sbv-5.12: SMT Based Verification: Symbolic Haskell theorem prover using SMT solving.

Data.SBV

Description

(The sbv library is hosted at http://github.com/LeventErkok/sbv. Comments, bug reports, and patches are always welcome.)

SBV: SMT Based Verification

Express properties about Haskell programs and automatically prove them using SMT solvers.

>>> prove $\x -> x shiftL 2 .== 4 * (x :: SWord8) Q.E.D.  >>> prove$ \x -> x shiftL 2 .== 2 * (x :: SWord8)
Falsifiable. Counter-example:
s0 = 32 :: Word8


The function prove has the following type:

    prove :: Provable a => a -> IO ThmResult


The class Provable comes with instances for n-ary predicates, for arbitrary n. The predicates are just regular Haskell functions over symbolic types listed below. Functions for checking satisfiability (sat and allSat) are also provided.

The sbv library introduces the following symbolic types:

• SBool: Symbolic Booleans (bits).
• SWord8, SWord16, SWord32, SWord64: Symbolic Words (unsigned).
• SInt8, SInt16, SInt32, SInt64: Symbolic Ints (signed).
• SInteger: Unbounded signed integers.
• SReal: Algebraic-real numbers
• SFloat: IEEE-754 single-precision floating point values
• SDouble: IEEE-754 double-precision floating point values
• SArray, SFunArray: Flat arrays of symbolic values.
• Symbolic polynomials over GF(2^n), polynomial arithmetic, and CRCs.
• Uninterpreted constants and functions over symbolic values, with user defined SMT-Lib axioms.
• Uninterpreted sorts, and proofs over such sorts, potentially with axioms.

The user can construct ordinary Haskell programs using these types, which behave very similar to their concrete counterparts. In particular these types belong to the standard classes Num, Bits, custom versions of Eq (EqSymbolic) and Ord (OrdSymbolic), along with several other custom classes for simplifying programming with symbolic values. The framework takes full advantage of Haskell's type inference to avoid many common mistakes.

Furthermore, predicates (i.e., functions that return SBool) built out of these types can also be:

• proven correct via an external SMT solver (the prove function)
• checked for satisfiability (the sat, allSat functions)
• used in synthesis (the sat function with existentials)
• quick-checked

If a predicate is not valid, prove will return a counterexample: An assignment to inputs such that the predicate fails. The sat function will return a satisfying assignment, if there is one. The allSat function returns all satisfying assignments, lazily.

The sbv library uses third-party SMT solvers via the standard SMT-Lib interface: http://smtlib.cs.uiowa.edu/

The SBV library is designed to work with any SMT-Lib compliant SMT-solver. Currently, we support the following SMT-Solvers out-of-the box:

SBV also allows calling these solvers in parallel, either getting results from multiple solvers or returning the fastest one. (See proveWithAll, proveWithAny, etc.)

Support for other compliant solvers can be added relatively easily, please get in touch if there is a solver you'd like to see included.

Synopsis

# Programming with symbolic values

The SBV library is really two things:

• A framework for writing symbolic programs in Haskell, i.e., programs operating on symbolic values along with the usual concrete counterparts.
• A framework for proving properties of such programs using SMT solvers.

The programming goal of SBV is to provide a seamless experience, i.e., let people program in the usual Haskell style without distractions of symbolic coding. While Haskell helps in some aspects (the Num and Bits classes simplify coding), it makes life harder in others. For instance, if-then-else only takes Bool as a test in Haskell, and comparisons (> etc.) only return Bools. Clearly we would like these values to be symbolic (i.e., SBool), thus stopping us from using some native Haskell constructs. When symbolic versions of operators are needed, they are typically obtained by prepending a dot, for instance == becomes .==. Care has been taken to make the transition painless. In particular, any Haskell program you build out of symbolic components is fully concretely executable within Haskell, without the need for any custom interpreters. (They are truly Haskell programs, not AST's built out of pieces of syntax.) This provides for an integrated feel of the system, one of the original design goals for SBV.

## Symbolic types

### Symbolic bit

type SBool = SBV Bool Source #

A symbolic boolean/bit

### Unsigned symbolic bit-vectors

8-bit unsigned symbolic value

16-bit unsigned symbolic value

32-bit unsigned symbolic value

64-bit unsigned symbolic value

### Signed symbolic bit-vectors

type SInt8 = SBV Int8 Source #

8-bit signed symbolic value, 2's complement representation

16-bit signed symbolic value, 2's complement representation

32-bit signed symbolic value, 2's complement representation

64-bit signed symbolic value, 2's complement representation

### Signed unbounded integers

The SBV library supports unbounded signed integers with the type SInteger, which are not subject to overflow/underflow as it is the case with the bounded types, such as SWord8, SInt16, etc. However, some bit-vector based operations are not supported for the SInteger type while in the verification mode. That is, you can use these operations on SInteger values during normal programming/simulation. but the SMT translation will not support these operations since there corresponding operations are not supported in SMT-Lib. Note that this should rarely be a problem in practice, as these operations are mostly meaningful on fixed-size bit-vectors. The operations that are restricted to bounded word/int sizes are:

• Rotations and shifts: rotateL, rotateR, shiftL, shiftR
• Bitwise logical ops: .&., .|., xor, complement
• Extraction and concatenation: split, #, and extend (see the Splittable class)

Usual arithmetic (+, -, *, sQuotRem, sQuot, sRem, sDivMod, sDiv, sMod) and logical operations (.<, .<=, .>, .>=, .==, ./=) operations are supported for SInteger fully, both in programming and verification modes.

Infinite precision signed symbolic value

### IEEE-floating point numbers

Floating point numbers are defined by the IEEE-754 standard; and correspond to Haskell's Float and Double types. For SMT support with floating-point numbers, see the paper by Rummer and Wahl: http://www.philipp.ruemmer.org/publications/smt-fpa.pdf.

IEEE-754 single-precision floating point numbers

IEEE-754 double-precision floating point numbers

class (SymWord a, RealFloat a) => IEEEFloating a where Source #

A class of floating-point (IEEE754) operations, some of which behave differently based on rounding modes. Note that unless the rounding mode is concretely RoundNearestTiesToEven, we will not concretely evaluate these, but rather pass down to the SMT solver.

Methods

fpAbs :: SBV a -> SBV a Source #

Compute the floating point absolute value.

fpNeg :: SBV a -> SBV a Source #

Compute the unary negation. Note that 0 - x is not equivalent to -x for floating-point, since -0 and 0 are different.

fpAdd :: SRoundingMode -> SBV a -> SBV a -> SBV a Source #

Add two floating point values, using the given rounding mode

fpSub :: SRoundingMode -> SBV a -> SBV a -> SBV a Source #

Subtract two floating point values, using the given rounding mode

fpMul :: SRoundingMode -> SBV a -> SBV a -> SBV a Source #

Multiply two floating point values, using the given rounding mode

fpDiv :: SRoundingMode -> SBV a -> SBV a -> SBV a Source #

Divide two floating point values, using the given rounding mode

fpFMA :: SRoundingMode -> SBV a -> SBV a -> SBV a -> SBV a Source #

Fused-multiply-add three floating point values, using the given rounding mode. fpFMA x y z = x*y+z but with only one rounding done for the whole operation; not two. Note that we will never concretely evaluate this function since Haskell lacks an FMA implementation.

fpSqrt :: SRoundingMode -> SBV a -> SBV a Source #

Compute the square-root of a float, using the given rounding mode

fpRem :: SBV a -> SBV a -> SBV a Source #

Compute the remainder: x - y * n, where n is the truncated integer nearest to x/y. The rounding mode is implicitly assumed to be RoundNearestTiesToEven.

Round to the nearest integral value, using the given rounding mode.

fpMin :: SBV a -> SBV a -> SBV a Source #

Compute the minimum of two floats, respects infinity and NaN values

fpMax :: SBV a -> SBV a -> SBV a Source #

Compute the maximum of two floats, respects infinity and NaN values

fpIsEqualObject :: SBV a -> SBV a -> SBool Source #

Are the two given floats exactly the same. That is, NaN will compare equal to itself, +0 will not compare equal to -0 etc. This is the object level equality, as opposed to the semantic equality. (For the latter, just use .==.)

Is the floating-point number a normal value. (i.e., not denormalized.)

Is the floating-point number a subnormal value. (Also known as denormal.)

fpIsZero :: SBV a -> SBool Source #

Is the floating-point number 0? (Note that both +0 and -0 will satisfy this predicate.)

Is the floating-point number infinity? (Note that both +oo and -oo will satisfy this predicate.)

fpIsNaN :: SBV a -> SBool Source #

Is the floating-point number a NaN value?

Is the floating-point number negative? Note that -0 satisfies this predicate but +0 does not.

Is the floating-point number positive? Note that +0 satisfies this predicate but -0 does not.

Is the floating point number -0?

Is the floating point number +0?

fpIsPoint :: SBV a -> SBool Source #

Is the floating-point number a regular floating point, i.e., not NaN, nor +oo, nor -oo. Normals or denormals are allowed.

Instances

 Source # SDouble instance Methods Source # SFloat instance Methods

class IEEEFloatConvertable a where Source #

Capture convertability from/to FloatingPoint representations NB. fromSFloat and fromSDouble are underspecified when given when given a NaN, +oo, or -oo value that cannot be represented in the target domain. For these inputs, we define the result to be +0, arbitrarily.

Minimal complete definition

Methods

Instances

 Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods

Rounding mode to be used for the IEEE floating-point operations. Note that Haskell's default is RoundNearestTiesToEven. If you use a different rounding mode, then the counter-examples you get may not match what you observe in Haskell.

Constructors

 RoundNearestTiesToEven Round 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.) RoundNearestTiesToAway Round 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.) RoundTowardPositive Round towards positive infinity. (Also known as rounding-up or ceiling.) RoundTowardNegative Round towards negative infinity. (Also known as rounding-down or floor.) RoundTowardZero Round towards zero. (Also known as truncation.)

Instances

 Source # Methods Source # Methods Source # Methods Source # Methodsgfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> RoundingMode -> c RoundingMode #gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c RoundingMode #dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c RoundingMode) #dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c RoundingMode) #gmapT :: (forall b. Data b => b -> b) -> RoundingMode -> RoundingMode #gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> RoundingMode -> r #gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> RoundingMode -> r #gmapQ :: (forall d. Data d => d -> u) -> RoundingMode -> [u] #gmapQi :: Int -> (forall d. Data d => d -> u) -> RoundingMode -> u #gmapM :: Monad m => (forall d. Data d => d -> m d) -> RoundingMode -> m RoundingMode #gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> RoundingMode -> m RoundingMode #gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> RoundingMode -> m RoundingMode # Source # Methods Source # Methods Source # MethodsshowList :: [RoundingMode] -> ShowS # Source # RoundingMode kind Methods Source # RoundingMode can be used symbolically Methods Source # A rounding mode, extracted from a model. (Default definition suffices) MethodsparseCWs :: [CW] -> Maybe (RoundingMode, [CW]) Source #cvtModel :: (RoundingMode -> Maybe b) -> Maybe (RoundingMode, [CW]) -> Maybe (b, [CW]) Source #

The symbolic variant of RoundingMode

nan :: Floating a => a Source #

Not-A-Number for Double and Float. Surprisingly, Haskell Prelude doesn't have this value defined, so we provide it here.

infinity :: Floating a => a Source #

Infinity for Double and Float. Surprisingly, Haskell Prelude doesn't have this value defined, so we provide it here.

sNaN :: (Floating a, SymWord a) => SBV a Source #

Symbolic variant of Not-A-Number. This value will inhabit both SDouble and SFloat.

sInfinity :: (Floating a, SymWord a) => SBV a Source #

Symbolic variant of infinity. This value will inhabit both SDouble and SFloat.

#### Rounding modes

Symbolic variant of RoundNearestTiesToEven

Symbolic variant of RoundNearestTiesToAway

Symbolic variant of RoundNearestPositive

Symbolic variant of RoundTowardNegative

Symbolic variant of RoundTowardZero

Alias for sRoundNearestTiesToEven

Alias for sRoundNearestTiesToAway

Alias for sRoundTowardPositive

Alias for sRoundTowardNegative

Alias for sRoundTowardZero

#### Bit-pattern conversions

Convert an SFloat to an SWord32, preserving the bit-correspondence. Note that since the representation for NaNs are not unique, this function will return a symbolic value when given a concrete NaN.

Implementation note: Since there's no corresponding function in SMTLib for conversion to bit-representation due to partiality, we use a translation trick by allocating a new word variable, converting it to float, and requiring it to be equivalent to the input. In code-generation mode, we simply map it to a simple conversion.

Reinterpret the bits in a 32-bit word as a single-precision floating point number

Convert an SDouble to an SWord64, preserving the bit-correspondence. Note that since the representation for NaNs are not unique, this function will return a symbolic value when given a concrete NaN.

See the implementation note for sFloatAsSWord32, as it applies here as well.

Reinterpret the bits in a 32-bit word as a single-precision floating point number

blastSFloat :: SFloat -> (SBool, [SBool], [SBool]) Source #

Extract the sign/exponent/mantissa of a single-precision float. The output will have 8 bits in the second argument for exponent, and 23 in the third for the mantissa.

blastSDouble :: SDouble -> (SBool, [SBool], [SBool]) Source #

Extract the sign/exponent/mantissa of a single-precision float. The output will have 11 bits in the second argument for exponent, and 52 in the third for the mantissa.

### Signed algebraic reals

Algebraic reals are roots of single-variable polynomials with rational coefficients. (See http://en.wikipedia.org/wiki/Algebraic_number.) Note that algebraic reals are infinite precision numbers, but they do not cover all real numbers. (In particular, they cannot represent transcendentals.) Some irrational numbers are algebraic (such as sqrt 2), while others are not (such as pi and e).

SBV can deal with real numbers just fine, since the theory of reals is decidable. (See http://smtlib.cs.uiowa.edu/theories-Reals.shtml.) In addition, by leveraging backend solver capabilities, SBV can also represent and solve non-linear equations involving real-variables. (For instance, the Z3 SMT solver, supports polynomial constraints on reals starting with v4.0.)

Infinite precision symbolic algebraic real value

data AlgReal Source #

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 equation

Instances

 Source # Methods(==) :: AlgReal -> AlgReal -> Bool #(/=) :: AlgReal -> AlgReal -> Bool # Source # NB: Following the other types we have, we require a/0 to be 0 for all a. Methods Source # Methods Source # Methods(<) :: AlgReal -> AlgReal -> Bool #(<=) :: AlgReal -> AlgReal -> Bool #(>) :: AlgReal -> AlgReal -> Bool #(>=) :: AlgReal -> AlgReal -> Bool # Source # Methods Source # MethodsshowList :: [AlgReal] -> ShowS # Random AlgReal Source # MethodsrandomR :: RandomGen g => (AlgReal, AlgReal) -> g -> (AlgReal, g)random :: RandomGen g => g -> (AlgReal, g)randomRs :: RandomGen g => (AlgReal, AlgReal) -> g -> [AlgReal]randoms :: RandomGen g => g -> [AlgReal]randomRIO :: (AlgReal, AlgReal) -> IO AlgReal Arbitrary AlgReal Source # Methodsarbitrary :: Gen AlgRealshrink :: AlgReal -> [AlgReal] Source # Methods Source # AlgReal as extracted from a model MethodsparseCWs :: [CW] -> Maybe (AlgReal, [CW]) Source #cvtModel :: (AlgReal -> Maybe b) -> Maybe (AlgReal, [CW]) -> Maybe (b, [CW]) Source # Source # Methods

Promote an SInteger to an SReal

## Creating a symbolic variable

These functions simplify declaring symbolic variables of various types. Strictly speaking, they are just synonyms for free (specialized at the given type), but they might be easier to use.

Declare an SBool

Declare an SWord8

Declare an SWord16

Declare an SWord32

Declare an SWord64

Declare an SInt8

Declare an SInt16

Declare an SInt32

Declare an SInt64

Declare an SInteger

Declare an SReal

Declare an SFloat

Declare an SDouble

## Creating a list of symbolic variables

These functions simplify declaring a sequence symbolic variables of various types. Strictly speaking, they are just synonyms for mapM free (specialized at the given type), but they might be easier to use.

sBools :: [String] -> Symbolic [SBool] Source #

Declare a list of SBools

sWord8s :: [String] -> Symbolic [SWord8] Source #

Declare a list of SWord8s

sWord16s :: [String] -> Symbolic [SWord16] Source #

Declare a list of SWord16s

sWord32s :: [String] -> Symbolic [SWord32] Source #

Declare a list of SWord32s

sWord64s :: [String] -> Symbolic [SWord64] Source #

Declare a list of SWord64s

sInt8s :: [String] -> Symbolic [SInt8] Source #

Declare a list of SInt8s

sInt16s :: [String] -> Symbolic [SInt16] Source #

Declare a list of SInt16s

sInt32s :: [String] -> Symbolic [SInt32] Source #

Declare a list of SInt32s

sInt64s :: [String] -> Symbolic [SInt64] Source #

Declare a list of SInt64s

sIntegers :: [String] -> Symbolic [SInteger] Source #

Declare a list of SIntegers

sReals :: [String] -> Symbolic [SReal] Source #

Declare a list of SReals

sFloats :: [String] -> Symbolic [SFloat] Source #

Declare a list of SFloats

sDoubles :: [String] -> Symbolic [SDouble] Source #

Declare a list of SDoubles

### Abstract SBV type

data SBV a Source #

The Symbolic value. The parameter a is phantom, but is extremely important in keeping the user interface strongly typed.

Instances

### Arrays of symbolic values

class SymArray array where Source #

Flat arrays of symbolic values An array a b is an array indexed by the type SBV a, with elements of type SBV b If an initial value is not provided in newArray_ and newArray 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 SymArray, the SArray and SFunArray instances already provided should cover most use cases in practice. (There are some differences between these models, however, see the corresponding declaration.)

Minimal complete definition: All methods are required, no defaults.

Minimal complete definition

Methods

newArray_ :: (HasKind a, HasKind b) => Maybe (SBV b) -> Symbolic (array a b) Source #

Create a new array, with an optional initial value

newArray :: (HasKind a, HasKind b) => String -> Maybe (SBV b) -> Symbolic (array a b) Source #

Create a named new array, with an optional initial value

readArray :: array a b -> SBV a -> SBV b Source #

Read the array element at a

resetArray :: SymWord b => array a b -> SBV b -> array a b Source #

Reset all the elements of the array to the value b

writeArray :: SymWord b => array a b -> SBV a -> SBV b -> array a b Source #

Update the element at a to be b

mergeArrays :: SymWord b => SBV Bool -> array a b -> array a b -> array a b Source #

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

Instances

 Source # MethodsnewArray_ :: (HasKind a, HasKind b) => Maybe (SBV b) -> Symbolic (SArray a b) Source #newArray :: (HasKind a, HasKind b) => String -> Maybe (SBV b) -> Symbolic (SArray a b) Source #readArray :: SArray a b -> SBV a -> SBV b Source #resetArray :: SymWord b => SArray a b -> SBV b -> SArray a b Source #writeArray :: SymWord b => SArray a b -> SBV a -> SBV b -> SArray a b Source #mergeArrays :: SymWord b => SBV Bool -> SArray a b -> SArray a b -> SArray a b Source #

data SArray a b Source #

Arrays implemented in terms of SMT-arrays: http://smtlib.cs.uiowa.edu/theories-ArraysEx.shtml

• Maps directly to SMT-lib arrays
• Reading from an unintialized value is OK and yields an unspecified result
• Can check for equality of these arrays
• Cannot quick-check theorems using SArray values
• Typically slower as it heavily relies on SMT-solving for the array theory

Instances

 Source # MethodsnewArray_ :: (HasKind a, HasKind b) => Maybe (SBV b) -> Symbolic (SArray a b) Source #newArray :: (HasKind a, HasKind b) => String -> Maybe (SBV b) -> Symbolic (SArray a b) Source #readArray :: SArray a b -> SBV a -> SBV b Source #resetArray :: SymWord b => SArray a b -> SBV b -> SArray a b Source #writeArray :: SymWord b => SArray a b -> SBV a -> SBV b -> SArray a b Source #mergeArrays :: SymWord b => SBV Bool -> SArray a b -> SArray a b -> SArray a b Source # (HasKind a, HasKind b) => Show (SArray a b) Source # MethodsshowsPrec :: Int -> SArray a b -> ShowS #show :: SArray a b -> String #showList :: [SArray a b] -> ShowS # (HasKind a, HasKind b, Provable p) => Provable (SArray a b -> p) Source # MethodsforAll_ :: (SArray a b -> p) -> Predicate Source #forAll :: [String] -> (SArray a b -> p) -> Predicate Source #forSome_ :: (SArray a b -> p) -> Predicate Source #forSome :: [String] -> (SArray a b -> p) -> Predicate Source # SymWord b => Mergeable (SArray a b) Source # MethodssymbolicMerge :: Bool -> SBool -> SArray a b -> SArray a b -> SArray a b Source #select :: (SymWord b, Num b) => [SArray a b] -> SArray a b -> SBV b -> SArray a b Source # EqSymbolic (SArray a b) Source # Methods(.==) :: SArray a b -> SArray a b -> SBool Source #(./=) :: SArray a b -> SArray a b -> SBool Source #

data SFunArray a b Source #

Arrays implemented internally as functions

• Internally handled by the library and not mapped to SMT-Lib
• Reading 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 translation

Instances

 (HasKind a, HasKind b) => Show (SFunArray a b) Source # MethodsshowsPrec :: Int -> SFunArray a b -> ShowS #show :: SFunArray a b -> String #showList :: [SFunArray a b] -> ShowS # (HasKind a, HasKind b, Provable p) => Provable (SFunArray a b -> p) Source # MethodsforAll_ :: (SFunArray a b -> p) -> Predicate Source #forAll :: [String] -> (SFunArray a b -> p) -> Predicate Source #forSome_ :: (SFunArray a b -> p) -> Predicate Source #forSome :: [String] -> (SFunArray a b -> p) -> Predicate Source # SymWord b => Mergeable (SFunArray a b) Source # MethodssymbolicMerge :: Bool -> SBool -> SFunArray a b -> SFunArray a b -> SFunArray a b Source #select :: (SymWord b, Num b) => [SFunArray a b] -> SFunArray a b -> SBV b -> SFunArray a b Source #

mkSFunArray :: (SBV a -> SBV b) -> SFunArray a b Source #

Lift a function to an array. Useful for creating arrays in a pure context. (Otherwise use newArray.)

### Full binary trees

type STree i e = STreeInternal (SBV i) (SBV e) Source #

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. STree structures provide logarithmic time reads and writes.

readSTree :: (Num i, Bits i, SymWord i, SymWord e) => STree i e -> SBV i -> SBV e Source #

Reading a value. We bit-blast the index and descend down the full tree according to bit-values.

writeSTree :: (Mergeable (SBV e), Num i, Bits i, SymWord i, SymWord e) => STree i e -> SBV i -> SBV e -> STree i e Source #

Writing a value, similar to how reads are done. The important thing is that the tree representation keeps updates to a minimum.

mkSTree :: forall i e. HasKind i => [SBV e] -> STree i e Source #

Construct the fully balanced initial tree using the given values.

## Operations on symbolic values

### Word level

sTestBit :: SBV a -> Int -> SBool Source #

Replacement for testBit. Since testBit requires a Bool to be returned, we cannot implement it for symbolic words. Index 0 is the least-significant bit.

sExtractBits :: SBV a -> [Int] -> [SBool] Source #

Variant of sTestBit, where we want to extract multiple bit positions.

sPopCount :: (Num a, Bits a, SymWord a) => SBV a -> SWord8 Source #

Replacement for popCount. Since popCount returns an Int, we cannot implement it for symbolic words. Here, we return an SWord8, which can overflow when used on quantities that have more than 255 bits. Currently, that's only the SInteger type that SBV supports, all other types are safe. Even with SInteger, 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 sPopCount for unbounded symbolic integers, as the only possible implementation wouldn't symbolically terminate. So the only overflow issue is with really-really large concrete SInteger values.

sShiftLeft :: (SIntegral a, SIntegral b) => SBV a -> SBV b -> SBV a Source #

Generalization of shiftL, when the shift-amount is symbolic. Since Haskell's shiftL only takes an Int as the shift amount, it cannot be used when we have a symbolic amount to shift with. The first argument should be a bounded quantity.

sShiftRight :: (SIntegral a, SIntegral b) => SBV a -> SBV b -> SBV a Source #

Generalization of shiftR, when the shift-amount is symbolic. Since Haskell's shiftR only takes an Int as the shift amount, it cannot be used when we have a symbolic amount to shift with. The first argument should be a bounded quantity.

NB. If the shiftee is signed, then this is an arithmetic shift; otherwise it's logical, following the usual Haskell convention. See sSignedShiftArithRight for a variant that explicitly uses the msb as the sign bit, even for unsigned underlying types.

sRotateLeft :: (SIntegral a, SIntegral b, SDivisible (SBV b)) => SBV a -> SBV b -> SBV a Source #

Generalization of rotateL, when the shift-amount is symbolic. Since Haskell's rotateL only takes an Int as the shift amount, it cannot be used when we have a symbolic amount to shift with. The first argument should be a bounded quantity.

sRotateRight :: (SIntegral a, SIntegral b, SDivisible (SBV b)) => SBV a -> SBV b -> SBV a Source #

Generalization of rotateR, when the shift-amount is symbolic. Since Haskell's rotateR only takes an Int as the shift amount, it cannot be used when we have a symbolic amount to shift with. The first argument should be a bounded quantity.

sSignedShiftArithRight :: (SIntegral a, SIntegral b) => SBV a -> SBV b -> SBV a Source #

Arithmetic shift-right with a symbolic unsigned shift amount. This is equivalent to sShiftRight 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.

sFromIntegral :: forall a b. (Integral a, HasKind a, Num a, Bits a, SymWord a, HasKind b, Num b, Bits b, SymWord b) => SBV a -> SBV b Source #

Conversion between integral-symbolic values, akin to Haskell's fromIntegral

setBitTo :: (Num a, Bits a, SymWord a) => SBV a -> Int -> SBool -> SBV a Source #

Generalization of setBit based on a symbolic boolean. Note that setBit and clearBit are still available on Symbolic words, this operation comes handy when the condition to set/clear happens to be symbolic.

oneIf :: (Num a, SymWord a) => SBool -> SBV a Source #

Returns 1 if the boolean is true, otherwise 0.

lsb :: SBV a -> SBool Source #

Least significant bit of a word, always stored at index 0.

msb :: (Num a, Bits a, SymWord a) => SBV a -> SBool Source #

Most significant bit of a word, always stored at the last position.

label :: SymWord a => String -> SBV a -> SBV a Source #

label: Label the result of an expression. This is essentially a no-op, but useful as it generates a comment in the generated C/SMT-Lib code. Note that if the argument is a constant, then the label is dropped completely, per the usual constant folding strategy.

### Predicates

allEqual :: EqSymbolic a => [a] -> SBool Source #

Returns (symbolic) true if all the elements of the given list are the same.

allDifferent :: EqSymbolic a => [a] -> SBool Source #

Returns (symbolic) true if all the elements of the given list are different.

inRange :: OrdSymbolic a => a -> (a, a) -> SBool Source #

Returns (symbolic) true if the argument is in range

sElem :: EqSymbolic a => a -> [a] -> SBool Source #

Symbolic membership test

### Addition and Multiplication with high-bits

fullAdder :: SIntegral a => SBV a -> SBV a -> (SBool, SBV a) Source #

N.B. Only works for unsigned types. Signed arguments will be rejected.

fullMultiplier :: SIntegral a => SBV a -> SBV a -> (SBV a, SBV a) Source #

Full multiplier: Returns both the high-order and the low-order bits in a tuple, thus fully accounting for the overflow.

N.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.

### Exponentiation

(.^) :: (Mergeable b, Num b, SIntegral e) => b -> SBV e -> b Source #

Symbolic exponentiation using bit blasting and repeated squaring.

N.B. The exponent must be unsigned. Signed exponents will be rejected.

### Blasting/Unblasting

blastBE :: (Num a, Bits a, SymWord a) => SBV a -> [SBool] Source #

Big-endian blasting of a word into its bits. Also see the FromBits class.

blastLE :: (Num a, Bits a, SymWord a) => SBV a -> [SBool] Source #

Little-endian blasting of a word into its bits. Also see the FromBits class.

class FromBits a where Source #

Unblasting 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 representations

Minimal complete definition: fromBitsLE

Minimal complete definition

fromBitsLE

Methods

fromBitsLE, fromBitsBE :: [SBool] -> a Source #

Instances

 Source # MethodsfromBitsLE :: [SBool] -> SInt64 Source #fromBitsBE :: [SBool] -> SInt64 Source # Source # MethodsfromBitsLE :: [SBool] -> SInt32 Source #fromBitsBE :: [SBool] -> SInt32 Source # Source # MethodsfromBitsLE :: [SBool] -> SInt16 Source #fromBitsBE :: [SBool] -> SInt16 Source # Source # MethodsfromBitsLE :: [SBool] -> SInt8 Source #fromBitsBE :: [SBool] -> SInt8 Source # Source # MethodsfromBitsLE :: [SBool] -> SWord64 Source #fromBitsBE :: [SBool] -> SWord64 Source # Source # MethodsfromBitsLE :: [SBool] -> SWord32 Source #fromBitsBE :: [SBool] -> SWord32 Source # Source # MethodsfromBitsLE :: [SBool] -> SWord16 Source #fromBitsBE :: [SBool] -> SWord16 Source # Source # MethodsfromBitsLE :: [SBool] -> SWord8 Source #fromBitsBE :: [SBool] -> SWord8 Source # Source # MethodsfromBitsLE :: [SBool] -> SBool Source #fromBitsBE :: [SBool] -> SBool Source # Source # Conversion from bits MethodsfromBitsLE :: [SBool] -> SWord4 Source #fromBitsBE :: [SBool] -> SWord4 Source #

### Splitting, joining, and extending

class Splittable a b | b -> a where Source #

Splitting an a into two b's and joining back. Intuitively, a is a larger bit-size word than b, typically double. The extend operation captures embedding of a b value into an a without changing its semantic value.

Minimal complete definition: All, no defaults.

Minimal complete definition

Methods

split :: a -> (b, b) Source #

(#) :: b -> b -> a infixr 5 Source #

extend :: b -> a Source #

Instances

 Source # Joiningsplitting tofrom Word8 Methodssplit :: Word8 -> (Word4, Word4) Source # Source # Methodssplit :: Word16 -> (Word8, Word8) Source # Source # Methodssplit :: Word32 -> (Word16, Word16) Source # Source # Methodssplit :: Word64 -> (Word32, Word32) Source # Source # Methods Source # Methods Source # Methodssplit :: SWord16 -> (SWord8, SWord8) Source #

## Polynomial arithmetic and CRCs

class (Num a, Bits a) => Polynomial a where Source #

Implements polynomial addition, multiplication, division, and modulus operations over GF(2^n). NB. Similar to sQuotRem, division by 0 is interpreted as follows:

x pDivMod 0 = (0, x)

for all x (including 0)

Minimal complete definition: pMult, pDivMod, showPolynomial

Minimal complete definition

Methods

polynomial :: [Int] -> a Source #

Given bit-positions to be set, create a polynomial For instance

polynomial [0, 1, 3] :: SWord8

will evaluate to 11, since it sets the bits 0, 1, and 3. Mathematicans would write this polynomial as x^3 + x + 1. And in fact, showPoly will show it like that.

pAdd :: a -> a -> a Source #

pMult :: (a, a, [Int]) -> a Source #

Multiply 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.

pDiv :: a -> a -> a Source #

Divide two polynomials in GF(2^n), see above note for division by 0.

pMod :: a -> a -> a Source #

Compute modulus of two polynomials in GF(2^n), see above note for modulus by 0.

pDivMod :: a -> a -> (a, a) Source #

Division and modulus packed together.

showPoly :: a -> String Source #

Display a polynomial like a mathematician would (over the monomial x), with a type.

showPolynomial :: Bool -> a -> String Source #

Display a polynomial like a mathematician would (over the monomial x), the first argument controls if the final type is shown as well.

Instances

 Source # Methodspolynomial :: [Int] -> Word8 Source #pMult :: (Word8, Word8, [Int]) -> Word8 Source #pDivMod :: Word8 -> Word8 -> (Word8, Word8) Source # Source # Methodspolynomial :: [Int] -> Word16 Source #pMult :: (Word16, Word16, [Int]) -> Word16 Source #pDivMod :: Word16 -> Word16 -> (Word16, Word16) Source # Source # Methodspolynomial :: [Int] -> Word32 Source #pMult :: (Word32, Word32, [Int]) -> Word32 Source #pDivMod :: Word32 -> Word32 -> (Word32, Word32) Source # Source # Methodspolynomial :: [Int] -> Word64 Source #pMult :: (Word64, Word64, [Int]) -> Word64 Source #pDivMod :: Word64 -> Word64 -> (Word64, Word64) Source # Source # Methodspolynomial :: [Int] -> SWord64 Source #pMult :: (SWord64, SWord64, [Int]) -> SWord64 Source # Source # Methodspolynomial :: [Int] -> SWord32 Source #pMult :: (SWord32, SWord32, [Int]) -> SWord32 Source # Source # Methodspolynomial :: [Int] -> SWord16 Source #pMult :: (SWord16, SWord16, [Int]) -> SWord16 Source # Source # Methodspolynomial :: [Int] -> SWord8 Source #pMult :: (SWord8, SWord8, [Int]) -> SWord8 Source #pDivMod :: SWord8 -> SWord8 -> (SWord8, SWord8) Source #

crcBV :: Int -> [SBool] -> [SBool] -> [SBool] Source #

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 n specifies how many bits of CRC is needed. Note that n is actually the degree of the polynomial p, and thus it seems redundant to pass it in. However, in a typical proof context, the polynomial can be symbolic, so we cannot compute the degree easily. While this can be worked-around by generating code that accounts for all possible degrees, the resulting code would be unnecessarily big and complicated, and much harder to reason with. (Also note that a CRC is just the remainder from the polynomial division, but this routine is much faster in practice.)

NB. The nth bit of the polynomial p must be set for the CRC to be computed correctly. Note that the polynomial argument p will not even have this bit present most of the time, as it will typically contain bits 0 through n-1 as usual in the CRC literature. The higher order nth bit is simply assumed to be set, as it does not make sense to use a polynomial of a lesser degree. This is usually not a problem since CRC polynomials are designed and expressed this way.

NB. The literature on CRC's has many variants on how CRC's are computed. We follow the following simple procedure:

• Extend the message m by adding n 0 bits on the right
• Divide the polynomial thus obtained by the p
• The 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.

crc :: (FromBits (SBV a), FromBits (SBV b), Num a, Num b, Bits a, Bits b, SymWord a, SymWord b) => Int -> SBV a -> SBV b -> SBV b Source #

Compute CRC's over polynomials, i.e., symbolic words. The first Int argument plays the same role as the one in the crcBV function.

## Conditionals: Mergeable values

class Mergeable a where Source #

Symbolic conditionals are modeled by the Mergeable class, describing how to merge the results of an if-then-else call with a symbolic test. SBV provides all basic types as instances of this class, so users only need to declare instances for custom data-types of their programs as needed.

A Mergeable instance may be automatically derived for a custom data-type with a single constructor where the type of each field is an instance of Mergeable, such as a record of symbolic values. Users only need to add Generic and Mergeable to the deriving clause for the data-type. See Status for an example and an illustration of what the instance would look like if written by hand.

The function select 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 ite function.

Minimal complete definition: None, if the type is instance of Generic. Otherwise symbolicMerge. Note that most types subject to merging are likely to be trivial instances of Generic.

Methods

symbolicMerge :: Bool -> SBool -> a -> a -> a Source #

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.

select :: (SymWord b, Num b) => [a] -> a -> SBV b -> a Source #

Total indexing operation. select xs default index is intuitively the same as xs !! index, except it evaluates to default if index underflows/overflows.

symbolicMerge :: (Generic a, GMergeable (Rep a)) => Bool -> SBool -> a -> a -> a Source #

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.

Instances

 Source # MethodssymbolicMerge :: Bool -> SBool -> () -> () -> () Source #select :: (SymWord b, Num b) => [()] -> () -> SBV b -> () Source # Source # Methodsselect :: (SymWord b, Num b) => [Mostek] -> Mostek -> SBV b -> Mostek Source # Source # Methodsselect :: (SymWord b, Num b) => [Status] -> Status -> SBV b -> Status Source # Mergeable a => Mergeable [a] Source # MethodssymbolicMerge :: Bool -> SBool -> [a] -> [a] -> [a] Source #select :: (SymWord b, Num b) => [[a]] -> [a] -> SBV b -> [a] Source # Mergeable a => Mergeable (Maybe a) Source # MethodssymbolicMerge :: Bool -> SBool -> Maybe a -> Maybe a -> Maybe a Source #select :: (SymWord b, Num b) => [Maybe a] -> Maybe a -> SBV b -> Maybe a Source # SymWord a => Mergeable (SBV a) Source # MethodssymbolicMerge :: Bool -> SBool -> SBV a -> SBV a -> SBV a Source #select :: (SymWord b, Num b) => [SBV a] -> SBV a -> SBV b -> SBV a Source # Mergeable a => Mergeable (Move a) Source # Mergeable instance for Move simply pushes the merging the data after run of each branch starting from the same state. MethodssymbolicMerge :: Bool -> SBool -> Move a -> Move a -> Move a Source #select :: (SymWord b, Num b) => [Move a] -> Move a -> SBV b -> Move a Source # Mergeable b => Mergeable (a -> b) Source # MethodssymbolicMerge :: Bool -> SBool -> (a -> b) -> (a -> b) -> a -> b Source #select :: (SymWord b, Num b) => [a -> b] -> (a -> b) -> SBV b -> a -> b Source # (Mergeable a, Mergeable b) => Mergeable (Either a b) Source # MethodssymbolicMerge :: Bool -> SBool -> Either a b -> Either a b -> Either a b Source #select :: (SymWord b, Num b) => [Either a b] -> Either a b -> SBV b -> Either a b Source # (Mergeable a, Mergeable b) => Mergeable (a, b) Source # MethodssymbolicMerge :: Bool -> SBool -> (a, b) -> (a, b) -> (a, b) Source #select :: (SymWord b, Num b) => [(a, b)] -> (a, b) -> SBV b -> (a, b) Source # (Ix a, Mergeable b) => Mergeable (Array a b) Source # MethodssymbolicMerge :: Bool -> SBool -> Array a b -> Array a b -> Array a b Source #select :: (SymWord b, Num b) => [Array a b] -> Array a b -> SBV b -> Array a b Source # SymWord b => Mergeable (SFunArray a b) Source # MethodssymbolicMerge :: Bool -> SBool -> SFunArray a b -> SFunArray a b -> SFunArray a b Source #select :: (SymWord b, Num b) => [SFunArray a b] -> SFunArray a b -> SBV b -> SFunArray a b Source # SymWord b => Mergeable (SArray a b) Source # MethodssymbolicMerge :: Bool -> SBool -> SArray a b -> SArray a b -> SArray a b Source #select :: (SymWord b, Num b) => [SArray a b] -> SArray a b -> SBV b -> SArray a b Source # (SymWord e, Mergeable (SBV e)) => Mergeable (STree i e) Source # MethodssymbolicMerge :: Bool -> SBool -> STree i e -> STree i e -> STree i e Source #select :: (SymWord b, Num b) => [STree i e] -> STree i e -> SBV b -> STree i e Source # (Mergeable a, Mergeable b, Mergeable c) => Mergeable (a, b, c) Source # MethodssymbolicMerge :: Bool -> SBool -> (a, b, c) -> (a, b, c) -> (a, b, c) Source #select :: (SymWord b, Num b) => [(a, b, c)] -> (a, b, c) -> SBV b -> (a, b, c) Source # (Mergeable a, Mergeable b, Mergeable c, Mergeable d) => Mergeable (a, b, c, d) Source # MethodssymbolicMerge :: Bool -> SBool -> (a, b, c, d) -> (a, b, c, d) -> (a, b, c, d) Source #select :: (SymWord b, Num b) => [(a, b, c, d)] -> (a, b, c, d) -> SBV b -> (a, b, c, d) Source # (Mergeable a, Mergeable b, Mergeable c, Mergeable d, Mergeable e) => Mergeable (a, b, c, d, e) Source # MethodssymbolicMerge :: Bool -> SBool -> (a, b, c, d, e) -> (a, b, c, d, e) -> (a, b, c, d, e) Source #select :: (SymWord b, Num b) => [(a, b, c, d, e)] -> (a, b, c, d, e) -> SBV b -> (a, b, c, d, e) Source # (Mergeable a, Mergeable b, Mergeable c, Mergeable d, Mergeable e, Mergeable f) => Mergeable (a, b, c, d, e, f) Source # MethodssymbolicMerge :: Bool -> SBool -> (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> (a, b, c, d, e, f) Source #select :: (SymWord b, Num b) => [(a, b, c, d, e, f)] -> (a, b, c, d, e, f) -> SBV b -> (a, b, c, d, e, f) Source # (Mergeable a, Mergeable b, Mergeable c, Mergeable d, Mergeable e, Mergeable f, Mergeable g) => Mergeable (a, b, c, d, e, f, g) Source # MethodssymbolicMerge :: Bool -> SBool -> (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) Source #select :: (SymWord b, Num b) => [(a, b, c, d, e, f, g)] -> (a, b, c, d, e, f, g) -> SBV b -> (a, b, c, d, e, f, g) Source #

ite :: Mergeable a => SBool -> a -> a -> a Source #

If-then-else. This is by definition symbolicMerge with both branches forced. This is typically the desired behavior, but also see iteLazy should you need more laziness.

iteLazy :: Mergeable a => SBool -> a -> a -> a Source #

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.

## Symbolic equality

class EqSymbolic a where Source #

Symbolic Equality. Note that we can't use Haskell's Eq class since Haskell insists on returning Bool Comparing symbolic values will necessarily return a symbolic value.

Minimal complete definition: .==

Minimal complete definition

(.==)

Methods

(.==), (./=) :: a -> a -> SBool infix 4 .==, ./= Source #

Instances

 Source # Methods EqSymbolic a => EqSymbolic [a] Source # Methods(.==) :: [a] -> [a] -> SBool Source #(./=) :: [a] -> [a] -> SBool Source # EqSymbolic a => EqSymbolic (Maybe a) Source # Methods(.==) :: Maybe a -> Maybe a -> SBool Source #(./=) :: Maybe a -> Maybe a -> SBool Source # Source # Methods(.==) :: SBV a -> SBV a -> SBool Source #(./=) :: SBV a -> SBV a -> SBool Source # (EqSymbolic a, EqSymbolic b) => EqSymbolic (Either a b) Source # Methods(.==) :: Either a b -> Either a b -> SBool Source #(./=) :: Either a b -> Either a b -> SBool Source # (EqSymbolic a, EqSymbolic b) => EqSymbolic (a, b) Source # Methods(.==) :: (a, b) -> (a, b) -> SBool Source #(./=) :: (a, b) -> (a, b) -> SBool Source # EqSymbolic (SArray a b) Source # Methods(.==) :: SArray a b -> SArray a b -> SBool Source #(./=) :: SArray a b -> SArray a b -> SBool Source # (EqSymbolic a, EqSymbolic b, EqSymbolic c) => EqSymbolic (a, b, c) Source # Methods(.==) :: (a, b, c) -> (a, b, c) -> SBool Source #(./=) :: (a, b, c) -> (a, b, c) -> SBool Source # (EqSymbolic a, EqSymbolic b, EqSymbolic c, EqSymbolic d) => EqSymbolic (a, b, c, d) Source # Methods(.==) :: (a, b, c, d) -> (a, b, c, d) -> SBool Source #(./=) :: (a, b, c, d) -> (a, b, c, d) -> SBool Source # (EqSymbolic a, EqSymbolic b, EqSymbolic c, EqSymbolic d, EqSymbolic e) => EqSymbolic (a, b, c, d, e) Source # Methods(.==) :: (a, b, c, d, e) -> (a, b, c, d, e) -> SBool Source #(./=) :: (a, b, c, d, e) -> (a, b, c, d, e) -> SBool Source # (EqSymbolic a, EqSymbolic b, EqSymbolic c, EqSymbolic d, EqSymbolic e, EqSymbolic f) => EqSymbolic (a, b, c, d, e, f) Source # Methods(.==) :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> SBool Source #(./=) :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> SBool Source # (EqSymbolic a, EqSymbolic b, EqSymbolic c, EqSymbolic d, EqSymbolic e, EqSymbolic f, EqSymbolic g) => EqSymbolic (a, b, c, d, e, f, g) Source # Methods(.==) :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> SBool Source #(./=) :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> SBool Source #

## Symbolic ordering

class (Mergeable a, EqSymbolic a) => OrdSymbolic a where Source #

Symbolic Comparisons. Similar to Eq, we cannot implement Haskell's Ord class since there is no way to return an Ordering value from a symbolic comparison. Furthermore, OrdSymbolic requires Mergeable to implement if-then-else, for the benefit of implementing symbolic versions of max and min functions.

Minimal complete definition: .<

Minimal complete definition

(.<)

Methods

(.<), (.<=), (.>), (.>=) :: a -> a -> SBool infix 4 .<, .<=, .>, .>= Source #

smin, smax :: a -> a -> a Source #

Instances

 OrdSymbolic a => OrdSymbolic [a] Source # Methods(.<) :: [a] -> [a] -> SBool Source #(.<=) :: [a] -> [a] -> SBool Source #(.>) :: [a] -> [a] -> SBool Source #(.>=) :: [a] -> [a] -> SBool Source #smin :: [a] -> [a] -> [a] Source #smax :: [a] -> [a] -> [a] Source # OrdSymbolic a => OrdSymbolic (Maybe a) Source # Methods(.<) :: Maybe a -> Maybe a -> SBool Source #(.<=) :: Maybe a -> Maybe a -> SBool Source #(.>) :: Maybe a -> Maybe a -> SBool Source #(.>=) :: Maybe a -> Maybe a -> SBool Source #smin :: Maybe a -> Maybe a -> Maybe a Source #smax :: Maybe a -> Maybe a -> Maybe a Source # SymWord a => OrdSymbolic (SBV a) Source # Methods(.<) :: SBV a -> SBV a -> SBool Source #(.<=) :: SBV a -> SBV a -> SBool Source #(.>) :: SBV a -> SBV a -> SBool Source #(.>=) :: SBV a -> SBV a -> SBool Source #smin :: SBV a -> SBV a -> SBV a Source #smax :: SBV a -> SBV a -> SBV a Source # (OrdSymbolic a, OrdSymbolic b) => OrdSymbolic (Either a b) Source # Methods(.<) :: Either a b -> Either a b -> SBool Source #(.<=) :: Either a b -> Either a b -> SBool Source #(.>) :: Either a b -> Either a b -> SBool Source #(.>=) :: Either a b -> Either a b -> SBool Source #smin :: Either a b -> Either a b -> Either a b Source #smax :: Either a b -> Either a b -> Either a b Source # (OrdSymbolic a, OrdSymbolic b) => OrdSymbolic (a, b) Source # Methods(.<) :: (a, b) -> (a, b) -> SBool Source #(.<=) :: (a, b) -> (a, b) -> SBool Source #(.>) :: (a, b) -> (a, b) -> SBool Source #(.>=) :: (a, b) -> (a, b) -> SBool Source #smin :: (a, b) -> (a, b) -> (a, b) Source #smax :: (a, b) -> (a, b) -> (a, b) Source # (OrdSymbolic a, OrdSymbolic b, OrdSymbolic c) => OrdSymbolic (a, b, c) Source # Methods(.<) :: (a, b, c) -> (a, b, c) -> SBool Source #(.<=) :: (a, b, c) -> (a, b, c) -> SBool Source #(.>) :: (a, b, c) -> (a, b, c) -> SBool Source #(.>=) :: (a, b, c) -> (a, b, c) -> SBool Source #smin :: (a, b, c) -> (a, b, c) -> (a, b, c) Source #smax :: (a, b, c) -> (a, b, c) -> (a, b, c) Source # (OrdSymbolic a, OrdSymbolic b, OrdSymbolic c, OrdSymbolic d) => OrdSymbolic (a, b, c, d) Source # Methods(.<) :: (a, b, c, d) -> (a, b, c, d) -> SBool Source #(.<=) :: (a, b, c, d) -> (a, b, c, d) -> SBool Source #(.>) :: (a, b, c, d) -> (a, b, c, d) -> SBool Source #(.>=) :: (a, b, c, d) -> (a, b, c, d) -> SBool Source #smin :: (a, b, c, d) -> (a, b, c, d) -> (a, b, c, d) Source #smax :: (a, b, c, d) -> (a, b, c, d) -> (a, b, c, d) Source # (OrdSymbolic a, OrdSymbolic b, OrdSymbolic c, OrdSymbolic d, OrdSymbolic e) => OrdSymbolic (a, b, c, d, e) Source # Methods(.<) :: (a, b, c, d, e) -> (a, b, c, d, e) -> SBool Source #(.<=) :: (a, b, c, d, e) -> (a, b, c, d, e) -> SBool Source #(.>) :: (a, b, c, d, e) -> (a, b, c, d, e) -> SBool Source #(.>=) :: (a, b, c, d, e) -> (a, b, c, d, e) -> SBool Source #smin :: (a, b, c, d, e) -> (a, b, c, d, e) -> (a, b, c, d, e) Source #smax :: (a, b, c, d, e) -> (a, b, c, d, e) -> (a, b, c, d, e) Source # (OrdSymbolic a, OrdSymbolic b, OrdSymbolic c, OrdSymbolic d, OrdSymbolic e, OrdSymbolic f) => OrdSymbolic (a, b, c, d, e, f) Source # Methods(.<) :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> SBool Source #(.<=) :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> SBool Source #(.>) :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> SBool Source #(.>=) :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> SBool Source #smin :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> (a, b, c, d, e, f) Source #smax :: (a, b, c, d, e, f) -> (a, b, c, d, e, f) -> (a, b, c, d, e, f) Source # (OrdSymbolic a, OrdSymbolic b, OrdSymbolic c, OrdSymbolic d, OrdSymbolic e, OrdSymbolic f, OrdSymbolic g) => OrdSymbolic (a, b, c, d, e, f, g) Source # Methods(.<) :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> SBool Source #(.<=) :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> SBool Source #(.>) :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> SBool Source #(.>=) :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> SBool Source #smin :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) Source #smax :: (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) -> (a, b, c, d, e, f, g) Source #

## Symbolic integral numbers

class (SymWord a, Num a, Bits a) => SIntegral a Source #

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 SWord8, SInt8 etc. For instance, we can write a generic list-minimum function as follows:

   mm :: SIntegral a => [SBV a] -> SBV a
mm = foldr1 (a b -> ite (a .<= b) a b)


It is similar to the standard Integral class, except ranging over symbolic instances.

Instances

 Source # Source # Source # Source # Source # Source # Source # Source # Source # Source # SIntegral instance, using default methods

## Division

class SDivisible a where Source #

The SDivisible class captures the essence of division. Unfortunately we cannot use Haskell's Integral class since the Real and Enum superclasses are not implementable for symbolic bit-vectors. However, quotRem and divMod makes perfect sense, and the SDivisible class captures this operation. One issue is how division by 0 behaves. The verification technology requires total functions, and there are several design choices here. We follow Isabelle/HOL approach of assigning the value 0 for division by 0. Therefore, we impose the following pair of laws:

     x sQuotRem 0 = (0, x)
x sDivMod  0 = (0, x)


Note that our instances implement this law even when x is 0 itself.

NB. quot truncates toward zero, while div truncates toward negative infinity.

Minimal complete definition: sQuotRem, sDivMod

Minimal complete definition

Methods

sQuotRem :: a -> a -> (a, a) Source #

sDivMod :: a -> a -> (a, a) Source #

sQuot :: a -> a -> a Source #

sRem :: a -> a -> a Source #

sDiv :: a -> a -> a Source #

sMod :: a -> a -> a Source #

Instances

 Source # MethodssQuotRem :: Int8 -> Int8 -> (Int8, Int8) Source #sDivMod :: Int8 -> Int8 -> (Int8, Int8) Source #sQuot :: Int8 -> Int8 -> Int8 Source #sRem :: Int8 -> Int8 -> Int8 Source #sDiv :: Int8 -> Int8 -> Int8 Source #sMod :: Int8 -> Int8 -> Int8 Source # Source # MethodssQuotRem :: Int16 -> Int16 -> (Int16, Int16) Source #sDivMod :: Int16 -> Int16 -> (Int16, Int16) Source # Source # MethodssQuotRem :: Int32 -> Int32 -> (Int32, Int32) Source #sDivMod :: Int32 -> Int32 -> (Int32, Int32) Source # Source # MethodssQuotRem :: Int64 -> Int64 -> (Int64, Int64) Source #sDivMod :: Int64 -> Int64 -> (Int64, Int64) Source # Source # Methods Source # MethodssQuotRem :: Word8 -> Word8 -> (Word8, Word8) Source #sDivMod :: Word8 -> Word8 -> (Word8, Word8) Source # Source # MethodssQuotRem :: Word16 -> Word16 -> (Word16, Word16) Source #sDivMod :: Word16 -> Word16 -> (Word16, Word16) Source # Source # MethodssQuotRem :: Word32 -> Word32 -> (Word32, Word32) Source #sDivMod :: Word32 -> Word32 -> (Word32, Word32) Source # Source # MethodssQuotRem :: Word64 -> Word64 -> (Word64, Word64) Source #sDivMod :: Word64 -> Word64 -> (Word64, Word64) Source # Source # MethodssQuotRem :: CW -> CW -> (CW, CW) Source #sDivMod :: CW -> CW -> (CW, CW) Source #sQuot :: CW -> CW -> CW Source #sRem :: CW -> CW -> CW Source #sDiv :: CW -> CW -> CW Source #sMod :: CW -> CW -> CW Source # Source # Methods Source # MethodssQuotRem :: SInt64 -> SInt64 -> (SInt64, SInt64) Source #sDivMod :: SInt64 -> SInt64 -> (SInt64, SInt64) Source # Source # MethodssQuotRem :: SInt32 -> SInt32 -> (SInt32, SInt32) Source #sDivMod :: SInt32 -> SInt32 -> (SInt32, SInt32) Source # Source # MethodssQuotRem :: SInt16 -> SInt16 -> (SInt16, SInt16) Source #sDivMod :: SInt16 -> SInt16 -> (SInt16, SInt16) Source # Source # MethodssQuotRem :: SInt8 -> SInt8 -> (SInt8, SInt8) Source #sDivMod :: SInt8 -> SInt8 -> (SInt8, SInt8) Source # Source # Methods Source # Methods Source # Methods Source # MethodssQuotRem :: SWord8 -> SWord8 -> (SWord8, SWord8) Source #sDivMod :: SWord8 -> SWord8 -> (SWord8, SWord8) Source # Source # SDvisible instance, using default methods MethodssQuotRem :: SWord4 -> SWord4 -> (SWord4, SWord4) Source #sDivMod :: SWord4 -> SWord4 -> (SWord4, SWord4) Source # Source # SDvisible instance, using 0-extension MethodssQuotRem :: Word4 -> Word4 -> (Word4, Word4) Source #sDivMod :: Word4 -> Word4 -> (Word4, Word4) Source #

## The Boolean class

class Boolean b where Source #

The Boolean class: a generalization of Haskell's Bool type Haskell Bool and SBV's SBool are instances of this class, unifying the treatment of boolean values.

Minimal complete definition: true, bnot, &&& However, it's advisable to define false, and ||| as well (typically), for clarity.

Minimal complete definition

Methods

true :: b Source #

logical true

false :: b Source #

logical false

bnot :: b -> b Source #

complement

(&&&) :: b -> b -> b infixr 3 Source #

and

(|||) :: b -> b -> b infixr 2 Source #

or

(~&) :: b -> b -> b infixr 3 Source #

nand

(~|) :: b -> b -> b infixr 2 Source #

nor

(<+>) :: b -> b -> b infixl 6 Source #

xor

(==>) :: b -> b -> b infixr 1 Source #

implies

(<=>) :: b -> b -> b infixr 1 Source #

equivalence

fromBool :: Bool -> b Source #

cast from Bool

Instances

 Source # Methods(&&&) :: Bool -> Bool -> Bool Source #(|||) :: Bool -> Bool -> Bool Source #(~&) :: Bool -> Bool -> Bool Source #(~|) :: Bool -> Bool -> Bool Source #(<+>) :: Bool -> Bool -> Bool Source #(==>) :: Bool -> Bool -> Bool Source #(<=>) :: Bool -> Bool -> Bool Source #

### Generalizations of boolean operations

bAnd :: Boolean b => [b] -> b Source #

Generalization of and

bOr :: Boolean b => [b] -> b Source #

Generalization of or

bAny :: Boolean b => (a -> b) -> [a] -> b Source #

Generalization of any

bAll :: Boolean b => (a -> b) -> [a] -> b Source #

Generalization of all

## Pretty-printing and reading numbers in Hex & Binary

class PrettyNum a where Source #

PrettyNum class captures printing of numbers in hex and binary formats; also supporting negative numbers.

Minimal complete definition: hexS and binS

Minimal complete definition

Methods

hexS :: a -> String Source #

Show a number in hexadecimal (starting with 0x and type.)

binS :: a -> String Source #

Show a number in binary (starting with 0b and type.)

hex :: a -> String Source #

Show a number in hex, without prefix, or types.

bin :: a -> String Source #

Show a number in bin, without prefix, or types.

Instances

 Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods Source # Methods (SymWord a, PrettyNum a) => PrettyNum (SBV a) Source # MethodshexS :: SBV a -> String Source #binS :: SBV a -> String Source #hex :: SBV a -> String Source #bin :: SBV a -> String Source #

readBin :: Num a => String -> a Source #

A more convenient interface for reading binary numbers, also supports negative numbers

# Checking satisfiability in path conditions

Check if a boolean condition is satisfiable in the current state. This function can be useful in contexts where an interpreter implemented on top of SBV needs to decide if a particular stae (represented by the boolean) is reachable in the current if-then-else paths implied by the ite calls.

# Uninterpreted sorts, constants, and functions

Users can introduce new uninterpreted sorts simply by defining a data-type in Haskell and registering it as such. The following example demonstrates:

    data B = B () deriving (Eq, Ord, Show, Read, Data, SymWord, HasKind, SatModel)


(Note that you'll also need to use the language pragmas DeriveDataTypeable, DeriveAnyClass, and import Data.Generics for the above to work.)

This is all it takes to introduce B as an uninterpreted sort in SBV, which makes the type SBV B automagically become available as the type of symbolic values that ranges over B values. Note that the () argument is important to distinguish it from enumerations.

Uninterpreted functions over both uninterpreted and regular sorts can be declared using the facilities introduced by the Uninterpreted class.

class Uninterpreted a where Source #

Uninterpreted 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 irrelevant for the purposes of the proof; i.e., when the proofs can be performed without any knowledge about the function itself.

Minimal complete definition: sbvUninterpret. However, most instances in practice are already provided by SBV, so end-users should not need to define their own instances.

Minimal complete definition

sbvUninterpret

Methods

uninterpret :: String -> a Source #

cgUninterpret :: String -> [String] -> a -> a Source #

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.

sbvUninterpret :: Maybe ([String], a) -> String -> a Source #

Most generalized form of uninterpretation, this function should not be needed by end-user-code, but is rather useful for the library development.

Instances

addAxiom :: String -> [String] -> Symbolic () Source #

Add 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.

# Enumerations

If the uninterpreted sort definition takes the form of an enumeration (i.e., a simple data type with all nullary constructors), then SBV will actually translate that as just such a data-type to SMT-Lib, and will use the constructors as the inhabitants of the said sort. A simple example is:

   data X = A | B | C deriving (Eq, Ord, Show, Read, Data, SymWord, HasKind, SatModel)


Now, the user can define

   type SX = SBV X


and treat SX as a regular symbolic type ranging over the values A, B, and C. Such values can be compared for equality, and with the usual other comparison operators, such as .==, ./=, .>, .>=, <, and <=.

Note that in this latter case the type is no longer uninterpreted, but is properly represented as a simple enumeration of the said elements. A simple query would look like:

    allSat $x -> x .== (x :: SX)  which would list all three elements of this domain as satisfying solutions.  Solution #1: s0 = A :: X Solution #2: s0 = B :: X Solution #3: s0 = C :: X Found 3 different solutions.  Note that the result is properly typed as X elements; these are not mere strings. So, in a getModel scenario, the user can recover actual elements of the domain and program further with those values as usual. # Properties, proofs, satisfiability, and safety The SBV library provides a "push-button" verification system via automated SMT solving. The design goal is to let SMT solvers be used without any knowledge of how SMT solvers work or how different logics operate. The details are hidden behind the SBV framework, providing Haskell programmers with a clean API that is unencumbered by the details of individual solvers. To that end, we use the SMT-Lib standard (http://smtlib.cs.uiowa.edu/) to communicate with arbitrary SMT solvers. ## Predicates A 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 Symbolic 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 Predicate type when necessary. class Provable a where Source # A type a 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.) Minimal complete definition Methods forAll_ :: a -> Predicate Source # 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.

forAll :: [String] -> a -> Predicate Source #

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:

 forAll ["x", "y"] $\(x::SWord8) y -> x shiftL 2 .== y This is the same as above, except the variables will be named x and y respectively, simplifying the counter-examples when they are printed. forSome_ :: a -> Predicate Source # Turns a value into an existentially quantified predicate. (Indeed, exists would have been a better choice here for the name, but alas it's already taken.) forSome :: [String] -> a -> Predicate Source # Version of forSome that allows user defined names Instances  Source # MethodsforAll :: [String] -> SBool -> Predicate Source #forSome :: [String] -> SBool -> Predicate Source # Source # MethodsforAll :: [String] -> Predicate -> Predicate Source #forSome :: [String] -> Predicate -> Predicate Source # (SymWord a, SymWord b, Provable p) => Provable ((SBV a, SBV b) -> p) Source # MethodsforAll_ :: ((SBV a, SBV b) -> p) -> Predicate Source #forAll :: [String] -> ((SBV a, SBV b) -> p) -> Predicate Source #forSome_ :: ((SBV a, SBV b) -> p) -> Predicate Source #forSome :: [String] -> ((SBV a, SBV b) -> p) -> Predicate Source # (SymWord a, SymWord b, SymWord c, Provable p) => Provable ((SBV a, SBV b, SBV c) -> p) Source # MethodsforAll_ :: ((SBV a, SBV b, SBV c) -> p) -> Predicate Source #forAll :: [String] -> ((SBV a, SBV b, SBV c) -> p) -> Predicate Source #forSome_ :: ((SBV a, SBV b, SBV c) -> p) -> Predicate Source #forSome :: [String] -> ((SBV a, SBV b, SBV c) -> p) -> Predicate Source # (SymWord a, SymWord b, SymWord c, SymWord d, Provable p) => Provable ((SBV a, SBV b, SBV c, SBV d) -> p) Source # MethodsforAll_ :: ((SBV a, SBV b, SBV c, SBV d) -> p) -> Predicate Source #forAll :: [String] -> ((SBV a, SBV b, SBV c, SBV d) -> p) -> Predicate Source #forSome_ :: ((SBV a, SBV b, SBV c, SBV d) -> p) -> Predicate Source #forSome :: [String] -> ((SBV a, SBV b, SBV c, SBV d) -> p) -> Predicate Source # (SymWord a, SymWord b, SymWord c, SymWord d, SymWord e, Provable p) => Provable ((SBV a, SBV b, SBV c, SBV d, SBV e) -> p) Source # MethodsforAll_ :: ((SBV a, SBV b, SBV c, SBV d, SBV e) -> p) -> Predicate Source #forAll :: [String] -> ((SBV a, SBV b, SBV c, SBV d, SBV e) -> p) -> Predicate Source #forSome_ :: ((SBV a, SBV b, SBV c, SBV d, SBV e) -> p) -> Predicate Source #forSome :: [String] -> ((SBV a, SBV b, SBV c, SBV d, SBV e) -> p) -> Predicate Source # (SymWord a, SymWord b, SymWord c, SymWord d, SymWord e, SymWord f, Provable p) => Provable ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f) -> p) Source # MethodsforAll_ :: ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f) -> p) -> Predicate Source #forAll :: [String] -> ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f) -> p) -> Predicate Source #forSome_ :: ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f) -> p) -> Predicate Source #forSome :: [String] -> ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f) -> p) -> Predicate Source # (SymWord a, SymWord b, SymWord c, SymWord d, SymWord e, SymWord f, SymWord g, Provable p) => Provable ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f, SBV g) -> p) Source # MethodsforAll_ :: ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f, SBV g) -> p) -> Predicate Source #forAll :: [String] -> ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f, SBV g) -> p) -> Predicate Source #forSome_ :: ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f, SBV g) -> p) -> Predicate Source #forSome :: [String] -> ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f, SBV g) -> p) -> Predicate Source # (HasKind a, HasKind b, Provable p) => Provable (SFunArray a b -> p) Source # MethodsforAll_ :: (SFunArray a b -> p) -> Predicate Source #forAll :: [String] -> (SFunArray a b -> p) -> Predicate Source #forSome_ :: (SFunArray a b -> p) -> Predicate Source #forSome :: [String] -> (SFunArray a b -> p) -> Predicate Source # (HasKind a, HasKind b, Provable p) => Provable (SArray a b -> p) Source # MethodsforAll_ :: (SArray a b -> p) -> Predicate Source #forAll :: [String] -> (SArray a b -> p) -> Predicate Source #forSome_ :: (SArray a b -> p) -> Predicate Source #forSome :: [String] -> (SArray a b -> p) -> Predicate Source # (SymWord a, Provable p) => Provable (SBV a -> p) Source # MethodsforAll_ :: (SBV a -> p) -> Predicate Source #forAll :: [String] -> (SBV a -> p) -> Predicate Source #forSome_ :: (SBV a -> p) -> Predicate Source #forSome :: [String] -> (SBV a -> p) -> Predicate Source # class Equality a where Source # Equality as a proof method. Allows for very concise construction of equivalence proofs, which is very typical in bit-precise proofs. Minimal complete definition (===) Methods (===) :: a -> a -> IO ThmResult infix 4 Source # Instances  (SymWord a, SymWord b, EqSymbolic z) => Equality ((SBV a, SBV b) -> z) Source # Methods(===) :: ((SBV a, SBV b) -> z) -> ((SBV a, SBV b) -> z) -> IO ThmResult Source # (SymWord a, SymWord b, SymWord c, EqSymbolic z) => Equality ((SBV a, SBV b, SBV c) -> z) Source # Methods(===) :: ((SBV a, SBV b, SBV c) -> z) -> ((SBV a, SBV b, SBV c) -> z) -> IO ThmResult Source # (SymWord a, SymWord b, SymWord c, SymWord d, EqSymbolic z) => Equality ((SBV a, SBV b, SBV c, SBV d) -> z) Source # Methods(===) :: ((SBV a, SBV b, SBV c, SBV d) -> z) -> ((SBV a, SBV b, SBV c, SBV d) -> z) -> IO ThmResult Source # (SymWord a, SymWord b, SymWord c, SymWord d, SymWord e, EqSymbolic z) => Equality ((SBV a, SBV b, SBV c, SBV d, SBV e) -> z) Source # Methods(===) :: ((SBV a, SBV b, SBV c, SBV d, SBV e) -> z) -> ((SBV a, SBV b, SBV c, SBV d, SBV e) -> z) -> IO ThmResult Source # (SymWord a, SymWord b, SymWord c, SymWord d, SymWord e, SymWord f, EqSymbolic z) => Equality ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f) -> z) Source # Methods(===) :: ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f) -> z) -> ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f) -> z) -> IO ThmResult Source # (SymWord a, SymWord b, SymWord c, SymWord d, SymWord e, SymWord f, SymWord g, EqSymbolic z) => Equality ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f, SBV g) -> z) Source # Methods(===) :: ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f, SBV g) -> z) -> ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f, SBV g) -> z) -> IO ThmResult Source # (SymWord a, SymWord b, SymWord c, SymWord d, SymWord e, SymWord f, SymWord g, EqSymbolic z) => Equality (SBV a -> SBV b -> SBV c -> SBV d -> SBV e -> SBV f -> SBV g -> z) Source # Methods(===) :: (SBV a -> SBV b -> SBV c -> SBV d -> SBV e -> SBV f -> SBV g -> z) -> (SBV a -> SBV b -> SBV c -> SBV d -> SBV e -> SBV f -> SBV g -> z) -> IO ThmResult Source # (SymWord a, SymWord b, SymWord c, SymWord d, SymWord e, SymWord f, EqSymbolic z) => Equality (SBV a -> SBV b -> SBV c -> SBV d -> SBV e -> SBV f -> z) Source # Methods(===) :: (SBV a -> SBV b -> SBV c -> SBV d -> SBV e -> SBV f -> z) -> (SBV a -> SBV b -> SBV c -> SBV d -> SBV e -> SBV f -> z) -> IO ThmResult Source # (SymWord a, SymWord b, SymWord c, SymWord d, SymWord e, EqSymbolic z) => Equality (SBV a -> SBV b -> SBV c -> SBV d -> SBV e -> z) Source # Methods(===) :: (SBV a -> SBV b -> SBV c -> SBV d -> SBV e -> z) -> (SBV a -> SBV b -> SBV c -> SBV d -> SBV e -> z) -> IO ThmResult Source # (SymWord a, SymWord b, SymWord c, SymWord d, EqSymbolic z) => Equality (SBV a -> SBV b -> SBV c -> SBV d -> z) Source # Methods(===) :: (SBV a -> SBV b -> SBV c -> SBV d -> z) -> (SBV a -> SBV b -> SBV c -> SBV d -> z) -> IO ThmResult Source # (SymWord a, SymWord b, SymWord c, EqSymbolic z) => Equality (SBV a -> SBV b -> SBV c -> z) Source # Methods(===) :: (SBV a -> SBV b -> SBV c -> z) -> (SBV a -> SBV b -> SBV c -> z) -> IO ThmResult Source # (SymWord a, SymWord b, EqSymbolic z) => Equality (SBV a -> SBV b -> z) Source # Methods(===) :: (SBV a -> SBV b -> z) -> (SBV a -> SBV b -> z) -> IO ThmResult Source # (SymWord a, EqSymbolic z) => Equality (SBV a -> z) Source # Methods(===) :: (SBV a -> z) -> (SBV a -> z) -> IO ThmResult Source # ## Proving properties prove :: Provable a => a -> IO ThmResult Source # Prove a predicate, equivalent to proveWith defaultSMTCfg proveWith :: Provable a => SMTConfig -> a -> IO ThmResult Source # Proves the predicate using the given SMT-solver isTheorem :: Provable a => Maybe Int -> a -> IO (Maybe Bool) Source # Checks theoremhood within the given optional time limit of i seconds. Returns Nothing if times out, or the result wrapped in a Just otherwise. isTheoremWith :: Provable a => SMTConfig -> Maybe Int -> a -> IO (Maybe Bool) Source # Check 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. ## Checking satisfiability sat :: Provable a => a -> IO SatResult Source # Find a satisfying assignment for a predicate, equivalent to satWith defaultSMTCfg satWith :: Provable a => SMTConfig -> a -> IO SatResult Source # Find a satisfying assignment using the given SMT-solver isSatisfiable :: Provable a => Maybe Int -> a -> IO (Maybe Bool) Source # Checks satisfiability within the given optional time limit of i seconds. Returns Nothing if times out, or the result wrapped in a Just otherwise. isSatisfiableWith :: Provable a => SMTConfig -> Maybe Int -> a -> IO (Maybe Bool) Source # Check 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. ## Checking safety The sAssert function allows users to introduce invariants to make sure certain properties hold at all times. This is another mechanism to provide further documentation/contract info into SBV code. The functions safe and safeWith can be used to statically discharge these proof assumptions. If a violation is found, SBV will print a model showing which inputs lead to the invariant being violated. Here's a simple example. Let's assume we have a function that does subtraction, and requires its first argument to be larger than the second: >>> let sub x y = sAssert Nothing "sub: x >= y must hold!" (x .>= y) (x - y)  Clearly, this function is not safe, as there's nothing that stops us from passing it a larger second argument. We can use safe to statically see if such a violation is possible before we use this function elsewhere. >>> safe (sub :: SInt8 -> SInt8 -> SInt8) [sub: x >= y must hold!: Violated. Model: s0 = -128 :: Int8 s1 = -127 :: Int8]  What happens if we make sure to arrange for this invariant? Consider this version: >>> let safeSub x y = ite (x .>= y) (sub x y) 0  Clearly, safeSub must be safe. And indeed, SBV can prove that: >>> safe (safeSub :: SInt8 -> SInt8 -> SInt8) [sub: x >= y must hold!: No violations detected]  Note how we used sub and safeSub polymorphically. We only need to monomorphise our types when a proof attempt is done, as we did in the safe calls. If required, the user can pass a CallStack through the first argument to sAssert, which will be used by SBV to print a diagnostic info to pinpoint the failure. Also see Data.SBV.Examples.Misc.NoDiv0 for the classic div-by-zero example. sAssert :: Maybe CallStack -> String -> SBool -> SBV a -> SBV a Source # Symbolic assert. Check that the given boolean condition is always true in the given path. The optional first argument can be used to provide call-stack info via GHC's location facilities. safe :: SExecutable a => a -> IO [SafeResult] Source # Check that all the sAssert calls are safe, equivalent to safeWith defaultSMTCfg safeWith :: SExecutable a => SMTConfig -> a -> IO [SafeResult] Source # Check if any of the assertions can be violated Check if a safe-call was safe or not, turning a SafeResult to a Bool. class SExecutable a where Source # Symbolically executable program fragments. This class is mainly used for safe calls, and is sufficently populated internally to cover most use cases. Users can extend it as they wish to allow safe checks for SBV programs that return/take types that are user-defined. Minimal complete definition Methods sName_ :: a -> Symbolic () Source # sName :: [String] -> a -> Symbolic () Source # Instances  Source # MethodssName_ :: () -> Symbolic () Source #sName :: [String] -> () -> Symbolic () Source # Source # MethodssName_ :: [SBV a] -> Symbolic () Source #sName :: [String] -> [SBV a] -> Symbolic () Source # NFData a => SExecutable (Symbolic a) Source # MethodssName_ :: Symbolic a -> Symbolic () Source #sName :: [String] -> Symbolic a -> Symbolic () Source # Source # MethodssName_ :: SBV a -> Symbolic () Source #sName :: [String] -> SBV a -> Symbolic () Source # (SymWord a, SymWord b, SExecutable p) => SExecutable ((SBV a, SBV b) -> p) Source # MethodssName_ :: ((SBV a, SBV b) -> p) -> Symbolic () Source #sName :: [String] -> ((SBV a, SBV b) -> p) -> Symbolic () Source # (SymWord a, SymWord b, SymWord c, SExecutable p) => SExecutable ((SBV a, SBV b, SBV c) -> p) Source # MethodssName_ :: ((SBV a, SBV b, SBV c) -> p) -> Symbolic () Source #sName :: [String] -> ((SBV a, SBV b, SBV c) -> p) -> Symbolic () Source # (SymWord a, SymWord b, SymWord c, SymWord d, SExecutable p) => SExecutable ((SBV a, SBV b, SBV c, SBV d) -> p) Source # MethodssName_ :: ((SBV a, SBV b, SBV c, SBV d) -> p) -> Symbolic () Source #sName :: [String] -> ((SBV a, SBV b, SBV c, SBV d) -> p) -> Symbolic () Source # (SymWord a, SymWord b, SymWord c, SymWord d, SymWord e, SExecutable p) => SExecutable ((SBV a, SBV b, SBV c, SBV d, SBV e) -> p) Source # MethodssName_ :: ((SBV a, SBV b, SBV c, SBV d, SBV e) -> p) -> Symbolic () Source #sName :: [String] -> ((SBV a, SBV b, SBV c, SBV d, SBV e) -> p) -> Symbolic () Source # (SymWord a, SymWord b, SymWord c, SymWord d, SymWord e, SymWord f, SExecutable p) => SExecutable ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f) -> p) Source # MethodssName_ :: ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f) -> p) -> Symbolic () Source #sName :: [String] -> ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f) -> p) -> Symbolic () Source # (SymWord a, SymWord b, SymWord c, SymWord d, SymWord e, SymWord f, SymWord g, SExecutable p) => SExecutable ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f, SBV g) -> p) Source # MethodssName_ :: ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f, SBV g) -> p) -> Symbolic () Source #sName :: [String] -> ((SBV a, SBV b, SBV c, SBV d, SBV e, SBV f, SBV g) -> p) -> Symbolic () Source # (SymWord a, SExecutable p) => SExecutable (SBV a -> p) Source # MethodssName_ :: (SBV a -> p) -> Symbolic () Source #sName :: [String] -> (SBV a -> p) -> Symbolic () Source # (NFData a, SymWord a, NFData b, SymWord b) => SExecutable (SBV a, SBV b) Source # MethodssName_ :: (SBV a, SBV b) -> Symbolic () Source #sName :: [String] -> (SBV a, SBV b) -> Symbolic () Source # (NFData a, SymWord a, NFData b, SymWord b, NFData c, SymWord c) => SExecutable (SBV a, SBV b, SBV c) Source # MethodssName_ :: (SBV a, SBV b, SBV c) -> Symbolic () Source #sName :: [String] -> (SBV a, SBV b, SBV c) -> Symbolic () Source # (NFData a, SymWord a, NFData b, SymWord b, NFData c, SymWord c, NFData d, SymWord d) => SExecutable (SBV a, SBV b, SBV c, SBV d) Source # MethodssName_ :: (SBV a, SBV b, SBV c, SBV d) -> Symbolic () Source #sName :: [String] -> (SBV a, SBV b, SBV c, SBV d) -> Symbolic () Source # (NFData a, SymWord a, NFData b, SymWord b, NFData c, SymWord c, NFData d, SymWord d, NFData e, SymWord e) => SExecutable (SBV a, SBV b, SBV c, SBV d, SBV e) Source # MethodssName_ :: (SBV a, SBV b, SBV c, SBV d, SBV e) -> Symbolic () Source #sName :: [String] -> (SBV a, SBV b, SBV c, SBV d, SBV e) -> Symbolic () Source # (NFData a, SymWord a, NFData b, SymWord b, NFData c, SymWord c, NFData d, SymWord d, NFData e, SymWord e, NFData f, SymWord f) => SExecutable (SBV a, SBV b, SBV c, SBV d, SBV e, SBV f) Source # MethodssName_ :: (SBV a, SBV b, SBV c, SBV d, SBV e, SBV f) -> Symbolic () Source #sName :: [String] -> (SBV a, SBV b, SBV c, SBV d, SBV e, SBV f) -> Symbolic () Source # (NFData a, SymWord a, NFData b, SymWord b, NFData c, SymWord c, NFData d, SymWord d, NFData e, SymWord e, NFData f, SymWord f, NFData g, SymWord g) => SExecutable (SBV a, SBV b, SBV c, SBV d, SBV e, SBV f, SBV g) Source # MethodssName_ :: (SBV a, SBV b, SBV c, SBV d, SBV e, SBV f, SBV g) -> Symbolic () Source #sName :: [String] -> (SBV a, SBV b, SBV c, SBV d, SBV e, SBV f, SBV g) -> Symbolic () Source # ## Finding all satisfying assignments allSat :: Provable a => a -> IO AllSatResult Source # Return all satisfying assignments for a predicate, equivalent to allSatWith defaultSMTCfg. Satisfying assignments are constructed lazily, so they will be available as returned by the solver and on demand. NB. Uninterpreted constant/function values and counter-examples for array values are ignored for the purposes of allSat. 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. allSatWith :: Provable a => SMTConfig -> a -> IO AllSatResult Source # Find all satisfying assignments using the given SMT-solver ## Satisfying a sequence of boolean conditions solve :: [SBool] -> Symbolic SBool Source # Form the symbolic conjunction of a given list of boolean conditions. Useful in expressing problems with constraints, like the following:  do [x, y, z] <- sIntegers ["x", "y", "z"] solve [x .> 5, y + z .< x]  ## Adding constraints A constraint is a means for restricting the input domain of a formula. Here's a simple example:  do x <- exists "x" y <- exists "y" constrain$ x .> y
constrain $x + y .>= 12 constrain$ y .>= 3
...


The first constraint requires x to be larger than y. The scond one says that sum of x and y must be at least 12, and the final one says that y to be at least 3. Constraints provide an easy way to assert additional properties on the input domain, right at the point of the introduction of variables.

Note that the proper reading of a constraint depends on the context:

• In a sat (or allSat) call: The constraint added is asserted conjunctively. That is, the resulting satisfying model (if any) will always satisfy all the constraints given.
• In a prove call: In this case, the constraint acts as an implication. The property is proved under the assumption that the constraint holds. In other words, the constraint says that we only care about the input space that satisfies the constraint.
• In a quickCheck call: The constraint acts as a filter for quickCheck; if the constraint does not hold, then the input value is considered to be irrelevant and is skipped. Note that this is similar to prove, but is stronger: We do not accept a test case to be valid just because the constraints fail on them, although semantically the implication does hold. We simply skip that test case as a bad test vector.
• In a genTest call: Similar to quickCheck and prove: If a constraint does not hold, the input value is ignored and is not included in the test set.

A good use case (in fact the motivating use case) for constrain is attaching a constraint to a forall or exists variable at the time of its creation. Also, the conjunctive semantics for sat and the implicative semantics for prove simplify programming by choosing the correct interpretation automatically. However, one should be aware of the semantic difference. For instance, in the presence of constraints, formulas that are provable are not necessarily satisfiable. To wit, consider:

   do x <- exists "x"
constrain $x .< x return$ x .< (x :: SWord8)


This predicate is unsatisfiable since no element of SWord8 is less than itself. But it's (vacuously) true, since it excludes the entire domain of values, thus making the proof trivial. Hence, this predicate is provable, but is not satisfiable. To make sure the given constraints are not vacuous, the functions isVacuous (and isVacuousWith) can be used.

Also note that this semantics imply that test case generation (genTest) and quick-check can take arbitrarily long in the presence of constraints, if the random input values generated rarely satisfy the constraints. (As an extreme case, consider constrain false.)

A probabilistic constraint (see pConstrain) attaches a probability threshold for the constraint to be considered. For instance:

    pConstrain 0.8 c


will make sure that the condition c is satisfied 80% of the time (and correspondingly, falsified 20% of the time), in expectation. This variant is useful for genTest and quickCheck functions, where we want to filter the test cases according to some probability distribution, to make sure that the test-vectors are drawn from interesting subsets of the input space. For instance, if we were to generate 100 test cases with the above constraint, we'd expect about 80 of them to satisfy the condition c, while about 20 of them will fail it.

The following properties hold:

   constrain      = pConstrain 1
pConstrain t c = pConstrain (1-t) (not c)


Note that while constrain can be used freely, pConstrain is only allowed in the contexts of genTest or quickCheck. Calls to pConstrain in a prove/sat call will be rejected as SBV does not deal with probabilistic constraints when it comes to satisfiability and proofs. Also, both constrain and pConstrain calls during code-generation will also be rejected, for similar reasons.

Adding arbitrary constraints. When adding constraints, one has to be careful about making sure they are not inconsistent. The function isVacuous can be use for this purpose. Here is an example. Consider the following predicate:

>>> let 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 pred
Q.E.D.


We can use isVacuous to make sure to see that the pass was vacuous:

>>> isVacuous pred
True


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:

>>> let 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'
False


Adding a probabilistic constraint. The Double 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.

## Checking constraint vacuity

isVacuous :: Provable a => a -> IO Bool Source #

Check if the given constraints are satisfiable, equivalent to isVacuousWith defaultSMTCfg. See the function constrain for an example use of isVacuous.

isVacuousWith :: Provable a => SMTConfig -> a -> IO Bool Source #

Determine if the constraints are vacuous using the given SMT-solver

## Quick-checking

Quick check an SBV property. Note that a regular quickCheck call will work just as well. Use this variant if you want to receive the boolean result.

# Proving properties using multiple solvers

On a multi-core machine, it might be desirable to try a given property using multiple SMT solvers, using parallel threads. Even with machines with single-cores, threading can be helpful if you want to try out multiple-solvers but do not know which one would work the best for the problem at hand ahead of time.

The functions in this section allow proving/satisfiability-checking with multiple backends at the same time. Each function comes in two variants, one that returns the results from all solvers, the other that returns the fastest one.

The All variants, (i.e., proveWithAll, satWithAll) run all solvers and return all the results. SBV internally makes sure that the result is lazily generated; so, the order of solvers given does not matter. In other words, the order of results will follow the order of the solvers as they finish, not as given by the user. These variants are useful when you want to make sure multiple-solvers agree (or disagree!) on a given problem.

The Any variants, (i.e., proveWithAny, satWithAny) will run all the solvers in parallel, and return the results of the first one finishing. The other threads will then be killed. These variants are useful when you do not care if the solvers produce the same result, but rather want to get the solution as quickly as possible, taking advantage of modern many-core machines.

Note that the function sbvAvailableSolvers will return all the installed solvers, which can be used as the first argument to all these functions, if you simply want to try all available solvers on a machine.

proveWithAll :: Provable a => [SMTConfig] -> a -> IO [(Solver, ThmResult)] Source #

Prove a property with multiple solvers, running them in separate threads. The results will be returned in the order produced.

proveWithAny :: Provable a => [SMTConfig] -> a -> IO (Solver, ThmResult) Source #

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.

satWithAll :: Provable a => [SMTConfig] -> a -> IO [(Solver, SatResult)] Source #

Find a satisfying assignment to a property with multiple solvers, running them in separate threads. The results will be returned in the order produced.

satWithAny :: Provable a => [SMTConfig] -> a -> IO (Solver, SatResult) Source #

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.

# Optimization

Symbolic optimization. A call of the form:

minimize Quantified cost n valid

returns Just xs, such that:

• xs has precisely n elements
• valid xs holds
• cost xs is minimal. That is, for all sequences ys that satisfy the first two criteria above, cost xs .<= cost ys holds.

If there is no such sequence, then minimize will return Nothing.

The function maximize is similar, except the comparator is .>=. So the value returned has the largest cost (or value, in that case).

The function optimize allows the user to give a custom comparison function.

The OptimizeOpts argument controls how the optimization is done. If Quantified is used, then the SBV optimization engine satisfies the following predicate:

exists xs. forall ys. valid xs && (valid ys implies (cost xs cmp cost ys))

Note that this may cause efficiency problems as it involves alternating quantifiers. If OptimizeOpts is set to Iterative True, then SBV will programmatically search for an optimal solution, by repeatedly calling the solver appropriately. (The boolean argument controls whether progress reports are given. Use False for quiet operation.)

### Quantified vs Iterative

Note that the quantified and iterative versions are two different optimization approaches and may not necessarily yield the same results. In particular, the quantified version can tell us no such solution exists if there is no global optimum value, while the iterative version might simply loop forever for such a problem. To wit, consider the example:

 maximize Quantified head 1 (const true :: [SInteger] -> SBool)

which asks for the largest SInteger value. The SMT solver will happily answer back saying there is no such value with the Quantified call, but the Iterative variant will simply loop forever as it would search through an infinite chain of ascending SInteger values.

In practice, however, the iterative version is usually the more effective choice since alternating quantifiers are hard to deal with for many SMT-solvers and thus will likely result in an unknown result. While the Iterative variant can loop for a long time, one can simply use the boolean flag True and see how the search is progressing.

minimize :: (SatModel a, SymWord a, Show a, SymWord c, Show c) => OptimizeOpts -> ([SBV a] -> SBV c) -> Int -> ([SBV a] -> SBool) -> IO (Maybe [a]) Source #

Minimizes a cost function with respect to a constraint. Examples:

>>> minimize Quantified sum 3 (bAll (.> (10 :: SInteger)))
Just [11,11,11]


maximize :: (SatModel a, SymWord a, Show a, SymWord c, Show c) => OptimizeOpts -> ([SBV a] -> SBV c) -> Int -> ([SBV a] -> SBool) -> IO (Maybe [a]) Source #

Maximizes a cost function with respect to a constraint. Examples:

>>> maximize Quantified sum 3 (bAll (.< (10 :: SInteger)))
Just [9,9,9]


optimize :: (SatModel a, SymWord a, Show a, SymWord c, Show c) => OptimizeOpts -> (SBV c -> SBV c -> SBool) -> ([SBV a] -> SBV c) -> Int -> ([SBV a] -> SBool) -> IO (Maybe [a]) Source #

Variant of optimizeWith using the default solver. See optimizeWith for parameter descriptions.

minimizeWith :: (SatModel a, SymWord a, Show a, SymWord c, Show c) => SMTConfig -> OptimizeOpts -> ([SBV a] -> SBV c) -> Int -> ([SBV a] -> SBool) -> IO (Maybe [a]) Source #

Variant of minimize allowing the use of a user specified solver. See optimizeWith for parameter descriptions.

maximizeWith :: (SatModel a, SymWord a, Show a, SymWord c, Show c) => SMTConfig -> OptimizeOpts -> ([SBV a] -> SBV c) -> Int -> ([SBV a] -> SBool) -> IO (Maybe [a]) Source #

Variant of maximize allowing the use of a user specified solver. See optimizeWith for parameter descriptions.

Arguments

 :: (SatModel a, SymWord a, Show a, SymWord c, Show c) => SMTConfig SMT configuration -> OptimizeOpts Optimization options -> (SBV c -> SBV c -> SBool) comparator -> ([SBV a] -> SBV c) cost function -> Int how many elements? -> ([SBV a] -> SBool) validity constraint -> IO (Maybe [a])

Symbolic optimization. Generalization on minimize and maximize that allows arbitrary cost functions and comparisons.

# Computing expected values

expectedValue :: Outputtable a => Symbolic a -> IO [Double] Source #

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 expectedValueWith the following parameters: verbose, warm-up round count of 10000, no maximum iteration count, and with convergence margin 0.0001.

expectedValueWith :: Outputtable a => Bool -> Int -> Maybe Int -> Double -> Symbolic a -> IO [Double] Source #

Generalized version of expectedValue, 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.

# Model extraction

The default Show instances for prover calls provide all the counter-example information in a human-readable form and should be sufficient for most casual uses of sbv. However, tools built on top of sbv will inevitably need to look into the constructed models more deeply, programmatically extracting their results and performing actions based on them. The API provided in this section aims at simplifying this task.

## Inspecting proof results

ThmResult, SatResult, and AllSatResult are simple newtype wrappers over SMTResult. Their main purpose is so that we can provide custom Show instances to print results accordingly.

newtype ThmResult Source #

A prove call results in a ThmResult

Constructors

 ThmResult SMTResult

Instances

 Source # User friendly way of printing theorem results MethodsshowList :: [ThmResult] -> ShowS # Source # ThmResult as a generic model provider MethodsgetModel :: SatModel b => ThmResult -> Either String (Bool, b) Source #

newtype SatResult Source #

A sat call results in a SatResult The reason for having a separate SatResult is to have a more meaningful Show instance.

Constructors

 SatResult SMTResult

Instances

 Source # User friendly way of printing satisfiablity results MethodsshowList :: [SatResult] -> ShowS # Source # SatResult as a generic model provider MethodsgetModel :: SatModel b => SatResult -> Either String (Bool, b) Source #

newtype SafeResult Source #

A safe call results in a SafeResult

Constructors

 SafeResult (Maybe String, String, SMTResult)

Instances

 Source # User friendly way of printing safety results MethodsshowList :: [SafeResult] -> ShowS #

newtype AllSatResult Source #

An allSat call results in a AllSatResult. The boolean says whether we should warn the user about prefix-existentials.

Constructors

 AllSatResult (Bool, [SMTResult])

Instances

 Source # 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. MethodsshowList :: [AllSatResult] -> ShowS #

data SMTResult Source #

The result of an SMT solver call. Each constructor is tagged with the SMTConfig 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.)

Constructors

 Unsatisfiable SMTConfig Unsatisfiable Satisfiable SMTConfig SMTModel Satisfiable with model Unknown SMTConfig SMTModel Prover returned unknown, with a potential (possibly bogus) model ProofError SMTConfig [String] Prover errored out TimeOut SMTConfig Computation timed out (see the timeout combinator)

Instances

 Source # Methodsrnf :: SMTResult -> () # Source # SMTResult as a generic model provider MethodsgetModel :: SatModel b => SMTResult -> Either String (Bool, b) Source #

## Programmable model extraction

While default Show instances are sufficient for most use cases, it is sometimes desirable (especially for library construction) that the SMT-models are reinterpreted in terms of domain types. Programmable extraction allows getting arbitrarily typed models out of SMT models.

class SatModel a where Source #

Instances of SatModel can be automatically extracted from models returned by the solvers. The idea is that the sbv infrastructure provides a stream of CW's (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: parseCWs

Methods

parseCWs :: [CW] -> Maybe (a, [CW]) Source #

Given 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 Nothing

cvtModel :: (a -> Maybe b) -> Maybe (a, [CW]) -> Maybe (b, [CW]) Source #

Given a parsed model instance, transform it using f, and return the result. The default definition for this method should be sufficient in most use cases.

parseCWs :: Read a => [CW] -> Maybe (a, [CW]) Source #

Given 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 Nothing

Instances

 Source # Bool as extracted from a model MethodsparseCWs :: [CW] -> Maybe (Bool, [CW]) Source #cvtModel :: (Bool -> Maybe b) -> Maybe (Bool, [CW]) -> Maybe (b, [CW]) Source # Source # Double as extracted from a model MethodsparseCWs :: [CW] -> Maybe (Double, [CW]) Source #cvtModel :: (Double -> Maybe b) -> Maybe (Double, [CW]) -> Maybe (b, [CW]) Source # Source # Float as extracted from a model MethodsparseCWs :: [CW] -> Maybe (Float, [CW]) Source #cvtModel :: (Float -> Maybe b) -> Maybe (Float, [CW]) -> Maybe (b, [CW]) Source # Source # Int8 as extracted from a model MethodsparseCWs :: [CW] -> Maybe (Int8, [CW]) Source #cvtModel :: (Int8 -> Maybe b) -> Maybe (Int8, [CW]) -> Maybe (b, [CW]) Source # Source # Int16 as extracted from a model MethodsparseCWs :: [CW] -> Maybe (Int16, [CW]) Source #cvtModel :: (Int16 -> Maybe b) -> Maybe (Int16, [CW]) -> Maybe (b, [CW]) Source # Source # Int32 as extracted from a model MethodsparseCWs :: [CW] -> Maybe (Int32, [CW]) Source #cvtModel :: (Int32 -> Maybe b) -> Maybe (Int32, [CW]) -> Maybe (b, [CW]) Source # Source # Int64 as extracted from a model MethodsparseCWs :: [CW] -> Maybe (Int64, [CW]) Source #cvtModel :: (Int64 -> Maybe b) -> Maybe (Int64, [CW]) -> Maybe (b, [CW]) Source # Source # Integer as extracted from a model MethodsparseCWs :: [CW] -> Maybe (Integer, [CW]) Source #cvtModel :: (Integer -> Maybe b) -> Maybe (Integer, [CW]) -> Maybe (b, [CW]) Source # Source # Word8 as extracted from a model MethodsparseCWs :: [CW] -> Maybe (Word8, [CW]) Source #cvtModel :: (Word8 -> Maybe b) -> Maybe (Word8, [CW]) -> Maybe (b, [CW]) Source # Source # Word16 as extracted from a model MethodsparseCWs :: [CW] -> Maybe (Word16, [CW]) Source #cvtModel :: (Word16 -> Maybe b) -> Maybe (Word16, [CW]) -> Maybe (b, [CW]) Source # Source # Word32 as extracted from a model MethodsparseCWs :: [CW] -> Maybe (Word32, [CW]) Source #cvtModel :: (Word32 -> Maybe b) -> Maybe (Word32, [CW]) -> Maybe (b, [CW]) Source # Source # Word64 as extracted from a model MethodsparseCWs :: [CW] -> Maybe (Word64, [CW]) Source #cvtModel :: (Word64 -> Maybe b) -> Maybe (Word64, [CW]) -> Maybe (b, [CW]) Source # SatModel () Source # Base case for SatModel at unit type. Comes in handy if there are no real variables. MethodsparseCWs :: [CW] -> Maybe ((), [CW]) Source #cvtModel :: (() -> Maybe b) -> Maybe ((), [CW]) -> Maybe (b, [CW]) Source # Source # AlgReal as extracted from a model MethodsparseCWs :: [CW] -> Maybe (AlgReal, [CW]) Source #cvtModel :: (AlgReal -> Maybe b) -> Maybe (AlgReal, [CW]) -> Maybe (b, [CW]) Source # Source # CW as extracted from a model; trivial definition MethodsparseCWs :: [CW] -> Maybe (CW, [CW]) Source #cvtModel :: (CW -> Maybe b) -> Maybe (CW, [CW]) -> Maybe (b, [CW]) Source # Source # A rounding mode, extracted from a model. (Default definition suffices) MethodsparseCWs :: [CW] -> Maybe (RoundingMode, [CW]) Source #cvtModel :: (RoundingMode -> Maybe b) -> Maybe (RoundingMode, [CW]) -> Maybe (b, [CW]) Source # Source # SatModel instance, merely uses the generic parsing method. MethodsparseCWs :: [CW] -> Maybe (Word4, [CW]) Source #cvtModel :: (Word4 -> Maybe b) -> Maybe (Word4, [CW]) -> Maybe (b, [CW]) Source # Source # MethodsparseCWs :: [CW] -> Maybe (Location, [CW]) Source #cvtModel :: (Location -> Maybe b) -> Maybe (Location, [CW]) -> Maybe (b, [CW]) Source # Source # MethodsparseCWs :: [CW] -> Maybe (U2Member, [CW]) Source #cvtModel :: (U2Member -> Maybe b) -> Maybe (U2Member, [CW]) -> Maybe (b, [CW]) Source # SatModel a => SatModel [a] Source # A 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! MethodsparseCWs :: [CW] -> Maybe ([a], [CW]) Source #cvtModel :: ([a] -> Maybe b) -> Maybe ([a], [CW]) -> Maybe (b, [CW]) Source # (SatModel a, SatModel b) => SatModel (a, b) Source # Tuples extracted from a model MethodsparseCWs :: [CW] -> Maybe ((a, b), [CW]) Source #cvtModel :: ((a, b) -> Maybe b) -> Maybe ((a, b), [CW]) -> Maybe (b, [CW]) Source # (SatModel a, SatModel b, SatModel c) => SatModel (a, b, c) Source # 3-Tuples extracted from a model MethodsparseCWs :: [CW] -> Maybe ((a, b, c), [CW]) Source #cvtModel :: ((a, b, c) -> Maybe b) -> Maybe ((a, b, c), [CW]) -> Maybe (b, [CW]) Source # (SatModel a, SatModel b, SatModel c, SatModel d) => SatModel (a, b, c, d) Source # 4-Tuples extracted from a model MethodsparseCWs :: [CW] -> Maybe ((a, b, c, d), [CW]) Source #cvtModel :: ((a, b, c, d) -> Maybe b) -> Maybe ((a, b, c, d), [CW]) -> Maybe (b, [CW]) Source # (SatModel a, SatModel b, SatModel c, SatModel d, SatModel e) => SatModel (a, b, c, d, e) Source # 5-Tuples extracted from a model MethodsparseCWs :: [CW] -> Maybe ((a, b, c, d, e), [CW]) Source #cvtModel :: ((a, b, c, d, e) -> Maybe b) -> Maybe ((a, b, c, d, e), [CW]) -> Maybe (b, [CW]) Source # (SatModel a, SatModel b, SatModel c, SatModel d, SatModel e, SatModel f) => SatModel (a, b, c, d, e, f) Source # 6-Tuples extracted from a model MethodsparseCWs :: [CW] -> Maybe ((a, b, c, d, e, f), [CW]) Source #cvtModel :: ((a, b, c, d, e, f) -> Maybe b) -> Maybe ((a, b, c, d, e, f), [CW]) -> Maybe (b, [CW]) Source # (SatModel a, SatModel b, SatModel c, SatModel d, SatModel e, SatModel f, SatModel g) => SatModel (a, b, c, d, e, f, g) Source # 7-Tuples extracted from a model MethodsparseCWs :: [CW] -> Maybe ((a, b, c, d, e, f, g), [CW]) Source #cvtModel :: ((a, b, c, d, e, f, g) -> Maybe b) -> Maybe ((a, b, c, d, e, f, g), [CW]) -> Maybe (b, [CW]) Source #

class Modelable a where Source #

Various SMT results that we can extract models out of.

Minimal complete definition

Methods

modelExists :: a -> Bool Source #

Is there a model?

getModel :: SatModel b => a -> Either String (Bool, b) Source #

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 getModelDictionaries.

getModelValue :: SymWord b => String -> a -> Maybe b Source #

Extract a model value for a given element. Also see getModelValues.

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 getModelUninterpretedValues.

extractModel :: SatModel b => a -> Maybe b Source #

A simpler variant of getModel to get a model out without the fuss.

Instances

 Source # SMTResult as a generic model provider MethodsgetModel :: SatModel b => SMTResult -> Either String (Bool, b) Source # Source # SatResult as a generic model provider MethodsgetModel :: SatModel b => SatResult -> Either String (Bool, b) Source # Source # ThmResult as a generic model provider MethodsgetModel :: SatModel b => ThmResult -> Either String (Bool, b) Source #

displayModels :: SatModel a => (Int -> (Bool, a) -> IO ()) -> AllSatResult -> IO Int Source #

Given an allSat call, we typically want to iterate over it and print the results in sequence. The displayModels function automates this task by calling disp on each result, consecutively. The first Int 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)

extractModels :: SatModel a => AllSatResult -> [a] Source #

Return all the models from an allSat call, similar to extractModel but is suitable for the case of multiple results.

Get dictionaries from an all-sat call. Similar to getModelDictionary.

getModelValues :: SymWord b => String -> AllSatResult -> [Maybe b] Source #

Extract value of a variable from an all-sat call. Similar to getModelValue.

Extract value of an uninterpreted variable from an all-sat call. Similar to getModelUninterpretedValue.

# SMT Interface: Configurations and solvers

data SMTConfig Source #

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 printRealPrec 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.

The printBase field can be used to print numbers in base 2, 10, or 16. If base 2 or 16 is used, then floating-point values will be printed in their internal memory-layout format as well, which can come in handy for bit-precise analysis.

Constructors

 SMTConfig Fieldsverbose :: BoolDebug modetiming :: BoolPrint timing information on how long different phases took (construction, solving, etc.)sBranchTimeOut :: Maybe IntHow much time to give to the solver for each call of sBranch check. (In seconds. Default: No limit.)timeOut :: Maybe IntHow much time to give to the solver. (In seconds. Default: No limit.)printBase :: IntPrint integral literals in this base (2, 10, and 16 are supported.)printRealPrec :: IntPrint algebraic real values with this precision. (SReal, default: 16)solverTweaks :: [String]Additional lines of script to give to the solver (user specified)satCmd :: StringUsually "(check-sat)". However, users might tweak it based on solver characteristics.isNonModelVar :: String -> BoolWhen constructing a model, ignore variables whose name satisfy this predicate. (Default: (const False), i.e., don't ignore anything)smtFile :: Maybe FilePathIf Just, the generated SMT script will be put in this file (for debugging purposes mostly)smtLibVersion :: SMTLibVersionWhat version of SMT-lib we use for the toolsolver :: SMTSolverThe actual SMT solver.roundingMode :: RoundingModeRounding mode to use for floating-point conversionsuseLogic :: Maybe LogicIf Nothing, pick automatically. Otherwise, either use the given one, or use the custom string.

Instances

 Source # MethodsshowList :: [SMTConfig] -> ShowS #

Representation of SMTLib Program versions. As of June 2015, we're dropping support for SMTLib1, and supporting SMTLib2 only. We keep this data-type around in case SMTLib3 comes along and we want to support 2 and 3 simultaneously.

Constructors

 SMTLib2

Instances

 Source # Methods Source # Methods Source # Methods Source # MethodsshowList :: [SMTLibVersion] -> ShowS # Source # Methodsrnf :: SMTLibVersion -> () #

SMT-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 useLogic parameter to the configuration. This is especially handy if one is experimenting with custom logics that might be supported on new solvers. See http://smtlib.cs.uiowa.edu/logics.shtml for the official list.

Constructors

 AUFLIA Formulas over the theory of linear integer arithmetic and arrays extended with free sort and function symbols but restricted to arrays with integer indices and values AUFLIRA Linear formulas with free sort and function symbols over one- and two-dimentional arrays of integer index and real value AUFNIRA Formulas with free function and predicate symbols over a theory of arrays of arrays of integer index and real value LRA Linear formulas in linear real arithmetic QF_ABV Quantifier-free formulas over the theory of bitvectors and bitvector arrays QF_AUFBV Quantifier-free formulas over the theory of bitvectors and bitvector arrays extended with free sort and function symbols QF_AUFLIA Quantifier-free linear formulas over the theory of integer arrays extended with free sort and function symbols QF_AX Quantifier-free formulas over the theory of arrays with extensionality QF_BV Quantifier-free formulas over the theory of fixed-size bitvectors QF_IDL Difference Logic over the integers. Boolean combinations of inequations of the form x - y < b where x and y are integer variables and b is an integer constant QF_LIA Unquantified linear integer arithmetic. In essence, Boolean combinations of inequations between linear polynomials over integer variables QF_LRA Unquantified linear real arithmetic. In essence, Boolean combinations of inequations between linear polynomials over real variables. QF_NIA Quantifier-free integer arithmetic. QF_NRA Quantifier-free real arithmetic. QF_RDL Difference 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. QF_UF Unquantified formulas built over a signature of uninterpreted (i.e., free) sort and function symbols. QF_UFBV Unquantified formulas over bitvectors with uninterpreted sort function and symbols. QF_UFIDL Difference Logic over the integers (in essence) but with uninterpreted sort and function symbols. QF_UFLIA Unquantified linear integer arithmetic with uninterpreted sort and function symbols. QF_UFLRA Unquantified linear real arithmetic with uninterpreted sort and function symbols. QF_UFNRA Unquantified non-linear real arithmetic with uninterpreted sort and function symbols. UFLRA Linear real arithmetic with uninterpreted sort and function symbols. UFNIA Non-linear integer arithmetic with uninterpreted sort and function symbols. QF_FPBV Quantifier-free formulas over the theory of floating point numbers, arrays, and bit-vectors QF_FP Quantifier-free formulas over the theory of floating point numbers

Instances

 Source # MethodsshowList :: [SMTLibLogic] -> ShowS #

data Logic Source #

Chosen logic for the solver

Constructors

 PredefinedLogic SMTLibLogic Use one of the logics as defined by the standard CustomLogic String Use this name for the logic

Instances

 Source # MethodsshowsPrec :: Int -> Logic -> ShowS #show :: Logic -> String #showList :: [Logic] -> ShowS #

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.

Constructors

 Iterative Bool Iteratively search. if True, it will be reporting progress Quantified Use quantifiers

data Solver Source #

Solvers that SBV is aware of

Constructors

 Z3 Yices Boolector CVC4 MathSAT ABC

Instances

 Source # Methods Source # MethodstoEnum :: Int -> Solver #enumFrom :: Solver -> [Solver] #enumFromThen :: Solver -> Solver -> [Solver] #enumFromTo :: Solver -> Solver -> [Solver] #enumFromThenTo :: Solver -> Solver -> Solver -> [Solver] # Source # MethodsshowsPrec :: Int -> Solver -> ShowS #showList :: [Solver] -> ShowS #

data SMTSolver Source #

An SMT solver

Constructors

 SMTSolver Fieldsname :: SolverThe solver in useexecutable :: StringThe path to its executableoptions :: [String]Options to provide to the solverengine :: SMTEngineThe solver engine, responsible for interpreting solver outputcapabilities :: SolverCapabilitiesVarious capabilities of the solver

Instances

 Source # MethodsshowList :: [SMTSolver] -> ShowS #

Default 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 solver

Default configuration for the MathSAT SMT solver

Default configuration for the ABC synthesis and verification tool.

The default configs corresponding to supported SMT solvers

The currently active solver, obtained by importing Data.SBV. To have other solvers current, import one of the bridge modules Data.SBV.Bridge.ABC, Data.SBV.Bridge.Boolector, Data.SBV.Bridge.CVC4, Data.SBV.Bridge.Yices, or Data.SBV.Bridge.Z3 directly.

The default solver used by SBV. This is currently set to z3.

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.

Return the known available solver configs, installed on your machine.

# Symbolic computations

data Symbolic a Source #

A 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.

Instances

 Source # Methods(>>=) :: Symbolic a -> (a -> Symbolic b) -> Symbolic b #(>>) :: Symbolic a -> Symbolic b -> Symbolic b #return :: a -> Symbolic a #fail :: String -> Symbolic a # Source # Methodsfmap :: (a -> b) -> Symbolic a -> Symbolic b #(<\$) :: a -> Symbolic b -> Symbolic a # Source # Methodspure :: a -> Symbolic a #(<*>) :: Symbolic (a -> b) -> Symbolic a -> Symbolic b #(*>) :: Symbolic a -> Symbolic b -> Symbolic b #(<*) :: Symbolic a -> Symbolic b -> Symbolic a # Source # MethodsliftIO :: IO a -> Symbolic a # Source # MethodsforAll :: [String] -> Predicate -> Predicate Source #forSome :: [String] -> Predicate -> Predicate Source # NFData a => SExecutable (Symbolic a) Source # MethodssName_ :: Symbolic a -> Symbolic () Source #sName :: [String] -> Symbolic a -> Symbolic () Source #

output :: Outputtable a => a -> Symbolic a Source #

Mark an interim result as an output. Useful when constructing Symbolic programs that return multiple values, or when the result is programmatically computed.

class (HasKind a, Ord a) => SymWord a where Source #

A SymWord 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, allSat etc, as default instances automatically provide the necessary bits.

Methods

Create a user named input (universal)

Create an automatically named input

Get a bunch of new words

Create an existential variable

Create an automatically named existential variable

Create a bunch of existentials

free :: String -> Symbolic (SBV a) Source #

Create a free variable, universal in a proof, existential in sat

Create an unnamed free variable, universal in proof, existential in sat

Create a bunch of free vars

Similar to free; Just a more convenient name

symbolics :: [String] -> Symbolic [SBV a] Source #

Similar to mkFreeVars; but automatically gives names based on the strings

literal :: a -> SBV a Source #

Turn a literal constant to symbolic

unliteral :: SBV a -> Maybe a Source #

Extract a literal, if the value is concrete

fromCW :: CW -> a Source #

Extract a literal, from a CW representation

isConcrete :: SBV a -> Bool Source #

Is the symbolic word concrete?

isSymbolic :: SBV a -> Bool Source #

Is the symbolic word really symbolic?

isConcretely :: SBV a -> (a -> Bool) -> Bool Source #

Does it concretely satisfy the given predicate?

One stop allocator

literal :: Show a => a -> SBV a Source #

Turn a literal constant to symbolic

fromCW :: Read a => CW -> a Source #

Extract a literal, from a CW representation

mkSymWord :: (Read a, Data a) => Maybe Quantifier -> Maybe String -> Symbolic (SBV a) Source #

One stop allocator

Instances

 Source # RoundingMode can be used symbolically Methods