----------------------------------------------------------------------------- -- | -- Module : Data.SBV.BitVectors.Operations -- Copyright : (c) Levent Erkok -- License : BSD3 -- Maintainer : erkokl@gmail.com -- Stability : experimental -- -- Constructors and basic operations on symbolic values ----------------------------------------------------------------------------- {-# LANGUAGE BangPatterns #-} module Data.SBV.BitVectors.Operations ( -- ** Basic constructors svTrue, svFalse, svBool , svInteger, svFloat, svDouble, svReal, svEnumFromThenTo -- ** Basic destructors , svAsBool, svAsInteger, svNumerator, svDenominator -- ** Basic operations , svPlus, svTimes, svMinus, svUNeg, svAbs , svDivide, svQuot, svRem , svEqual, svNotEqual , svLessThan, svGreaterThan, svLessEq, svGreaterEq , svAnd, svOr, svXOr, svNot , svShl, svShr, svRol, svRor , svExtract, svJoin , svUninterpreted , svIte, svLazyIte, svSymbolicMerge , svSelect , svSign, svUnsign, svSetBit, svWordFromBE, svWordFromLE , svExp, svFromIntegral -- ** Derived operations , svToWord1, svFromWord1, svTestBit , svShiftLeft, svShiftRight , svRotateLeft, svRotateRight , svBlastLE, svBlastBE , svAddConstant, svIncrement, svDecrement ) where import Data.Bits (Bits(..)) import Data.List (genericIndex, genericLength, genericTake) import Data.SBV.BitVectors.AlgReals import Data.SBV.BitVectors.Kind import Data.SBV.BitVectors.Concrete import Data.SBV.BitVectors.Symbolic import Data.Ratio -------------------------------------------------------------------------------- -- Basic constructors -- | Boolean True. svTrue :: SVal svTrue = SVal KBool (Left trueCW) -- | Boolean False. svFalse :: SVal svFalse = SVal KBool (Left falseCW) -- | Convert from a Boolean. svBool :: Bool -> SVal svBool b = if b then svTrue else svFalse -- | Convert from an Integer. svInteger :: Kind -> Integer -> SVal svInteger k n = SVal k (Left $! mkConstCW k n) -- | Convert from a Float svFloat :: Float -> SVal svFloat f = SVal KFloat (Left $! CW KFloat (CWFloat f)) -- | Convert from a Float svDouble :: Double -> SVal svDouble d = SVal KDouble (Left $! CW KDouble (CWDouble d)) -- | Convert from a Rational svReal :: Rational -> SVal svReal d = SVal KReal (Left $! CW KReal (CWAlgReal (fromRational d))) -------------------------------------------------------------------------------- -- Basic destructors -- | Extract a bool, by properly interpreting the integer stored. svAsBool :: SVal -> Maybe Bool svAsBool (SVal _ (Left cw)) = Just (cwToBool cw) svAsBool _ = Nothing -- | Extract an integer from a concrete value. svAsInteger :: SVal -> Maybe Integer svAsInteger (SVal _ (Left (CW _ (CWInteger n)))) = Just n svAsInteger _ = Nothing -- | Grab the numerator of an SReal, if available svNumerator :: SVal -> Maybe Integer svNumerator (SVal KReal (Left (CW KReal (CWAlgReal (AlgRational True r))))) = Just $ numerator r svNumerator _ = Nothing -- | Grab the denominator of an SReal, if available svDenominator :: SVal -> Maybe Integer svDenominator (SVal KReal (Left (CW KReal (CWAlgReal (AlgRational True r))))) = Just $ denominator r svDenominator _ = Nothing ------------------------------------------------------------------------------------- -- | Constructing [x, y, .. z] and [x .. y]. Only works when all arguments are concrete and integral and the result is guaranteed finite -- Note that the it isn't "obviously" clear why the following works; after all we're doing the construction over Integer's and mapping -- it back to other types such as SIntN/SWordN. The reason is that the values we receive are guaranteed to be in their domains; and thus -- the lifting to Integers preserves the bounds; and then going back is just fine. So, things like @[1, 5 .. 200] :: [SInt8]@ work just -- fine (end evaluate to empty list), since we see @[1, 5 .. -56]@ in the @Integer@ domain. Also note the explicit check for @s /= f@ -- below to make sure we don't stutter and produce an infinite list. svEnumFromThenTo :: SVal -> Maybe SVal -> SVal -> Maybe [SVal] svEnumFromThenTo bf mbs bt | Just bs <- mbs, Just f <- svAsInteger bf, Just s <- svAsInteger bs, Just t <- svAsInteger bt, s /= f = Just $ map (svInteger (kindOf bf)) [f, s .. t] | Nothing <- mbs, Just f <- svAsInteger bf, Just t <- svAsInteger bt = Just $ map (svInteger (kindOf bf)) [f .. t] | True = Nothing ------------------------------------------------------------------------------------- -- Basic operations -- | Addition. svPlus :: SVal -> SVal -> SVal svPlus x y | isConcreteZero x = y | isConcreteZero y = x | True = liftSym2 (mkSymOp Plus) rationalCheck (+) (+) (+) (+) x y -- | Multiplication. svTimes :: SVal -> SVal -> SVal svTimes x y | isConcreteZero x = x | isConcreteZero y = y | isConcreteOne x = y | isConcreteOne y = x | True = liftSym2 (mkSymOp Times) rationalCheck (*) (*) (*) (*) x y -- | Subtraction. svMinus :: SVal -> SVal -> SVal svMinus x y | isConcreteZero y = x | True = liftSym2 (mkSymOp Minus) rationalCheck (-) (-) (-) (-) x y -- | Unary minus. svUNeg :: SVal -> SVal svUNeg = liftSym1 (mkSymOp1 UNeg) negate negate negate negate -- | Absolute value. svAbs :: SVal -> SVal svAbs = liftSym1 (mkSymOp1 Abs) abs abs abs abs -- | Division. svDivide :: SVal -> SVal -> SVal svDivide = liftSym2 (mkSymOp Quot) rationalCheck (/) die (/) (/) where -- should never happen die = error "impossible: integer valued data found in Fractional instance" -- | Exponentiation. svExp :: SVal -> SVal -> SVal svExp b e | hasSign (kindOf e) = error "svExp: exponentiation only works with unsigned exponents" | True = prod $ zipWith (\use n -> svIte use n one) (svBlastLE e) (iterate (\x -> svTimes x x) b) where prod = foldr svTimes one one = svInteger (kindOf b) 1 -- | Bit-blast: Little-endian. Assumes the input is a bit-vector. svBlastLE :: SVal -> [SVal] svBlastLE x = map (svTestBit x) [0 .. intSizeOf x - 1] -- | Set a given bit at index svSetBit :: SVal -> Int -> SVal svSetBit x i = x `svXOr` svInteger (kindOf x) (bit i :: Integer) -- | Bit-blast: Big-endian. Assumes the input is a bit-vector. svBlastBE :: SVal -> [SVal] svBlastBE = reverse . svBlastLE -- | Un-bit-blast from big-endian representation to a word of the right size. -- The input is assumed to be unsigned. svWordFromLE :: [SVal] -> SVal svWordFromLE bs = go zero 0 bs where zero = svInteger (KBounded False (length bs)) 0 go !acc _ [] = acc go !acc !i (x:xs) = go (svIte x (svSetBit acc i) acc) (i+1) xs -- | Un-bit-blast from little-endian representation to a word of the right size. -- The input is assumed to be unsigned. svWordFromBE :: [SVal] -> SVal svWordFromBE = svWordFromLE . reverse -- | Add a constant value: svAddConstant :: Integral a => SVal -> a -> SVal svAddConstant x i = x `svPlus` svInteger (kindOf x) (fromIntegral i) -- | Increment: svIncrement :: SVal -> SVal svIncrement x = svAddConstant x (1::Integer) -- | Decrement: svDecrement :: SVal -> SVal svDecrement x = svAddConstant x (-1 :: Integer) -- | Quotient: Overloaded operation whose meaning depends on the kind at which -- it is used: For unbounded integers, it corresponds to the SMT-Lib -- "div" operator ("Euclidean" division, which always has a -- non-negative remainder). For unsigned bitvectors, it is "bvudiv"; -- and for signed bitvectors it is "bvsdiv", which rounds toward zero. -- All operations have unspecified semantics in case @y = 0@. svQuot :: SVal -> SVal -> SVal svQuot x y | isConcreteZero x = x | isConcreteOne y = x | True = liftSym2 (mkSymOp Quot) nonzeroCheck (noReal "quot") quot' (noFloat "quot") (noDouble "quot") x y where quot' a b | kindOf x == KUnbounded = div a (abs b) * signum b | otherwise = quot a b -- | Remainder: Overloaded operation whose meaning depends on the kind at which -- it is used: For unbounded integers, it corresponds to the SMT-Lib -- "mod" operator (always non-negative). For unsigned bitvectors, it -- is "bvurem"; and for signed bitvectors it is "bvsrem", which rounds -- toward zero (sign of remainder matches that of @x@). All operations -- have unspecified semantics in case @y = 0@. svRem :: SVal -> SVal -> SVal svRem x y | isConcreteZero x = x | isConcreteOne y = svInteger (kindOf x) 0 | True = liftSym2 (mkSymOp Rem) nonzeroCheck (noReal "rem") rem' (noFloat "rem") (noDouble "rem") x y where rem' a b | kindOf x == KUnbounded = mod a (abs b) | otherwise = rem a b -- | Optimize away x == true and x /= false to x; otherwise just do eqOpt eqOptBool :: Op -> SW -> SW -> SW -> Maybe SW eqOptBool op w x y | k == KBool && op == Equal && x == trueSW = Just y -- true .== y --> y | k == KBool && op == Equal && y == trueSW = Just x -- x .== true --> x | k == KBool && op == NotEqual && x == falseSW = Just y -- false ./= y --> y | k == KBool && op == NotEqual && y == falseSW = Just x -- x ./= false --> x | True = eqOpt w x y -- fallback where k = swKind x -- | Equality. svEqual :: SVal -> SVal -> SVal svEqual = liftSym2B (mkSymOpSC (eqOptBool Equal trueSW) Equal) rationalCheck (==) (==) (==) (==) (==) -- | Inequality. svNotEqual :: SVal -> SVal -> SVal svNotEqual = liftSym2B (mkSymOpSC (eqOptBool NotEqual falseSW) NotEqual) rationalCheck (/=) (/=) (/=) (/=) (/=) -- | Less than. svLessThan :: SVal -> SVal -> SVal svLessThan x y | isConcreteMax x = svFalse | isConcreteMin y = svFalse | True = liftSym2B (mkSymOpSC (eqOpt falseSW) LessThan) rationalCheck (<) (<) (<) (<) (uiLift "<" (<)) x y -- | Greater than. svGreaterThan :: SVal -> SVal -> SVal svGreaterThan x y | isConcreteMin x = svFalse | isConcreteMax y = svFalse | True = liftSym2B (mkSymOpSC (eqOpt falseSW) GreaterThan) rationalCheck (>) (>) (>) (>) (uiLift ">" (>)) x y -- | Less than or equal to. svLessEq :: SVal -> SVal -> SVal svLessEq x y | isConcreteMin x = svTrue | isConcreteMax y = svTrue | True = liftSym2B (mkSymOpSC (eqOpt trueSW) LessEq) rationalCheck (<=) (<=) (<=) (<=) (uiLift "<=" (<=)) x y -- | Greater than or equal to. svGreaterEq :: SVal -> SVal -> SVal svGreaterEq x y | isConcreteMax x = svTrue | isConcreteMin y = svTrue | True = liftSym2B (mkSymOpSC (eqOpt trueSW) GreaterEq) rationalCheck (>=) (>=) (>=) (>=) (uiLift ">=" (>=)) x y -- | Bitwise and. svAnd :: SVal -> SVal -> SVal svAnd x y | isConcreteZero x = x | isConcreteOnes x = y | isConcreteZero y = y | isConcreteOnes y = x | True = liftSym2 (mkSymOpSC opt And) (const (const True)) (noReal ".&.") (.&.) (noFloat ".&.") (noDouble ".&.") x y where opt a b | a == falseSW || b == falseSW = Just falseSW | a == trueSW = Just b | b == trueSW = Just a | True = Nothing -- | Bitwise or. svOr :: SVal -> SVal -> SVal svOr x y | isConcreteZero x = y | isConcreteOnes x = x | isConcreteZero y = x | isConcreteOnes y = y | True = liftSym2 (mkSymOpSC opt Or) (const (const True)) (noReal ".|.") (.|.) (noFloat ".|.") (noDouble ".|.") x y where opt a b | a == trueSW || b == trueSW = Just trueSW | a == falseSW = Just b | b == falseSW = Just a | True = Nothing -- | Bitwise xor. svXOr :: SVal -> SVal -> SVal svXOr x y | isConcreteZero x = y | isConcreteOnes x = svNot y | isConcreteZero y = x | isConcreteOnes y = svNot x | True = liftSym2 (mkSymOpSC opt XOr) (const (const True)) (noReal "xor") xor (noFloat "xor") (noDouble "xor") x y where opt a b | a == b && swKind a == KBool = Just falseSW | a == falseSW = Just b | b == falseSW = Just a | True = Nothing -- | Bitwise complement. svNot :: SVal -> SVal svNot = liftSym1 (mkSymOp1SC opt Not) (noRealUnary "complement") complement (noFloatUnary "complement") (noDoubleUnary "complement") where opt a | a == falseSW = Just trueSW | a == trueSW = Just falseSW | True = Nothing -- | Shift left by a constant amount. Translates to the "bvshl" -- operation in SMT-Lib. svShl :: SVal -> Int -> SVal svShl x i | i < 0 = svShr x (-i) | i == 0 = x | True = liftSym1 (mkSymOp1 (Shl i)) (noRealUnary "shiftL") (`shiftL` i) (noFloatUnary "shiftL") (noDoubleUnary "shiftL") x -- | Shift right by a constant amount. Translates to either "bvlshr" -- (logical shift right) or "bvashr" (arithmetic shift right) in -- SMT-Lib, depending on whether @x@ is a signed bitvector. svShr :: SVal -> Int -> SVal svShr x i | i < 0 = svShl x (-i) | i == 0 = x | True = liftSym1 (mkSymOp1 (Shr i)) (noRealUnary "shiftR") (`shiftR` i) (noFloatUnary "shiftR") (noDoubleUnary "shiftR") x -- | Rotate-left, by a constant svRol :: SVal -> Int -> SVal svRol x i | i < 0 = svRor x (-i) | i == 0 = x | True = case kindOf x of KBounded _ sz -> liftSym1 (mkSymOp1 (Rol (i `mod` sz))) (noRealUnary "rotateL") (rot True sz i) (noFloatUnary "rotateL") (noDoubleUnary "rotateL") x _ -> svShl x i -- for unbounded Integers, rotateL is the same as shiftL in Haskell -- | Rotate-right, by a constant svRor :: SVal -> Int -> SVal svRor x i | i < 0 = svRol x (-i) | i == 0 = x | True = case kindOf x of KBounded _ sz -> liftSym1 (mkSymOp1 (Ror (i `mod` sz))) (noRealUnary "rotateR") (rot False sz i) (noFloatUnary "rotateR") (noDoubleUnary "rotateR") x _ -> svShr x i -- for unbounded integers, rotateR is the same as shiftR in Haskell -- | Generic rotation. Since the underlying representation is just Integers, rotations has to be -- careful on the bit-size. rot :: Bool -> Int -> Int -> Integer -> Integer rot toLeft sz amt x | sz < 2 = x | True = norm x y' `shiftL` y .|. norm (x `shiftR` y') y where (y, y') | toLeft = (amt `mod` sz, sz - y) | True = (sz - y', amt `mod` sz) norm v s = v .&. ((1 `shiftL` s) - 1) -- | Extract bit-sequences. svExtract :: Int -> Int -> SVal -> SVal svExtract i j x@(SVal (KBounded s _) _) | i < j = SVal k (Left $! CW k (CWInteger 0)) | SVal _ (Left (CW _ (CWInteger v))) <- x = SVal k (Left $! normCW (CW k (CWInteger (v `shiftR` j)))) | True = SVal k (Right (cache y)) where k = KBounded s (i - j + 1) y st = do sw <- svToSW st x newExpr st k (SBVApp (Extract i j) [sw]) svExtract _ _ _ = error "extract: non-bitvector type" -- | Join two words, by concataneting svJoin :: SVal -> SVal -> SVal svJoin x@(SVal (KBounded s i) a) y@(SVal (KBounded _ j) b) | i == 0 = y | j == 0 = x | Left (CW _ (CWInteger m)) <- a, Left (CW _ (CWInteger n)) <- b = SVal k (Left $! CW k (CWInteger (m `shiftL` j .|. n))) | True = SVal k (Right (cache z)) where k = KBounded s (i + j) z st = do xsw <- svToSW st x ysw <- svToSW st y newExpr st k (SBVApp Join [xsw, ysw]) svJoin _ _ = error "svJoin: non-bitvector type" -- | 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. svUninterpreted :: Kind -> String -> Maybe [String] -> [SVal] -> SVal svUninterpreted k nm code args = SVal k $ Right $ cache result where result st = do let ty = SBVType (map kindOf args ++ [k]) newUninterpreted st nm ty code sws <- mapM (svToSW st) args mapM_ forceSWArg sws newExpr st k $ SBVApp (Uninterpreted nm) sws -- | If-then-else. This one will force branches. svIte :: SVal -> SVal -> SVal -> SVal svIte t a b = svSymbolicMerge (kindOf a) True t a b -- | Lazy If-then-else. This one will delay forcing the branches unless it's really necessary. svLazyIte :: Kind -> SVal -> SVal -> SVal -> SVal svLazyIte k t a b = svSymbolicMerge k False t a b -- | Merge two symbolic values, at kind @k@, possibly @force@'ing the branches to make -- sure they do not evaluate to the same result. svSymbolicMerge :: Kind -> Bool -> SVal -> SVal -> SVal -> SVal svSymbolicMerge k force t a b | Just r <- svAsBool t = if r then a else b | force, rationalSBVCheck a b, areConcretelyEqual a b = a | True = SVal k $ Right $ cache c where c st = do swt <- svToSW st t case () of () | swt == trueSW -> svToSW st a -- these two cases should never be needed as we expect symbolicMerge to be () | swt == falseSW -> svToSW st b -- called with symbolic tests, but just in case.. () -> do {- It is tempting to record the choice of the test expression here as we branch down to the 'then' and 'else' branches. That is, when we evaluate 'a', we can make use of the fact that the test expression is True, and similarly we can use the fact that it is False when b is evaluated. In certain cases this can cut down on symbolic simulation significantly, for instance if repetitive decisions are made in a recursive loop. Unfortunately, the implementation of this idea is quite tricky, due to our sharing based implementation. As the 'then' branch is evaluated, we will create many expressions that are likely going to be "reused" when the 'else' branch is executed. But, it would be *dead wrong* to share those values, as they were "cached" under the incorrect assumptions. To wit, consider the following: foo x y = ite (y .== 0) k (k+1) where k = ite (y .== 0) x (x+1) When we reduce the 'then' branch of the first ite, we'd record the assumption that y is 0. But while reducing the 'then' branch, we'd like to share 'k', which would evaluate (correctly) to 'x' under the given assumption. When we backtrack and evaluate the 'else' branch of the first ite, we'd see 'k' is needed again, and we'd look it up from our sharing map to find (incorrectly) that its value is 'x', which was stored there under the assumption that y was 0, which no longer holds. Clearly, this is unsound. A sound implementation would have to precisely track which assumptions were active at the time expressions get shared. That is, in the above example, we should record that the value of 'k' was cached under the assumption that 'y' is 0. While sound, this approach unfortunately leads to significant loss of valid sharing when the value itself had nothing to do with the assumption itself. To wit, consider: foo x y = ite (y .== 0) k (k+1) where k = x+5 If we tracked the assumptions, we would recompute 'k' twice, since the branch assumptions would differ. Clearly, there is no need to re-compute 'k' in this case since its value is independent of y. Note that the whole SBV performance story is based on agressive sharing, and losing that would have other significant ramifications. The "proper" solution would be to track, with each shared computation, precisely which assumptions it actually *depends* on, rather than blindly recording all the assumptions present at that time. SBV's symbolic simulation engine clearly has all the info needed to do this properly, but the implementation is not straightforward at all. For each subexpression, we would need to chase down its dependencies transitively, which can require a lot of scanning of the generated program causing major slow-down; thus potentially defeating the whole purpose of sharing in the first place. Design choice: Keep it simple, and simply do not track the assumption at all. This will maximize sharing, at the cost of evaluating unreachable branches. I think the simplicity is more important at this point than efficiency. Also note that the user can avoid most such issues by properly combining if-then-else's with common conditions together. That is, the first program above should be written like this: foo x y = ite (y .== 0) x (x+2) In general, the following transformations should be done whenever possible: ite e1 (ite e1 e2 e3) e4 --> ite e1 e2 e4 ite e1 e2 (ite e1 e3 e4) --> ite e1 e2 e4 This is in accordance with the general rule-of-thumb stating conditionals should be avoided as much as possible. However, we might prefer the following: ite e1 (f e2 e4) (f e3 e5) --> f (ite e1 e2 e3) (ite e1 e4 e5) especially if this expression happens to be inside 'f's body itself (i.e., when f is recursive), since it reduces the number of recursive calls. Clearly, programming with symbolic simulation in mind is another kind of beast alltogether. -} let sta = st `extendSValPathCondition` svAnd t let stb = st `extendSValPathCondition` svAnd (svNot t) swa <- svToSW sta a -- evaluate 'then' branch swb <- svToSW stb b -- evaluate 'else' branch case () of -- merge: () | swa == swb -> return swa () | swa == trueSW && swb == falseSW -> return swt () | swa == falseSW && swb == trueSW -> newExpr st k (SBVApp Not [swt]) () -> newExpr st k (SBVApp Ite [swt, swa, swb]) -- | Total indexing operation. @svSelect xs default index@ is -- intuitively the same as @xs !! index@, except it evaluates to -- @default@ if @index@ overflows. Translates to SMT-Lib tables. svSelect :: [SVal] -> SVal -> SVal -> SVal svSelect xs err ind | SVal _ (Left c) <- ind = case cwVal c of CWInteger i -> if i < 0 || i >= genericLength xs then err else xs `genericIndex` i _ -> error $ "SBV.select: unsupported " ++ show (kindOf ind) ++ " valued select/index expression" svSelect xsOrig err ind = xs `seq` SVal kElt (Right (cache r)) where kInd = kindOf ind kElt = kindOf err -- Based on the index size, we need to limit the elements. For -- instance if the index is 8 bits, but there are 257 elements, -- that last element will never be used and we can chop it off. xs = case kInd of KBounded False i -> genericTake ((2::Integer) ^ i) xsOrig KBounded True i -> genericTake ((2::Integer) ^ (i-1)) xsOrig KUnbounded -> xsOrig _ -> error $ "SBV.select: unsupported " ++ show kInd ++ " valued select/index expression" r st = do sws <- mapM (svToSW st) xs swe <- svToSW st err if all (== swe) sws -- off-chance that all elts are the same then return swe else do idx <- getTableIndex st kInd kElt sws swi <- svToSW st ind let len = length xs -- NB. No need to worry here that the index -- might be < 0; as the SMTLib translation -- takes care of that automatically newExpr st kElt (SBVApp (LkUp (idx, kInd, kElt, len) swi swe) []) svChangeSign :: Bool -> SVal -> SVal svChangeSign s x | Just n <- svAsInteger x = svInteger k n | True = SVal k (Right (cache y)) where k = KBounded s (intSizeOf x) y st = do xsw <- svToSW st x newExpr st k (SBVApp (Extract (intSizeOf x - 1) 0) [xsw]) -- | Convert a symbolic bitvector from unsigned to signed. svSign :: SVal -> SVal svSign = svChangeSign True -- | Convert a symbolic bitvector from signed to unsigned. svUnsign :: SVal -> SVal svUnsign = svChangeSign False -- | Convert a symbolic bitvector from one integral kind to another. svFromIntegral :: Kind -> SVal -> SVal svFromIntegral kTo x | Just v <- svAsInteger x = svInteger kTo v | True = result where result = SVal kTo (Right (cache y)) kFrom = kindOf x y st = do xsw <- svToSW st x newExpr st kTo (SBVApp (KindCast kFrom kTo) [xsw]) -------------------------------------------------------------------------------- -- Derived operations -- | Convert an SVal from kind Bool to an unsigned bitvector of size 1. svToWord1 :: SVal -> SVal svToWord1 b = svSymbolicMerge k True b (svInteger k 1) (svInteger k 0) where k = KBounded False 1 -- | Convert an SVal from a bitvector of size 1 (signed or unsigned) to kind Bool. svFromWord1 :: SVal -> SVal svFromWord1 x = svNotEqual x (svInteger k 0) where k = kindOf x -- | Test the value of a bit. Note that we do an extract here -- as opposed to masking and checking against zero, as we found -- extraction to be much faster with large bit-vectors. svTestBit :: SVal -> Int -> SVal svTestBit x i | i < intSizeOf x = svFromWord1 (svExtract i i x) | True = svFalse -- | Generalization of 'svShl', where the shift-amount is symbolic. -- The first argument should be a bounded quantity. svShiftLeft :: SVal -> SVal -> SVal svShiftLeft x i | not (isBounded x) = error "SBV.svShiftLeft: Shifted amount should be a bounded quantity!" | True = svIte (svLessThan i zi) (svSelect [svShr x k | k <- [0 .. intSizeOf x - 1]] z (svUNeg i)) (svSelect [svShl x k | k <- [0 .. intSizeOf x - 1]] z i) where z = svInteger (kindOf x) 0 zi = svInteger (kindOf i) 0 -- | Generalization of 'svShr', where the shift-amount is symbolic. -- The first argument should be a bounded quantity. -- -- NB. If the shiftee is signed, then this is an arithmetic shift; -- otherwise it's logical. svShiftRight :: SVal -> SVal -> SVal svShiftRight x i | not (isBounded x) = error "SBV.svShiftLeft: Shifted amount should be a bounded quantity!" | True = svIte (svLessThan i zi) (svSelect [svShl x k | k <- [0 .. intSizeOf x - 1]] z (svUNeg i)) (svSelect [svShr x k | k <- [0 .. intSizeOf x - 1]] z i) where z = svInteger (kindOf x) 0 zi = svInteger (kindOf i) 0 -- | Generalization of 'svRol', where the rotation amount is symbolic. -- The first argument should be a bounded quantity. svRotateLeft :: SVal -> SVal -> SVal svRotateLeft x i | not (isBounded x) = svShiftLeft x i | isBounded i && bit si <= toInteger sx -- wrap-around not possible = svIte (svLessThan i zi) (svSelect [x `svRor` k | k <- [0 .. bit si - 1]] z (svUNeg i)) (svSelect [x `svRol` k | k <- [0 .. bit si - 1]] z i) | True = svIte (svLessThan i zi) (svSelect [x `svRor` k | k <- [0 .. sx - 1]] z (svUNeg i `svRem` n)) (svSelect [x `svRol` k | k <- [0 .. sx - 1]] z ( i `svRem` n)) where sx = intSizeOf x si = intSizeOf i z = svInteger (kindOf x) 0 zi = svInteger (kindOf i) 0 n = svInteger (kindOf i) (toInteger sx) -- | Generalization of 'svRor', where the rotation amount is symbolic. -- The first argument should be a bounded quantity. svRotateRight :: SVal -> SVal -> SVal svRotateRight x i | not (isBounded x) = svShiftRight x i | isBounded i && bit si <= toInteger sx -- wrap-around not possible = svIte (svLessThan i zi) (svSelect [x `svRol` k | k <- [0 .. bit si - 1]] z (svUNeg i)) (svSelect [x `svRor` k | k <- [0 .. bit si - 1]] z i) | True = svIte (svLessThan i zi) (svSelect [x `svRol` k | k <- [0 .. sx - 1]] z (svUNeg i `svRem` n)) (svSelect [x `svRor` k | k <- [0 .. sx - 1]] z ( i `svRem` n)) where sx = intSizeOf x si = intSizeOf i z = svInteger (kindOf x) 0 zi = svInteger (kindOf i) 0 n = svInteger (kindOf i) (toInteger sx) -------------------------------------------------------------------------------- -- Utility functions noUnint :: (Maybe Int, String) -> a noUnint x = error $ "Unexpected operation called on uninterpreted/enumerated value: " ++ show x noUnint2 :: (Maybe Int, String) -> (Maybe Int, String) -> a noUnint2 x y = error $ "Unexpected binary operation called on uninterpreted/enumerated values: " ++ show (x, y) liftSym1 :: (State -> Kind -> SW -> IO SW) -> (AlgReal -> AlgReal) -> (Integer -> Integer) -> (Float -> Float) -> (Double -> Double) -> SVal -> SVal liftSym1 _ opCR opCI opCF opCD (SVal k (Left a)) = SVal k . Left $! mapCW opCR opCI opCF opCD noUnint a liftSym1 opS _ _ _ _ a@(SVal k _) = SVal k $ Right $ cache c where c st = do swa <- svToSW st a opS st k swa liftSW2 :: (State -> Kind -> SW -> SW -> IO SW) -> Kind -> SVal -> SVal -> Cached SW liftSW2 opS k a b = cache c where c st = do sw1 <- svToSW st a sw2 <- svToSW st b opS st k sw1 sw2 liftSym2 :: (State -> Kind -> SW -> SW -> IO SW) -> (CW -> CW -> Bool) -> (AlgReal -> AlgReal -> AlgReal) -> (Integer -> Integer -> Integer) -> (Float -> Float -> Float) -> (Double -> Double -> Double) -> SVal -> SVal -> SVal liftSym2 _ okCW opCR opCI opCF opCD (SVal k (Left a)) (SVal _ (Left b)) | okCW a b = SVal k . Left $! mapCW2 opCR opCI opCF opCD noUnint2 a b liftSym2 opS _ _ _ _ _ a@(SVal k _) b = SVal k $ Right $ liftSW2 opS k a b liftSym2B :: (State -> Kind -> SW -> SW -> IO SW) -> (CW -> CW -> Bool) -> (AlgReal -> AlgReal -> Bool) -> (Integer -> Integer -> Bool) -> (Float -> Float -> Bool) -> (Double -> Double -> Bool) -> ((Maybe Int, String) -> (Maybe Int, String) -> Bool) -> SVal -> SVal -> SVal liftSym2B _ okCW opCR opCI opCF opCD opUI (SVal _ (Left a)) (SVal _ (Left b)) | okCW a b = svBool (liftCW2 opCR opCI opCF opCD opUI a b) liftSym2B opS _ _ _ _ _ _ a b = SVal KBool $ Right $ liftSW2 opS KBool a b mkSymOpSC :: (SW -> SW -> Maybe SW) -> Op -> State -> Kind -> SW -> SW -> IO SW mkSymOpSC shortCut op st k a b = maybe (newExpr st k (SBVApp op [a, b])) return (shortCut a b) mkSymOp :: Op -> State -> Kind -> SW -> SW -> IO SW mkSymOp = mkSymOpSC (const (const Nothing)) mkSymOp1SC :: (SW -> Maybe SW) -> Op -> State -> Kind -> SW -> IO SW mkSymOp1SC shortCut op st k a = maybe (newExpr st k (SBVApp op [a])) return (shortCut a) mkSymOp1 :: Op -> State -> Kind -> SW -> IO SW mkSymOp1 = mkSymOp1SC (const Nothing) -- | eqOpt says the references are to the same SW, thus we can optimize. Note that -- we explicitly disallow KFloat/KDouble here. Why? Because it's *NOT* true that -- NaN == NaN, NaN >= NaN, and so-forth. So, we have to make sure we don't optimize -- floats and doubles, in case the argument turns out to be NaN. eqOpt :: SW -> SW -> SW -> Maybe SW eqOpt w x y = case swKind x of KFloat -> Nothing KDouble -> Nothing _ -> if x == y then Just w else Nothing -- For uninterpreted/enumerated values, we carefully lift through the constructor index for comparisons: uiLift :: String -> (Int -> Int -> Bool) -> (Maybe Int, String) -> (Maybe Int, String) -> Bool uiLift _ cmp (Just i, _) (Just j, _) = i `cmp` j uiLift w _ a b = error $ "Data.SBV.BitVectors.Model: Impossible happened while trying to lift " ++ w ++ " over " ++ show (a, b) -- | Predicate for optimizing word operations like (+) and (*). isConcreteZero :: SVal -> Bool isConcreteZero (SVal _ (Left (CW _ (CWInteger n)))) = n == 0 isConcreteZero (SVal KReal (Left (CW KReal (CWAlgReal v)))) = isExactRational v && v == 0 isConcreteZero _ = False -- | Predicate for optimizing word operations like (+) and (*). isConcreteOne :: SVal -> Bool isConcreteOne (SVal _ (Left (CW _ (CWInteger 1)))) = True isConcreteOne (SVal KReal (Left (CW KReal (CWAlgReal v)))) = isExactRational v && v == 1 isConcreteOne _ = False -- | Predicate for optimizing bitwise operations. isConcreteOnes :: SVal -> Bool isConcreteOnes (SVal _ (Left (CW (KBounded b w) (CWInteger n)))) = n == if b then -1 else bit w - 1 isConcreteOnes (SVal _ (Left (CW KUnbounded (CWInteger n)))) = n == -1 isConcreteOnes (SVal _ (Left (CW KBool (CWInteger n)))) = n == 1 isConcreteOnes _ = False -- | Predicate for optimizing comparisons. isConcreteMax :: SVal -> Bool isConcreteMax (SVal _ (Left (CW (KBounded False w) (CWInteger n)))) = n == bit w - 1 isConcreteMax (SVal _ (Left (CW (KBounded True w) (CWInteger n)))) = n == bit (w - 1) - 1 isConcreteMax (SVal _ (Left (CW KBool (CWInteger n)))) = n == 1 isConcreteMax _ = False -- | Predicate for optimizing comparisons. isConcreteMin :: SVal -> Bool isConcreteMin (SVal _ (Left (CW (KBounded False _) (CWInteger n)))) = n == 0 isConcreteMin (SVal _ (Left (CW (KBounded True w) (CWInteger n)))) = n == - bit (w - 1) isConcreteMin (SVal _ (Left (CW KBool (CWInteger n)))) = n == 0 isConcreteMin _ = False -- | Predicate for optimizing conditionals. areConcretelyEqual :: SVal -> SVal -> Bool areConcretelyEqual (SVal _ (Left a)) (SVal _ (Left b)) = a == b areConcretelyEqual _ _ = False -- | Most operations on concrete rationals require a compatibility check to avoid faulting -- on algebraic reals. rationalCheck :: CW -> CW -> Bool rationalCheck a b = case (cwVal a, cwVal b) of (CWAlgReal x, CWAlgReal y) -> isExactRational x && isExactRational y _ -> True -- | Quot/Rem operations require a nonzero check on the divisor. -- nonzeroCheck :: CW -> CW -> Bool nonzeroCheck _ b = cwVal b /= CWInteger 0 -- | Same as rationalCheck, except for SBV's rationalSBVCheck :: SVal -> SVal -> Bool rationalSBVCheck (SVal KReal (Left a)) (SVal KReal (Left b)) = rationalCheck a b rationalSBVCheck _ _ = True noReal :: String -> AlgReal -> AlgReal -> AlgReal noReal o a b = error $ "SBV.AlgReal." ++ o ++ ": Unexpected arguments: " ++ show (a, b) noFloat :: String -> Float -> Float -> Float noFloat o a b = error $ "SBV.Float." ++ o ++ ": Unexpected arguments: " ++ show (a, b) noDouble :: String -> Double -> Double -> Double noDouble o a b = error $ "SBV.Double." ++ o ++ ": Unexpected arguments: " ++ show (a, b) noRealUnary :: String -> AlgReal -> AlgReal noRealUnary o a = error $ "SBV.AlgReal." ++ o ++ ": Unexpected argument: " ++ show a noFloatUnary :: String -> Float -> Float noFloatUnary o a = error $ "SBV.Float." ++ o ++ ": Unexpected argument: " ++ show a noDoubleUnary :: String -> Double -> Double noDoubleUnary o a = error $ "SBV.Double." ++ o ++ ": Unexpected argument: " ++ show a {-# ANN svIte ("HLint: ignore Eta reduce" :: String) #-} {-# ANN svLazyIte ("HLint: ignore Eta reduce" :: String) #-} {-# ANN module ("HLint: ignore Reduce duplication" :: String) #-}