w a^       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~                                                                               !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~                !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"""""""############$$$$%%%%%%%%%%%%&&&&&&&'''''' 'B((c) Levent ErkokBSD3erkokl@gmail.com experimental Safe-Inferred "If selected, runs the computation mS, and prints the time it took to run it. The return type should be an instance of  0 to ensure the correct elapsed time is printed.     )(c) Levent ErkokBSD3erkokl@gmail.com experimental Safe-Inferred Monadic lift over 2-tuplesMonadic lift over 3-tuplesMonadic lift over 4-tuplesMonadic lift over 5-tuplesMonadic lift over 6-tuplesMonadic lift over 7-tuplesMonadic lift over 8-tuples   *(c) Levent ErkokBSD3erkokl@gmail.com experimental Safe-InferredThe & class: a generalization of Haskell's  type Haskell  and SBV's SBoolG are instances of this class, unifying the treatment of boolean values.Minimal complete definition: , , $ However, it's advisable to define , and " as well (typically), for clarity. logical true logical false complementandornandnorxor implies  equivalence cast from Bool Generalization of  Generalization of Generalization of Generalization of       +(c) Levent ErkokBSD3erkokl@gmail.com experimental Safe-Inferred24 A univariate polynomial, represented simply as a coefficient list. For instance, "5x^3 + 2x - 5" is represented as [(5, 3), (2, 1), (-5, 0)]Algebraic reals. Note that the representation is left abstract. We represent rational results explicitly, while the roots-of-polynomials are represented implicitly by their defining equation2Check wheter a given argument is an exact rationalcConstruct a poly-root real with a given approximate value (either as a decimal, or polynomial-root)AStructural equality for AlgReal; used when constants are Map keysDStructural comparisons for AlgReal; used when constants are Map keys Render an A as an SMTLib2 value. Only supports rationals for the time being.  Render an  as a Haskell value. Only supports rationals, since there is no corresponding standard Haskell type that can represent root-of-polynomial variety.!cMerge 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)."#$ %!&'()*+,-./  !"#$ %!&'()*+,-./,(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone 1246BJKM An SMT solverThe solver in useThe path to its executable Options to provide to the solver=The solver engine, responsible for interpreting solver outputlShould we re-interpret exit codes. Most solvers behave rationally, i.e., id will do. Some (like CVC4) don't."Various capabilities of the solverSolvers that SBV is aware of0 An SMT engine1%A script, to be passed to the solver.2 Initial feed32Optional continuation script, if the result is sat!GThe 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.)"Computation timed out (see the timeout combinator)#Prover errored out$@Prover returned unknown, with a potential (possibly bogus) model%Satisfiable with model& Unsatisfiable4 A model, as returned by a solver5(Mapping of symbolic values to constants.6*Arrays, very crude; only works with Yices.77Uninterpreted funcs; very crude; only works with Yices.'Solver configuration. See also z3, yices, cvc4,  boolector, mathSAT, etc. which are instantiations of this type for those solvers, with reasonable defaults. In particular, custom configuration can be created by varying those values. (Such as z3{verbose=True}.)Most fields are self explanatory. The notion of precision for printing algebraic reals stems from the fact that such values does not necessarily have finite decimal representations, and hence we have to stop printing at some depth. It is important to emphasize that such values always have infinite precision internally. The issue is merely with how we print such an infinite precision value on the screen. The field . controls the printing precision, by specifying the number of digits after the decimal point. The default value is 16, but it can be set to any positive integer.'When printing, SBV will add the suffix ... at the and of a real-value, if the given bound is not sufficient to represent the real-value exactly. Otherwise, the number will be written out in standard decimal notation. Note that SBV will always print the whole value if it is precise (i.e., if it fits in a finite number of digits), regardless of the precision limit. The limit only applies if the representation of the real value is not finite, i.e., if it is not rational.) Debug mode*XPrint timing information on how long different phases took (construction, solving, etc.)+5How much time to give to the solver for each call of sBranch( check. (In seconds. Default: No limit.),EHow much time to give to the solver. (In seconds. Default: No limit.)-FPrint integral literals in this base (2, 8, 10, and 16 are supported.).EPrint algebraic real values with this precision. (SReal, default: 16)/AAdditional lines of script to give to the solver (user specified)0UUsually "(check-sat)". However, users might tweak it based on solver characteristics.1ZIf Just, the generated SMT script will be put in this file (for debugging purposes mostly)2QIf True, we'll treat the solver as using SMTLib2 input format. Otherwise, SMTLib13The actual SMT solver.43Rounding mode to use for floating-point conversions5^If Nothing, pick automatically. Otherwise, either use the given one, or use the custom string.8KTranslation tricks needed for specific capabilities afforded by each solver9Name of the solver:Eset-logic string to use in case not automatically determined (if any);+Does the solver understand SMT-Lib2 macros?<8Does the solver understand produce-models option setting=6Does the solver understand SMT-Lib2 style quantifiers?>=Does the solver understand SMT-Lib2 style uninterpreted-sorts?+Does the solver support unbounded integers?@Does the solver support reals?A@Does the solver support single-precision floating point numbers?B@Does the solver support double-precision floating point numbers?6Chosen logic for the solver7Use this name for the logic80Use one of the logics as defined by the standard9SMT-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 the 5 parameter to the configuration. This is especially handy if one is experimenting with custom logics that might be supported on new solvers.:BQuantifier-free formulas over the theory of floating point numbers;[Quantifier-free formulas over the theory of floating point numbers, arrays, and bit-vectors<VUnquantified non-linear real arithmetic with uninterpreted sort and function symbols. =RUnquantified linear real arithmetic with uninterpreted sort and function symbols. >UUnquantified linear integer arithmetic with uninterpreted sort and function symbols. ?bDifference Logic over the integers (in essence) but with uninterpreted sort and function symbols. @TUnquantified formulas over bitvectors with uninterpreted sort function and symbols. AfUnquantified formulas built over a signature of uninterpreted (i.e., free) sort and function symbols. BDifference 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. C!Quantifier-free real arithmetic. D$Quantifier-free integer arithmetic. EUnquantified linear real arithmetic. In essence, Boolean combinations of inequations between linear polynomials over real variables. FUnquantified linear integer arithmetic. In essence, Boolean combinations of inequations between linear polynomials over integer variablesGDifference 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 constantHAQuantifier-free formulas over the theory of fixed-size bitvectorsIFQuantifier-free formulas over the theory of arrays with extensionalityJnQuantifier-free linear formulas over the theory of integer arrays extended with free sort and function symbolsKxQuantifier-free formulas over the theory of bitvectors and bitvector arrays extended with free sort and function symbolsLKQuantifier-free formulas over the theory of bitvectors and bitvector arraysMLNon-linear integer arithmetic with uninterpreted sort and function symbols. NELinear real arithmetic with uninterpreted sort and function symbols. O)Linear formulas in linear real arithmeticPsFormulas with free function and predicate symbols over a theory of arrays of arrays of integer index and real valueQxLinear formulas with free sort and function symbols over one- and two-dimentional arrays of integer index and real valueRFormulas over the theory of linear integer arithmetic and arrays extended with free sort and function symbols but restricted to arrays with integer indices and valuesCURepresentation of an SMT-Lib program. In between pre and post goes the refuted modelsDRepresentation of SMTLib Program versions, currently we only know of versions 1 and 2. (NB. Eventually, we should just drop SMTLib1.)EWe implement a peculiar caching mechanism, applicable to the use case in implementation of SBV's. Whenever we do a state based computation, we do not want to keep on evaluating it in the then-current state. That will produce essentially a semantically equivalent value. Thus, we want to run it only once, and reuse that result, capturing the sharing at the Haskell level. This is similar to the "type-safe observable sharing" work, but also takes into the account of how symbolic simulation executes.^See Andy Gill's type-safe obervable sharing trick for the inspiration behind this technique: 9http://ittc.ku.edu/~andygill/paper.php?label=DSLExtract09/Note that this is *not* a general memo utility!S*Arrays implemented internally as functions;Internally handled by the library and not mapped to SMT-LibLReading an uninitialized value is considered an error (will throw exception)?Cannot check for equality (internally represented as functions)Can quick-check<Typically faster as it gets compiled away during translationF%An array index is simple an int valueT+Arrays implemented in terms of SMT-arrays: 8http://goedel.cs.uiowa.edu/smtlib/theories/ArraysEx.smt2Maps directly to SMT-lib arraysKReading from an unintialized value is OK and yields an uninterpreted result&Can check for equality of these arrays"Cannot quick-check theorems using SArray valuesITypically slower as it heavily relies on SMT-solving for the array theoryU#Flat arrays of symbolic values An  array a b! is an array indexed by the type  a, with elements of type  b) If an initial value is not provided in V and W methods, then the elements are left unspecified, i.e., the solver is free to choose any value. This is the right thing to do if arrays are used as inputs to functions to be verified, typically. >While it's certainly possible for user to create instances of U, the T and S instances already provided should cover most use cases in practice. (There are some differences between these models, however, see the corresponding declaration.)CMinimal complete definition: All methods are required, no defaults.V2Create a new array, with an optional initial valueW8Create a named new array, with an optional initial valueXRead the array element at aY1Reset all the elements of the array to the value bZUpdate the element at a to be b[?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 elements\A \ is a potential symbolic bitvector that can be created instances of to be fed to a symbolic program. Note that these methods are typically not needed in casual uses with prove, sat, allSatE etc, as default instances automatically provide the necessary bits.]%Create a user named input (universal)^#Create an automatically named input_Get a bunch of new words`Create an existential variablea2Create an automatically named existential variablebCreate a bunch of existentialsc@Create a free variable, universal in a proof, existential in satdGCreate an unnamed free variable, universal in proof, existential in sateCreate a bunch of free varsf,Similar to free; Just a more convenient namegISimilar to mkFreeVars; but automatically gives names based on the stringsh#Turn a literal constant to symbolici+Extract a literal, if the value is concretej+Extract a literal, from a CW representationkIs the symbolic word concrete?l%Is the symbolic word really symbolic?m/Does it concretely satisfy the given predicate?n@max/minbounds, if available. Note that we don't want to impose Bounded; on our class as Integer is not Bounded but it is a SymWordo@max/minbounds, if available. Note that we don't want to impose Bounded; on our class as Integer is not Bounded but it is a SymWordpOne stop allocatorGFA class representing what can be returned from a symbolic computation.qMark an interim result as an output. Useful when constructing Symbolic programs that return multiple values, or when the result is programmatically computed.rA Symbolic computation. Represented by a reader monad carrying the state of the computation, layered on top of IO for creating unique references to hold onto intermediate results.saRounding mode to be used for the IEEE floating-point operations. Note that Haskell's default is xv. If you use a different rounding mode, then the counter-examples you get may not match what you observe in Haskell.t/Round towards zero. (Also known as truncation.)uHRound towards negative infinity. (Also known as rounding-down or floor.)vHRound towards positive infinity. (Also known as rounding-up or ceiling.)wRound 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.)xyRound 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.)y0IEEE-754 double-precision floating point numbersz0IEEE-754 single-precision floating point numbers{0Infinite precision symbolic algebraic real value|(Infinite precision signed symbolic value};64-bit signed symbolic value, 2's complement representation~;32-bit signed symbolic value, 2's complement representation;16-bit signed symbolic value, 2's complement representation:8-bit signed symbolic value, 2's complement representation64-bit unsigned symbolic value32-bit unsigned symbolic value16-bit unsigned symbolic value8-bit unsigned symbolic valueA symbolic boolean/bitThe Symbolic value. Either a constant (Left) or a symbolic value ( Right CachedW). Note that caching is essential for making sure sharing is preserved. The parameter aV is phantom, but is extremely important in keeping the user interface strongly typed.H%The state of the symbolic interpreter3Different means of running a symbolic piece of codeSConcrete simulation mode. The StdGen is for the pConstrain acceptance in cross runsCode generation modeiSymbolic simulation mode, for proof purposes. Bool is True if it's a sat instance. SMTConfig is used for sBranch calls.I#Cached values, implementing sharingJ?Code-segments for Uninterpreted-constants, as given by the userK7Uninterpreted-constants generated during a symbolic runL&Arrays generated during a symbolic runM"Representation for symbolic arraysN&Tables generated during a symbolic runOOKinds used in the program; used for determining the final SMT-Lib logic to pickP/Constants are stored in a map, for hash-consingQ%Expression map, used for hash-consingR*The context of a symbolic array as createdS9An array created by symbolically merging two other arraysT:An array created by mutating another array at a given cellUKAn array created from another array by fixing each element to another valueV5A new array, with potential initializer for each cell(Result of running a symbolic computationWWk pairs array names and uninterpreted constants with their "kinds" used mainly for printing counterexamplesXXB pairs symbolic words and user given/automatically generated namesY&A program is a sequence of assignmentsA class for capturing values that have a sign and a size (finite or infinite) minimal complete definition: kindOf. This class can be automatically derived for data-types that have a Z< instance; this is useful for creating uninterpreted sorts.[A symbolic expression\Symbolic operations]A simple type for SBV computations, used mainly for uninterpreted constants. We keep track of the signedness/size of the arguments. A non-function will have just one entry in the list.^FQuantifiers: forall or exists. Note that we allow arbitrary nestings._3A symbolic word, tracking it's signedness and size.`A symbolic node idKind of symbolic valueW represents a concrete word of a fixed size: Endianness is mostly irrelevant (see the FromBitsT class). For signed words, the most significant digit is considered to be the sign.A constant valuevalue of an uninterpreted kinddoublefloatbit-vector/unbounded integeralgebraic realaAre two CW's of the same type?bIs this a bit?DConvert a CW to a Haskell boolean (NB. Assumes input is well-kinded)c5Normalize a CW. 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.)dForcing an argument; this is a necessary evil to make sure all the arguments to an uninterpreted function and sBranch test conditions are evaluated before called; the semantics of uinterpreted functions is necessarily strict; deviating from Haskell'se&Are there any existential quantifiers?fEConstant False as a SW. Note that this value always occupies slot -2.gEConstant False as a SW. Note that this value always occupies slot -1.hBConstant False as a CW. We represent it using the integer value 0.iAConstant True as a CW. We represent it using the integer value 1.j&how many arguments does the type take?kSMT-Lib's square-root over floats/doubles. We piggy back on to the uninterpreted function mechanism to implement these; which is not a terrible idea; although the use of the constructor l might be confusing. This function will *not* be uninterpreted in reality, as QF_FPA will define it. It's a bit of a shame, but much easier to implement it this way.m6SMT-Lib's fusedMA over floats/doubles. Similar to the k. Note that we cannot implement this function in Haskell as precision loss would be inevitable. Maybe Haskell will eventually add this op to the Num class.n#Lift a unary function thruough a CWo#Lift a binary function through a CWp!Map a unary function through a CWq"Map a binary function through a CWr`To improve hash-consing, take advantage of commutative operators by reordering their arguments.s%Extract the constraints from a resultt5Extract the traced-values from a result (quick-check)uDConvert an SBV-type to the kind-of uninterpreted value it representsvLConvert an array value type to the kind-of uninterpreted value it representswCIs this a concrete run? (i.e., quick-check or test-generation like)xGet the current path conditiony4Extend the path condition with the given test value.zAre we running in proof mode?{=If in proof mode, get the underlying configuration (used for sBranch)Not-A-Number for | and }X. Surprisingly, Haskell Prelude doesn't have this value defined, so we provide it here. Infinity for | and }X. Surprisingly, Haskell Prelude doesn't have this value defined, so we provide it here.@Symbolic variant of Not-A-Number. This value will inhabit both y and z.<Symbolic variant of infinity. This value will inhabit both y and z.~Increment the variable counter>Generate a random value, for quick-check and test-gen purposes@Create a new uninterpreted symbol, possibly with user given codeCreate a new SW-Create a new constant; hash-cons as necessary*Create a new table; hash-cons as necessary'Create a constant word from an integral/Create a new expression; hash-cons as necessary+Convert a symbolic value to a symbolic-wordCreate a symbolic value, based on the quantifier we have. If an explicit quantifier is given, we just use that. If not, then we pick existential for SAT calls and universal for everything else.<Convert a symbolic value to an SW, inside the Symbolic monadAdd a user specified axiom to the generated SMT-Lib file. The first argument is a mere string, use for commenting purposes. The second argument is intended to hold the multiple-lines of the axiom text as expressed in SMT-Lib notation. Note that we perform no checks on the axiom itself, to see whether it's actually well-formed or is sensical by any means. A separate formalization of SMT-Lib would be very useful here.6Run a symbolic computation in Proof mode and return a C. The boolean argument indicates if this is a sat instance or not.HRun a symbolic computation, and return a extra value paired up with the Grab the program from a running symbolic simulation state. This is useful for internal purposes, for instance when implementing sBranch.<Declare a new symbolic array, with a potential initial valueZLift a function to an array. Useful for creating arrays in a pure context. (Otherwise use W.)Handling constraints)Add a constraint with a given probabilityCache a state-based computation'Uncache a previously cached computation!Uncache, retrieving array indexesSGeneric uncaching. Note that this is entirely safe, since we do it in the IO monad. 0123!"#$%&4567'()*+,-./01234589:;<=>?@AB6789:;<=>?@ABCDEFGHIJKLMNOPQRCDESFTUVWXYZ[\]^_`abcdefghijklmnopGqrstuvwxyz{|}~HIJKLMNOPQRSTUVWXY[\l]^_`abcdefghijkmnopqrstuvwxyz{~      123!"#$%&4567'()*+,-./01234589:;<=>?@AB6789:;<=>?@ABCDEFGHIJKLMNOPQRCDESTUVWXYZ[\]^_`abcdefghijklmnopGqrstuvwxyz{|}~HMRSTUVWXY[\l]^_`abcdefghikmopqstuvxyz{ 0123!&%$#"4567'()*+,-./0123458 9:;<=>?@AB6879RQPONMLKJIHGFEDCBA@?>=<;:CDESFTUVWXYZ[\]^_`abcdefghijklmnopGqrsxwvutyz{|}~HIJKLMNOPQRVUTSWXY [\l]^_`abcdefghijkmnopqrstuvwxyz{~     -(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone2MiPrettyNum class captures printing of numbers in hex and binary formats; also supporting negative numbers.Minimal complete definition:  and ,Show a number in hexadecimal (starting with 0x and type.)'Show a number in binary (starting with 0b and type.)/Show a number in hex, without prefix, or types./Show a number in bin, without prefix, or types.Show as a hexadecimal value. First bool controls whether type info is printed while the second boolean controls wether 0x prefix is printed. The tuple is the signedness and the bit-length of the input. The length of the string will not0 depend on the value, but rather the bit-length.Show as a hexadecimal value, integer version. Almost the same as shex above except we don't have a bit-length so the length of the string will depend on the actual value. Similar to ; except in binary. Similar to ; except in binary.UPad a string to a given length. If the string is longer, then we don't drop anything.Binary printer Hex printerVA more convenient interface for reading binary numbers, also supports negative numbersvA version of show for floats that generates correct C literals for nan/infinite. NB. Requires "math.h" to be included.wA version of show for doubles that generates correct C literals for nan/infinite. NB. Requires "math.h" to be included.UA version of show for floats that generates correct Haskell literals for nan/infiniteVA version of show for doubles that generates correct Haskell literals for nan/infinite[A version of show for floats that generates correct SMTLib literals using the rounding mode\A version of show for doubles that generates correct SMTLib literals using the rounding mode Show a rational in SMTLib format;Convert a rounding mode to the format SMT-Lib2 understands.! !"#$%&'()*+ !"#$%&'()*+.(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone24B+CDifferent kinds of "files" we can produce. Currently this is quite C specific.5Representation of a collection of generated programs.Possible mappings for the {C 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.  long double double floatThe code-generation monad. Allows for precise layout of input values reference parameters (for returning composite values in languages such as C), and return values.,Code-generation state-%Abstraction of target language values.Options for code-generation./If 0R, perform run-time-checks for index-out-of-bounds or shifting-by-large values etc.12Bit-size to use for representing SInteger (if any)2+Type to use for representing SReal (if any)3YValues to use for the driver program generated, useful for generating non-random drivers.4If 0 , will generate a driver program5If 0, will generate a makefile65Abstract over code generation for different languages7uDefault options for code generation. The run-time checks are turned-off, and the driver values are completely random.8)Initial configuration for code-generation9.Reach into symbolic monad from code-generation:JReach into symbolic monad and output a value. Returns the corresponding SWaSets RTC (run-time-checks) for index-out-of-bounds, shift-with-large value etc. on/off. Default: ;.4Sets 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.0Sets 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..Should we generate a driver program? Default: 0l. When a library is generated, it will have a driver if any of the contituent functions has a driver. (See  compileToCLib.)(Should we generate a Makefile? Default: 0.Sets driver program run time values, useful for generating programs with fixed drivers for testing. Default: None, i.e., use random values.oAdds the given lines to the header file generated, useful for generating programs with uninterpreted functions.pAdds the given lines to the program file generated, useful for generating programs with uninterpreted functions.jAdds the given words to the compiler options in the generated Makefile, useful for linking extra stuff in..Creates an atomic input in the generated code.-Creates an array input in the generated code./Creates an atomic output in the generated code..Creates an array output in the generated code.9Creates a returned (unnamed) value in the generated code.?Creates a returned (unnamed) array value in the generated code.<Is this a driver program?=Is this a make file?>}Generate code for a symbolic program, returning a Code-gen bundle, i.e., collection of makefiles, source code, headers, etc.?4Render a code-gen bundle to a directory or to stdout@An alternative to Pretty's renderb, which might have "leading" white-space in empty lines. This version eliminates such whitespace.A=A simple way to print bundles, mostly for debugging purposes.BC instance for 5 displays values as they would be used in a C program>D,EFGHIJKL-MN.O/123456PQ789:<=>?@AB<D,EFGHIJKL-MN.O/123456PQ789:<=>?@"D,EFGHIJKL-NM.O/123456PQ789:<=>?@AB/(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonegGiven 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 R ".". Use S 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 Makefile3, a header file, and a driver (test) program, etc.Lower level version of , producing a Create 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 R ".". Use S 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.Lower level version of , producing a TPretty 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.UTRenders as "const SWord8 s0", etc. the first parameter is the width of the typefieldV3Renders as "s0", etc, or the corresponding constantW!Words as it would map to a C wordX!The printf specifier for the typeYMake 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 8-bit or less constants using decimal; otherwise we use hex. Note that this automatically takes care of the boolean (1-bit) value problem, since it shows the result as an integer, which is OK as far as C is concerned.ZDGenerate a makefile. The first argument is True if we have a driver.[Generate the header\"Generate an example driver program]Generate the C program^7Merge a bunch of bundles to generate code for a library_!Create a Makefile for the library`Create a driver for a library abcdeTfgUVWhXYiZ[j\]klmn^_`oabcdeTfgUVWhXYiZ[j\]klmn^_`o0(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneGeneralized version of , allowing the user to specify the warm-up count and the convergence factor. Maximum iteration count can also be specified, at which point convergence won't be sought. The boolean controls verbosity.Given a symbolic computation that produces a value, compute the expected value that value would take if this computation is run with its free variables drawn from uniform distributions of its respective values, satisfying the given constraints specified by  constrain and  pConstrain' calls. This is equivalent to calling < the following parameters: verbose, warm-up round count of 10000;, no maximum iteration count, and with convergence margin 0.0001.1(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonepAdd constraints to generate new\ models. This function is used to query the SMT-solver, while disallowing a previous model.q*Translate a problem into an SMTLib1 scriptprsqDUser selected rounding mode to be used for floating point arithmetic'SMT-Lib logic, if requested by the user"capabilities of the current solver kinds usedis this a sat problem?extra comments to place on topinputs skolemized version of the inputs constants auto-generated tables user specified arrays !uninterpreted functions/constants user given axioms  assignmentsextra constraintsoutput variabletuvwxyz{|}~pqprsqtuvwxyz{|}~2(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneTest output styleAs a Forte/Verilog value with given name. If the boolean is True then vectors are blasted big-endian, otherwise little-endian The indices are the split points on bit-vectors for input and output values'As a C array of structs with given name"As a Haskell value with given nameType of test vectors (abstract)ZRetrieve the test vectors for further processing. This function is useful in cases where K is not sufficient and custom output (or further preprocessing) is needed.Generate a set of concrete test values from a symbolic program. The output can be rendered as test vectors in different languages as necessary. Use the function qG call to indicate what fields should be in the test result. (Also see  constrain and  pConstrain' for filtering acceptable test values.)7Render the test as a Haskell value with the given name n. 3(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneM96Various SMT results that we can extract models out of.Is there a model?Extract 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.)Extract a model dictionary. Extract a dictionary mapping the variables to their respective values as returned by the SMT solver. Also see .4Extract a model value for a given element. Also see .Extract a representative name for the model value of an uninterpreted kind. This is supposed to correspond to the value as computed internally by the SMT solver; and is unportable from solver to solver. Also see .A simpler variant of % to get a model out without the fuss. Instances of  can be automatically extracted from models returned by the solvers. The idea is that the sbv infrastructure provides a stream of CW'9s (constant-words) coming from the solver, and the type a is interpreted based on these constants. Many typical instances are already provided, so new instances can be declared with relative ease.Minimum complete definition: EGiven a sequence of constant-words, extract one instance of the type a{, returning the remaining elements untouched. If the next element is not what's expected for this type you should return S2Given a parsed model instance, transform it using fh, and return the result. The default definition for this method should be sufficient in most use cases.An allSat call results in a N. The boolean says whether we should warn the user about prefix-existentials.A sat call results in a # The reason for having a separate  is to have a more meaningful C instance.A prove call results in a -Extract the final configuration from a result1Parse a signed/sized value from a sequence of CWsReturn all the models from an allSat call, similar to 3 but is suitable for the case of multiple results.2Get dictionaries from an all-sat call. Similar to .=Extract value of a variable from an all-sat call. Similar to .LExtract value of an uninterpreted variable from an all-sat call. Similar to .@Extract a model out, will throw error if parsing is unsuccessful Given an allSatT call, we typically want to iterate over it and print the results in sequence. The ) function automates this task by calling disp+ on each result, consecutively. The first  argument to disp 'is the current model number. The second argument is a tuple, where the first element indicates whether the model is alleged (i.e., if the solver is not sure, returing Unknown)"Show an SMTResult; generic version#Show a model in human readable form1Show a constant value, in the user-specified baseCPrint uninterpreted function values from models. Very, very crude..@Print uninterpreted array values from models. Very, very crude..-Helper function to spin off to an SMT solver.}A standard solver interface. If the solver is SMT-Lib compliant, then this function should suffice in communicating with it. A variant of readProcessWithExitCodeU; except it knows about continuation strings and can speak SMT-Lib2 (just a little).In case the SMT-Lib solver returns a response over multiple lines, compress them so we have each S-Expression spanning only a single line. We'll ignore things line parentheses inside quotes etc., as it should not be an issue! as a generic model provider as a generic model provider as a generic model provider7-Tuples extracted from a model6-Tuples extracted from a model5-Tuples extracted from a model4-Tuples extracted from a model3-Tuples extracted from a modelTuples extracted from a modelA list of values as extracted from a model. When reading a list, we go as long as we can (maximal-munch). Note that this never fails, as we can always return the empty list!| as extracted from a model} as extracted from a model as extracted from a model as extracted from a model as extracted from a model as extracted from a model as extracted from a model as extracted from a model as extracted from a model as extracted from a model as extracted from a model as extracted from a model as extracted from a modelBase case for = at unit type. Comes in handy if there are no real variables.The Show instance of AllSatResults. Note that we have to be careful in being lazy enough as the typical use case is to pull results out as they become available.3User friendly way of printing satisfiablity results-User friendly way of printing theorem results<!14(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneNADT S-Expression format, suitable for representing get-model output of SMT-LibGParse a string into an SExpr, potentially failing with an error messageMParses the Z3 floating point formatted numbers like so: 1.321p5/1.2123e9 etc.  5(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneAdd constraints to generate new\ models. This function is used to query the SMT-solver, while disallowing a previous model.*Translate a problem into an SMTLib2 scriptDUser selected rounding mode to be used for floating point arithmetic'SMT-Lib logic, if requested by the user"capabilities of the current solver kinds usedis this a sat problem?extra comments to place on topinputsskolemized version inputs constants auto-generated tables user specified arrays !uninterpreted functions/constants user given axioms  assignmentsextra constraintsoutput variable6(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneyAn instance of SMT-Lib converter; instantiated for SMT-Lib v1 and v2. (And potentially for newer versions in the future.)Convert to SMTLib-1 formatConvert to SMTLib-2 formatIAdd constraints generated from older models, used for querying new modelsBInterpret solver output based on SMT-Lib standard output responses,Get a counter-example from an SMT-Lib2 like model output line This routing is necessarily fragile as SMT solvers tend to print output in whatever form they deem convenient for them.. Currently, it's tuned to work with Z3 and CVC4; if new solvers are added, we might need to rework the logic here.DUser selected rounding mode to be used for floating point arithmetic;User selected logic to use. If Nothing, pick automatically.+Capabilities of the backend solver targetedKinds used in the problemis this a sat problem?extra comments to place on topinputs and aliasing namesskolemized inputs constants auto-generated tables user specified arrays !uninterpreted functions/constants user given axioms  assignmentsextra constraintsoutput variableC7(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneMCThe description of the Yices SMT solver The default executable is  "yices-smt"., which must be in your path. You can use the  SBV_YICESZ environment variable to point to the executable on your system. The default options are "-m -f"7, which is valid for Yices 2.1 series. You can use the SBV_YICES_OPTIONS. environment variable to override the options.8(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneMGThe description of the Boolector SMT solver The default executable is  "boolector"., which must be in your path. You can use the  SBV_BOOLECTORZ environment variable to point to the executable on your system. The default options are  "-m --smt2". You can use the SBV_BOOLECTOR_OPTIONS. environment variable to override the options.LSimilar to CVC4, Boolector uses different exit codes to indicate its status.9(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneMBThe description of the CVC4 SMT solver The default executable is "cvc4"., which must be in your path. You can use the SBV_CVC4Z environment variable to point to the executable on your system. The default options are  "--lang smt". You can use the SBV_CVC4_OPTIONS. environment variable to override the options.CVC4 uses different exit codes to indicate its status, rather than the standard 0 being success and non-0 being failure. Make it palatable to SBV. See  3http://cvc4.cs.nyu.edu/wiki/User_Manual#Exit_status for details.:(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneMEThe description of the MathSAT SMT solver The default executable is  "mathsat"., which must be in your path. You can use the  SBV_MATHSATZ environment variable to point to the executable on your system. The default options are  "-input=smt2". You can use the SBV_MATHSAT_OPTIONS. environment variable to override the options.;(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneM@The description of the Z3 SMT solver The default executable is "z3"., which must be in your path. You can use the SBV_Z3Z environment variable to point to the executable on your system. The default options are  "-in -smt2"-, which is valid for Z3 4.1. You can use the SBV_Z3_OPTIONS. environment variable to override the options.<(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone24A type aa is provable if we can turn it into a predicate. Note that a predicate can be made from a curried function of arbitrary arity, where each element is either a symbolic type or up-to a 7-tuple of symbolic-types. So predicates can be constructed from almost arbitrary Haskell functions that have arbitrary shapes. (See the instance declarations below.)Turns a value into a universally quantified predicate, internally naming the inputs. In this case the sbv library will use names of the form s1, s2(, etc. to name these variables Example: / forAll_ $ \(x::SWord8) y -> x `shiftL` 2 .== y]is a predicate with two arguments, captured using an ordinary Haskell function. Internally, x will be named s0 and y will be named s1.]Turns a value into a predicate, allowing users to provide names for the inputs. If the user does not provide enough number of names for the variables, the remaining ones will be internally generated. Note that the names are only used for printing models and has no other significance; in particular, we do not check that they are unique. Example: 9 forAll ["x", "y"] $ \(x::SWord8) y -> x `shiftL` 2 .== y>This is the same as above, except the variables will be named x and yG respectively, simplifying the counter-examples when they are printed.CTurns a value into an existentially quantified predicate. (Indeed, `R would have been a better choice here for the name, but alas it's already taken.) Version of  that allows user defined namesA 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 r 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.2Default configuration for the Boolector SMT solver.Default configuration for the CVC4 SMT Solver./Default configuration for the Yices SMT Solver.+Default configuration for the Z3 SMT solver0Default configuration for the MathSAT SMT solver<The default solver used by SBV. This is currently set to z3.!Prove a predicate, equivalent to  <Find a satisfying assignment for a predicate, equivalent to  AReturn all satisfying assignments for a predicate, equivalent to  u. Satisfying assignments are constructed lazily, so they will be available as returned by the solver and on demand.rNB. Uninterpreted constant/function values and counter-examples for array values are ignored for the purposes of . That is, only the satisfying assignments modulo uninterpreted functions and array inputs will be returned. This is due to the limitation of not having a robust means of getting a function counter-example back from the SMT solver.>Check if the given constraints are satisfiable, equivalent to  . See the function  constrain for an example use of . fCheck whether a given property is a theorem, with an optional time out and the given solver. Returns Nothing* if times out, or the result wrapped in a Just otherwise. hCheck whether a given property is satisfiable, with an optional time out and the given solver. Returns Nothing* if times out, or the result wrapped in a Just otherwise. ;Checks theoremhood within the given optional time limit of i seconds. Returns Nothing* if times out, or the result wrapped in a Just otherwise. >Checks satisfiability within the given optional time limit of i seconds. Returns Nothing* if times out, or the result wrapped in a Just otherwise. Compiles to SMT-Lib and returns the resulting program as a string. Useful for saving the result to a file for off-line analysis, for instance if you have an SMT solver that's not natively supported out-of-the box by the SBV library. It takes two booleans: smtLib2: If 0:, will generate SMT-Lib2 output, otherwise SMT-Lib1 output isSat : If 0>, will translate it as a SAT query, i.e., in the positive. If ;, will translate as a PROVE query, i.e., it will negate the result. (In this case, the check-sat call to the SMT solver will produce UNSAT if the input is a theorem, as usual.)Create both SMT-Lib1 and SMT-Lib2 benchmarks. The first argument is the basename of the file, SMT-Lib1 version will be written with suffix ".smt1" and SMT-Lib2 version will be written with suffix ".smt2". The  argument controls whether this is a SAT instance, i.e., translate the query directly, or a PROVE instance, i.e., translate the negated query. (See the second boolean argument to   for details.)/Proves the predicate using the given SMT-solver7Find a satisfying assignment using the given SMT-solverCDetermine if the constraints are vacuous using the given SMT-solver:Find all satisfying assignments using the given SMT-solver<Check if a branch condition is feasible in the current statepCheck if a boolean condition is satisfiable in the current state. If so, it returns such a satisfying assignmentMCheck the boolean SAT of an internal condition in the current execution state,     ,If True, output SMT-Lib2, otherwise SMT-Lib1lIf True, translate directly, otherwise negate the goal. (Use True for SAT queries, False for PROVE queries.)     N!"#$%&'()*+,-./012345     (          =(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone 2346HMMUninterpreted constants and functions. An uninterpreted constant is a value that is indexed by its name. The only property the prover assumes about these values are that they are equivalent to themselves; i.e., (for functions) they return the same results when applied to same arguments. We support uninterpreted-functions as a general means of black-box'ing operations that are  irrelevantx for the purposes of the proof; i.e., when the proofs can be performed without any knowledge about the function itself.Minimal complete definition: . However, most instances in practice are already provided by SBV, so end-users should not need to define their own instances.Uninterpret a value, receiving an object that can be used instead. Use this version when you do not need to add an axiom about this value. Uninterpret 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.Most generalized form of uninterpretation, this function should not be needed by end-user-code, but is rather useful for the library development.)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. 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: 'Merge 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.Total indexing operation. select xs default index is intuitively the same as  xs !! index, except it evaluates to default if index overflowsThe P class captures the essence of division. Unfortunately we cannot use Haskell's  class since the  and H superclasses are not implementable for symbolic bit-vectors. However,  and  makes 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 law: 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.Minimal complete definition: , !Symbolic 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 , M etc. For instance, we can write a generic list-minimum function as follows: S 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."!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 R to implement if-then-else, for the benefit of implementing symbolic versions of  and  functions.Minimal complete definition: #)4Symbolic Equality. Note that we can't use Haskell's s class since Haskell insists on returning Bool Comparing symbolic values will necessarily return a symbolic value.Minimal complete definition: *Newer versions of GHC (Starting with 7.8 I think), distinguishes between FiniteBits and Bits classes. We should really use FiniteBitSize for SBV which would make things better. In the interim, just work around pesky warnings..,+Generate a finite symbolic bitvector, named--Generate a finite symbolic bitvector, unnamed$Generate a finite constant bitvector'Convert a constant to an integral valueGenerically make a symbolic var. Declare an /Declare a list of s0 Declare an 1Declare a list of s2 Declare an 3Declare a list of s4 Declare an 5Declare a list of s6 Declare an 7Declare a list of s8 Declare an 9Declare a list of s: Declare an ;Declare a list of s< Declare an ~=Declare a list of ~s> Declare an }?Declare a list of }s@ Declare an |ADeclare a list of |sB Declare an {CDeclare a list of {sD Declare an zEDeclare a list of zsF Declare an yGDeclare a list of ysHPromote an SInteger to an SReal-eqOpt says the references are to the same SW, thus we can optimize. Note that we explicitly disallow KFloat/KDouble 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.ILReturns (symbolic) true if all the elements of the given list are different.JKReturns (symbolic) true if all the elements of the given list are the same.K3Returns (symbolic) true if the argument is in rangeLSymbolic membership testM.Returns 1 if the boolean is true, otherwise 0.NFused-multiply add. fusedMA a b c = a * b + c5, for double and floating point values. Note that a N call will *never* be concrete, even if all the arguments are constants; since we cannot guarantee the precision requirements, which is the whole reason why N" exists in the first place. (NB. NW only rounds once, even though it does two operations, and hence the extra precision.) Lift a float/double unary function, using a corresponding function in SMT-lib. We piggy-back on the uninterpreted function mechanism here, as it essentially is the same as introducing this as a new function.!7Lift a float/double unary function, only over constants"8Lift a float/double binary function, only over constantsOReplacement for #. Since # requires a b to be returned, we cannot implement it for symbolic words. Index 0 is the least-significant bit.PReplacement for $. Since $ returns an A, we cannot implement it for symbolic words. Here, we return an g, which can overflow when used on quantities that have more than 255 bits. Currently, that's only the |> type that SBV supports, all other types are safe. Even with |, this will only overflow if there are at least 256-bits set in the number, and the smallest such number is 2^256-1, which is a pretty darn big number to worry about for practical purposes. In any case, we do not support P for unbounded symbolic integers, as the only possible implementation wouldn't symbolically terminate. So the only overflow issue is with really-really large concrete | values.QGeneralization of %( based on a symbolic boolean. Note that % and &{ are still available on Symbolic words, this operation comes handy when the condition to set/clear happens to be symbolic.RGeneralization 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 shift amount must be an unsigned quantity.SGeneralization 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 shift amount must be an unsigned quantity.NB. If the shiftee is signed, then this is an arithmetic shift; otherwise it's logical, following the usual Haskell convention. See Ta for a variant that explicitly uses the msb as the sign bit, even for unsigned underlying types.TUArithmetic shift-right with a symbolic unsigned shift amount. This is equivalent to SH 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.U4Full adder. Returns the carry-out from the addition.FN.B. Only works for unsigned types. Signed arguments will be rejected.VxFull multiplier: Returns both the high-order and the low-order bits in a tuple, thus fully accounting for the overflow.FN.B. Only works for unsigned types. Signed arguments will be rejected.N.B. The higher-order bits are determined using a simple shift-add multiplier, thus involving bit-blasting. It'd be naive to expect SMT solvers to deal efficiently with properties involving this function, at least with the current state of the art.W=Little-endian blasting of a word into its bits. Also see the FromBits class.X:Big-endian blasting of a word into its bits. Also see the FromBits class.Y:Least significant bit of a word, always stored at index 0.ZCMost significant bit of a word, always stored at the last position.)*Helper function for use in enum operations[$If-then-else. This is by definition S with both branches forced. This is typically the desired behavior, but also see \ should you need more laziness.\A 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.]!Branch on a condition, much like [. The exception is that SBV will check to make sure if the test condition is feasible by making an external call to the SMT solver. Note that this can be expensive, thus we shall use a time-out value (+=). There might be zero, one, or two such external calls per ] call::If condition is statically known to be True/False: 0 calls'In this case, we simply constant fold..9If condition is determined to be unsatisfiable : 1 callMIn this case, we know then-branch is infeasible, so just take the else-branch:If condition is determined to be satisfable : 2 calls_In this case, we know then-branch is feasible, but we still have to check if the else-branch is In summary, ]a calls can be expensive, but they can help with the so-called symbolic-termination problem. See Data.SBV.Examples.Misc.SBranch for an example.^Symbolic assert. Check that the given boolean condition is always true in the given path. Otherwise symbolic simulation will stop with a run-time error._Symbolic assert with a programmable continuation. Check that the given boolean condition is always true in the given path. Otherwise symbolic simulation will transfer the failing model to the given continuation. The continuation takes the  SMTConfig', and a possible model: If it receives Nothingb, then it means that the condition fails for all assignments to inputs. Otherwise, it'll receive Just@ a dictionary that maps the input variables to the appropriate CW values that exhibit the failure. Note that the continuation has no option but to display the result in some fashion and call error, due to its restricted type.`Adding arbitrary constraints. When adding constraints, one has to be careful about making sure they are not inconsistent. The function T can be use for this purpose. Here is an example. Consider the following predicate:Slet pred = do { x <- forall "x"; constrain $ x .< x; return $ x .>= (5 :: SWord8) }~This predicate asserts that all 8-bit values are larger than 5, subject to the constraint that the values considered satisfy x .< x~, i.e., they are less than themselves. Since there are no values that satisfy this constraint, the proof will pass vacuously: prove predQ.E.D. We can use / to make sure to see that the pass was vacuous:isVacuous predTrue While the above example is trivial, things can get complicated if there are multiple constraints with non-straightforward relations; so if constraints are used one should make sure to check the predicate is not vacuously true. Here's an example that is not vacuous:Tlet pred' = do { x <- forall "x"; constrain $ x .> 6; return $ x .>= (5 :: SWord8) }'This time the proof passes as expected: prove pred'Q.E.D.And the proof is not vacuous:isVacuous pred'Falsea'Adding a probabilistic constraint. The |R argument is the probability threshold. Probabilistic constraints are useful for genTest and  quickCheck+ calls where we restrict our attention to  interesting parts of the input domain.*Boolean symbolic reduction. See if we can reduce a boolean condition to true/false using the path context information, by making external calls to the SMT solvers. Used in the implementation of ].bExplicit sharing combinator. The SBV library has internal caching/hash-consing mechanisms built in, based on Andy Gill's type-safe obervable sharing technique (see:  9http://ittc.ku.edu/~andygill/paper.php?label=DSLExtract09u). However, there might be times where being explicit on the sharing can help, especially in experimental code. The b 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.+'Define Floating instance on SBV's; only for base types that are already floating; i.e., SFloat and SDouble Note that most of the fields are "undefined" for symbolic values, we add methods as they are supported by SMTLib. Currently, the only symbolicly available function in this class is sqrt. !"#$%&'()*+,-./01234567,-./0123456789:;<=>?@ABCDEFGHIJKLMN !"89:;<=>?@OPQRSTUVWXYZ)AB[\]^_`a*bCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|+}~T]^`a !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`ab !"#$%&'()*+,-./01234567,-./0123456789:;<=>?@ABCDEFGHIJKLMN !"89:;<=>?@OPQRSTUVWXYZ)AB[\]^_`a*bCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|+}~#$%&*+>(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone2468cSign casting a value into another. This essentially means forgetting the sign bit and reinterpreting the bits accordingly when converting a signed value to an unsigned one. Similarly, when an unsigned quantity is converted to a signed one, the most significant bit is interpreted as the sign. We only define instances when the source and target types are precisely the same size. The idea is that d and e2 must form an isomorphism pair between the types a and b8, i.e., we expect the following two properties to hold: < signCast . unsignCast = id unsingCast . signCast = id QNote that one naive way to implement both these operations is simply to compute fromBitsLE . blastLE[, i.e., first get all the bits of the word and then reconstruct in the target type. While this is semantically correct, it generates a lot of code (both during proofs via SMT-Lib, and when compiled to C). The goal of this class is to avoid that cost, so these operations can be compiled very efficiently, they will essentially become no-op's..Minimal complete definition: All, no defaults.dInterpret as a signed wordeInterpret as an unsigned word cdecde cde?(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone2468fUnblasting a value from symbolic-bits. The bits can be given little-endian or big-endian. For a signed number in little-endian, we assume the very last bit is the sign digit. This is a bit awkward, but it is more consistent with the "reverse" view of little-big-endian representationsMinimal complete definition: gi Splitting an a into two b#'s and joining back. Intuitively, a is a larger bit-size word than b, typically double. The l# operation captures embedding of a b value into an a& without changing its semantic value..Minimal complete definition: All, no defaults.>Construct a symbolic word from its bits given in little-endianPerform a sanity check that we should receive precisely the same number of bits as required by the resulting type. The input is little-endianfghijklfghijklfghijklk@(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone234Mm]A 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. m7 structures provide logarithmic time reads and writes.n`Reading a value. We bit-blast the index and descend down the full tree according to bit-values.oWriting a value, similar to how reads are done. The important thing is that the tree representation keeps updates to a minimum.pAConstruct the fully balanced initial tree using the given values.mnopmnopmnopA(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone234qpImplements polynomial addition, multiplication, division, and modulus operations over GF(2^n). NB. Similar to , division by 0 is interpreted as follows: x w 0 = (0, x)for all x (including 0)Minimal complete definition: t, w, yr@Given 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 30. Mathematicans would write this polynomial as  x^3 + x + 1. And in fact, x will show it like that.sAdd two polynomials in GF(2^n).tMultiply 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 usally 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.uDDivide two polynomials in GF(2^n), see above note for division by 0.vOCompute modulus of two polynomials in GF(2^n), see above note for modulus by 0.w%Division and modulus packed together.xCDisplay a polynomial like a mathematician would (over the monomial x), with a type.yCDisplay a polynomial like a mathematician would (over the monomial xC), the first argument controls if the final type is shown as well.Pretty print as a polynomialAdd two polynomials}Multiply two polynomials and reduce by the third (concrete) irreducible, given by its coefficients. See the remarks for the t function for this design choicez(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 mustQ 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.hNB. The literature on CRC's has many variants on how CRC's are computed. We follow the painless guide ( +http://www.ross.net/crc/download/crc_v3.txt") and compute the CRC as follows: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.{ACompute CRC's over polynomials, i.e., symbolic words. The first 0 argument plays the same role as the one in the z function.qrstuvwxyz{ qrstuvwxyz{qrstuvwxyz{(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone+,-b+b,-B(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone2M |<Optimizer configuration. Note that iterative and quantified approaches are in general not interchangeable. For instance, iterative solutions will loop infinitely when there is no optimal value, but quantified solutions can handle such problems. Of course, quantified problems are harder for SMT solvers, naturally.}Use quantifiers~:Iteratively search. if True, it will be reporting progress)Symbolic optimization. Generalization on  and 7 that allows arbitrary cost functions and comparisons. Variant of  using the default solver. See  for parameter descriptions. Variant of 2 allowing the use of a user specified solver. See  for parameter descriptions.AMaximizes a cost function with respect to a constraint. Examples:6maximize Quantified sum 3 (bAll (.< (10 :: SInteger))) Just [9,9,9] Variant of 2 allowing the use of a user specified solver. See  for parameter descriptions.AMinimizes a cost function with respect to a constraint. Examples:6minimize Quantified sum 3 (bAll (.> (10 :: SInteger)))Just [11,11,11]Optimization using quantifiersOptimization using iteration |}~SMT configurationOptimization options comparator cost functionhow many elements?validity constraint |}~ |~}(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone24Equality as a proof method. Allows for very concise construction of equivalence proofs, which is very typical in bit-precise proofs.3The currently active solver, obtained by importing Data.SBV. To have other solvers current$, import one of the bridge modules Data.SBV.Bridge.CVC4, Data.SBV.Bridge.Yices, or Data.SBV.Bridge.Z3 directly.Note that the floating point value NaN does not compare equal to itself, so we need a special recognizer for that. Haskell provides the isNaN predicate with the . class, which unfortunately is not currently implementable for symbolic cases. (Requires trigonometric functions etc.) Thus, we provide this recognizer separately. Note that the definition simply tests equality against itself, which fails for NaN. Who said equality for floating point was reflexive?CWe call a FP number FPPoint if it is neither NaN, nor +/- infinity.Form the symbolic conjunction of a given list of boolean conditions. Useful in expressing problems with constraints, like the following: L do [x, y, z] <- sIntegers ["x", "y", "z"] solve [x .> 5, y + z .< x] Check 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.:The default configs corresponding to supported SMT solversEReturn the known available solver configs, installed on your machine.~Prove a property with multiple solvers, running them in separate threads. The results will be returned in the order produced.Prove 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.Find a satisfying assignment to a property with multiple solvers, running them in separate threads. The results will be returned in the order produced.Find 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.Find all satisfying assignments 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.Find all satisfying assignments 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.$('#&%       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`acdefghijklmnopqrstuvwxyz{|}~w~}|zysxwvutN{H.02468:<>@BDF/13579;=?ACEGUVWXYZ[TSmnopOPRSTQMYZJIKLUVXWfghijklcdeqrstuvwxyz{[\]^_)*+"#$%&'(!      `a!&%$#"'()*+,-./0123459RQPONMLKJIHGFEDCBA@?>=<;:687|~} rq\]^_`abcdefghijklmnop (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone *Current solver instance, pointing to cvc4.)Prove theorems, using the CVC4 SMT solver4Find satisfying solutions, using the CVC4 SMT solver8Find all satisfying solutions, using the CVC4 SMT solverECheck vacuity of the explicit constraints introduced by calls to the `$ function, using the CVC4 SMT solverdCheck if the statement is a theorem, with an optional time-out in seconds, using the CVC4 SMT solverfCheck if the statement is satisfiable, with an optional time-out in seconds, using the CVC4 SMT solver2Optimize cost functions, using the CVC4 SMT solver2Minimize cost functions, using the CVC4 SMT solver2Maximize cost functions, using the CVC4 SMT solver Property to checkEResponse from the SMT solver, containing the counter-example if foundProperty to check9Response of the SMT Solver, containing the model if foundProperty to checkList of all satisfying modelsProperty to check*True if the constraints are unsatisifiable%Optional time-out, specify in secondsProperty to check#Returns Nothing if time-out expires%Optional time-out, specify in secondsProperty to check#Returns Nothing if time-out expiers8Parameters to optimization (Iterative, Quantified, etc.)CBetterness check: This is the comparison predicate for optimization Cost functionNumber of inputsValidity functionLReturns Nothing if there is no valid solution, otherwise an optimal solution8Parameters to optimization (Iterative, Quantified, etc.)Cost function to minimizeNumber of inputsValidity functionLReturns Nothing if there is no valid solution, otherwise an optimal solution8Parameters to optimization (Iterative, Quantified, etc.)Cost function to maximizeNumber of inputsValidity functionLReturns Nothing if there is no valid solution, otherwise an optimal solution$('#&%       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~    !"#$%&'()*+./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`acdefghijklmnopqrstuvwxyz{|}~  (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone *Current solver instance, pointing to cvc4.)Prove theorems, using the CVC4 SMT solver4Find satisfying solutions, using the CVC4 SMT solver8Find all satisfying solutions, using the CVC4 SMT solverECheck vacuity of the explicit constraints introduced by calls to the `$ function, using the CVC4 SMT solverdCheck if the statement is a theorem, with an optional time-out in seconds, using the CVC4 SMT solverfCheck if the statement is satisfiable, with an optional time-out in seconds, using the CVC4 SMT solver2Optimize cost functions, using the CVC4 SMT solver2Minimize cost functions, using the CVC4 SMT solver2Maximize cost functions, using the CVC4 SMT solver Property to checkEResponse from the SMT solver, containing the counter-example if foundProperty to check9Response of the SMT Solver, containing the model if foundProperty to checkList of all satisfying modelsProperty to check*True if the constraints are unsatisifiable%Optional time-out, specify in secondsProperty to check#Returns Nothing if time-out expires%Optional time-out, specify in secondsProperty to check#Returns Nothing if time-out expiers8Parameters to optimization (Iterative, Quantified, etc.)CBetterness check: This is the comparison predicate for optimization Cost functionNumber of inputsValidity functionLReturns Nothing if there is no valid solution, otherwise an optimal solution8Parameters to optimization (Iterative, Quantified, etc.)Cost function to minimizeNumber of inputsValidity functionLReturns Nothing if there is no valid solution, otherwise an optimal solution8Parameters to optimization (Iterative, Quantified, etc.)Cost function to maximizeNumber of inputsValidity functionLReturns Nothing if there is no valid solution, otherwise an optimal solution$('#&%       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~    !"#$%&'()*+./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`acdefghijklmnopqrstuvwxyz{|}~  (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone *Current solver instance, pointing to cvc4.)Prove theorems, using the CVC4 SMT solver4Find satisfying solutions, using the CVC4 SMT solver8Find all satisfying solutions, using the CVC4 SMT solverECheck vacuity of the explicit constraints introduced by calls to the `$ function, using the CVC4 SMT solverdCheck if the statement is a theorem, with an optional time-out in seconds, using the CVC4 SMT solverfCheck if the statement is satisfiable, with an optional time-out in seconds, using the CVC4 SMT solver2Optimize cost functions, using the CVC4 SMT solver2Minimize cost functions, using the CVC4 SMT solver2Maximize cost functions, using the CVC4 SMT solver Property to checkEResponse from the SMT solver, containing the counter-example if foundProperty to check9Response of the SMT Solver, containing the model if foundProperty to checkList of all satisfying modelsProperty to check*True if the constraints are unsatisifiable%Optional time-out, specify in secondsProperty to check#Returns Nothing if time-out expires%Optional time-out, specify in secondsProperty to check#Returns Nothing if time-out expiers8Parameters to optimization (Iterative, Quantified, etc.)CBetterness check: This is the comparison predicate for optimization Cost functionNumber of inputsValidity functionLReturns Nothing if there is no valid solution, otherwise an optimal solution8Parameters to optimization (Iterative, Quantified, etc.)Cost function to minimizeNumber of inputsValidity functionLReturns Nothing if there is no valid solution, otherwise an optimal solution8Parameters to optimization (Iterative, Quantified, etc.)Cost function to maximizeNumber of inputsValidity functionLReturns Nothing if there is no valid solution, otherwise an optimal solution$('#&%       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~    !"#$%&'()*+./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`acdefghijklmnopqrstuvwxyz{|}~  (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone +Current solver instance, pointing to yices.*Prove theorems, using the Yices SMT solver5Find satisfying solutions, using the Yices SMT solver9Find all satisfying solutions, using the Yices SMT solverECheck vacuity of the explicit constraints introduced by calls to the `% function, using the Yices SMT solvereCheck if the statement is a theorem, with an optional time-out in seconds, using the Yices SMT solvergCheck if the statement is satisfiable, with an optional time-out in seconds, using the Yices SMT solver3Optimize cost functions, using the Yices SMT solver3Minimize cost functions, using the Yices SMT solver3Maximize cost functions, using the Yices SMT solver Property to checkEResponse from the SMT solver, containing the counter-example if foundProperty to check9Response of the SMT Solver, containing the model if foundProperty to checkList of all satisfying modelsProperty to check*True if the constraints are unsatisifiable%Optional time-out, specify in secondsProperty to check#Returns Nothing if time-out expires%Optional time-out, specify in secondsProperty to check#Returns Nothing if time-out expiers8Parameters to optimization (Iterative, Quantified, etc.)CBetterness check: This is the comparison predicate for optimization Cost functionNumber of inputsValidity functionLReturns Nothing if there is no valid solution, otherwise an optimal solution8Parameters to optimization (Iterative, Quantified, etc.)Cost function to minimizeNumber of inputsValidity functionLReturns Nothing if there is no valid solution, otherwise an optimal solution8Parameters to optimization (Iterative, Quantified, etc.)Cost function to maximizeNumber of inputsValidity functionLReturns Nothing if there is no valid solution, otherwise an optimal solution$('#&%       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~    !"#$%&'()*+./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`acdefghijklmnopqrstuvwxyz{|}~  (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone (Current solver instance, pointing to z3.'Prove theorems, using the Z3 SMT solver2Find satisfying solutions, using the Z3 SMT solver6Find all satisfying solutions, using the Z3 SMT solverECheck vacuity of the explicit constraints introduced by calls to the `" function, using the Z3 SMT solverbCheck if the statement is a theorem, with an optional time-out in seconds, using the Z3 SMT solverdCheck if the statement is satisfiable, with an optional time-out in seconds, using the Z3 SMT solver0Optimize cost functions, using the Z3 SMT solver0Minimize cost functions, using the Z3 SMT solver0Maximize cost functions, using the Z3 SMT solver Property to checkEResponse from the SMT solver, containing the counter-example if foundProperty to check9Response of the SMT Solver, containing the model if foundProperty to checkList of all satisfying modelsProperty to check*True if the constraints are unsatisifiable%Optional time-out, specify in secondsProperty to check#Returns Nothing if time-out expires%Optional time-out, specify in secondsProperty to check#Returns Nothing if time-out expiers8Parameters to optimization (Iterative, Quantified, etc.)CBetterness check: This is the comparison predicate for optimization Cost functionNumber of inputsValidity functionLReturns Nothing if there is no valid solution, otherwise an optimal solution8Parameters to optimization (Iterative, Quantified, etc.)Cost function to minimizeNumber of inputsValidity functionLReturns Nothing if there is no valid solution, otherwise an optimal solution8Parameters to optimization (Iterative, Quantified, etc.)Cost function to maximizeNumber of inputsValidity functionLReturns Nothing if there is no valid solution, otherwise an optimal solution$('#&%       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~    !"#$%&'()*+./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`acdefghijklmnopqrstuvwxyz{|}~  (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone Formalizes Chttp://graphics.stanford.edu/~seander/bithacks.html#IntegerMinOrMax Formalizes Chttp://graphics.stanford.edu/~seander/bithacks.html#IntegerMinOrMax Formalizes Ghttp://graphics.stanford.edu/~seander/bithacks.html#DetectOppositeSigns Formalizes ]http://graphics.stanford.edu/~seander/bithacks.html#ConditionalSetOrClearBitsWithoutBranching Formalizes Ghttp://graphics.stanford.edu/~seander/bithacks.html#DetermineIfPowerOf2Collection of queries (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone'LChoose the appropriate array model to be used for modeling the memory. (See .) The S is the function based model. T$ is the SMT-Lib array's based model.4Helper synonym for capturing relevant bits of MostekAn instruction is modeled as a M transformer. We model mostek programs in direct continuation passing style.BPrograms are essentially state transformers (on the machine state)0Given a machine state, compute a value out of itAbstraction 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!)2The memory maps 32-bit words to 8-bit words. (The H data-type is defined later, depending on the verification model used.) Flag bank Register bank-Convenient synonym for symbolic machine bits.Mostek was an 8-bit machine.The carry flag () and the zero flag ()XWe model only two registers of Mostek that is used in the above algorithm, can add more.(The memory is addressed by 32-bit words.!Get the value of a given register!Set the value of a given registerGet the value of a flagSet the value of a flag Read memoryWrite to memory-Checking overflow. In Legato's multipler 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 .Correctness theorem for our  implementation.We have:checkOverflowCorrectQ.E.D.LDX: Set register X to value vLDA: Set register A to value vCLC: Clear the carry flag9ROR, memory version: Rotate the value at memory location aP 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.ROR, register version: Same as , except through register r.BCC: branch to label l if the carry flag is false%ADC: Increment the value of register A- by the value of memory contents at address a7, using the carry-bit as the carry-in for the addition.%DEX: Decrement the value of register X%BNE: Branch if the zero-flag is falseThe U combinator "stops" our program, providing the final continuation that does nothing.<Parameterized by the addresses of locations of the factors (F1 and F2f), the following program multiplies them, storing the low-byte of the result in the memory location lowAddr , and the high-byte in register An. The implementation is a direct transliteration of Legato's algorithm given at the top, using our notation.jGiven address/value pairs for F1 and F2, and the location of where the low-byte of the result should go,  runLegato" takes an arbitrary machine state m; and returns the high and low bytes of the multiplication.{Create an instance of the Mostek machine, initialized by the memory and the relevant values of the registers and the flagsNThe 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._The correctness theorem. On a decent MacBook Pro, this proof takes about 3 minutes with the S/ memory model and about 30 minutes with the T% model, using yices as the SMT solverFGenerate a C program that implements Legato's algorithm automatically. instance of . simply pushes the merging into record fields././' (c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneIElement type of lists we'd like to sort. For simplicity, we'll just use ) here, but we can pick any symbolic type.5Merging two given sorted lists, preserving the order.Simple merge-sort implementation. We simply divide the input list in two two halves so long as it has at least two elements, sort each half on its own, and then merge.1Check whether a given sequence is non-decreasing.Check 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.Asserting 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 nY increases. A value around 5 or 6 should be fairly easy to prove. For instance, we have: correctness 5Q.E.D.3Generate 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 it's Haskell counterpart. (c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneHMIA poor man's representation of powerlists and basic operations on them:  0http://www.cs.utexas.edu/users/psp/powerlist.pdf5. We merely represent power-lists by ordinary lists. The tie operator, concatenation.aThe zip operator, zips the power-lists of the same size, returns a powerlist of double the size.Inverse of zipping.Reference prefix sum (ps) is simply Haskell's scanl1 function.The Ladner-Fischer (lf$) implementation of prefix-sum. See  Jhttp://www.cs.utexas.edu/~plaxton/c/337/05f/slides/ParallelRecursion-4.pdf or pg. 16 of  0http://www.cs.utexas.edu/users/psp/powerlist.pdf.gCorrectness theorem, for a powerlist of given size, an associative operator, and its left-unit element. NProves Ladner-Fischer is equivalent to reference specification for addition. 0; is the left-unit element, and we use a power-list of size 8. PProves 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. CTry proving correctness for an arbitrary operator. This proof will not{ go through since the SMT solver does not know that the operator associative and has the given left-unit element. We have:thm3Falsifiable. Counter-example: s0 = 0 :: SWord32 s1 = 0 :: SWord32 s2 = 0 :: SWord32 s3 = 0 :: SWord32 s4 = 1073741824 :: SWord32 s5 = 0 :: SWord32 s6 = 0 :: SWord32 s7 = 0 :: SWord32 -- uninterpreted: u u = 0 -- uninterpreted: flOp. flOp 0 0 = 2147483648. flOp 0 1073741824 = 3221225472. flOp 2147483648 0 = 3221225472. flOp 2147483648 1073741824 = 1073741824% flOp _ _ = 0!You can verify that the function flOp is indeed not associative: { ghci> flOp 3221225472 (flOp 2147483648 1073741824) 0 ghci> flOp (flOp 3221225472 2147483648) 1073741824 3221225472 Also, the unit 0 is clearly not a left-unit for flOp, as the last equation for flOp" will simply map many elements to 0. (NB. We need to use yices for this proof as the uninterpreted function examples are only supported through the yices interface currently.) Generate an instance of the prefix-sum problem for an arbitrary operator, by telling the SMT solver the necessary axioms for associativity and left-unit. The first argument states how wide the power list should be. yProve the generic problem for powerlists of given sizes. Note that this will only work for Yices-1. This is due to the fact that Yices-2 follows the SMT-Lib standard and does not accept bit-vector problems with quantified axioms in them, while Yices-1 did allow for that. The crux of the problem is that there are no SMT-Lib logics that combine BV's and quantifiers, see:  -http://goedel.cs.uiowa.edu/smtlib/logics.htmlB. So we are stuck until new powerful logics are added to SMT-Lib.Here, we explicitly tell SBV to use Yices-1 that did not have that limitation. Tweak the executable location accordingly below for your platform..We have: prefixSum 2Q.E.D. prefixSum 4Q.E.D.tNote that these proofs tend to run long. Also, Yices ran out of memory and crashed on my box when I tried for size 8', after running for about 2.5 minutes..@Old version of Yices that supports quantified axioms in SMT-Lib1FAnother old version of yices, suitable for the non-axiom based problemA symbolic trace can help illustrate the action of Ladner-Fischer. This generator produces the actions of Ladner-Fischer for addition, showing how the computation proceeds:ladnerFischerTrace 8'INPUTS s0 :: SWord8 s1 :: SWord8 s2 :: SWord8 s3 :: SWord8 s4 :: SWord8 s5 :: SWord8 s6 :: SWord8 s7 :: SWord8 CONSTANTS s_2 = False s_1 = TrueTABLESARRAYSUNINTERPRETED CONSTANTSUSER GIVEN CODE SEGMENTSAXIOMSDEFINE s8 :: SWord8 = s0 + s1 s9 :: SWord8 = s2 + s8 s10 :: SWord8 = s2 + s3 s11 :: SWord8 = s8 + s10 s12 :: SWord8 = s4 + s11 s13 :: SWord8 = s4 + s5 s14 :: SWord8 = s11 + s13 s15 :: SWord8 = s6 + s14 s16 :: SWord8 = s6 + s7 s17 :: SWord8 = s13 + s16 s18 :: SWord8 = s11 + s17 CONSTRAINTSOUTPUTS s0 s8 s9 s11 s12 s14 s15 s18Trace generator for the reference spec. It clearly demonstrates that the reference implementation fewer operations, but is not parallelizable at all: scanlTrace 8#INPUTS s0 :: SWord8 s1 :: SWord8 s2 :: SWord8 s3 :: SWord8 s4 :: SWord8 s5 :: SWord8 s6 :: SWord8 s7 :: SWord8 CONSTANTS s_2 = False s_1 = TrueTABLESARRAYSUNINTERPRETED CONSTANTSUSER GIVEN CODE SEGMENTSAXIOMSDEFINE s8 :: SWord8 = s0 + s1 s9 :: SWord8 = s2 + s8 s10 :: SWord8 = s3 + s9 s11 :: SWord8 = s4 + s10 s12 :: SWord8 = s5 + s11 s13 :: SWord8 = s6 + s12 s14 :: SWord8 = s7 + s13 CONSTRAINTSOUTPUTS s0 s8 s9 s10 s11 s12 s13 s14                     (c) Levent ErkokBSD3erkokl@gmail.com experimentalNone,Simple function that returns add/sum of argsKGenerate C code for addSub. Here's the output showing the generated C code: genAddSubn%== BEGIN: "Makefile" ================C# 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 $@clean: rm -f *.overyclean: clean rm -f addSub_driver%== END: "Makefile" ==================%== BEGIN: "addSub.h" ================J/* Header file for addSub. Automatically generated by SBV. Do not edit! */##ifndef __addSub__HEADER_INCLUDED__##define __addSub__HEADER_INCLUDED__#include <inttypes.h>#include <stdint.h>#include <stdbool.h>#include <math.h>/* The boolean type */typedef bool SBool;/* The float type */typedef float SFloat;/* The double type */typedef double SDouble;/* Unsigned bit-vectors */typedef uint8_t SWord8 ;typedef uint16_t SWord16;typedef uint32_t SWord32;typedef uint64_t SWord64;/* Signed bit-vectors */typedef int8_t SInt8 ;typedef int16_t SInt16;typedef int32_t SInt32;typedef int64_t SInt64;/* Entry point prototype: */8void addSub(const SWord8 x, const SWord8 y, SWord8 *sum, SWord8 *dif);(#endif /* __addSub__HEADER_INCLUDED__ */%== END: "addSub.h" ==================,== BEGIN: "addSub_driver.c" ================(/* Example driver program for addSub. */:/* Automatically generated by SBV. Edit as you see fit! */#include <inttypes.h>#include <stdint.h>#include <stdbool.h>#include <math.h>#include <stdio.h>#include "addSub.h"int main(void){ SWord8 sum; SWord8 dif; addSub(132, 241, &sum, &dif);. printf("addSub(132, 241, &sum, &dif) ->\n");$ printf(" sum = %"PRIu8"\n", sum);$ printf(" dif = %"PRIu8"\n", dif); return 0;},== END: "addSub_driver.c" ==================%== BEGIN: "addSub.c" ================D/* File: "addSub.c". Automatically generated by SBV. Do not edit! */#include <inttypes.h>#include <stdint.h>#include <stdbool.h>#include <math.h>#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 experimentalNoneThe 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. Given 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.(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.Prove that the custom z^ function is equivalent to the mathematical definition of CRC's for 11 bit messages. We have:crcGoodQ.E.D.OGenerate a C function to compute the USB CRC, using the internal CRC function.Generate a C function to compute the USB CRC, using the mathematical definition of the CRCs. Whule this version generates functionally eqivalent 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 experimentalNonecThis 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.)cThe 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]KThe 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..We can generate code for  using the T action. Note that the generated code will grow larger as we pick larger values of topI, but only linearly, thanks to the accumulating parameter trick used by D. 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; },Compute the fibonacci numbers statically at code-generation0 time and put them in a table, accessed by the  call.  Once we have s, 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.) SSWord64 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 experimentalNone>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 terminatingE, 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. We have:prove sgcdIsCorrectQ.E.D.!`This 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 <inttypes.h> #include <stdint.h> #include <stdbool.h> #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 experimentalNone"(Given 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 :: SWord8#Faster 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.$Look-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.%States the correctness of faster population-count algorithm, with respect to the reference slow version. (We use yices here as it's quite fast for this problem. Z3 seems to take much longer.) We have:%proveWith yices fastPopCountIsCorrectQ.E.D.&Not 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" ================E# 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" ================L/* Header file for popCount. Automatically generated by SBV. Do not edit! */%#ifndef __popCount__HEADER_INCLUDED__%#define __popCount__HEADER_INCLUDED__#include <inttypes.h>#include <stdint.h>#include <stdbool.h>#include <math.h>/* 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 <inttypes.h>#include <stdint.h>#include <stdbool.h>#include <math.h>#include <stdio.h>#include "popCount.h"int main(void){: const SWord8 __result = popCount(0x1b02e143e4f0e0e5ULL);C printf("popCount(0x1b02e143e4f0e0e5ULL) = %"PRIu8"\n", __result); return 0;}.== END: "popCount_driver.c" =================='== BEGIN: "popCount.c" ================F/* File: "popCount.c". Automatically generated by SBV. Do not edit! */#include <inttypes.h>#include <stdint.h>#include <stdbool.h>#include <math.h>#include "popCount.h" SWord8 popCount(const SWord64 x){ const SWord64 s0 = x;" static const SWord8 table0[] = {G 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 2, 3,G 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4,G 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 1, 2,G 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5,G 3, 4, 4, 5, 4, 5, 5, 6, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5,G 5, 6, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 1, 2, 2, 3,G 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4,G 4, 5, 4, 5, 5, 6, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6,G 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 5, 6, 6, 7, 2, 3, 3, 4, 3, 4,G 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6,G 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 s13 = s0 >> 8;2 const SWord64 s14 = 0x00000000000000ffULL & s13;" const SWord8 s15 = table0[s14]; const SWord8 s16 = s12 + s15; const SWord64 s17 = s13 >> 8;2 const SWord64 s18 = 0x00000000000000ffULL & s17;" const SWord8 s19 = table0[s18]; const SWord8 s20 = s16 + s19; const SWord64 s21 = s17 >> 8;2 const SWord64 s22 = 0x00000000000000ffULL & s21;" const SWord8 s23 = table0[s22]; const SWord8 s24 = s20 + s23; const SWord64 s25 = s21 >> 8;2 const SWord64 s26 = 0x00000000000000ffULL & s25;" const SWord8 s27 = table0[s26]; const SWord8 s28 = s24 + s27; const SWord64 s29 = s25 >> 8;2 const SWord64 s30 = 0x00000000000000ffULL & s29;" const SWord8 s31 = table0[s30]; const SWord8 s32 = s28 + s31; const SWord64 s33 = s29 >> 8;2 const SWord64 s34 = 0x00000000000000ffULL & s33;" const SWord8 s35 = table0[s34]; const SWord8 s36 = s32 + s35; const SWord64 s37 = s33 >> 8;2 const SWord64 s38 = 0x00000000000000ffULL & s37;" const SWord8 s39 = table0[s38]; const SWord8 s40 = s36 + s39; return s40;}'== END: "popCount.c" =================="#$%&"#$%&"#$%&"#$%&(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone'UA definition of shiftLeft that can deal with variable length shifts. (Note that the ``shiftL`` 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.(Test 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.)Generate 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 experimentalNone?-*,The 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.+WThe key, which can be 128, 192, or 256 bits. Represented as a sequence of 32-bit words.,AES 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.-An 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..lMultiplication in GF(2^8). This is simple polynomial multipliation, followed by the irreducible polynomial x^8+x^4+x^3+x^1+1. We simply use the t0 function exported by SBV to do the operation. /Exponentiation by a constant in GF(2^8). The implementation uses the usual square-and-multiply trick to speed up the computation.0yComputing 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.14Conversion from 32-bit words to 4 constituent bytes.2:Conversion from 4 bytes, back to a 32-bit row, inverse of 1R above. We have the following simple theorems stating this relationship formally:Eprove $ \a b c d -> toBytes (fromBytes [a, b, c, d]) .== [a, b, c, d]Q.E.D.)prove $ \r -> fromBytes (toBytes r) .== rQ.E.D.34Rotating a state row by a fixed amount to the right.4ODefinition of round-constants, as specified in Section 5.2 of the AES standard.5The  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.6Key expansion. Starting with the given key, returns an infinite sequence of words, as described by the AES standard, Section 5.2, Figure 11.77The 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".8wThe sbox transformation. We simply select from the sbox table. Note that we are obliged to give a default value (here 0A) to be used if the index is out-of-bounds as required by SBV's W function. However, that will never happen since the table has all 256 elements in it.9OThe values of the inverse S-box table. Again, the construction is programmatic.:!The inverse s-box transformation.;Prove that the 8 and : are inverses. We have:prove sboxInverseCorrectQ.E.D.<Adding the round-key to the current state. We simply exploit the fact that addition is just xor in implementing this transformation.=.T-box table generation function for encryption>&First look-up table used in encryption?'Second look-up table used in encryption@&Third look-up table used in encryptionA'Fourth look-up table used in encryptionB.T-box table generating function for decryptionC&First look-up table used in decryptionD'Second look-up table used in decryptionE&Third look-up table used in decryptionF'Fourth look-up table used in decryptionGGeneric round function. Given the function to perform one round, a key-schedule, and a starting state, it performs the AES rounds.HOne encryption round. The first argument indicates whether this is the final round or not, in which case the construction is slightly different.I|One decryption round. Similar to the encryption round, the first argument indicates whether this is the final round or not.JKey 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.)KBlock 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 JH. The output will always have 4 32-bit words, which is the cipher-text.L3Block decryption. The arguments are the same as in K`, except the first argument is the cipher-text and the output is the corresponding plain-text.M?128-bit encryption test, from Appendix C.1 of the AES standard:map hex t128Enc-["69c4e0d8","6a7b0430","d8cdb780","70b4c55a"]N?128-bit decryption test, from Appendix C.1 of the AES standard:map hex t128Dec-["00112233","44556677","8899aabb","ccddeeff"]O?192-bit encryption test, from Appendix C.2 of the AES standard:map hex t192Enc-["dda97ca4","864cdfe0","6eaf70a0","ec0d7191"]P?192-bit decryption test, from Appendix C.2 of the AES standard:map hex t192Dec-["00112233","44556677","8899aabb","ccddeeff"]Q:256-bit encryption, from Appendix C.3 of the AES standard:map hex t256Enc-["8ea2b7ca","516745bf","eafc4990","4b496089"]R:256-bit decryption, from Appendix C.3 of the AES standard:map hex t256Dec-["00112233","44556677","8899aabb","ccddeeff"]S;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.T+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; xThe GNU C-compiler does a fine job of optimizing this straightline code to generate a fairly efficient C implementation.UKComponents of the AES-128 implementation that the library is generated fromVEGenerate 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.)-*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSplain-text words key-words+True if round-trip gives us plain-text backTUV-*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUV--./0,+*123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUV-*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUV(c) Austin SeippBSD3erkokl@gmail.com experimentalNoneM W: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.XThe key is a stream of  values.YZRC4 State contains 256 8-bit values. We use the symbolically accessible full-binary type m~ to represent the state, since RC4 needs access to the array via a symbolic index and it's important to minimize access time.ZSConstruct the fully balanced initial tree, where the leaves are simply the numbers 0 through 255.[$Swaps two elements in the RC4 array.\ZImplements the PRGA used in RC4. We return the new state and the next key value generated.]@Constructs the state to be used by the PRGA using the given key.^CThe key-schedule. Note that this function returns an infinite list._0Generate a key-schedule from a given key-string.`pRC4 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 hex $ encrypt "Key" "Plaintext""bbf316e8d940af0ad3"&concatMap hex $ encrypt "Wiki" "pedia" "1021bf0420"1concatMap hex $ encrypt "Secret" "Attack at dawn""45a01f645fc35b383552544b9bf5"aWRC4 decryption. Essentially the same as decryption. For the above test vectors we have:Ddecrypt "Key" [0xbb, 0xf3, 0x16, 0xe8, 0xd9, 0x40, 0xaf, 0x0a, 0xd3] "Plaintext"-decrypt "Wiki" [0x10, 0x21, 0xbf, 0x04, 0x20]"pedia"edecrypt "Secret" [0x45, 0xa0, 0x1f, 0x64, 0x5f, 0xc3, 0x5b, 0x38, 0x35, 0x52, 0x54, 0x4b, 0x9b, 0xf5]"Attack at dawn"bProve 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 = xk). However, the proof takes quite a while to complete, as it gives rise to a fairly large symbolic trace. WXYZ[\]^_`ab WXYZ[\]^_`ab YZXW[\]^_`ab WXYZ[\]^_`ab(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneckSBV doesn't support 48 bit words natively. So, we represent them as a tuple, 32 high-bits and 16 low-bits.dFCompute the 16 bit CRC of a 48 bit message, using the given polynomialeACount the differing bits in the message and the corresponding CRCfGiven a hamming distance value hd, f returns trueJ if the 16 bit polynomial can distinguish all messages that has at most hd? different bits. Note that we express this conversely: If the sent and receivedy messages are different, then it must be the case that that must differ from each other (including CRCs), in more than hd bits.gKGenerate good CRC polynomials for 48-bit words, given the hamming distance hd.haFind and display all degree 16 polynomials with hamming distance at least 4, for 48 bit messages.When run, this function prints:  Polynomial #1. x^16 + x^2 + x + 1 Polynomial #2. x^16 + x^15 + x^2 + 1 Polynomial #3. x^16 + x^15 + x^14 + 1 Polynomial #4. x^16 + x^15 + x^2 + x + 1 Polynomial #5. x^16 + x^14 + x + 1 ... Note that different runs can produce different results, depending on the random numbers used by the solver, solver version, etc. (Also, the solver will take some time to generate these results. On my machine, the first five polynomials were generated in about 5 minutes.)cdefghcdefghcdefghcdefgh(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneiFor 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.lldn: 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.mFind 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 lexicographic order using the ordinary less-than relation. (NB. We explicitly tell z3 to use the logic AUFLIA for this problem, as the BV solver that is chosen automatically has a performance issue. See: %https://z3.codeplex.com/workitem/88.)nSolve the equation: 2x + y - z = 2We have:test2NonHomogeneous [[0,2,0],[1,0,0]] [[1,0,2],[0,1,1]]/which means that the solutions are of the form: 9(1, 0, 0) + k (0, 1, 1) + k' (1, 0, 2) = (1+k', k, k+2k')OR 9(0, 2, 0) + k (0, 1, 1) + k' (1, 0, 2) = (k', 2+k, k+2k')for arbitrary k, k'. It's easy to see that these are really solutions to the equation given. It's harder to see that they cover all possibilities, but a moments thought reveals that is indeed the case.oA 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 5We need to solve for x_0, over the naturals. We have:sailors%[15621,3124,2499,1999,1599,1279,1023]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.ijklmnoijklmnoikjlmnoikjlmno(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonep?Prove that floating point addition is not associative. We have:prove assocPlusFalsifiable. Counter-example: s0 = -Infinity :: SFloat s1 = Infinity :: SFloat s2 = -9.403955e-38 :: SFloatIndeed:let i = 1/0 :: Float%((-i) + i) + (-9.403955e-38) :: FloatNaN%(-i) + (i + (-9.403955e-38)) :: FloatNaNBut keep in mind that NaN< does not equal itself in the floating point world! We have:$let nan = 0/0 :: Float in nan == nanFalseq:Prove that addition is not associative, even if we ignore NaN/Infinity+ values. To do this, we use the predicate -, which is true of a floating point number (z or y) if it is neither NaN nor InfinityA. (That is, it's a representable point in the real-number line.)We have:assocPlusRegularFalsifiable. Counter-example: x = 1.5775295e-30 :: SFloat y = 1.92593e-34 :: SFloat z = -2.1521e-41 :: SFloatIndeed, we have:6(1.5775295e-30 + 1.92593e-34) + (-2.1521e-41) :: Float 1.5777222e-3061.5775295e-30 + (1.92593e-34 + (-2.1521e-41)) :: Float 1.577722e-304Note the loss of precision in the second expression.rDemonstrate that a+b = a does not necessarily mean b is 0O in the floating point world, even when we disallow the obvious solution when a and b are  Infinity. We have:nonZeroAdditionFalsifiable. Counter-example: a = -4.0 :: SFloat b = 4.5918e-41 :: SFloatIndeed, we have:$-4.0 + 4.5918e-41 == (-4.0 :: Float)TrueBut:4.5918e-41 == (0 :: Float)Falses"The last 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 = 1.3625818045773776e-308 :: SDoubleIndeed, we have:)let a = 1.3625818045773776e-308 :: Double a * (1/a)0.9999999999999999pqrspqrspqrspqrs(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonetbA fast implementation of population-count. Note that SBV already provides this functionality via P2, using simple expansion and counting algorithm. Po is linear in the size of the input, i.e., a 32-bit word would take 32 additions. This implementation here is fasterW in the sense that it takes as many additions as there are set-bits in the given word.YOf course, the issue is that this definition is recursive, and the usual definition via [j would never symbolically terminate: Recursion is done on the input argument: In each recursive call, we reduce the value n to  n .&. (n-1)d. This eliminates one set-bit in the input. However, this claim is far from obvious. By the use of ] we tell SBV to call the SMT solver in each test to ensure we only evaluate the branches we need, thus avoiding the symbolic-termination issue. In a sense, the SMT solvers proves that the implementation terminates for all valid inputs.Note that replacing ] in this implementation with [I would cause symbolic-termination to loop forever. Of course, this does not mean that ]_ is fast: It is costly to make external calls to the solver for each branch, so use with care.uProve that the t[ function implemented here is equivalent to the internal "slower" implementation. We have:propQ.E.D.vhIllustrates the use of path-conditions in avoiding infeasible paths in symbolic simulation. If we used [ instead of ]. in the else-branch of the implementation of v0 symbolic simulation would have encountered the ) call, and hence would have failed. But ] keeps track of the path condition, and can successfully determine that this path will never be taken, and hence avoids the problem. Note that we can freely mix/match [ and ]H calls; path conditions will be tracked in both cases. In fact, use of [f is advisable if we know for a fact that both branches are feasible, as it avoids the external call. ]4 will have the same result, albeit it'll cost more.w Prove that v always produces either 10 or 206, i.e., symbolic simulation will not fail due to the  call. We have: pathCheckQ.E.D.Were we to use [ instead of ] in the implementation of v[, this expression would have caused an exception to be raised at symbolic simulation time.tuvwtuvwtuvwtuvw(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonexA 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-negativeyWe 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]xyxyxyxy(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonezHelper synonym for representing GF(2^8); which are merely 8-bit unsigned words. Largest term in such a polynomial has degree 7.{Multiplication 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:  Nhttp://en.wikipedia.org/wiki/Finite_field_arithmetic#Rijndael.27s_finite_fieldI. Note that the irreducible itself is not in GF28! It has a degree of 8.NB. You can use the xF function to print polynomials nicely, as a mathematician would write.| States that the unit polynomial 1, is the unit element})States that multiplication is commutative~sStates that multiplication is associative, note that associativity proofs are notoriously hard for SAT/SMT solversQStates that the usual multiplication rule holds over GF(2^n) polynomials Checks:  if (a, b) = x w y then x = y t a + b being careful about y = 0z. 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).)Queriesz{|}~z{|}~z{|}~z{|}~(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone QWe will represent coins with 16-bit words (more than enough precision for coins).Create 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.0Return all combinations of a sequence of values..Constraint 1: Cannot make change for a dollar.3Constraint 2: Cannot make change for half a dollar./Constraint 3: Cannot make change for a quarter.,Constraint 4: Cannot make change for a dime.-Constraint 5: Cannot make change for a nickelConstraint 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.Solve the puzzle. We have:puzzleSatisfiable. Model: c1 = 50 :: SWord16 c2 = 25 :: SWord16 c3 = 10 :: SWord16 c4 = 10 :: SWord16 c5 = 10 :: SWord16 c6 = 10 :: SWord16<i.e., your friend has 4 dimes, a quarter, and a half dollar.    (c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneYWe will assume each number can be represented by an 8-bit word, i.e., can be at most 128.OGiven a number, increment the count array depending on the digits of the numberEncoding of the puzzle. The solution is a sequence of 10 numbers for the occurrences of the digits such that if we count each digit, we find these numbers.6Finds all two known solutions to this puzzle. We have:counts Solution #1In this sentence, the number of occurrences of 0 is 1, of 1 is 11, of 2 is 2, of 3 is 1, of 4 is 1, of 5 is 1, of 6 is 1, of 7 is 1, of 8 is 1, of 9 is 1. Solution #2In this sentence, the number of occurrences of 0 is 1, of 1 is 7, of 2 is 3, of 3 is 2, of 4 is 1, of 5 is 1, of 6 is 1, of 7 is 2, of 8 is 1, of 9 is 1.Found: 2 solution(s).(c) Levent ErkokBSD3erkokl@gmail.com experimentalNonePrints the only solution:puzzle Solution #1: dog = 3 :: SInteger cat = 41 :: SInteger mouse = 56 :: SIntegerThis is the only solution.(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneIThe given guesses and the correct digit counts, encoded as a simple list.$Encode 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.'Print out the solution nicely. We have: solveEuler1854640261571849533Number of solutions: 1(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone"The puzzle board is a list of rowsA row is a list of elementsUse 32-bit words for elements.4Checks that all elements in a list are within bounds#Get the diagonal of a square matrix'Test if a given board is a magic square3Group a list of elements in the sublists of length iGiven n, magic n prints all solutions to the nxn magic square problem(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneEA solution is a sequence of row-numbers where queens should be placed Checks that a given solution of n5-queens is valid, i.e., no queen captures any other.Given n, it solves the n-queens) puzzle, printing all possible solutions. (c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneA puzzle is a pair: First is the number of missing elements, second is a function that given that many elements returns the final board.&A Sudoku board is a sequence of 9 rows\A row is a sequence of 8-bit words, too large indeed for representing 1-9, but does not harmfGiven a series of elements, make sure they are all different and they all are numbers between 1 and 91Given a full Sudoku board, check that it is valid*Solve a given puzzle and print the resultsTHelper function to display results nicely, not really needed, but helps presentationFind all solutions to a puzzleFind an arbitrary good board(A random puzzle, found on the internet...Another random puzzle, found on the internet...Another random puzzle, found on the internet..dAccording to the web, this is the toughest sudoku puzzle ever.. It even has a name: Al Escargot: Jhttp://zonkedyak.blogspot.com/2006/11/worlds-hardest-sudoku-puzzle-al.html/This one has been called diabolical, apparently)The following is nefarious according to %http://haskell.org/haskellwiki/SudokuFSolve them all, this takes a fraction of a second to run for each case!(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone24&A 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 together/A puzzle move is modeled as a state-transformer(The status of the puzzle after each move elapsed timelocation of the flashlocation of Bonolocation of Edgelocation of Adamlocation of LarryLocation of the flashEach member gets an 8-bit idModel time using 32 bitsU2 band members Bono's ID Edge's ID Adam's ID Larry's IDIs this a valid person?*Crossing times for each member of the band(We represent this side of the bridge as , and arbitrarily as )We represent other side of the bridge as , and arbitrarily as 8Start configuration, time elapsed is 0 and everybody is 'Read the state via an accessor function.Given an arbitrary member, return his location(Transferring the flash to the other side'Transferring a person to the other side0Increment the time, when only one person crosses2Increment the time, when two people cross togetherSymbolic version of when.Move one member, remembering to take the flash&Move two members, again with the flash Run a sequence of given actions.Check if a given sequence of actions is valid, i.e., they must all cross the bridge according to the rules and in less than 17 seconds.See if there is a solution that has precisely n stepsSolve the U2-bridge crossing puzzle, starting by testing solutions with increasing number of steps, until we find one. We have:solveU2#Checking for solutions with 1 move.$Checking for solutions with 2 moves.$Checking for solutions with 3 moves.$Checking for solutions with 4 moves.$Checking for solutions with 5 moves. Solution #1:  0 --> Edge, Bono 2 <-- Edge 4 --> Larry, Adam 14 <-- Bono15 --> Edge, BonoTotal time: 17 Solution #2:  0 --> Edge, Bono 2 <-- Bono 3 --> Larry, Adam 13 <-- Edge15 --> Edge, BonoTotal time: 17 Found: 2 solutions with 5 moves.-Finding all possible solutions to the puzzle.kThe SatModel instance makes it easy to build models, mapping words to U2 members in the way we designated.Mergeable instance for [ simply pushes the merging the data after run of each branch starting from the same state.Mergeable instance for 9 simply walks down the structure fields and merges them.+(+ "(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneThis version directly uses SMT-arrays and hence does not need an initializer. Reading an element before writing to it returns an arbitrary value.The array type, takes symbolic 32-bit unsigned indexes and stores 32-bit unsigned symbolic values. These are functional arrays where reading before writing a cell throws an exception.%Uninterpreted function in the theorem3Correctness theorem. We state it for all values of x, y, and the array a6. We also take an arbitrary initializer for the array.#Prints Q.E.D. when run, as expected proveThm1Q.E.D.Same as /, except we don't need an initializer with the T model.$Prints Q.E.D. when run, as expected: proveThm2Q.E.D.#(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone+ 4Handy shortcut for the type of symbolic values over The uninterpreted sort , corresponding to the carrier.!Uninterpreted logical connective !Uninterpreted logical connective !Uninterpreted logical connective Distributivity of OR over AND, as an axiom in terms of the uninterpreted functions we have introduced. Note how variables range over the uninterpreted sort .]One of De Morgan's laws, again as an axiom in terms of our uninterpeted logical connectives.,Double negation axiom, similar to the above.Proves 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.!Default instance declaration for !Default instance declaration for \    $(c) Levent ErkokBSD3erkokl@gmail.com experimentalNoneAn uninterpreted function Asserts that f x z == f (y+2) z whenever x == y+2. Naturally correct: prove thmGoodQ.E.D. Asserts that f is commutative; which is not necessarily true! Indeed, the SMT solver returns a counter-example function that is not commutative. (Note that we have to use Yices as Z3 function counterexamples are not yet supported by sbv.) We have:/proveWith yicesSMT09 $ forAll ["x", "y"] thmBadFalsifiable. Counter-example: x = 0 :: SWord8 y = 128 :: SWord8 -- uninterpreted: f f 128 0 = 32768 f _ _ = 0%Note how the counterexample function fk returned by Yices violates commutativity; thus providing evidence that the asserted theorem is not valid.MOld version of Yices, which supports nice output for uninterpreted functions.%(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone A binary boolean functionA ternary boolean functionTPositive Shannon cofactor of a boolean function, with respect to its first argumentTNegative Shannon cofactor of a boolean function, with respect to its first argumentCShannon's expansion over the first argument of a function. We have:shannonQ.E.D.WAlternative form of Shannon's expansion over the first argument of a function. We have:shannon2Q.E.D.Computing the derivative of a boolean function (boolean difference). Defined as exclusive-or of Shannon cofactors with respect to that variable.fThe 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.Universal quantification of a boolean function with respect to a variable. Simply defined as the conjunction of the Shannon cofactors.-Show 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.Existential quantification of a boolean function with respect to a variable. Simply defined as the conjunction of the Shannon cofactors.0Show 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.    &(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone+nA new data-type that we expect to use in an uninterpreted fashion in the backend SMT solver. Note the custom deriving6 clause, which takes care of most of the boilerplate.5Declare an uninterpreted function that works over Q'sGA satisfiable example, stating that there is an element of the domain  such that L 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 :: QkThis is a variant on the first example, except we also add an axiom for the sort, stating that the domain R has only one element. In this case the problem naturally becomes unsat. We have:t2 UnsatisfiableF instance is again straightforward, no specific implementation needed.We need \ and  instances, but default definitions are always sufficient for uninterpreted sorts, so all we do is to declare them as such. Note that, starting with GHC 7.6.1, we will be able to simply derive these classes as well. (See  /http://hackage.haskell.org/trac/ghc/ticket/5462.)'(c) Levent ErkokBSD3erkokl@gmail.com experimentalNone+=A "list-like" data type, but one we plan to uninterpret at the SMT level. The actual shape is really immaterial for us, but could be used as a proxy to generate test cases or explore data-space in some other part of a program. Note that we neither rely on the shape of this data, nor need the actual constructors.yAn uninterpreted "classify" function. Really, we only care about the fact that such a function exists, not what it does._Formulate 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:genLs Solution #1: l = L!val!0 :: L l0 = L!val!0 :: L l1 = L!val!1 :: L l2 = L!val!2 :: L Solution #2: l = L!val!2 :: L l0 = L!val!0 :: L l1 = L!val!1 :: L l2 = L!val!2 :: L Solution #3: l = L!val!1 :: L l0 = L!val!0 :: L l1 = L!val!1 :: L l2 = L!val!2 :: LFound 3 different solutions. Similarly, 's default implementation is sufficient. Declare instances to make 1 a usable uninterpreted sort. First we need the \1 instance, with the default definition sufficing.   *C*D*E*F*G*H*I*J*K*L*M*N*O*P*Q*R+S+T+U,V,V,W,X,Y,Z,[,\,],^,_,`,a,b,c,d,e,f,g,h,i,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z,{,|,},~,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,------................. . . . . ......////002222222 2!3"3#3$3%3&3'3(3)3*3+3,3,3-3-3.3.3/30313233<4<5<6<7<8<9<:<;<<<=<><?<@<A<B<C<D<E<F<G<H<I<J<K<L<M=N=O=P=Q=R=S=T=U=V=W=X=Y=Z=[=\=]=^=_=`=a=b=c=d=e=f=g=h=i=j=k=l=m=n=o=p=q=r=s=t=u=v=w=x=y=z={=|=}=~===============================>>>???????@@@@AAAAAAAAAAABBBBBBBBB@ABCFG@ABCFG@ABCFG@ABCFG@ABCFG                                                                              !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_3`abcdefghijklmnopqrstuvwxyz{|}~5o                !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"""""""##########t##$$$$%%%%%%%%%%%%&&&&G&H&&'''''''(()))))))           *++++++++++++++++++++ +!+",#,$,%,&,',(,),*,+,,,-,.,/,0,1,2,3,4,5,6,7,8,9,:,4,;,<,=,>,?,@,A,B,C,D,E,F,G,H,I,J,K LM,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,[,\,],^,N,_,`,a,b,c,d,e,f,g,h,i,j,k,l,mno,p,q,r,s,t,u,v,w,x,y,z,{,|,},~,,,,$,',+,6,,,8,,,,4,,,,,,,,,,,,,,,,,,,,,,,K,,,,,,,,,,,,,,,,,,,,,,,,,,,,P,,,R,S,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,---------------------- - - - - -................ . .!.".#.$ %&...'.(.).*.+.,.-.../..0.1 23 24/5/6/7/8/9/:/;/</=/>/?/@/A/B/B/C/D/E/F/G/H/I/J/K/L/M/N/O1P1Q1R1S1T1C1U1V1W1X1Y1Z1[1\1]1^1_1`1a1b1c2d2e2f2g3h3i3jk3l3m3n3o3p3q3r3s3t3u3v3w3x3y3z3{3|3}3~33333 3 3 3 3 3 3 3 333334444444445P5Q5@55R5S5D5W5V5555U5555b555\5]5^55a5_5`6666P667<777777777777778:889;99:>:;=;;<<<<<<<<<<<<<<<<<<       ========      ===================== = = =_= = =================== =!="=#=$=%=&='=(=)=*=+=,=-=.=/=0=1=2=3=4=5=6=7=8=9=:=;=<===>=?=@=A=B=C=D=E=F=G=H=I=J=K=L=M=N=O=P=Q=R=S=T=U=V=W=X=Y=Z=[=\=]=^=_=`=a=b=c=d=e=f=g=h=i=j=k=l=m=n=o=p=q=r=s=t=u>v>w>x>y>z>{>|>}>~>?????????????????????@@@@AAAAAAAAAAAAAAAAAAAABB                                 sbv-3.2Data.SBVData.SBV.InternalsData.SBV.Bridge.BoolectorData.SBV.Bridge.CVC4Data.SBV.Bridge.MathSATData.SBV.Bridge.YicesData.SBV.Bridge.Z3&Data.SBV.Examples.BitPrecise.BitTricks#Data.SBV.Examples.BitPrecise.Legato&Data.SBV.Examples.BitPrecise.MergeSort&Data.SBV.Examples.BitPrecise.PrefixSum'Data.SBV.Examples.CodeGeneration.AddSub)Data.SBV.Examples.CodeGeneration.CRC_USB5*Data.SBV.Examples.CodeGeneration.Fibonacci$Data.SBV.Examples.CodeGeneration.GCD0Data.SBV.Examples.CodeGeneration.PopulationCount.Data.SBV.Examples.CodeGeneration.UninterpretedData.SBV.Examples.Crypto.AESData.SBV.Examples.Crypto.RC4,Data.SBV.Examples.Existentials.CRCPolynomial*Data.SBV.Examples.Existentials.DiophantineData.SBV.Examples.Misc.FloatingData.SBV.Examples.Misc.SBranch#Data.SBV.Examples.Misc.ModelExtract)Data.SBV.Examples.Polynomials.PolynomialsData.SBV.Examples.Puzzles.Coins Data.SBV.Examples.Puzzles.Counts%Data.SBV.Examples.Puzzles.DogCatMouse"Data.SBV.Examples.Puzzles.Euler185%Data.SBV.Examples.Puzzles.MagicSquare!Data.SBV.Examples.Puzzles.NQueens Data.SBV.Examples.Puzzles.Sudoku"Data.SBV.Examples.Puzzles.U2Bridge#Data.SBV.Examples.Uninterpreted.AUF&Data.SBV.Examples.Uninterpreted.Deduce(Data.SBV.Examples.Uninterpreted.Function'Data.SBV.Examples.Uninterpreted.Shannon$Data.SBV.Examples.Uninterpreted.Sort,Data.SBV.Examples.Uninterpreted.UISortAllSatData.SBV.Utils.TDiffData.SBV.Utils.LibData.SBV.Utils.BooleanData.SBV.BitVectors.AlgRealsData.SBV.BitVectors.DataData.SBV.BitVectors.PrettyNumData.SBV.Compilers.CodeGenData.SBV.Compilers.CData.SBV.Tools.ExpectedValueData.SBV.SMT.SMTLib1Data.SBV.Tools.GenTestData.SBV.SMT.SMTData.SBV.Provers.SExprData.SBV.SMT.SMTLib2Data.SBV.SMT.SMTLibData.SBV.Provers.YicesData.SBV.Provers.BoolectorData.SBV.Provers.CVC4Data.SBV.Provers.MathSATData.SBV.Provers.Z3Data.SBV.Provers.ProverData.SBV.BitVectors.ModelData.SBV.BitVectors.SignCastData.SBV.BitVectors.SplittableData.SBV.BitVectors.STreeData.SBV.Tools.PolynomialData.SBV.Tools.OptimizeBooleantruefalsebnot&&&|||~&~|<+>==><=>fromBoolbAndbOrbAnybAllAlgReal AlgPolyRoot AlgRational SMTSolvername executableoptionsengine xformExitCode capabilitiesSolverMathSATCVC4 BoolectorYicesZ3 SMTResultTimeOut ProofErrorUnknown Satisfiable Unsatisfiable SMTConfigverbosetimingsBranchTimeOuttimeOut printBase printRealPrec solverTweakssatCmdsmtFile useSMTLib2solver roundingModeuseLogicLogic CustomLogicPredefinedLogic SMTLibLogicQF_FPAQF_FPABVQF_UFNRAQF_UFLRAQF_UFLIAQF_UFIDLQF_UFBVQF_UFQF_RDLQF_NRAQF_NIAQF_LRAQF_LIAQF_IDLQF_BVQF_AX QF_AUFLIAQF_AUFBVQF_ABVUFNIAUFLRALRAAUFNIRAAUFLIRAAUFLIA SFunArraySArraySymArray newArray_newArray readArray resetArray writeArray mergeArraysSymWordforallforall_ mkForallVarsexistsexists_ mkExistVarsfreefree_ mkFreeVarssymbolic symbolicsliteral unliteralfromCW isConcrete isSymbolic isConcretely mbMaxBound mbMinBound mkSymWordoutputSymbolic RoundingModeRoundTowardZeroRoundTowardNegativeRoundTowardPositiveRoundNearestTiesToAwayRoundNearestTiesToEvenSDoubleSFloatSRealSIntegerSInt64SInt32SInt16SInt8SWord64SWord32SWord16SWord8SBoolSBV 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extractModelsgetModelDictionariesgetModelValuesgetModelUninterpretedValues displayModelsProvableforAll_forAllforSome_forSome Predicate boolectorcvc4yicesz3mathSAT defaultSMTCfgprovesatallSat isVacuous isTheoremWithisSatisfiableWith isTheorem isSatisfiablecompileToSMTLibgenerateSMTBenchmarks proveWithsatWith isVacuousWith allSatWith Uninterpreted uninterpret cgUninterpretsbvUninterpret Mergeable symbolicMergeselect SDivisiblesQuotRemsDivModsQuotsRemsDivsMod SIntegral OrdSymbolic.<.>=.>.<=sminsmax EqSymbolic.==./=genVargenVar_sBoolsBoolssWord8sWord8ssWord16sWord16ssWord32sWord32ssWord64sWord64ssInt8sInt8ssInt16sInt16ssInt32sInt32ssInt64sInt64ssInteger sIntegerssRealsRealssFloatsFloatssDoublesDoublestoSReal allDifferentallEqualinRangesElemoneIffusedMA sbvTestBit sbvPopCountsetBitTo sbvShiftLeft sbvShiftRightsbvSignedShiftArithRight fullAdderfullMultiplierblastLEblastBElsbmsbiteiteLazysBranchsAssert sAssertCont constrain pConstrainsletSignCastsignCast unsignCastFromBits fromBitsLE fromBitsBE Splittablesplit#extendSTree readSTree writeSTreemkSTree Polynomial polynomialpAddpMultpDivpModpDivModshowPolyshowPolynomialcrcBVcrc OptimizeOpts Quantified Iterative optimizeWithoptimize maximizeWithmaximize minimizeWithminimizeEquality===sbvCurrentSolverisSNaN isFPPointsolvesbvCheckSolverInstallationdefaultSolverConfigsbvAvailableSolvers proveWithAll proveWithAny satWithAll satWithAny allSatWithAll allSatWithAnyfastMinCorrectfastMaxCorrectoppositeSignsCorrectconditionalSetClearCorrectpowerOfTwoCorrectqueriesModelInitVals InstructionProgramExtractMostekmemory registersflagsMemoryFlags RegistersBitValueFlagFlagZFlagCRegisterRegARegXAddressgetRegsetReggetFlagsetFlagpeekpoke checkOverflowcheckOverflowCorrectldxldaclcrorMrorRbccadcdexbneendlegato runLegato initMachinelegatoIsCorrectcorrectnessTheorem legatoInC$fMergeableMostekEmerge mergeSort nonDecreasingisPermutationOf correctnesscodeGen PowerListtiePLzipPLunzipPLpslf flIsCorrectthm1thm2thm3genPrefixSumInstance prefixSum yices1029 yicesSMT09ladnerFischerTrace scanlTraceaddSub genAddSubusb5crcUSBcrcUSB'crcGoodcg1cg2fib0fib1genFib1fib2genFib2sgcd sgcdIsCorrect genGCDInC popCountSlow popCountFastpop8fastPopCountIsCorrectgenPopCountInC shiftLeft tstShiftLeftgenCCodeKSKeyStateGF28gf28Multgf28Pow gf28InversetoBytes fromBytesrotRroundConstants invMixColumns keyExpansion sboxTablesbox unSBoxTableunSBoxsboxInverseCorrect addRoundKeyt0Funct0t1t2t3u0Funcu0u1u2u3doRoundsaesRound aesInvRoundaesKeySchedule aesEncrypt aesDecryptt128Enct128Dect192Enct192Dect256Enct256Decaes128IsCorrectcgAES128BlockEncryptaes128LibComponentscgAES128LibraryRC4SinitSswapprgainitRC4 keySchedulekeyScheduleStringencryptdecrypt rc4IsCorrectSWord48 crc_48_16 diffCountgenPolyfindHD4PolynomialsSolutionNonHomogeneous Homogeneousldnbasistestsailors assocPlusassocPlusRegularnonZeroAddition multInversebitCountproppath pathCheckoutsidegenValsgfMultmultUnitmultComm multAssoc polyDivModtestGF28CoinmkCoin 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$fProvableSBV$fProvableSymbolicGHC.RealIntegralRealGHC.EnumEnumquotRemdivModquotdiv GHC.ClassesEqOrdOrderingmaxmin ghcBitSize genLiteral genFromCW genMkSymVareqOptlift1Flift1FNSlift2FNS Data.BitstestBitpopCountsetBitclearBitshiftLshiftRenumCvtreduceInPathCondition $fFloatingSBVnoUnintnoUnint2liftSym1liftSW2liftSym2 liftSym2B liftSym1Bool liftSym2Bool mkSymOpSCmkSymOp mkSymOp1SCmkSymOp1 rationalCheckrationalSBVChecknoRealnoFloatnoDouble noRealUnary noFloatUnary noDoubleUnaryliftQRemliftDMod__unused$fTestableSymbolic $fTestableSBV$fUninterpreted(->)$fUninterpreted(->)0$fUninterpreted(->)1$fUninterpreted(->)2$fUninterpreted(->)3$fUninterpreted(->)4$fUninterpreted(->)5$fUninterpreted(->)6$fUninterpreted(->)7$fUninterpreted(->)8$fUninterpreted(->)9$fUninterpreted(->)10$fUninterpreted(->)11$fUninterpretedSBV$fMergeableSFunArray$fSymArraySFunArray$fMergeableSArray$fEqSymbolicSArray 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$fFromBitsSBV$fFromBitsSBV0$fFromBitsSBV1$fFromBitsSBV2$fFromBitsSBV3$fFromBitsSBV4$fFromBitsSBV5$fFromBitsSBV6$fFromBitsSBV7$fSplittableSBVSBV$fSplittableSBVSBV0$fSplittableSBVSBV1$fSplittableWord16Word8$fSplittableWord32Word16$fSplittableWord64Word32 STreeInternalSBinSLeaf$fMergeableSTreeInternalspaddPolypolyMultliftliftCliftSitesdegreemdpidxdivx$fPolynomialSBV$fPolynomialSBV0$fPolynomialSBV1$fPolynomialSBV2$fPolynomialWord64$fPolynomialWord32$fPolynomialWord16$fPolynomialWord8 quantOptimize iterOptimizeRealFrac sbvWithAny sbvWithAll$fEquality(->)$fEquality(->)0$fEquality(->)1$fEquality(->)2$fEquality(->)3$fEquality(->)4$fEquality(->)5$fEquality(->)6$fEquality(->)7$fEquality(->)8$fEquality(->)9$fEquality(->)10$fEquality(->)11RatioRationalWord Data.RatioapproxRational byteSwap64 byteSwap32 byteSwap16popCountDefaulttestBitDefault bitDefaultrotateRrotateL unsafeShiftR unsafeShiftLisSignedbitSize bitSizeMaybe complementBitbitzeroBitsrotateshift complementxor.|..&.Bits finiteBitSize 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