----------------------------------------------------------------------------- -- | -- Module : Data.SBV.Provers.Prover -- Copyright : (c) Levent Erkok -- License : BSD3 -- Maintainer : erkokl@gmail.com -- Stability : experimental -- -- Provable abstraction and the connection to SMT solvers ----------------------------------------------------------------------------- {-# LANGUAGE TypeSynonymInstances #-} {-# LANGUAGE FlexibleInstances #-} {-# LANGUAGE OverlappingInstances #-} {-# LANGUAGE BangPatterns #-} module Data.SBV.Provers.Prover ( SMTSolver(..), SMTConfig(..), Predicate, Provable(..) , ThmResult(..), SatResult(..), AllSatResult(..), SMTResult(..) , isSatisfiable, isSatisfiableWith, isTheorem, isTheoremWith , Equality(..) , prove, proveWith , sat, satWith , allSat, allSatWith , isVacuous, isVacuousWith , solve , SatModel(..), Modelable(..), displayModels, extractModels , yices, z3, cvc4, defaultSMTCfg , compileToSMTLib, generateSMTBenchmarks , sbvCheckSolverInstallation ) where import qualified Control.Exception as E import Control.Concurrent (forkIO, newChan, writeChan, getChanContents) import Control.Monad (when, unless, void) import Data.List (intercalate) import Data.Maybe (fromJust, isJust, mapMaybe) import System.FilePath (addExtension) import System.Time (getClockTime) import Data.SBV.BitVectors.Data import Data.SBV.BitVectors.Model import Data.SBV.SMT.SMT import Data.SBV.SMT.SMTLib import qualified Data.SBV.Provers.CVC4 as CVC4 import qualified Data.SBV.Provers.Yices as Yices import qualified Data.SBV.Provers.Z3 as Z3 import Data.SBV.Utils.TDiff import Data.SBV.Utils.Boolean mkConfig :: SMTSolver -> Bool -> [String] -> SMTConfig mkConfig s isSMTLib2 tweaks = SMTConfig { verbose = False , timing = False , timeOut = Nothing , printBase = 10 , printRealPrec = 16 , smtFile = Nothing , solver = s , solverTweaks = tweaks , useSMTLib2 = isSMTLib2 , satCmd = "(check-sat)" } -- | Default configuration for the CVC4 SMT Solver. cvc4 :: SMTConfig cvc4 = mkConfig CVC4.cvc4 True [] -- | Default configuration for the Yices SMT Solver. yices :: SMTConfig yices = mkConfig Yices.yices False [] -- | Default configuration for the Z3 SMT solver z3 :: SMTConfig z3 = mkConfig Z3.z3 True ["(set-option :mbqi true) ; use model based quantifier instantiation"] -- | The default solver used by SBV. This is currently set to z3. defaultSMTCfg :: SMTConfig defaultSMTCfg = z3 -- | 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. type Predicate = Symbolic SBool -- | 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.) class Provable a where -- | 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_ :: a -> Predicate -- | 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. forAll :: [String] -> a -> Predicate -- | 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_ :: a -> Predicate -- | Version of 'forSome' that allows user defined names forSome :: [String] -> a -> Predicate instance Provable Predicate where forAll_ = id forAll [] = id forAll xs = error $ "SBV.forAll: Extra unmapped name(s) in predicate construction: " ++ intercalate ", " xs forSome_ = id forSome [] = id forSome xs = error $ "SBV.forSome: Extra unmapped name(s) in predicate construction: " ++ intercalate ", " xs instance Provable SBool where forAll_ = return forAll _ = return forSome_ = return forSome _ = return {- -- The following works, but it lets us write properties that -- are not useful.. Such as: prove $ \x y -> (x::SInt8) == y -- Running that will throw an exception since Haskell's equality -- is not be supported by symbolic things. (Needs .==). instance Provable Bool where forAll_ x = forAll_ (if x then true else false :: SBool) forAll s x = forAll s (if x then true else false :: SBool) forSome_ x = forSome_ (if x then true else false :: SBool) forSome s x = forSome s (if x then true else false :: SBool) -} -- Functions instance (SymWord a, Provable p) => Provable (SBV a -> p) where forAll_ k = forall_ >>= \a -> forAll_ $ k a forAll (s:ss) k = forall s >>= \a -> forAll ss $ k a forAll [] k = forAll_ k forSome_ k = exists_ >>= \a -> forSome_ $ k a forSome (s:ss) k = exists s >>= \a -> forSome ss $ k a forSome [] k = forSome_ k -- Arrays (memory), only supported universally for the time being instance (HasKind a, HasKind b, SymArray array, Provable p) => Provable (array a b -> p) where forAll_ k = newArray_ Nothing >>= \a -> forAll_ $ k a forAll (s:ss) k = newArray s Nothing >>= \a -> forAll ss $ k a forAll [] k = forAll_ k forSome_ _ = error "SBV.forSome: Existential arrays are not currently supported." forSome _ _ = error "SBV.forSome: Existential arrays are not currently supported." -- 2 Tuple instance (SymWord a, SymWord b, Provable p) => Provable ((SBV a, SBV b) -> p) where forAll_ k = forall_ >>= \a -> forAll_ $ \b -> k (a, b) forAll (s:ss) k = forall s >>= \a -> forAll ss $ \b -> k (a, b) forAll [] k = forAll_ k forSome_ k = exists_ >>= \a -> forSome_ $ \b -> k (a, b) forSome (s:ss) k = exists s >>= \a -> forSome ss $ \b -> k (a, b) forSome [] k = forSome_ k -- 3 Tuple instance (SymWord a, SymWord b, SymWord c, Provable p) => Provable ((SBV a, SBV b, SBV c) -> p) where forAll_ k = forall_ >>= \a -> forAll_ $ \b c -> k (a, b, c) forAll (s:ss) k = forall s >>= \a -> forAll ss $ \b c -> k (a, b, c) forAll [] k = forAll_ k forSome_ k = exists_ >>= \a -> forSome_ $ \b c -> k (a, b, c) forSome (s:ss) k = exists s >>= \a -> forSome ss $ \b c -> k (a, b, c) forSome [] k = forSome_ k -- 4 Tuple instance (SymWord a, SymWord b, SymWord c, SymWord d, Provable p) => Provable ((SBV a, SBV b, SBV c, SBV d) -> p) where forAll_ k = forall_ >>= \a -> forAll_ $ \b c d -> k (a, b, c, d) forAll (s:ss) k = forall s >>= \a -> forAll ss $ \b c d -> k (a, b, c, d) forAll [] k = forAll_ k forSome_ k = exists_ >>= \a -> forSome_ $ \b c d -> k (a, b, c, d) forSome (s:ss) k = exists s >>= \a -> forSome ss $ \b c d -> k (a, b, c, d) forSome [] k = forSome_ k -- 5 Tuple instance (SymWord a, SymWord b, SymWord c, SymWord d, SymWord e, Provable p) => Provable ((SBV a, SBV b, SBV c, SBV d, SBV e) -> p) where forAll_ k = forall_ >>= \a -> forAll_ $ \b c d e -> k (a, b, c, d, e) forAll (s:ss) k = forall s >>= \a -> forAll ss $ \b c d e -> k (a, b, c, d, e) forAll [] k = forAll_ k forSome_ k = exists_ >>= \a -> forSome_ $ \b c d e -> k (a, b, c, d, e) forSome (s:ss) k = exists s >>= \a -> forSome ss $ \b c d e -> k (a, b, c, d, e) forSome [] k = forSome_ k -- 6 Tuple instance (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) where forAll_ k = forall_ >>= \a -> forAll_ $ \b c d e f -> k (a, b, c, d, e, f) forAll (s:ss) k = forall s >>= \a -> forAll ss $ \b c d e f -> k (a, b, c, d, e, f) forAll [] k = forAll_ k forSome_ k = exists_ >>= \a -> forSome_ $ \b c d e f -> k (a, b, c, d, e, f) forSome (s:ss) k = exists s >>= \a -> forSome ss $ \b c d e f -> k (a, b, c, d, e, f) forSome [] k = forSome_ k -- 7 Tuple instance (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) where forAll_ k = forall_ >>= \a -> forAll_ $ \b c d e f g -> k (a, b, c, d, e, f, g) forAll (s:ss) k = forall s >>= \a -> forAll ss $ \b c d e f g -> k (a, b, c, d, e, f, g) forAll [] k = forAll_ k forSome_ k = exists_ >>= \a -> forSome_ $ \b c d e f g -> k (a, b, c, d, e, f, g) forSome (s:ss) k = exists s >>= \a -> forSome ss $ \b c d e f g -> k (a, b, c, d, e, f, g) forSome [] k = forSome_ k -- | Prove a predicate, equivalent to @'proveWith' 'defaultSMTCfg'@ prove :: Provable a => a -> IO ThmResult prove = proveWith defaultSMTCfg -- | Find a satisfying assignment for a predicate, equivalent to @'satWith' 'defaultSMTCfg'@ sat :: Provable a => a -> IO SatResult sat = satWith defaultSMTCfg -- | 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] -- @ solve :: [SBool] -> Symbolic SBool solve = return . bAnd -- | 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. allSat :: Provable a => a -> IO AllSatResult allSat = allSatWith defaultSMTCfg -- | Check if the given constraints are satisfiable, equivalent to @'isVacuousWith' 'defaultSMTCfg'@. This -- call can be used to ensure that the specified constraints (via 'constrain') are satisfiable, i.e., that -- the proof involving these constraints is not passing vacuously. 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 isVacuous :: Provable a => a -> IO Bool isVacuous = isVacuousWith defaultSMTCfg -- Decision procedures (with optional timeout) -- | 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. isTheoremWith :: Provable a => SMTConfig -> Maybe Int -> a -> IO (Maybe Bool) isTheoremWith cfg mbTo p = do r <- proveWith cfg{timeOut = mbTo} p case r of ThmResult (Unsatisfiable _) -> return $ Just True ThmResult (Satisfiable _ _) -> return $ Just False ThmResult (TimeOut _) -> return Nothing _ -> error $ "SBV.isTheorem: Received:\n" ++ show r -- | 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. isSatisfiableWith :: Provable a => SMTConfig -> Maybe Int -> a -> IO (Maybe Bool) isSatisfiableWith cfg mbTo p = do r <- satWith cfg{timeOut = mbTo} p case r of SatResult (Satisfiable _ _) -> return $ Just True SatResult (Unsatisfiable _) -> return $ Just False SatResult (TimeOut _) -> return Nothing _ -> error $ "SBV.isSatisfiable: Received: " ++ show r -- | Checks theoremhood within the given optional time limit of @i@ seconds. -- Returns @Nothing@ if times out, or the result wrapped in a @Just@ otherwise. isTheorem :: Provable a => Maybe Int -> a -> IO (Maybe Bool) isTheorem = isTheoremWith defaultSMTCfg -- | Checks satisfiability within the given optional time limit of @i@ seconds. -- Returns @Nothing@ if times out, or the result wrapped in a @Just@ otherwise. isSatisfiable :: Provable a => Maybe Int -> a -> IO (Maybe Bool) isSatisfiable = isSatisfiableWith defaultSMTCfg -- | Compiles to SMT-Lib and returns the resulting program as a string. Useful for saving -- the result to a file for off-line analysis, for instance if you have an SMT solver that's not natively -- supported out-of-the box by the SBV library. It takes two booleans: -- -- * smtLib2: If 'True', will generate SMT-Lib2 output, otherwise SMT-Lib1 output -- -- * isSat : If 'True', will translate it as a SAT query, i.e., in the positive. If 'False', will -- translate as a PROVE query, i.e., it will negate the result. (In this case, the check-sat -- call to the SMT solver will produce UNSAT if the input is a theorem, as usual.) compileToSMTLib :: Provable a => Bool -- ^ If True, output SMT-Lib2, otherwise SMT-Lib1 -> Bool -- ^ If True, translate directly, otherwise negate the goal. (Use True for SAT queries, False for PROVE queries.) -> a -> IO String compileToSMTLib smtLib2 isSat a = do t <- getClockTime let comments = ["Created on " ++ show t] cvt = if smtLib2 then toSMTLib2 else toSMTLib1 (_, _, _, _, smtLibPgm) <- simulate cvt defaultSMTCfg isSat comments a let out = show smtLibPgm if smtLib2 -- append check-sat in case of smtLib2 then return $ out ++ "\n(check-sat)\n" else return $ out ++ "\n" -- | Create both SMT-Lib1 and SMT-Lib2 benchmarks. The first argument is the basename of the file, -- SMT-Lib1 version will be written with suffix ".smt1" and SMT-Lib2 version will be written with -- suffix ".smt2". The 'Bool' argument controls whether this is a SAT instance, i.e., translate the query -- directly, or a PROVE instance, i.e., translate the negated query. (See the second boolean argument to -- 'compileToSMTLib' for details.) generateSMTBenchmarks :: Provable a => Bool -> FilePath -> a -> IO () generateSMTBenchmarks isSat f a = gen False smt1 >> gen True smt2 where smt1 = addExtension f "smt1" smt2 = addExtension f "smt2" gen b fn = do s <- compileToSMTLib b isSat a writeFile fn s putStrLn $ "Generated SMT benchmark " ++ show fn ++ "." -- | Proves the predicate using the given SMT-solver proveWith :: Provable a => SMTConfig -> a -> IO ThmResult proveWith config a = simulate cvt config False [] a >>= callSolver False "Checking Theoremhood.." ThmResult config where cvt = if useSMTLib2 config then toSMTLib2 else toSMTLib1 -- | Find a satisfying assignment using the given SMT-solver satWith :: Provable a => SMTConfig -> a -> IO SatResult satWith config a = simulate cvt config True [] a >>= callSolver True "Checking Satisfiability.." SatResult config where cvt = if useSMTLib2 config then toSMTLib2 else toSMTLib1 -- | Determine if the constraints are vacuous using the given SMT-solver isVacuousWith :: Provable a => SMTConfig -> a -> IO Bool isVacuousWith config a = do Result ub us tr uic is cs ts as uis ax asgn cstr _ <- runSymbolic True $ forAll_ a >>= output case cstr of [] -> return False -- no constraints, no need to check _ -> do let is' = [(EX, i) | (_, i) <- is] -- map all quantifiers to "exists" for the constraint check res' = Result ub us tr uic is' cs ts as uis ax asgn cstr [trueSW] cvt = if useSMTLib2 config then toSMTLib2 else toSMTLib1 SatResult result <- runProofOn cvt config True [] res' >>= callSolver True "Checking Satisfiability.." SatResult config case result of Unsatisfiable{} -> return True -- constraints are unsatisfiable! Satisfiable{} -> return False -- constraints are satisfiable! Unknown{} -> error "SBV: isVacuous: Solver returned unknown!" ProofError _ ls -> error $ "SBV: isVacuous: error encountered:\n" ++ unlines ls TimeOut _ -> error "SBV: isVacuous: time-out." -- | Find all satisfying assignments using the given SMT-solver allSatWith :: Provable a => SMTConfig -> a -> IO AllSatResult allSatWith config p = do let converter = if useSMTLib2 config then toSMTLib2 else toSMTLib1 msg "Checking Satisfiability, all solutions.." sbvPgm@(qinps, _, _, usorts, _) <- simulate converter config True [] p unless (null usorts) $ error $ "SBV.allSat: All-sat calls are not supported in the presence of uninterpreted sorts: " ++ unwords usorts ++ "\n Only 'sat' and 'prove' calls are available when uninterpreted sorts are used." resChan <- newChan let add = writeChan resChan . Just stop = writeChan resChan Nothing final r = add r >> stop die m = final (ProofError config [m]) -- only fork if non-verbose.. otherwise stdout gets garbled fork io = if verbose config then io else void (forkIO io) fork $ E.catch (go sbvPgm add stop final (1::Int) []) (\e -> die (show (e::E.SomeException))) results <- getChanContents resChan -- See if there are any existentials below any universals -- If such is the case, then the solutions are unique upto prefix existentials let w = ALL `elem` map fst qinps return $ AllSatResult (w, map fromJust (takeWhile isJust results)) where msg = when (verbose config) . putStrLn . ("** " ++) go sbvPgm add stop final = loop where loop !n nonEqConsts = do curResult <- invoke nonEqConsts n sbvPgm case curResult of Nothing -> stop Just (SatResult r) -> case r of Satisfiable _ (SMTModel [] _ _) -> final r Unknown _ (SMTModel [] _ _) -> final r ProofError _ _ -> final r TimeOut _ -> stop Unsatisfiable _ -> stop Satisfiable _ model -> add r >> loop (n+1) (modelAssocs model : nonEqConsts) Unknown _ model -> add r >> loop (n+1) (modelAssocs model : nonEqConsts) invoke nonEqConsts n (qinps, modelMap, skolemMap, _, smtLibPgm) = do msg $ "Looking for solution " ++ show n case addNonEqConstraints qinps nonEqConsts smtLibPgm of Nothing -> -- no new constraints added, stop return Nothing Just finalPgm -> do msg $ "Generated SMTLib program:\n" ++ finalPgm smtAnswer <- engine (solver config) config True qinps modelMap skolemMap finalPgm msg "Done.." return $ Just $ SatResult smtAnswer type SMTProblem = ( [(Quantifier, NamedSymVar)] -- inputs , [(String, UnintKind)] -- model-map , [Either SW (SW, [SW])] -- skolem-map , [String] -- uninterpreted sorts , SMTLibPgm -- SMTLib representation ) callSolver :: Bool -> String -> (SMTResult -> b) -> SMTConfig -> SMTProblem -> IO b callSolver isSat checkMsg wrap config (qinps, modelMap, skolemMap, _, smtLibPgm) = do let msg = when (verbose config) . putStrLn . ("** " ++) msg checkMsg let finalPgm = intercalate "\n" (pre ++ post) where SMTLibPgm _ (_, pre, post) = smtLibPgm msg $ "Generated SMTLib program:\n" ++ finalPgm smtAnswer <- engine (solver config) config isSat qinps modelMap skolemMap finalPgm msg "Done.." return $ wrap smtAnswer simulate :: Provable a => SMTLibConverter -> SMTConfig -> Bool -> [String] -> a -> IO SMTProblem simulate converter config isSat comments predicate = do let msg = when (verbose config) . putStrLn . ("** " ++) isTiming = timing config msg "Starting symbolic simulation.." res <- timeIf isTiming "problem construction" $ runSymbolic isSat $ (if isSat then forSome_ else forAll_) predicate >>= output msg $ "Generated symbolic trace:\n" ++ show res msg "Translating to SMT-Lib.." runProofOn converter config isSat comments res runProofOn :: SMTLibConverter -> SMTConfig -> Bool -> [String] -> Result -> IO SMTProblem runProofOn converter config isSat comments res = let isTiming = timing config defLogic = defaultLogic (solver config) in case res of Result boundInfo usorts _qcInfo _codeSegs is consts tbls arrs uis axs pgm cstrs [o@(SW (KBounded False 1) _)] -> timeIf isTiming "translation" $ let uiMap = mapMaybe arrayUIKind arrs ++ map unintFnUIKind uis skolemMap = skolemize (if isSat then is else map flipQ is) where flipQ (ALL, x) = (EX, x) flipQ (EX, x) = (ALL, x) skolemize :: [(Quantifier, NamedSymVar)] -> [Either SW (SW, [SW])] skolemize qinps = go qinps ([], []) where go [] (_, sofar) = reverse sofar go ((ALL, (v, _)):rest) (us, sofar) = go rest (v:us, Left v : sofar) go ((EX, (v, _)):rest) (us, sofar) = go rest (us, Right (v, reverse us) : sofar) in return (is, uiMap, skolemMap, usorts, converter boundInfo defLogic isSat comments usorts is skolemMap consts tbls arrs uis axs pgm cstrs o) Result _boundInfo _us _qcInfo _codeSegs _is _consts _tbls _arrs _uis _axs _pgm _cstrs os -> case length os of 0 -> error $ "Impossible happened, unexpected non-outputting result\n" ++ show res 1 -> error $ "Impossible happened, non-boolean output in " ++ show os ++ "\nDetected while generating the trace:\n" ++ show res _ -> error $ "User error: Multiple output values detected: " ++ show os ++ "\nDetected while generating the trace:\n" ++ show res ++ "\n*** Check calls to \"output\", they are typically not needed!" -- | 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. sbvCheckSolverInstallation :: SMTConfig -> IO Bool sbvCheckSolverInstallation cfg = do ThmResult r <- proveWith cfg $ \x -> (x+x) .== ((x*2) :: SWord8) case r of Unsatisfiable _ -> return True _ -> return False -- | Equality as a proof method. Allows for -- very concise construction of equivalence proofs, which is very typical in -- bit-precise proofs. infix 4 === class Equality a where (===) :: a -> a -> IO ThmResult instance (SymWord a, EqSymbolic z) => Equality (SBV a -> z) where k === l = prove $ \a -> k a .== l a instance (SymWord a, SymWord b, EqSymbolic z) => Equality (SBV a -> SBV b -> z) where k === l = prove $ \a b -> k a b .== l a b instance (SymWord a, SymWord b, EqSymbolic z) => Equality ((SBV a, SBV b) -> z) where k === l = prove $ \a b -> k (a, b) .== l (a, b) instance (SymWord a, SymWord b, SymWord c, EqSymbolic z) => Equality (SBV a -> SBV b -> SBV c -> z) where k === l = prove $ \a b c -> k a b c .== l a b c instance (SymWord a, SymWord b, SymWord c, EqSymbolic z) => Equality ((SBV a, SBV b, SBV c) -> z) where k === l = prove $ \a b c -> k (a, b, c) .== l (a, b, c) instance (SymWord a, SymWord b, SymWord c, SymWord d, EqSymbolic z) => Equality (SBV a -> SBV b -> SBV c -> SBV d -> z) where k === l = prove $ \a b c d -> k a b c d .== l a b c d instance (SymWord a, SymWord b, SymWord c, SymWord d, EqSymbolic z) => Equality ((SBV a, SBV b, SBV c, SBV d) -> z) where k === l = prove $ \a b c d -> k (a, b, c, d) .== l (a, b, c, d) instance (SymWord a, SymWord b, SymWord c, SymWord d, SymWord e, EqSymbolic z) => Equality (SBV a -> SBV b -> SBV c -> SBV d -> SBV e -> z) where k === l = prove $ \a b c d e -> k a b c d e .== l a b c d e instance (SymWord a, SymWord b, SymWord c, SymWord d, SymWord e, EqSymbolic z) => Equality ((SBV a, SBV b, SBV c, SBV d, SBV e) -> z) where k === l = prove $ \a b c d e -> k (a, b, c, d, e) .== l (a, b, c, d, e) instance (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) where k === l = prove $ \a b c d e f -> k a b c d e f .== l a b c d e f instance (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) where k === l = prove $ \a b c d e f -> k (a, b, c, d, e, f) .== l (a, b, c, d, e, f) instance (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) where k === l = prove $ \a b c d e f g -> k a b c d e f g .== l a b c d e f g instance (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) where k === l = prove $ \a b c d e f g -> k (a, b, c, d, e, f, g) .== l (a, b, c, d, e, f, g)