "popCount.c". Automatically generated by SBV. Do not edit! */#include "popCount.h" SWord8 popCount(const SWord64 x){ const SWord64 s0 = x;" static const SWord8 table0[] = { 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 4, 5,. 5, 6, 5, 6, 6, 7, 5, 6, 6, 7, 6, 7, 7, 8 };1 const SWord64 s11 = s0 & 0x00000000000000ffULL;" const SWord8 s12 = table0[s11]; const SWord64 s14 = s0 >> 8;2 const SWord64 s15 = 0x00000000000000ffULL & s14;" const SWord8 s16 = table0[s15]; const SWord8 s17 = s12 + s16; const SWord64 s18 = s14 >> 8;2 const SWord64 s19 = 0x00000000000000ffULL & s18;" const SWord8 s20 = table0[s19]; const SWord8 s21 = s17 + s20; const SWord64 s22 = s18 >> 8;2 const SWord64 s23 = 0x00000000000000ffULL & s22;" const SWord8 s24 = table0[s23]; const SWord8 s25 = s21 + s24; const SWord64 s26 = s22 >> 8;2 const SWord64 s27 = 0x00000000000000ffULL & s26;" const SWord8 s28 = table0[s27]; const SWord8 s29 = s25 + s28; const SWord64 s30 = s26 >> 8;2 const SWord64 s31 = 0x00000000000000ffULL & s30;" const SWord8 s32 = table0[s31]; const SWord8 s33 = s29 + s32; const SWord64 s34 = s30 >> 8;2 const SWord64 s35 = 0x00000000000000ffULL & s34;" const SWord8 s36 = table0[s35]; const SWord8 s37 = s33 + s36; const SWord64 s38 = s34 >> 8;2 const SWord64 s39 = 0x00000000000000ffULL & s38;" const SWord8 s40 = table0[s39]; const SWord8 s41 = s37 + s40; return s41;}'== END: "popCount.c" ==================((c) Levent ErkokBSD3erkokl@gmail.com experimentalNone =sbvA definition of shiftLeft that can deal with variable length shifts. (Note that the   method from the ! class requires an  shift amount.) Unfortunately, this'll generate rather clumsy C code due to the use of tables etc., so we uninterpret it for code generation purposes using the  function.sbvTest function that uses shiftLeft defined above. When used as a normal Haskell function or in verification the definition is fully used, i.e., no uninterpretation happens. To wit, we have:tstShiftLeft 3 4 5224 :: SWord32,prove $ \x y -> tstShiftLeft x y 0 .== x + yQ.E.D.sbvGenerate C code for "tstShiftLeft". In this case, SBV will *use* the user given definition verbatim, instead of generating code for it. (Also see the functions  ,  , and  .))(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone1L9sbvThe key schedule. AES executes in rounds, and it treats first and last round keys slightly differently than the middle ones. We reflect that choice by being explicit about it in our type. The length of the middle list of keys depends on the key-size, which in turn determines the number of rounds.sbvThe key, which can be 128, 192, or 256 bits. Represented as a sequence of 32-bit words.sbvAES state. The state consists of four 32-bit words, each of which is in turn treated as four GF28's, i.e., 4 bytes. The T-Box implementation keeps the four-bytes together for efficient representation.sbvAn element of the Galois Field 2^8, which are essentially polynomials with maximum degree 7. They are conveniently represented as values between 0 and 255.sbvMultiplication in GF(2^8). This is simple polynomial multiplication, followed by the irreducible polynomial x^8+x^4+x^3+x^1+1. We simply use the  / function exported by SBV to do the operation.sbvExponentiation by a constant in GF(2^8). The implementation uses the usual square-and-multiply trick to speed up the computation.sbvComputing inverses in GF(2^8). By the mathematical properties of GF(2^8) and the particular irreducible polynomial used x^8+x^5+x^3+x^1+1, it turns out that raising to the 254 power gives us the multiplicative inverse. Of course, we can prove this using SBV::prove $ \x -> x ./= 0 .=> x `gf28Mult` gf28Inverse x .== 1Q.E.D.Note that we exclude 0? in our theorem, as it does not have a multiplicative inverse.sbv4Rotating a state row by a fixed amount to the right.sbvDefinition of round-constants, as specified in Section 5.2 of the AES standard.sbvThe  InvMixColumns transformation, as described in Section 5.3.3 of the standard. Note that this transformation is only used explicitly during key-expansion in the T-Box implementation of AES.sbvKey expansion. Starting with the given key, returns an infinite sequence of words, as described by the AES standard, Section 5.2, Figure 11.sbvThe values of the AES S-box table. Note that we describe the S-box programmatically using the mathematical construction given in Section 5.1.1 of the standard. However, the code-generation will turn this into a mere look-up table, as it is just a constant table, all computation being done at "compile-time".sbvThe sbox transformation. We simply select from the sbox table. Note that we are obliged to give a default value (here 0) to be used if the index is out-of-bounds as required by SBV's  function. However, that will never happen since the table has all 256 elements in it.sbvThe values of the inverse S-box table. Again, the construction is programmatic.sbv!The inverse s-box transformation.sbvProve that the  and  are inverses. We have:prove sboxInverseCorrectQ.E.D.sbvAdding the round-key to the current state. We simply exploit the fact that addition is just xor in implementing this transformation.sbv.T-box table generation function for encryptionsbv&First look-up table used in encryptionsbv'Second look-up table used in encryptionsbv&Third look-up table used in encryptionsbv'Fourth look-up table used in encryptionsbv.T-box table generating function for decryptionsbv&First look-up table used in decryptionsbv'Second look-up table used in decryptionsbv&Third look-up table used in decryptionsbv'Fourth look-up table used in decryptionsbvGeneric round function. Given the function to perform one round, a key-schedule, and a starting state, it performs the AES rounds.sbvOne encryption round. The first argument indicates whether this is the final round or not, in which case the construction is slightly different.sbvOne decryption round. Similar to the encryption round, the first argument indicates whether this is the final round or not.sbvKey schedule. Given a 128, 192, or 256 bit key, expand it to get key-schedules for encryption and decryption. The key is given as a sequence of 32-bit words. (4 elements for 128-bits, 6 for 192, and 8 for 256.) Compare this function to  which can calculate the key-expansion for decryption on the fly, as opposed to calculating the forward key-expansion first.sbvBlock encryption. The first argument is the plain-text, which must have precisely 4 elements, for a total of 128-bits of input. The second argument is the key-schedule to be used, obtained by a call to . The output will always have 4 32-bit words, which is the cipher-text.sbv3Block decryption. The arguments are the same as in , except the first argument is the cipher-text and the output is the corresponding plain-text.sbvInverse key expansion. Starting from the final round key, unwinds key generation operation to construct keys for the previous rounds. Used in on-the-fly decryption.sbvAES inverse key schedule. Starting from the last-round key, construct the sequence of keys that can be used for doing on-the-fly decryption. Compare this function to  which returns both encryption and decryption schedules: In this case, we don't calculate the encryption sequence, hence we can fuse this function with the decryption operation.sbvBlock decryption, starting from the unwound key. That is, start from the final key. Also; we don't use the T-box implementation. Just pure AES inverse cipher.sbv"Common plain text for test vectorssbvKey for 128-bit encryption testsbvKey for 192-bit encryption testsbvKey for 256-bit encryption testsbv+Expected cipher-text for 128-bit encryptionsbv+Expected cipher-text for 192-bit encryptionsbv+Expected cipher-text for 256-bit encryptionsbvCalculate the 128-bit final-round key from on-the-fly decryption key schedulesbvCalculate the 192-bit final-round key from on-the-fly decryption key schedulesbvCalculate the 192-bit final-round key from on-the-fly decryption key schedule. Compare this to : Typically we just need the final 6-blocks, but it is advantageous to have the entire last 8-blocks even for 192-bit keys. That is, e store the final 256-bits of key-expansion for speed purposes for both 192 and 256 bit versions. (But only the final 128 bits for the 128-bit version.)sbvCalculate the 256-bit final-round key from on-the-fly decryption key schedulesbvExtract the final key for on-the-fly decryption. This will extract exactly the number of blocks we need.sbvExtract the extended key for on-the-fly decryption. This will extract 4-blocks for 128-bit decryption, but 256 bit for both 192 and 256-bit variantssbv?128-bit encryption test, from Appendix C.1 of the AES standard:map hex8 t128Enc-["69c4e0d8","6a7b0430","d8cdb780","70b4c55a"]sbv?128-bit decryption test, from Appendix C.1 of the AES standard:map hex8 t128Dec-["00112233","44556677","8899aabb","ccddeeff"]sbv?192-bit encryption test, from Appendix C.2 of the AES standard:map hex8 t192Enc-["dda97ca4","864cdfe0","6eaf70a0","ec0d7191"]sbv?192-bit decryption test, from Appendix C.2 of the AES standard:map hex8 t192Dec-["00112233","44556677","8899aabb","ccddeeff"]sbv:256-bit encryption, from Appendix C.3 of the AES standard:map hex8 t256Enc-["8ea2b7ca","516745bf","eafc4990","4b496089"]sbv:256-bit decryption, from Appendix C.3 of the AES standard:map hex8 t256Dec-["00112233","44556677","8899aabb","ccddeeff"]sbv1Various tests for round-trip properties. We have:runAESTests False GOOD: Key generation AES128GOOD: Key generation AES192GOOD: Key generation AES256GOOD: Encryption AES128GOOD: Decryption AES128GOOD: Decryption-OTF AES128GOOD: Encryption AES192GOOD: Decryption AES192GOOD: Decryption-OTF AES192GOOD: Encryption AES256GOOD: Decryption AES256GOOD: Decryption-OTF AES256sbv;Correctness theorem for 128-bit AES. Ideally, we would run:  prove aes128IsCorrect to get a proof automatically. Unfortunately, while SBV will successfully generate the proof obligation for this theorem and ship it to the SMT solver, it would be naive to expect the SMT-solver to finish that proof in any reasonable time with the currently available SMT solving technologies. Instead, we can issue:  quickCheck aes128IsCorrect and get some degree of confidence in our code. Similar predicates can be easily constructed for 192, and 256 bit cases as well.sbv128-bit encryption, that takes 128-bit values, instead of chunks. We have:hex8 $ aes128Enc 0x000102030405060708090a0b0c0d0e0f 0x00112233445566778899aabbccddeeff""69c4e0d86a7b0430d8cdb78070b4c55a"You can also render this function as a stand-alone function using: . sbv2smt (smtFunction "aes128Enc" aes128Enc) sbv+Code generation for 128-bit AES encryption.The following sample from the generated code-lines show how T-Boxes are rendered as C arrays:  static const SWord32 table1[] = { 0xc66363a5UL, 0xf87c7c84UL, 0xee777799UL, 0xf67b7b8dUL, 0xfff2f20dUL, 0xd66b6bbdUL, 0xde6f6fb1UL, 0x91c5c554UL, 0x60303050UL, 0x02010103UL, 0xce6767a9UL, 0x562b2b7dUL, 0xe7fefe19UL, 0xb5d7d762UL, 0x4dababe6UL, 0xec76769aUL, ... } The generated program has 5 tables (one sbox table, and 4-Tboxes), all converted to fast C arrays. Here is a sample of the generated straightline C-code:  const SWord8 s1915 = (SWord8) s1912; const SWord8 s1916 = table0[s1915]; const SWord16 s1917 = (((SWord16) s1914) << 8) | ((SWord16) s1916); const SWord32 s1918 = (((SWord32) s1911) << 16) | ((SWord32) s1917); const SWord32 s1919 = s1844 ^ s1918; const SWord32 s1920 = s1903 ^ s1919; The GNU C-compiler does a fine job of optimizing this straightline code to generate a fairly efficient C implementation.sbvComponents of the AES implementation that the library is generated from. For each case, we provide the driver values from the AES test-vectors.sbv8Generate code for AES functionality; given the key size.sbvGenerate a C library, containing functions for performing 128-bit encdeckey-expansion. A note on performance: In a very rough speed test, the generated code was able to do 6.3 million block encryptions per second on a decent MacBook Pro. On the same machine, OpenSSL reports 8.2 million block encryptions per second. So, the generated code is about 25% slower as compared to the highly optimized OpenSSL implementation. (Note that the speed test was done somewhat simplistically, so these numbers should be considered very rough estimates.)sbvFor doctest purposes onlysbvChunk in groups of 4. (This function must be in some standard library, where?)sbvplain-text wordssbv key-wordssbv+True if round-trip gives us plain-text back*(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone1T sbvA nibble is 4-bits. Ideally, we would like to represent a nibble by SWord 4; and indeed SBV can do that for verification purposes just fine. Unfortunately, the SBV's C compiler doesn't support 4-bit bit-vectors, as there's nothing meaningful in the C-land that we can map it to. Thus, we represent a nibble with 8-bits. The top 4 bits will always be 0.sbv$Cypher text is another 64-bit block.sbvKey is again a 64-bit block.sbvPlantext is simply a block.sbvSection 2: Prince is essentially a 64-bit cipher, with 128-bit key, coming in two parts.sbv.Expanding a key, from Section 3.4 of the spec.sbv0expandKey(x) = x has a unique solution. We have:prop_ExpandKeyQ.E.D.sbvSection 2: Encryptionsbv DecryptionsbvBasic prince algorithmsbvCore prince. It's essentially folding of 12 rounds stitched together:sbvForward round.sbvBackend round.sbvM transformation.sbv Inverse of M.sbvSR.sbvInverse of SR:sbv)Prove sr and srInv are inverses: We have: prove prop_srQ.E.D.sbvM' transformationsbv&The matrix as described in Section 3.3sbvMultiplication.sbv$Non-linear transformation of a blocksbvSBox transformation.sbvInverse SBox transformation.sbv2Prove that sbox and sBoxInv are inverses: We have:prove prop_SBoxQ.E.D.sbvRound constantssbvRound-constants property: rc_i ! rc_{11-i} is constant. We have:prop_RoundKeysTruesbv Convert a 64 bit word to nibblessbv%Convert from nibbles to a 64 bit wordsbv%From Appendix A of the spec. We have: testVectorsTruesbvNicely show a concrete block.sbv+Generating C code for the encryption block.  +(c) Austin SeippBSD3erkokl@gmail.com experimentalNone^I sbv:Represents the current state of the RC4 stream: it is the S array along with the i and j index values used by the PRGA.sbvThe key is a stream of  values.sbvRC4 State contains 256 8-bit values. We use the symbolically accessible full-binary type   to represent the state, since RC4 needs access to the array via a symbolic index and it's important to minimize access time.sbvConstruct the fully balanced initial tree, where the leaves are simply the numbers 0 through 255.sbv$Swaps two elements in the RC4 array.sbvImplements the PRGA used in RC4. We return the new state and the next key value generated.sbvConstructs the state to be used by the PRGA using the given key.sbvThe key-schedule. Note that this function returns an infinite list.sbv0Generate a key-schedule from a given key-string.sbvRC4 encryption. We generate key-words and xor it with the input. The following test-vectors are from Wikipedia  http://en.wikipedia.org/wiki/RC4:*concatMap hex2 $ encrypt "Key" "Plaintext""bbf316e8d940af0ad3"'concatMap hex2 $ encrypt "Wiki" "pedia" "1021bf0420"2concatMap hex2 $ encrypt "Secret" "Attack at dawn""45a01f645fc35b383552544b9bf5"sbvRC4 decryption. Essentially the same as decryption. For the above test vectors we have:decrypt "Key" [0xbb, 0xf3, 0x16, 0xe8, 0xd9, 0x40, 0xaf, 0x0a, 0xd3] "Plaintext"-decrypt "Wiki" [0x10, 0x21, 0xbf, 0x04, 0x20]"pedia"decrypt "Secret" [0x45, 0xa0, 0x1f, 0x64, 0x5f, 0xc3, 0x5b, 0x38, 0x35, 0x52, 0x54, 0x4b, 0x9b, 0xf5]"Attack at dawn"sbvProve that round-trip encryption/decryption leaves the plain-text unchanged. The theorem is stated parametrically over key and plain-text sizes. The expression performs the proof for a 40-bit key (5 bytes) and 40-bit plaintext (again 5 bytes).Note that this theorem is trivial to prove, since it is essentially establishing xor'in the same value twice leaves a word unchanged (i.e., x ! y ! y = x). However, the proof takes quite a while to complete, as it gives rise to a fairly large symbolic trace.sbvFor doctest purposes only  ,(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone'1i%sbv5 is a synonym for lists, but makes the intent clear.sbvParameterized SHA representation, that captures all the differences between variants of the algorithm. w is the word-size type.sbvSection 6.2.2, 6.4.2: How many iterations are there in the inner loopsbv(Section 5.3.2-6 : Initial hash valuesbv)Section 4.2.2-3 : Magic SHA constantssbv9Section 4.1.2-3 : Coefficients of the sigma1 functionsbv9Section 4.1.2-3 : Coefficients of the sigma0 functionsbv7Section 4.1.2-3 : Coefficients of the Sum1 functionsbv7Section 4.1.2-3 : Coefficients of the Sum0 functionsbv-Section 1 : Block size for messagessbv/Section 1 : Word size we operate withsbvThe choose function.sbvThe majority function.sbvThe sum-0 function. We parameterize over the rotation amounts as different variants of SHA use different rotation amounts.sbv)The sum-1 function. Again, parameterized.sbv#The sigma0 function. Parameterized.sbv#The sigma1 function. Parameterized.sbvParameterization for SHA224.sbv9Parameterization for SHA256. Inherits mostly from SHA224.sbvParameterization for SHA384.sbv9Parameterization for SHA512. Inherits mostly from SHA384.sbv 2^64 (or 2^128), and you'd run out of memory first!sbvHash one block of message, starting from a previous hash. This function corresponds to body of the for-loop in the spec. This function always produces a list of length 8, corresponding to the final 8 values of the H.sbvCompute the hash of a given string using the specified parameterized hash algorithm.sbvSHA224 digest.sbvSHA256 digest.sbvSHA384 digest.sbvSHA512 digest.sbvSHA512_224 digest.sbvSHA512_256 digest.sbvCollection of known answer tests for SHA. Since these tests take too long during regular regression runs, we pass as an argument how many to run. Increase the below number to 24 to run all tests. We have:knownAnswerTests 1TruesbvGenerate code for one block of SHA256 in action, starting from an arbitrary hash value.sbvGenerate code for one block of SHA512 in action, starting from an arbitrary hash value.sbvHelper for chunking a list by given lengths and combining each chunk with a functionsbv'Nicely lay out a hash value as a string''-(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonek4sbv)Encode the delta-sat problem as given in  http://dreal.github.io/ We have:flyspeck Unsatisfiable.(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone1x0sbvFor a homogeneous problem, the solution is any linear combination of the resulting vectors. For a non-homogeneous problem, the solution is any linear combination of the vectors in the second component plus one of the vectors in the first component.sbvldn: Solve a (L)inear (D)iophantine equation, returning minimal solutions over (N)aturals. The input is given as a rows of equations, with rhs values separated into a tuple. The first argument must be a proxy of a natural, must be total number of columns in the system. (i.e., #of variables + 1). The second parameter limits the search to bound: In case there are too many solutions, you might want to limit your search space.sbvFind the basis solution. By definition, the basis has all non-trivial (i.e., non-0) solutions that cannot be written as the sum of two other solutions. We use the mathematically equivalent statement that a solution is in the basis if it's least according to the natural partial order using the ordinary less-than relation.sbvSolve the equation: 2x + y - z = 2We have:test(k, 2+k', 2k+k')(1+k, k', 2k+k')That is, for arbitrary k and k', we have two different solutions. (An infinite family.) You can verify these solutuions by substituting the values for x, y and z in the above, for each choice. It's harder to see that they cover all possibilities, but a moments thought reveals that is indeed the case.sbvA puzzle: Five sailors and a monkey escape from a naufrage and reach an island with coconuts. Before dawn, they gather a few of them and decide to sleep first and share the next day. At night, however, one of them awakes, counts the nuts, makes five parts, gives the remaining nut to the monkey, saves his share away, and sleeps. All other sailors do the same, one by one. When they all wake up in the morning, they again make 5 shares, and give the last remaining nut to the monkey. How many nuts were there at the beginning?7We can model this as a series of diophantine equations:  x_0 = 5 x_1 + 1 4 x_1 = 5 x_2 + 1 4 x_2 = 5 x_3 + 1 4 x_3 = 5 x_4 + 1 4 x_4 = 5 x_5 + 1 4 x_5 = 5 x_6 + 1 We need to solve for x_0, over the naturals. If you run this program, z3 takes its time (quite long!) but, it eventually computes: [15621,3124,2499,1999,1599,1279,1023] as the answer.That is:  * There was a total of 15621 coconuts * 1st sailor: 15621 = 3124*5+1, leaving 15621-3124-1 = 12496 * 2nd sailor: 12496 = 2499*5+1, leaving 12496-2499-1 = 9996 * 3rd sailor: 9996 = 1999*5+1, leaving 9996-1999-1 = 7996 * 4th sailor: 7996 = 1599*5+1, leaving 7996-1599-1 = 6396 * 5th sailor: 6396 = 1279*5+1, leaving 6396-1279-1 = 5116 * In the morning, they had: 5116 = 1023*5+1. Note that this is the minimum solution, that is, we are guaranteed that there's no solution with less number of coconuts. In fact, any member of [15625*k-4 | k <- [1..]] is a solution, i.e., so are 31246, 46871, 62496, 78121, etc.Note that we iteratively deepen our search by requesting increasing number of solutions to avoid the all-sat pitfall./(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone#67> sbv,Each agent can be in one of the three statessbv Regular worksbv!Intention to enter critical statesbvIn the critical statesbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvA bounded mutex property holds for two sequences of state transitions, if they are not in their critical section at the same time up to that given bound.sbv1A sequence is valid upto a bound if it starts at ', and follows the mutex rules. That is:From  it can switch to  or stay From  it can switch to  if it's its turnFrom  it can either stay in  or go back to  The variable me identifies the agent id.sbvThe mutex algorithm, coded implicitly as an assignment to turns. Turns start at 1, and at each stage is either 1 or 2; giving preference to that process. The only condition is that if either process is in its critical section, then the turn value stays the same. Note that this is sufficient to satisfy safety (i.e., mutual exclusion), though it does not guarantee liveness.sbv1Check that we have the mutex property so long as  and  holds; i.e., so long as both the agents and the arbiter act according to the rules. The check is bounded up-to-the given concrete bound; so this is an example of a bounded-model-checking style proof. We have: checkMutex 20 All is good!sbvOur algorithm is correct, but it is not fair. It does not guarantee that a process that wants to enter its critical-section will always do so eventually. Demonstrate this by trying to show a bounded trace of length 10, such that the second process is ready but never transitions to critical. We have: ghci> notFair 10 Fairness is violated at bound: 10 P1: [Idle,Idle,Ready,Critical,Idle,Idle,Ready,Critical,Idle,Idle] P2: [Idle,Ready,Ready,Ready,Ready,Ready,Ready,Ready,Ready,Ready] Ts: [1,2,1,1,1,1,1,1,1,1]As expected, P2 gets ready but never goes critical since the arbiter keeps picking P1 unfairly. (You might get a different trace depending on what z3 happens to produce!)$Exercise for the reader: Change the  function so that it alternates the turns from the previous value if neither process is in critical. Show that this makes the  function below no longer exhibits the issue. Is this sufficient? Concurrent programming is tricky!  0(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneKsbvA deck is simply a list of integers. Note that a regular deck will have distinct cards, we do not impose this in our proof. That is, the proof works regardless whether we put duplicates into the deck, which generalizes the theorem.sbv$Count-out-and-transfer (COAT): Take k cards from top, reverse it, and put it at the bottom of a deck.sbv COAT 4 times.sbv4Key property of COATing. If you take a deck of size n, and COAT it 4 times, then the deck remains in the same order. The COAT factor, k6, must be greater than half the size of the deck size.6Note that the proof time increases significantly with n.. Here's a proof for deck size of 6, for all k >= 3. coatCheck 6Q.E.D.It's interesting to note that one can also express this theorem by making n symbolic as well. However, doing so definitely requires an inductive proof, and the SMT-solver doesn't handle this case out-of-the-box, running forever.1(c) Joel BurgetBSD3erkokl@gmail.com experimentalNone{sbv3Compute a prefix of the fibonacci numbers. We have: mkFibs 10[1,1,2,3,5,8,13,21,34,55]sbvGenerate fibonacci numbers as a sequence. Note that we constrain only the first 200 entries.2(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone"sbv+A simple predicate, based on two variables x and y , true when  0 <= x <= 1 and  x - abs y is 0.sbv=Generate all satisfying assignments for our problem. We have: allModels Solution #1: x = 1 :: Integer y = -1 :: Integer Solution #2: x = 1 :: Integer y = 1 :: Integer Solution #3: x = 0 :: Integer y = 0 :: IntegerFound 3 different solutions.Note that solutions 2 and 3 share the value x = 1&, since there are multiple values of y% that make this particular choice of x satisfy our constraint.sbvGenerate all satisfying assignments, but we first tell SBV that y should not be considered as a model problem, i.e., it's auxiliary. We have:modelsWithYAux Solution #1: x = 1 :: Integer Solution #2: x = 0 :: IntegerFound 2 different solutions.Note that we now have only two solutions, one for each unique value of x that satisfy our constraint.3(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone#sbvAdd one to an argumentsbvReverse run the add1 function. Note that the generated SMTLib will have the function add1 itself defined. You can verify this by running the below in verbose mode. add1ExampleSatisfiable. Model: x = 4 :: IntegersbvSum of numbers from 0 to the given number. Since this is a recursive definition, we cannot simply symbolically simulate it as it wouldn't terminat. So, we use the function generation facilities to define it directly in SMTLib. Note how the function itself takes a "recursive version" of itself, and all recursive calls are made with this name.sbv$Prove that sumToN works as expected.We have: sumToNExampleSatisfiable. Model: s0 = 5 :: Integer s1 = 15 :: IntegersbvCoding list-length recursively. Again, we map directly to an SMTLib function.sbv:Calculate the length of a list, using recursive functions.We have: lenExampleSatisfiable. Model: s0 = [1,2,3] :: [Integer] s1 = 3 :: IntegersbvA simple mutual-recursion example, from the z3 documentation. We have:pingPongSatisfiable. Model: s0 = 1 :: IntegersbvUsual way to define even-odd mutually recursively. Unfortunately, while this goes through, the backend solver does not terminate on this example. See  for an alternative technique to handle such definitions, which seems to be more solver friendly.sbvAnother technique to handle mutually definitions is to define the functions together, and pull the results out individually. This usually works better than defining the functions separately, from a solver perspective.sbv+Extract the isEven function for easier use.sbv*Extract the isOdd function for easier use.sbv7We can prove 20 is even and definitely not odd, thusly:evenOdd2Satisfiable. Model: s0 = 20 :: Integer s1 = True :: Bool s2 = False :: Boolsbv3Ackermann function, demonstrating nested recursion.sbv8We can prove constant-folding instances of the equality ack 1 y == y + 2:ack1ySatisfiable. Model: s0 = 5 :: Integer s1 = 7 :: Integer>Expecting the prover to handle the general case for arbitrary y is beyond the current scope of what SMT solvers do out-of-the-box for the time being.4(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone67> sbvA simple enumerated type, that we'd like to translate to SMT-Lib intact; i.e., this type will not be uninterpreted but rather preserved and will be just like any other symbolic type SBV provides.-Also note that we need to have the following LANGUAGE options defined: TemplateHaskell, StandaloneDeriving, DeriveDataTypeable, DeriveAnyClass for this to work.sbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvHave the SMT solver enumerate the elements of the domain. We have:elts Solution #1: s0 = C :: E Solution #2: s0 = B :: E Solution #3: s0 = A :: EFound 3 different solutions.sbv9Shows that if we require 4 distinct elements of the type , we shall fail; as the domain only has three elements. We have:four UnsatisfiablesbvEnumerations are automatically ordered, so we can ask for the maximum element. Note the use of quantification. We have:maxESatisfiable. Model: maxE = C :: Esbv/Similarly, we get the minimum element. We have:minESatisfiable. Model: minE = A :: E  5(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone 167>sbv/An uninterpreted sort for demo purposes, named sbv/An uninterpreted sort for demo purposes, named sbvSymbolic version of the type .sbv,An enumerated type for demo purposes, named sbvSymbolic version of the type .sbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvHelper to turn quantified formula to a regular boolean. We can think of this as quantifier elimination, hence the name .sbvConsider the formula \forall x\,\exists y\, x \ge y;, over bit-vectors of size 8. We can ask SBV to satisfy it: sat skolemEx1 SatisfiableBut this isn't really illimunating. We can first skolemize, and then ask to satisfy:sat $ skolemize skolemEx1Satisfiable. Model: y :: Word8 -> Word8 y x = xwhich is much better We are told that we can have the witness as the value given for each choice of x.sbvConsider the formula ;\forall a\,\exists b\,\forall c\,\exists d\, a + b >= c + d;, over bit-vectors of size 8. We can ask SBV to satisfy it: sat skolemEx2 SatisfiableAgain, we're left in the dark as to why this is satisfiable. Let's skolemize first, and then call   on it:sat $ skolemize skolemEx2Satisfiable. Model: b :: Word8 -> Word8 b _ = 0 d :: Word8 -> Word8 -> Word8 d a c = a + 255 * cLet's see what the solver said. It suggested we should use the value of 0 for b, regardless of the choice of a . (Note how b) is a function of one variable, i.e., of a) And it suggested using  a + (255 * c) for d, for whatever we choose for a and c-. Why does this work? Well, given arbitrary a and c, we end up with:  a + b >= c + d --> substitute b = 0 and d = a + 255c as suggested by the solver a + 0 >= c + a + 255c a >= 256c + a a >= a showing the formula is satisfiable for whatever values you pick for a and c . Note that 256 is simply 0 when interpreted modulo 2^8 . Clever!sbvA common proof technique to show validity is to show that the negation is unsatisfiable. Note that if you want to skolemize during this process, you should first negate and then skolemize!4This example demonstrates the possible pitfall. The  function encodes \exists x\, \forall y\, y \ge x9 for 8-bit bitvectors; which is a valid statement since x = 08 acts as the witness. We can directly prove this in SBV:prove skolemEx3Q.E.D.0Or, we can ask if the negation is unsatisfiable:sat (qNot skolemEx3) Unsatisfiable5If we want, we can skolemize after the negation step: sat (skolemize (qNot skolemEx3)) Unsatisfiable.and get the same result. However, it would be unsound$ to skolemize first and then negate: sat (qNot (skolemize skolemEx3))Satisfiable. Model: x = 1 :: Word8And that would be the incorrect conclusion that our formula is invalid with a counter-example! You can see the same by doing:prove (skolemize skolemEx3)Falsifiable. Counter-example: x = 1 :: Word8So, if you want to check validity and want to also perform skolemization; you should negate your formula first and then skolemize, not the other way around!sbvIf you skolemize different formulas that share the same name for their existentials, then SBV will get confused and will think those represent the same skolem function. This is unfortunate, but it follows the requirement that uninterpreted function names should be unique. In this particular case, however, since SBV creates these functions, it is harder to control the internal names. In such cases, use the function  to provide a name to prefix the skolem functions. As demonstrated by  . We get: skolemEx4Satisfiable. Model: c1_y :: Integer -> Integer c1_y x = x c2_y :: Integer -> Integer c2_y x = x + 1Note how the internal skolem functions are named according to the tag given. If you use regular  this program will essentially do the wrong thing by assuming the skolem functions for both predicates are the same, and will return unsat. Beware! All skolem functions should be named differently in your program for your deductions to be sound.sbv/Demonstrates creating a partial order. We have: poExampleQ.E.D.sbvCreate a transitive relation of a simple relation and show that transitive connections are respected. We have: tcExample1Q.E.D.sbvAnother transitive-closure example, this time we show the transitive closure is the smallest relation, i.e., doesn't have extra connections. We have: tcExample2Q.E.D.sbvDemonstrates computing the transitive closure of existing relations. We have: tcExample3Q.E.D.6(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone1ŨsbvProve that floating point addition is not associative. For illustration purposes, we will require one of the inputs to be a NaN . We have:prove $ assocPlus (0/0)Falsifiable. Counter-example: s0 = 0.0 :: Float s1 = 0.0 :: FloatIndeed:let i = 0/0 :: Floati + (0.0 + 0.0)NaN((i + 0.0) + 0.0)NaNBut keep in mind that NaN< does not equal itself in the floating point world! We have:$let nan = 0/0 :: Float in nan == nanFalsesbv:Prove that addition is not associative, even if we ignore NaN/Infinity+ values. To do this, we use the predicate  -, which is true of a floating point number ( or ) if it is neither NaN nor Infinity. (That is, it's a representable point in the real-number line.)We have:assocPlusRegularFalsifiable. Counter-example: x = -1.7478492e-21 :: Float y = 9.693523e-27 :: Float z = 5.595795e-20 :: FloatIndeed, we have:let x = -1.7478492e-21 :: Floatlet y = 9.693523e-27 :: Floatlet z = 5.595795e-20 :: Float x + (y + z) 5.4210105e-20 (x + y) + z 5.421011e-20#Note the difference in the results!sbvDemonstrate that a+b = a does not necessarily mean b is 0 in the floating point world, even when we disallow the obvious solution when a and b are  Infinity. We have:nonZeroAdditionFalsifiable. Counter-example: a = 7.135861e-8 :: Float b = 8.57579e-39 :: FloatIndeed, we have:let a = 7.135861e-8 :: Floatlet b = 8.57579e-39 :: Float a + b == aTrueb == 0FalsesbvThis example illustrates that  a * (1/a) does not necessarily equal 1). Again, we protect against division by 0 and NaN/Infinity.We have: multInverseFalsifiable. Counter-example: a = -1.4991268e38 :: FloatIndeed, we have:let a = -1.4991268e38 :: Float a * (1/a) 0.99999994sbvOne interesting aspect of floating-point is that the chosen rounding-mode can effect the results of a computation if the exact result cannot be precisely represented. SBV exports the functions  ,  ,  ,  ,   and  2 which allows users to specify the IEEE supported b for the operation. This example illustrates how SBV can be used to find rounding-modes where, for instance, addition can produce different results. We have: roundingAddSatisfiable. Model:& rm = RoundTowardZero :: RoundingMode x = 1.7499695 :: Float y = 1.2539366 :: Float(Note that depending on your version of Z3, you might get a different result.) Unfortunately Haskell floats do not allow computation with arbitrary rounding modes, but SBV's  type does. We have:7fpAdd sRoundTowardZero 1.7499695 1.2539366 :: SFPSingle!3.00390601 :: SFloatingPoint 8 24>fpAdd sRoundNearestTiesToEven 1.7499695 1.2539366 :: SFPSingle!3.00390625 :: SFloatingPoint 8 24;We can see why these two results are indeed different: The g (which rounds towards the origin) produces a smaller result, closer to 0. Indeed, if we treat these numbers as  values, we get:1.7499695 + 1.2539366 :: Double 3.0039061?we see that the "more precise" result is smaller than what the  value is, justifying the smaller value with 'RoundTowardZero. A more detailed study is beyond our current scope, so we'll merely note that floating point representation and semantics is indeed a thorny subject, and point to  http://ece.uwaterloo.ca/~dwharder/NumericalAnalysis/02Numerics/Double/paper.pdf as an excellent guide.sbvArbitrary precision floating-point numbers. SBV can talk about floating point numbers with arbitrary exponent and significand sizes as well. Here is a simple example demonstrating the minimum non-zero positive and maximum floating point values with exponent width 5 and significand width 4, which is actually 3 bits for the significand explicitly stored, includes the hidden bit. We have: fp54BoundsObjective "max": Optimal model:" x = 61400 :: FloatingPoint 5 4 max = 503 :: WordN 9 min = 503 :: WordN 9Objective "min": Optimal model:' x = 0.00000763 :: FloatingPoint 5 4 max = 257 :: WordN 9 min = 257 :: WordN 90The careful reader will notice that the numbers 61400 and  0.00000763 are quite suspicious, but the metric space equivalents are correct. The reason for this is due to the sparcity of floats. The "computed" value of the maximum in this bound is actually 61440 , however in FloatingPoint 5 4% representation all numbers between 57344 and 61440 collapse to the same bit-pattern, and the pretty-printer picks a string representation in decimal that falls within range that it considers is the "simplest." (Printing floats precisely is a thorny subject!) Likewise, the minimum value we're looking for is actually 2^-17, but any number between 2^-16 and 2^-17 will map to this number. It turns out that 0.00000763 in decimal is one such value. Moral of the story is that when reading floating-point numbers in decimal notation one should be very careful about the printed representation and the numeric value; while they will match in vsalue (if there are no bugs!), they can print quite differently! (Also keep in mind the rounding modes that impact how the conversion is done.)7(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonesbvGiven an array, and bounds on it, initialize it within the bounds to the element given. Otherwise, leave it untouched.sbvProve a simple property: If we read from the initialized region, we get the initial value. We have: memsetExampleQ.E.D.sbvGet an example of reading a value out of range. The value returned should be out-of-range for lo/hi outOfInitSatisfiable. Model: Read = 1 :: Integer lo = 0 :: Integer hi = 0 :: Integer zero = 0 :: Integer idx = 1 :: Integer8(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneʟsbvA simple function to generate a new integer value, that is not in the given set of values. We also require the value to be non-negativesbvWe now use "outside" repeatedly to generate 10 integers, such that we not only disallow previously generated elements, but also any value that differs from previous solutions by less than 5. Here, we use the < function. We could have also extracted the dictionary via < and did fancier programming as well, as necessary. We have:genVals[45,40,35,30,25,20,15,10,5,0]9(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone˔sbvModel a nested array that is indexed by integers, and we store another integer to integer array in each index. We have: nestedArray(0,10):2(c) Curran McConnell Levent ErkokBSD3erkokl@gmail.com experimentalNoneG sbvSymbolic version of .sbvSimilarly, we can create another newtype, this time wrapping over . As an example, consider measuring the human height in centimetres? The tallest person in history, Robert Wadlow, was 272 cm. We don't need negative values, so , is the smallest type that suits our needs.sbvSymbolic version of .sbvA  is a newtype wrapper around .sbv5The tallest human ever was 272 cm. We can simply use # to lift it to the symbolic space.sbvGiven a distance between a floor and a ceiling, we can see whether the human can stand in that room. Comparison is expressed using  .sbvNow, suppose we want to see whether we could design a room with a ceiling high enough that any human could stand in it. We have: sat problemSatisfiable. Model:! floorToCeiling = 3 :: Integer humanheight = 272 :: Word16sbvThe  instance simply uses stock definitions. This is always possible for newtypes that simply wrap over an existing symbolic type.sbvTo use  symbolically, we associate it with the underlying symbolic type's kind.sbvSimilarly here, for the  instance.sbvSymbolic instance simply follows the underlying type, just like .  ;(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonesbvA simple variant of division, where we explicitly require the caller to make sure the divisor is not 0.sbv#Check whether an arbitrary call to 5 is safe. Clearly, we do not expect this to be safe:test1[./Documentation/SBV/Examples/Misc/NoDiv0.hs:38:14:checkedDiv: Divisor should not be 0: Violated. Model: s0 = 0 :: Int32 s1 = 0 :: Int32]sbvRepeat the test, except this time we explicitly protect against the bad case. We have:test2[./Documentation/SBV/Examples/Misc/NoDiv0.hs:46:41:checkedDiv: Divisor should not be 0: No violations detected]<(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone|sbvHelper synonym for representing GF(2^8); which are merely 8-bit unsigned words. Largest term in such a polynomial has degree 7.sbvMultiplication in Rijndael's field; usual polynomial multiplication followed by reduction by the irreducible polynomial. The irreducible used by Rijndael's field is the polynomial x^8 + x^4 + x^3 + x + 1 , which we write by giving it's  exponents in SBV. See:  http://en.wikipedia.org/wiki/Finite_field_arithmetic#Rijndael.27s_finite_field. Note that the irreducible itself is not in GF28! It has a degree of 8.NB. You can use the   function to print polynomials nicely, as a mathematician would write.sbv States that the unit polynomial 1, is the unit elementsbv)States that multiplication is commutativesbvStates that multiplication is associative, note that associativity proofs are notoriously hard for SAT/SMT solverssbvStates that the usual multiplication rule holds over GF(2^n) polynomials Checks:  if (a, b) = x   y then x = y   a + b being careful about y = 0. When divisor is 0, then quotient is defined to be 0 and the remainder is the numerator. (Note that addition is simply ! in GF(2^8).)sbvQueries=(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonesbvSymbolic version of #d1 x y = if y < x - 2 then 7 else 2sbvSymbolic version of #d2 x y = if y > 3 then 10 else 50sbvSymbolic version of (d3 x y = if y < -x + 3 then 100 else 200sbvSymbolic version of !d4 x y = d1 x y + d2 x y + d3 x ysbv5Compute all possible program paths. Note the call to  , which causes   to find models that generate differing values for the given expression. We have:paths Solution #1: x = -2 :: Integer y = 4 :: Integer r = 112 :: Integer Solution #2: x = 0 :: Integer y = 3 :: Integer r = 252 :: Integer Solution #3: x = -1 :: Integer y = 4 :: Integer r = 212 :: Integer Solution #4: x = 3 :: Integer y = 0 :: Integer r = 257 :: Integer Solution #5: x = 2 :: Integer y = -1 :: Integer r = 157 :: Integer Solution #6: x = 7 :: Integer y = 4 :: Integer r = 217 :: Integer Solution #7: x = 0 :: Integer y = 0 :: Integer r = 152 :: IntegerFound 7 different solutions.>(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone޽sbvAbbreviation for set of integers. For convenience only in monomorphising the properties.?(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone"<sbvCreate two strings, requiring one to be a particular value, constraining the other to be different than another constant string. But also add soft constraints to indicate our preferences for each of these variables. We get:exampleSatisfiable. Model:( x = "x-must-really-be-hello" :: String( y = "default-y-value" :: StringNote how the value of x is constrained properly and thus the default value doesn't kick in, but y takes the default value since it is acceptable by all the other hard constraints.@-(c) Joel Burget Levent ErkokBSD3erkokl@gmail.com experimentalNone"sbvA dictionary is a list of lookup values. Note that we store the type [(a, b)]8 as a symbolic value here, mixing sequences and tuples.sbvCreate a dictionary of length 5, such that each element has an string key and each value is the length of the key. We impose a few more constraints to make the output interesting. For instance, you might get:  ghci> example [("nt_",3),("dHAk",4),("kzkk0",5),("mZxs9s",6),("c32'dPM",7)] Depending on your version of z3, a different answer might be provided. Here, we check that it satisfies our length conditions: import Data.List (genericLength)example >>= \ex -> return (length ex == 5 && all (\(l, i) -> genericLength l == i) ex)TrueA(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone 67>sbvA simple enumerationsbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvIdentify weekend dayssbvUsing optimization, find the latest day that is not a weekend. We have: almostWeekendOptimal model: almostWeekend = Fri :: Day last-day = 4 :: Word8sbvUsing optimization, find the first day after the weekend. We have:weekendJustOverOptimal model: weekendJustOver = Mon :: Day first-day = 0 :: Word8sbv9Using optimization, find the first weekend day: We have: firstWeekendOptimal model: firstWeekend = Sat :: Day first-weekend = 5 :: Word8sbv0Make day an optimizable value, by mapping it to  in the most obvious way. We can map it to any value the underlying solver can optimize, but ( is the simplest and it'll fit the bill.B(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonesbvOptimization goals where min/max values might require assignments to values that are infinite (integer case), or infinite/epsilon (real case). This simple example demonstrates how SBV can be used to extract such values.We have:optimize Independent problem 1Objective "one-x": Optimal in an extension field: one-x = oo :: Integer min_y = 7.0 :: Real min_z = 5.0 :: Real1Objective "min_y": Optimal in an extension field: one-x = oo :: Integer min_y = 7.0 :: Real min_z = 5.0 :: Real1Objective "min_z": Optimal in an extension field: one-x = oo :: Integer min_y = 7.0 :: Real min_z = 5.0 :: RealC(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonensbv Taken from 6http://people.brunel.ac.uk/~mastjjb/jeb/or/morelp.htmlmaximize 5x1 + 6x2 subject to  x1 + x2 <= 10 x1 - x2 >= 35x1 + 4x2 <= 35x1 >= 0x2 >= 0optimize Lexicographic problemOptimal model: x1 = 47 % 9 :: Real x2 = 20 % 9 :: Real goal = 355 % 9 :: RealD(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneSsbv Taken from 6http://people.brunel.ac.uk/~mastjjb/jeb/or/morelp.htmlA company makes two products (X and Y) using two machines (A and B).Each unit of X that is produced requires 50 minutes processing time on machine A and 30 minutes processing time on machine B.Each unit of Y that is produced requires 24 minutes processing time on machine A and 33 minutes processing time on machine B.At the start of the current week there are 30 units of X and 90 units of Y in stock. Available processing time on machine A is forecast to be 40 hours and on machine B is forecast to be 35 hours.The demand for X in the current week is forecast to be 75 units and for Y is forecast to be 95 units.Company policy is to maximise the combined sum of the units of X and the units of Y in stock at the end of the week. 0 \Rightarrow inv (x, y, z, i)sbvSecond verification condition: If the loop body executes, invariant must still hold at the end:=inv (x, y, z, i) \land i < y \Rightarrow inv (x, y, z+1, i+1)sbvThird verification condition: Once the loop exits, invariant and the negation of the loop condition must establish the final assertion:6inv (x, y, z, i) \land i \geq y \Rightarrow z == x + ysbvSynthesize the invariant. We use an uninterpreted function for the SMT solver to synthesize. We get: synthesizeSatisfiable. Model:; invariant :: (Integer, Integer, Integer, Integer) -> Bool invariant (x, y, z, i) = x + (-z) + i > (-1) && x + (-z) + i < 1 && x + y + (-z) > (-1)This is a bit hard to read, but you can convince yourself it is equivalent to x + i .== z .&& x + y .>= z:let f (x, y, z, i) = x + (-z) + i .> (-1) .&& x + (-z) + i .< 1 .&& x + y + (-z) .> (-1)0let g (x, y, z, i) = x + i .== z .&& x + y .>= zf === (g :: Inv)Q.E.D.sbvVerify that the synthesized function does indeed work. To do so, we simply prove that the invariant found satisfies all the vcs:verifyQ.E.D.G(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone'9:;sbv*System state, containing the two integers.sbvWe parameterize over the initial state for different variations.sbvExample 1: We start from x=0, y=10, and search up to depth 10 . We have:ex1BMC: Iteration: 0BMC: Iteration: 1BMC: Iteration: 2BMC: Iteration: 3"BMC: Solution found at iteration 3$Right (3,[(0,10),(0,6),(0,2),(2,2)])As expected, there's a solution in this case. Furthermore, since the BMC engine found a solution at depth 34, we also know that there is no solution at depths 0, 1, or 2; i.e., this is "a" shortest solution. (That is, it may not be unique, but there isn't a shorter sequence to get us to our goal.)sbvExample 2: We start from x=0, y=11, and search up to depth 10 . We have:ex2 BMC: Iteration: 0BMC: Iteration: 1BMC: Iteration: 2BMC: Iteration: 3BMC: Iteration: 4BMC: Iteration: 5BMC: Iteration: 6BMC: Iteration: 7BMC: Iteration: 8BMC: Iteration: 9Left "BMC limit of 10 reached"As expected, there's no solution in this case. While SBV (and BMC) cannot establish that there is no solution at a larger depth, you can see that this will never be the case: In each step we do not change the parity of either variable. That is, x will remain even, and y will remain odd. So, there will never be a solution at any depth. This isn't the only way to see this result of course, but the point remains that BMC is just not capable of establishing inductive facts.sbv instance for our statesbvSymbolic equality for S.sbvShow the state as a pairH(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone '9:;<>sbvSystem state. We simply have two components, parameterized over the type so we can put in both concrete and symbolic values.sbv;Encoding partial correctness of the sum algorithm. We have: fibCorrectQ.E.D.NB. In my experiments, I found that this proof is quite fragile due to the use of quantifiers: If you make a mistake in your algorithm or the coding, z3 pretty much spins forever without finding a counter-example. However, with the correct coding, the proof is almost instantaneous!sbv instance for our stateI(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone'9:;G sbvSystem state. We simply have two components, parameterized over the type so we can put in both concrete and symbolic values.sbvWe parameterize over the transition relation and the strengthenings to investigate various combinations.sbv2The first program, coded as a transition relation:sbv3The second program, coded as a transition relation:sbv:Example 1: First program, with no strengthenings. We have:ex1&Failed while establishing consecution.!Counter-example to inductiveness: S {x = -1, y = 1}sbv,Example 2: First program, strengthened with x >= 0 . We have:ex2Q.E.D.sbv;Example 3: Second program, with no strengthenings. We have:ex3&Failed while establishing consecution.!Counter-example to inductiveness: S {x = -1, y = 1}sbv-Example 4: Second program, strengthened with x >= 0 . We have:ex4Failed while establishing consecution for strengthening "x >= 0".!Counter-example to inductiveness: S {x = 0, y = -1}sbv-Example 5: Second program, strengthened with x >= 0 and y >= 1 separately. We have:ex5Failed while establishing consecution for strengthening "x >= 0".!Counter-example to inductiveness: S {x = 0, y = -1} Note how this was sufficient in  to establish the invariant for the first program, but fails for the second.sbv-Example 6: Second program, strengthened with x >= 0 /\ y >= 1 simultaneously. We have:ex6Q.E.D.Compare this to .. As pointed out by Bradley, this shows that a conjunction of assertions can be inductive when none of its components, on its own, is inductive. It remains an art to find proper loop invariants, though the science is improving!sbv instance for our state  J(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone '9:;<>sbvSystem state. We simply have two components, parameterized over the type so we can put in both concrete and symbolic values.sbv;Encoding partial correctness of the sum algorithm. We have: sumCorrectQ.E.D.sbv instance for our stateK(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone'VsbvThe ALU is simply a state transformer, manipulating the state, wrapped around SBV's  monad.sbvThe state of the machine. We keep track of the data values, along with the input parameters.sbv$Input parameters, stored in reverse.sbvValues of registerssbvAn item of data to be processed. We can either be referring to a named register, or an immediate value.sbvWe operate on 64-bit signed integers. It is also possible to use the unbounded integers here as the problem description doesn't mention any size limitations. But performance isn't as good with unbounded integers, and 64-bit signed bit-vectors seem to fit the bill just fine, much like any other modern processor these days.sbv2A Register in the machine, identified by its name.sbvShorthand for the w register.sbvShorthand for the x register.sbvShorthand for the y register.sbvShorthand for the z register.sbvReading a value. For a register, we simply look it up in the environment. For an immediate, we simply return it.sbv)Writing a value. We update the registers.sbvReading an input value. In this version, we simply write a free symbolic value to the specified register, and keep track of the inputs explicitly.sbv Addition.sbvMultiplication.sbv Division.sbvModulus.sbv Equality.sbvRun a given program, returning the final state. We simply start with the initial environment mapping all registers to zero, as specified in the problem specification.sbvWe simply run the  program, and specify the constraints at the end. We take a boolean as a parameter, choosing whether we want to minimize or maximize the model-number. Note that this test takes rather long to run. We get: ghci> puzzle True Optimal model: Maximum model number = 96918996924991 :: Int64 ghci> puzzle False Optimal model: Minimum model number = 91811241911641 :: Int64 sbvThe program we need to crack. Note that different users get different programs on the Advent-Of-Code site, so this is simply one example. You can simply cut-and-paste your version instead. (Don't forget the pragma NegativeLiterals to GHC so add x -1 parses correctly as  add x (-1).)sbv instance for . This is merely there for us to be able to represent programs in a natural way, i.e, lifting integers (positive and negative). Other methods are neither implemented nor needed.L(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone67>!sbv3Days. Again, only the ones mentioned in the puzzle.sbvMonths. We only put in the months involved in the puzzle for simplicitysbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv"Represent the birthday as a recordsbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvMake a valid symbolic birthdaysbvIs this a valid birthday? i.e., one that was declared by Cheryl to be a possibility.sbv/Encode the conversation as given in the puzzle.NB. Lee Pike pointed out that not all the constraints are actually necessary! (Private communication.) The puzzle still has a unique solution if the statements a1 and b1 (i.e., Albert and Bernard saying they themselves do not know the answer) are removed. To experiment you can simply comment out those statements and observe that there still is a unique solution. Thanks to Lee for pointing this out! In fact, it is instructive to assert the conversation line-by-line, and see how the search-space gets reduced in each step.sbv4Find all solutions to the birthday problem. We have:cheryl Solution #1: birthMonth = Jul :: Month birthDay = D16 :: DayThis is the only solution.M(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone& sbvWe will represent coins with 16-bit words (more than enough precision for coins).sbvCreate a coin. The argument Int argument just used for naming the coin. Note that we constrain the value to be one of the valid U.S. coin values as we create it.sbv0Return all combinations of a sequence of values.sbv.Constraint 1: Cannot make change for a dollar.sbv3Constraint 2: Cannot make change for half a dollar.sbv/Constraint 3: Cannot make change for a quarter.sbv,Constraint 4: Cannot make change for a dime.sbv-Constraint 5: Cannot make change for a nickelsbvConstraint 6: Cannot buy the candy either. Here's where we need to have the extra knowledge that the vending machines do not take 50 cent coins.sbvSolve the puzzle. We have:puzzleSatisfiable. Model: c1 = 50 :: Word16 c2 = 25 :: Word16 c3 = 10 :: Word16 c4 = 10 :: Word16 c5 = 10 :: Word16 c6 = 10 :: Word16-sbvDeclare a carrier data-type in Haskell named P, representing all the people in a bar.sbvSymbolic version of the type .sbv!Declare the uninterpret function , standing for drinking. For each person, this function assigns whether they are drinking; but is otherwise completely uninterpreted. (i.e., our theorem will be true for all such functions.)sbvFormulate the drinkers paradox, if some one is drinking, then everyone is!drinkerQ.E.D.Q(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone/sbvThe given guesses and the correct digit counts, encoded as a simple list.sbvEncode the problem, note that we check digits are within 0-9 as we use 8-bit words to represent them. Otherwise, the constraints are simply generated by zipping the alleged solution with each guess, and making sure the number of matching digits match what's given in the problem statement.sbv'Print out the solution nicely. We have: solveEuler1854640261571849533Number of solutions: 1R(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone67>8$sbvColors of housessbvNationalities of the occupantssbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvBeverage choicessbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvPets they keepsbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvSports they engage insbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvWe have: fishOwnerGermanIt's not hard to modify this program to grab the values of all the assignments, i.e., the full solution to the puzzle. We leave that as an exercise to the interested reader! NB. We use the  configuration to indicate that the uninterpreted function changes do not matter for generating different values. All we care is that the fishOwner changes!==S(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone"67>>> sbvColors of the flowerssbv&Represent flowers by symbolic integerssbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvThe uninterpreted function  assigns a color to each flower.sbv'Describe a valid pick of three flowers i, j, k, assuming we have n flowers to start with. Essentially the numbers should be within bounds and distinct.sbvCount the number of flowers that occur in a given set of flowers.sbvSmullyan's puzzle.sbvSolve the puzzle. We have: flowerCount Solution #1: N = 3 :: IntegerThis is the only solution.So, a garden with 3 flowers is the only solution. (Note that we simply skip over the prefix existentials and the assignments to uninterpreted function  for model purposes here, as they don't represent a different solution.)T(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone67>CS sbvColors we're allowedsbv6The grid is an array mapping each button to its color.sbvSymbolic version of button.sbvUse 8-bit words for button numbers, even though we only have 1 to 19.sbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvGiven a button press, and the current grid, compute the next grid. If the button is "unpressable", i.e., if it is not one of the center buttons or it is currently colored black, we return the grid unchanged.sbvIteratively search at increasing depths of button-presses to see if we can transform from the initial board position to a final board position.sbv"A particular example run. We have:example Searching at depth: 0Searching at depth: 1Searching at depth: 2Searching at depth: 3Searching at depth: 4Searching at depth: 5Searching at depth: 6Found: [10,10,9,11,14,6]Found: [10,10,11,9,14,6]There are no more solutions.U(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone<>HCsbvA Jug has a capacity (i.e., maximum amount of water it can hold), and content, showing how much it currently has. The invariant is that content is always non-negative and is at most the capacity.sbvTransfer from one jug to another. By definition, we transfer to fill the second jug, which may end up filling it fully, or leaving some in the first jug.sbvAt the beginning, we have an full 8-gallon jug, and two empty jugs, 5 and 3 gallons each.sbvWe've solved the puzzle if 8 and 5 gallon jugs have 4 gallons each, and the third one is empty.sbvExecute a bunch of moves.sbvSolve the puzzle. We have:puzzle # of moves: 0 # of moves: 1 # of moves: 2 # of moves: 3 # of moves: 4 # of moves: 5 # of moves: 6 # of moves: 71 --> 22 --> 33 --> 12 --> 31 --> 22 --> 33 --> 1Here's the contents in terms of gallons after each move: (8, 0, 0) (3, 5, 0) (3, 2, 3) (6, 2, 0) (6, 0, 2) (1, 5, 2) (1, 4, 3) (4, 4, 0)Note that by construction this is the minimum length solution. (Though our construction does not guarantee that it is unique.)  V(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone67>N|sbv3Inhabitants of the island, as an uninterpreted sortsbv-Each inhabitant is either a knave or a knightsbvSymbolic version of the type .sbv8Statements are utterances which are either true or falsesbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$John is an inhabitant of the island.sbv$Bill is an inhabitant of the island.sbvThe connective  makes a statement about an inhabitant regarding his/her identity.sbvThe connective 1 makes a predicate from what an inhabitant statessbvThe connective ) is will be true if the statement is truesbvThe connective * creates the conjunction of two statementssbvThe connective  negates a statementsbvThe connective  creates a statement that equates the truth values of its argument statementssbv"Encode Smullyan's puzzle. We have:puzzle Question 1. John says, We are both knaves Then, John is: Knave And, Bill is: Knight Question 2. John says If (and only if) Bill is a knave, then I am a knave.& Bill says We are of different kinds. Then, John is: Knave And, Bill is: KnightW(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonePjsbvPrints the only solution: ladyAndTigers Solution #1: sign1 = False :: Bool sign2 = False :: Bool sign3 = True :: Bool tiger1 = False :: Bool tiger2 = True :: Bool tiger3 = True :: BoolThis is the only solution.That is, the lady is in room 1, and only the third room's sign is true.X(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneRsbv"The puzzle board is a list of rowssbvA row is a list of elementssbvUse 32-bit words for elements.sbv4Checks that all elements in a list are within boundssbv#Get the diagonal of a square matrixsbv'Test if a given board is a magic squaresbv3Group a list of elements in the sublists of length isbvGiven n, magic n prints all solutions to the nxn magic square problemY(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone'67>XsbvRolessbvSexessbv LocationssbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvHelper functorsbvA person has a name, age, together with location and sex. We parameterize over a function so we can use this struct both in a concrete context and a symbolic context. Note that the name is always concrete.sbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvCreate a new symbolic personsbv1Get the concrete value of the person in the modelsbvSolve the puzzle. We have:killer&Alice 48 Bar Female Bystander#Husband 47 Beach Male Killer#Brother 48 Beach Male Victim&Daughter 21 Alone Female Bystander&Son 20 Bar Male BystanderThat is, Alice's brother was the victim and Alice's husband was the killer.sbvConstraints of the puzzle, coded following the English description.sbv Show a person$$Z(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneZ~sbvA solution is a sequence of row-numbers where queens should be placedsbv Checks that a given solution of n5-queens is valid, i.e., no queen captures any other.sbvGiven n, it solves the n-queens) puzzle, printing all possible solutions.[(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone\sbvEncoding of the constraints.sbv,Print all solutions to the problem. We have: solvePuzzle Solution #1: a = 144 :: Integer b = 2 :: Integer c = True :: Bool d = 2 :: Integer e = False :: Bool f = 24 :: Integer g = False :: Bool h = -12 :: Integer i = True :: Bool( j = Right (-16) :: Either Bool IntegerThis is the only solution.\(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone67<>cJsbvLocation for each orangutan.sbvHandlers for each orangutan.sbvOrangutans in the puzzle.sbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv'An assignment is solution to the puzzlesbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv4Create a symbolic assignment, using symbolic fields.sbvWe get: allSat puzzle Solution #1:( Merah.handler = Gracie :: Handler) Merah.location = Tarakan :: Location( Merah.age = 10 :: Integer( Ofallo.handler = Eva :: Handler) Ofallo.location = Kendisi :: Location( Ofallo.age = 13 :: Integer( Quirrel.handler = Dolly :: Handler) Quirrel.location = Basahan :: Location( Quirrel.age = 4 :: Integer( Shamir.handler = Francine :: Handler) Shamir.location = Ambalat :: Location( Shamir.age = 7 :: IntegerThis is the only solution.&&](c) Levent ErkokBSD3erkokl@gmail.com experimentalNone67>f sbvA universe of rabbitssbvSymbolic version of the type .sbvIdentify those rabbits that are greedy. Note that we leave the predicate uninterpreted.sbvIdentify those rabbits that are black. Note that we leave the predicate uninterpreted.sbvIdentify those rabbits that are old. Note that we leave the predicate uninterpreted.sbvExpress the puzzle.sbvProve the claim. We have: rabbitsAreOKQ.E.D.^(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonegsbvSolve the puzzle. We have: sendMoreMoney Solution #1: s = 9 :: Integer e = 5 :: Integer n = 6 :: Integer d = 7 :: Integer m = 1 :: Integer o = 0 :: Integer r = 8 :: Integer y = 2 :: IntegerThis is the only solution.That is:9567 + 1085 == 10652True_(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonelAsbvy&sbvU2 band members. We want to translate this to SMT-Lib as a data-type, and hence the call to mkSymbolicEnumeration.sbvLocation of the flashsbvSymbolic variant for timesbvModel time using 32 bitssbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv*Crossing times for each member of the bandsbv6The symbolic variant.. The duplication is unfortunate.sbvA move action is a sequence of triples. The first component is symbolically True if only one member crosses. (In this case the third element of the triple is irrelevant.) If the first component is (symbolically) False, then both members move togethersbv/A puzzle move is modeled as a state-transformersbv(The status of the puzzle after each move4This type is equipped with an automatically derived ! instance because each field is . A ": instance must also be derived for this to work, and the DeriveAnyClass2 language extension must be enabled. The derived  instance simply walks down the structure field by field and merges each one. An equivalent hand-written * instance is provided in a comment below.sbvlocation of Larrysbvlocation of Adamsbvlocation of Edgesbvlocation of Bonosbvlocation of the flashsbv elapsed timesbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv Edge, Bono 2 <-- Bono 3 --> Larry, Adam 13 <-- Edge15 --> Edge, BonoTotal time: 17 Solution #2: 0 --> Edge, Bono 2 <-- Edge 4 --> Larry, Adam 14 <-- Bono15 --> Edge, BonoTotal time: 17 Found: 2 solutions with 5 moves.-Finding all possible solutions to the puzzle.sbvMergeable instance for  simply pushes the merging the data after run of each branch starting from the same state.,,a(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone|sbv2Abduct extraction example. We have the constraint x >= 0 and we want to make  x + y >= 2 . We have:example6Got: (define-fun abd () Bool (and (= s0 s1) (= s0 1)))5Got: (define-fun abd () Bool (and (= 2 s1) (= s0 1)))7Got: (define-fun abd () Bool (and (= s0 s1) (<= 1 s1)))&Got: (define-fun abd () Bool (= 2 s1)) Note that s0 refers to x and s1 refers to y= above. You can verify that adding any of these will ensure  x + y >= 2.b(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonesbvFind all solutions to  x + y .== 10 for positive x and y3. This is rather silly to do in the query mode as   can do this automatically, but it demonstrates how we can dynamically query the result and put in new constraints based on those.sbvRun the query. We have:demoStarting the all-sat engine! Iteration: 1Current solution is: (0,10) Iteration: 2Current solution is: (1,9) Iteration: 3Current solution is: (2,8) Iteration: 4Current solution is: (3,7) Iteration: 5Current solution is: (4,6) Iteration: 6Current solution is: (5,5) Iteration: 7Current solution is: (6,4) Iteration: 8Current solution is: (7,3) Iteration: 9Current solution is: (8,2) Iteration: 10Current solution is: (9,1) Iteration: 11Current solution is: (10,0) Iteration: 12No other solution![(0,10),(1,9),(2,8),(3,7),(4,6),(5,5),(6,4),(7,3),(8,2),(9,1),(10,0)]c(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneUsbvA simple floating-point problem, but we do the sat-analysis via a case-split. Due to the nature of floating-point numbers, a case-split on the characteristics of the number (such as NaN, negative-zero, etc. is most suitable.)We have:csDemo1 Case fpIsNegativeZero: Starting$Case fpIsNegativeZero: UnsatisfiableCase fpIsPositiveZero: Starting$Case fpIsPositiveZero: UnsatisfiableCase fpIsNormal: StartingCase fpIsNormal: UnsatisfiableCase fpIsSubnormal: Starting!Case fpIsSubnormal: UnsatisfiableCase fpIsPoint: StartingCase fpIsPoint: UnsatisfiableCase fpIsNaN: StartingCase fpIsNaN: Satisfiable("fpIsNaN",NaN)sbv!Demonstrates the "coverage" case.We have:csDemo2Case negative: StartingCase negative: UnsatisfiableCase less than 8: StartingCase less than 8: UnsatisfiableCase Coverage: StartingCase Coverage: Satisfiable("Coverage",10)d/(c) Jeffrey Young Levent ErkokBSD3erkokl@gmail.com experimentalNone.sbvFind all solutions to  x + y .== 10 for positive x and y, but at each iteration we would like to ensure that the value of x we get is at least twice as large as the previous one. This is rather silly, but demonstrates how we can dynamically query the result and put in new constraints based on those.sbvIn our first query we'll define a constraint that will not be known to the shared or second query and then solve for an answer that will differ from the first query. Note that we need to pass an MVar in so that we can operate on the shared variables. In general, the variables you want to operate on should be defined in the shared part of the query and then passed to the children queries via channels, MVars, or TVars. In this query we constrain x to be less than y and then return the sum of the values. We add a threadDelay just for demonstration purposessbvIn the second query we constrain for an answer where y is smaller than x, and then return the product of the found values.sbvRun the demo several times to see that the children threads will change ordering.sbvExample computation.sbvIn our first query we will make a constrain, solve the constraint and return the values for our variables, then we'll mutate the MVar sending information to the second query. Note that you could use channels, or TVars, or TMVars, whatever you need here, we just use MVars for demonstration purposes. Also note that this effectively creates an ordering between the children queriessbvIn the second query we create a new variable z, and then a symbolic query using information from the first query and return a solution that uses the new variable and the old variables. Each child query is run in a separate instance of z3 so you can think of this query as driving to a point in the search space, then waiting for more information, once it gets that information it will run a completely separate computation from the first one and return its results.sbvIn our second demonstration we show how through the use of concurrency constructs the user can have children queries communicate with one another. Note that the children queries are independent and so anything side-effectual like a push or a pop will be isolated to that child thread, unless of course it happens in shared.e(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone67> sbv0Days of the week. We make it symbolic using the   splice.sbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvA trivial query to find three consecutive days that's all before . The point here is that we can perform queries on such enumerated values and use   on them and return their values from queries just like any other value. We have:findDays[Monday,Tuesday,Wednesday]f(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone 67>sbvSupported binary operators. To keep the search-space small, we will only allow division by 2 or 40, and exponentiation will only be to the power 0. This does restrict the search space, but is sufficient to solve all the instances.sbv&Supported unary operators. Similar to  case, we will restrict square-root and factorial to be only applied to the value @4.sbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvThe shape of a tree, either a binary node, or a unary node, or the number 4', represented hear by the constructor F. We parameterize by the operator type: When doing symbolic computations, we'll fill those with  and . When finding the shapes, we will simply put unit values, i.e., holes.sbvSymbolic version of the type .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbv$Symbolic version of the constructor .sbvConstruct all possible tree shapes. The argument here follows the logic in  !http://www.gigamonkeys.com/trees/: We simply construct all possible shapes and extend with the operators. The number of such trees is:length allPossibleTrees640Note that this is a lot# smaller than what is generated by  !http://www.gigamonkeys.com/trees/. (There, the number of trees is 10240000: 16000 times more than what we have to consider!)sbvGiven a tree with hols, fill it with symbolic operators. This is the trick that allows us to consider only 640 trees as opposed to over 10 million.sbvMinor helper for writing "symbolic" case statements. Simply walks down a list of values to match against a symbolic version of the key.sbvEvaluate a symbolic tree, obtaining a symbolic value. Note how we structure this evaluation so we impose extra constraints on what values square-root, divide etc. can take. This is the power of the symbolic approach: We can put arbitrary symbolic constraints as we evaluate the tree.sbv8In the query mode, find a filling of a given tree shape t2, such that it evaluates to the requested number i+. Note that we return back a concrete tree.sbvGiven an integer, walk through all possible tree shapes (at most 640 of them), and find a filling that solves the puzzle.sbvSolution to the puzzle. When you run this puzzle, the solver can produce different results than what's shown here, but the expressions should still be all valid! ghci> puzzle 0 [OK]: (4 - (4 + (4 - 4))) 1 [OK]: (4 / (4 + (4 - 4))) 2 [OK]: sqrt((4 + (4 * (4 - 4)))) 3 [OK]: (4 - (4 ^ (4 - 4))) 4 [OK]: (4 + (4 * (4 - 4))) 5 [OK]: (4 + (4 ^ (4 - 4))) 6 [OK]: (4 + sqrt((4 * (4 / 4)))) 7 [OK]: (4 + (4 - (4 / 4))) 8 [OK]: (4 - (4 - (4 + 4))) 9 [OK]: (4 + (4 + (4 / 4))) 10 [OK]: (4 + (4 + (4 - sqrt(4)))) 11 [OK]: (4 + ((4 + 4!) / 4)) 12 [OK]: (4 * (4 - (4 / 4))) 13 [OK]: (4! + ((sqrt(4) - 4!) / sqrt(4))) 14 [OK]: (4 + (4 + (4 + sqrt(4)))) 15 [OK]: (4 + ((4! - sqrt(4)) / sqrt(4))) 16 [OK]: (4 * (4 * (4 / 4))) 17 [OK]: (4 + ((sqrt(4) + 4!) / sqrt(4))) 18 [OK]: -(4 + (4 - (sqrt(4) + 4!))) 19 [OK]: -(4 - (4! - (4 / 4))) 20 [OK]: (4 * (4 + (4 / 4))) sbvA rudimentary # instance for trees, nothing fancy.g(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonesbvUse the backend solver to guess the number given as argument. The number is assumed to be between 0 and 1000, and we use a simple binary search. Returns the sequence of guesses we performed during the search process.sbvPlay a round of the game, making the solver guess the secret number 42. Note that you can generate a random-number and make the solver guess it too! We have:play Current bounds: (0,1000)Current bounds: (21,1000)Current bounds: (31,1000)Current bounds: (36,1000)Current bounds: (39,1000)Current bounds: (40,1000)Current bounds: (41,1000)Current bounds: (42,1000)Solved in: 8 guesses: 8 21 31 36 39 40 41 42h(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone]sbvMathSAT example. Compute the interpolant for the following sets of formulas: {x - 3y >= -1, x + y >= 0}AND {z - 2x >= 3, 2z <= 1}where the variables are integers. Note that these sets of formulas are themselves satisfiable, but not taken all together. The pair (x, y) = (0, 0)# satisfies the first set. The pair (x, z) = (-2, 0)3 satisfies the second. However, there's no triple  (x, y, z) that satisfies all these four formulas together. We can use SBV to check this fact:sat $ \x y z -> sAnd [x - 3*y .>= -1, x + y .>= 0, z - 2*x .>= 3, 2 * z .<= (1::SInteger)] UnsatisfiableAn interpolant for these sets would only talk about the variable x" that is common to both. We have:!runSMTWith mathSAT exampleMathSAT "(<= 0 s0)"Notice that we get a string back, not a term; so there's some back-translation we need to do. We know that s0 is x through our translation mechanism, so the interpolant is saying that x >= 0 is entailed by the first set of formulas, and is inconsistent with the second. Let's use SBV to indeed show that this is the case:prove $ \x y -> (x - 3*y .>= -1 .&& x + y .>= 0) .=> (x .>= (0::SInteger))Q.E.D.And:prove $ \x z -> (z - 2*x .>= 3 .&& 2 * z .<= 1) .=> sNot (x .>= (0::SInteger))Q.E.D.4This establishes that we indeed have an interpolant!sbv1Z3 example. Compute the interpolant for formulas y = 2x and y = 2z+1.These formulas are not satisfiable together since it would mean y is both even and odd at the same time. An interpolant for this pair of formulas is a formula that's expressed only in terms of y6, which is the only common symbol among them. We have:runSMT evenOdd"(let (a!1 (= (mod (+ (* (- 1) s1) 0) 2) 0)) (or (= s1 0) a!1))"This is a bit hard to read unfortunately, due to translation artifacts and use of strings. To analyze, we need to know that s1 is y through SBV's translation. Let's express it in regular infix notation with y for s1(, and substitute the let-bound variable: (y == 0) || ((-y) " 2 == 0)Notice that the only symbol is y, as required. To establish that this is indeed an interpolant, we should establish that when y is even, this formula is True ; and if y is odd, then it should be False. You can argue mathematically that this indeed the case, but let's just use SBV to prove the required relationships:prove $ \(y :: SInteger) -> (y `sMod` 2 .== 0) .=> ((y .== 0) .|| ((-y) `sMod` 2 .== 0))Q.E.D.And:prove $ \(y :: SInteger) -> (y `sMod` 2 .== 1) .=> sNot ((y .== 0) .|| ((-y) `sMod` 2 .== 0))Q.E.D.4This establishes that we indeed have an interpolant!i(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone`sbvA simple goal with three constraints, two of which are conflicting with each other. The third is irrelevant, in the sense that it does not contribute to the fact that the goal is unsatisfiable.sbvExtract the unsat-core of  . We have:ucCore-Unsat core is: ["less than 5","more than 10"]"Demonstrating that the constraint a .> b is not) needed for unsatisfiablity in this case.j(c) Joel BurgetBSD3erkokl@gmail.com experimentalNone"Osbv9Solve a given crossword, returning the corresponding rowssbvSolve ;http://regexcrossword.com/challenges/intermediate/puzzles/1puzzle1 ["ATO","WEL"]sbvSolve ;http://regexcrossword.com/challenges/intermediate/puzzles/2puzzle2["WA","LK","ER"]sbvSolve ;http://regexcrossword.com/challenges/palindromeda/puzzles/3puzzle3["RATS","ABUT","TUBA","STAR"]k(c) Joel BurgetBSD3erkokl@gmail.com experimentalNone"p sbvEvaluation monad. The state argument is the environment to store variables as we evaluate.sbvSimple expression languagesbv-Given an expression, symbolically evaluate itsbvA simple program to query all messages with a given topic id. In SQL like notation:  query ("SELECT msg FROM msgs where topicid='" ++ my_topicid ++ "'") sbv findInjection exampleProgram "kg'; DROP TABLE 'users" though the topic might change obviously. Indeed, if we substitute the suggested string, we get the program: query ("SELECT msg FROM msgs WHERE topicid='kg'; DROP TABLE 'users'")which would query for topic kg! and then delete the users table!Here, we make sure that the injection ends with the malicious string:("'; DROP TABLE 'users" `Data.List.isSuffixOf`) <$> findInjection exampleProgramTruesbv6Literals strings can be lifted to be constant programsl(c) Brian SchroederBSD3erkokl@gmail.com experimentalNone)*9»sbvMonad for querying a solver.sbvThe result of ing the combination of a  and a .sbv$A property describes a quality of a  . It is a  yields a boolean value.sbvA symbolic value representing the result of running a program -- its output.sbv2A program that can reference two input variables, x and y.sbv)Monad for performing symbolic evaluation.sbvsbvThe uninterpreted sort , corresponding to the carrier.sbvSymbolic version of the type .sbv!Uninterpreted logical connective sbv!Uninterpreted logical connective sbv!Uninterpreted logical connective sbvProves the equivalence >NOT (p OR (q AND r)) == (NOT p AND NOT q) OR (NOT p AND NOT r)>, following from the axioms we have specified above. We have:testQ.E.D.o(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone sbvAn uninterpreted functionsbv Asserts that f x z == f (y+2) z whenever x == y+2. Naturally correct: prove thmGoodQ.E.D.p(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneҴsbvThe uninterpreted implementation of our 2x2 multiplier. We simply receive two 2-bit values, and return the high and the low bit of the resulting multiplication via two uninterpreted functions that we called mul22_hi and mul22_lo. Note that there is absolutely no computation going on here, aside from simply passing the arguments to the uninterpreted functions and stitching it back together.NB. While defining mul22_lo we used our domain knowledge that the low-bit of the multiplication only depends on the low bits of the inputs. However, this is merely a simplifying assumption; we could have passed all the arguments as well.sbvSynthesize a 2x2 multiplier. We use 8-bit inputs merely because that is the lowest bit-size SBV supports but that is more or less irrelevant. (Larger sizes would work too.) We simply assert this for all input values, extract the bottom two bits, and assert that our "uninterpreted" implementation in ! is precisely the same. We have:sat synthMul22 Satisfiable. Model:2 mul22_hi :: Bool -> Bool -> Bool -> Bool -> Bool) mul22_hi False True True True = True) mul22_hi True True False True = True) mul22_hi True False True True = True) mul22_hi True False False True = True) mul22_hi False True True False = True) mul22_hi True True True False = True* mul22_hi _ _ _ _ = False" mul22_lo :: Bool -> Bool -> Bool mul22_lo True True = True mul22_lo _ _ = FalseIt is easy to see that the low bit is simply the logical-and of the low bits. It takes a moment of staring, but you can see that the high bit is correct as well: The logical formula is  a1b xor a0b1, and if you work out the truth-table presented, you'll see that it is exactly that. Of course, you can use SBV to prove this. First, let's define the function we have synthesized into a symbolic function::{ 1mul22_hi :: (SBool, SBool, SBool, SBool) -> SBoolmul22_hi params = params `sElem` [ (sFalse, sTrue, sTrue, sTrue) , (sTrue, sTrue, sFalse, sTrue) , (sTrue, sFalse, sTrue, sTrue) , (sTrue, sFalse, sFalse, sTrue) , (sFalse, sTrue, sTrue, sFalse) , (sTrue, sTrue, sTrue, sFalse)" ]:}Now we can say:prove $ \a1 a0 b1 b0 -> mul22_hi (a1, a0, b1, b0) .== (a1 .&& b0) .<+> (a0 .&& b1)Q.E.D.>and rest assured that we have a correctly synthesized circuit!q(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone sbvA binary boolean functionsbvA ternary boolean functionsbvPositive Shannon cofactor of a boolean function, with respect to its first argumentsbvNegative Shannon cofactor of a boolean function, with respect to its first argumentsbvShannon's expansion over the first argument of a function. We have:shannonQ.E.D.sbvAlternative form of Shannon's expansion over the first argument of a function. We have:shannon2Q.E.D.sbvComputing the derivative of a boolean function (boolean difference). Defined as exclusive-or of Shannon cofactors with respect to that variable.sbvThe no-wiggle theorem: If the derivative of a function with respect to a variable is constant False, then that variable does not "wiggle" the function; i.e., any changes to it won't affect the result of the function. In fact, we have an equivalence: The variable only changes the result of the function iff the derivative with respect to it is not False:noWiggleQ.E.D.sbvUniversal quantification of a boolean function with respect to a variable. Simply defined as the conjunction of the Shannon cofactors.sbvShow that universal quantification is really meaningful: That is, if the universal quantification with respect to a variable is True, then both cofactors are true for those arguments. Of course, this is a trivial theorem if you think about it for a moment, or you can just let SBV prove it for you:univOKQ.E.D.sbvExistential quantification of a boolean function with respect to a variable. Simply defined as the conjunction of the Shannon cofactors.sbvShow that existential quantification is really meaningful: That is, if the existential quantification with respect to a variable is True, then one of the cofactors must be true for those arguments. Again, this is a trivial theorem if you think about it for a moment, but we will just let SBV prove it:existsOKQ.E.D.  r(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone67>SsbvA new data-type that we expect to use in an uninterpreted fashion in the backend SMT solver.sbvSymbolic version of the type .sbv5Declare an uninterpreted function that works over Q'ssbvA satisfiable example, stating that there is an element of the domain  such that  returns a different element. Note that this is valid only when the domain $ has at least two elements. We have:t1Satisfiable. Model: x = Q!val!0 :: Q f :: Q -> Q f _ = Q!val!1sbvThis is a variant on the first example, except we also add an axiom for the sort, stating that the domain  has only one element. In this case the problem naturally becomes unsat. We have:t2 Unsatisfiables(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone67>sbvA "list-like" data type, but one we plan to uninterpret at the SMT level. The actual shape is really immaterial for us.sbvSymbolic version of the type .sbvAn uninterpreted "classify" function. Really, we only care about the fact that such a function exists, not what it does.sbvFormulate a query that essentially asserts a cardinality constraint on the uninterpreted sort . The goal is to say there are precisely 3 such things, as it might be the case. We manage this by declaring four elements, and asserting that for a free variable of this sort, the shape of the data matches one of these three instances. That is, we assert that all the instances of the data  can be classified into 3 equivalence classes. Then, allSat returns all the possible instances, which of course are all uninterpreted.As expected, we have: allSat genLs Solution #1: l = L!val!2 :: L l0 = L!val!0 :: L l1 = L!val!1 :: L l2 = L!val!2 :: L classify :: L -> Integer classify L!val!2 = 2 classify L!val!1 = 1 classify _ = 0 Solution #2: l = L!val!1 :: L l0 = L!val!0 :: L l1 = L!val!1 :: L l2 = L!val!2 :: L classify :: L -> Integer classify L!val!2 = 2 classify L!val!1 = 1 classify _ = 0 Solution #3: l = L!val!0 :: L l0 = L!val!0 :: L l1 = L!val!1 :: L l2 = L!val!2 :: L classify :: L -> Integer classify L!val!2 = 2 classify L!val!1 = 1 classify _ = 0Found 3 different solutions.t Levent ErkokBSD3erkokl@gmail.com experimentalNone #'<> sbvHelper type synonymsbvThe concrete counterpart of 2. Again, we can't simply use the duality between SBV a and a due to the difference between SList a and [a].sbvThe state of the length program, paramaterized over the element type asbvOutputsbvTemporary variablesbvThe second input listsbvThe first input listsbv The imperative append algorithm:  zs = [] ts = xs while not (null ts) zs = zs ++ [head ts] ts = tail ts ts = ys while not (null ts) zs = zs ++ [head ts] ts = tail ts sbvA program is the algorithm, together with its pre- and post-conditions.sbvWe check that zs is xs ++ ys upon termination. correctness!Total correctness is established.Q.E.D.sbvShow instance for . The above deriving clause would work just as well, but we want it to be a little prettier here, and hence the OVERLAPS directive.sbv instance for the program statesbv>Show instance, a bit more prettier than what would be derived:  u(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone '9:;<> sbvHelper type synonymsbv?The state for the swap program, parameterized over a base type a.sbvOutputsbv Input valuesbvThe increment algorithm:  y = x+1 The point here isn't really that this program is interesting, but we want to demonstrate various aspects of WP proofs. So, we take a before and after program to annotate our algorithm so we can experiment later.sbvPrecondition for our program. Strictly speaking, we don't really need any preconditions, but for example purposes, we'll require x to be non-negative.sbvPostcondition for our program: y must equal x+1.sbv Stability: x must remain unchanged.sbvA program is the algorithm, together with its pre- and post-conditions.sbvState the correctness with respect to before/after programs. In the simple case of nothing prior/after, we have the obvious proof:correctness Skip Skip!Total correctness is established.Q.E.D.sbv instance for the program statesbvShow instance for . The above deriving clause would work just as well, but we want it to be a little prettier here, and hence the OVERLAPS directive.  v(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone '9:;<>sbvHelper type synonymsbv>The state for the sum program, parameterized over a base type a.sbvtracks fib isbvtracks  fib (i+1)sbv Loop countersbvThe input valuesbv#The imperative fibonacci algorithm:  i = 0 k = 1 m = 0 while i < n: m, k = k, m + k i++ When the loop terminates, m contains fib(n).sbvSymbolic fibonacci as our specification. Note that we cannot really implement the fibonacci function since it is not symbolically terminating. So, we instead uninterpret and axiomatize it below.NB. The concrete part of the definition is only used in calls to = and is not needed for the proof. If you don't need to call <, you can simply ignore that part and directly uninterpret.sbvConstraints and axioms we need to state explicitly to tell the SMT solver about our specification for fibonacci.sbvPrecondition for our program: n must be non-negative.sbvPostcondition for our program:  m = fib nsbv(Stability condition: Program must leave n unchanged.sbvA program is the algorithm, together with its pre- and post-conditions.sbvWith the axioms in place, it is trivial to establish correctness: correctness!Total correctness is established.Q.E.D.Note that I found this proof to be quite fragile: If you do not get the algorithm right or the axioms aren't in place, z3 simply goes to an infinite loop, instead of providing counter-examples. Of course, this is to be expected with the quantifiers present.sbv instance for the program statesbvShow instance for . The above deriving clause would work just as well, but we want it to be a little prettier here, and hence the OVERLAPS directive.w(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone '9:;<>sbvHelper type synonymsbv>The state for the sum program, parameterized over a base type a.sbvCopy of y to be modifiedsbvCopy of x to be modifiedsbv Second valuesbv First valuesbv9The imperative GCD algorithm, assuming strictly positive x and y:  i = x j = y while i != j -- While not equal if i > j i = i - j -- i is greater; reduce it by j else j = j - i -- j is greater; reduce it by i When the loop terminates, i equals j and contains  GCD(x, y).sbvSymbolic GCD as our specification. Note that we cannot really implement the GCD function since it is not symbolically terminating. So, we instead uninterpret and axiomatize it below.NB. The concrete part of the definition is only used in calls to = and is not needed for the proof. If you don't need to call , you can simply ignore that part and directly uninterpret. In that case, we simply use Prelude's version.sbvConstraints and axioms we need to state explicitly to tell the SMT solver about our specification for GCD.sbvPrecondition for our program: x and y must be strictly positivesbvPostcondition for our program: i == j and  i = gcd x ysbv(Stability condition: Program must leave x and y unchanged.sbvA program is the algorithm, together with its pre- and post-conditions.sbvWith the axioms in place, it is trivial to establish correctness: correctness!Total correctness is established.Q.E.D.Note that I found this proof to be quite fragile: If you do not get the algorithm right or the axioms aren't in place, z3 simply goes to an infinite loop, instead of providing counter-examples. Of course, this is to be expected with the quantifiers present.sbv instance for the program statesbvShow instance for . The above deriving clause would work just as well, but we want it to be a little prettier here, and hence the OVERLAPS directive.x(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone '9:;<>jsbvHelper type synonymsbvThe state for the division program, parameterized over a base type a.sbv The remaindersbv The quotientsbv The divisorsbv The dividendsbv9The imperative division algorithm, assuming non-negative x and strictly positive y:  r = x -- set remainder to x q = 0 -- set quotient to 0 while y <= r -- while we can still subtract r = r - y -- reduce the remainder q = q + 1 -- increase the quotient Note that we need to explicitly annotate each loop with its invariant and the termination measure. For convenience, we take those two as parameters for simplicity.sbvPrecondition for our program: x must non-negative and y must be strictly positive. Note that there is an explicit call to  in our program to protect against this case, so if we do not have this precondition, all programs will fail.sbvPostcondition for our program: Remainder must be non-negative and less than y, and it must hold that  x = q*y + r:sbv Stability: x and y must remain unchanged.sbvA program is the algorithm, together with its pre- and post-conditions.sbvThe invariant is simply that  x = q * y + r holds at all times and r$ is strictly positive. We need the y > 0 part of the invariant to establish the measure decreases, which is guaranteed by our precondition.sbvThe measure. In each iteration r0 decreases, but always remains positive. Since y is strictly positive, r% can serve as a measure for the loop.sbvCheck that the program terminates and the post condition holds. We have: correctness!Total correctness is established.Q.E.D.sbv instance for the program statesbvShow instance for . The above deriving clause would work just as well, but we want it to be a little prettier here, and hence the OVERLAPS directive.y(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone '9:;<>}sbvHelper type synonymsbvThe state for the division program, parameterized over a base type a.sbvSuccessive oddssbv%Successive squares, as the sum of j'ssbvThe floor of the square rootsbv The inputsbv 0. part is needed to establish the termination.sbvThe measure. In each iteration i4 strictly increases, thus reducing the differential x - isbvCheck that the program terminates and the post condition holds. We have: correctness!Total correctness is established.Q.E.D.sbv instance for the program statesbvShow instance for . The above deriving clause would work just as well, but we want it to be a little prettier here, and hence the OVERLAPS directive.z(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone '<>sbvHelper type synonymsbvThe concrete counterpart to 6. Note that we can no longer use the duality between SBV a and a as in other examples and just use one datatype for both. This is because SList a and [a] are fundamentally different types. This can be a bit confusing at first, but the point is that it is the list that is symbolic in case of an SList a, not that we have a concrete list with symbolic elements in it. Subtle difference, but it is important to keep these two separate.sbvThe state of the length program, paramaterized over the element type asbvRunning lengthsbv Copy of inputsbvThe input listsbv The imperative length algorithm:  ys = xs l = 0 while not (null ys) l = l+1 ys = tail ys Note that we need to explicitly annotate each loop with its invariant and the termination measure. For convenience, we take those two as parameters, so we can experiment later.sbv>Precondition for our program. Nothing! It works for all lists.sbvPostcondition for our program: l& must be the length of the input list.sbv(Stability condition: Program must leave xs unchanged.sbvA program is the algorithm, together with its pre- and post-conditions.sbvThe invariant simply relates the length of the input to the length of the current suffix and the length of the prefix traversed so far.sbv'The measure is obviously the length of ys1, as we peel elements off of it through the loop.sbvWe check that l! is the length of the input list xs upon termination. Note that even though this is an inductive proof, it is fairly easy to prove with our SMT based technology, which doesn't really handle induction at all! The usual inductive proof steps are baked into the invariant establishment phase of the WP proof. We have: correctness!Total correctness is established.Q.E.D.sbvShow instance: A simplified version of what would otherwise be generated.sbv'We have to write the bijection between  and  explicitly. Luckily, the definition is more or less boilerplate.sbvShow instance: Similarly, we want to be a bit more concise here.{(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone '9:;<>' sbvHelper type synonymsbv>The state for the sum program, parameterized over a base type a.sbv Running sumsbv Loop countersbvThe input valuesbv#The imperative summation algorithm: ; i = 0 s = 0 while i < n i = i+1 s = s+i Note that we need to explicitly annotate each loop with its invariant and the termination measure. For convenience, we take those two as parameters, so we can experiment later.sbvPrecondition for our program: n? must be non-negative. Note that there is an explicit call to  in our program to protect against this case, so if we do not have this precondition, all programs will fail.sbvPostcondition for our program: s5 must be the sum of all numbers up to and including n.sbv(Stability condition: Program must leave n unchanged.sbvA program is the algorithm, together with its pre- and post-conditions.sbv&Check that the program terminates and s equals  n*(n+1)/26 upon termination, i.e., the sum of all numbers upto n . Note that this only holds if n >= 0 to start with, as guaranteed by the precondition of our program.#The correct termination measure is n-i9: It goes down in each iteration provided we start with n >= 0 and it always remains non-negative while the loop is executing. Note that we do not need a lexicographic measure in this case, hence we simply return a list of one element.?@@AAAAAAAAAAAAAAAAAAAAAAAAAAAAABCDEFFFFFFFFGGGGGGGGGGGGGHHHHHHHHHHHHHHIIIIIIIIIIIIIIIIIIJJJJJJJJJJJJJKKKKKKKKKKKKKKKKKKKKKKKKKKKLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLMMMMMMMMMMNNNNOPPPPPPPPPPQQQRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRSSSSSSSSSSSSSSSSSSSSSSTTTTTTTTTTTTTTTTTTTTTTTTUUUUUUUUUUUVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVVWXXXXXXXXYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYZZZ[[\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\]]]]]]]]]]]]]^______________```````````````````````````````````````````````````````````````abbccddddddddeeeeeeeeeeeeeeeeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffgghhiijjjjkkkkkkkkkkkkkkkklllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllmmmnnnnnnnnnnnnooppqqqqqqqqq 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modelUIFuns modelAssocs modelBindingsmodelObjectives SMTConfigignoreExitCodesolverSetOptions roundingMode extraArgssolver dsatPrecision smtLibVersion transcriptoptimizeValidateConstraints validateModel isNonModelVarallSatTrackUFsallSatPrintAlongsatCmdcrackNum printRealPrec printBasetimingSolverCapabilitiessupportsFlattenedModelssupportsDirectAccessorssupportsSpecialRelssupportsFoldAndMapsupportsDataTypessupportsGlobalDeclssupportsCustomQueriessupportsPseudoBooleanssupportsOptimization supportsSetssupportsIEEE754supportsDeltaSatsupportsApproxReals supportsRealssupportsInt2bvsupportsUnboundedIntssupportsUninterpretedSortssupportsBitVectorssupportsDistinctsupportsDefineFunsupportsQuantifiers SMTLibPgm SMTLibVersionSMTLib2Cached SymbolicT MonadSymbolic symbolicEnv UICodeKindUINoneUISMTUICgCSValState SBVRunModeSMTModeCodeGen LambdaGenConcreteIStageISetupISafeIRun ArrayInfo ArrayContext ArrayFree ArrayMutate ArrayMergeResult resOutputs resAssertionsresConstraintsresAsgnsresDefinitions 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startTimesvStringsvChar eqOptBool svImplies svSetEqualrotsvShiftsvRotate svMkOverflow1 mkSymOpSCmkSymOpeqOptisConcreteZero isConcreteOneisConcreteOnes ghc-bignumGHC.Num.IntegerIntegerGHC.BitstestBit isConcreteMax isConcreteMin rationalCheck nonzeroCheckrationalSBVChecktupleLTmaybeLTeitherLTsvFloatAsSWord32svDoubleAsSWord64svFloatingPointAsSWord svMkOverflow2 GEqSymbolicMaybeEitherFalseBool Data.FoldablemkQArgsymbolicEqDefaultrewriteExistsUnique$fEqSBV $fShowSBVGHC.ShowShow$fQuantifiedBoolSBV $fLambdamSBV$fConstraintmSBV $fLambdamFUN$fConstraintmFUN$fConstraintmFUN0$fConstraintmFUN1$fConstraintmFUN2$fSymValRoundingMode $fIsListSBV$fConstraintmFUN3$fSkolemizeFUN$fSkolemizeFUN0$fSkolemizeFUN1$fSkolemizeFUN2$fSkolemizeFUN3$fSkolemizeSBV $fQNotFUN $fQNotFUN0 $fQNotFUN1 $fQNotFUN2 $fQNotFUN3 $fQNotSBVSExprtokenize parenDeficit parseSExprrdFPgetTripleFloatgetTripleDouble constantMapparseSExprFunctionparseSetLambdaparseLambdaExpressionparseStoreAssociations 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recordEndTimestartTranscriptfinalizeTranscript$fSatModelTuple7$fSatModelTuple6$fSatModelTuple5$fSatModelTuple4$fSatModelTuple3$fSatModelTuple2$fSatModelList$fSatModelCharChar$fSatModelList0GHC.BaseString$fSatModelRoundingMode $fSatModelCV$fSatModelFloatingPoint$fSatModelDouble$fSatModelFloat$fSatModelAlgReal$fSatModelInteger$fSatModelInt64GHC.IntInt64$fSatModelWord64$fSatModelInt32Int32$fSatModelWord32$fSatModelInt16Int16$fSatModelWord16Word16$fSatModelInt8Int8$fSatModelWord8Word8$fSatModelBool$fSatModelUnit$fModelableSMTResult$fModelableSatResult$fModelableThmResult inSubState lambdaGennamedLambdaGen constraintGenconverttoLambda$fQuantifiedBoola SMTFunctionaddQueryConstraint getConfig getObjectives getSBVPgmgetSBVAssertions syncUpSolver getQueryState mkFreshArraymkFreshLambdaArray askIgnoringpointWiseExtractmkSaturatingArggetValueCVHelperdefaultKindedValue sexprToValrecoverKindedValue extractValuegetTopLevelInputsgetAllSatResultparse runProofOn $fQueriablemt$fQueriablemSBV$fSolverContextQueryT$fSMTFunctionFUNTuple8r$fSMTFunctionFUNTuple7r$fSMTFunctionFUNTuple6r$fSMTFunctionFUNTuple5r$fSMTFunctionFUNTuple4r$fSMTFunctionFUNTuple3r$fSMTFunctionFUNTuple2r$fSMTFunctionFUNTuple8r0$fSMTFunctionFUNTuple7r0$fSMTFunctionFUNTuple6r0$fSMTFunctionFUNTuple5r0$fSMTFunctionFUNTuple4r0$fSMTFunctionFUNTuple3r0$fSMTFunctionFUNTuple2r0$fSMTFunctionFUNar sexprToArg sexprToFun smtFunDefault smtFunNamesmtFunSaturate smtFunType classifyModelgetModelAtIndexgetObjectiveValuescheckSatAssumingHelperrestoreTablesAndArraysgetUnsatCoreIfRequested allOnStdOutvalidatedefs2smtgenerateSMTBenchMarkGenmkArg runWithQuery runInThread sbvWithAny sbvWithAllUIKindUIFreeUIFunUICodeC GMergeablesymbolicMergeDefaultIntegralRealGHC.EnumEnumquotRemdivModquotpopCountsetBitclearBitOrderingmaxmingenVargenVar_checkObservableNameliftPB pbToIntegerlift1Flift1FNSlift2FNS lift1SReal lift2SReal fromIntegral liftViaSValshiftLshiftRrotateLrotateRenumCvtsymbolicMergeWithKind 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$fMetricFloat$fIEEEFloatingDouble$fIEEEFloatingFloat liftFunL1 liftPredL1__unuseddeclareSymbolicensureEnumerationensureEmptyDatarender' pprCFunHeaderdeclSV declSVNoConstshowSVpprCWord showCType specifiermkConstgenMake genHeader genDrivergenCProg mergeToLib genLibMake mergeDrivers ToSizedBV ToSizedCstr FromSizedBV FromSizedCstr ToSizedErr FromSizedErr!<<.!>>..<<..>>..^.oneBits Data.RatioapproxRational bitDefaultpopCountDefaulttestBitDefaulttoIntegralSized% denominator numerator bitReverse16 bitReverse32 bitReverse64 bitReverse8 byteSwap16 byteSwap32 byteSwap64getAndIffgetIffIorgetIorXorgetXor.&..|.bitbitSize bitSizeMaybe complementBitisSignedrotateshift unsafeShiftL unsafeShiftRxorzeroBits FiniteBitscountLeadingZeroscountTrailingZeros finiteBitSizeRatioWordsanitizeRelNamecheckSpecialRelationLocisTotaldispVCisClosedlcasebparabinsert GHC.GenericsData.Functor.Identity