sbv-12.2: Data/SBV/Core/Model.hs
-----------------------------------------------------------------------------
-- |
-- Module : Data.SBV.Core.Model
-- Copyright : (c) Levent Erkok
-- License : BSD3
-- Maintainer: erkokl@gmail.com
-- Stability : experimental
--
-- Instance declarations for our symbolic world
-----------------------------------------------------------------------------
{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE CPP #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE DefaultSignatures #-}
{-# LANGUAGE DeriveFunctor #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE InstanceSigs #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE NamedFieldPuns #-}
{-# LANGUAGE Rank2Types #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE UndecidableInstances #-}
{-# OPTIONS_GHC -Wall -Werror -fno-warn-orphans -Wno-incomplete-uni-patterns #-}
module Data.SBV.Core.Model (
Mergeable(..), Equality(..), EqSymbolic(..), OrdSymbolic(..), SDivisible(..), SMTDefinable(..), QSaturate, qSaturateSavingObservables
, Metric(..), minimize, maximize, assertWithPenalty, SIntegral, SFiniteBits(..)
, ite, iteLazy, sFromIntegral, sShiftLeft, sShiftRight, sRotateLeft, sBarrelRotateLeft, sRotateRight, sBarrelRotateRight, sSignedShiftArithRight, (.^)
, some
, oneIf, genVar, genVar_
, pbAtMost, pbAtLeast, pbExactly, pbLe, pbGe, pbEq, pbMutexed, pbStronglyMutexed
, sBool, sBool_, sBools, sWord8, sWord8_, sWord8s, sWord16, sWord16_, sWord16s, sWord32, sWord32_, sWord32s
, sWord64, sWord64_, sWord64s, sInt8, sInt8_, sInt8s, sInt16, sInt16_, sInt16s, sInt32, sInt32_, sInt32s, sInt64, sInt64_
, sInt64s, sInteger, sInteger_, sIntegers, sReal, sReal_, sReals, sFloat, sFloat_, sFloats, sDouble, sDouble_, sDoubles
, sWord, sWord_, sWords, sInt, sInt_, sInts
, sFPHalf, sFPHalf_, sFPHalfs, sFPBFloat, sFPBFloat_, sFPBFloats, sFPSingle, sFPSingle_, sFPSingles, sFPDouble, sFPDouble_, sFPDoubles, sFPQuad, sFPQuad_, sFPQuads, sArray, sArray_, sArrays
, sFloatingPoint, sFloatingPoint_, sFloatingPoints
, sChar, sChar_, sChars, sString, sString_, sStrings, sList, sList_, sLists
, sRational, sRational_, sRationals
, SymTuple, sTuple, sTuple_, sTuples
, sEither, sEither_, sEithers, sMaybe, sMaybe_, sMaybes
, sSet, sSet_, sSets
, sEDivMod, sEDiv, sEMod
, sDivides
, solve
, slet
, sRealToSInteger, sRealToSIntegerTruncate, label, observe, observeIf, sObserve
, sAssert
, liftQRem, liftDMod, symbolicMergeWithKind
, genLiteral, genFromCV, genMkSymVar
, zeroExtend, signExtend
, sbvQuickCheck
, readArray, writeArray, lambdaArray, listArray
, FromSized, ToSized, FromSizedBV(..), ToSizedBV(..)
, smtHOFunction, Closure(..)
)
where
import Control.Applicative (ZipList(ZipList))
import Control.Monad (when, unless, mplus, replicateM)
import Control.Monad.IO.Class (MonadIO, liftIO)
import qualified Control.Exception as C
import GHC.Generics (M1(..), U1(..), (:*:)(..), K1(..))
import qualified GHC.Generics as G
import GHC.Stack
import GHC.TypeLits
#if MIN_VERSION_base(4,18,0)
hiding(SChar)
#endif
import Data.Array (Array, Ix, elems, bounds, rangeSize)
import qualified Data.Array as DA (listArray)
import Data.Bits (Bits(..))
import Data.Int (Int8, Int16, Int32, Int64)
import Data.Kind (Type, Constraint)
import Data.List (genericLength, genericIndex, genericTake, unzip4, unzip5, unzip6, unzip7, intercalate)
import Data.Maybe (fromMaybe, mapMaybe)
import Data.String (IsString(..))
import Data.Word (Word8, Word16, Word32, Word64)
import Data.List.NonEmpty (NonEmpty(..))
import qualified Data.List.NonEmpty as NE
import qualified Data.Set as Set
import Data.Proxy
import Data.Dynamic (fromDynamic, toDyn, Typeable)
import Test.QuickCheck (Testable(..), Arbitrary(..))
import qualified Test.QuickCheck.Test as QC (isSuccess)
import qualified Test.QuickCheck as QC (quickCheckResult, counterexample)
import qualified Test.QuickCheck.Monadic as QC (monadicIO, run, assert, pre, monitor)
import qualified Data.Foldable as F (toList)
import Data.SBV.Core.AlgReals
import Data.SBV.Core.Sized
import Data.SBV.Core.SizedFloats
import Data.SBV.Core.Data hiding (Constraint)
import Data.SBV.Core.Symbolic
import Data.SBV.Core.Operations
import Data.SBV.Core.Kind
import Data.SBV.Lambda
import Data.SBV.Utils.ExtractIO(ExtractIO)
import Data.SBV.Provers.Prover (defaultSMTCfg, SafeResult(..), defs2smt, prove)
import Data.SBV.SMT.SMT (ThmResult, showModel)
import Data.SBV.Utils.Numeric (fpIsEqualObjectH)
import Data.IORef (readIORef, writeIORef)
import Data.SBV.Utils.Lib
import Data.Char
-- Symbolic-Word class instances
import Crypto.Hash.SHA512 (hash)
import qualified Data.ByteString.Base16 as B
import qualified Data.ByteString.Char8 as BC
-- | Generate a variable, named
genVar :: MonadSymbolic m => VarContext -> Kind -> String -> m (SBV a)
genVar q k = mkSymSBV q k . Just
-- | Generate an unnamed variable
genVar_ :: MonadSymbolic m => VarContext -> Kind -> m (SBV a)
genVar_ q k = mkSymSBV q k Nothing
-- | Generate a finite constant bitvector
genLiteral :: Integral a => Kind -> a -> SBV b
genLiteral k = SBV . SVal k . Left . mkConstCV k
-- | Convert a constant to an integral value
genFromCV :: Integral a => CV -> a
genFromCV (CV _ (CInteger x)) = fromInteger x
genFromCV c = error $ "genFromCV: Unsupported non-integral value: " ++ show c
-- | Generalization of 'Data.SBV.genMkSymVar'
genMkSymVar :: MonadSymbolic m => Kind -> VarContext -> Maybe String -> m (SBV a)
genMkSymVar k mbq Nothing = genVar_ mbq k
genMkSymVar k mbq (Just s) = genVar mbq k s
instance SymVal Bool where
mkSymVal = genMkSymVar KBool
literal = SBV . svBool
fromCV = cvToBool
instance SymVal Word8 where
mkSymVal = genMkSymVar (KBounded False 8)
literal = genLiteral (KBounded False 8)
fromCV = genFromCV
instance SymVal Int8 where
mkSymVal = genMkSymVar (KBounded True 8)
literal = genLiteral (KBounded True 8)
fromCV = genFromCV
instance SymVal Word16 where
mkSymVal = genMkSymVar (KBounded False 16)
literal = genLiteral (KBounded False 16)
fromCV = genFromCV
instance SymVal Int16 where
mkSymVal = genMkSymVar (KBounded True 16)
literal = genLiteral (KBounded True 16)
fromCV = genFromCV
instance SymVal Word32 where
mkSymVal = genMkSymVar (KBounded False 32)
literal = genLiteral (KBounded False 32)
fromCV = genFromCV
instance SymVal Int32 where
mkSymVal = genMkSymVar (KBounded True 32)
literal = genLiteral (KBounded True 32)
fromCV = genFromCV
instance SymVal Word64 where
mkSymVal = genMkSymVar (KBounded False 64)
literal = genLiteral (KBounded False 64)
fromCV = genFromCV
instance SymVal Int64 where
mkSymVal = genMkSymVar (KBounded True 64)
literal = genLiteral (KBounded True 64)
fromCV = genFromCV
instance SymVal Integer where
mkSymVal = genMkSymVar KUnbounded
literal = SBV . SVal KUnbounded . Left . mkConstCV KUnbounded
fromCV = genFromCV
minMaxBound = Nothing
instance SymVal Rational where
mkSymVal = genMkSymVar KRational
literal = SBV . SVal KRational . Left . CV KRational . CRational
fromCV (CV _ (CRational r)) = r
fromCV c = error $ "SymVal.Rational: Unexpected non-rational value: " ++ show c
minMaxBound = Nothing
instance SymVal AlgReal where
mkSymVal = genMkSymVar KReal
literal = SBV . SVal KReal . Left . CV KReal . CAlgReal
fromCV (CV _ (CAlgReal a)) = a
fromCV c = error $ "SymVal.AlgReal: Unexpected non-real value: " ++ show c
minMaxBound = Nothing
-- AlgReal needs its own definition of isConcretely
-- to make sure we avoid using unimplementable Haskell functions
isConcretely (SBV (SVal KReal (Left (CV KReal (CAlgReal v))))) p
| isExactRational v = p v
isConcretely _ _ = False
instance SymVal Float where
mkSymVal = genMkSymVar KFloat
literal = SBV . SVal KFloat . Left . CV KFloat . CFloat
fromCV (CV _ (CFloat a)) = a
fromCV c = error $ "SymVal.Float: Unexpected non-float value: " ++ show c
minMaxBound = Nothing
-- For Float, we conservatively return 'False' for isConcretely. The reason is that
-- this function is used for optimizations when only one of the argument is concrete,
-- and in the presence of NaN's it would be incorrect to do any optimization
isConcretely _ _ = False
instance SymVal Double where
mkSymVal = genMkSymVar KDouble
literal = SBV . SVal KDouble . Left . CV KDouble . CDouble
fromCV (CV _ (CDouble a)) = a
fromCV c = error $ "SymVal.Double: Unexpected non-double value: " ++ show c
minMaxBound = Nothing
-- For Double, we conservatively return 'False' for isConcretely. The reason is that
-- this function is used for optimizations when only one of the argument is concrete,
-- and in the presence of NaN's it would be incorrect to do any optimization
isConcretely _ _ = False
instance SymVal Char where
mkSymVal = genMkSymVar KChar
literal c = SBV . SVal KChar . Left . CV KChar $ CChar c
fromCV (CV _ (CChar a)) = a
fromCV c = error $ "SymVal.String: Unexpected non-char value: " ++ show c
instance SymVal a => SymVal [a] where
mkSymVal
| isKString @[a] undefined = genMkSymVar KString
| True = genMkSymVar (KList (kindOf (Proxy @a)))
literal as
| isKString @[a] undefined = case fromDynamic (toDyn as) of
Just s -> SBV . SVal KString . Left . CV KString . CString $ s
Nothing -> error "SString: Cannot construct literal string!"
| True = let k = KList (kindOf (Proxy @a))
in SBV $ SVal k $ Left $ CV k $ CList $ map toCV as
fromCV (CV _ (CString a)) = fromMaybe (error "SString: Cannot extract a literal string!")
(fromDynamic (toDyn a))
fromCV (CV _ (CList a)) = fromCV . CV (kindOf (Proxy @a)) <$> a
fromCV c = error $ "SymVal.fromCV: Unexpected non-list value: " ++ show c
minMaxBound = Nothing
instance ValidFloat eb sb => HasKind (FloatingPoint eb sb) where
kindOf _ = KFP (intOfProxy (Proxy @eb)) (intOfProxy (Proxy @sb))
instance ValidFloat eb sb => SymVal (FloatingPoint eb sb) where
mkSymVal = genMkSymVar (KFP (intOfProxy (Proxy @eb)) (intOfProxy (Proxy @sb)))
literal (FloatingPoint r) = let k = KFP (intOfProxy (Proxy @eb)) (intOfProxy (Proxy @sb))
in SBV $ SVal k $ Left $ CV k (CFP r)
fromCV (CV _ (CFP r)) = FloatingPoint r
fromCV c = error $ "SymVal.FPR: Unexpected non-arbitrary-precision value: " ++ show c
minMaxBound = Nothing
-- | 'SymVal' instance for 'WordN'
instance (KnownNat n, BVIsNonZero n) => SymVal (WordN n) where
literal x = genLiteral (kindOf x) x
mkSymVal = genMkSymVar (kindOf (undefined :: WordN n))
fromCV = genFromCV
-- | 'SymVal' instance for 'IntN'
instance (KnownNat n, BVIsNonZero n) => SymVal (IntN n) where
literal x = genLiteral (kindOf x) x
mkSymVal = genMkSymVar (kindOf (undefined :: IntN n))
fromCV = genFromCV
toCV :: SymVal a => a -> CVal
toCV a = case literal a of
SBV (SVal _ (Left cv)) -> cvVal cv
_ -> error "SymVal.toCV: Impossible happened, couldn't produce a concrete value"
mkCVTup :: Int -> Kind -> [CVal] -> SBV a
mkCVTup i k@(KTuple ks) cs
| lks == lcs && lks == i
= SBV $ SVal k $ Left $ CV k $ CTuple cs
| True
= error $ "SymVal.mkCVTup: Impossible happened. Malformed tuple received: " ++ show (i, k)
where lks = length ks
lcs = length cs
mkCVTup i k _
= error $ "SymVal.mkCVTup: Impossible happened. Non-tuple received: " ++ show (i, k)
fromCVTup :: Int -> CV -> [CV]
fromCVTup i inp@(CV (KTuple ks) (CTuple cs))
| lks == lcs && lks == i
= zipWith CV ks cs
| True
= error $ "SymVal.fromCTup: Impossible happened. Malformed tuple received: " ++ show (i, inp)
where lks = length ks
lcs = length cs
fromCVTup i inp = error $ "SymVal.fromCVTup: Impossible happened. Non-tuple received: " ++ show (i, inp)
instance (SymVal a, SymVal b) => SymVal (Either a b) where
mkSymVal = genMkSymVar (kindOf (Proxy @(Either a b)))
literal s
| Left a <- s = mk $ Left (toCV a)
| Right b <- s = mk $ Right (toCV b)
where k = kindOf (Proxy @(Either a b))
mk = SBV . SVal k . Left . CV k . CEither
fromCV (CV (KEither k1 _ ) (CEither (Left c))) = Left $ fromCV $ CV k1 c
fromCV (CV (KEither _ k2) (CEither (Right c))) = Right $ fromCV $ CV k2 c
fromCV bad = error $ "SymVal.fromCV (Either): Malformed either received: " ++ show bad
minMaxBound = Nothing
instance SymVal a => SymVal (Maybe a) where
mkSymVal = genMkSymVar (kindOf (Proxy @(Maybe a)))
literal s
| Nothing <- s = mk Nothing
| Just a <- s = mk $ Just (toCV a)
where k = kindOf (Proxy @(Maybe a))
mk = SBV . SVal k . Left . CV k . CMaybe
fromCV (CV (KMaybe _) (CMaybe Nothing)) = Nothing
fromCV (CV (KMaybe k) (CMaybe (Just x))) = Just $ fromCV $ CV k x
fromCV bad = error $ "SymVal.fromCV (Maybe): Malformed sum received: " ++ show bad
minMaxBound = Nothing
instance (HasKind a, HasKind b, SymVal a, SymVal b) => SymVal (ArrayModel a b) where
mkSymVal = genMkSymVar (KArray (kindOf (Proxy @a)) (kindOf (Proxy @b)))
-- If the table has duplicate entries for keys, then the first one takes precedence.
-- That is, [(a, v1), (a, v2)] is equivalent to [(a, v1)]. The best way to think about
-- this is as a "stack" of writes. [(a, v1), (a, v2)] means we first "wrote" v2 at
-- a, and then wrote v1 at the same address; so the first write of v2 got overwritten.
literal (ArrayModel tbl def) = SBV . SVal knd . Left . CV knd $ CArray $ ArrayModel [(toCV k, toCV v) | (k, v) <- tbl] (toCV def)
where knd = kindOf (Proxy @(ArrayModel a b))
fromCV (CV (KArray k1 k2) (CArray (ArrayModel assocs def))) = ArrayModel [(fromCV (CV k1 a), fromCV (CV k2 b)) | (a, b) <- assocs]
(fromCV (CV k2 def))
fromCV bad = error $ "SymVal.fromCV (SArray): Malformed array received: " ++ show bad
minMaxBound = Nothing
instance (Arbitrary a, Arbitrary b) => Arbitrary (ArrayModel a b) where
arbitrary = ArrayModel <$> arbitrary <*> arbitrary
instance (Ord a, SymVal a) => SymVal (RCSet a) where
mkSymVal = genMkSymVar (kindOf (Proxy @(RCSet a)))
literal eur = SBV $ SVal k $ Left $ CV k $ CSet $ dir $ Set.map toCV s
where (dir, s) = case eur of
RegularSet x -> (RegularSet, x)
ComplementSet x -> (ComplementSet, x)
k = kindOf (Proxy @(RCSet a))
fromCV (CV (KSet a) (CSet (RegularSet s))) = RegularSet $ Set.map (fromCV . CV a) s
fromCV (CV (KSet a) (CSet (ComplementSet s))) = ComplementSet $ Set.map (fromCV . CV a) s
fromCV bad = error $ "SymVal.fromCV (Set): Malformed set received: " ++ show bad
minMaxBound = Nothing
-- | SymVal for 0-tuple (i.e., unit)
instance SymVal () where
mkSymVal = genMkSymVar (KTuple [])
literal () = mkCVTup 0 (kindOf (Proxy @())) []
fromCV cv = fromCVTup 0 cv `seq` ()
-- | SymVal for 2-tuples
instance (SymVal a, SymVal b) => SymVal (a, b) where
mkSymVal = genMkSymVar (kindOf (Proxy @(a, b)))
literal (v1, v2) = mkCVTup 2 (kindOf (Proxy @(a, b))) [toCV v1, toCV v2]
fromCV cv = let ~[v1, v2] = fromCVTup 2 cv
in (fromCV v1, fromCV v2)
minMaxBound = Nothing
-- | SymVal for 3-tuples
instance (SymVal a, SymVal b, SymVal c) => SymVal (a, b, c) where
mkSymVal = genMkSymVar (kindOf (Proxy @(a, b, c)))
literal (v1, v2, v3) = mkCVTup 3 (kindOf (Proxy @(a, b, c))) [toCV v1, toCV v2, toCV v3]
fromCV cv = let ~[v1, v2, v3] = fromCVTup 3 cv
in (fromCV v1, fromCV v2, fromCV v3)
minMaxBound = Nothing
-- | SymVal for 4-tuples
instance (SymVal a, SymVal b, SymVal c, SymVal d) => SymVal (a, b, c, d) where
mkSymVal = genMkSymVar (kindOf (Proxy @(a, b, c, d)))
literal (v1, v2, v3, v4) = mkCVTup 4 (kindOf (Proxy @(a, b, c, d))) [toCV v1, toCV v2, toCV v3, toCV v4]
fromCV cv = let ~[v1, v2, v3, v4] = fromCVTup 4 cv
in (fromCV v1, fromCV v2, fromCV v3, fromCV v4)
minMaxBound = Nothing
-- | SymVal for 5-tuples
instance (SymVal a, SymVal b, SymVal c, SymVal d, SymVal e) => SymVal (a, b, c, d, e) where
mkSymVal = genMkSymVar (kindOf (Proxy @(a, b, c, d, e)))
literal (v1, v2, v3, v4, v5) = mkCVTup 5 (kindOf (Proxy @(a, b, c, d, e))) [toCV v1, toCV v2, toCV v3, toCV v4, toCV v5]
fromCV cv = let ~[v1, v2, v3, v4, v5] = fromCVTup 5 cv
in (fromCV v1, fromCV v2, fromCV v3, fromCV v4, fromCV v5)
minMaxBound = Nothing
-- | SymVal for 6-tuples
instance (SymVal a, SymVal b, SymVal c, SymVal d, SymVal e, SymVal f) => SymVal (a, b, c, d, e, f) where
mkSymVal = genMkSymVar (kindOf (Proxy @(a, b, c, d, e, f)))
literal (v1, v2, v3, v4, v5, v6) = mkCVTup 6 (kindOf (Proxy @(a, b, c, d, e, f))) [toCV v1, toCV v2, toCV v3, toCV v4, toCV v5, toCV v6]
fromCV cv = let ~[v1, v2, v3, v4, v5, v6] = fromCVTup 6 cv
in (fromCV v1, fromCV v2, fromCV v3, fromCV v4, fromCV v5, fromCV v6)
minMaxBound = Nothing
-- | SymVal for 7-tuples
instance (SymVal a, SymVal b, SymVal c, SymVal d, SymVal e, SymVal f, SymVal g) => SymVal (a, b, c, d, e, f, g) where
mkSymVal = genMkSymVar (kindOf (Proxy @(a, b, c, d, e, f, g)))
literal (v1, v2, v3, v4, v5, v6, v7) = mkCVTup 7 (kindOf (Proxy @(a, b, c, d, e, f, g))) [toCV v1, toCV v2, toCV v3, toCV v4, toCV v5, toCV v6, toCV v7]
fromCV cv = let ~[v1, v2, v3, v4, v5, v6, v7] = fromCVTup 7 cv
in (fromCV v1, fromCV v2, fromCV v3, fromCV v4, fromCV v5, fromCV v6, fromCV v7)
minMaxBound = Nothing
-- | SymVal for 8-tuples
instance (SymVal a, SymVal b, SymVal c, SymVal d, SymVal e, SymVal f, SymVal g, SymVal h) => SymVal (a, b, c, d, e, f, g, h) where
mkSymVal = genMkSymVar (kindOf (Proxy @(a, b, c, d, e, f, g, h)))
literal (v1, v2, v3, v4, v5, v6, v7, v8) = mkCVTup 8 (kindOf (Proxy @(a, b, c, d, e, f, g, h))) [toCV v1, toCV v2, toCV v3, toCV v4, toCV v5, toCV v6, toCV v7, toCV v8]
fromCV cv = let ~[v1, v2, v3, v4, v5, v6, v7, v8] = fromCVTup 8 cv
in (fromCV v1, fromCV v2, fromCV v3, fromCV v4, fromCV v5, fromCV v6, fromCV v7, fromCV v8)
minMaxBound = Nothing
instance IsString SString where
fromString = literal
------------------------------------------------------------------------------------
-- * Smart constructors for creating symbolic values. These are not strictly
-- necessary, as they are mere aliases for 'symbolic' and 'symbolics', but
-- they nonetheless make programming easier.
------------------------------------------------------------------------------------
-- | Generalization of 'Data.SBV.sBool'
sBool :: MonadSymbolic m => String -> m SBool
sBool = symbolic
-- | Generalization of 'Data.SBV.sBool_'
sBool_ :: MonadSymbolic m => m SBool
sBool_ = free_
-- | Generalization of 'Data.SBV.sBools'
sBools :: MonadSymbolic m => [String] -> m [SBool]
sBools = symbolics
-- | Generalization of 'Data.SBV.sWord8'
sWord8 :: MonadSymbolic m => String -> m SWord8
sWord8 = symbolic
-- | Generalization of 'Data.SBV.sWord8_'
sWord8_ :: MonadSymbolic m => m SWord8
sWord8_ = free_
-- | Generalization of 'Data.SBV.sWord8s'
sWord8s :: MonadSymbolic m => [String] -> m [SWord8]
sWord8s = symbolics
-- | Generalization of 'Data.SBV.sWord16'
sWord16 :: MonadSymbolic m => String -> m SWord16
sWord16 = symbolic
-- | Generalization of 'Data.SBV.sWord16_'
sWord16_ :: MonadSymbolic m => m SWord16
sWord16_ = free_
-- | Generalization of 'Data.SBV.sWord16s'
sWord16s :: MonadSymbolic m => [String] -> m [SWord16]
sWord16s = symbolics
-- | Generalization of 'Data.SBV.sWord32'
sWord32 :: MonadSymbolic m => String -> m SWord32
sWord32 = symbolic
-- | Generalization of 'Data.SBV.sWord32_'
sWord32_ :: MonadSymbolic m => m SWord32
sWord32_ = free_
-- | Generalization of 'Data.SBV.sWord32s'
sWord32s :: MonadSymbolic m => [String] -> m [SWord32]
sWord32s = symbolics
-- | Generalization of 'Data.SBV.sWord64'
sWord64 :: MonadSymbolic m => String -> m SWord64
sWord64 = symbolic
-- | Generalization of 'Data.SBV.sWord64_'
sWord64_ :: MonadSymbolic m => m SWord64
sWord64_ = free_
-- | Generalization of 'Data.SBV.sWord64s'
sWord64s :: MonadSymbolic m => [String] -> m [SWord64]
sWord64s = symbolics
-- | Generalization of 'Data.SBV.sInt8'
sInt8 :: MonadSymbolic m => String -> m SInt8
sInt8 = symbolic
-- | Generalization of 'Data.SBV.sInt8_'
sInt8_ :: MonadSymbolic m => m SInt8
sInt8_ = free_
-- | Generalization of 'Data.SBV.sInt8s'
sInt8s :: MonadSymbolic m => [String] -> m [SInt8]
sInt8s = symbolics
-- | Generalization of 'Data.SBV.sInt16'
sInt16 :: MonadSymbolic m => String -> m SInt16
sInt16 = symbolic
-- | Generalization of 'Data.SBV.sInt16_'
sInt16_ :: MonadSymbolic m => m SInt16
sInt16_ = free_
-- | Generalization of 'Data.SBV.sInt16s'
sInt16s :: MonadSymbolic m => [String] -> m [SInt16]
sInt16s = symbolics
-- | Generalization of 'Data.SBV.sInt32'
sInt32 :: MonadSymbolic m => String -> m SInt32
sInt32 = symbolic
-- | Generalization of 'Data.SBV.sInt32_'
sInt32_ :: MonadSymbolic m => m SInt32
sInt32_ = free_
-- | Generalization of 'Data.SBV.sInt32s'
sInt32s :: MonadSymbolic m => [String] -> m [SInt32]
sInt32s = symbolics
-- | Generalization of 'Data.SBV.sInt64'
sInt64 :: MonadSymbolic m => String -> m SInt64
sInt64 = symbolic
-- | Generalization of 'Data.SBV.sInt64_'
sInt64_ :: MonadSymbolic m => m SInt64
sInt64_ = free_
-- | Generalization of 'Data.SBV.sInt64s'
sInt64s :: MonadSymbolic m => [String] -> m [SInt64]
sInt64s = symbolics
-- | Generalization of 'Data.SBV.sInteger'
sInteger:: MonadSymbolic m => String -> m SInteger
sInteger = symbolic
-- | Generalization of 'Data.SBV.sInteger_'
sInteger_:: MonadSymbolic m => m SInteger
sInteger_ = free_
-- | Generalization of 'Data.SBV.sIntegers'
sIntegers :: MonadSymbolic m => [String] -> m [SInteger]
sIntegers = symbolics
-- | Generalization of 'Data.SBV.sReal'
sReal:: MonadSymbolic m => String -> m SReal
sReal = symbolic
-- | Generalization of 'Data.SBV.sReal_'
sReal_:: MonadSymbolic m => m SReal
sReal_ = free_
-- | Generalization of 'Data.SBV.sReals'
sReals :: MonadSymbolic m => [String] -> m [SReal]
sReals = symbolics
-- | Generalization of 'Data.SBV.sFloat'
sFloat :: MonadSymbolic m => String -> m SFloat
sFloat = symbolic
-- | Generalization of 'Data.SBV.sFloat_'
sFloat_ :: MonadSymbolic m => m SFloat
sFloat_ = free_
-- | Generalization of 'Data.SBV.sFloats'
sFloats :: MonadSymbolic m => [String] -> m [SFloat]
sFloats = symbolics
-- | Generalization of 'Data.SBV.sDouble'
sDouble :: MonadSymbolic m => String -> m SDouble
sDouble = symbolic
-- | Generalization of 'Data.SBV.sDouble_'
sDouble_ :: MonadSymbolic m => m SDouble
sDouble_ = free_
-- | Generalization of 'Data.SBV.sDoubles'
sDoubles :: MonadSymbolic m => [String] -> m [SDouble]
sDoubles = symbolics
-- | Generalization of 'Data.SBV.sFPHalf'
sFPHalf :: String -> Symbolic SFPHalf
sFPHalf = symbolic
-- | Generalization of 'Data.SBV.sFPHalf_'
sFPHalf_ :: Symbolic SFPHalf
sFPHalf_ = free_
-- | Generalization of 'Data.SBV.sFPHalfs'
sFPHalfs :: [String] -> Symbolic [SFPHalf]
sFPHalfs = symbolics
-- | Generalization of 'Data.SBV.sFPBFloat'
sFPBFloat :: String -> Symbolic SFPBFloat
sFPBFloat = symbolic
-- | Generalization of 'Data.SBV.sFPBFloat_'
sFPBFloat_ :: Symbolic SFPBFloat
sFPBFloat_ = free_
-- | Generalization of 'Data.SBV.sFPBFloats'
sFPBFloats :: [String] -> Symbolic [SFPBFloat]
sFPBFloats = symbolics
-- | Generalization of 'Data.SBV.sFPSingle'
sFPSingle :: String -> Symbolic SFPSingle
sFPSingle = symbolic
-- | Generalization of 'Data.SBV.sFPSingle_'
sFPSingle_ :: Symbolic SFPSingle
sFPSingle_ = free_
-- | Generalization of 'Data.SBV.sFPSingles'
sFPSingles :: [String] -> Symbolic [SFPSingle]
sFPSingles = symbolics
-- | Generalization of 'Data.SBV.sFPDouble'
sFPDouble :: String -> Symbolic SFPDouble
sFPDouble = symbolic
-- | Generalization of 'Data.SBV.sFPDouble_'
sFPDouble_ :: Symbolic SFPDouble
sFPDouble_ = free_
-- | Generalization of 'Data.SBV.sFPDoubles'
sFPDoubles :: [String] -> Symbolic [SFPDouble]
sFPDoubles = symbolics
-- | Generalization of 'Data.SBV.sFPQuad'
sFPQuad :: String -> Symbolic SFPQuad
sFPQuad = symbolic
-- | Generalization of 'Data.SBV.sFPQuad_'
sFPQuad_ :: Symbolic SFPQuad
sFPQuad_ = free_
-- | Generalization of 'Data.SBV.sFPQuads'
sFPQuads :: [String] -> Symbolic [SFPQuad]
sFPQuads = symbolics
-- | Generalization of 'Data.SBV.sFloatingPoint'
sFloatingPoint :: ValidFloat eb sb => String -> Symbolic (SFloatingPoint eb sb)
sFloatingPoint = symbolic
-- | Generalization of 'Data.SBV.sFloatingPoint_'
sFloatingPoint_ :: ValidFloat eb sb => Symbolic (SFloatingPoint eb sb)
sFloatingPoint_ = free_
-- | Generalization of 'Data.SBV.sFloatingPoints'
sFloatingPoints :: ValidFloat eb sb => [String] -> Symbolic [SFloatingPoint eb sb]
sFloatingPoints = symbolics
-- | Generalization of 'Data.SBV.sWord'
sWord :: (KnownNat n, BVIsNonZero n) => MonadSymbolic m => String -> m (SWord n)
sWord = symbolic
-- | Generalization of 'Data.SBV.sWord_'
sWord_ :: (KnownNat n, BVIsNonZero n) => MonadSymbolic m => m (SWord n)
sWord_ = free_
-- | Generalization of 'Data.SBV.sWord64s'
sWords :: (KnownNat n, BVIsNonZero n) => MonadSymbolic m => [String] -> m [SWord n]
sWords = symbolics
-- | Generalization of 'Data.SBV.sInt'
sInt :: (KnownNat n, BVIsNonZero n) => MonadSymbolic m => String -> m (SInt n)
sInt = symbolic
-- | Generalization of 'Data.SBV.sInt_'
sInt_ :: (KnownNat n, BVIsNonZero n) => MonadSymbolic m => m (SInt n)
sInt_ = free_
-- | Generalization of 'Data.SBV.sInts'
sInts :: (KnownNat n, BVIsNonZero n) => MonadSymbolic m => [String] -> m [SInt n]
sInts = symbolics
-- | Generalization of 'Data.SBV.sChar'
sChar :: MonadSymbolic m => String -> m SChar
sChar = symbolic
-- | Generalization of 'Data.SBV.sChar_'
sChar_ :: MonadSymbolic m => m SChar
sChar_ = free_
-- | Generalization of 'Data.SBV.sChars'
sChars :: MonadSymbolic m => [String] -> m [SChar]
sChars = symbolics
-- | Generalization of 'Data.SBV.sString'
sString :: MonadSymbolic m => String -> m SString
sString = symbolic
-- | Generalization of 'Data.SBV.sString_'
sString_ :: MonadSymbolic m => m SString
sString_ = free_
-- | Generalization of 'Data.SBV.sStrings'
sStrings :: MonadSymbolic m => [String] -> m [SString]
sStrings = symbolics
-- | Generalization of 'Data.SBV.sList'
sList :: (SymVal a, MonadSymbolic m) => String -> m (SList a)
sList = symbolic
-- | Generalization of 'Data.SBV.sList_'
sList_ :: (SymVal a, MonadSymbolic m) => m (SList a)
sList_ = free_
-- | Generalization of 'Data.SBV.sLists'
sLists :: (SymVal a, MonadSymbolic m) => [String] -> m [SList a]
sLists = symbolics
-- | Generalization of 'Data.SBV.sAray'
sArray :: (SymVal a, SymVal b, MonadSymbolic m) => String -> m (SArray a b)
sArray = symbolic
-- | Generalization of 'Data.SBV.sList_'
sArray_ :: (SymVal a, SymVal b, MonadSymbolic m) => m (SArray a b)
sArray_ = free_
-- | Generalization of 'Data.SBV.sLists'
sArrays :: (SymVal a, SymVal b, MonadSymbolic m) => [String] -> m [SArray a b]
sArrays = symbolics
-- | Identify tuple like things. Note that there are no methods, just instances to control type inference
class SymTuple a
instance SymTuple ()
instance SymTuple (a, b)
instance SymTuple (a, b, c)
instance SymTuple (a, b, c, d)
instance SymTuple (a, b, c, d, e)
instance SymTuple (a, b, c, d, e, f)
instance SymTuple (a, b, c, d, e, f, g)
instance SymTuple (a, b, c, d, e, f, g, h)
-- | Generalization of 'Data.SBV.sTuple'
sTuple :: (SymTuple tup, SymVal tup, MonadSymbolic m) => String -> m (SBV tup)
sTuple = symbolic
-- | Generalization of 'Data.SBV.sTuple_'
sTuple_ :: (SymTuple tup, SymVal tup, MonadSymbolic m) => m (SBV tup)
sTuple_ = free_
-- | Generalization of 'Data.SBV.sTuples'
sTuples :: (SymTuple tup, SymVal tup, MonadSymbolic m) => [String] -> m [SBV tup]
sTuples = symbolics
-- | Generalization of 'Data.SBV.sRational'
sRational :: MonadSymbolic m => String -> m SRational
sRational = symbolic
-- | Generalization of 'Data.SBV.sRational_'
sRational_ :: MonadSymbolic m => m SRational
sRational_ = free_
-- | Generalization of 'Data.SBV.sRationals'
sRationals :: MonadSymbolic m => [String] -> m [SRational]
sRationals = symbolics
-- | Generalization of 'Data.SBV.sEither'
sEither :: (SymVal a, SymVal b, MonadSymbolic m) => String -> m (SEither a b)
sEither = symbolic
-- | Generalization of 'Data.SBV.sEither_'
sEither_ :: (SymVal a, SymVal b, MonadSymbolic m) => m (SEither a b)
sEither_ = free_
-- | Generalization of 'Data.SBV.sEithers'
sEithers :: (SymVal a, SymVal b, MonadSymbolic m) => [String] -> m [SEither a b]
sEithers = symbolics
-- | Generalization of 'Data.SBV.sMaybe'
sMaybe :: (SymVal a, MonadSymbolic m) => String -> m (SMaybe a)
sMaybe = symbolic
-- | Generalization of 'Data.SBV.sMaybe_'
sMaybe_ :: (SymVal a, MonadSymbolic m) => m (SMaybe a)
sMaybe_ = free_
-- | Generalization of 'Data.SBV.sMaybes'
sMaybes :: (SymVal a, MonadSymbolic m) => [String] -> m [SMaybe a]
sMaybes = symbolics
-- | Generalization of 'Data.SBV.sSet'
sSet :: (Ord a, SymVal a, MonadSymbolic m) => String -> m (SSet a)
sSet = symbolic
-- | Generalization of 'Data.SBV.sMaybe_'
sSet_ :: (Ord a, SymVal a, MonadSymbolic m) => m (SSet a)
sSet_ = free_
-- | Generalization of 'Data.SBV.sMaybes'
sSets :: (Ord a, SymVal a, MonadSymbolic m) => [String] -> m [SSet a]
sSets = symbolics
-- | Generalization of 'Data.SBV.solve'
solve :: MonadSymbolic m => [SBool] -> m SBool
solve = return . sAnd
-- | Convert an SReal to an SInteger. That is, it computes the
-- largest integer @n@ that satisfies @sIntegerToSReal n <= r@
-- essentially giving us the @floor@.
--
-- For instance, @1.3@ will be @1@, but @-1.3@ will be @-2@.
sRealToSInteger :: SReal -> SInteger
sRealToSInteger x
| Just i <- unliteral x, isExactRational i
= literal $ floor (toRational i)
| True
= SBV (SVal KUnbounded (Right (cache y)))
where y st = do xsv <- sbvToSV st x
newExpr st KUnbounded (SBVApp (KindCast KReal KUnbounded) [xsv])
-- | Convert an SReal to an SInteger, truncating version.
sRealToSIntegerTruncate :: SReal -> SInteger
sRealToSIntegerTruncate x = ite (x .< 0) (sRealToSInteger x) (- (sRealToSInteger (- x)))
-- | 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. Compare this to 'observe'
-- which is good for printing counter-examples.
label :: SymVal a => String -> SBV a -> SBV a
label m x
| Just _ <- unliteral x = x
| True = SBV $ SVal k $ Right $ cache r
where k = kindOf x
r st = do xsv <- sbvToSV st x
newExpr st k (SBVApp (Label m) [xsv])
-- | Observe the value of an expression, if the given condition holds. Such values are useful in model construction, as they are printed part of a satisfying model, or a
-- counter-example. The same works for quick-check as well. Useful when we want to see intermediate values, or expected/obtained
-- pairs in a particular run. Note that an observed expression is always symbolic, i.e., it won't be constant folded. Compare this to 'label'
-- which is used for putting a label in the generated SMTLib-C code.
--
-- NB. If the observed expression happens under a SBV-lambda expression, then it is silently ignored; since
-- there's no way to access the value of such a value.
observeIf :: SymVal a => (a -> Bool) -> String -> SBV a -> SBV a
observeIf cond m x
| Just bad <- checkObservableName m
= error bad
| True
= SBV $ SVal k $ Right $ cache r
where k = kindOf x
r st = do xsv <- sbvToSV st (label ("Observing: " ++ m) x)
recordObservable st m (cond . fromCV) xsv
return xsv
-- | Observe the value of an expression, unconditionally. See 'observeIf' for a generalized version.
observe :: SymVal a => String -> SBV a -> SBV a
observe = observeIf (const True)
-- | 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.
infix 4 .<, .<=, .>, .>=
class (Mergeable a, EqSymbolic a) => OrdSymbolic a where
-- | Symbolic less than.
(.<) :: a -> a -> SBool
-- | Symbolic less than or equal to.
(.<=) :: a -> a -> SBool
-- | Symbolic greater than.
(.>) :: a -> a -> SBool
-- | Symbolic greater than or equal to.
(.>=) :: a -> a -> SBool
-- | Symbolic minimum.
smin :: a -> a -> a
-- | Symbolic maximum.
smax :: a -> a -> a
-- | Is the value within the allowed /inclusive/ range?
inRange :: a -> (a, a) -> SBool
{-# MINIMAL (.<) #-}
a .<= b = a .< b .|| a .== b
a .> b = b .< a
a .>= b = b .<= a
a `smin` b = ite (a .<= b) a b
a `smax` b = ite (a .<= b) b a
inRange x (y, z) = x .>= y .&& x .<= z
{- We can't have a generic instance of the form:
instance Eq a => EqSymbolic a where
x .== y = if x == y then true else sFalse
even if we're willing to allow Flexible/undecidable instances..
This is because if we allow this it would imply EqSymbolic (SBV a);
since (SBV a) has to be Eq as it must be a Num. But this wouldn't be
the right choice obviously; as the Eq instance is bogus for SBV
for natural reasons..
-}
-- It is tempting to put in an @Eq a@ superclass here. But doing so
-- is complicated, as it requires all underlying types to have equality,
-- which is at best shaky for algebraic reals and sets. So, leave it out.
instance (HasKind a, SymVal a) => EqSymbolic (SBV a) where
SBV x .== SBV y = SBV (svEqual x y)
SBV x ./= SBV y = SBV (svNotEqual x y)
SBV x .=== SBV y = SBV (svStrongEqual x y)
-- Custom version of distinct that generates better code for base types
distinct [] = sTrue
distinct [_] = sTrue
distinct xs | all isConc xs = checkDiff xs
| [SBV a, SBV b] <- xs, a `is` svBool True = SBV $ svNot b
| [SBV a, SBV b] <- xs, b `is` svBool True = SBV $ svNot a
| [SBV a, SBV b] <- xs, a `is` svBool False = SBV b
| [SBV a, SBV b] <- xs, b `is` svBool False = SBV a
-- 3 booleans can't be distinct!
| (x : _ : _ : _) <- xs, isBool x = sFalse
| True = SBV (SVal KBool (Right (cache r)))
where r st = do xsv <- mapM (sbvToSV st) xs
newExpr st KBool (SBVApp NotEqual xsv)
-- We call this in case all are concrete, which will
-- reduce to a constant and generate no code at all!
-- Note that this is essentially the same as the default
-- definition, which unfortunately we can no longer call!
checkDiff [] = sTrue
checkDiff (a:as) = sAll (a ./=) as .&& checkDiff as
-- Sigh, we can't use isConcrete since that requires SymVal
-- constraint that we don't have here. (To support SBools.)
isConc (SBV (SVal _ (Left _))) = True
isConc _ = False
-- Likewise here; need to go lower.
SVal k1 (Left c1) `is` SVal k2 (Left c2) = (k1, c1) == (k2, c2)
_ `is` _ = False
isBool (SBV (SVal KBool _)) = True
isBool _ = False
-- Custom version of distinctExcept that generates better code for base types
distinctExcept [] _ = sTrue
distinctExcept [_] _ = sTrue
distinctExcept es ignored
| all isConc (es ++ ignored)
= distinct (filter ignoreConc es)
| True
= SBV (SVal KBool (Right (cache r)))
where ignoreConc x = case x `sElem` ignored of
SBV (SVal KBool (Left cv)) -> cvToBool cv
_ -> error $ "distinctExcept: Impossible happened, concrete sElem failed: " ++ show (es, ignored, x)
r st = do let incr x table = ite (x `sElem` ignored) (0 :: SInteger) (1 + readArrayNoEq table x)
initArray :: SArray a Integer
initArray = lambdaArray (const 0)
finalArray = foldl (\table x -> writeArrayNoKnd table x (incr x table)) initArray es
sbvToSV st $ sAll (\e -> readArrayNoEq finalArray e .<= (1 :: SInteger)) es
-- Sigh, we can't use isConcrete since that requires SymVal
-- constraint that we don't have here. (To support SBools.)
isConc (SBV (SVal _ (Left _))) = True
isConc _ = False
-- Version of readArray that doesn't have the Eq constraint, since we don't have it here
readArrayNoEq array key = SBV . SVal KUnbounded . Right $ cache g
where g st = do f <- sbvToSV st array
k <- sbvToSV st key
newExpr st KUnbounded (SBVApp ReadArray [f, k])
writeArrayNoKnd :: forall key. HasKind key => SArray key Integer -> SBV key -> SInteger -> SArray key Integer
writeArrayNoKnd array key value = SBV . SVal k . Right $ cache g
where k = KArray (kindOf (Proxy @key)) KUnbounded
g st = do arr <- sbvToSV st array
keyVal <- sbvToSV st key
val <- sbvToSV st value
newExpr st k (SBVApp WriteArray [arr, keyVal, val])
-- We don't want to do a generic OrdSymbolic (SBV a) instance; since that would be dangerous, like the case
-- for Num. So, we explicitly define for each type we care about.
#define MKSORD(CSTR, TYPE) \
instance CSTR => OrdSymbolic TYPE where { \
a@(SBV x) .< b@(SBV y) | smtComparable "<" a b = SBV (svLessThan x y) \
| True = SBV (svStructuralLessThan x y); \
\
a@(SBV x) .<= b@(SBV y) | smtComparable ".<=" a b = SBV (svLessEq x y) \
| True = a .< b .|| a .== b; \
\
a@(SBV x) .> b@(SBV y) | smtComparable ">" a b = SBV (svGreaterThan x y) \
| True = b .< a; \
\
a@(SBV x) .>= b@(SBV y) | smtComparable ">=" a b = SBV (svGreaterEq x y) \
| True = b .<= a; \
} \
-- Derive basic instances we need. NB. We don't give the SRational instance here. It's handled
-- in Data/SBV/Rational due to representation issues.
MKSORD((), SInteger)
MKSORD((), SWord8)
MKSORD((), SWord16)
MKSORD((), SWord32)
MKSORD((), SWord64)
MKSORD((), SInt8)
MKSORD((), SInt16)
MKSORD((), SInt32)
MKSORD((), SInt64)
MKSORD((), SFloat)
MKSORD((), SChar)
MKSORD((SymVal a), (SMaybe a))
MKSORD((SymVal a), (SList a))
MKSORD((SymVal a, SymVal b), (SEither a b))
MKSORD((), SDouble)
MKSORD((), SReal)
MKSORD((KnownNat n, BVIsNonZero n), (SWord n))
MKSORD((KnownNat n, BVIsNonZero n), (SInt n))
MKSORD((ValidFloat eb sb), (SFloatingPoint eb sb))
-- Tuples
MKSORD((SymVal a, SymVal b), (SBV (a, b)))
MKSORD((SymVal a, SymVal b, SymVal c), (SBV (a, b, c)))
MKSORD((SymVal a, SymVal b, SymVal c, SymVal d), (SBV (a, b, c, d)))
MKSORD((SymVal a, SymVal b, SymVal c, SymVal d, SymVal e), (SBV (a, b, c, d, e)))
MKSORD((SymVal a, SymVal b, SymVal c, SymVal d, SymVal e, SymVal f), (SBV (a, b, c, d, e, f)))
MKSORD((SymVal a, SymVal b, SymVal c, SymVal d, SymVal e, SymVal f, SymVal g), (SBV (a, b, c, d, e, f, g)))
MKSORD((SymVal a, SymVal b, SymVal c, SymVal d, SymVal e, SymVal f, SymVal g, SymVal h), (SBV (a, b, c, d, e, f, g, h)))
#undef MKSORD
-- Is this a type that's comparable by underlying translation to SMTLib?
-- Note that we allow concrete versions to go through unless the type is a set, as there's really no reason not to.
smtComparable :: (SymVal a, HasKind a) => String -> SBV a -> SBV a -> Bool
smtComparable op x y
| isConcrete x && isConcrete y && not (isSet k)
= True
| True
= case k of
KBool -> True
KBounded {} -> True
KUnbounded {} -> True
KReal {} -> True
KUserSort {} -> True
KFloat -> True
KDouble -> True
KRational {} -> True
KFP {} -> True
KChar -> True
KString -> True
KList {} -> nope -- Unfortunately, no way for us to desugar this
KSet {} -> nope -- Ditto here..
KTuple {} -> False
KMaybe {} -> False
KEither {} -> False
KArray {} -> True
where k = kindOf x
nope = error $ "Data.SBV.OrdSymbolic: SMTLib does not support " ++ op ++ " for " ++ show k
-- Bool
instance EqSymbolic Bool where
x .== y = fromBool $ x == y
-- Lists
instance EqSymbolic a => EqSymbolic [a] where
[] .== [] = sTrue
(x:xs) .== (y:ys) = x .== y .&& xs .== ys
_ .== _ = sFalse
[] .=== [] = sTrue
(x:xs) .=== (y:ys) = x .=== y .&& xs .=== ys
_ .=== _ = sFalse
instance OrdSymbolic a => OrdSymbolic [a] where
[] .< [] = sFalse
[] .< _ = sTrue
_ .< [] = sFalse
(x:xs) .< (y:ys) = x .< y .|| (x .== y .&& xs .< ys)
-- NonEmpty
instance EqSymbolic a => EqSymbolic (NonEmpty a) where
(x :| xs) .== (y :| ys) = x : xs .== y : ys
(x :| xs) .=== (y :| ys) = x : xs .=== y : ys
instance OrdSymbolic a => OrdSymbolic (NonEmpty a) where
(x :| xs) .< (y :| ys) = x : xs .< y : ys
-- Maybe
instance EqSymbolic a => EqSymbolic (Maybe a) where
Nothing .== Nothing = sTrue
Just a .== Just b = a .== b
_ .== _ = sFalse
instance OrdSymbolic a => OrdSymbolic (Maybe a) where
Nothing .< Nothing = sFalse
Nothing .< _ = sTrue
Just _ .< Nothing = sFalse
Just a .< Just b = a .< b
-- Either
instance (EqSymbolic a, EqSymbolic b) => EqSymbolic (Either a b) where
Left a .== Left b = a .== b
Right a .== Right b = a .== b
_ .== _ = sFalse
Left a .=== Left b = a .=== b
Right a .=== Right b = a .=== b
_ .=== _ = sFalse
instance (OrdSymbolic a, OrdSymbolic b) => OrdSymbolic (Either a b) where
Left a .< Left b = a .< b
Left _ .< Right _ = sTrue
Right _ .< Left _ = sFalse
Right a .< Right b = a .< b
-- 2-Tuple
instance (EqSymbolic a, EqSymbolic b) => EqSymbolic (a, b) where
(a0, b0) .== (a1, b1) = a0 .== a1 .&& b0 .== b1
(a0, b0) .=== (a1, b1) = a0 .=== a1 .&& b0 .=== b1
instance (OrdSymbolic a, OrdSymbolic b) => OrdSymbolic (a, b) where
(a0, b0) .< (a1, b1) = a0 .< a1 .|| (a0 .== a1 .&& b0 .< b1)
-- 3-Tuple
instance (EqSymbolic a, EqSymbolic b, EqSymbolic c) => EqSymbolic (a, b, c) where
(a0, b0, c0) .== (a1, b1, c1) = (a0, b0) .== (a1, b1) .&& c0 .== c1
(a0, b0, c0) .=== (a1, b1, c1) = (a0, b0) .=== (a1, b1) .&& c0 .=== c1
instance (OrdSymbolic a, OrdSymbolic b, OrdSymbolic c) => OrdSymbolic (a, b, c) where
(a0, b0, c0) .< (a1, b1, c1) = (a0, b0) .< (a1, b1) .|| ((a0, b0) .== (a1, b1) .&& c0 .< c1)
-- 4-Tuple
instance (EqSymbolic a, EqSymbolic b, EqSymbolic c, EqSymbolic d) => EqSymbolic (a, b, c, d) where
(a0, b0, c0, d0) .== (a1, b1, c1, d1) = (a0, b0, c0) .== (a1, b1, c1) .&& d0 .== d1
(a0, b0, c0, d0) .=== (a1, b1, c1, d1) = (a0, b0, c0) .=== (a1, b1, c1) .&& d0 .=== d1
instance (OrdSymbolic a, OrdSymbolic b, OrdSymbolic c, OrdSymbolic d) => OrdSymbolic (a, b, c, d) where
(a0, b0, c0, d0) .< (a1, b1, c1, d1) = (a0, b0, c0) .< (a1, b1, c1) .|| ((a0, b0, c0) .== (a1, b1, c1) .&& d0 .< d1)
-- 5-Tuple
instance (EqSymbolic a, EqSymbolic b, EqSymbolic c, EqSymbolic d, EqSymbolic e) => EqSymbolic (a, b, c, d, e) where
(a0, b0, c0, d0, e0) .== (a1, b1, c1, d1, e1) = (a0, b0, c0, d0) .== (a1, b1, c1, d1) .&& e0 .== e1
(a0, b0, c0, d0, e0) .=== (a1, b1, c1, d1, e1) = (a0, b0, c0, d0) .=== (a1, b1, c1, d1) .&& e0 .=== e1
instance (OrdSymbolic a, OrdSymbolic b, OrdSymbolic c, OrdSymbolic d, OrdSymbolic e) => OrdSymbolic (a, b, c, d, e) where
(a0, b0, c0, d0, e0) .< (a1, b1, c1, d1, e1) = (a0, b0, c0, d0) .< (a1, b1, c1, d1) .|| ((a0, b0, c0, d0) .== (a1, b1, c1, d1) .&& e0 .< e1)
-- 6-Tuple
instance (EqSymbolic a, EqSymbolic b, EqSymbolic c, EqSymbolic d, EqSymbolic e, EqSymbolic f) => EqSymbolic (a, b, c, d, e, f) where
(a0, b0, c0, d0, e0, f0) .== (a1, b1, c1, d1, e1, f1) = (a0, b0, c0, d0, e0) .== (a1, b1, c1, d1, e1) .&& f0 .== f1
(a0, b0, c0, d0, e0, f0) .=== (a1, b1, c1, d1, e1, f1) = (a0, b0, c0, d0, e0) .=== (a1, b1, c1, d1, e1) .&& f0 .=== f1
instance (OrdSymbolic a, OrdSymbolic b, OrdSymbolic c, OrdSymbolic d, OrdSymbolic e, OrdSymbolic f) => OrdSymbolic (a, b, c, d, e, f) where
(a0, b0, c0, d0, e0, f0) .< (a1, b1, c1, d1, e1, f1) = (a0, b0, c0, d0, e0) .< (a1, b1, c1, d1, e1)
.|| ((a0, b0, c0, d0, e0) .== (a1, b1, c1, d1, e1) .&& f0 .< f1)
-- 7-Tuple
instance (EqSymbolic a, EqSymbolic b, EqSymbolic c, EqSymbolic d, EqSymbolic e, EqSymbolic f, EqSymbolic g) => EqSymbolic (a, b, c, d, e, f, g) where
(a0, b0, c0, d0, e0, f0, g0) .== (a1, b1, c1, d1, e1, f1, g1) = (a0, b0, c0, d0, e0, f0) .== (a1, b1, c1, d1, e1, f1) .&& g0 .== g1
(a0, b0, c0, d0, e0, f0, g0) .=== (a1, b1, c1, d1, e1, f1, g1) = (a0, b0, c0, d0, e0, f0) .=== (a1, b1, c1, d1, e1, f1) .&& g0 .=== g1
instance (OrdSymbolic a, OrdSymbolic b, OrdSymbolic c, OrdSymbolic d, OrdSymbolic e, OrdSymbolic f, OrdSymbolic g) => OrdSymbolic (a, b, c, d, e, f, g) where
(a0, b0, c0, d0, e0, f0, g0) .< (a1, b1, c1, d1, e1, f1, g1) = (a0, b0, c0, d0, e0, f0) .< (a1, b1, c1, d1, e1, f1)
.|| ((a0, b0, c0, d0, e0, f0) .== (a1, b1, c1, d1, e1, f1) .&& g0 .< g1)
-- | Regular expressions can be compared for equality. Note that we diverge here from the equality
-- in the concrete sense; i.e., the Eq instance does not match the symbolic case. This is a bit unfortunate,
-- but unavoidable with the current design of how we "distinguish" operators. Hopefully shouldn't be a big deal,
-- though one should be careful.
instance EqSymbolic RegExp where
r1 .== r2 = SBV $ SVal KBool $ Right $ cache r
where r st = newExpr st KBool $ SBVApp (RegExOp (RegExEq r1 r2)) []
r1 ./= r2 = SBV $ SVal KBool $ Right $ cache r
where r st = newExpr st KBool $ SBVApp (RegExOp (RegExNEq r1 r2)) []
-- | 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.
class (SymVal a, Num a, Num (SBV a), Bits a, Integral a) => SIntegral a
-- 'SIntegral' Instances, skips Real/Float/Bool
instance SIntegral Word8
instance SIntegral Word16
instance SIntegral Word32
instance SIntegral Word64
instance SIntegral Int8
instance SIntegral Int16
instance SIntegral Int32
instance SIntegral Int64
instance SIntegral Integer
instance (KnownNat n, BVIsNonZero n) => SIntegral (WordN n)
instance (KnownNat n, BVIsNonZero n) => SIntegral (IntN n)
-- | Zero extend a bit-vector.
zeroExtend :: forall n m bv. ( KnownNat n, BVIsNonZero n, SymVal (bv n)
, KnownNat m, BVIsNonZero m, SymVal (bv m)
, n + 1 <= m
, SIntegral (bv (m - n))
, BVIsNonZero (m - n)
) => SBV (bv n) -- ^ Input, of size @n@
-> SBV (bv m) -- ^ Output, of size @m@. @n < m@ must hold
zeroExtend n = SBV $ svZeroExtend i (unSBV n)
where nv = intOfProxy (Proxy @n)
mv = intOfProxy (Proxy @m)
i = fromIntegral (mv - nv)
-- | Sign extend a bit-vector.
signExtend :: forall n m bv. ( KnownNat n, BVIsNonZero n, SymVal (bv n)
, KnownNat m, BVIsNonZero m, SymVal (bv m)
, n + 1 <= m
, SFiniteBits (bv n)
, SIntegral (bv (m - n))
, BVIsNonZero (m - n)
) => SBV (bv n) -- ^ Input, of size @n@
-> SBV (bv m) -- ^ Output, of size @m@. @n < m@ must hold
signExtend n = SBV $ svSignExtend i (unSBV n)
where nv = intOfProxy (Proxy @n)
mv = intOfProxy (Proxy @m)
i = fromIntegral (mv - nv)
-- | Finite bit-length symbolic values. Essentially the same as 'SIntegral', but further leaves out 'Integer'. Loosely
-- based on Haskell's @FiniteBits@ class, but with more methods defined and structured differently to fit into the
-- symbolic world view. Minimal complete definition: 'sFiniteBitSize'.
class (Ord a, SymVal a, Num a, Num (SBV a), OrdSymbolic (SBV a), Bits a) => SFiniteBits a where
-- | Bit size.
sFiniteBitSize :: SBV a -> Int
-- | Least significant bit of a word, always stored at index 0.
lsb :: SBV a -> SBool
-- | Most significant bit of a word, always stored at the last position.
msb :: SBV a -> SBool
-- | Big-endian blasting of a word into its bits.
blastBE :: SBV a -> [SBool]
-- | Little-endian blasting of a word into its bits.
blastLE :: SBV a -> [SBool]
-- | Reconstruct from given bits, given in little-endian.
fromBitsBE :: [SBool] -> SBV a
-- | Reconstruct from given bits, given in little-endian.
fromBitsLE :: [SBool] -> SBV a
-- | Replacement for 'testBit', returning 'SBool' instead of 'Bool'.
sTestBit :: SBV a -> Int -> SBool
-- | Variant of 'sTestBit', where we want to extract multiple bit positions.
sExtractBits :: SBV a -> [Int] -> [SBool]
-- | Variant of 'popCount', returning a symbolic value.
sPopCount :: SBV a -> SWord8
-- | A combo of 'setBit' and 'clearBit', when the bit to be set is symbolic.
setBitTo :: SBV a -> Int -> SBool -> SBV a
-- | Variant of 'setBitTo' when the index is symbolic. If the index it out-of-bounds,
-- then the result is underspecified.
sSetBitTo :: Integral a => SBV a -> SBV a -> SBool -> SBV a
-- | Full adder, returns carry-out from the addition. Only for unsigned quantities.
fullAdder :: SBV a -> SBV a -> (SBool, SBV a)
-- | Full multiplier, returns both high and low-order bits. Only for unsigned quantities.
fullMultiplier :: SBV a -> SBV a -> (SBV a, SBV a)
-- | Count leading zeros in a word, big-endian interpretation.
sCountLeadingZeros :: SBV a -> SWord8
-- | Count trailing zeros in a word, big-endian interpretation.
sCountTrailingZeros :: SBV a -> SWord8
{-# MINIMAL sFiniteBitSize #-}
-- Default implementations
lsb (SBV v) = SBV (svTestBit v 0)
msb x = sTestBit x (sFiniteBitSize x - 1)
blastBE = reverse . blastLE
blastLE x = map (sTestBit x) [0 .. intSizeOf x - 1]
fromBitsBE = fromBitsLE . reverse
fromBitsLE bs
| length bs /= w
= error $ "SBV.SFiniteBits.fromBitsLE/BE: Expected: " ++ show w ++ " bits, received: " ++ show (length bs)
| True
= result
where w = sFiniteBitSize result
result = go 0 0 bs
go !acc _ [] = acc
go !acc !i (x:xs) = go (ite x (setBit acc i) acc) (i+1) xs
sTestBit (SBV x) i = SBV (svTestBit x i)
sExtractBits x = map (sTestBit x)
-- NB. 'sPopCount' returns an 'SWord8', which can overflow when used on quantities that have
-- more than 255 bits. For the regular interface, this suffices for all types we support.
-- For the Dynamic interface, if we ever implement this, this will fail for bit-vectors
-- larger than that many bits. The alternative would be to return SInteger here, but that
-- seems a total overkill for most use cases. If such is required, users are encouraged
-- to define their own variants, which is rather easy.
sPopCount x
| Just v <- unliteral x = go 0 v
| True = sum [ite b 1 0 | b <- blastLE x]
where -- concrete case
go !c 0 = c
go !c w = go (c+1) (w .&. (w-1))
setBitTo x i b = ite b (setBit x i) (clearBit x i)
sSetBitTo x idx b
| Just i <- unliteral idx, Just index <- safe i
= setBitTo x index b
| True
= go x [0 .. sFiniteBitSize x - 1]
where -- paranoia check: make sure index can fit in an int
safe i = let asInteger = toInteger i
asInt = fromIntegral asInteger
backInteger = toInteger asInt
in if backInteger == asInteger
then Just asInt
else Nothing
go v [] = v
go v (i:is) = go (ite (idx .== literal (fromIntegral i)) (setBitTo v (fromIntegral i) b) v) is
fullAdder a b
| isSigned a = error "fullAdder: only works on unsigned numbers"
| True = (a .> s .|| b .> s, s)
where s = a + b
-- 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.
fullMultiplier a b
| isSigned a = error "fullMultiplier: only works on unsigned numbers"
| True = (go (sFiniteBitSize a) 0 a, a*b)
where go 0 p _ = p
go n p x = let (c, p') = ite (lsb x) (fullAdder p b) (sFalse, p)
(o, p'') = shiftIn c p'
(_, x') = shiftIn o x
in go (n-1) p'' x'
shiftIn k v = (lsb v, mask .|. (v `shiftR` 1))
where mask = ite k (bit (sFiniteBitSize v - 1)) 0
-- See the note for 'sPopCount' for a comment on why we return 'SWord8'
sCountLeadingZeros x = fromIntegral m - go m
where m = sFiniteBitSize x - 1
-- NB. When i is 0 below, which happens when x is 0 as we count all the way down,
-- we return -1, which is equal to 2^n-1, giving us: n-1-(2^n-1) = n-2^n = n, as required, i.e., the bit-size.
go :: Int -> SWord8
go i | i < 0 = i8
| True = ite (sTestBit x i) i8 (go (i-1))
where i8 = literal (fromIntegral i :: Word8)
-- See the note for 'sPopCount' for a comment on why we return 'SWord8'
sCountTrailingZeros x = go 0
where m = sFiniteBitSize x
go :: Int -> SWord8
go i | i >= m = i8
| True = ite (sTestBit x i) i8 (go (i+1))
where i8 = literal (fromIntegral i :: Word8)
-- 'SFiniteBits' Instances, skips Real/Float/Bool/Integer
instance SFiniteBits Word8 where sFiniteBitSize _ = 8
instance SFiniteBits Word16 where sFiniteBitSize _ = 16
instance SFiniteBits Word32 where sFiniteBitSize _ = 32
instance SFiniteBits Word64 where sFiniteBitSize _ = 64
instance SFiniteBits Int8 where sFiniteBitSize _ = 8
instance SFiniteBits Int16 where sFiniteBitSize _ = 16
instance SFiniteBits Int32 where sFiniteBitSize _ = 32
instance SFiniteBits Int64 where sFiniteBitSize _ = 64
instance (KnownNat n, BVIsNonZero n) => SFiniteBits (WordN n) where sFiniteBitSize _ = intOfProxy (Proxy @n)
instance (KnownNat n, BVIsNonZero n) => SFiniteBits (IntN n) where sFiniteBitSize _ = intOfProxy (Proxy @n)
-- | Returns 1 if the boolean is 'sTrue', otherwise 0.
oneIf :: (Ord a, Num (SBV a), SymVal a) => SBool -> SBV a
oneIf t = ite t 1 0
-- | Lift a pseudo-boolean op, performing checks
liftPB :: String -> PBOp -> [SBool] -> SBool
liftPB w o xs
| Just e <- check o
= error $ "SBV." ++ w ++ ": " ++ e
| True
= result
where check (PB_AtMost k) = pos k
check (PB_AtLeast k) = pos k
check (PB_Exactly k) = pos k
check (PB_Le cs k) = pos k `mplus` match cs
check (PB_Ge cs k) = pos k `mplus` match cs
check (PB_Eq cs k) = pos k `mplus` match cs
pos k
| k < 0 = Just $ "comparison value must be positive, received: " ++ show k
| True = Nothing
match cs
| any (< 0) cs = Just $ "coefficients must be non-negative. Received: " ++ show cs
| lxs /= lcs = Just $ "coefficient length must match number of arguments. Received: " ++ show (lcs, lxs)
| True = Nothing
where lxs = length xs
lcs = length cs
result = SBV (SVal KBool (Right (cache r)))
r st = do xsv <- mapM (sbvToSV st) xs
-- PseudoBoolean's implicitly require support for integers, so make sure to register that kind!
registerKind st KUnbounded
newExpr st KBool (SBVApp (PseudoBoolean o) xsv)
-- | 'sTrue' if at most @k@ of the input arguments are 'sTrue'
pbAtMost :: [SBool] -> Int -> SBool
pbAtMost xs k
| k < 0 = error $ "SBV.pbAtMost: Non-negative value required, received: " ++ show k
| all isConcrete xs = literal $ sum (map (pbToInteger "pbAtMost" 1) xs) <= fromIntegral k
| True = liftPB "pbAtMost" (PB_AtMost k) xs
-- | 'sTrue' if at least @k@ of the input arguments are 'sTrue'
pbAtLeast :: [SBool] -> Int -> SBool
pbAtLeast xs k
| k < 0 = error $ "SBV.pbAtLeast: Non-negative value required, received: " ++ show k
| all isConcrete xs = literal $ sum (map (pbToInteger "pbAtLeast" 1) xs) >= fromIntegral k
| True = liftPB "pbAtLeast" (PB_AtLeast k) xs
-- | 'sTrue' if exactly @k@ of the input arguments are 'sTrue'
pbExactly :: [SBool] -> Int -> SBool
pbExactly xs k
| k < 0 = error $ "SBV.pbExactly: Non-negative value required, received: " ++ show k
| all isConcrete xs = literal $ sum (map (pbToInteger "pbExactly" 1) xs) == fromIntegral k
| True = liftPB "pbExactly" (PB_Exactly k) xs
-- | 'sTrue' if the sum of coefficients for 'sTrue' elements is at most @k@. Generalizes 'pbAtMost'.
pbLe :: [(Int, SBool)] -> Int -> SBool
pbLe xs k
| k < 0 = error $ "SBV.pbLe: Non-negative value required, received: " ++ show k
| all (isConcrete . snd) xs = literal $ sum [pbToInteger "pbLe" c b | (c, b) <- xs] <= fromIntegral k
| True = liftPB "pbLe" (PB_Le (map fst xs) k) (map snd xs)
-- | 'sTrue' if the sum of coefficients for 'sTrue' elements is at least @k@. Generalizes 'pbAtLeast'.
pbGe :: [(Int, SBool)] -> Int -> SBool
pbGe xs k
| k < 0 = error $ "SBV.pbGe: Non-negative value required, received: " ++ show k
| all (isConcrete . snd) xs = literal $ sum [pbToInteger "pbGe" c b | (c, b) <- xs] >= fromIntegral k
| True = liftPB "pbGe" (PB_Ge (map fst xs) k) (map snd xs)
-- | 'sTrue' if the sum of coefficients for 'sTrue' elements is exactly least @k@. Useful for coding
-- /exactly K-of-N/ constraints, and in particular mutex constraints.
pbEq :: [(Int, SBool)] -> Int -> SBool
pbEq xs k
| k < 0 = error $ "SBV.pbEq: Non-negative value required, received: " ++ show k
| all (isConcrete . snd) xs = literal $ sum [pbToInteger "pbEq" c b | (c, b) <- xs] == fromIntegral k
| True = liftPB "pbEq" (PB_Eq (map fst xs) k) (map snd xs)
-- | 'sTrue' if there is at most one set bit
pbMutexed :: [SBool] -> SBool
pbMutexed xs = pbAtMost xs 1
-- | 'sTrue' if there is exactly one set bit
pbStronglyMutexed :: [SBool] -> SBool
pbStronglyMutexed xs = pbExactly xs 1
-- | Convert a concrete pseudo-boolean to given int; converting to integer
pbToInteger :: String -> Int -> SBool -> Integer
pbToInteger w c b
| c < 0 = error $ "SBV." ++ w ++ ": Non-negative coefficient required, received: " ++ show c
| Just v <- unliteral b = if v then fromIntegral c else 0
| True = error $ "SBV.pbToInteger: Received a symbolic boolean: " ++ show (c, b)
-- | Predicate for optimizing word operations like (+) and (*).
isConcreteZero :: SBV a -> Bool
isConcreteZero (SBV (SVal _ (Left (CV _ (CInteger n))))) = n == 0
isConcreteZero (SBV (SVal KReal (Left (CV KReal (CAlgReal v))))) = isExactRational v && v == 0
isConcreteZero _ = False
-- | Predicate for optimizing word operations like (+) and (*).
isConcreteOne :: SBV a -> Bool
isConcreteOne (SBV (SVal _ (Left (CV _ (CInteger 1))))) = True
isConcreteOne (SBV (SVal KReal (Left (CV KReal (CAlgReal v))))) = isExactRational v && v == 1
isConcreteOne _ = False
-- | Symbolic exponentiation using bit blasting and repeated squaring.
--
-- N.B. The exponent must be unsigned/bounded if symbolic. Signed exponents will be rejected.
(.^) :: (Mergeable b, Num b, SIntegral e) => b -> SBV e -> b
b .^ e
| isConcrete e, Just (x :: Integer) <- unliteral (sFromIntegral e)
= if x >= 0 then let go n v
| n == 0 = 1
| even n = go (n `div` 2) (v * v)
| True = v * go (n `div` 2) (v * v)
in go x b
else error $ "(.^): exponentiation: negative exponent: " ++ show x
| not (isBounded e) || isSigned e
= error $ "(.^): exponentiation only works with unsigned bounded symbolic exponents, kind: " ++ show (kindOf e)
| True
= -- NB. We can't simply use sTestBit and blastLE since they have SFiniteBit requirement
-- but we want to have SIntegral here only.
let SBV expt = e
expBit i = SBV (svTestBit expt i)
blasted = map expBit [0 .. intSizeOf e - 1]
in product $ zipWith (\use n -> ite use n 1)
blasted
(iterate (\x -> x*x) b)
infixr 8 .^
instance (Ord a, Num (SBV a), SymVal a, Fractional a) => Fractional (SBV a) where
fromRational = literal . fromRational
SBV x / sy@(SBV y) | div0 = ite (sy .== 0) 0 res
| True = res
where res = SBV (svDivide x y)
-- Identify those kinds where we have a div-0 equals 0 exception
div0 = case kindOf sy of
KFloat -> False
KDouble -> False
KFP{} -> False
KReal -> True
KRational -> True
-- Following cases should not happen since these types should *not* be instances of Fractional
k@KBounded{} -> error $ "Unexpected Fractional case for: " ++ show k
k@KUnbounded -> error $ "Unexpected Fractional case for: " ++ show k
k@KBool -> error $ "Unexpected Fractional case for: " ++ show k
k@KString -> error $ "Unexpected Fractional case for: " ++ show k
k@KChar -> error $ "Unexpected Fractional case for: " ++ show k
k@KList{} -> error $ "Unexpected Fractional case for: " ++ show k
k@KSet{} -> error $ "Unexpected Fractional case for: " ++ show k
k@KUserSort{} -> error $ "Unexpected Fractional case for: " ++ show k
k@KTuple{} -> error $ "Unexpected Fractional case for: " ++ show k
k@KMaybe{} -> error $ "Unexpected Fractional case for: " ++ show k
k@KEither{} -> error $ "Unexpected Fractional case for: " ++ show k
k@KArray{} -> error $ "Unexpected Fractional case for: " ++ show k
-- | Define Floating instance on SBV's; only for base types that are already floating; i.e., 'SFloat', 'SDouble', and 'SReal'.
-- (See the separate definition below for 'SFloatingPoint'.) Note that unless you use delta-sat via 'Data.SBV.Provers.dReal' on 'SReal', most
-- of the fields are "undefined" for symbolic values. We will add methods as they are supported by SMTLib. Currently, the
-- only symbolically available function in this class is 'sqrt' for 'SFloat', 'SDouble' and 'SFloatingPoint'.
instance (Ord a, Num (SBV a), SymVal a, Fractional a, Floating a) => Floating (SBV a) where
pi = fromRational . toRational $ (pi :: Double)
exp = lift1FNS "exp" exp
log = lift1FNS "log" log
sqrt = lift1F FP_Sqrt sqrt
sin = lift1FNS "sin" sin
cos = lift1FNS "cos" cos
tan = lift1FNS "tan" tan
asin = lift1FNS "asin" asin
acos = lift1FNS "acos" acos
atan = lift1FNS "atan" atan
sinh = lift1FNS "sinh" sinh
cosh = lift1FNS "cosh" cosh
tanh = lift1FNS "tanh" tanh
asinh = lift1FNS "asinh" asinh
acosh = lift1FNS "acosh" acosh
atanh = lift1FNS "atanh" atanh
(**) = lift2FNS "**" (**)
logBase = lift2FNS "logBase" logBase
unsupported :: String -> a
unsupported w = error $ "Data.SBV.FloatingPoint: Unsupported operation: " ++ w ++ ". Please request this as a feature!"
-- | We give a specific instance for 'SFloatingPoint', because the underlying floating-point type doesn't support
-- fromRational directly. The overlap with the above instance is unfortunate.
instance {-# OVERLAPPING #-} ValidFloat eb sb => Floating (SFloatingPoint eb sb) where
-- Try from double; if there's enough precision this'll work, otherwise will bail out.
pi
| ei > 11 || si > 53 = unsupported $ "Floating.SFloatingPoint.pi (not-enough-precision for " ++ show (ei, si) ++ ")"
| True = literal $ FloatingPoint $ fpFromRational ei si (toRational (pi :: Double))
where ei = intOfProxy (Proxy @eb)
si = intOfProxy (Proxy @sb)
-- Likewise, exponentiation is again limited to precision of double
exp i
| ei > 11 || si > 53 = unsupported $ "Floating.SFloatingPoint.exp (not-enough-precision for " ++ show (ei, si) ++ ")"
| True = literal e ** i
where ei = intOfProxy (Proxy @eb)
si = intOfProxy (Proxy @sb)
e = FloatingPoint $ fpFromRational ei si (toRational (exp 1 :: Double))
log = lift1FNS "log" log
sqrt = lift1F FP_Sqrt sqrt
sin = lift1FNS "sin" sin
cos = lift1FNS "cos" cos
tan = lift1FNS "tan" tan
asin = lift1FNS "asin" asin
acos = lift1FNS "acos" acos
atan = lift1FNS "atan" atan
sinh = lift1FNS "sinh" sinh
cosh = lift1FNS "cosh" cosh
tanh = lift1FNS "tanh" tanh
asinh = lift1FNS "asinh" asinh
acosh = lift1FNS "acosh" acosh
atanh = lift1FNS "atanh" atanh
(**) = lift2FNS "**" (**)
logBase = lift2FNS "logBase" logBase
-- | Lift a 1 arg FP-op, using sRNE default
lift1F :: SymVal a => FPOp -> (a -> a) -> SBV a -> SBV a
lift1F w op a
| Just v <- unliteral a
= literal $ op v
| True
= SBV $ SVal k $ Right $ cache r
where k = kindOf a
r st = do swa <- sbvToSV st a
swm <- sbvToSV st sRNE
newExpr st k (SBVApp (IEEEFP w) [swm, swa])
-- | Lift a float/double unary function, only over constants
lift1FNS :: (SymVal a, Floating a) => String -> (a -> a) -> SBV a -> SBV a
lift1FNS nm f sv
| Just v <- unliteral sv = literal $ f v
| True = error $ "SBV." ++ nm ++ ": not supported for symbolic values of type " ++ show (kindOf sv)
-- | Lift a float/double binary function, only over constants
lift2FNS :: (SymVal a, Floating a) => String -> (a -> a -> a) -> SBV a -> SBV a -> SBV a
lift2FNS nm f sv1 sv2
| Just v1 <- unliteral sv1
, Just v2 <- unliteral sv2 = literal $ f v1 v2
| True = error $ "SBV." ++ nm ++ ": not supported for symbolic values of type " ++ show (kindOf sv1)
-- | SReal Floating instance, used in conjunction with the dReal solver for delta-satisfiability. Note that
-- we do not constant fold these values (except for pi), as Haskell doesn't really have any means of computing
-- them for arbitrary rationals.
instance {-# OVERLAPPING #-} Floating SReal where
pi = fromRational . toRational $ (pi :: Double) -- Perhaps not good enough?
exp = lift1SReal NR_Exp
log = lift1SReal NR_Log
sqrt = lift1SReal NR_Sqrt
sin = lift1SReal NR_Sin
cos = lift1SReal NR_Cos
tan = lift1SReal NR_Tan
asin = lift1SReal NR_ASin
acos = lift1SReal NR_ACos
atan = lift1SReal NR_ATan
sinh = lift1SReal NR_Sinh
cosh = lift1SReal NR_Cosh
tanh = lift1SReal NR_Tanh
asinh = error "Data.SBV.SReal: asinh is currently not supported. Please request this as a feature!"
acosh = error "Data.SBV.SReal: acosh is currently not supported. Please request this as a feature!"
atanh = error "Data.SBV.SReal: atanh is currently not supported. Please request this as a feature!"
(**) = lift2SReal NR_Pow
logBase x y = log y / log x
-- | Lift an sreal unary function
lift1SReal :: NROp -> SReal -> SReal
lift1SReal w a = SBV $ SVal k $ Right $ cache r
where k = kindOf a
r st = do swa <- sbvToSV st a
newExpr st k (SBVApp (NonLinear w) [swa])
-- | Lift an sreal binary function
lift2SReal :: NROp -> SReal -> SReal -> SReal
lift2SReal w a b = SBV $ SVal k $ Right $ cache r
where k = kindOf a
r st = do swa <- sbvToSV st a
swb <- sbvToSV st b
newExpr st k (SBVApp (NonLinear w) [swa, swb])
-- Bail out nicely.
noEquals :: String -> String -> (String, String) -> a
noEquals o n (l, r) = error $ unlines [ ""
, "*** Data.SBV: Comparing symbolic values using Haskell's Eq class!"
, "***"
, "*** Received: (" ++ l ++ ") " ++ o ++ " (" ++ r ++ ")"
, "*** Instead use: (" ++ l ++ ") " ++ n ++ " (" ++ r ++ ")"
, "***"
, "*** The Eq instance for symbolic values are necessiated only because"
, "*** of the Bits class requirement. You must use symbolic equality"
, "*** operators instead. (And complain to Haskell folks that they"
, "*** remove the 'Eq' superclass from 'Bits'!.)"
]
-- | This instance is only defined so that we can define an instance for
-- 'Data.Bits.Bits'. '==' and '/=' simply throw an error. Use
-- 'Data.SBV.EqSymbolic' instead.
instance SymVal a => Eq (SBV a) where
a == b = fromMaybe (noEquals "==" ".==" (show a, show b)) (unliteral (a .== b))
a /= b = fromMaybe (noEquals "/=" "./=" (show a, show b)) (unliteral (a ./= b))
-- NB. In the optimizations below, use of -1 is valid as
-- -1 has all bits set to True for both signed and unsigned values
-- | Using 'popCount' or 'testBit' on non-concrete values will result in an
-- error. Use 'sPopCount' or 'sTestBit' instead.
instance (Ord a, Num (SBV a), Num a, Bits a, SymVal a) => Bits (SBV a) where
SBV x .&. SBV y = SBV (svAnd x y)
SBV x .|. SBV y = SBV (svOr x y)
SBV x `xor` SBV y = SBV (svXOr x y)
complement (SBV x) = SBV (svNot x)
bitSize x = intSizeOf x
bitSizeMaybe x = Just $ intSizeOf x
isSigned x = hasSign x
bit i = 1 `shiftL` i
setBit x i = x .|. genLiteral (kindOf x) (bit i :: Integer)
clearBit x i = x .&. genLiteral (kindOf x) (complement (bit i) :: Integer)
complementBit x i = x `xor` genLiteral (kindOf x) (bit i :: Integer)
shiftL (SBV x) i = SBV (svShl x i)
shiftR (SBV x) i = SBV (svShr x i)
rotateL (SBV x) i = SBV (svRol x i)
rotateR (SBV x) i = SBV (svRor x i)
-- NB. testBit is *not* implementable on non-concrete symbolic words
x `testBit` i
| SBV (SVal _ (Left (CV _ (CInteger n)))) <- x
= testBit n i
| True
= error $ "SBV.testBit: Called on symbolic value: " ++ show x ++ ". Use sTestBit instead."
-- NB. popCount is *not* implementable on non-concrete symbolic words
popCount x
| SBV (SVal _ (Left (CV (KBounded _ w) (CInteger n)))) <- x
= popCount (n .&. (bit w - 1))
| True
= error $ "SBV.popCount: Called on symbolic value: " ++ show x ++ ". Use sPopCount instead."
-- | Conversion between integral-symbolic values, akin to Haskell's `fromIntegral`
sFromIntegral :: forall a b. (Integral a, HasKind a, Num a, SymVal a, HasKind b, Num b, SymVal b) => SBV a -> SBV b
sFromIntegral x
| kFrom == kTo
= SBV (unSBV x)
| isReal x
= error "SBV.sFromIntegral: Called on a real value" -- can't really happen due to types, but being overcautious
| Just v <- unliteral x
= literal (fromIntegral v)
| True
= result
where result = SBV (SVal kTo (Right (cache y)))
kFrom = kindOf x
kTo = kindOf (Proxy @b)
y st = do xsv <- sbvToSV st x
newExpr st kTo (SBVApp (KindCast kFrom kTo) [xsv])
-- | Lift a binary operation thru it's dynamic counterpart. Note that
-- we still want the actual functions here as differ in their type
-- compared to their dynamic counterparts, but the implementations
-- are the same.
liftViaSVal :: (SVal -> SVal -> SVal) -> SBV a -> SBV b -> SBV c
liftViaSVal f (SBV a) (SBV b) = SBV $ f a b
-- | 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.
sShiftLeft :: (SIntegral a, SIntegral b) => SBV a -> SBV b -> SBV a
sShiftLeft = liftViaSVal svShiftLeft
-- | 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.
--
-- 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.
sShiftRight :: (SIntegral a, SIntegral b) => SBV a -> SBV b -> SBV a
sShiftRight = liftViaSVal svShiftRight
-- | 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.
sSignedShiftArithRight:: (SFiniteBits a, SIntegral b) => SBV a -> SBV b -> SBV a
sSignedShiftArithRight x i
| isSigned i = error "sSignedShiftArithRight: shift amount should be unsigned"
| isSigned x = ssa x i
| True = ite (msb x)
(complement (ssa (complement x) i))
(ssa x i)
where ssa = liftViaSVal svShiftRight
-- | 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.
sRotateLeft :: (SIntegral a, SIntegral b) => SBV a -> SBV b -> SBV a
sRotateLeft = liftViaSVal svRotateLeft
-- | An implementation of rotate-left, using a barrel shifter like design. Only works when both
-- arguments are finite bitvectors, and furthermore when the second argument is unsigned.
-- The first condition is enforced by the type, but the second is dynamically checked.
-- We provide this implementation as an alternative to `sRotateLeft` since SMTLib logic
-- does not support variable argument rotates (as opposed to shifts), and thus this
-- implementation can produce better code for verification compared to `sRotateLeft`.
sBarrelRotateLeft :: (SFiniteBits a, SFiniteBits b) => SBV a -> SBV b -> SBV a
sBarrelRotateLeft = liftViaSVal svBarrelRotateLeft
-- | 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.
sRotateRight :: (SIntegral a, SIntegral b) => SBV a -> SBV b -> SBV a
sRotateRight = liftViaSVal svRotateRight
-- | An implementation of rotate-right, using a barrel shifter like design. See comments
-- for `sBarrelRotateLeft` for details.
sBarrelRotateRight :: (SFiniteBits a, SFiniteBits b) => SBV a -> SBV b -> SBV a
sBarrelRotateRight = liftViaSVal svBarrelRotateRight
-- | Capturing non-matching instances for better error messages, conversions from sized
type FromSizedErr (arg :: Type) = 'Text "fromSized: Cannot convert from type: " ':<>: 'ShowType arg
':$$: 'Text " Source type must be one of SInt N, SWord N, IntN N, WordN N"
':$$: 'Text " where N is 8, 16, 32, or 64."
-- | Capturing non-matching instances for better error messages, conversions to sized
type ToSizedErr (arg :: Type) = 'Text "toSized: Cannot convert from type: " ':<>: 'ShowType arg
':$$: 'Text " Source type must be one of Int8/16/32/64"
':$$: 'Text " OR Word8/16/32/64"
':$$: 'Text " OR their symbolic variants."
-- | Capture the correspondence between sized and fixed-sized BVs
type family FromSized (t :: Type) :: Type where
FromSized (WordN 8) = Word8
FromSized (WordN 16) = Word16
FromSized (WordN 32) = Word32
FromSized (WordN 64) = Word64
FromSized (IntN 8) = Int8
FromSized (IntN 16) = Int16
FromSized (IntN 32) = Int32
FromSized (IntN 64) = Int64
FromSized (SWord 8) = SWord8
FromSized (SWord 16) = SWord16
FromSized (SWord 32) = SWord32
FromSized (SWord 64) = SWord64
FromSized (SInt 8) = SInt8
FromSized (SInt 16) = SInt16
FromSized (SInt 32) = SInt32
FromSized (SInt 64) = SInt64
-- | Capture the correspondence, in terms of a constraint
type family FromSizedCstr (t :: Type) :: Constraint where
FromSizedCstr (WordN 8) = ()
FromSizedCstr (WordN 16) = ()
FromSizedCstr (WordN 32) = ()
FromSizedCstr (WordN 64) = ()
FromSizedCstr (IntN 8) = ()
FromSizedCstr (IntN 16) = ()
FromSizedCstr (IntN 32) = ()
FromSizedCstr (IntN 64) = ()
FromSizedCstr (SWord 8) = ()
FromSizedCstr (SWord 16) = ()
FromSizedCstr (SWord 32) = ()
FromSizedCstr (SWord 64) = ()
FromSizedCstr (SInt 8) = ()
FromSizedCstr (SInt 16) = ()
FromSizedCstr (SInt 32) = ()
FromSizedCstr (SInt 64) = ()
FromSizedCstr arg = TypeError (FromSizedErr arg)
-- | Conversion from a sized BV to a fixed-sized bit-vector.
class FromSizedBV a where
-- | Convert a sized bit-vector to the corresponding fixed-sized bit-vector,
-- for instance 'SWord 16' to 'SWord16'. See also 'toSized'.
fromSized :: a -> FromSized a
default fromSized :: (Num (FromSized a), Integral a) => a -> FromSized a
fromSized = fromIntegral
instance {-# OVERLAPPING #-} FromSizedBV (WordN 8)
instance {-# OVERLAPPING #-} FromSizedBV (WordN 16)
instance {-# OVERLAPPING #-} FromSizedBV (WordN 32)
instance {-# OVERLAPPING #-} FromSizedBV (WordN 64)
instance {-# OVERLAPPING #-} FromSizedBV (IntN 8)
instance {-# OVERLAPPING #-} FromSizedBV (IntN 16)
instance {-# OVERLAPPING #-} FromSizedBV (IntN 32)
instance {-# OVERLAPPING #-} FromSizedBV (IntN 64)
instance {-# OVERLAPPING #-} FromSizedBV (SWord 8) where fromSized = sFromIntegral
instance {-# OVERLAPPING #-} FromSizedBV (SWord 16) where fromSized = sFromIntegral
instance {-# OVERLAPPING #-} FromSizedBV (SWord 32) where fromSized = sFromIntegral
instance {-# OVERLAPPING #-} FromSizedBV (SWord 64) where fromSized = sFromIntegral
instance {-# OVERLAPPING #-} FromSizedBV (SInt 8) where fromSized = sFromIntegral
instance {-# OVERLAPPING #-} FromSizedBV (SInt 16) where fromSized = sFromIntegral
instance {-# OVERLAPPING #-} FromSizedBV (SInt 32) where fromSized = sFromIntegral
instance {-# OVERLAPPING #-} FromSizedBV (SInt 64) where fromSized = sFromIntegral
instance {-# OVERLAPPABLE #-} FromSizedCstr arg => FromSizedBV arg where fromSized = error "unreachable"
-- | Capture the correspondence between fixed-sized and sized BVs
type family ToSized (t :: Type) :: Type where
ToSized Word8 = WordN 8
ToSized Word16 = WordN 16
ToSized Word32 = WordN 32
ToSized Word64 = WordN 64
ToSized Int8 = IntN 8
ToSized Int16 = IntN 16
ToSized Int32 = IntN 32
ToSized Int64 = IntN 64
ToSized SWord8 = SWord 8
ToSized SWord16 = SWord 16
ToSized SWord32 = SWord 32
ToSized SWord64 = SWord 64
ToSized SInt8 = SInt 8
ToSized SInt16 = SInt 16
ToSized SInt32 = SInt 32
ToSized SInt64 = SInt 64
-- | Capture the correspondence in terms of a constraint
type family ToSizedCstr (t :: Type) :: Constraint where
ToSizedCstr Word8 = ()
ToSizedCstr Word16 = ()
ToSizedCstr Word32 = ()
ToSizedCstr Word64 = ()
ToSizedCstr Int8 = ()
ToSizedCstr Int16 = ()
ToSizedCstr Int32 = ()
ToSizedCstr Int64 = ()
ToSizedCstr SWord8 = ()
ToSizedCstr SWord16 = ()
ToSizedCstr SWord32 = ()
ToSizedCstr SWord64 = ()
ToSizedCstr SInt8 = ()
ToSizedCstr SInt16 = ()
ToSizedCstr SInt32 = ()
ToSizedCstr SInt64 = ()
ToSizedCstr arg = TypeError (ToSizedErr arg)
-- | Conversion from a fixed-sized BV to a sized bit-vector.
class ToSizedBV a where
-- | Convert a fixed-sized bit-vector to the corresponding sized bit-vector,
-- for instance 'SWord16' to 'SWord 16'. See also 'fromSized'.
toSized :: a -> ToSized a
default toSized :: (Num (ToSized a), Integral a) => (a -> ToSized a)
toSized = fromIntegral
instance {-# OVERLAPPING #-} ToSizedBV Word8
instance {-# OVERLAPPING #-} ToSizedBV Word16
instance {-# OVERLAPPING #-} ToSizedBV Word32
instance {-# OVERLAPPING #-} ToSizedBV Word64
instance {-# OVERLAPPING #-} ToSizedBV Int8
instance {-# OVERLAPPING #-} ToSizedBV Int16
instance {-# OVERLAPPING #-} ToSizedBV Int32
instance {-# OVERLAPPING #-} ToSizedBV Int64
instance {-# OVERLAPPING #-} ToSizedBV SWord8 where toSized = sFromIntegral
instance {-# OVERLAPPING #-} ToSizedBV SWord16 where toSized = sFromIntegral
instance {-# OVERLAPPING #-} ToSizedBV SWord32 where toSized = sFromIntegral
instance {-# OVERLAPPING #-} ToSizedBV SWord64 where toSized = sFromIntegral
instance {-# OVERLAPPING #-} ToSizedBV SInt8 where toSized = sFromIntegral
instance {-# OVERLAPPING #-} ToSizedBV SInt16 where toSized = sFromIntegral
instance {-# OVERLAPPING #-} ToSizedBV SInt32 where toSized = sFromIntegral
instance {-# OVERLAPPING #-} ToSizedBV SInt64 where toSized = sFromIntegral
instance {-# OVERLAPPABLE #-} ToSizedCstr arg => ToSizedBV arg where toSized = error "unreachable"
-- | 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' both make 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.
--
-- === C code generation of division operations
--
-- In the case of division or modulo of a minimal signed value (e.g. @-128@ for
-- 'SInt8') by @-1@, SMTLIB and Haskell agree on what the result should be.
-- Unfortunately the result in C code depends on CPU architecture and compiler
-- settings, as this is undefined behaviour in C. **SBV does not guarantee**
-- what will happen in generated C code in this corner case.
class SDivisible a where
sQuotRem :: a -> a -> (a, a)
sDivMod :: a -> a -> (a, a)
sQuot :: a -> a -> a
sRem :: a -> a -> a
sDiv :: a -> a -> a
sMod :: a -> a -> a
{-# MINIMAL sQuotRem, sDivMod #-}
x `sQuot` y = fst $ x `sQuotRem` y
x `sRem` y = snd $ x `sQuotRem` y
x `sDiv` y = fst $ x `sDivMod` y
x `sMod` y = snd $ x `sDivMod` y
instance SDivisible Word64 where
sQuotRem x 0 = (0, x)
sQuotRem x y = x `quotRem` y
sDivMod x 0 = (0, x)
sDivMod x y = x `divMod` y
instance SDivisible Int64 where
sQuotRem x 0 = (0, x)
sQuotRem x y = x `quotRem` y
sDivMod x 0 = (0, x)
sDivMod x y = x `divMod` y
instance SDivisible Word32 where
sQuotRem x 0 = (0, x)
sQuotRem x y = x `quotRem` y
sDivMod x 0 = (0, x)
sDivMod x y = x `divMod` y
instance SDivisible Int32 where
sQuotRem x 0 = (0, x)
sQuotRem x y = x `quotRem` y
sDivMod x 0 = (0, x)
sDivMod x y = x `divMod` y
instance SDivisible Word16 where
sQuotRem x 0 = (0, x)
sQuotRem x y = x `quotRem` y
sDivMod x 0 = (0, x)
sDivMod x y = x `divMod` y
instance SDivisible Int16 where
sQuotRem x 0 = (0, x)
sQuotRem x y = x `quotRem` y
sDivMod x 0 = (0, x)
sDivMod x y = x `divMod` y
instance SDivisible Word8 where
sQuotRem x 0 = (0, x)
sQuotRem x y = x `quotRem` y
sDivMod x 0 = (0, x)
sDivMod x y = x `divMod` y
instance SDivisible Int8 where
sQuotRem x 0 = (0, x)
sQuotRem x y = x `quotRem` y
sDivMod x 0 = (0, x)
sDivMod x y = x `divMod` y
instance SDivisible Integer where
sQuotRem x 0 = (0, x)
sQuotRem x y = x `quotRem` y
sDivMod x 0 = (0, x)
sDivMod x y = x `divMod` y
instance SDivisible CV where
sQuotRem a b
| CInteger x <- cvVal a, CInteger y <- cvVal b
= let (r1, r2) = sQuotRem x y in (normCV a{ cvVal = CInteger r1 }, normCV b{ cvVal = CInteger r2 })
sQuotRem a b = error $ "SBV.sQuotRem: impossible, unexpected args received: " ++ show (a, b)
sDivMod a b
| CInteger x <- cvVal a, CInteger y <- cvVal b
= let (r1, r2) = sDivMod x y in (normCV a{ cvVal = CInteger r1 }, normCV b{ cvVal = CInteger r2 })
sDivMod a b = error $ "SBV.sDivMod: impossible, unexpected args received: " ++ show (a, b)
instance SDivisible SWord64 where {sQuotRem = liftQRem; sDivMod = liftDMod}
instance SDivisible SWord32 where {sQuotRem = liftQRem; sDivMod = liftDMod}
instance SDivisible SWord16 where {sQuotRem = liftQRem; sDivMod = liftDMod}
instance SDivisible SWord8 where {sQuotRem = liftQRem; sDivMod = liftDMod}
instance SDivisible SInt64 where {sQuotRem = liftQRem; sDivMod = liftDMod}
instance SDivisible SInt32 where {sQuotRem = liftQRem; sDivMod = liftDMod}
instance SDivisible SInt16 where {sQuotRem = liftQRem; sDivMod = liftDMod}
instance SDivisible SInt8 where {sQuotRem = liftQRem; sDivMod = liftDMod}
-- | 'SDivisible' instance for 'WordN'
instance (KnownNat n, BVIsNonZero n) => SDivisible (WordN n) where
sQuotRem x 0 = (0, x)
sQuotRem x y = x `quotRem` y
sDivMod x 0 = (0, x)
sDivMod x y = x `divMod` y
-- | 'SDivisible' instance for 'IntN'
instance (KnownNat n, BVIsNonZero n) => SDivisible (IntN n) where
sQuotRem x 0 = (0, x)
sQuotRem x y = x `quotRem` y
sDivMod x 0 = (0, x)
sDivMod x y = x `divMod` y
-- | 'SDivisible' instance for 'SWord'
instance (KnownNat n, BVIsNonZero n) => SDivisible (SWord n) where
sQuotRem = liftQRem
sDivMod = liftDMod
-- | 'SDivisible' instance for 'SInt'
instance (KnownNat n, BVIsNonZero n) => SDivisible (SInt n) where
sQuotRem = liftQRem
sDivMod = liftDMod
-- | Does the concrete positive number n divide the given integer?
sDivides :: Integer -> SInteger -> SBool
sDivides n v
| n < 0
= error $ "svDivides: First argument must be a strictly positive integer. Received: " ++ show n
| Just x <- unliteral v
= if x `mod` n == 0 then sTrue else sFalse
| True
= SBV $ svDivides n (unSBV v)
-- | Lift 'quotRem' to symbolic words. Division by 0 is defined s.t. @x/0 = 0@; which
-- holds even when @x@ is @0@ itself.
liftQRem :: (Eq a, SymVal a) => SBV a -> SBV a -> (SBV a, SBV a)
liftQRem x y
| isConcreteZero x
= (x, x)
| isConcreteOne y
= (x, z)
{-------------------------------
- N.B. The seemingly innocuous variant when y == -1 only holds if the type is signed;
- and also is problematic around the minBound.. So, we refrain from that optimization
| isConcreteOnes y
= (-x, z)
--------------------------------}
| True
= ite (y .== z) (z, x) (qr x y)
where qr (SBV (SVal sgnsz (Left a))) (SBV (SVal _ (Left b))) = let (q, r) = sQuotRem a b in (SBV (SVal sgnsz (Left q)), SBV (SVal sgnsz (Left r)))
qr a@(SBV (SVal sgnsz _)) b = (SBV (SVal sgnsz (Right (cache (mk Quot)))), SBV (SVal sgnsz (Right (cache (mk Rem)))))
where mk o st = do sw1 <- sbvToSV st a
sw2 <- sbvToSV st b
mkSymOp o st sgnsz sw1 sw2
z = genLiteral (kindOf x) (0::Integer)
-- | Lift 'divMod' to symbolic words. Division by 0 is defined s.t. @x/0 = 0@; which
-- holds even when @x@ is @0@ itself. Essentially, this is conversion from quotRem
-- (truncate to 0) to divMod (truncate towards negative infinity)
liftDMod :: (Ord a, SymVal a, Num a, Num (SBV a), SDivisible (SBV a)) => SBV a -> SBV a -> (SBV a, SBV a)
liftDMod x y
| isConcreteZero x
= (x, x)
| isConcreteOne y
= (x, z)
{-------------------------------
- N.B. The seemingly innocuous variant when y == -1 only holds if the type is signed;
- and also is problematic around the minBound.. So, we refrain from that optimization
| isConcreteOnes y
= (-x, z)
--------------------------------}
| True
= ite (y .== z) (z, x) $ ite (signum r .== negate (signum y)) (q-i, r+y) qr
where qr@(q, r) = x `sQuotRem` y
z = genLiteral (kindOf x) (0::Integer)
i = genLiteral (kindOf x) (1::Integer)
-- SInteger instance for quotRem/divMod are tricky!
-- SMT-Lib only has Euclidean operations, but Haskell
-- uses "truncate to 0" for quotRem, and "truncate to negative infinity" for divMod.
-- So, we cannot just use the above liftings directly.
instance SDivisible SInteger where
sDivMod x y = ite (y .> 0) (sEDivMod x y) (liftDMod x y)
sQuotRem x y
| not (isSymbolic x || isSymbolic y)
= liftQRem x y
| True
= ite (y .== 0) (0, x) (qE+i, rE-i*y)
where (qE, rE) = liftQRem x y -- for integers, this is euclidean due to SMTLib semantics
i = ite (x .>= 0 .|| rE .== 0) 0
$ ite (y .> 0) 1 (-1)
-- | Euclidian division and modulus.
sEDivMod :: SInteger -> SInteger -> (SInteger, SInteger)
sEDivMod a b = (a `sEDiv` b, a `sEMod` b)
-- | Euclidian division. Note that unlike regular division, Euclidian division by @0@
-- is unconstrained. i.e., it can take any value whatsoever.
sEDiv :: SInteger -> SInteger -> SInteger
sEDiv (SBV a) (SBV b) = SBV $ a `svQuot` b
-- | Euclidian modulus. Note that unlike regular modulus, Euclidian division by @0@
-- is unconstrained. i.e., it can take any value whatsoever.
sEMod :: SInteger -> SInteger -> SInteger
sEMod (SBV a) (SBV b) = SBV $ a `svRem` b
-- Quickcheck interface
instance (SymVal a, Arbitrary a) => Arbitrary (SBV a) where
arbitrary = literal `fmap` arbitrary
-- | 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
-- 'G.Generic' and 'Mergeable' to the @deriving@ clause for the data-type. See
-- 'Documentation.SBV.Examples.Puzzles.U2Bridge.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@.
class Mergeable a where
-- | 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.
symbolicMerge :: Bool -> SBool -> a -> a -> a
-- | Total indexing operation. @select xs default index@ is intuitively
-- the same as @xs !! index@, except it evaluates to @default@ if @index@
-- underflows/overflows.
select :: (Ord b, SymVal b, Num b, Num (SBV b), OrdSymbolic (SBV b)) => [a] -> a -> SBV b -> a
-- NB. Earlier implementation of select used the binary-search trick
-- on the index to chop down the search space. While that is a good trick
-- in general, it doesn't work for SBV since we do not have any notion of
-- "concrete" subwords: If an index is symbolic, then all its bits are
-- symbolic as well. So, the binary search only pays off only if the indexed
-- list is really humongous, which is not very common in general. (Also,
-- for the case when the list is bit-vectors, we use SMT tables anyhow.)
select xs err ind
| isReal ind = bad "real"
| isFloat ind = bad "float"
| isDouble ind = bad "double"
| hasSign ind = ite (ind .< 0) err (walk xs ind err)
| True = walk xs ind err
where bad w = error $ "SBV.select: unsupported " ++ w ++ " valued select/index expression"
walk [] _ acc = acc
walk (e:es) i acc = walk es (i-1) (ite (i .== 0) e acc)
-- Default implementation for 'symbolicMerge' if the type is 'Generic'
default symbolicMerge :: (G.Generic a, GMergeable (G.Rep a)) => Bool -> SBool -> a -> a -> a
symbolicMerge = symbolicMergeDefault
-- | 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.
ite :: Mergeable a => SBool -> a -> a -> a
ite t a b
| Just r <- unliteral t = if r then a else b
| True = symbolicMerge True t a b
-- | 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.
iteLazy :: Mergeable a => SBool -> a -> a -> a
iteLazy t a b
| Just r <- unliteral t = if r then a else b
| True = symbolicMerge False t a b
-- | Symbolic assert. Check that the given boolean condition is always 'sTrue' in the given path. The
-- optional first argument can be used to provide call-stack info via GHC's location facilities.
sAssert :: HasKind a => Maybe CallStack -> String -> SBool -> SBV a -> SBV a
sAssert cs msg cond x
| Just mustHold <- unliteral cond
= if mustHold
then x
else error $ show $ SafeResult ((locInfo . getCallStack) `fmap` cs, msg, Satisfiable defaultSMTCfg (SMTModel [] Nothing [] []))
| True
= SBV $ SVal k $ Right $ cache r
where k = kindOf x
r st = do xsv <- sbvToSV st x
let pc = getPathCondition st
-- We're checking if there are any cases where the path-condition holds, but not the condition
-- Any violations of this, should be signaled, i.e., whenever the following formula is satisfiable
mustNeverHappen = pc .&& sNot cond
cnd <- sbvToSV st mustNeverHappen
addAssertion st cs msg cnd
return xsv
locInfo ps = intercalate ",\n " (map loc ps)
where loc (f, sl) = concat [srcLocFile sl, ":", show (srcLocStartLine sl), ":", show (srcLocStartCol sl), ":", f]
-- | Merge two symbolic values, at kind @k@, possibly @force@'ing the branches to make
-- sure they do not evaluate to the same result. This should only be used for internal purposes;
-- as default definitions provided should suffice in many cases. (i.e., End users should
-- only need to define 'symbolicMerge' when needed; which should be rare to start with.)
symbolicMergeWithKind :: Kind -> Bool -> SBool -> SBV a -> SBV a -> SBV a
symbolicMergeWithKind k force (SBV t) (SBV a) (SBV b) = SBV (svSymbolicMerge k force t a b)
instance SymVal a => Mergeable (SBV a) where
symbolicMerge force t x y
-- Carefully use the kindOf instance to avoid strictness issues.
| force = symbolicMergeWithKind (kindOf x) True t x y
| True = symbolicMergeWithKind (kindOf (Proxy @a)) False t x y
-- Custom version of select that translates to SMT-Lib tables at the base type of words
select xs err ind
| SBV (SVal _ (Left c)) <- ind = case cvVal c of
CInteger i -> if i < 0 || i >= genericLength xs
then err
else xs `genericIndex` i
_ -> error $ "SBV.select: unsupported " ++ show (kindOf ind) ++ " valued select/index expression"
select xsOrig err ind = xs `seq` SBV (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 of..
xs = case kindOf ind of
KBounded False i -> genericTake ((2::Integer) ^ (fromIntegral i :: Integer)) xsOrig
KBounded True i -> genericTake ((2::Integer) ^ (fromIntegral (i-1) :: Integer)) xsOrig
KUnbounded -> xsOrig
_ -> error $ "SBV.select: unsupported " ++ show (kindOf ind) ++ " valued select/index expression"
r st = do sws <- mapM (sbvToSV st) xs
swe <- sbvToSV st err
if all (== swe) sws -- off-chance that all elts are the same. Note that this also correctly covers the case when list is empty.
then return swe
else do idx <- getTableIndex st kInd kElt sws
swi <- sbvToSV 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) [])
-- | Construct a useful error message if we hit an unmergeable case.
cannotMerge :: String -> String -> String -> a
cannotMerge typ why hint = error $ unlines [ ""
, "*** Data.SBV.Mergeable: Cannot merge instances of " ++ typ ++ "."
, "*** While trying to do a symbolic if-then-else with incompatible branch results."
, "***"
, "*** " ++ why
, "*** "
, "*** Hint: " ++ hint
]
-- | Merge concrete values that can be checked for equality
concreteMerge :: Show a => String -> String -> (a -> a -> Bool) -> a -> a -> a
concreteMerge t st eq x y
| x `eq` y = x
| True = cannotMerge t
("Concrete values can only be merged when equal. Got: " ++ show x ++ " vs. " ++ show y)
("Use an " ++ st ++ " field if the values can differ.")
-- Mergeable instances for List/Maybe/Either/Array are useful, but can
-- throw exceptions if there is no structural matching of the results
-- It's a question whether we should really keep them..
-- Lists
instance Mergeable a => Mergeable [a] where
symbolicMerge f t xs ys
| lxs == lys = zipWith (symbolicMerge f t) xs ys
| True = cannotMerge "lists"
("Branches produce different sizes: " ++ show lxs ++ " vs " ++ show lys ++ ". Must have the same length.")
"Use the 'SList' type (and Data.SBV.List routines) to model fully symbolic lists."
where (lxs, lys) = (length xs, length ys)
-- NonEmpty
instance Mergeable a => Mergeable (NonEmpty a) where
symbolicMerge f t xs ys
| lxs == lys = NE.zipWith (symbolicMerge f t) xs ys
| True = cannotMerge "non-empty lists"
("Branches produce different sizes: " ++ show lxs ++ " vs " ++ show lys ++ ". Must have the same length.")
"Use the 'SList' type (and Data.SBV.List routines) to model fully symbolic lists."
where (lxs, lys) = (length xs, length ys)
-- ZipList
instance Mergeable a => Mergeable (ZipList a) where
symbolicMerge force test (ZipList xs) (ZipList ys)
= ZipList (symbolicMerge force test xs ys)
-- Maybe
instance Mergeable a => Mergeable (Maybe a) where
symbolicMerge _ _ Nothing Nothing = Nothing
symbolicMerge f t (Just a) (Just b) = Just $ symbolicMerge f t a b
symbolicMerge _ _ a b = cannotMerge "'Maybe' values"
("Branches produce different constructors: " ++ show (k a, k b))
"Instead of an option type, try using a valid bit to indicate when a result is valid."
where k Nothing = "Nothing"
k _ = "Just"
-- Either
instance (Mergeable a, Mergeable b) => Mergeable (Either a b) where
symbolicMerge f t (Left a) (Left b) = Left $ symbolicMerge f t a b
symbolicMerge f t (Right a) (Right b) = Right $ symbolicMerge f t a b
symbolicMerge _ _ a b = cannotMerge "'Either' values"
("Branches produce different constructors: " ++ show (k a, k b))
"Consider using a product type by a tag instead."
where k (Left _) = "Left"
k (Right _) = "Right"
-- Arrays
instance (Ix a, Mergeable b) => Mergeable (Array a b) where
symbolicMerge f t a b
| ba == bb = DA.listArray ba (zipWith (symbolicMerge f t) (elems a) (elems b))
| True = cannotMerge "'Array' values"
("Branches produce different ranges: " ++ show (k ba, k bb))
"Consider using SBV's native 'SArray' abstraction."
where [ba, bb] = map bounds [a, b]
k = rangeSize
-- Functions
instance Mergeable b => Mergeable (a -> b) where
symbolicMerge f t g h x = symbolicMerge f t (g x) (h x)
{- Following definition, while correct, is utterly inefficient. Since the
application is delayed, this hangs on to the inner list and all the
impending merges, even when ind is concrete. Thus, it's much better to
simply use the default definition for the function case.
-}
-- select xs err ind = \x -> select (map ($ x) xs) (err x) ind
-- 2-Tuple
instance (Mergeable a, Mergeable b) => Mergeable (a, b) where
symbolicMerge f t (i0, i1) (j0, j1) = ( symbolicMerge f t i0 j0
, symbolicMerge f t i1 j1
)
select xs (err1, err2) ind = ( select as err1 ind
, select bs err2 ind
)
where (as, bs) = unzip xs
-- 3-Tuple
instance (Mergeable a, Mergeable b, Mergeable c) => Mergeable (a, b, c) where
symbolicMerge f t (i0, i1, i2) (j0, j1, j2) = ( symbolicMerge f t i0 j0
, symbolicMerge f t i1 j1
, symbolicMerge f t i2 j2
)
select xs (err1, err2, err3) ind = ( select as err1 ind
, select bs err2 ind
, select cs err3 ind
)
where (as, bs, cs) = unzip3 xs
-- 4-Tuple
instance (Mergeable a, Mergeable b, Mergeable c, Mergeable d) => Mergeable (a, b, c, d) where
symbolicMerge f t (i0, i1, i2, i3) (j0, j1, j2, j3) = ( symbolicMerge f t i0 j0
, symbolicMerge f t i1 j1
, symbolicMerge f t i2 j2
, symbolicMerge f t i3 j3
)
select xs (err1, err2, err3, err4) ind = ( select as err1 ind
, select bs err2 ind
, select cs err3 ind
, select ds err4 ind
)
where (as, bs, cs, ds) = unzip4 xs
-- 5-Tuple
instance (Mergeable a, Mergeable b, Mergeable c, Mergeable d, Mergeable e) => Mergeable (a, b, c, d, e) where
symbolicMerge f t (i0, i1, i2, i3, i4) (j0, j1, j2, j3, j4) = ( symbolicMerge f t i0 j0
, symbolicMerge f t i1 j1
, symbolicMerge f t i2 j2
, symbolicMerge f t i3 j3
, symbolicMerge f t i4 j4
)
select xs (err1, err2, err3, err4, err5) ind = ( select as err1 ind
, select bs err2 ind
, select cs err3 ind
, select ds err4 ind
, select es err5 ind
)
where (as, bs, cs, ds, es) = unzip5 xs
-- 6-Tuple
instance (Mergeable a, Mergeable b, Mergeable c, Mergeable d, Mergeable e, Mergeable f) => Mergeable (a, b, c, d, e, f) where
symbolicMerge f t (i0, i1, i2, i3, i4, i5) (j0, j1, j2, j3, j4, j5) = ( symbolicMerge f t i0 j0
, symbolicMerge f t i1 j1
, symbolicMerge f t i2 j2
, symbolicMerge f t i3 j3
, symbolicMerge f t i4 j4
, symbolicMerge f t i5 j5
)
select xs (err1, err2, err3, err4, err5, err6) ind = ( select as err1 ind
, select bs err2 ind
, select cs err3 ind
, select ds err4 ind
, select es err5 ind
, select fs err6 ind
)
where (as, bs, cs, ds, es, fs) = unzip6 xs
-- 7-Tuple
instance (Mergeable a, Mergeable b, Mergeable c, Mergeable d, Mergeable e, Mergeable f, Mergeable g) => Mergeable (a, b, c, d, e, f, g) where
symbolicMerge f t (i0, i1, i2, i3, i4, i5, i6) (j0, j1, j2, j3, j4, j5, j6) = ( symbolicMerge f t i0 j0
, symbolicMerge f t i1 j1
, symbolicMerge f t i2 j2
, symbolicMerge f t i3 j3
, symbolicMerge f t i4 j4
, symbolicMerge f t i5 j5
, symbolicMerge f t i6 j6
)
select xs (err1, err2, err3, err4, err5, err6, err7) ind = ( select as err1 ind
, select bs err2 ind
, select cs err3 ind
, select ds err4 ind
, select es err5 ind
, select fs err6 ind
, select gs err7 ind
)
where (as, bs, cs, ds, es, fs, gs) = unzip7 xs
-- Base types are mergeable so long as they are equal
instance Mergeable () where symbolicMerge _ _ = concreteMerge "()" "()" (==)
instance Mergeable Integer where symbolicMerge _ _ = concreteMerge "Integer" "SInteger" (==)
instance Mergeable Bool where symbolicMerge _ _ = concreteMerge "Bool" "SBool" (==)
instance Mergeable Char where symbolicMerge _ _ = concreteMerge "Char" "SChar" (==)
instance Mergeable Float where symbolicMerge _ _ = concreteMerge "Float" "SFloat" fpIsEqualObjectH
instance Mergeable Double where symbolicMerge _ _ = concreteMerge "Double" "SDouble" fpIsEqualObjectH
instance Mergeable Word8 where symbolicMerge _ _ = concreteMerge "Word8" "SWord8" (==)
instance Mergeable Word16 where symbolicMerge _ _ = concreteMerge "Word16" "SWord16" (==)
instance Mergeable Word32 where symbolicMerge _ _ = concreteMerge "Word32" "SWord32" (==)
instance Mergeable Word64 where symbolicMerge _ _ = concreteMerge "Word64" "SWord64" (==)
instance Mergeable Int8 where symbolicMerge _ _ = concreteMerge "Int8" "SInt8" (==)
instance Mergeable Int16 where symbolicMerge _ _ = concreteMerge "Int16" "SInt16" (==)
instance Mergeable Int32 where symbolicMerge _ _ = concreteMerge "Int32" "SInt32" (==)
instance Mergeable Int64 where symbolicMerge _ _ = concreteMerge "Int64" "SInt64" (==)
-- Arbitrary product types, using GHC.Generics
--
-- NB: Because of the way GHC.Generics works, the implementation of
-- symbolicMerge' is recursive. The derived instance for @data T a = T a a a a@
-- resembles that for (a, (a, (a, a))), not the flat 4-tuple (a, a, a, a). This
-- difference should have no effect in practice. Note also that, unlike the
-- hand-rolled tuple instances, the generic instance does not provide a custom
-- 'select' implementation, and so does not benefit from the SMT-table
-- implementation in the 'SBV a' instance.
-- | Not exported. Symbolic merge using the generic representation provided by
-- 'G.Generics'.
symbolicMergeDefault :: (G.Generic a, GMergeable (G.Rep a)) => Bool -> SBool -> a -> a -> a
symbolicMergeDefault force t x y = G.to $ symbolicMerge' force t (G.from x) (G.from y)
-- | Not exported. Used only in 'symbolicMergeDefault'. Instances are provided for
-- the generic representations of product types where each element is Mergeable.
class GMergeable f where
symbolicMerge' :: Bool -> SBool -> f a -> f a -> f a
{-
- N.B. A V1 instance like the below would be wrong!
- Why? Because inSBV, we use empty data to mean "uninterpreted" sort; not
- something that has no constructors. Perhaps that was a bad design
- decision. So, do not allow merging of such values!
instance GMergeable V1 where
symbolicMerge' _ _ x _ = x
-}
instance GMergeable U1 where
symbolicMerge' _ _ _ _ = U1
instance (Mergeable c) => GMergeable (K1 i c) where
symbolicMerge' force t (K1 x) (K1 y) = K1 $ symbolicMerge force t x y
instance (GMergeable f) => GMergeable (M1 i c f) where
symbolicMerge' force t (M1 x) (M1 y) = M1 $ symbolicMerge' force t x y
instance (GMergeable f, GMergeable g) => GMergeable (f :*: g) where
symbolicMerge' force t (x1 :*: y1) (x2 :*: y2) = symbolicMerge' force t x1 x2 :*: symbolicMerge' force t y1 y2
{- A mergeable instance for sum-types isn't possible. Why? It would something like:
instance (GMergeable f, GMergeable g) => GMergeable (f :+: g) where
symbolicMerge' force t (L1 x) (L1 y) = L1 $ symbolicMerge' force t x y
symbolicMerge' force t (R1 x) (R1 y) = R1 $ symbolicMerge' force t x y
symbolicMerge' force t l r
| Just tv <- unliteral t = if tv then l else r
| True = ????
There's really no good code to put in ????. We have no way to ask the SMT solver to merge composite values that
have different constructors. Calling "error" here would pass the type-checker, but that simply postpones the problem
to run-time. If you need mergeable on sum-types, you better write one yourself, possibly using the SEither type yourself.
As we have it, you'll get a type-error; which can be hard to read, but is preferable.
NB. This isn't a problem with the generic version of symbolic equality; since we can simply return sFalse if we
see different constructors. Such isn't the case when merging.
-}
-- Bounded instances
instance {-# OVERLAPPABLE #-} (SymVal a, Bounded a) => Bounded (SBV a) where
minBound = literal minBound
maxBound = literal maxBound
-- Haskell and SMTLib differ in their default char ranges. In Haskell, maxbound is a lot larger.
-- But in SMTLib, we only go upto 0x2FFFF. So, we adopt the SMTLib variant here. This is hardly
-- an issue in practice, but the discrepancy is disconcerting.
instance {-# OVERLAPPING #-} Bounded SChar where
minBound = literal (chr 0)
maxBound = literal (chr 0x2FFFF)
-- | Choose a value that satisfies the given predicate. This is Hillbert's choice, essentially. Note that
-- if the predicate given is not satisfiable (for instance @const sFalse@), then the element returned will be arbitrary.
-- The only guarantee is that if there's at least one element that satisfies the predicate, then the returned
-- element will be one of those that do. The returned element is not guaranteed to be unique, least, greatest etc, unless
-- there happens to be exactly one satisfying element.
some :: forall a. (SymVal a, HasKind a) => String -> (SBV a -> SBool) -> SBV a
some inpName cond = mk f
where mk = SBV . SVal k . Right . cache
k = kindOf (Proxy @a)
f st = do ctr <- incrementFreshNameCounter st
let pre = atProxy (Proxy @a) inpName
nm | ctr == 0 = pre
| True = pre ++ "_" ++ show ctr
nm' <- newUninterpreted st (UIGiven nm) Nothing (SBVType [k]) (UINone False)
chosen <- newExpr st k $ SBVApp (Uninterpreted nm') []
let ifExists = quantifiedBool $ \(Exists ex) -> cond ex
internalConstraint st False [] (unSBV (ifExists .=> cond (mk (pure (pure chosen)))))
pure chosen
-- | SMT definable constants and functions, which can also be uninterpeted.
-- This class captures functions that we can generate standalone-code for
-- in the SMT solver. Note that we also allow uninterpreted constants and
-- functions too. 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: 'sbvDefineValue'. However, most instances in
-- practice are already provided by SBV, so end-users should not need to define their
-- own instances.
class SMTDefinable a where
-- | Generate the code for this value as an SMTLib function, instead of
-- the usual unrolling semantics. This is useful for generating sub-functions
-- in generated SMTLib problem, or handling recursive (and mutually-recursive)
-- definitions that wouldn't terminate in an unrolling symbolic simulation context.
--
-- __IMPORTANT NOTE__ The string argument names this function. Note that SBV will identify
-- this function with that name, i.e., if you use this function twice (or use it recursively),
-- it will simply assume this name uniquely identifies the function being defined. Hence,
-- the user has to assure that this string is unique amongst all the functions you use.
-- Furthermore, if the call to 'smtFunction' happens in the scope of a parameter, you
-- must make sure the string is chosen to keep it unique per parameter value. For instance,
-- if you have:
--
-- @
-- bar :: SInteger -> SInteger -> SInteger
-- bar k = smtFunction "bar" (\x -> x+k) -- Note the capture of k!
-- @
--
-- and you call @bar 2@ and @bar 3@, you *will* get the same SMTLib function. Obviously
-- this is unsound. The reason is that the parameter value isn't captured by the name. In general,
-- you should simply not do this, but if you must, have a concrete argument to make sure you can
-- create a unique name. Something like:
--
-- @
-- bar :: String -> SInteger -> SInteger -> SInteger
-- bar tag k = smtFunction ("bar_" ++ tag) (\x -> x+k) -- Tag should make the name unique!
-- @
--
-- Then, make sure you use @bar "two" 2@ and @bar "three" 3@ etc. to preserve the invariant.
--
-- Furthermore, the function argument must not capture any non-constant variables in the context.
-- You can also define higher-order functions, see 'smtHOFunction' for that purpose.
--
-- Note that this is a design choice, to keep function creation as easy to use as possible. SBV
-- could've made 'smtFunction' a monadic call and generated the name itself to avoid all these issues.
-- But the ergonomics of that is worse, and doesn't fit with the general design philosophy. If you
-- can think of a solution (perhaps using some nifty GHC tricks?) to avoid this issue without making
-- 'smtFunction' return a monadic result, please get in touch!
smtFunction :: (Typeable a, Lambda Symbolic a) => String -> a -> a
-- | Register a function. This function is typically not needed as SBV will register functions used
-- automatically upon first use. However, there are scenarios (in particular query contexts)
-- where the definition isn't used before query-mode starts, and SBV (for historical reasons)
-- requires functions to be known before query-mode starts executing. In such cases, use this function
-- to register them with the system.
registerFunction :: a -> Symbolic ()
-- | Uninterpret a value, i.e., add this value as a completely undefined value/function that
-- the solver is free to instantiate to satisfy other constraints.
--
-- __Known issues__
--
-- Usually using an uninterpret function will register itself to the solver, but sometimes the lazyness
-- of the evaluation might render this unreliable.
--
-- For example, when working with quantifiers and uninterpreted functions with the following code:
--
-- > runSMTWith z3 $ do
-- > let f = uninterpret "f" :: SInteger -> SInteger
-- > query $ do
-- > constrain $ \(Forall (b :: SInteger)) -> f b .== f b
-- > checkSat
--
-- The solver will complain about the unknown constant @f (Int)@.
--
-- A workaround of this is to explicit register them with 'Data.SBV.Control.registerUISMTFunction':
--
-- > runSMTWith z3 $ do
-- > let f = uninterpret "f" :: SInteger -> SInteger
-- > registerUISMTFunction f
-- > query $ do
-- > constrain $ \(Forall (b :: SInteger)) -> f b .== f b
-- > checkSat
--
-- See https://github.com/LeventErkok/sbv/issues/711 for more info.
uninterpret :: String -> a
-- | Uninterpret a value, with named arguments in case of functions. SBV will use these
-- names when it shows the values for the arguments. If the given names are more than needed
-- we ignore the excess. If not enough, we add from a stock set of variables.
uninterpretWithArgs :: String -> [String] -> a
-- | 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.
cgUninterpret :: String -> [String] -> a -> a
-- | Most generalized form of uninterpretation, this function should not be needed
-- by end-user-code, but is rather useful for the library development.
sbvDefineValue :: UIName -> Maybe [String] -> UIKind a -> a
-- | A synonym for 'uninterpret'. Allows us to create variables without
-- having to call 'free' explicitly, i.e., without being in the symbolic monad.
sym :: String -> a
-- | Render an uninterpeted value as an SMTLib definition
sbv2smt :: ExtractIO m => a -> m String
{-# MINIMAL sbvDefineValue, sbv2smt #-}
-- defaults:
uninterpret nm = sbvDefineValue (UIGiven nm) Nothing $ UIFree True
uninterpretWithArgs nm as = sbvDefineValue (UIGiven nm) (Just as) $ UIFree True
cgUninterpret nm code v = sbvDefineValue (UIGiven nm) Nothing $ UICodeC (v, code)
sym = uninterpret
smtFunction nm v = sbvDefineValue (UIGiven (atProxy (Proxy @a) nm)) Nothing $ UIFun (v, \st fk -> lambda st TopLevel fk v)
default registerFunction :: forall b c. (a ~ (SBV b -> c), SymVal b, SMTDefinable c) => a -> Symbolic ()
registerFunction f = do let k = kindOf (Proxy @b)
st <- symbolicEnv
v <- liftIO $ newInternalVariable st k
let b = SBV $ SVal k $ Right $ cache (const (pure v))
registerFunction $ f b
-- | Kind of uninterpretation
data UIKind a = UIFree Bool -- ^ completely uninterpreted. If Bool is true, then this is curried.
| UIFun (a, State -> Kind -> IO SMTDef) -- ^ has code for SMTLib, with final type of kind (note this is the result
-- , not the arguments), which can be generated by calling the function on the state.
| UICodeC (a, [String]) -- ^ has code for code-generation, i.e., C
deriving Functor
-- Get the code associated with the UI, unless we've already did this once. (To support recursive defs.)
retrieveUICode :: UIName -> State -> Kind -> UIKind a -> IO UICodeKind
retrieveUICode _ _ _ (UIFree c) = pure $ UINone c
retrieveUICode (UIGiven nm) st fk (UIFun (_, f)) = do userFuncs <- readIORef (rUserFuncs st)
if nm `Set.member` userFuncs
then pure $ UINone True
else do modifyState st rUserFuncs (Set.insert nm) (pure ())
UISMT <$> f st fk
retrieveUICode _ _ _ (UICodeC (_, c)) = pure $ UICgC c
-- Get the constant value associated with the UI
retrieveConstCode :: UIKind a -> Maybe a
retrieveConstCode UIFree{} = Nothing
retrieveConstCode (UIFun (v, _)) = Just v
retrieveConstCode (UICodeC (v, _)) = Just v
-- Plain constants
instance (SymVal a, HasKind a) => SMTDefinable (SBV a) where
sbv2smt a = do st <- mkNewState defaultSMTCfg (LambdaGen (Just 0))
s <- lambdaStr st TopLevel (kindOf a) a
pure $ intercalate "\n" [ "; Automatically generated by SBV. Do not modify!"
, "; Type: " ++ smtType (kindOf a)
, show s
]
registerFunction x = constrain $ x .== x
sbvDefineValue nm mbArgs uiKind
| Just v <- retrieveConstCode uiKind
= v
| True
= SBV $ SVal ka $ Right $ cache result
where ka = kindOf (Proxy @a)
result st = do isSMT <- inSMTMode st
case (isSMT, uiKind) of
(True, UICodeC (v, _)) -> sbvToSV st v
_ -> do nm' <- newUninterpreted st nm mbArgs (SBVType [ka]) =<< retrieveUICode nm st ka uiKind
newExpr st ka $ SBVApp (Uninterpreted nm') []
-- Functions of one argument
instance (SymVal b, SymVal a, HasKind a) => SMTDefinable (SBV b -> SBV a) where
sbv2smt fn = defs2smt $ \b -> fn b .== fn b
sbvDefineValue nm mbArgs uiKind = f
where f arg0
| Just v <- retrieveConstCode uiKind, isConcrete arg0
= v arg0
| True
= SBV $ SVal ka $ Right $ cache result
where ka = kindOf (Proxy @a)
kb = kindOf (Proxy @b)
result st = do isSMT <- inSMTMode st
case (isSMT, uiKind) of
(True, UICodeC (v, _)) -> sbvToSV st (v arg0)
_ -> do nm' <- newUninterpreted st nm mbArgs (SBVType [kb, ka]) =<< retrieveUICode nm st ka uiKind
sw0 <- sbvToSV st arg0
mapM_ forceSVArg [sw0]
newExpr st ka $ SBVApp (Uninterpreted nm') [sw0]
-- Functions of two arguments
instance (SymVal c, SymVal b, SymVal a, HasKind a) => SMTDefinable (SBV c -> SBV b -> SBV a) where
sbv2smt fn = defs2smt $ \c b -> fn c b .== fn c b
sbvDefineValue nm mbArgs uiKind = f
where f arg0 arg1
| Just v <- retrieveConstCode uiKind, isConcrete arg0, isConcrete arg1
= v arg0 arg1
| True
= SBV $ SVal ka $ Right $ cache result
where ka = kindOf (Proxy @a)
kb = kindOf (Proxy @b)
kc = kindOf (Proxy @c)
result st = do isSMT <- inSMTMode st
case (isSMT, uiKind) of
(True, UICodeC (v, _)) -> sbvToSV st (v arg0 arg1)
_ -> do nm' <- newUninterpreted st nm mbArgs (SBVType [kc, kb, ka]) =<< retrieveUICode nm st ka uiKind
sw0 <- sbvToSV st arg0
sw1 <- sbvToSV st arg1
mapM_ forceSVArg [sw0, sw1]
newExpr st ka $ SBVApp (Uninterpreted nm') [sw0, sw1]
-- Functions of three arguments
instance (SymVal d, SymVal c, SymVal b, SymVal a, HasKind a) => SMTDefinable (SBV d -> SBV c -> SBV b -> SBV a) where
sbv2smt fn = defs2smt $ \d c b -> fn d c b .== fn d c b
sbvDefineValue nm mbArgs uiKind = f
where f arg0 arg1 arg2
| Just v <- retrieveConstCode uiKind, isConcrete arg0, isConcrete arg1, isConcrete arg2
= v arg0 arg1 arg2
| True
= SBV $ SVal ka $ Right $ cache result
where ka = kindOf (Proxy @a)
kb = kindOf (Proxy @b)
kc = kindOf (Proxy @c)
kd = kindOf (Proxy @d)
result st = do isSMT <- inSMTMode st
case (isSMT, uiKind) of
(True, UICodeC (v, _)) -> sbvToSV st (v arg0 arg1 arg2)
_ -> do nm' <- newUninterpreted st nm mbArgs (SBVType [kd, kc, kb, ka]) =<< retrieveUICode nm st ka uiKind
sw0 <- sbvToSV st arg0
sw1 <- sbvToSV st arg1
sw2 <- sbvToSV st arg2
mapM_ forceSVArg [sw0, sw1, sw2]
newExpr st ka $ SBVApp (Uninterpreted nm') [sw0, sw1, sw2]
-- Functions of four arguments
instance (SymVal e, SymVal d, SymVal c, SymVal b, SymVal a, HasKind a) => SMTDefinable (SBV e -> SBV d -> SBV c -> SBV b -> SBV a) where
sbv2smt fn = defs2smt $ \e d c b -> fn e d c b .== fn e d c b
sbvDefineValue nm mbArgs uiKind = f
where f arg0 arg1 arg2 arg3
| Just v <- retrieveConstCode uiKind, isConcrete arg0, isConcrete arg1, isConcrete arg2, isConcrete arg3
= v arg0 arg1 arg2 arg3
| True
= SBV $ SVal ka $ Right $ cache result
where ka = kindOf (Proxy @a)
kb = kindOf (Proxy @b)
kc = kindOf (Proxy @c)
kd = kindOf (Proxy @d)
ke = kindOf (Proxy @e)
result st = do isSMT <- inSMTMode st
case (isSMT, uiKind) of
(True, UICodeC (v, _)) -> sbvToSV st (v arg0 arg1 arg2 arg3)
_ -> do nm' <- newUninterpreted st nm mbArgs (SBVType [ke, kd, kc, kb, ka]) =<< retrieveUICode nm st ka uiKind
sw0 <- sbvToSV st arg0
sw1 <- sbvToSV st arg1
sw2 <- sbvToSV st arg2
sw3 <- sbvToSV st arg3
mapM_ forceSVArg [sw0, sw1, sw2, sw3]
newExpr st ka $ SBVApp (Uninterpreted nm') [sw0, sw1, sw2, sw3]
-- Functions of five arguments
instance (SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, SymVal a, HasKind a) => SMTDefinable (SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) where
sbv2smt fn = defs2smt $ \f e d c b -> fn f e d c b .== fn f e d c b
sbvDefineValue nm mbArgs uiKind = f
where f arg0 arg1 arg2 arg3 arg4
| Just v <- retrieveConstCode uiKind, isConcrete arg0, isConcrete arg1, isConcrete arg2, isConcrete arg3, isConcrete arg4
= v arg0 arg1 arg2 arg3 arg4
| True
= SBV $ SVal ka $ Right $ cache result
where ka = kindOf (Proxy @a)
kb = kindOf (Proxy @b)
kc = kindOf (Proxy @c)
kd = kindOf (Proxy @d)
ke = kindOf (Proxy @e)
kf = kindOf (Proxy @f)
result st = do isSMT <- inSMTMode st
case (isSMT, uiKind) of
(True, UICodeC (v, _)) -> sbvToSV st (v arg0 arg1 arg2 arg3 arg4)
_ -> do nm' <- newUninterpreted st nm mbArgs (SBVType [kf, ke, kd, kc, kb, ka]) =<< retrieveUICode nm st ka uiKind
sw0 <- sbvToSV st arg0
sw1 <- sbvToSV st arg1
sw2 <- sbvToSV st arg2
sw3 <- sbvToSV st arg3
sw4 <- sbvToSV st arg4
mapM_ forceSVArg [sw0, sw1, sw2, sw3, sw4]
newExpr st ka $ SBVApp (Uninterpreted nm') [sw0, sw1, sw2, sw3, sw4]
-- Functions of six arguments
instance (SymVal g, SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, SymVal a, HasKind a) => SMTDefinable (SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) where
sbv2smt fn = defs2smt $ \f g e d c b -> fn g f e d c b .== fn g f e d c b
sbvDefineValue nm mbArgs uiKind = f
where f arg0 arg1 arg2 arg3 arg4 arg5
| Just v <- retrieveConstCode uiKind, isConcrete arg0, isConcrete arg1, isConcrete arg2, isConcrete arg3, isConcrete arg4, isConcrete arg5
= v arg0 arg1 arg2 arg3 arg4 arg5
| True
= SBV $ SVal ka $ Right $ cache result
where ka = kindOf (Proxy @a)
kb = kindOf (Proxy @b)
kc = kindOf (Proxy @c)
kd = kindOf (Proxy @d)
ke = kindOf (Proxy @e)
kf = kindOf (Proxy @f)
kg = kindOf (Proxy @g)
result st = do isSMT <- inSMTMode st
case (isSMT, uiKind) of
(True, UICodeC (v, _)) -> sbvToSV st (v arg0 arg1 arg2 arg3 arg4 arg5)
_ -> do nm' <- newUninterpreted st nm mbArgs (SBVType [kg, kf, ke, kd, kc, kb, ka]) =<< retrieveUICode nm st ka uiKind
sw0 <- sbvToSV st arg0
sw1 <- sbvToSV st arg1
sw2 <- sbvToSV st arg2
sw3 <- sbvToSV st arg3
sw4 <- sbvToSV st arg4
sw5 <- sbvToSV st arg5
mapM_ forceSVArg [sw0, sw1, sw2, sw3, sw4, sw5]
newExpr st ka $ SBVApp (Uninterpreted nm') [sw0, sw1, sw2, sw3, sw4, sw5]
-- Functions of seven arguments
instance (SymVal h, SymVal g, SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, SymVal a, HasKind a)
=> SMTDefinable (SBV h -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) where
sbv2smt fn = defs2smt $ \h g f e d c b -> fn h g f e d c b .== fn h g f e d c b
sbvDefineValue nm mbArgs uiKind = f
where f arg0 arg1 arg2 arg3 arg4 arg5 arg6
| Just v <- retrieveConstCode uiKind, isConcrete arg0, isConcrete arg1, isConcrete arg2, isConcrete arg3, isConcrete arg4, isConcrete arg5, isConcrete arg6
= v arg0 arg1 arg2 arg3 arg4 arg5 arg6
| True
= SBV $ SVal ka $ Right $ cache result
where ka = kindOf (Proxy @a)
kb = kindOf (Proxy @b)
kc = kindOf (Proxy @c)
kd = kindOf (Proxy @d)
ke = kindOf (Proxy @e)
kf = kindOf (Proxy @f)
kg = kindOf (Proxy @g)
kh = kindOf (Proxy @h)
result st = do isSMT <- inSMTMode st
case (isSMT, uiKind) of
(True, UICodeC (v, _)) -> sbvToSV st (v arg0 arg1 arg2 arg3 arg4 arg5 arg6)
_ -> do nm' <- newUninterpreted st nm mbArgs (SBVType [kh, kg, kf, ke, kd, kc, kb, ka]) =<< retrieveUICode nm st ka uiKind
sw0 <- sbvToSV st arg0
sw1 <- sbvToSV st arg1
sw2 <- sbvToSV st arg2
sw3 <- sbvToSV st arg3
sw4 <- sbvToSV st arg4
sw5 <- sbvToSV st arg5
sw6 <- sbvToSV st arg6
mapM_ forceSVArg [sw0, sw1, sw2, sw3, sw4, sw5, sw6]
newExpr st ka $ SBVApp (Uninterpreted nm') [sw0, sw1, sw2, sw3, sw4, sw5, sw6]
-- Functions of eight arguments
instance (SymVal i, SymVal h, SymVal g, SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, SymVal a, SymVal a, HasKind a)
=> SMTDefinable (SBV i -> SBV h -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) where
sbv2smt fn = defs2smt $ \i h g f e d c b -> fn i h g f e d c b .== fn i h g f e d c b
sbvDefineValue nm mbArgs uiKind = f
where f arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7
| Just v <- retrieveConstCode uiKind, isConcrete arg0, isConcrete arg1, isConcrete arg2, isConcrete arg3, isConcrete arg4, isConcrete arg5, isConcrete arg6, isConcrete arg7
= v arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7
| True
= SBV $ SVal ka $ Right $ cache result
where ka = kindOf (Proxy @a)
kb = kindOf (Proxy @b)
kc = kindOf (Proxy @c)
kd = kindOf (Proxy @d)
ke = kindOf (Proxy @e)
kf = kindOf (Proxy @f)
kg = kindOf (Proxy @g)
kh = kindOf (Proxy @h)
ki = kindOf (Proxy @i)
result st = do isSMT <- inSMTMode st
case (isSMT, uiKind) of
(True, UICodeC (v, _)) -> sbvToSV st (v arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7)
_ -> do nm' <- newUninterpreted st nm mbArgs (SBVType [ki, kh, kg, kf, ke, kd, kc, kb, ka]) =<< retrieveUICode nm st ka uiKind
sw0 <- sbvToSV st arg0
sw1 <- sbvToSV st arg1
sw2 <- sbvToSV st arg2
sw3 <- sbvToSV st arg3
sw4 <- sbvToSV st arg4
sw5 <- sbvToSV st arg5
sw6 <- sbvToSV st arg6
sw7 <- sbvToSV st arg7
mapM_ forceSVArg [sw0, sw1, sw2, sw3, sw4, sw5, sw6, sw7]
newExpr st ka $ SBVApp (Uninterpreted nm') [sw0, sw1, sw2, sw3, sw4, sw5, sw6, sw7]
-- Functions of nine arguments
instance (SymVal j, SymVal i, SymVal h, SymVal g, SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, SymVal a, HasKind a)
=> SMTDefinable (SBV j -> SBV i -> SBV h -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) where
sbv2smt fn = defs2smt $ \j i h g f e d c b -> fn j i h g f e d c b .== fn j i h g f e d c b
sbvDefineValue nm mbArgs uiKind = f
where f arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7 arg8
| Just v <- retrieveConstCode uiKind, isConcrete arg0, isConcrete arg1, isConcrete arg2, isConcrete arg3, isConcrete arg4, isConcrete arg5, isConcrete arg6, isConcrete arg7, isConcrete arg8
= v arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7 arg8
| True
= SBV $ SVal ka $ Right $ cache result
where ka = kindOf (Proxy @a)
kb = kindOf (Proxy @b)
kc = kindOf (Proxy @c)
kd = kindOf (Proxy @d)
ke = kindOf (Proxy @e)
kf = kindOf (Proxy @f)
kg = kindOf (Proxy @g)
kh = kindOf (Proxy @h)
ki = kindOf (Proxy @i)
kj = kindOf (Proxy @j)
result st = do isSMT <- inSMTMode st
case (isSMT, uiKind) of
(True, UICodeC (v, _)) -> sbvToSV st (v arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7 arg8)
_ -> do nm' <- newUninterpreted st nm mbArgs (SBVType [kj, ki, kh, kg, kf, ke, kd, kc, kb, ka]) =<< retrieveUICode nm st ka uiKind
sw0 <- sbvToSV st arg0
sw1 <- sbvToSV st arg1
sw2 <- sbvToSV st arg2
sw3 <- sbvToSV st arg3
sw4 <- sbvToSV st arg4
sw5 <- sbvToSV st arg5
sw6 <- sbvToSV st arg6
sw7 <- sbvToSV st arg7
sw8 <- sbvToSV st arg8
mapM_ forceSVArg [sw0, sw1, sw2, sw3, sw4, sw5, sw6, sw7, sw8]
newExpr st ka $ SBVApp (Uninterpreted nm') [sw0, sw1, sw2, sw3, sw4, sw5, sw6, sw7, sw8]
-- Functions of ten arguments
instance (SymVal k, SymVal j, SymVal i, SymVal h, SymVal g, SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, SymVal a, HasKind a)
=> SMTDefinable (SBV k -> SBV j -> SBV i -> SBV h -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) where
sbv2smt fn = defs2smt $ \k j i h g f e d c b -> fn k j i h g f e d c b .== fn k j i h g f e d c b
sbvDefineValue nm mbArgs uiKind = f
where f arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7 arg8 arg9
| Just v <- retrieveConstCode uiKind, isConcrete arg0, isConcrete arg1, isConcrete arg2, isConcrete arg3, isConcrete arg4, isConcrete arg5, isConcrete arg6, isConcrete arg7, isConcrete arg8, isConcrete arg9
= v arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7 arg8 arg9
| True
= SBV $ SVal ka $ Right $ cache result
where ka = kindOf (Proxy @a)
kb = kindOf (Proxy @b)
kc = kindOf (Proxy @c)
kd = kindOf (Proxy @d)
ke = kindOf (Proxy @e)
kf = kindOf (Proxy @f)
kg = kindOf (Proxy @g)
kh = kindOf (Proxy @h)
ki = kindOf (Proxy @i)
kj = kindOf (Proxy @j)
kk = kindOf (Proxy @k)
result st = do isSMT <- inSMTMode st
case (isSMT, uiKind) of
(True, UICodeC (v, _)) -> sbvToSV st (v arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7 arg8 arg9)
_ -> do nm' <- newUninterpreted st nm mbArgs (SBVType [kk, kj, ki, kh, kg, kf, ke, kd, kc, kb, ka]) =<< retrieveUICode nm st ka uiKind
sw0 <- sbvToSV st arg0
sw1 <- sbvToSV st arg1
sw2 <- sbvToSV st arg2
sw3 <- sbvToSV st arg3
sw4 <- sbvToSV st arg4
sw5 <- sbvToSV st arg5
sw6 <- sbvToSV st arg6
sw7 <- sbvToSV st arg7
sw8 <- sbvToSV st arg8
sw9 <- sbvToSV st arg9
mapM_ forceSVArg [sw0, sw1, sw2, sw3, sw4, sw5, sw6, sw7, sw8, sw9]
newExpr st ka $ SBVApp (Uninterpreted nm') [sw0, sw1, sw2, sw3, sw4, sw5, sw6, sw7, sw8, sw9]
-- Functions of eleven arguments
instance (SymVal l, SymVal k, SymVal j, SymVal i, SymVal h, SymVal g, SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, SymVal a, HasKind a)
=> SMTDefinable (SBV l -> SBV k -> SBV j -> SBV i -> SBV h -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) where
sbv2smt fn = defs2smt $ \l k j i h g f e d c b -> fn l k j i h g f e d c b .== fn l k j i h g f e d c b
sbvDefineValue nm mbArgs uiKind = f
where f arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7 arg8 arg9 arg10
| Just v <- retrieveConstCode uiKind, isConcrete arg0, isConcrete arg1, isConcrete arg2, isConcrete arg3, isConcrete arg4, isConcrete arg5, isConcrete arg6, isConcrete arg7, isConcrete arg8, isConcrete arg9, isConcrete arg10
= v arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7 arg8 arg9 arg10
| True
= SBV $ SVal ka $ Right $ cache result
where ka = kindOf (Proxy @a)
kb = kindOf (Proxy @b)
kc = kindOf (Proxy @c)
kd = kindOf (Proxy @d)
ke = kindOf (Proxy @e)
kf = kindOf (Proxy @f)
kg = kindOf (Proxy @g)
kh = kindOf (Proxy @h)
ki = kindOf (Proxy @i)
kj = kindOf (Proxy @j)
kk = kindOf (Proxy @k)
kl = kindOf (Proxy @l)
result st = do isSMT <- inSMTMode st
case (isSMT, uiKind) of
(True, UICodeC (v, _)) -> sbvToSV st (v arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7 arg8 arg9 arg10)
_ -> do nm' <- newUninterpreted st nm mbArgs (SBVType [kl, kk, kj, ki, kh, kg, kf, ke, kd, kc, kb, ka]) =<< retrieveUICode nm st ka uiKind
sw0 <- sbvToSV st arg0
sw1 <- sbvToSV st arg1
sw2 <- sbvToSV st arg2
sw3 <- sbvToSV st arg3
sw4 <- sbvToSV st arg4
sw5 <- sbvToSV st arg5
sw6 <- sbvToSV st arg6
sw7 <- sbvToSV st arg7
sw8 <- sbvToSV st arg8
sw9 <- sbvToSV st arg9
sw10 <- sbvToSV st arg10
mapM_ forceSVArg [sw0, sw1, sw2, sw3, sw4, sw5, sw6, sw7, sw8, sw9, sw10]
newExpr st ka $ SBVApp (Uninterpreted nm') [sw0, sw1, sw2, sw3, sw4, sw5, sw6, sw7, sw8, sw9, sw10]
-- Functions of twelve arguments
instance (SymVal m, SymVal l, SymVal k, SymVal j, SymVal i, SymVal h, SymVal g, SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, SymVal a, HasKind a)
=> SMTDefinable (SBV m -> SBV l -> SBV k -> SBV j -> SBV i -> SBV h -> SBV g -> SBV f -> SBV e -> SBV d -> SBV c -> SBV b -> SBV a) where
sbv2smt fn = defs2smt $ \m l k j i h g f e d c b -> fn m l k j i h g f e d c b .== fn m l k j i h g f e d c b
sbvDefineValue nm mbArgs uiKind = f
where f arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7 arg8 arg9 arg10 arg11
| Just v <- retrieveConstCode uiKind, isConcrete arg0, isConcrete arg1, isConcrete arg2, isConcrete arg3, isConcrete arg4, isConcrete arg5, isConcrete arg6, isConcrete arg7, isConcrete arg8, isConcrete arg9, isConcrete arg10, isConcrete arg11
= v arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7 arg8 arg9 arg10 arg11
| True
= SBV $ SVal ka $ Right $ cache result
where ka = kindOf (Proxy @a)
kb = kindOf (Proxy @b)
kc = kindOf (Proxy @c)
kd = kindOf (Proxy @d)
ke = kindOf (Proxy @e)
kf = kindOf (Proxy @f)
kg = kindOf (Proxy @g)
kh = kindOf (Proxy @h)
ki = kindOf (Proxy @i)
kj = kindOf (Proxy @j)
kk = kindOf (Proxy @k)
kl = kindOf (Proxy @l)
km = kindOf (Proxy @m)
result st = do isSMT <- inSMTMode st
case (isSMT, uiKind) of
(True, UICodeC (v, _)) -> sbvToSV st (v arg0 arg1 arg2 arg3 arg4 arg5 arg6 arg7 arg8 arg9 arg10 arg11)
_ -> do nm' <- newUninterpreted st nm mbArgs (SBVType [km, kl, kk, kj, ki, kh, kg, kf, ke, kd, kc, kb, ka]) =<< retrieveUICode nm st ka uiKind
sw0 <- sbvToSV st arg0
sw1 <- sbvToSV st arg1
sw2 <- sbvToSV st arg2
sw3 <- sbvToSV st arg3
sw4 <- sbvToSV st arg4
sw5 <- sbvToSV st arg5
sw6 <- sbvToSV st arg6
sw7 <- sbvToSV st arg7
sw8 <- sbvToSV st arg8
sw9 <- sbvToSV st arg9
sw10 <- sbvToSV st arg10
sw11 <- sbvToSV st arg11
mapM_ forceSVArg [sw0, sw1, sw2, sw3, sw4, sw5, sw6, sw7, sw8, sw9, sw10, sw11]
newExpr st ka $ SBVApp (Uninterpreted nm') [sw0, sw1, sw2, sw3, sw4, sw5, sw6, sw7, sw8, sw9, sw10, sw11]
-- Mark the UIKind as uncurried
mkUncurried :: UIKind a -> UIKind a
mkUncurried (UIFree _) = UIFree False
mkUncurried (UIFun a) = UIFun a
mkUncurried (UICodeC a) = UICodeC a
-- Uncurried functions of two arguments
instance (SymVal c, SymVal b, SymVal a, HasKind a) => SMTDefinable ((SBV c, SBV b) -> SBV a) where
sbv2smt fn = defs2smt $ \p -> fn p .== fn p
registerFunction = registerFunction . curry2
sbvDefineValue nm mbArgs uiKind = let f = sbvDefineValue nm mbArgs (curry2 <$> mkUncurried uiKind) in uncurry2 f
-- Uncurried functions of three arguments
instance (SymVal d, SymVal c, SymVal b, SymVal a, HasKind a) => SMTDefinable ((SBV d, SBV c, SBV b) -> SBV a) where
sbv2smt fn = defs2smt $ \p -> fn p .== fn p
registerFunction = registerFunction . curry3
sbvDefineValue nm mbArgs uiKind = let f = sbvDefineValue nm mbArgs (curry3 <$> mkUncurried uiKind) in uncurry3 f
-- Uncurried functions of four arguments
instance (SymVal e, SymVal d, SymVal c, SymVal b, SymVal a, HasKind a) => SMTDefinable ((SBV e, SBV d, SBV c, SBV b) -> SBV a) where
sbv2smt fn = defs2smt $ \p -> fn p .== fn p
registerFunction = registerFunction . curry4
sbvDefineValue nm mbArgs uiKind = let f = sbvDefineValue nm mbArgs (curry4 <$> mkUncurried uiKind) in uncurry4 f
-- Uncurried functions of five arguments
instance (SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, SymVal a, HasKind a) => SMTDefinable ((SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) where
sbv2smt fn = defs2smt $ \p -> fn p .== fn p
registerFunction = registerFunction . curry5
sbvDefineValue nm mbArgs uiKind = let f = sbvDefineValue nm mbArgs (curry5 <$> mkUncurried uiKind) in uncurry5 f
-- Uncurried functions of six arguments
instance (SymVal g, SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, SymVal a, HasKind a) => SMTDefinable ((SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) where
sbv2smt fn = defs2smt $ \p -> fn p .== fn p
registerFunction = registerFunction . curry6
sbvDefineValue nm mbArgs uiKind = let f = sbvDefineValue nm mbArgs (curry6 <$> mkUncurried uiKind) in uncurry6 f
-- Uncurried functions of seven arguments
instance (SymVal h, SymVal g, SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, SymVal a, HasKind a) => SMTDefinable ((SBV h, SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) where
sbv2smt fn = defs2smt $ \p -> fn p .== fn p
registerFunction = registerFunction . curry7
sbvDefineValue nm mbArgs uiKind = let f = sbvDefineValue nm mbArgs (curry7 <$> mkUncurried uiKind) in uncurry7 f
-- Uncurried functions of eight arguments
instance (SymVal i, SymVal h, SymVal g, SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, SymVal a, HasKind a) => SMTDefinable ((SBV i, SBV h, SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) where
sbv2smt fn = defs2smt $ \p -> fn p .== fn p
registerFunction = registerFunction . curry8
sbvDefineValue nm mbArgs uiKind = let f = sbvDefineValue nm mbArgs (curry8 <$> mkUncurried uiKind) in uncurry8 f
-- Uncurried functions of nine arguments
instance (SymVal j, SymVal i, SymVal h, SymVal g, SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, SymVal a, HasKind a) => SMTDefinable ((SBV j, SBV i, SBV h, SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) where
sbv2smt fn = defs2smt $ \p -> fn p .== fn p
registerFunction = registerFunction . curry9
sbvDefineValue nm mbArgs uiKind = let f = sbvDefineValue nm mbArgs (curry9 <$> mkUncurried uiKind) in uncurry9 f
-- Uncurried functions of ten arguments
instance (SymVal k, SymVal j, SymVal i, SymVal h, SymVal g, SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, SymVal a, HasKind a) => SMTDefinable ((SBV k, SBV j, SBV i, SBV h, SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) where
sbv2smt fn = defs2smt $ \p -> fn p .== fn p
registerFunction = registerFunction . curry10
sbvDefineValue nm mbArgs uiKind = let f = sbvDefineValue nm mbArgs (curry10 <$> mkUncurried uiKind) in uncurry10 f
-- Uncurried functions of eleven arguments
instance (SymVal l, SymVal k, SymVal j, SymVal i, SymVal h, SymVal g, SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, SymVal a, HasKind a) => SMTDefinable ((SBV l, SBV k, SBV j, SBV i, SBV h, SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) where
sbv2smt fn = defs2smt $ \p -> fn p .== fn p
registerFunction = registerFunction . curry11
sbvDefineValue nm mbArgs uiKind = let f = sbvDefineValue nm mbArgs (curry11 <$> mkUncurried uiKind) in uncurry11 f
-- Uncurried functions of twelve arguments
instance (SymVal m, SymVal l, SymVal k, SymVal j, SymVal i, SymVal h, SymVal g, SymVal f, SymVal e, SymVal d, SymVal c, SymVal b, SymVal a, HasKind a) => SMTDefinable ((SBV m, SBV l, SBV k, SBV j, SBV i, SBV h, SBV g, SBV f, SBV e, SBV d, SBV c, SBV b) -> SBV a) where
sbv2smt fn = defs2smt $ \p -> fn p .== fn p
registerFunction = registerFunction . curry12
sbvDefineValue nm mbArgs uiKind = let f = sbvDefineValue nm mbArgs (curry12 <$> mkUncurried uiKind) in uncurry12 f
-- | Symbolic computations provide a context for writing symbolic programs.
instance MonadIO m => SolverContext (SymbolicT m) where
constrain = imposeConstraint False [] . unSBV . quantifiedBool
softConstrain = imposeConstraint True [] . unSBV . quantifiedBool
namedConstraint nm = imposeConstraint False [(":named", nm)] . unSBV . quantifiedBool
constrainWithAttribute atts = imposeConstraint False atts . unSBV . quantifiedBool
contextState = symbolicEnv
setOption o = addNewSMTOption o
internalVariable k = contextState >>= \st -> liftIO $ do
sv <- newInternalVariable st k
pure $ SBV $ SVal k (Right (cache (const (pure sv))))
-- | Generalization of 'Data.SBV.assertWithPenalty'
assertWithPenalty :: MonadSymbolic m => String -> SBool -> Penalty -> m ()
assertWithPenalty nm o p = addSValOptGoal $ unSBV `fmap` AssertWithPenalty nm o p
-- | Class of metrics we can optimize for. Currently, booleans,
-- bounded signed/unsigned bit-vectors, unbounded integers,
-- algebraic reals and floats can be optimized. You can add
-- your instances, but bewared that the 'MetricSpace' should
-- map your type to something the backend solver understands, which
-- are limited to unsigned bit-vectors, reals, and unbounded integers
-- for z3.
--
-- A good reference on these features is given in the following paper:
-- <http://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/nbjorner-scss2014.pdf>.
--
-- Minimal completion: None. However, if @MetricSpace@ is not identical to the type, you want
-- to define 'toMetricSpace'/'annotateForMS', and possibly 'minimize'/'maximize' to add extra constraints as necessary.
class Metric a where
-- | The metric space we optimize the goal over. Usually the same as the type itself, but not always!
-- For instance, signed bit-vectors are optimized over their unsigned counterparts, floats are
-- optimized over their 'Word32' comparable counterparts, etc.
type MetricSpace a :: Type
type MetricSpace a = a
-- | Compute the metric value to optimize.
toMetricSpace :: SBV a -> SBV (MetricSpace a)
-- | Compute the value itself from the metric corresponding to it.
fromMetricSpace :: SBV (MetricSpace a) -> SBV a
-- | Annotate for the metric space, to clarify the new name. If this result is not identity,
-- we will add an sObserve on the original.
annotateForMS :: Proxy a -> String -> String
-- | Minimizing a metric space
msMinimize :: (MonadSymbolic m, SolverContext m) => String -> SBV a -> m ()
msMinimize nm o = do let nm' = annotateForMS (Proxy @a) nm
when (nm' /= nm) $ sObserve nm (unSBV o)
addSValOptGoal $ unSBV `fmap` Minimize nm' (toMetricSpace o)
-- | Maximizing a metric space
msMaximize :: (MonadSymbolic m, SolverContext m) => String -> SBV a -> m ()
msMaximize nm o = do let nm' = annotateForMS (Proxy @a) nm
when (nm' /= nm) $ sObserve nm (unSBV o)
addSValOptGoal $ unSBV `fmap` Maximize nm' (toMetricSpace o)
-- if MetricSpace is the same, we can give a default definition
default toMetricSpace :: (a ~ MetricSpace a) => SBV a -> SBV (MetricSpace a)
toMetricSpace = id
default fromMetricSpace :: (a ~ MetricSpace a) => SBV (MetricSpace a) -> SBV a
fromMetricSpace = id
-- Annotations to indicate if the metric space transition was needed
default annotateForMS :: (a ~ MetricSpace a) => Proxy a -> String -> String
annotateForMS _ s = s
-- Booleans assume True is greater than False
instance Metric Bool where
type MetricSpace Bool = Word8
toMetricSpace t = ite t 1 0
fromMetricSpace w = w ./= 0
annotateForMS _ s = "toMetricSpace(" ++ s ++ ")"
-- | Generalization of 'Data.SBV.minimize'
minimize :: (Metric a, MonadSymbolic m, SolverContext m) => String -> SBV a -> m ()
minimize = msMinimize
-- | Generalization of 'Data.SBV.maximize'
maximize :: (Metric a, MonadSymbolic m, SolverContext m) => String -> SBV a -> m ()
maximize = msMaximize
-- Unsigned types, integers, and reals directly optimize
instance Metric Word8
instance Metric Word16
instance Metric Word32
instance Metric Word64
instance Metric Integer
instance Metric AlgReal
-- To optimize signed bounded values, we have to adjust to the range
instance Metric Int8 where
type MetricSpace Int8 = Word8
toMetricSpace x = sFromIntegral x + 128 -- 2^7
fromMetricSpace x = sFromIntegral x - 128
annotateForMS _ s = "toMetricSpace(" ++ s ++ ")"
instance Metric Int16 where
type MetricSpace Int16 = Word16
toMetricSpace x = sFromIntegral x + 32768 -- 2^15
fromMetricSpace x = sFromIntegral x - 32768
annotateForMS _ s = "toMetricSpace(" ++ s ++ ")"
instance Metric Int32 where
type MetricSpace Int32 = Word32
toMetricSpace x = sFromIntegral x + 2147483648 -- 2^31
fromMetricSpace x = sFromIntegral x - 2147483648
annotateForMS _ s = "toMetricSpace(" ++ s ++ ")"
instance Metric Int64 where
type MetricSpace Int64 = Word64
toMetricSpace x = sFromIntegral x + 9223372036854775808 -- 2^63
fromMetricSpace x = sFromIntegral x - 9223372036854775808
annotateForMS _ s = "toMetricSpace(" ++ s ++ ")"
-- | Optimizing 'WordN'
instance (KnownNat n, BVIsNonZero n) => Metric (WordN n)
-- | Optimizing 'IntN'
instance (KnownNat n, BVIsNonZero n) => Metric (IntN n) where
type MetricSpace (IntN n) = WordN n
toMetricSpace x = sFromIntegral x + 2 ^ (intOfProxy (Proxy @n) - 1)
fromMetricSpace x = sFromIntegral x - 2 ^ (intOfProxy (Proxy @n) - 1)
annotateForMS _ s = "toMetricSpace(" ++ s ++ ")"
-- Quickcheck interface on symbolic-booleans..
instance Testable SBool where
property (SBV (SVal _ (Left b))) = property (cvToBool b)
property s = cantQuickCheck $ "Result did not evaluate to a concrete boolean: " ++ show s
instance Testable (Symbolic SBool) where
property prop = QC.monadicIO $ do (cond, r, modelVals) <- QC.run test
QC.pre cond
unless (r || null modelVals) $ QC.monitor (QC.counterexample (complain modelVals))
QC.assert r
where test = do (r, Result{resTraces=tvals, resObservables=ovals, resConsts=(_, cs), resConstraints=cstrs, resUIConsts=unints}) <-
C.catch (runSymbolic defaultSMTCfg (Concrete Nothing) prop)
(\(e :: C.SomeException) -> cantQuickCheck (show e))
let cval = fromMaybe (cantQuickCheck "A constraint did not evaluate to a concrete boolean") . (`lookup` cs)
cond = -- Only pick-up "hard" constraints, as indicated by False in the fist component
and [cvToBool (cval v) | (False, _, v) <- F.toList cstrs]
getObservable (nm, f, v) = case v `lookup` cs of
Just cv -> if f cv then Just (nm, cv) else Nothing
Nothing -> cantQuickCheck "An observable did not evaluate to a concrete value"
case map fst unints of
[] -> case unliteral r of
Nothing -> cantQuickCheck "The result did not evaluate to a concrete value"
Just b -> return (cond, b, tvals ++ mapMaybe getObservable ovals)
uis -> cantQuickCheck $ "Uninterpreted constants remain: " ++ unwords uis
complain qcInfo = showModel defaultSMTCfg (SMTModel [] Nothing qcInfo [])
-- Complain if what we got isn't something we can quick-check
cantQuickCheck :: String -> a
cantQuickCheck why = error $ unlines [ "*** Data.SBV: Cannot quickcheck the given property."
, "***"
, "*** Certain SBV properties cannot be quick-checked. In particular,"
, "*** SBV can't quick-check in the presence of:"
, "***"
, "*** - Uninterpreted constants."
, "*** - Uninterpreted types."
, "*** - Floating point operations with rounding modes other than RNE."
, "*** - Floating point FMA operation, regardless of rounding mode."
, "*** - Quantified booleans, i.e., uses of Forall/Exists/ExistsUnique."
, "*** - Uses of quantifiedBool"
, "*** - Calls to 'observe' (use 'sObserve' instead)"
, "***"
, "*** If you can't avoid the above features or run into an issue with"
, "*** quickcheck even though you haven't used these features, please report this as a bug!"
, "***"
, "*** Origin:"
, "***"
, why
]
-- | 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.
sbvQuickCheck :: Symbolic SBool -> IO Bool
sbvQuickCheck prop = QC.isSuccess `fmap` QC.quickCheckResult prop
-- Quickcheck interface on dynamically-typed values. A run-time check
-- ensures that the value has boolean type.
instance Testable (Symbolic SVal) where
property m = property $ do s <- m
when (kindOf s /= KBool) $ error "Cannot quickcheck non-boolean value"
return (SBV s :: SBool)
-- | Explicit sharing combinator. The SBV library has internal caching/hash-consing mechanisms
-- built in, based on Andy Gill's type-safe observable sharing technique (see: <http://ku-fpg.github.io/files/Gill-09-TypeSafeReification.pdf>).
-- However, there might be times where being explicit on the sharing can help, especially in experimental code. The 'slet' combinator
-- ensures that its first argument is computed once and passed on to its continuation, explicitly indicating the intent of sharing. Most
-- use cases of the SBV library should simply use Haskell's @let@ construct for this purpose.
slet :: forall a b. (HasKind a, HasKind b) => SBV a -> (SBV a -> SBV b) -> SBV b
slet x f = SBV $ SVal k $ Right $ cache r
where k = kindOf (Proxy @b)
r st = do xsv <- sbvToSV st x
let xsbv = SBV $ SVal (kindOf x) (Right (cache (const (return xsv))))
res = f xsbv
sbvToSV st res
-- | Class of things that we can logically reduce to a boolean, by saturating and then asserting equivalance to itself
class QSaturate m a where
qSaturate :: a -> m ()
-- | Base case; simple variable in the symbolic monad
instance SolverContext m => QSaturate m SBool where
qSaturate b = constrain $ b .== b
-- | Saturate over a universal quantifier
instance (HasKind a, Monad m, SolverContext m, QSaturate m r) => QSaturate m (Forall nm a -> r) where
qSaturate f = qSaturate . f . Forall =<< internalVariable (kindOf (Proxy @a))
-- | Saturate over a pair of universal quantifiers
instance (HasKind a, HasKind b, Monad m, SolverContext m, QSaturate m r) => QSaturate m ((Forall na a, Forall nb b) -> r) where
qSaturate = qSaturate . curry
-- | Saturate over a pair of existential quantifiers
instance (HasKind a, HasKind b, Monad m, SolverContext m, QSaturate m r) => QSaturate m ((Exists na a, Exists nb b) -> r) where
qSaturate = qSaturate . curry
-- | Saturate over a number of universal quantifiers
instance (KnownNat n, HasKind a, Monad m, SolverContext m, QSaturate m r) => QSaturate m (ForallN n nm a -> r) where
qSaturate f = qSaturate . f . ForallN =<< replicateM (intOfProxy (Proxy @n)) (internalVariable (kindOf (Proxy @a)))
-- | Saturate over an existential quantifier
instance (HasKind a, Monad m, SolverContext m, QSaturate m r) => QSaturate m (Exists nm a -> r) where
qSaturate f = qSaturate . f . Exists =<< internalVariable (kindOf (Proxy @a))
-- | Saturate over an a number of existential quantifiers
instance (KnownNat n, HasKind a, Monad m, SolverContext m, QSaturate m r) => QSaturate m (ExistsN n nm a -> r) where
qSaturate f = qSaturate . f . ExistsN =<< replicateM (intOfProxy (Proxy @n)) (internalVariable (kindOf (Proxy @a)))
-- | Saturate over a unique-exists variable
instance (HasKind a, Monad m, SolverContext m, QSaturate m r) => QSaturate m (ExistsUnique nm a -> r) where
qSaturate f = qSaturate . f . ExistsUnique =<< internalVariable (kindOf (Proxy @a))
-- | Saturate a predicate, but save/restore observables so they're not messed up.
qSaturateSavingObservables :: (Monad m, MonadIO m, SolverContext m, QSaturate m a) => a -> m ()
qSaturateSavingObservables p = do State{rObservables} <- contextState
curObservables <- liftIO $ readIORef rObservables
qSaturate p
liftIO $ writeIORef rObservables curObservables
-- | 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 {-# OVERLAPPABLE #-} (SymVal a, EqSymbolic z) => Equality (SBV a -> z) where
k === l = prove $ \a -> k a .== l a
instance {-# OVERLAPPABLE #-} (SymVal a, SymVal b, EqSymbolic z) => Equality (SBV a -> SBV b -> z) where
k === l = prove $ \a b -> k a b .== l a b
instance {-# OVERLAPPABLE #-} (SymVal a, SymVal b, EqSymbolic z) => Equality ((SBV a, SBV b) -> z) where
k === l = prove $ \a b -> k (a, b) .== l (a, b)
instance {-# OVERLAPPABLE #-} (SymVal a, SymVal b, SymVal 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 {-# OVERLAPPABLE #-} (SymVal a, SymVal b, SymVal 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 {-# OVERLAPPABLE #-} (SymVal a, SymVal b, SymVal c, SymVal 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 {-# OVERLAPPABLE #-} (SymVal a, SymVal b, SymVal c, SymVal 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 {-# OVERLAPPABLE #-} (SymVal a, SymVal b, SymVal c, SymVal d, SymVal 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 {-# OVERLAPPABLE #-} (SymVal a, SymVal b, SymVal c, SymVal d, SymVal 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 {-# OVERLAPPABLE #-} (SymVal a, SymVal b, SymVal c, SymVal d, SymVal e, SymVal 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 {-# OVERLAPPABLE #-}
(SymVal a, SymVal b, SymVal c, SymVal d, SymVal e, SymVal 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 {-# OVERLAPPABLE #-}
(SymVal a, SymVal b, SymVal c, SymVal d, SymVal e, SymVal f, SymVal 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 {-# OVERLAPPABLE #-} (SymVal a, SymVal b, SymVal c, SymVal d, SymVal e, SymVal f, SymVal 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)
-- | Reading a value from an array.
readArray :: forall key val. (Eq key, SymVal key, SymVal val, HasKind val) => SArray key val -> SBV key -> SBV val
readArray array key
| eqCheckIsObjectEq ka, Just (ArrayModel tbl def) <- unliteral array, Just k <- unliteral key
= literal $ fromMaybe def (k `lookup` tbl) -- return the first value, since we don't bother deleting previous writes
| True
= symRes
where symRes = SBV . SVal kb . Right $ cache g
ka = kindOf (Proxy @key)
kb = kindOf (Proxy @val)
g st = do f <- sbvToSV st array
k <- sbvToSV st key
newExpr st kb (SBVApp ReadArray [f, k])
-- | Writing a value to an array. For the concrete case, we don't bother deleting earlier entries, we keep a history. The earlier a value is in the list, the "later" it happened; in a stack fashion.
writeArray :: forall key val. (HasKind key, SymVal key, SymVal val, HasKind val) => SArray key val -> SBV key -> SBV val -> SArray key val
writeArray array key value
| Just (ArrayModel tbl def) <- unliteral array, Just keyVal <- unliteral key, Just val <- unliteral value
= literal $ ArrayModel ((keyVal, val) : tbl) def -- It's important that we "cons" the value here, since it takes precedence in a read
| True
= SBV . SVal k . Right $ cache g
where k = KArray (kindOf (Proxy @key)) (kindOf (Proxy @val))
g st = do arr <- sbvToSV st array
keyVal <- sbvToSV st key
val <- sbvToSV st value
newExpr st k (SBVApp WriteArray [arr, keyVal, val])
-- | Using a lambda as an array.
lambdaArray :: forall a b. (SymVal a, HasKind b) => (SBV a -> SBV b) -> SArray a b
lambdaArray f = SBV . SVal k . Right $ cache g
where k = KArray (kindOf (Proxy @a)) (kindOf (Proxy @b))
g st = do def <- lambdaStr st TopLevel (kindOf (Proxy @b)) f
newExpr st k (SBVApp (ArrayLambda def) [])
-- | Turn a constant association-list and a default into a symbolic array.
listArray :: (SymVal a, SymVal b) => [(a, b)] -> b -> SArray a b
listArray ascs def = literal $ ArrayModel ascs def
-- | Create a closure, wrapping the free variables together with the function. When using higher-order functions
-- in SBV (like map), the function passed must be closed, i.e., not have any free variables. If you need to call
-- such a function with a function capturing a free variable, you should create a closure instead.
data Closure env a = Closure { closureEnv :: env
, closureFun :: env -> a
}
-- | Define a higher-order function. Similar to 'smtFunction', but when we have a higher-order argument. Note that
-- the higher-order argument cannot have free variables. Also, if the function is recursive, you should call
-- the first argument of the defining function, which SBV uses to tie the recursive knot. (Note that recursive
-- functions defined via 'smtFunction' don't have this latter requirement as they can figure out the recursion
-- automatically. Higher-order functions, unfortunately, can't do this: They firstify their high-order argument,
-- giving the whole function a unique name; captured via the call to the recursive definition.)
smtHOFunction :: forall a b f.
( SMTDefinable (a -> SBV b)
, Lambda Symbolic f
, Lambda Symbolic (a -> SBV b)
, HasKind b
, HasKind f
, Typeable a
, Typeable b
, Typeable f
) => String -- prefix to use
-> f -- The higher-order argument. We're very generic here!
-> (a -> SBV b) -- The ho-function we're modeling
-> a -> SBV b -- The resulting function, that can be used as is, and will be rendered in SMTLib without unfolding
smtHOFunction nm f hof arg = SBV $ SVal (kindOf (Proxy @(SBV b))) $ Right $ cache r
where r st = do SMTLambda lam <- lambdaStr st HigherOrderArg (resKindOf (kindOf (Proxy @f))) f
let uniqLen = firstifyUniqueLen $ stCfg st
uniq = take uniqLen (BC.unpack (B.encode (hash (BC.pack (unwords (words lam))))))
sbvToSV st (smtFunction (atProxy (Proxy @f) nm <> "_" <> uniq) hof arg)
-- we get the functions as arrays here, so chase to find the result
resKindOf (KArray _ k) = resKindOf k
resKindOf k = k
{- HLint ignore module "Reduce duplication" -}
{- HLint ignore module "Eta reduce" -}
{- HLint ignore module "Avoid NonEmpty.unzip" -}
{- HLint ignore module "Redundant id" -}