what4-1.2: src/What4/Expr/Builder.hs
{-|
Module : What4.Expr.Builder
Description : Main definitions of the What4 expression representation
Copyright : (c) Galois Inc, 2015-2020
License : BSD3
Maintainer : jhendrix@galois.com
This module defines the canonical implementation of the solver interface
from "What4.Interface". Type @'ExprBuilder' t st@ is
an instance of the classes 'IsExprBuilder' and 'IsSymExprBuilder'.
Notes regarding concurrency: The expression builder datatype contains
a number of mutable storage locations. These are designed so they
may reasonably be used in a multithreaded context. In particular,
nonce values are generated atomically, and other IORefs used in this
module are modified or written atomically, so modifications should
propagate in the expected sequentially-consistent ways. Of course,
threads may still clobber state others have set (e.g., the current
program location) so the potential for truly multithreaded use is
somewhat limited.
-}
{-# LANGUAGE CPP #-}
{-# LANGUAGE BangPatterns #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE EmptyCase #-}
{-# LANGUAGE EmptyDataDecls #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE ImplicitParams #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE LambdaCase #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE MultiWayIf #-}
{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE PatternGuards #-}
{-# LANGUAGE PatternSynonyms #-}
{-# LANGUAGE PolyKinds #-}
{-# LANGUAGE RankNTypes #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TupleSections #-}
{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE TypeSynonymInstances #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE ViewPatterns #-}
module What4.Expr.Builder
( -- * ExprBuilder
ExprBuilder
, newExprBuilder
, getSymbolVarBimap
, sbMakeExpr
, sbNonceExpr
, curProgramLoc
, sbUnaryThreshold
, sbCacheStartSize
, sbBVDomainRangeLimit
, sbUserState
, exprCounter
, startCaching
, stopCaching
-- * Specialized representations
, bvUnary
, intSum
, realSum
, bvSum
, scalarMul
-- * configuration options
, unaryThresholdOption
, bvdomainRangeLimitOption
, cacheStartSizeOption
, cacheTerms
-- * Expr
, Expr(..)
, asApp
, asNonceApp
, iteSize
, exprLoc
, ppExpr
, ppExprTop
, exprMaybeId
, asConjunction
, asDisjunction
, Polarity(..)
, BM.negatePolarity
-- ** AppExpr
, AppExpr
, appExprId
, appExprLoc
, appExprApp
-- ** NonceAppExpr
, NonceAppExpr
, nonceExprId
, nonceExprLoc
, nonceExprApp
-- ** Type abbreviations
, BoolExpr
, IntegerExpr
, RealExpr
, FloatExpr
, BVExpr
, CplxExpr
, StringExpr
-- * App
, App(..)
, traverseApp
, appType
-- * NonceApp
, NonceApp(..)
, nonceAppType
-- * Bound Variable information
, ExprBoundVar
, bvarId
, bvarLoc
, bvarName
, bvarType
, bvarKind
, bvarAbstractValue
, VarKind(..)
, boundVars
, ppBoundVar
, evalBoundVars
-- * Symbolic Function
, ExprSymFn(..)
, SymFnInfo(..)
, symFnArgTypes
, symFnReturnType
-- * SymbolVarBimap
, SymbolVarBimap
, SymbolBinding(..)
, emptySymbolVarBimap
, lookupBindingOfSymbol
, lookupSymbolOfBinding
-- * IdxCache
, IdxCache
, newIdxCache
, lookupIdx
, lookupIdxValue
, insertIdxValue
, deleteIdxValue
, clearIdxCache
, idxCacheEval
, idxCacheEval'
-- * Flags
, type FloatMode
, FloatModeRepr(..)
, FloatIEEE
, FloatUninterpreted
, FloatReal
, Flags
-- * BV Or Set
, BVOrSet
, bvOrToList
, bvOrSingleton
, bvOrInsert
, bvOrUnion
, bvOrAbs
, traverseBVOrSet
-- * Re-exports
, SymExpr
, What4.Interface.bvWidth
, What4.Interface.exprType
, What4.Interface.IndexLit(..)
, What4.Interface.ArrayResultWrapper(..)
) where
import qualified Control.Exception as Ex
import Control.Lens hiding (asIndex, (:>), Empty)
import Control.Monad
import Control.Monad.IO.Class
import Control.Monad.ST
import Control.Monad.Trans.Writer.Strict (writer, runWriter)
import qualified Data.BitVector.Sized as BV
import Data.Bimap (Bimap)
import qualified Data.Bimap as Bimap
import qualified Data.Binary.IEEE754 as IEEE754
import Data.Hashable
import Data.IORef
import Data.Kind
import Data.List.NonEmpty (NonEmpty(..))
import Data.Map.Strict (Map)
import qualified Data.Map.Strict as Map
import Data.Maybe
import Data.Monoid (Any(..))
import Data.Parameterized.Classes
import Data.Parameterized.Context as Ctx
import qualified Data.Parameterized.HashTable as PH
import qualified Data.Parameterized.Map as PM
import Data.Parameterized.NatRepr
import Data.Parameterized.Nonce
import Data.Parameterized.Some
import Data.Parameterized.TraversableFC
import Data.Ratio (numerator, denominator)
import Data.Set (Set)
import qualified Data.Set as Set
import qualified LibBF as BF
import What4.BaseTypes
import What4.Concrete
import qualified What4.Config as CFG
import What4.Interface
import What4.InterpretedFloatingPoint
import What4.ProgramLoc
import qualified What4.SemiRing as SR
import What4.Symbol
import What4.Expr.App
import qualified What4.Expr.ArrayUpdateMap as AUM
import What4.Expr.BoolMap (BoolMap, Polarity(..), BoolMapView(..))
import qualified What4.Expr.BoolMap as BM
import What4.Expr.MATLAB
import What4.Expr.WeightedSum (WeightedSum, SemiRingProduct)
import qualified What4.Expr.WeightedSum as WSum
import qualified What4.Expr.StringSeq as SSeq
import What4.Expr.UnaryBV (UnaryBV)
import qualified What4.Expr.UnaryBV as UnaryBV
import What4.Utils.AbstractDomains
import What4.Utils.Arithmetic
import qualified What4.Utils.BVDomain as BVD
import What4.Utils.Complex
import What4.Utils.FloatHelpers
import What4.Utils.StringLiteral
------------------------------------------------------------------------
-- Utilities
toDouble :: Rational -> Double
toDouble = fromRational
cachedEval :: (HashableF k, TestEquality k)
=> PH.HashTable RealWorld k a
-> k tp
-> IO (a tp)
-> IO (a tp)
cachedEval tbl k action = do
mr <- stToIO $ PH.lookup tbl k
case mr of
Just r -> return r
Nothing -> do
r <- action
seq r $ do
stToIO $ PH.insert tbl k r
return r
------------------------------------------------------------------------
-- SymbolVarBimap
-- | A bijective map between vars and their canonical name for printing
-- purposes.
-- Parameter @t@ is a phantom type brand used to track nonces.
newtype SymbolVarBimap t = SymbolVarBimap (Bimap SolverSymbol (SymbolBinding t))
-- | This describes what a given SolverSymbol is associated with.
-- Parameter @t@ is a phantom type brand used to track nonces.
data SymbolBinding t
= forall tp . VarSymbolBinding !(ExprBoundVar t tp)
-- ^ Solver
| forall args ret . FnSymbolBinding !(ExprSymFn t args ret)
instance Eq (SymbolBinding t) where
VarSymbolBinding x == VarSymbolBinding y = isJust (testEquality x y)
FnSymbolBinding x == FnSymbolBinding y = isJust (testEquality (symFnId x) (symFnId y))
_ == _ = False
instance Ord (SymbolBinding t) where
compare (VarSymbolBinding x) (VarSymbolBinding y) =
toOrdering (compareF x y)
compare VarSymbolBinding{} _ = LT
compare _ VarSymbolBinding{} = GT
compare (FnSymbolBinding x) (FnSymbolBinding y) =
toOrdering (compareF (symFnId x) (symFnId y))
-- | Empty symbol var bimap
emptySymbolVarBimap :: SymbolVarBimap t
emptySymbolVarBimap = SymbolVarBimap Bimap.empty
lookupBindingOfSymbol :: SolverSymbol -> SymbolVarBimap t -> Maybe (SymbolBinding t)
lookupBindingOfSymbol s (SymbolVarBimap m) = Bimap.lookup s m
lookupSymbolOfBinding :: SymbolBinding t -> SymbolVarBimap t -> Maybe SolverSymbol
lookupSymbolOfBinding b (SymbolVarBimap m) = Bimap.lookupR b m
------------------------------------------------------------------------
-- MatlabSolverFn
-- Parameter @t@ is a phantom type brand used to track nonces.
data MatlabFnWrapper t c where
MatlabFnWrapper :: !(MatlabSolverFn (Expr t) a r) -> MatlabFnWrapper t (a::> r)
instance TestEquality (MatlabFnWrapper t) where
testEquality (MatlabFnWrapper f) (MatlabFnWrapper g) = do
Refl <- testSolverFnEq f g
return Refl
instance HashableF (MatlabFnWrapper t) where
hashWithSaltF s (MatlabFnWrapper f) = hashWithSalt s f
data ExprSymFnWrapper t c
= forall a r . (c ~ (a ::> r)) => ExprSymFnWrapper (ExprSymFn t a r)
data SomeSymFn sym = forall args ret . SomeSymFn (SymFn sym args ret)
------------------------------------------------------------------------
-- ExprBuilder
-- | Mode flag for how floating-point values should be interpreted.
data FloatMode where
FloatIEEE :: FloatMode
FloatUninterpreted :: FloatMode
FloatReal :: FloatMode
type FloatIEEE = 'FloatIEEE
type FloatUninterpreted = 'FloatUninterpreted
type FloatReal = 'FloatReal
data Flags (fi :: FloatMode)
data FloatModeRepr :: FloatMode -> Type where
FloatIEEERepr :: FloatModeRepr FloatIEEE
FloatUninterpretedRepr :: FloatModeRepr FloatUninterpreted
FloatRealRepr :: FloatModeRepr FloatReal
instance Show (FloatModeRepr fm) where
showsPrec _ FloatIEEERepr = showString "FloatIEEE"
showsPrec _ FloatUninterpretedRepr = showString "FloatUninterpreted"
showsPrec _ FloatRealRepr = showString "FloatReal"
instance ShowF FloatModeRepr
instance KnownRepr FloatModeRepr FloatIEEE where knownRepr = FloatIEEERepr
instance KnownRepr FloatModeRepr FloatUninterpreted where knownRepr = FloatUninterpretedRepr
instance KnownRepr FloatModeRepr FloatReal where knownRepr = FloatRealRepr
instance TestEquality FloatModeRepr where
testEquality FloatIEEERepr FloatIEEERepr = return Refl
testEquality FloatUninterpretedRepr FloatUninterpretedRepr = return Refl
testEquality FloatRealRepr FloatRealRepr = return Refl
testEquality _ _ = Nothing
-- | Cache for storing dag terms.
-- Parameter @t@ is a phantom type brand used to track nonces.
data ExprBuilder t (st :: Type -> Type) (fs :: Type)
= forall fm. (fs ~ (Flags fm)) =>
SB { sbTrue :: !(BoolExpr t)
, sbFalse :: !(BoolExpr t)
-- | Constant zero.
, sbZero :: !(RealExpr t)
-- | Configuration object for this symbolic backend
, sbConfiguration :: !CFG.Config
-- | Flag used to tell the backend whether to evaluate
-- ground rational values as double precision floats when
-- a function cannot be evaluated as a rational.
, sbFloatReduce :: !Bool
-- | The maximum number of distinct values a term may have and use the
-- unary representation.
, sbUnaryThreshold :: !(CFG.OptionSetting BaseIntegerType)
-- | The maximum number of distinct ranges in a BVDomain expression.
, sbBVDomainRangeLimit :: !(CFG.OptionSetting BaseIntegerType)
-- | The starting size when building a new cache
, sbCacheStartSize :: !(CFG.OptionSetting BaseIntegerType)
-- | Counter to generate new unique identifiers for elements and functions.
, exprCounter :: !(NonceGenerator IO t)
-- | Reference to current allocator for expressions.
, curAllocator :: !(IORef (ExprAllocator t))
-- | Number of times an 'Expr' for a non-linear operation has been
-- created.
, sbNonLinearOps :: !(IORef Integer)
-- | The current program location
, sbProgramLoc :: !(IORef ProgramLoc)
-- | Additional state maintained by the state manager
, sbUserState :: !(st t)
, sbVarBindings :: !(IORef (SymbolVarBimap t))
, sbUninterpFnCache :: !(IORef (Map (SolverSymbol, Some (Ctx.Assignment BaseTypeRepr)) (SomeSymFn (ExprBuilder t st fs))))
-- | Cache for Matlab functions
, sbMatlabFnCache
:: !(PH.HashTable RealWorld (MatlabFnWrapper t) (ExprSymFnWrapper t))
, sbSolverLogger
:: !(IORef (Maybe (SolverEvent -> IO ())))
-- | Flag dictating how floating-point values/operations are translated
-- when passed to the solver.
, sbFloatMode :: !(FloatModeRepr fm)
}
type instance SymFn (ExprBuilder t st fs) = ExprSymFn t
type instance SymExpr (ExprBuilder t st fs) = Expr t
type instance BoundVar (ExprBuilder t st fs) = ExprBoundVar t
type instance SymAnnotation (ExprBuilder t st fs) = Nonce t
------------------------------------------------------------------------
-- | ExprAllocator provides an interface for creating expressions from
-- an applications.
-- Parameter @t@ is a phantom type brand used to track nonces.
data ExprAllocator t
= ExprAllocator { appExpr :: forall tp
. ProgramLoc
-> App (Expr t) tp
-> AbstractValue tp
-> IO (Expr t tp)
, nonceExpr :: forall tp
. ProgramLoc
-> NonceApp t (Expr t) tp
-> AbstractValue tp
-> IO (Expr t tp)
}
------------------------------------------------------------------------
-- Uncached storage
-- | Create a new storage that does not do hash consing.
newStorage :: NonceGenerator IO t -> IO (ExprAllocator t)
newStorage g = do
return $! ExprAllocator { appExpr = uncachedExprFn g
, nonceExpr = uncachedNonceExpr g
}
uncachedExprFn :: NonceGenerator IO t
-> ProgramLoc
-> App (Expr t) tp
-> AbstractValue tp
-> IO (Expr t tp)
uncachedExprFn g pc a v = do
n <- freshNonce g
return $! mkExpr n pc a v
uncachedNonceExpr :: NonceGenerator IO t
-> ProgramLoc
-> NonceApp t (Expr t) tp
-> AbstractValue tp
-> IO (Expr t tp)
uncachedNonceExpr g pc p v = do
n <- freshNonce g
return $! NonceAppExpr $ NonceAppExprCtor { nonceExprId = n
, nonceExprLoc = pc
, nonceExprApp = p
, nonceExprAbsValue = v
}
------------------------------------------------------------------------
-- Cached storage
cachedNonceExpr :: NonceGenerator IO t
-> PH.HashTable RealWorld (NonceApp t (Expr t)) (Expr t)
-> ProgramLoc
-> NonceApp t (Expr t) tp
-> AbstractValue tp
-> IO (Expr t tp)
cachedNonceExpr g h pc p v = do
me <- stToIO $ PH.lookup h p
case me of
Just e -> return e
Nothing -> do
n <- freshNonce g
let e = NonceAppExpr $ NonceAppExprCtor { nonceExprId = n
, nonceExprLoc = pc
, nonceExprApp = p
, nonceExprAbsValue = v
}
seq e $ stToIO $ PH.insert h p e
return $! e
cachedAppExpr :: forall t tp
. NonceGenerator IO t
-> PH.HashTable RealWorld (App (Expr t)) (Expr t)
-> ProgramLoc
-> App (Expr t) tp
-> AbstractValue tp
-> IO (Expr t tp)
cachedAppExpr g h pc a v = do
me <- stToIO $ PH.lookup h a
case me of
Just e -> return e
Nothing -> do
n <- freshNonce g
let e = mkExpr n pc a v
seq e $ stToIO $ PH.insert h a e
return e
-- | Create a storage that does hash consing.
newCachedStorage :: forall t
. NonceGenerator IO t
-> Int
-> IO (ExprAllocator t)
newCachedStorage g sz = stToIO $ do
appCache <- PH.newSized sz
predCache <- PH.newSized sz
return $ ExprAllocator { appExpr = cachedAppExpr g appCache
, nonceExpr = cachedNonceExpr g predCache
}
------------------------------------------------------------------------
-- IdxCache
-- | An IdxCache is used to map expressions with type @Expr t tp@ to
-- values with a corresponding type @f tp@. It is a mutable map using
-- an 'IO' hash table. Parameter @t@ is a phantom type brand used to
-- track nonces.
newtype IdxCache t (f :: BaseType -> Type)
= IdxCache { cMap :: IORef (PM.MapF (Nonce t) f) }
-- | Create a new IdxCache
newIdxCache :: MonadIO m => m (IdxCache t f)
newIdxCache = liftIO $ IdxCache <$> newIORef PM.empty
{-# INLINE lookupIdxValue #-}
-- | Return the value associated to the expr in the index.
lookupIdxValue :: MonadIO m => IdxCache t f -> Expr t tp -> m (Maybe (f tp))
lookupIdxValue _ SemiRingLiteral{} = return Nothing
lookupIdxValue _ StringExpr{} = return Nothing
lookupIdxValue _ BoolExpr{} = return Nothing
lookupIdxValue _ FloatExpr{} = return Nothing
lookupIdxValue c (NonceAppExpr e) = lookupIdx c (nonceExprId e)
lookupIdxValue c (AppExpr e) = lookupIdx c (appExprId e)
lookupIdxValue c (BoundVarExpr i) = lookupIdx c (bvarId i)
{-# INLINE lookupIdx #-}
lookupIdx :: (MonadIO m) => IdxCache t f -> Nonce t tp -> m (Maybe (f tp))
lookupIdx c n = liftIO $ PM.lookup n <$> readIORef (cMap c)
{-# INLINE insertIdxValue #-}
-- | Bind the value to the given expr in the index.
insertIdxValue :: MonadIO m => IdxCache t f -> Nonce t tp -> f tp -> m ()
insertIdxValue c e v = seq v $ liftIO $ atomicModifyIORef' (cMap c) $ (\m -> (PM.insert e v m, ()))
{-# INLINE deleteIdxValue #-}
-- | Remove a value from the IdxCache
deleteIdxValue :: MonadIO m => IdxCache t f -> Nonce t (tp :: BaseType) -> m ()
deleteIdxValue c e = liftIO $ atomicModifyIORef' (cMap c) $ (\m -> (PM.delete e m, ()))
-- | Remove all values from the IdxCache
clearIdxCache :: MonadIO m => IdxCache t f -> m ()
clearIdxCache c = liftIO $ atomicWriteIORef (cMap c) PM.empty
exprMaybeId :: Expr t tp -> Maybe (Nonce t tp)
exprMaybeId SemiRingLiteral{} = Nothing
exprMaybeId StringExpr{} = Nothing
exprMaybeId BoolExpr{} = Nothing
exprMaybeId FloatExpr{} = Nothing
exprMaybeId (NonceAppExpr e) = Just $! nonceExprId e
exprMaybeId (AppExpr e) = Just $! appExprId e
exprMaybeId (BoundVarExpr e) = Just $! bvarId e
-- | Implements a cached evaluated using the given element. Given an element
-- this function returns the value of the element if bound, and otherwise
-- calls the evaluation function, stores the result in the cache, and
-- returns the value.
{-# INLINE idxCacheEval #-}
idxCacheEval :: (MonadIO m)
=> IdxCache t f
-> Expr t tp
-> m (f tp)
-> m (f tp)
idxCacheEval c e m = do
case exprMaybeId e of
Nothing -> m
Just n -> idxCacheEval' c n m
-- | Implements a cached evaluated using the given element. Given an element
-- this function returns the value of the element if bound, and otherwise
-- calls the evaluation function, stores the result in the cache, and
-- returns the value.
{-# INLINE idxCacheEval' #-}
idxCacheEval' :: (MonadIO m)
=> IdxCache t f
-> Nonce t tp
-> m (f tp)
-> m (f tp)
idxCacheEval' c n m = do
mr <- lookupIdx c n
case mr of
Just r -> return r
Nothing -> do
r <- m
insertIdxValue c n r
return r
------------------------------------------------------------------------
-- ExprBuilder operations
curProgramLoc :: ExprBuilder t st fs -> IO ProgramLoc
curProgramLoc sym = readIORef (sbProgramLoc sym)
-- | Create an element from a nonce app.
sbNonceExpr :: ExprBuilder t st fs
-> NonceApp t (Expr t) tp
-> IO (Expr t tp)
sbNonceExpr sym a = do
s <- readIORef (curAllocator sym)
pc <- curProgramLoc sym
nonceExpr s pc a (quantAbsEval exprAbsValue a)
semiRingLit :: ExprBuilder t st fs
-> SR.SemiRingRepr sr
-> SR.Coefficient sr
-> IO (Expr t (SR.SemiRingBase sr))
semiRingLit sb sr x = do
l <- curProgramLoc sb
return $! SemiRingLiteral sr x l
sbMakeExpr :: ExprBuilder t st fs -> App (Expr t) tp -> IO (Expr t tp)
sbMakeExpr sym a = do
s <- readIORef (curAllocator sym)
pc <- curProgramLoc sym
let v = abstractEval exprAbsValue a
when (isNonLinearApp a) $
atomicModifyIORef' (sbNonLinearOps sym) (\n -> (n+1,()))
case appType a of
-- Check if abstract interpretation concludes this is a constant.
BaseBoolRepr | Just b <- v -> return $ backendPred sym b
BaseIntegerRepr | Just c <- asSingleRange v -> intLit sym c
BaseRealRepr | Just c <- asSingleRange (ravRange v) -> realLit sym c
BaseBVRepr w | Just x <- BVD.asSingleton v -> bvLit sym w (BV.mkBV w x)
_ -> appExpr s pc a v
-- | Update the binding to point to the current variable.
updateVarBinding :: ExprBuilder t st fs
-> SolverSymbol
-> SymbolBinding t
-> IO ()
updateVarBinding sym nm v
| nm == emptySymbol = return ()
| otherwise =
atomicModifyIORef' (sbVarBindings sym) $ (\x -> v `seq` (ins nm v x, ()))
where ins n x (SymbolVarBimap m) = SymbolVarBimap (Bimap.insert n x m)
-- | Creates a new bound var.
sbMakeBoundVar :: ExprBuilder t st fs
-> SolverSymbol
-> BaseTypeRepr tp
-> VarKind
-> Maybe (AbstractValue tp)
-> IO (ExprBoundVar t tp)
sbMakeBoundVar sym nm tp k absVal = do
n <- sbFreshIndex sym
pc <- curProgramLoc sym
return $! BVar { bvarId = n
, bvarLoc = pc
, bvarName = nm
, bvarType = tp
, bvarKind = k
, bvarAbstractValue = absVal
}
-- | Create fresh index
sbFreshIndex :: ExprBuilder t st fs -> IO (Nonce t (tp::BaseType))
sbFreshIndex sb = freshNonce (exprCounter sb)
sbFreshSymFnNonce :: ExprBuilder t st fs -> IO (Nonce t (ctx:: Ctx BaseType))
sbFreshSymFnNonce sb = freshNonce (exprCounter sb)
------------------------------------------------------------------------
-- Configuration option for controlling the maximum number of value a unary
-- threshold may have.
-- | Maximum number of values in unary bitvector encoding.
--
-- This option is named \"backend.unary_threshold\"
unaryThresholdOption :: CFG.ConfigOption BaseIntegerType
unaryThresholdOption = CFG.configOption BaseIntegerRepr "backend.unary_threshold"
-- | The configuration option for setting the maximum number of
-- values a unary threshold may have.
unaryThresholdDesc :: CFG.ConfigDesc
unaryThresholdDesc = CFG.mkOpt unaryThresholdOption sty help (Just (ConcreteInteger 0))
where sty = CFG.integerWithMinOptSty (CFG.Inclusive 0)
help = Just "Maximum number of values in unary bitvector encoding."
------------------------------------------------------------------------
-- Configuration option for controlling how many disjoint ranges
-- should be allowed in bitvector domains.
-- | Maximum number of ranges in bitvector abstract domains.
--
-- This option is named \"backend.bvdomain_range_limit\"
bvdomainRangeLimitOption :: CFG.ConfigOption BaseIntegerType
bvdomainRangeLimitOption = CFG.configOption BaseIntegerRepr "backend.bvdomain_range_limit"
bvdomainRangeLimitDesc :: CFG.ConfigDesc
bvdomainRangeLimitDesc = CFG.mkOpt bvdomainRangeLimitOption sty help (Just (ConcreteInteger 2))
where sty = CFG.integerWithMinOptSty (CFG.Inclusive 0)
help = Just "Maximum number of ranges in bitvector domains."
------------------------------------------------------------------------
-- Cache start size
-- | Starting size for element cache when caching is enabled.
--
-- This option is named \"backend.cache_start_size\"
cacheStartSizeOption :: CFG.ConfigOption BaseIntegerType
cacheStartSizeOption = CFG.configOption BaseIntegerRepr "backend.cache_start_size"
-- | The configuration option for setting the size of the initial hash set
-- used by simple builder
cacheStartSizeDesc :: CFG.ConfigDesc
cacheStartSizeDesc = CFG.mkOpt cacheStartSizeOption sty help (Just (ConcreteInteger 100000))
where sty = CFG.integerWithMinOptSty (CFG.Inclusive 0)
help = Just "Starting size for element cache"
------------------------------------------------------------------------
-- Cache terms
-- | Indicates if we should cache terms. When enabled, hash-consing
-- is used to find and deduplicate common subexpressions.
--
-- This option is named \"use_cache\"
cacheTerms :: CFG.ConfigOption BaseBoolType
cacheTerms = CFG.configOption BaseBoolRepr "use_cache"
cacheOptStyle ::
NonceGenerator IO t ->
IORef (ExprAllocator t) ->
CFG.OptionSetting BaseIntegerType ->
CFG.OptionStyle BaseBoolType
cacheOptStyle gen storageRef szSetting =
CFG.boolOptSty & CFG.set_opt_onset
(\mb b -> f (fmap fromConcreteBool mb) (fromConcreteBool b) >> return CFG.optOK)
where
f :: Maybe Bool -> Bool -> IO ()
f mb b | mb /= Just b = if b then start else stop
| otherwise = return ()
stop = do s <- newStorage gen
atomicWriteIORef storageRef s
start = do sz <- CFG.getOpt szSetting
s <- newCachedStorage gen (fromInteger sz)
atomicWriteIORef storageRef s
cacheOptDesc ::
NonceGenerator IO t ->
IORef (ExprAllocator t) ->
CFG.OptionSetting BaseIntegerType ->
CFG.ConfigDesc
cacheOptDesc gen storageRef szSetting =
CFG.mkOpt
cacheTerms
(cacheOptStyle gen storageRef szSetting)
(Just "Use hash-consing during term construction")
(Just (ConcreteBool False))
newExprBuilder ::
FloatModeRepr fm
-- ^ Float interpretation mode (i.e., how are floats translated for the solver).
-> st t
-- ^ Initial state for the expression builder
-> NonceGenerator IO t
-- ^ Nonce generator for names
-> IO (ExprBuilder t st (Flags fm))
newExprBuilder floatMode st gen = do
es <- newStorage gen
let t = BoolExpr True initializationLoc
let f = BoolExpr False initializationLoc
let z = SemiRingLiteral SR.SemiRingRealRepr 0 initializationLoc
loc_ref <- newIORef initializationLoc
storage_ref <- newIORef es
bindings_ref <- newIORef emptySymbolVarBimap
uninterp_fn_cache_ref <- newIORef Map.empty
matlabFnCache <- stToIO $ PH.new
loggerRef <- newIORef Nothing
-- Set up configuration options
cfg <- CFG.initialConfig 0
[ unaryThresholdDesc
, bvdomainRangeLimitDesc
, cacheStartSizeDesc
]
unarySetting <- CFG.getOptionSetting unaryThresholdOption cfg
domainRangeSetting <- CFG.getOptionSetting bvdomainRangeLimitOption cfg
cacheStartSetting <- CFG.getOptionSetting cacheStartSizeOption cfg
CFG.extendConfig [cacheOptDesc gen storage_ref cacheStartSetting] cfg
nonLinearOps <- newIORef 0
return $! SB { sbTrue = t
, sbFalse = f
, sbZero = z
, sbConfiguration = cfg
, sbFloatReduce = True
, sbUnaryThreshold = unarySetting
, sbBVDomainRangeLimit = domainRangeSetting
, sbCacheStartSize = cacheStartSetting
, sbProgramLoc = loc_ref
, exprCounter = gen
, curAllocator = storage_ref
, sbNonLinearOps = nonLinearOps
, sbUserState = st
, sbVarBindings = bindings_ref
, sbUninterpFnCache = uninterp_fn_cache_ref
, sbMatlabFnCache = matlabFnCache
, sbSolverLogger = loggerRef
, sbFloatMode = floatMode
}
-- | Get current variable bindings.
getSymbolVarBimap :: ExprBuilder t st fs -> IO (SymbolVarBimap t)
getSymbolVarBimap sym = readIORef (sbVarBindings sym)
-- | Stop caching applications in backend.
stopCaching :: ExprBuilder t st fs -> IO ()
stopCaching sb = do
s <- newStorage (exprCounter sb)
atomicWriteIORef (curAllocator sb) s
-- | Restart caching applications in backend (clears cache if it is currently caching).
startCaching :: ExprBuilder t st fs -> IO ()
startCaching sb = do
sz <- CFG.getOpt (sbCacheStartSize sb)
s <- newCachedStorage (exprCounter sb) (fromInteger sz)
atomicWriteIORef (curAllocator sb) s
bvBinDivOp :: (1 <= w)
=> (NatRepr w -> BV.BV w -> BV.BV w -> BV.BV w)
-> (NatRepr w -> BVExpr t w -> BVExpr t w -> App (Expr t) (BaseBVType w))
-> ExprBuilder t st fs
-> BVExpr t w
-> BVExpr t w
-> IO (BVExpr t w)
bvBinDivOp f c sb x y = do
let w = bvWidth x
case (asBV x, asBV y) of
(Just i, Just j) | j /= BV.zero w -> bvLit sb w $ f w i j
_ -> sbMakeExpr sb $ c w x y
asConcreteIndices :: IsExpr e
=> Ctx.Assignment e ctx
-> Maybe (Ctx.Assignment IndexLit ctx)
asConcreteIndices = traverseFC f
where f :: IsExpr e => e tp -> Maybe (IndexLit tp)
f x =
case exprType x of
BaseIntegerRepr -> IntIndexLit <$> asInteger x
BaseBVRepr w -> BVIndexLit w <$> asBV x
_ -> Nothing
symbolicIndices :: forall sym ctx
. IsExprBuilder sym
=> sym
-> Ctx.Assignment IndexLit ctx
-> IO (Ctx.Assignment (SymExpr sym) ctx)
symbolicIndices sym = traverseFC f
where f :: IndexLit tp -> IO (SymExpr sym tp)
f (IntIndexLit n) = intLit sym n
f (BVIndexLit w i) = bvLit sym w i
-- | This evaluate a symbolic function against a set of arguments.
betaReduce :: ExprBuilder t st fs
-> ExprSymFn t args ret
-> Ctx.Assignment (Expr t) args
-> IO (Expr t ret)
betaReduce sym f args =
case symFnInfo f of
UninterpFnInfo{} ->
sbNonceExpr sym $! FnApp f args
DefinedFnInfo bound_vars e _ -> do
evalBoundVars sym e bound_vars args
MatlabSolverFnInfo fn_id _ _ -> do
evalMatlabSolverFn fn_id sym args
-- | This runs one action, and if it returns a value different from the input,
-- then it runs the second. Otherwise it returns the result value passed in.
--
-- It is used when an action may modify a value, and we only want to run a
-- second action if the value changed.
runIfChanged :: Eq e
=> e
-> (e -> IO e) -- ^ First action to run
-> r -- ^ Result if no change.
-> (e -> IO r) -- ^ Second action to run
-> IO r
runIfChanged x f unChanged onChange = do
y <- f x
if x == y then
return unChanged
else
onChange y
-- | This adds a binding from the variable to itself in the hashtable
-- to ensure it can't be rebound.
recordBoundVar :: PH.HashTable RealWorld (Expr t) (Expr t)
-> ExprBoundVar t tp
-> IO ()
recordBoundVar tbl v = do
let e = BoundVarExpr v
mr <- stToIO $ PH.lookup tbl e
case mr of
Just r -> do
when (r /= e) $ do
fail $ "Simulator internal error; do not support rebinding variables."
Nothing -> do
-- Bind variable to itself to ensure we catch when it is used again.
stToIO $ PH.insert tbl e e
-- | The CachedSymFn is used during evaluation to store the results of reducing
-- the definitions of symbolic functions.
--
-- For each function it stores a pair containing a 'Bool' that is true if the
-- function changed as a result of evaluating it, and the reduced function
-- after evaluation.
--
-- The second arguments contains the arguments with the return type appended.
data CachedSymFn t c
= forall a r
. (c ~ (a ::> r))
=> CachedSymFn Bool (ExprSymFn t a r)
-- | Data structure used for caching evaluation.
data EvalHashTables t
= EvalHashTables { exprTable :: !(PH.HashTable RealWorld (Expr t) (Expr t))
, fnTable :: !(PH.HashTable RealWorld (Nonce t) (CachedSymFn t))
}
-- | Evaluate a simple function.
--
-- This returns whether the function changed as a Boolean and the function itself.
evalSimpleFn :: EvalHashTables t
-> ExprBuilder t st fs
-> ExprSymFn t idx ret
-> IO (Bool,ExprSymFn t idx ret)
evalSimpleFn tbl sym f =
case symFnInfo f of
UninterpFnInfo{} -> return (False, f)
DefinedFnInfo vars e evalFn -> do
let n = symFnId f
let nm = symFnName f
CachedSymFn changed f' <-
cachedEval (fnTable tbl) n $ do
traverseFC_ (recordBoundVar (exprTable tbl)) vars
e' <- evalBoundVars' tbl sym e
if e == e' then
return $! CachedSymFn False f
else
CachedSymFn True <$> definedFn sym nm vars e' evalFn
return (changed, f')
MatlabSolverFnInfo{} -> return (False, f)
evalBoundVars' :: forall t st fs ret
. EvalHashTables t
-> ExprBuilder t st fs
-> Expr t ret
-> IO (Expr t ret)
evalBoundVars' tbls sym e0 =
case e0 of
SemiRingLiteral{} -> return e0
StringExpr{} -> return e0
BoolExpr{} -> return e0
FloatExpr{} -> return e0
AppExpr ae -> cachedEval (exprTable tbls) e0 $ do
let a = appExprApp ae
a' <- traverseApp (evalBoundVars' tbls sym) a
if a == a' then
return e0
else
reduceApp sym bvUnary a'
NonceAppExpr ae -> cachedEval (exprTable tbls) e0 $ do
case nonceExprApp ae of
Annotation tpr n a -> do
a' <- evalBoundVars' tbls sym a
if a == a' then
return e0
else
sbNonceExpr sym $ Annotation tpr n a'
Forall v e -> do
recordBoundVar (exprTable tbls) v
-- Regenerate forallPred if e is changed by evaluation.
runIfChanged e (evalBoundVars' tbls sym) e0 (forallPred sym v)
Exists v e -> do
recordBoundVar (exprTable tbls) v
-- Regenerate forallPred if e is changed by evaluation.
runIfChanged e (evalBoundVars' tbls sym) e0 (existsPred sym v)
ArrayFromFn f -> do
(changed, f') <- evalSimpleFn tbls sym f
if not changed then
return e0
else
arrayFromFn sym f'
MapOverArrays f _ args -> do
(changed, f') <- evalSimpleFn tbls sym f
let evalWrapper :: ArrayResultWrapper (Expr t) (idx ::> itp) utp
-> IO (ArrayResultWrapper (Expr t) (idx ::> itp) utp)
evalWrapper (ArrayResultWrapper a) =
ArrayResultWrapper <$> evalBoundVars' tbls sym a
args' <- traverseFC evalWrapper args
if not changed && args == args' then
return e0
else
arrayMap sym f' args'
ArrayTrueOnEntries f a -> do
(changed, f') <- evalSimpleFn tbls sym f
a' <- evalBoundVars' tbls sym a
if not changed && a == a' then
return e0
else
arrayTrueOnEntries sym f' a'
FnApp f a -> do
(changed, f') <- evalSimpleFn tbls sym f
a' <- traverseFC (evalBoundVars' tbls sym) a
if not changed && a == a' then
return e0
else
applySymFn sym f' a'
BoundVarExpr{} -> cachedEval (exprTable tbls) e0 $ return e0
initHashTable :: (HashableF key, TestEquality key)
=> Ctx.Assignment key args
-> Ctx.Assignment val args
-> ST s (PH.HashTable s key val)
initHashTable keys vals = do
let sz = Ctx.size keys
tbl <- PH.newSized (Ctx.sizeInt sz)
Ctx.forIndexM sz $ \i -> do
PH.insert tbl (keys Ctx.! i) (vals Ctx.! i)
return tbl
-- | This evaluates the term with the given bound variables rebound to
-- the given arguments.
--
-- The algorithm works by traversing the subterms in the term in a bottom-up
-- fashion while using a hash-table to memoize results for shared subterms. The
-- hash-table is pre-populated so that the bound variables map to the element,
-- so we do not need any extra map lookup when checking to see if a variable is
-- bound.
--
-- NOTE: This function assumes that variables in the substitution are not
-- themselves bound in the term (e.g. in a function definition or quantifier).
-- If this is not respected, then 'evalBoundVars' will call 'fail' with an
-- error message.
evalBoundVars :: ExprBuilder t st fs
-> Expr t ret
-> Ctx.Assignment (ExprBoundVar t) args
-> Ctx.Assignment (Expr t) args
-> IO (Expr t ret)
evalBoundVars sym e vars exprs = do
expr_tbl <- stToIO $ initHashTable (fmapFC BoundVarExpr vars) exprs
fn_tbl <- stToIO $ PH.new
let tbls = EvalHashTables { exprTable = expr_tbl
, fnTable = fn_tbl
}
evalBoundVars' tbls sym e
-- | This attempts to lookup an entry in a symbolic array.
--
-- It patterns maps on the array constructor.
sbConcreteLookup :: forall t st fs d tp range
. ExprBuilder t st fs
-- ^ Simple builder for creating terms.
-> Expr t (BaseArrayType (d::>tp) range)
-- ^ Array to lookup value in.
-> Maybe (Ctx.Assignment IndexLit (d::>tp))
-- ^ A concrete index that corresponds to the index or nothing
-- if the index is symbolic.
-> Ctx.Assignment (Expr t) (d::>tp)
-- ^ The index to lookup.
-> IO (Expr t range)
sbConcreteLookup sym arr0 mcidx idx
-- Try looking up a write to a concrete address.
| Just (ArrayMap _ _ entry_map def) <- asApp arr0
, Just cidx <- mcidx =
case AUM.lookup cidx entry_map of
Just v -> return v
Nothing -> sbConcreteLookup sym def mcidx idx
-- Evaluate function arrays on ground values.
| Just (ArrayFromFn f) <- asNonceApp arr0 = do
betaReduce sym f idx
-- Lookups on constant arrays just return value
| Just (ConstantArray _ _ v) <- asApp arr0 = do
return v
-- Lookups on mux arrays just distribute over mux.
| Just (BaseIte _ _ p x y) <- asApp arr0 = do
xv <- sbConcreteLookup sym x mcidx idx
yv <- sbConcreteLookup sym y mcidx idx
baseTypeIte sym p xv yv
| Just (MapOverArrays f _ args) <- asNonceApp arr0 = do
let eval :: ArrayResultWrapper (Expr t) (d::>tp) utp
-> IO (Expr t utp)
eval a = sbConcreteLookup sym (unwrapArrayResult a) mcidx idx
betaReduce sym f =<< traverseFC eval args
-- Create select index.
| otherwise = do
case exprType arr0 of
BaseArrayRepr _ range ->
sbMakeExpr sym (SelectArray range arr0 idx)
----------------------------------------------------------------------
-- Expression builder instances
-- | Evaluate a weighted sum of integer values.
intSum :: ExprBuilder t st fs -> WeightedSum (Expr t) SR.SemiRingInteger -> IO (IntegerExpr t)
intSum sym s = semiRingSum sym s
-- | Evaluate a weighted sum of real values.
realSum :: ExprBuilder t st fs -> WeightedSum (Expr t) SR.SemiRingReal -> IO (RealExpr t)
realSum sym s = semiRingSum sym s
bvSum :: ExprBuilder t st fs -> WeightedSum (Expr t) (SR.SemiRingBV flv w) -> IO (BVExpr t w)
bvSum sym s = semiRingSum sym s
conjPred :: ExprBuilder t st fs -> BoolMap (Expr t) -> IO (BoolExpr t)
conjPred sym bm =
case BM.viewBoolMap bm of
BoolMapUnit -> return $ truePred sym
BoolMapDualUnit -> return $ falsePred sym
BoolMapTerms ((x,p):|[]) ->
case p of
Positive -> return x
Negative -> notPred sym x
_ -> sbMakeExpr sym $ ConjPred bm
bvUnary :: (1 <= w) => ExprBuilder t st fs -> UnaryBV (BoolExpr t) w -> IO (BVExpr t w)
bvUnary sym u
-- BGS: We probably don't need to re-truncate the result, but
-- until we refactor UnaryBV to use BV w instead of integer,
-- that'll have to wait.
| Just v <- UnaryBV.asConstant u = bvLit sym w (BV.mkBV w v)
| otherwise = sbMakeExpr sym (BVUnaryTerm u)
where w = UnaryBV.width u
asUnaryBV :: (?unaryThreshold :: Int)
=> ExprBuilder t st fs
-> BVExpr t n
-> Maybe (UnaryBV (BoolExpr t) n)
asUnaryBV sym e
| Just (BVUnaryTerm u) <- asApp e = Just u
| ?unaryThreshold == 0 = Nothing
| SemiRingLiteral (SR.SemiRingBVRepr _ w) v _ <- e = Just $ UnaryBV.constant sym w (BV.asUnsigned v)
| otherwise = Nothing
-- | This create a unary bitvector representing if the size is not too large.
sbTryUnaryTerm :: (1 <= w, ?unaryThreshold :: Int)
=> ExprBuilder t st fs
-> Maybe (IO (UnaryBV (BoolExpr t) w))
-> IO (BVExpr t w)
-> IO (BVExpr t w)
sbTryUnaryTerm _sym Nothing fallback = fallback
sbTryUnaryTerm sym (Just mku) fallback =
do u <- mku
if UnaryBV.size u < ?unaryThreshold then
bvUnary sym u
else
fallback
semiRingProd ::
ExprBuilder t st fs ->
SemiRingProduct (Expr t) sr ->
IO (Expr t (SR.SemiRingBase sr))
semiRingProd sym pd
| WSum.nullProd pd = semiRingLit sym (WSum.prodRepr pd) (SR.one (WSum.prodRepr pd))
| Just v <- WSum.asProdVar pd = return v
| otherwise = sbMakeExpr sym $ SemiRingProd pd
semiRingSum ::
ExprBuilder t st fs ->
WeightedSum (Expr t) sr ->
IO (Expr t (SR.SemiRingBase sr))
semiRingSum sym s
| Just c <- WSum.asConstant s = semiRingLit sym (WSum.sumRepr s) c
| Just r <- WSum.asVar s = return r
| otherwise = sum' sym s
sum' ::
ExprBuilder t st fs ->
WeightedSum (Expr t) sr ->
IO (Expr t (SR.SemiRingBase sr))
sum' sym s = sbMakeExpr sym $ SemiRingSum s
{-# INLINE sum' #-}
scalarMul ::
ExprBuilder t st fs ->
SR.SemiRingRepr sr ->
SR.Coefficient sr ->
Expr t (SR.SemiRingBase sr) ->
IO (Expr t (SR.SemiRingBase sr))
scalarMul sym sr c x
| SR.eq sr (SR.zero sr) c = semiRingLit sym sr (SR.zero sr)
| SR.eq sr (SR.one sr) c = return x
| Just r <- asSemiRingLit sr x =
semiRingLit sym sr (SR.mul sr c r)
| Just s <- asSemiRingSum sr x =
sum' sym (WSum.scale sr c s)
| otherwise =
sum' sym (WSum.scaledVar sr c x)
semiRingIte ::
ExprBuilder t st fs ->
SR.SemiRingRepr sr ->
Expr t BaseBoolType ->
Expr t (SR.SemiRingBase sr) ->
Expr t (SR.SemiRingBase sr) ->
IO (Expr t (SR.SemiRingBase sr))
semiRingIte sym sr c x y
-- evaluate as constants
| Just True <- asConstantPred c = return x
| Just False <- asConstantPred c = return y
-- reduce negations
| Just (NotPred c') <- asApp c
= semiRingIte sym sr c' y x
-- remove the ite if the then and else cases are the same
| x == y = return x
-- Try to extract common sum information.
| (z, x',y') <- WSum.extractCommon (asWeightedSum sr x) (asWeightedSum sr y)
, not (WSum.isZero sr z) = do
xr <- semiRingSum sym x'
yr <- semiRingSum sym y'
let sz = 1 + iteSize xr + iteSize yr
r <- sbMakeExpr sym (BaseIte (SR.semiRingBase sr) sz c xr yr)
semiRingSum sym $! WSum.addVar sr z r
-- final fallback, create the ite term
| otherwise =
let sz = 1 + iteSize x + iteSize y in
sbMakeExpr sym (BaseIte (SR.semiRingBase sr) sz c x y)
mkIte ::
ExprBuilder t st fs ->
Expr t BaseBoolType ->
Expr t bt ->
Expr t bt ->
IO (Expr t bt)
mkIte sym c x y
-- evaluate as constants
| Just True <- asConstantPred c = return x
| Just False <- asConstantPred c = return y
-- reduce negations
| Just (NotPred c') <- asApp c
= mkIte sym c' y x
-- remove the ite if the then and else cases are the same
| x == y = return x
| otherwise =
let sz = 1 + iteSize x + iteSize y in
sbMakeExpr sym (BaseIte (exprType x) sz c x y)
semiRingLe ::
ExprBuilder t st fs ->
SR.OrderedSemiRingRepr sr ->
(Expr t (SR.SemiRingBase sr) -> Expr t (SR.SemiRingBase sr) -> IO (Expr t BaseBoolType))
{- ^ recursive call for simplifications -} ->
Expr t (SR.SemiRingBase sr) ->
Expr t (SR.SemiRingBase sr) ->
IO (Expr t BaseBoolType)
semiRingLe sym osr rec x y
-- Check for syntactic equality.
| x == y = return (truePred sym)
-- Strength reductions on a non-linear constraint to piecewise linear.
| Just c <- asSemiRingLit sr x
, SR.eq sr c (SR.zero sr)
, Just (SemiRingProd pd) <- asApp y
, Just Refl <- testEquality sr (WSum.prodRepr pd)
= prodNonneg sym osr pd
-- Another strength reduction
| Just c <- asSemiRingLit sr y
, SR.eq sr c (SR.zero sr)
, Just (SemiRingProd pd) <- asApp x
, Just Refl <- testEquality sr (WSum.prodRepr pd)
= prodNonpos sym osr pd
-- Push some comparisons under if/then/else
| SemiRingLiteral _ _ _ <- x
, Just (BaseIte _ _ c a b) <- asApp y
= join (itePred sym c <$> rec x a <*> rec x b)
-- Push some comparisons under if/then/else
| Just (BaseIte tp _ c a b) <- asApp x
, SemiRingLiteral _ _ _ <- y
, Just Refl <- testEquality tp (SR.semiRingBase sr)
= join (itePred sym c <$> rec a y <*> rec b y)
-- Try to extract common sum information.
| (z, x',y') <- WSum.extractCommon (asWeightedSum sr x) (asWeightedSum sr y)
, not (WSum.isZero sr z) = do
xr <- semiRingSum sym x'
yr <- semiRingSum sym y'
rec xr yr
-- Default case
| otherwise = sbMakeExpr sym $ SemiRingLe osr x y
where sr = SR.orderedSemiRing osr
semiRingEq ::
ExprBuilder t st fs ->
SR.SemiRingRepr sr ->
(Expr t (SR.SemiRingBase sr) -> Expr t (SR.SemiRingBase sr) -> IO (Expr t BaseBoolType))
{- ^ recursive call for simplifications -} ->
Expr t (SR.SemiRingBase sr) ->
Expr t (SR.SemiRingBase sr) ->
IO (Expr t BaseBoolType)
semiRingEq sym sr rec x y
-- Check for syntactic equality.
| x == y = return (truePred sym)
-- Push some equalities under if/then/else
| SemiRingLiteral _ _ _ <- x
, Just (BaseIte _ _ c a b) <- asApp y
= join (itePred sym c <$> rec x a <*> rec x b)
-- Push some equalities under if/then/else
| Just (BaseIte _ _ c a b) <- asApp x
, SemiRingLiteral _ _ _ <- y
= join (itePred sym c <$> rec a y <*> rec b y)
| (z, x',y') <- WSum.extractCommon (asWeightedSum sr x) (asWeightedSum sr y)
, not (WSum.isZero sr z) =
case (WSum.asConstant x', WSum.asConstant y') of
(Just a, Just b) -> return $! backendPred sym (SR.eq sr a b)
_ -> do xr <- semiRingSum sym x'
yr <- semiRingSum sym y'
sbMakeExpr sym $ BaseEq (SR.semiRingBase sr) (min xr yr) (max xr yr)
| otherwise =
sbMakeExpr sym $ BaseEq (SR.semiRingBase sr) (min x y) (max x y)
semiRingAdd ::
forall t st fs sr.
ExprBuilder t st fs ->
SR.SemiRingRepr sr ->
Expr t (SR.SemiRingBase sr) ->
Expr t (SR.SemiRingBase sr) ->
IO (Expr t (SR.SemiRingBase sr))
semiRingAdd sym sr x y =
case (viewSemiRing sr x, viewSemiRing sr y) of
(SR_Constant c, _) | SR.eq sr c (SR.zero sr) -> return y
(_, SR_Constant c) | SR.eq sr c (SR.zero sr) -> return x
(SR_Constant xc, SR_Constant yc) ->
semiRingLit sym sr (SR.add sr xc yc)
(SR_Constant xc, SR_Sum ys) ->
sum' sym (WSum.addConstant sr ys xc)
(SR_Sum xs, SR_Constant yc) ->
sum' sym (WSum.addConstant sr xs yc)
(SR_Constant xc, _)
| Just (BaseIte _ _ cond a b) <- asApp y
, isConstantSemiRingExpr a || isConstantSemiRingExpr b -> do
xa <- semiRingAdd sym sr x a
xb <- semiRingAdd sym sr x b
semiRingIte sym sr cond xa xb
| otherwise ->
sum' sym (WSum.addConstant sr (WSum.var sr y) xc)
(_, SR_Constant yc)
| Just (BaseIte _ _ cond a b) <- asApp x
, isConstantSemiRingExpr a || isConstantSemiRingExpr b -> do
ay <- semiRingAdd sym sr a y
by <- semiRingAdd sym sr b y
semiRingIte sym sr cond ay by
| otherwise ->
sum' sym (WSum.addConstant sr (WSum.var sr x) yc)
(SR_Sum xs, SR_Sum ys) -> semiRingSum sym (WSum.add sr xs ys)
(SR_Sum xs, _) -> semiRingSum sym (WSum.addVar sr xs y)
(_ , SR_Sum ys) -> semiRingSum sym (WSum.addVar sr ys x)
_ -> semiRingSum sym (WSum.addVars sr x y)
where isConstantSemiRingExpr :: Expr t (SR.SemiRingBase sr) -> Bool
isConstantSemiRingExpr (viewSemiRing sr -> SR_Constant _) = True
isConstantSemiRingExpr _ = False
semiRingMul ::
ExprBuilder t st fs ->
SR.SemiRingRepr sr ->
Expr t (SR.SemiRingBase sr) ->
Expr t (SR.SemiRingBase sr) ->
IO (Expr t (SR.SemiRingBase sr))
semiRingMul sym sr x y =
case (viewSemiRing sr x, viewSemiRing sr y) of
(SR_Constant c, _) -> scalarMul sym sr c y
(_, SR_Constant c) -> scalarMul sym sr c x
(SR_Sum (WSum.asAffineVar -> Just (c,x',o)), _) ->
do cxy <- scalarMul sym sr c =<< semiRingMul sym sr x' y
oy <- scalarMul sym sr o y
semiRingAdd sym sr cxy oy
(_, SR_Sum (WSum.asAffineVar -> Just (c,y',o))) ->
do cxy <- scalarMul sym sr c =<< semiRingMul sym sr x y'
ox <- scalarMul sym sr o x
semiRingAdd sym sr cxy ox
(SR_Prod px, SR_Prod py) -> semiRingProd sym (WSum.prodMul px py)
(SR_Prod px, _) -> semiRingProd sym (WSum.prodMul px (WSum.prodVar sr y))
(_, SR_Prod py) -> semiRingProd sym (WSum.prodMul (WSum.prodVar sr x) py)
_ -> semiRingProd sym (WSum.prodMul (WSum.prodVar sr x) (WSum.prodVar sr y))
prodNonneg ::
ExprBuilder t st fs ->
SR.OrderedSemiRingRepr sr ->
WSum.SemiRingProduct (Expr t) sr ->
IO (Expr t BaseBoolType)
prodNonneg sym osr pd =
do let sr = SR.orderedSemiRing osr
zero <- semiRingLit sym sr (SR.zero sr)
fst <$> computeNonnegNonpos sym osr zero pd
prodNonpos ::
ExprBuilder t st fs ->
SR.OrderedSemiRingRepr sr ->
WSum.SemiRingProduct (Expr t) sr ->
IO (Expr t BaseBoolType)
prodNonpos sym osr pd =
do let sr = SR.orderedSemiRing osr
zero <- semiRingLit sym sr (SR.zero sr)
snd <$> computeNonnegNonpos sym osr zero pd
computeNonnegNonpos ::
ExprBuilder t st fs ->
SR.OrderedSemiRingRepr sr ->
Expr t (SR.SemiRingBase sr) {- zero element -} ->
WSum.SemiRingProduct (Expr t) sr ->
IO (Expr t BaseBoolType, Expr t BaseBoolType)
computeNonnegNonpos sym osr zero pd =
fromMaybe (truePred sym, falsePred sym) <$> WSum.prodEvalM merge single pd
where
single x = (,) <$> reduceApp sym bvUnary (SemiRingLe osr zero x) -- nonnegative
<*> reduceApp sym bvUnary (SemiRingLe osr x zero) -- nonpositive
merge (nn1, np1) (nn2, np2) =
do nn <- join (orPred sym <$> andPred sym nn1 nn2 <*> andPred sym np1 np2)
np <- join (orPred sym <$> andPred sym nn1 np2 <*> andPred sym np1 nn2)
return (nn, np)
arrayResultIdxType :: BaseTypeRepr (BaseArrayType (idx ::> itp) d)
-> Ctx.Assignment BaseTypeRepr (idx ::> itp)
arrayResultIdxType (BaseArrayRepr idx _) = idx
-- | This decomposes A ExprBuilder array expression into a set of indices that
-- have been updated, and an underlying index.
data ArrayMapView i f tp
= ArrayMapView { _arrayMapViewIndices :: !(AUM.ArrayUpdateMap f i tp)
, _arrayMapViewExpr :: !(f (BaseArrayType i tp))
}
-- | Construct an 'ArrayMapView' for an element.
viewArrayMap :: Expr t (BaseArrayType i tp)
-> ArrayMapView i (Expr t) tp
viewArrayMap x
| Just (ArrayMap _ _ m c) <- asApp x = ArrayMapView m c
| otherwise = ArrayMapView AUM.empty x
-- | Construct an 'ArrayMapView' for an element.
underlyingArrayMapExpr :: ArrayResultWrapper (Expr t) i tp
-> ArrayResultWrapper (Expr t) i tp
underlyingArrayMapExpr x
| Just (ArrayMap _ _ _ c) <- asApp (unwrapArrayResult x) = ArrayResultWrapper c
| otherwise = x
-- | Return set of addresss in assignment that are written to by at least one expr
concreteArrayEntries :: forall t i ctx
. Ctx.Assignment (ArrayResultWrapper (Expr t) i) ctx
-> Set (Ctx.Assignment IndexLit i)
concreteArrayEntries = foldlFC' f Set.empty
where f :: Set (Ctx.Assignment IndexLit i)
-> ArrayResultWrapper (Expr t) i tp
-> Set (Ctx.Assignment IndexLit i)
f s e
| Just (ArrayMap _ _ m _) <- asApp (unwrapArrayResult e) =
Set.union s (AUM.keysSet m)
| otherwise = s
data IntLit tp = (tp ~ BaseIntegerType) => IntLit Integer
asIntBounds :: Ctx.Assignment (Expr t) idx -> Maybe (Ctx.Assignment IntLit idx)
asIntBounds = traverseFC f
where f :: Expr t tp -> Maybe (IntLit tp)
f (SemiRingLiteral SR.SemiRingIntegerRepr n _) = Just (IntLit n)
f _ = Nothing
foldBoundLeM :: (r -> Integer -> IO r) -> r -> Integer -> IO r
foldBoundLeM f r n
| n <= 0 = pure r
| otherwise =
do r' <- foldBoundLeM f r (n-1)
f r' n
foldIndicesInRangeBounds :: forall sym idx r
. IsExprBuilder sym
=> sym
-> (r -> Ctx.Assignment (SymExpr sym) idx -> IO r)
-> r
-> Ctx.Assignment IntLit idx
-> IO r
foldIndicesInRangeBounds sym f0 a0 bnds0 = do
case bnds0 of
Ctx.Empty -> f0 a0 Ctx.empty
bnds Ctx.:> IntLit b -> foldIndicesInRangeBounds sym (g f0) a0 bnds
where g :: (r -> Ctx.Assignment (SymExpr sym) (idx0 ::> BaseIntegerType) -> IO r)
-> r
-> Ctx.Assignment (SymExpr sym) idx0
-> IO r
g f a i = foldBoundLeM (h f i) a b
h :: (r -> Ctx.Assignment (SymExpr sym) (idx0 ::> BaseIntegerType) -> IO r)
-> Ctx.Assignment (SymExpr sym) idx0
-> r
-> Integer
-> IO r
h f i a j = do
je <- intLit sym j
f a (i Ctx.:> je)
-- | Examine the list of terms, and determine if any one of them
-- appears in the given @BoolMap@ with the same polarity.
checkAbsorption ::
BoolMap (Expr t) ->
[(BoolExpr t, Polarity)] ->
Bool
checkAbsorption _bm [] = False
checkAbsorption bm ((x,p):_)
| Just p' <- BM.contains bm x, p == p' = True
checkAbsorption bm (_:xs) = checkAbsorption bm xs
-- | If @tryAndAbsorption x y@ returns @True@, that means that @y@
-- implies @x@, so that the conjunction @x AND y = y@. A @False@
-- result gives no information.
tryAndAbsorption ::
BoolExpr t ->
BoolExpr t ->
Bool
tryAndAbsorption (asApp -> Just (NotPred (asApp -> Just (ConjPred as)))) (asConjunction -> bs)
= checkAbsorption (BM.reversePolarities as) bs
tryAndAbsorption _ _ = False
-- | If @tryOrAbsorption x y@ returns @True@, that means that @x@
-- implies @y@, so that the disjunction @x OR y = y@. A @False@
-- result gives no information.
tryOrAbsorption ::
BoolExpr t ->
BoolExpr t ->
Bool
tryOrAbsorption (asApp -> Just (ConjPred as)) (asDisjunction -> bs)
= checkAbsorption as bs
tryOrAbsorption _ _ = False
instance IsExprBuilder (ExprBuilder t st fs) where
getConfiguration = sbConfiguration
setSolverLogListener sb = atomicWriteIORef (sbSolverLogger sb)
getSolverLogListener sb = readIORef (sbSolverLogger sb)
logSolverEvent sb ev =
readIORef (sbSolverLogger sb) >>= \case
Nothing -> return ()
Just f -> f ev
getStatistics sb = do
allocs <- countNoncesGenerated (exprCounter sb)
nonLinearOps <- readIORef (sbNonLinearOps sb)
return $ Statistics { statAllocs = allocs
, statNonLinearOps = nonLinearOps }
annotateTerm sym e =
case e of
NonceAppExpr (nonceExprApp -> Annotation _ n _) -> return (n, e)
_ -> do
let tpr = exprType e
n <- sbFreshIndex sym
e' <- sbNonceExpr sym (Annotation tpr n e)
return (n, e')
getAnnotation _sym e =
case e of
NonceAppExpr (nonceExprApp -> Annotation _ n _) -> Just n
_ -> Nothing
----------------------------------------------------------------------
-- Program location operations
getCurrentProgramLoc = curProgramLoc
setCurrentProgramLoc sym l = atomicWriteIORef (sbProgramLoc sym) l
----------------------------------------------------------------------
-- Bool operations.
truePred = sbTrue
falsePred = sbFalse
notPred sym x
| Just b <- asConstantPred x
= return (backendPred sym $! not b)
| Just (NotPred x') <- asApp x
= return x'
| otherwise
= sbMakeExpr sym (NotPred x)
eqPred sym x y
| x == y
= return (truePred sym)
| Just (NotPred x') <- asApp x
= xorPred sym x' y
| Just (NotPred y') <- asApp y
= xorPred sym x y'
| otherwise
= case (asConstantPred x, asConstantPred y) of
(Just False, _) -> notPred sym y
(Just True, _) -> return y
(_, Just False) -> notPred sym x
(_, Just True) -> return x
_ -> sbMakeExpr sym $ BaseEq BaseBoolRepr (min x y) (max x y)
xorPred sym x y = notPred sym =<< eqPred sym x y
andPred sym x y =
case (asConstantPred x, asConstantPred y) of
(Just True, _) -> return y
(Just False, _) -> return x
(_, Just True) -> return x
(_, Just False) -> return y
_ | x == y -> return x -- and is idempotent
| otherwise -> go x y
where
go a b
| Just (ConjPred as) <- asApp a
, Just (ConjPred bs) <- asApp b
= conjPred sym $ BM.combine as bs
| tryAndAbsorption a b
= return b
| tryAndAbsorption b a
= return a
| Just (ConjPred as) <- asApp a
= conjPred sym $ uncurry BM.addVar (asPosAtom b) as
| Just (ConjPred bs) <- asApp b
= conjPred sym $ uncurry BM.addVar (asPosAtom a) bs
| otherwise
= conjPred sym $ BM.fromVars [asPosAtom a, asPosAtom b]
orPred sym x y =
case (asConstantPred x, asConstantPred y) of
(Just True, _) -> return x
(Just False, _) -> return y
(_, Just True) -> return y
(_, Just False) -> return x
_ | x == y -> return x -- or is idempotent
| otherwise -> go x y
where
go a b
| Just (NotPred (asApp -> Just (ConjPred as))) <- asApp a
, Just (NotPred (asApp -> Just (ConjPred bs))) <- asApp b
= notPred sym =<< conjPred sym (BM.combine as bs)
| tryOrAbsorption a b
= return b
| tryOrAbsorption b a
= return a
| Just (NotPred (asApp -> Just (ConjPred as))) <- asApp a
= notPred sym =<< conjPred sym (uncurry BM.addVar (asNegAtom b) as)
| Just (NotPred (asApp -> Just (ConjPred bs))) <- asApp b
= notPred sym =<< conjPred sym (uncurry BM.addVar (asNegAtom a) bs)
| otherwise
= notPred sym =<< conjPred sym (BM.fromVars [asNegAtom a, asNegAtom b])
itePred sb c x y
-- ite c c y = c || y
| c == x = orPred sb c y
-- ite c x c = c && x
| c == y = andPred sb c x
-- ite c x x = x
| x == y = return x
-- ite 1 x y = x
| Just True <- asConstantPred c = return x
-- ite 0 x y = y
| Just False <- asConstantPred c = return y
-- ite !c x y = ite c y x
| Just (NotPred c') <- asApp c = itePred sb c' y x
-- ite c 1 y = c || y
| Just True <- asConstantPred x = orPred sb c y
-- ite c 0 y = !c && y
| Just False <- asConstantPred x = andPred sb y =<< notPred sb c
-- ite c x 1 = !c || x
| Just True <- asConstantPred y = orPred sb x =<< notPred sb c
-- ite c x 0 = c && x
| Just False <- asConstantPred y = andPred sb c x
-- Default case
| otherwise =
let sz = 1 + iteSize x + iteSize y in
sbMakeExpr sb $ BaseIte BaseBoolRepr sz c x y
----------------------------------------------------------------------
-- Integer operations.
intLit sym n = semiRingLit sym SR.SemiRingIntegerRepr n
intNeg sym x = scalarMul sym SR.SemiRingIntegerRepr (-1) x
intAdd sym x y = semiRingAdd sym SR.SemiRingIntegerRepr x y
intMul sym x y = semiRingMul sym SR.SemiRingIntegerRepr x y
intIte sym c x y = semiRingIte sym SR.SemiRingIntegerRepr c x y
intEq sym x y
-- Use range check
| Just b <- rangeCheckEq (exprAbsValue x) (exprAbsValue y)
= return $ backendPred sym b
-- Reduce to bitvector equality, when possible
| Just (SBVToInteger xbv) <- asApp x
, Just (SBVToInteger ybv) <- asApp y
= let wx = bvWidth xbv
wy = bvWidth ybv
-- Sign extend to largest bitvector and compare.
in case testNatCases wx wy of
NatCaseLT LeqProof -> do
x' <- bvSext sym wy xbv
bvEq sym x' ybv
NatCaseEQ ->
bvEq sym xbv ybv
NatCaseGT LeqProof -> do
y' <- bvSext sym wx ybv
bvEq sym xbv y'
-- Reduce to bitvector equality, when possible
| Just (BVToInteger xbv) <- asApp x
, Just (BVToInteger ybv) <- asApp y
= let wx = bvWidth xbv
wy = bvWidth ybv
-- Zero extend to largest bitvector and compare.
in case testNatCases wx wy of
NatCaseLT LeqProof -> do
x' <- bvZext sym wy xbv
bvEq sym x' ybv
NatCaseEQ ->
bvEq sym xbv ybv
NatCaseGT LeqProof -> do
y' <- bvZext sym wx ybv
bvEq sym xbv y'
| Just (SBVToInteger xbv) <- asApp x
, Just yi <- asSemiRingLit SR.SemiRingIntegerRepr y
= let w = bvWidth xbv in
if yi < minSigned w || yi > maxSigned w
then return (falsePred sym)
else bvEq sym xbv =<< bvLit sym w (BV.mkBV w yi)
| Just xi <- asSemiRingLit SR.SemiRingIntegerRepr x
, Just (SBVToInteger ybv) <- asApp x
= let w = bvWidth ybv in
if xi < minSigned w || xi > maxSigned w
then return (falsePred sym)
else bvEq sym ybv =<< bvLit sym w (BV.mkBV w xi)
| Just (BVToInteger xbv) <- asApp x
, Just yi <- asSemiRingLit SR.SemiRingIntegerRepr y
= let w = bvWidth xbv in
if yi < minUnsigned w || yi > maxUnsigned w
then return (falsePred sym)
else bvEq sym xbv =<< bvLit sym w (BV.mkBV w yi)
| Just xi <- asSemiRingLit SR.SemiRingIntegerRepr x
, Just (BVToInteger ybv) <- asApp x
= let w = bvWidth ybv in
if xi < minUnsigned w || xi > maxUnsigned w
then return (falsePred sym)
else bvEq sym ybv =<< bvLit sym w (BV.mkBV w xi)
| otherwise = semiRingEq sym SR.SemiRingIntegerRepr (intEq sym) x y
intLe sym x y
-- Use abstract domains
| Just b <- rangeCheckLe (exprAbsValue x) (exprAbsValue y)
= return $ backendPred sym b
-- Check with two bitvectors.
| Just (SBVToInteger xbv) <- asApp x
, Just (SBVToInteger ybv) <- asApp y
= do let wx = bvWidth xbv
let wy = bvWidth ybv
-- Sign extend to largest bitvector and compare.
case testNatCases wx wy of
NatCaseLT LeqProof -> do
x' <- bvSext sym wy xbv
bvSle sym x' ybv
NatCaseEQ -> bvSle sym xbv ybv
NatCaseGT LeqProof -> do
y' <- bvSext sym wx ybv
bvSle sym xbv y'
-- Check with two bitvectors.
| Just (BVToInteger xbv) <- asApp x
, Just (BVToInteger ybv) <- asApp y
= do let wx = bvWidth xbv
let wy = bvWidth ybv
-- Zero extend to largest bitvector and compare.
case testNatCases wx wy of
NatCaseLT LeqProof -> do
x' <- bvZext sym wy xbv
bvUle sym x' ybv
NatCaseEQ -> bvUle sym xbv ybv
NatCaseGT LeqProof -> do
y' <- bvZext sym wx ybv
bvUle sym xbv y'
| Just (SBVToInteger xbv) <- asApp x
, Just yi <- asSemiRingLit SR.SemiRingIntegerRepr y
= let w = bvWidth xbv in
if | yi < minSigned w -> return (falsePred sym)
| yi > maxSigned w -> return (truePred sym)
| otherwise -> join (bvSle sym <$> pure xbv <*> bvLit sym w (BV.mkBV w yi))
| Just xi <- asSemiRingLit SR.SemiRingIntegerRepr x
, Just (SBVToInteger ybv) <- asApp x
= let w = bvWidth ybv in
if | xi < minSigned w -> return (truePred sym)
| xi > maxSigned w -> return (falsePred sym)
| otherwise -> join (bvSle sym <$> bvLit sym w (BV.mkBV w xi) <*> pure ybv)
| Just (BVToInteger xbv) <- asApp x
, Just yi <- asSemiRingLit SR.SemiRingIntegerRepr y
= let w = bvWidth xbv in
if | yi < minUnsigned w -> return (falsePred sym)
| yi > maxUnsigned w -> return (truePred sym)
| otherwise -> join (bvUle sym <$> pure xbv <*> bvLit sym w (BV.mkBV w yi))
| Just xi <- asSemiRingLit SR.SemiRingIntegerRepr x
, Just (BVToInteger ybv) <- asApp x
= let w = bvWidth ybv in
if | xi < minUnsigned w -> return (truePred sym)
| xi > maxUnsigned w -> return (falsePred sym)
| otherwise -> join (bvUle sym <$> bvLit sym w (BV.mkBV w xi) <*> pure ybv)
{- FIXME? how important are these reductions?
-- Compare to BV lower bound.
| Just (SBVToInteger xbv) <- x = do
let w = bvWidth xbv
l <- curProgramLoc sym
b_max <- realGe sym y (SemiRingLiteral SemiRingReal (toRational (maxSigned w)) l)
b_min <- realGe sym y (SemiRingLiteral SemiRingReal (toRational (minSigned w)) l)
orPred sym b_max =<< andPred sym b_min =<< (bvSle sym xbv =<< realToSBV sym w y)
-- Compare to SBV upper bound.
| SBVToReal ybv <- y = do
let w = bvWidth ybv
l <- curProgramLoc sym
b_min <- realLe sym x (SemiRingLiteral SemiRingReal (toRational (minSigned w)) l)
b_max <- realLe sym x (SemiRingLiteral SemiRingReal (toRational (maxSigned w)) l)
orPred sym b_min
=<< andPred sym b_max
=<< (\xbv -> bvSle sym xbv ybv) =<< realToSBV sym w x
-}
| otherwise
= semiRingLe sym SR.OrderedSemiRingIntegerRepr (intLe sym) x y
intAbs sym x
| Just i <- asInteger x = intLit sym (abs i)
| Just True <- rangeCheckLe (SingleRange 0) (exprAbsValue x) = return x
| Just True <- rangeCheckLe (exprAbsValue x) (SingleRange 0) = intNeg sym x
| otherwise = sbMakeExpr sym (IntAbs x)
intDiv sym x y
-- Div by 1.
| Just 1 <- asInteger y = return x
-- As integers.
| Just xi <- asInteger x, Just yi <- asInteger y, yi /= 0 =
if yi >= 0 then
intLit sym (xi `div` yi)
else
intLit sym (negate (xi `div` negate yi))
-- Return int div
| otherwise =
sbMakeExpr sym (IntDiv x y)
intMod sym x y
-- Mod by 1.
| Just 1 <- asInteger y = intLit sym 0
-- As integers.
| Just xi <- asInteger x, Just yi <- asInteger y, yi /= 0 =
intLit sym (xi `mod` abs yi)
| Just (SemiRingSum xsum) <- asApp x
, SR.SemiRingIntegerRepr <- WSum.sumRepr xsum
, Just yi <- asInteger y
, yi /= 0 =
case WSum.reduceIntSumMod xsum (abs yi) of
xsum' | Just xi <- WSum.asConstant xsum' ->
intLit sym xi
| otherwise ->
do x' <- intSum sym xsum'
sbMakeExpr sym (IntMod x' y)
-- Return int mod.
| otherwise =
sbMakeExpr sym (IntMod x y)
intDivisible sym x k
| k == 0 = intEq sym x =<< intLit sym 0
| k == 1 = return (truePred sym)
| Just xi <- asInteger x = return $ backendPred sym (xi `mod` (toInteger k) == 0)
| Just (SemiRingSum xsum) <- asApp x
, SR.SemiRingIntegerRepr <- WSum.sumRepr xsum =
case WSum.reduceIntSumMod xsum (toInteger k) of
xsum' | Just xi <- WSum.asConstant xsum' ->
return $ backendPred sym (xi == 0)
| otherwise ->
do x' <- intSum sym xsum'
sbMakeExpr sym (IntDivisible x' k)
| otherwise =
sbMakeExpr sym (IntDivisible x k)
---------------------------------------------------------------------
-- Bitvector operations
bvLit sym w bv =
semiRingLit sym (SR.SemiRingBVRepr SR.BVArithRepr w) bv
bvConcat sym x y =
case (asBV x, asBV y) of
-- both values are constants, just compute the concatenation
(Just xv, Just yv) -> do
let w' = addNat (bvWidth x) (bvWidth y)
LeqProof <- return (leqAddPos (bvWidth x) (bvWidth y))
bvLit sym w' (BV.concat (bvWidth x) (bvWidth y) xv yv)
-- reassociate to combine constants where possible
(Just _xv, _)
| Just (BVConcat _w a b) <- asApp y
, Just _av <- asBV a
, Just Refl <- testEquality (addNat (bvWidth x) (addNat (bvWidth a) (bvWidth b)))
(addNat (addNat (bvWidth x) (bvWidth a)) (bvWidth b))
, Just LeqProof <- isPosNat (addNat (bvWidth x) (bvWidth a)) -> do
xa <- bvConcat sym x a
bvConcat sym xa b
-- concat two adjacent sub-selects just makes a single select
_ | Just (BVSelect idx1 n1 a) <- asApp x
, Just (BVSelect idx2 n2 b) <- asApp y
, Just Refl <- sameTerm a b
, Just Refl <- testEquality idx1 (addNat idx2 n2)
, Just LeqProof <- isPosNat (addNat n1 n2)
, Just LeqProof <- testLeq (addNat idx2 (addNat n1 n2)) (bvWidth a) ->
bvSelect sym idx2 (addNat n1 n2) a
-- always reassociate to the right
_ | Just (BVConcat _w a b) <- asApp x
, Just _bv <- asBV b
, Just Refl <- testEquality (addNat (bvWidth a) (addNat (bvWidth b) (bvWidth y)))
(addNat (addNat (bvWidth a) (bvWidth b)) (bvWidth y))
, Just LeqProof <- isPosNat (addNat (bvWidth b) (bvWidth y)) -> do
by <- bvConcat sym b y
bvConcat sym a by
-- no special case applies, emit a basic concat expression
_ -> do
let wx = bvWidth x
let wy = bvWidth y
Just LeqProof <- return (isPosNat (addNat wx wy))
sbMakeExpr sym $ BVConcat (addNat wx wy) x y
-- bvSelect has a bunch of special cases that examine the form of the
-- bitvector being selected from. This can significantly reduce the size
-- of expressions that result from the very verbose packing and unpacking
-- operations that arise from byte-oriented memory models.
bvSelect sb idx n x
| Just xv <- asBV x = do
bvLit sb n (BV.select idx n xv)
-- nested selects can be collapsed
| Just (BVSelect idx' _n' b) <- asApp x
, let idx2 = addNat idx idx'
, Just LeqProof <- testLeq (addNat idx2 n) (bvWidth b) =
bvSelect sb idx2 n b
-- select the entire bitvector is the identity function
| Just _ <- testEquality idx (knownNat :: NatRepr 0)
, Just Refl <- testEquality n (bvWidth x) =
return x
| Just (BVShl w a b) <- asApp x
, Just diff <- asBV b
, Some diffRepr <- mkNatRepr (BV.asNatural diff)
, Just LeqProof <- testLeq diffRepr idx = do
Just LeqProof <- return $ testLeq (addNat (subNat idx diffRepr) n) w
bvSelect sb (subNat idx diffRepr) n a
| Just (BVShl _w _a b) <- asApp x
, Just diff <- asBV b
, Some diffRepr <- mkNatRepr (BV.asNatural diff)
, Just LeqProof <- testLeq (addNat idx n) diffRepr =
bvLit sb n (BV.zero n)
| Just (BVAshr w a b) <- asApp x
, Just diff <- asBV b
, Some diffRepr <- mkNatRepr (BV.asNatural diff)
, Just LeqProof <- testLeq (addNat (addNat idx diffRepr) n) w =
bvSelect sb (addNat idx diffRepr) n a
| Just (BVLshr w a b) <- asApp x
, Just diff <- asBV b
, Some diffRepr <- mkNatRepr (BV.asNatural diff)
, Just LeqProof <- testLeq (addNat (addNat idx diffRepr) n) w =
bvSelect sb (addNat idx diffRepr) n a
| Just (BVLshr w _a b) <- asApp x
, Just diff <- asBV b
, Some diffRepr <- mkNatRepr (BV.asNatural diff)
, Just LeqProof <- testLeq w (addNat idx diffRepr) =
bvLit sb n (BV.zero n)
-- select from a sign extension
| Just (BVSext w b) <- asApp x = do
-- Add dynamic check
Just LeqProof <- return $ testLeq (bvWidth b) w
let ext = subNat w (bvWidth b)
-- Add dynamic check
Just LeqProof <- return $ isPosNat w
Just LeqProof <- return $ isPosNat ext
zeros <- minUnsignedBV sb ext
ones <- maxUnsignedBV sb ext
c <- bvIsNeg sb b
hi <- bvIte sb c ones zeros
x' <- bvConcat sb hi b
-- Add dynamic check
Just LeqProof <- return $ testLeq (addNat idx n) (addNat ext (bvWidth b))
bvSelect sb idx n x'
-- select from a zero extension
| Just (BVZext w b) <- asApp x = do
-- Add dynamic check
Just LeqProof <- return $ testLeq (bvWidth b) w
let ext = subNat w (bvWidth b)
Just LeqProof <- return $ isPosNat w
Just LeqProof <- return $ isPosNat ext
hi <- bvLit sb ext (BV.zero ext)
x' <- bvConcat sb hi b
-- Add dynamic check
Just LeqProof <- return $ testLeq (addNat idx n) (addNat ext (bvWidth b))
bvSelect sb idx n x'
-- select is entirely within the less-significant bits of a concat
| Just (BVConcat _w _a b) <- asApp x
, Just LeqProof <- testLeq (addNat idx n) (bvWidth b) = do
bvSelect sb idx n b
-- select is entirely within the more-significant bits of a concat
| Just (BVConcat _w a b) <- asApp x
, Just LeqProof <- testLeq (bvWidth b) idx
, Just LeqProof <- isPosNat idx
, let diff = subNat idx (bvWidth b)
, Just LeqProof <- testLeq (addNat diff n) (bvWidth a) = do
bvSelect sb (subNat idx (bvWidth b)) n a
-- when the selected region overlaps a concat boundary we have:
-- select idx n (concat a b) =
-- concat (select 0 n1 a) (select idx n2 b)
-- where n1 + n2 = n and idx + n2 = width b
--
-- NB: this case must appear after the two above that check for selects
-- entirely within the first or second arguments of a concat, otherwise
-- some of the arithmetic checks below may fail
| Just (BVConcat _w a b) <- asApp x = do
Just LeqProof <- return $ testLeq idx (bvWidth b)
let n2 = subNat (bvWidth b) idx
Just LeqProof <- return $ testLeq n2 n
let n1 = subNat n n2
let z = knownNat :: NatRepr 0
Just LeqProof <- return $ isPosNat n1
Just LeqProof <- return $ testLeq (addNat z n1) (bvWidth a)
a' <- bvSelect sb z n1 a
Just LeqProof <- return $ isPosNat n2
Just LeqProof <- return $ testLeq (addNat idx n2) (bvWidth b)
b' <- bvSelect sb idx n2 b
Just Refl <- return $ testEquality (addNat n1 n2) n
bvConcat sb a' b'
-- Truncate a weighted sum: Remove terms with coefficients that
-- would become zero after truncation.
--
-- Truncation of w-bit words down to n bits respects congruence
-- modulo 2^n. Furthermore, w-bit addition and multiplication also
-- preserve congruence modulo 2^n. This means that it is sound to
-- replace coefficients in a weighted sum with new masked ones
-- that are congruent modulo 2^n: the final result after
-- truncation will be the same.
--
-- NOTE: This case is carefully designed to preserve sharing. Only
-- one App node (the SemiRingSum) is ever deconstructed. The
-- 'traverseCoeffs' call does not touch any other App nodes inside
-- the WeightedSum. Finally, we only reconstruct a new SemiRingSum
-- App node in the event that one of the coefficients has changed;
-- the writer monad tracks whether a change has occurred.
| Just (SemiRingSum s) <- asApp x
, SR.SemiRingBVRepr SR.BVArithRepr w <- WSum.sumRepr s
, Just Refl <- testEquality idx (knownNat :: NatRepr 0) =
do let mask = case testStrictLeq n w of
Left LeqProof -> BV.zext w (BV.maxUnsigned n)
Right Refl -> BV.maxUnsigned n
let reduce i
| i `BV.and` mask == BV.zero w = writer (BV.zero w, Any True)
| otherwise = writer (i, Any False)
let (s', Any changed) = runWriter $ WSum.traverseCoeffs reduce s
x' <- if changed then sbMakeExpr sb (SemiRingSum s') else return x
sbMakeExpr sb $ BVSelect idx n x'
{- Avoid doing work that may lose sharing...
-- Select from a weighted XOR: push down through the sum
| Just (SemiRingSum s) <- asApp x
, SR.SemiRingBVRepr SR.BVBitsRepr _w <- WSum.sumRepr s
= do let mask = maxUnsigned n
let shft = fromIntegral (natValue idx)
s' <- WSum.transformSum (SR.SemiRingBVRepr SR.BVBitsRepr n)
(\c -> return ((c `Bits.shiftR` shft) Bits..&. mask))
(bvSelect sb idx n)
s
semiRingSum sb s'
-- Select from a AND: push down through the AND
| Just (SemiRingProd pd) <- asApp x
, SR.SemiRingBVRepr SR.BVBitsRepr _w <- WSum.prodRepr pd
= do pd' <- WSum.prodEvalM
(bvAndBits sb)
(bvSelect sb idx n)
pd
maybe (bvLit sb n (maxUnsigned n)) return pd'
-- Select from an OR: push down through the OR
| Just (BVOrBits pd) <- asApp x
= do pd' <- WSum.prodEvalM
(bvOrBits sb)
(bvSelect sb idx n)
pd
maybe (bvLit sb n 0) return pd'
-}
-- Truncate from a unary bitvector
| Just (BVUnaryTerm u) <- asApp x
, Just Refl <- testEquality idx (knownNat @0) =
bvUnary sb =<< UnaryBV.trunc sb u n
-- if none of the above apply, produce a basic select term
| otherwise = sbMakeExpr sb $ BVSelect idx n x
testBitBV sym i y
| i < 0 || i >= natValue (bvWidth y) =
fail $ "Illegal bit index."
-- Constant evaluation
| Just yc <- asBV y
, i <= fromIntegral (maxBound :: Int)
= return $! backendPred sym (BV.testBit' (fromIntegral i) yc)
| Just (BVZext _w y') <- asApp y
= if i >= natValue (bvWidth y') then
return $ falsePred sym
else
testBitBV sym i y'
| Just (BVSext _w y') <- asApp y
= if i >= natValue (bvWidth y') then
testBitBV sym (natValue (bvWidth y') - 1) y'
else
testBitBV sym i y'
| Just (BVFill _ p) <- asApp y
= return p
| Just b <- BVD.testBit (bvWidth y) (exprAbsValue y) i
= return $! backendPred sym b
| Just (BaseIte _ _ c a b) <- asApp y
, isJust (asBV a) || isJust (asBV b) -- NB avoid losing sharing
= do a' <- testBitBV sym i a
b' <- testBitBV sym i b
itePred sym c a' b'
{- These rewrites can sometimes yield significant simplifications, but
also may lead to loss of sharing, so they are disabled...
| Just ws <- asSemiRingSum (SR.SemiRingBVRepr SR.BVBitsRepr (bvWidth y)) y
= let smul c x
| Bits.testBit c (fromIntegral i) = testBitBV sym i x
| otherwise = return (falsePred sym)
cnst c = return $! backendPred sym (Bits.testBit c (fromIntegral i))
in WSum.evalM (xorPred sym) smul cnst ws
| Just pd <- asSemiRingProd (SR.SemiRingBVRepr SR.BVBitsRepr (bvWidth y)) y
= fromMaybe (truePred sym) <$> WSum.prodEvalM (andPred sym) (testBitBV sym i) pd
| Just (BVOrBits pd) <- asApp y
= fromMaybe (falsePred sym) <$> WSum.prodEvalM (orPred sym) (testBitBV sym i) pd
-}
| otherwise = sbMakeExpr sym $ BVTestBit i y
bvFill sym w p
| Just True <- asConstantPred p = bvLit sym w (BV.maxUnsigned w)
| Just False <- asConstantPred p = bvLit sym w (BV.zero w)
| otherwise = sbMakeExpr sym $ BVFill w p
bvIte sym c x y
| Just (BVFill w px) <- asApp x
, Just (BVFill _w py) <- asApp y =
do z <- itePred sym c px py
bvFill sym w z
| Just (BVZext w x') <- asApp x
, Just (BVZext w' y') <- asApp y
, Just Refl <- testEquality (bvWidth x') (bvWidth y')
, Just Refl <- testEquality w w' =
do z <- bvIte sym c x' y'
bvZext sym w z
| Just (BVSext w x') <- asApp x
, Just (BVSext w' y') <- asApp y
, Just Refl <- testEquality (bvWidth x') (bvWidth y')
, Just Refl <- testEquality w w' =
do z <- bvIte sym c x' y'
bvSext sym w z
| Just (FloatToBinary fpp1 x') <- asApp x
, Just (FloatToBinary fpp2 y') <- asApp y
, Just Refl <- testEquality fpp1 fpp2 =
floatToBinary sym =<< floatIte sym c x' y'
| otherwise =
do ut <- CFG.getOpt (sbUnaryThreshold sym)
let ?unaryThreshold = fromInteger ut
sbTryUnaryTerm sym
(do ux <- asUnaryBV sym x
uy <- asUnaryBV sym y
return (UnaryBV.mux sym c ux uy))
(case inSameBVSemiRing x y of
Just (Some flv) ->
semiRingIte sym (SR.SemiRingBVRepr flv (bvWidth x)) c x y
Nothing ->
mkIte sym c x y)
bvEq sym x y
| x == y = return $! truePred sym
| Just (BVFill _ px) <- asApp x
, Just (BVFill _ py) <- asApp y =
eqPred sym px py
| Just b <- BVD.eq (exprAbsValue x) (exprAbsValue y) = do
return $! backendPred sym b
-- Push some equalities under if/then/else
| SemiRingLiteral _ _ _ <- x
, Just (BaseIte _ _ c a b) <- asApp y
, isJust (asBV a) || isJust (asBV b) -- avoid loss of sharing
= join (itePred sym c <$> bvEq sym x a <*> bvEq sym x b)
-- Push some equalities under if/then/else
| Just (BaseIte _ _ c a b) <- asApp x
, SemiRingLiteral _ _ _ <- y
, isJust (asBV a) || isJust (asBV b) -- avoid loss of sharing
= join (itePred sym c <$> bvEq sym a y <*> bvEq sym b y)
| Just (Some flv) <- inSameBVSemiRing x y
, let sr = SR.SemiRingBVRepr flv (bvWidth x)
, (z, x',y') <- WSum.extractCommon (asWeightedSum sr x) (asWeightedSum sr y)
, not (WSum.isZero sr z) =
case (WSum.asConstant x', WSum.asConstant y') of
(Just a, Just b) -> return $! backendPred sym (SR.eq sr a b)
_ -> do xr <- semiRingSum sym x'
yr <- semiRingSum sym y'
sbMakeExpr sym $ BaseEq (SR.semiRingBase sr) (min xr yr) (max xr yr)
| otherwise = do
ut <- CFG.getOpt (sbUnaryThreshold sym)
let ?unaryThreshold = fromInteger ut
if | Just ux <- asUnaryBV sym x
, Just uy <- asUnaryBV sym y
-> UnaryBV.eq sym ux uy
| otherwise
-> sbMakeExpr sym $ BaseEq (BaseBVRepr (bvWidth x)) (min x y) (max x y)
bvSlt sym x y
| Just xc <- asBV x
, Just yc <- asBV y =
return $! backendPred sym (BV.slt (bvWidth x) xc yc)
| Just b <- BVD.slt (bvWidth x) (exprAbsValue x) (exprAbsValue y) =
return $! backendPred sym b
| x == y = return (falsePred sym)
| otherwise = do
ut <- CFG.getOpt (sbUnaryThreshold sym)
let ?unaryThreshold = fromInteger ut
if | Just ux <- asUnaryBV sym x
, Just uy <- asUnaryBV sym y
-> UnaryBV.slt sym ux uy
| otherwise
-> sbMakeExpr sym $ BVSlt x y
bvUlt sym x y
| Just xc <- asBV x
, Just yc <- asBV y = do
return $! backendPred sym (BV.ult xc yc)
| Just b <- BVD.ult (exprAbsValue x) (exprAbsValue y) =
return $! backendPred sym b
| x == y =
return $! falsePred sym
| otherwise = do
ut <- CFG.getOpt (sbUnaryThreshold sym)
let ?unaryThreshold = fromInteger ut
if | Just ux <- asUnaryBV sym x
, Just uy <- asUnaryBV sym y
-> UnaryBV.ult sym ux uy
| otherwise
-> sbMakeExpr sym $ BVUlt x y
bvShl sym x y
-- shift by 0 is the identity function
| Just (BV.BV 0) <- asBV y
= pure x
-- shift by more than word width returns 0
| let (lo, _hi) = BVD.ubounds (exprAbsValue y)
, lo >= intValue (bvWidth x)
= bvLit sym (bvWidth x) (BV.zero (bvWidth x))
| Just xv <- asBV x, Just n <- asBV y
= bvLit sym (bvWidth x) (BV.shl (bvWidth x) xv (BV.asNatural n))
| otherwise
= sbMakeExpr sym $ BVShl (bvWidth x) x y
bvLshr sym x y
-- shift by 0 is the identity function
| Just (BV.BV 0) <- asBV y
= pure x
-- shift by more than word width returns 0
| let (lo, _hi) = BVD.ubounds (exprAbsValue y)
, lo >= intValue (bvWidth x)
= bvLit sym (bvWidth x) (BV.zero (bvWidth x))
| Just xv <- asBV x, Just n <- asBV y
= bvLit sym (bvWidth x) $ BV.lshr (bvWidth x) xv (BV.asNatural n)
| otherwise
= sbMakeExpr sym $ BVLshr (bvWidth x) x y
bvAshr sym x y
-- shift by 0 is the identity function
| Just (BV.BV 0) <- asBV y
= pure x
-- shift by more than word width returns either 0 (if x is nonnegative)
-- or 1 (if x is negative)
| let (lo, _hi) = BVD.ubounds (exprAbsValue y)
, lo >= intValue (bvWidth x)
= bvFill sym (bvWidth x) =<< bvIsNeg sym x
| Just xv <- asBV x, Just n <- asBV y
= bvLit sym (bvWidth x) $ BV.ashr (bvWidth x) xv (BV.asNatural n)
| otherwise
= sbMakeExpr sym $ BVAshr (bvWidth x) x y
bvRol sym x y
| Just xv <- asBV x, Just n <- asBV y
= bvLit sym (bvWidth x) $ BV.rotateL (bvWidth x) xv (BV.asNatural n)
| Just n <- asBV y
, n `BV.urem` BV.width (bvWidth y) == BV.zero (bvWidth y)
= return x
| Just (BVRol w x' n) <- asApp x
, isPow2 (natValue w)
= do z <- bvAdd sym n y
bvRol sym x' z
| Just (BVRol w x' n) <- asApp x
= do wbv <- bvLit sym w (BV.width w)
n' <- bvUrem sym n wbv
y' <- bvUrem sym y wbv
z <- bvAdd sym n' y'
bvRol sym x' z
| Just (BVRor w x' n) <- asApp x
, isPow2 (natValue w)
= do z <- bvSub sym n y
bvRor sym x' z
| Just (BVRor w x' n) <- asApp x
= do wbv <- bvLit sym w (BV.width w)
y' <- bvUrem sym y wbv
n' <- bvUrem sym n wbv
z <- bvAdd sym n' =<< bvSub sym wbv y'
bvRor sym x' z
| otherwise
= let w = bvWidth x in
sbMakeExpr sym $ BVRol w x y
bvRor sym x y
| Just xv <- asBV x, Just n <- asBV y
= bvLit sym (bvWidth x) $ BV.rotateR (bvWidth x) xv (BV.asNatural n)
| Just n <- asBV y
, n `BV.urem` BV.width (bvWidth y) == BV.zero (bvWidth y)
= return x
| Just (BVRor w x' n) <- asApp x
, isPow2 (natValue w)
= do z <- bvAdd sym n y
bvRor sym x' z
| Just (BVRor w x' n) <- asApp x
= do wbv <- bvLit sym w (BV.width w)
n' <- bvUrem sym n wbv
y' <- bvUrem sym y wbv
z <- bvAdd sym n' y'
bvRor sym x' z
| Just (BVRol w x' n) <- asApp x
, isPow2 (natValue w)
= do z <- bvSub sym n y
bvRol sym x' z
| Just (BVRol w x' n) <- asApp x
= do wbv <- bvLit sym w (BV.width w)
n' <- bvUrem sym n wbv
y' <- bvUrem sym y wbv
z <- bvAdd sym n' =<< bvSub sym wbv y'
bvRol sym x' z
| otherwise
= let w = bvWidth x in
sbMakeExpr sym $ BVRor w x y
bvZext sym w x
| Just xv <- asBV x = do
-- Add dynamic check for GHC typechecker.
Just LeqProof <- return $ isPosNat w
bvLit sym w (BV.zext w xv)
-- Concatenate unsign extension.
| Just (BVZext _ y) <- asApp x = do
-- Add dynamic check for GHC typechecker.
Just LeqProof <- return $ testLeq (incNat (bvWidth y)) w
Just LeqProof <- return $ testLeq (knownNat :: NatRepr 1) w
sbMakeExpr sym $ BVZext w y
-- Extend unary representation.
| Just (BVUnaryTerm u) <- asApp x = do
-- Add dynamic check for GHC typechecker.
Just LeqProof <- return $ isPosNat w
bvUnary sym $ UnaryBV.uext u w
| otherwise = do
Just LeqProof <- return $ testLeq (knownNat :: NatRepr 1) w
sbMakeExpr sym $ BVZext w x
bvSext sym w x
| Just xv <- asBV x = do
-- Add dynamic check for GHC typechecker.
Just LeqProof <- return $ isPosNat w
bvLit sym w (BV.sext (bvWidth x) w xv)
-- Concatenate sign extension.
| Just (BVSext _ y) <- asApp x = do
-- Add dynamic check for GHC typechecker.
Just LeqProof <- return $ testLeq (incNat (bvWidth y)) w
Just LeqProof <- return $ testLeq (knownNat :: NatRepr 1) w
sbMakeExpr sym (BVSext w y)
-- Extend unary representation.
| Just (BVUnaryTerm u) <- asApp x = do
-- Add dynamic check for GHC typechecker.
Just LeqProof <- return $ isPosNat w
bvUnary sym $ UnaryBV.sext u w
| otherwise = do
Just LeqProof <- return $ testLeq (knownNat :: NatRepr 1) w
sbMakeExpr sym (BVSext w x)
bvXorBits sym x y
| x == y = bvLit sym (bvWidth x) (BV.zero (bvWidth x)) -- special case: x `xor` x = 0
| otherwise
= let sr = SR.SemiRingBVRepr SR.BVBitsRepr (bvWidth x)
in semiRingAdd sym sr x y
bvAndBits sym x y
| x == y = return x -- Special case: idempotency of and
| Just (BVOrBits _ bs) <- asApp x
, bvOrContains y bs
= return y -- absorption law
| Just (BVOrBits _ bs) <- asApp y
, bvOrContains x bs
= return x -- absorption law
| otherwise
= let sr = SR.SemiRingBVRepr SR.BVBitsRepr (bvWidth x)
in semiRingMul sym sr x y
-- XOR by the all-1 constant of the bitwise semiring.
-- This is equivalant to negation
bvNotBits sym x
| Just xv <- asBV x
= bvLit sym (bvWidth x) $ xv `BV.xor` (BV.maxUnsigned (bvWidth x))
| otherwise
= let sr = (SR.SemiRingBVRepr SR.BVBitsRepr (bvWidth x))
in semiRingSum sym $ WSum.addConstant sr (asWeightedSum sr x) (BV.maxUnsigned (bvWidth x))
bvOrBits sym x y =
case (asBV x, asBV y) of
(Just xv, Just yv) -> bvLit sym (bvWidth x) (xv `BV.or` yv)
(Just xv , _)
| xv == BV.zero (bvWidth x) -> return y
| xv == BV.maxUnsigned (bvWidth x) -> return x
(_, Just yv)
| yv == BV.zero (bvWidth y) -> return x
| yv == BV.maxUnsigned (bvWidth x) -> return y
_
| x == y
-> return x -- or is idempotent
| Just (SemiRingProd xs) <- asApp x
, SR.SemiRingBVRepr SR.BVBitsRepr _w <- WSum.prodRepr xs
, WSum.prodContains xs y
-> return y -- absorption law
| Just (SemiRingProd ys) <- asApp y
, SR.SemiRingBVRepr SR.BVBitsRepr _w <- WSum.prodRepr ys
, WSum.prodContains ys x
-> return x -- absorption law
| Just (BVOrBits w xs) <- asApp x
, Just (BVOrBits _ ys) <- asApp y
-> sbMakeExpr sym $ BVOrBits w $ bvOrUnion xs ys
| Just (BVOrBits w xs) <- asApp x
-> sbMakeExpr sym $ BVOrBits w $ bvOrInsert y xs
| Just (BVOrBits w ys) <- asApp y
-> sbMakeExpr sym $ BVOrBits w $ bvOrInsert x ys
-- (or (shl x n) (zext w y)) is equivalent to (concat (trunc (w - n) x) y) when n is
-- the number of bits of y. Notice that the low bits of a shl expression are 0 and
-- the high bits of a zext expression are 0, thus the or expression is equivalent to
-- the concatenation between the high bits of the shl expression and the low bits of
-- the zext expression.
| Just (BVShl w x' n) <- asApp x
, Just (BVZext _ lo) <- asApp y
, Just ni <- BV.asUnsigned <$> asBV n
, intValue (bvWidth lo) == ni
, Just LeqProof <- testLeq (bvWidth lo) w -- dynamic check for GHC typechecker
, w' <- subNat w (bvWidth lo)
, Just LeqProof <- testLeq (knownNat @1) w' -- dynamic check for GHC typechecker
, Just LeqProof <- testLeq (addNat w' (knownNat @1)) w -- dynamic check for GHC typechecker
, Just Refl <- testEquality w (addNat w' (bvWidth lo)) -- dynamic check for GHC typechecker
-> do
hi <- bvTrunc sym w' x'
bvConcat sym hi lo
| Just (BVShl w y' n) <- asApp y
, Just (BVZext _ lo) <- asApp x
, Just ni <- BV.asUnsigned <$> asBV n
, intValue (bvWidth lo) == ni
, Just LeqProof <- testLeq (bvWidth lo) w -- dynamic check for GHC typechecker
, w' <- subNat w (bvWidth lo)
, Just LeqProof <- testLeq (knownNat @1) w' -- dynamic check for GHC typechecker
, Just LeqProof <- testLeq (addNat w' (knownNat @1)) w -- dynamic check for GHC typechecker
, Just Refl <- testEquality w (addNat w' (bvWidth lo)) -- dynamic check for GHC typechecker
-> do
hi <- bvTrunc sym w' y'
bvConcat sym hi lo
| otherwise
-> sbMakeExpr sym $ BVOrBits (bvWidth x) $ bvOrInsert x $ bvOrSingleton y
bvAdd sym x y = semiRingAdd sym sr x y
where sr = SR.SemiRingBVRepr SR.BVArithRepr (bvWidth x)
bvMul sym x y = semiRingMul sym sr x y
where sr = SR.SemiRingBVRepr SR.BVArithRepr (bvWidth x)
bvNeg sym x
| Just xv <- asBV x = bvLit sym (bvWidth x) (BV.negate (bvWidth x) xv)
| otherwise =
do ut <- CFG.getOpt (sbUnaryThreshold sym)
let ?unaryThreshold = fromInteger ut
sbTryUnaryTerm sym
(do ux <- asUnaryBV sym x
Just (UnaryBV.neg sym ux))
(do let sr = SR.SemiRingBVRepr SR.BVArithRepr (bvWidth x)
scalarMul sym sr (BV.mkBV (bvWidth x) (-1)) x)
bvIsNonzero sym x
| Just (BaseIte _ _ p t f) <- asApp x
, isJust (asBV t) || isJust (asBV f) -- NB, avoid losing possible sharing
= do t' <- bvIsNonzero sym t
f' <- bvIsNonzero sym f
itePred sym p t' f'
| Just (BVConcat _ a b) <- asApp x
, isJust (asBV a) || isJust (asBV b) -- NB, avoid losing possible sharing
= do pa <- bvIsNonzero sym a
pb <- bvIsNonzero sym b
orPred sym pa pb
| Just (BVZext _ y) <- asApp x =
bvIsNonzero sym y
| Just (BVSext _ y) <- asApp x =
bvIsNonzero sym y
| Just (BVFill _ p) <- asApp x =
return p
| Just (BVUnaryTerm ubv) <- asApp x =
UnaryBV.sym_evaluate
(\i -> return $! backendPred sym (i/=0))
(itePred sym)
ubv
| otherwise = do
let w = bvWidth x
zro <- bvLit sym w (BV.zero w)
notPred sym =<< bvEq sym x zro
bvUdiv = bvBinDivOp (const BV.uquot) BVUdiv
bvUrem sym x y
| Just True <- BVD.ult (exprAbsValue x) (exprAbsValue y) = return x
| otherwise = bvBinDivOp (const BV.urem) BVUrem sym x y
bvSdiv = bvBinDivOp BV.squot BVSdiv
bvSrem = bvBinDivOp BV.srem BVSrem
bvPopcount sym x
| Just xv <- asBV x = bvLit sym w (BV.popCount xv)
| otherwise = sbMakeExpr sym $ BVPopcount w x
where w = bvWidth x
bvCountTrailingZeros sym x
| Just xv <- asBV x = bvLit sym w (BV.ctz w xv)
| otherwise = sbMakeExpr sym $ BVCountTrailingZeros w x
where w = bvWidth x
bvCountLeadingZeros sym x
| Just xv <- asBV x = bvLit sym w (BV.clz w xv)
| otherwise = sbMakeExpr sym $ BVCountLeadingZeros w x
where w = bvWidth x
mkStruct sym args = do
sbMakeExpr sym $ StructCtor (fmapFC exprType args) args
structField sym s i
| Just (StructCtor _ args) <- asApp s = return $! args Ctx.! i
| otherwise = do
case exprType s of
BaseStructRepr flds ->
sbMakeExpr sym $ StructField s i (flds Ctx.! i)
structIte sym p x y
| Just True <- asConstantPred p = return x
| Just False <- asConstantPred p = return y
| x == y = return x
| otherwise = mkIte sym p x y
--------------------------------------------------------------------
-- String operations
stringEmpty sym si = stringLit sym (stringLitEmpty si)
stringLit sym s =
do l <- curProgramLoc sym
return $! StringExpr s l
stringEq sym x y
| Just x' <- asString x
, Just y' <- asString y
= return $! backendPred sym (isJust (testEquality x' y'))
stringEq sym x y
= sbMakeExpr sym $ BaseEq (BaseStringRepr (stringInfo x)) x y
stringIte _sym c x y
| Just c' <- asConstantPred c
= if c' then return x else return y
stringIte _sym _c x y
| Just x' <- asString x
, Just y' <- asString y
, isJust (testEquality x' y')
= return x
stringIte sym c x y
= mkIte sym c x y
stringIndexOf sym x y k
| Just x' <- asString x
, Just y' <- asString y
, Just k' <- asInteger k
= intLit sym $! stringLitIndexOf x' y' k'
stringIndexOf sym x y k
= sbMakeExpr sym $ StringIndexOf x y k
stringContains sym x y
| Just x' <- asString x
, Just y' <- asString y
= return $! backendPred sym (stringLitContains x' y')
| Just b <- stringAbsContains (getAbsValue x) (getAbsValue y)
= return $! backendPred sym b
| otherwise
= sbMakeExpr sym $ StringContains x y
stringIsPrefixOf sym x y
| Just x' <- asString x
, Just y' <- asString y
= return $! backendPred sym (stringLitIsPrefixOf x' y')
| Just b <- stringAbsIsPrefixOf (getAbsValue x) (getAbsValue y)
= return $! backendPred sym b
| otherwise
= sbMakeExpr sym $ StringIsPrefixOf x y
stringIsSuffixOf sym x y
| Just x' <- asString x
, Just y' <- asString y
= return $! backendPred sym (stringLitIsSuffixOf x' y')
| Just b <- stringAbsIsSuffixOf (getAbsValue x) (getAbsValue y)
= return $! backendPred sym b
| otherwise
= sbMakeExpr sym $ StringIsSuffixOf x y
stringSubstring sym x off len
| Just x' <- asString x
, Just off' <- asInteger off
, Just len' <- asInteger len
= stringLit sym $! stringLitSubstring x' off' len'
| otherwise
= sbMakeExpr sym $ StringSubstring (stringInfo x) x off len
stringConcat sym x y
| Just x' <- asString x, stringLitNull x'
= return y
| Just y' <- asString y, stringLitNull y'
= return x
| Just x' <- asString x
, Just y' <- asString y
= stringLit sym (x' <> y')
| Just (StringAppend si xs) <- asApp x
, Just (StringAppend _ ys) <- asApp y
= sbMakeExpr sym $ StringAppend si (SSeq.append xs ys)
| Just (StringAppend si xs) <- asApp x
= sbMakeExpr sym $ StringAppend si (SSeq.append xs (SSeq.singleton si y))
| Just (StringAppend si ys) <- asApp y
= sbMakeExpr sym $ StringAppend si (SSeq.append (SSeq.singleton si x) ys)
| otherwise
= let si = stringInfo x in
sbMakeExpr sym $ StringAppend si (SSeq.append (SSeq.singleton si x) (SSeq.singleton si y))
stringLength sym x
| Just x' <- asString x
= intLit sym (stringLitLength x')
| Just (StringAppend _si xs) <- asApp x
= do let f sm (SSeq.StringSeqLiteral l) = intAdd sym sm =<< intLit sym (stringLitLength l)
f sm (SSeq.StringSeqTerm t) = intAdd sym sm =<< sbMakeExpr sym (StringLength t)
z <- intLit sym 0
foldM f z (SSeq.toList xs)
| otherwise
= sbMakeExpr sym $ StringLength x
--------------------------------------------------------------------
-- Symbolic array operations
constantArray sym idxRepr v =
sbMakeExpr sym $ ConstantArray idxRepr (exprType v) v
arrayFromFn sym fn = do
sbNonceExpr sym $ ArrayFromFn fn
arrayMap sym f arrays
-- Cancel out integerToReal (realToInteger a)
| Just IntegerToRealFn <- asMatlabSolverFn f
, Just (MapOverArrays g _ args) <- asNonceApp (unwrapArrayResult (arrays^._1))
, Just RealToIntegerFn <- asMatlabSolverFn g =
return $! unwrapArrayResult (args^._1)
-- Cancel out realToInteger (integerToReal a)
| Just RealToIntegerFn <- asMatlabSolverFn f
, Just (MapOverArrays g _ args) <- asNonceApp (unwrapArrayResult (arrays^._1))
, Just IntegerToRealFn <- asMatlabSolverFn g =
return $! unwrapArrayResult (args^._1)
-- When the array is an update of concrete entries, map over the entries.
| s <- concreteArrayEntries arrays
, not (Set.null s) = do
-- Distribute over base values.
--
-- The underlyingArrayMapElf function strings a top-level arrayMap value.
--
-- It is ok because we don't care what the value of base is at any index
-- in s.
base <- arrayMap sym f (fmapFC underlyingArrayMapExpr arrays)
BaseArrayRepr _ ret <- return (exprType base)
-- This lookups a given index in an array used as an argument.
let evalArgs :: Ctx.Assignment IndexLit (idx ::> itp)
-- ^ A representatio of the concrete index (if defined).
-> Ctx.Assignment (Expr t) (idx ::> itp)
-- ^ The index to use.
-> ArrayResultWrapper (Expr t) (idx ::> itp) d
-- ^ The array to get the value at.
-> IO (Expr t d)
evalArgs const_idx sym_idx a = do
sbConcreteLookup sym (unwrapArrayResult a) (Just const_idx) sym_idx
let evalIndex :: ExprSymFn t ctx ret
-> Ctx.Assignment (ArrayResultWrapper (Expr t) (i::>itp)) ctx
-> Ctx.Assignment IndexLit (i::>itp)
-> IO (Expr t ret)
evalIndex g arrays0 const_idx = do
sym_idx <- traverseFC (indexLit sym) const_idx
applySymFn sym g =<< traverseFC (evalArgs const_idx sym_idx) arrays0
m <- AUM.fromAscList ret <$> mapM (\k -> (k,) <$> evalIndex f arrays k) (Set.toAscList s)
arrayUpdateAtIdxLits sym m base
-- When entries are constants, then just evaluate constant.
| Just cns <- traverseFC (\a -> asConstantArray (unwrapArrayResult a)) arrays = do
r <- betaReduce sym f cns
case exprType (unwrapArrayResult (Ctx.last arrays)) of
BaseArrayRepr idxRepr _ -> do
constantArray sym idxRepr r
| otherwise = do
let idx = arrayResultIdxType (exprType (unwrapArrayResult (Ctx.last arrays)))
sbNonceExpr sym $ MapOverArrays f idx arrays
arrayUpdate sym arr i v
-- Update at concrete index.
| Just ci <- asConcreteIndices i =
case asApp arr of
Just (ArrayMap idx tp m def) -> do
let new_map =
case asApp def of
Just (ConstantArray _ _ cns) | v == cns -> AUM.delete ci m
_ -> AUM.insert tp ci v m
sbMakeExpr sym $ ArrayMap idx tp new_map def
_ -> do
let idx = fmapFC exprType i
let bRepr = exprType v
let new_map = AUM.singleton bRepr ci v
sbMakeExpr sym $ ArrayMap idx bRepr new_map arr
| otherwise = do
let bRepr = exprType v
sbMakeExpr sym (UpdateArray bRepr (fmapFC exprType i) arr i v)
arrayLookup sym arr idx =
sbConcreteLookup sym arr (asConcreteIndices idx) idx
-- | Create an array from a map of concrete indices to values.
arrayUpdateAtIdxLits sym m def_map = do
BaseArrayRepr idx_tps baseRepr <- return $ exprType def_map
let new_map
| Just (ConstantArray _ _ default_value) <- asApp def_map =
AUM.filter (/= default_value) m
| otherwise = m
if AUM.null new_map then
return def_map
else
sbMakeExpr sym $ ArrayMap idx_tps baseRepr new_map def_map
arrayIte sym p x y
-- Extract all concrete updates out.
| ArrayMapView mx x' <- viewArrayMap x
, ArrayMapView my y' <- viewArrayMap y
, not (AUM.null mx) || not (AUM.null my) = do
case exprType x of
BaseArrayRepr idxRepr bRepr -> do
let both_fn _ u v = baseTypeIte sym p u v
left_fn idx u = do
v <- sbConcreteLookup sym y' (Just idx) =<< symbolicIndices sym idx
both_fn idx u v
right_fn idx v = do
u <- sbConcreteLookup sym x' (Just idx) =<< symbolicIndices sym idx
both_fn idx u v
mz <- AUM.mergeM bRepr both_fn left_fn right_fn mx my
z' <- arrayIte sym p x' y'
sbMakeExpr sym $ ArrayMap idxRepr bRepr mz z'
| otherwise = mkIte sym p x y
arrayEq sym x y
| x == y =
return $! truePred sym
| otherwise =
sbMakeExpr sym $! BaseEq (exprType x) x y
arrayTrueOnEntries sym f a
| Just True <- exprAbsValue a =
return $ truePred sym
| Just (IndicesInRange _ bnds) <- asMatlabSolverFn f
, Just v <- asIntBounds bnds = do
let h :: Expr t (BaseArrayType (i::>it) BaseBoolType)
-> BoolExpr t
-> Ctx.Assignment (Expr t) (i::>it)
-> IO (BoolExpr t)
h a0 p i = andPred sym p =<< arrayLookup sym a0 i
foldIndicesInRangeBounds sym (h a) (truePred sym) v
| otherwise =
sbNonceExpr sym $! ArrayTrueOnEntries f a
----------------------------------------------------------------------
-- Lossless (injective) conversions
integerToReal sym x
| SemiRingLiteral SR.SemiRingIntegerRepr i l <- x = return $! SemiRingLiteral SR.SemiRingRealRepr (toRational i) l
| Just (RealToInteger y) <- asApp x = return y
| otherwise = sbMakeExpr sym (IntegerToReal x)
realToInteger sym x
-- Ground case
| SemiRingLiteral SR.SemiRingRealRepr r l <- x = return $! SemiRingLiteral SR.SemiRingIntegerRepr (floor r) l
-- Match integerToReal
| Just (IntegerToReal xi) <- asApp x = return xi
-- Static case
| otherwise =
sbMakeExpr sym (RealToInteger x)
bvToInteger sym x
| Just xv <- asBV x =
intLit sym (BV.asUnsigned xv)
-- bvToInteger (integerToBv x w) == mod x (2^w)
| Just (IntegerToBV xi w) <- asApp x =
intMod sym xi =<< intLit sym (2^natValue w)
| otherwise =
sbMakeExpr sym (BVToInteger x)
sbvToInteger sym x
| Just xv <- asBV x =
intLit sym (BV.asSigned (bvWidth x) xv)
-- sbvToInteger (integerToBv x w) == mod (x + 2^(w-1)) (2^w) - 2^(w-1)
| Just (IntegerToBV xi w) <- asApp x =
do halfmod <- intLit sym (2 ^ (natValue w - 1))
modulus <- intLit sym (2 ^ natValue w)
x' <- intAdd sym xi halfmod
z <- intMod sym x' modulus
intSub sym z halfmod
| otherwise =
sbMakeExpr sym (SBVToInteger x)
predToBV sym p w
| Just b <- asConstantPred p =
if b then bvLit sym w (BV.one w) else bvLit sym w (BV.zero w)
| otherwise =
case testNatCases w (knownNat @1) of
NatCaseEQ -> sbMakeExpr sym (BVFill (knownNat @1) p)
NatCaseGT LeqProof -> bvZext sym w =<< sbMakeExpr sym (BVFill (knownNat @1) p)
NatCaseLT LeqProof -> fail "impossible case in predToBV"
integerToBV sym xr w
| SemiRingLiteral SR.SemiRingIntegerRepr i _ <- xr =
bvLit sym w (BV.mkBV w i)
| Just (BVToInteger r) <- asApp xr =
case testNatCases (bvWidth r) w of
NatCaseLT LeqProof -> bvZext sym w r
NatCaseEQ -> return r
NatCaseGT LeqProof -> bvTrunc sym w r
| Just (SBVToInteger r) <- asApp xr =
case testNatCases (bvWidth r) w of
NatCaseLT LeqProof -> bvSext sym w r
NatCaseEQ -> return r
NatCaseGT LeqProof -> bvTrunc sym w r
| otherwise =
sbMakeExpr sym (IntegerToBV xr w)
realRound sym x
-- Ground case
| SemiRingLiteral SR.SemiRingRealRepr r l <- x = return $ SemiRingLiteral SR.SemiRingIntegerRepr (roundAway r) l
-- Match integerToReal
| Just (IntegerToReal xi) <- asApp x = return xi
-- Static case
| Just True <- ravIsInteger (exprAbsValue x) =
sbMakeExpr sym (RealToInteger x)
-- Unsimplified case
| otherwise = sbMakeExpr sym (RoundReal x)
realRoundEven sym x
-- Ground case
| SemiRingLiteral SR.SemiRingRealRepr r l <- x = return $ SemiRingLiteral SR.SemiRingIntegerRepr (round r) l
-- Match integerToReal
| Just (IntegerToReal xi) <- asApp x = return xi
-- Static case
| Just True <- ravIsInteger (exprAbsValue x) =
sbMakeExpr sym (RealToInteger x)
-- Unsimplified case
| otherwise = sbMakeExpr sym (RoundEvenReal x)
realFloor sym x
-- Ground case
| SemiRingLiteral SR.SemiRingRealRepr r l <- x = return $ SemiRingLiteral SR.SemiRingIntegerRepr (floor r) l
-- Match integerToReal
| Just (IntegerToReal xi) <- asApp x = return xi
-- Static case
| Just True <- ravIsInteger (exprAbsValue x) =
sbMakeExpr sym (RealToInteger x)
-- Unsimplified case
| otherwise = sbMakeExpr sym (FloorReal x)
realCeil sym x
-- Ground case
| SemiRingLiteral SR.SemiRingRealRepr r l <- x = return $ SemiRingLiteral SR.SemiRingIntegerRepr (ceiling r) l
-- Match integerToReal
| Just (IntegerToReal xi) <- asApp x = return xi
-- Static case
| Just True <- ravIsInteger (exprAbsValue x) =
sbMakeExpr sym (RealToInteger x)
-- Unsimplified case
| otherwise = sbMakeExpr sym (CeilReal x)
----------------------------------------------------------------------
-- Real operations
realLit sb r = do
l <- curProgramLoc sb
return (SemiRingLiteral SR.SemiRingRealRepr r l)
realZero = sbZero
realEq sym x y
-- Use range check
| Just b <- ravCheckEq (exprAbsValue x) (exprAbsValue y)
= return $ backendPred sym b
-- Reduce to integer equality, when possible
| Just (IntegerToReal xi) <- asApp x
, Just (IntegerToReal yi) <- asApp y
= intEq sym xi yi
| Just (IntegerToReal xi) <- asApp x
, SemiRingLiteral SR.SemiRingRealRepr yr _ <- y
= if denominator yr == 1
then intEq sym xi =<< intLit sym (numerator yr)
else return (falsePred sym)
| SemiRingLiteral SR.SemiRingRealRepr xr _ <- x
, Just (IntegerToReal yi) <- asApp y
= if denominator xr == 1
then intEq sym yi =<< intLit sym (numerator xr)
else return (falsePred sym)
| otherwise
= semiRingEq sym SR.SemiRingRealRepr (realEq sym) x y
realLe sym x y
-- Use range check
| Just b <- ravCheckLe (exprAbsValue x) (exprAbsValue y)
= return $ backendPred sym b
-- Reduce to integer inequality, when possible
| Just (IntegerToReal xi) <- asApp x
, Just (IntegerToReal yi) <- asApp y
= intLe sym xi yi
-- if the upper range is a constant, do an integer comparison
-- with @floor(y)@
| Just (IntegerToReal xi) <- asApp x
, SemiRingLiteral SR.SemiRingRealRepr yr _ <- y
= join (intLe sym <$> pure xi <*> intLit sym (floor yr))
-- if the lower range is a constant, do an integer comparison
-- with @ceiling(x)@
| SemiRingLiteral SR.SemiRingRealRepr xr _ <- x
, Just (IntegerToReal yi) <- asApp y
= join (intLe sym <$> intLit sym (ceiling xr) <*> pure yi)
| otherwise
= semiRingLe sym SR.OrderedSemiRingRealRepr (realLe sym) x y
realIte sym c x y = semiRingIte sym SR.SemiRingRealRepr c x y
realNeg sym x = scalarMul sym SR.SemiRingRealRepr (-1) x
realAdd sym x y = semiRingAdd sym SR.SemiRingRealRepr x y
realMul sym x y = semiRingMul sym SR.SemiRingRealRepr x y
realDiv sym x y
| Just 0 <- asRational x =
return x
| Just xd <- asRational x, Just yd <- asRational y, yd /= 0 = do
realLit sym (xd / yd)
-- Handle division by a constant.
| Just yd <- asRational y, yd /= 0 = do
scalarMul sym SR.SemiRingRealRepr (1 / yd) x
| otherwise =
sbMakeExpr sym $ RealDiv x y
isInteger sb x
| Just r <- asRational x = return $ backendPred sb (denominator r == 1)
| Just b <- ravIsInteger (exprAbsValue x) = return $ backendPred sb b
| otherwise = sbMakeExpr sb $ RealIsInteger x
realSqrt sym x = do
let sqrt_dbl :: Double -> Double
sqrt_dbl = sqrt
case x of
SemiRingLiteral SR.SemiRingRealRepr r _
| r < 0 -> sbMakeExpr sym (RealSqrt x)
| Just w <- tryRationalSqrt r -> realLit sym w
| sbFloatReduce sym -> realLit sym (toRational (sqrt_dbl (fromRational r)))
_ -> sbMakeExpr sym (RealSqrt x)
realPi sym = do
if sbFloatReduce sym then
realLit sym (toRational (pi :: Double))
else
sbMakeExpr sym Pi
realSin sym x =
case asRational x of
Just 0 -> realLit sym 0
Just c | sbFloatReduce sym -> realLit sym (toRational (sin (toDouble c)))
_ -> sbMakeExpr sym (RealSin x)
realCos sym x =
case asRational x of
Just 0 -> realLit sym 1
Just c | sbFloatReduce sym -> realLit sym (toRational (cos (toDouble c)))
_ -> sbMakeExpr sym (RealCos x)
realAtan2 sb y x = do
case (asRational y, asRational x) of
(Just 0, _) -> realLit sb 0
(Just yc, Just xc) | xc /= 0, sbFloatReduce sb -> do
realLit sb (toRational (atan2 (toDouble yc) (toDouble xc)))
_ -> sbMakeExpr sb (RealATan2 y x)
realSinh sb x =
case asRational x of
Just 0 -> realLit sb 0
Just c | sbFloatReduce sb -> realLit sb (toRational (sinh (toDouble c)))
_ -> sbMakeExpr sb (RealSinh x)
realCosh sb x =
case asRational x of
Just 0 -> realLit sb 1
Just c | sbFloatReduce sb -> realLit sb (toRational (cosh (toDouble c)))
_ -> sbMakeExpr sb (RealCosh x)
realExp sym x
| Just 0 <- asRational x = realLit sym 1
| Just c <- asRational x, sbFloatReduce sym = realLit sym (toRational (exp (toDouble c)))
| otherwise = sbMakeExpr sym (RealExp x)
realLog sym x =
case asRational x of
Just c | c > 0, sbFloatReduce sym -> realLit sym (toRational (log (toDouble c)))
_ -> sbMakeExpr sym (RealLog x)
----------------------------------------------------------------------
-- IEEE-754 floating-point operations
floatLit sym fpp f =
do l <- curProgramLoc sym
return $! FloatExpr fpp f l
floatPZero sym fpp = floatLit sym fpp BF.bfPosZero
floatNZero sym fpp = floatLit sym fpp BF.bfNegZero
floatNaN sym fpp = floatLit sym fpp BF.bfNaN
floatPInf sym fpp = floatLit sym fpp BF.bfPosInf
floatNInf sym fpp = floatLit sym fpp BF.bfNegInf
floatNeg sym (FloatExpr fpp x _) = floatLit sym fpp (BF.bfNeg x)
floatNeg sym x = floatIEEEArithUnOp FloatNeg sym x
floatAbs sym (FloatExpr fpp x _) = floatLit sym fpp (BF.bfAbs x)
floatAbs sym x = floatIEEEArithUnOp FloatAbs sym x
floatSqrt sym r (FloatExpr fpp x _) =
floatLit sym fpp (bfStatus (BF.bfSqrt (fppOpts fpp r) x))
floatSqrt sym r x = floatIEEEArithUnOpR FloatSqrt sym r x
floatAdd sym r (FloatExpr fpp x _) (FloatExpr _ y _) =
floatLit sym fpp (bfStatus (BF.bfAdd (fppOpts fpp r) x y))
floatAdd sym r x y = floatIEEEArithBinOpR FloatAdd sym r x y
floatSub sym r (FloatExpr fpp x _) (FloatExpr _ y _) =
floatLit sym fpp (bfStatus (BF.bfSub (fppOpts fpp r) x y ))
floatSub sym r x y = floatIEEEArithBinOpR FloatSub sym r x y
floatMul sym r (FloatExpr fpp x _) (FloatExpr _ y _) =
floatLit sym fpp (bfStatus (BF.bfMul (fppOpts fpp r) x y))
floatMul sym r x y = floatIEEEArithBinOpR FloatMul sym r x y
floatDiv sym r (FloatExpr fpp x _) (FloatExpr _ y _) =
floatLit sym fpp (bfStatus (BF.bfDiv (fppOpts fpp r) x y))
floatDiv sym r x y = floatIEEEArithBinOpR FloatDiv sym r x y
floatRem sym (FloatExpr fpp x _) (FloatExpr _ y _) =
floatLit sym fpp (bfStatus (BF.bfRem (fppOpts fpp RNE) x y))
floatRem sym x y = floatIEEEArithBinOp FloatRem sym x y
floatFMA sym r (FloatExpr fpp x _) (FloatExpr _ y _) (FloatExpr _ z _) =
floatLit sym fpp (bfStatus (BF.bfFMA (fppOpts fpp r) x y z))
floatFMA sym r x y z =
let BaseFloatRepr fpp = exprType x in sbMakeExpr sym $ FloatFMA fpp r x y z
floatEq sym (FloatExpr _ x _) (FloatExpr _ y _) =
pure . backendPred sym $! (BF.bfCompare x y == EQ)
floatEq sym x y
| x == y = return $! truePred sym
| otherwise = floatIEEELogicBinOp (BaseEq (exprType x)) sym x y
floatNe sym x y = notPred sym =<< floatEq sym x y
floatFpEq sym (FloatExpr _ x _) (FloatExpr _ y _) =
pure . backendPred sym $! (x == y)
floatFpEq sym x y
| x == y = notPred sym =<< floatIsNaN sym x
| otherwise = floatIEEELogicBinOp FloatFpEq sym x y
floatLe sym (FloatExpr _ x _) (FloatExpr _ y _) =
pure . backendPred sym $! (x <= y)
floatLe sym x y
| x == y = notPred sym =<< floatIsNaN sym x
| otherwise = floatIEEELogicBinOp FloatLe sym x y
floatLt sym (FloatExpr _ x _) (FloatExpr _ y _) =
pure . backendPred sym $! (x < y)
floatLt sym x y
| x == y = return $ falsePred sym
| otherwise = floatIEEELogicBinOp FloatLt sym x y
floatGe sym x y = floatLe sym y x
floatGt sym x y = floatLt sym y x
floatIte sym c x y = mkIte sym c x y
floatIsNaN sym (FloatExpr _ x _) =
pure . backendPred sym $! BF.bfIsNaN x
floatIsNaN sym x = floatIEEELogicUnOp FloatIsNaN sym x
floatIsInf sym (FloatExpr _ x _) =
pure . backendPred sym $! BF.bfIsInf x
floatIsInf sym x = floatIEEELogicUnOp FloatIsInf sym x
floatIsZero sym (FloatExpr _ x _) =
pure . backendPred sym $! BF.bfIsZero x
floatIsZero sym x = floatIEEELogicUnOp FloatIsZero sym x
floatIsPos sym (FloatExpr _ x _) =
pure . backendPred sym $! BF.bfIsPos x
floatIsPos sym x = floatIEEELogicUnOp FloatIsPos sym x
floatIsNeg sym (FloatExpr _ x _) =
pure . backendPred sym $! BF.bfIsNeg x
floatIsNeg sym x = floatIEEELogicUnOp FloatIsNeg sym x
floatIsSubnorm sym (FloatExpr fpp x _) =
pure . backendPred sym $! BF.bfIsSubnormal (fppOpts fpp RNE) x
floatIsSubnorm sym x = floatIEEELogicUnOp FloatIsSubnorm sym x
floatIsNorm sym (FloatExpr fpp x _) =
pure . backendPred sym $! BF.bfIsNormal (fppOpts fpp RNE) x
floatIsNorm sym x = floatIEEELogicUnOp FloatIsNorm sym x
floatCast sym fpp r (FloatExpr _ x _) =
floatLit sym fpp (bfStatus (BF.bfRoundFloat (fppOpts fpp r) x))
floatCast sym fpp r x
| FloatingPointPrecisionRepr eb sb <- fpp
, Just (FloatCast (FloatingPointPrecisionRepr eb' sb') _ fval) <- asApp x
, natValue eb <= natValue eb'
, natValue sb <= natValue sb'
, Just Refl <- testEquality (BaseFloatRepr fpp) (exprType fval)
= return fval
| otherwise = sbMakeExpr sym $ FloatCast fpp r x
floatRound sym r (FloatExpr fpp x _) =
floatLit sym fpp (floatRoundToInt fpp r x)
floatRound sym r x = floatIEEEArithUnOpR FloatRound sym r x
floatFromBinary sym fpp x
| Just bv <- asBV x
= floatLit sym fpp (BF.bfFromBits (fppOpts fpp RNE) (BV.asUnsigned bv))
| Just (FloatToBinary fpp' fval) <- asApp x
, Just Refl <- testEquality fpp fpp'
= return fval
| otherwise = sbMakeExpr sym $ FloatFromBinary fpp x
floatToBinary sym (FloatExpr fpp@(FloatingPointPrecisionRepr eb sb) x _)
| Just LeqProof <- isPosNat (addNat eb sb) =
bvLit sym (addNat eb sb) (BV.mkBV (addNat eb sb) (BF.bfToBits (fppOpts fpp RNE) x))
floatToBinary sym x = case exprType x of
BaseFloatRepr fpp | LeqProof <- lemmaFloatPrecisionIsPos fpp ->
sbMakeExpr sym $ FloatToBinary fpp x
floatMin sym x y =
iteList floatIte sym
[ (floatIsNaN sym x, pure y)
, (floatIsNaN sym y, pure x)
, (floatLt sym x y , pure x)
, (floatLt sym y x , pure y)
, (floatEq sym x y , pure x) -- NB logical equality, not IEEE 754 equality
]
-- The only way to get here is if x and y are zeros
-- with different sign.
-- Return one of the two values nondeterministicly.
(do b <- freshConstant sym emptySymbol BaseBoolRepr
floatIte sym b x y)
floatMax sym x y =
iteList floatIte sym
[ (floatIsNaN sym x, pure y)
, (floatIsNaN sym y, pure x)
, (floatLt sym x y , pure y)
, (floatLt sym y x , pure x)
, (floatEq sym x y , pure x) -- NB logical equality, not IEEE 754 equality
]
-- The only way to get here is if x and y are zeros
-- with different sign.
-- Return one of the two values nondeterministicly.
(do b <- freshConstant sym emptySymbol BaseBoolRepr
floatIte sym b x y)
bvToFloat sym fpp r x
| Just bv <- asBV x = floatLit sym fpp (floatFromInteger (fppOpts fpp r) (BV.asUnsigned bv))
| otherwise = sbMakeExpr sym (BVToFloat fpp r x)
sbvToFloat sym fpp r x
| Just bv <- asBV x = floatLit sym fpp (floatFromInteger (fppOpts fpp r) (BV.asSigned (bvWidth x) bv))
| otherwise = sbMakeExpr sym (SBVToFloat fpp r x)
realToFloat sym fpp r x
| Just x' <- asRational x = floatLit sym fpp (floatFromRational (fppOpts fpp r) x')
| otherwise = sbMakeExpr sym (RealToFloat fpp r x)
floatToBV sym w r x
| FloatExpr _ bf _ <- x
, Just i <- floatToInteger r bf
, 0 <= i && i <= maxUnsigned w
= bvLit sym w (BV.mkBV w i)
| otherwise = sbMakeExpr sym (FloatToBV w r x)
floatToSBV sym w r x
| FloatExpr _ bf _ <- x
, Just i <- floatToInteger r bf
, minSigned w <= i && i <= maxSigned w
= bvLit sym w (BV.mkBV w i)
| otherwise = sbMakeExpr sym (FloatToSBV w r x)
floatToReal sym x
| FloatExpr _ bf _ <- x
, Just q <- floatToRational bf
= realLit sym q
| otherwise = sbMakeExpr sym (FloatToReal x)
----------------------------------------------------------------------
-- Cplx operations
mkComplex sym c = sbMakeExpr sym (Cplx c)
getRealPart _ e
| Just (Cplx (r :+ _)) <- asApp e = return r
getRealPart sym x =
sbMakeExpr sym (RealPart x)
getImagPart _ e
| Just (Cplx (_ :+ i)) <- asApp e = return i
getImagPart sym x =
sbMakeExpr sym (ImagPart x)
cplxGetParts _ e
| Just (Cplx c) <- asApp e = return c
cplxGetParts sym x =
(:+) <$> sbMakeExpr sym (RealPart x)
<*> sbMakeExpr sym (ImagPart x)
inSameBVSemiRing :: Expr t (BaseBVType w) -> Expr t (BaseBVType w) -> Maybe (Some SR.BVFlavorRepr)
inSameBVSemiRing x y
| Just (SemiRingSum s1) <- asApp x
, Just (SemiRingSum s2) <- asApp y
, SR.SemiRingBVRepr flv1 _w <- WSum.sumRepr s1
, SR.SemiRingBVRepr flv2 _w <- WSum.sumRepr s2
, Just Refl <- testEquality flv1 flv2
= Just (Some flv1)
| otherwise
= Nothing
floatIEEEArithBinOp
:: (e ~ Expr t)
=> ( FloatPrecisionRepr fpp
-> e (BaseFloatType fpp)
-> e (BaseFloatType fpp)
-> App e (BaseFloatType fpp)
)
-> ExprBuilder t st fs
-> e (BaseFloatType fpp)
-> e (BaseFloatType fpp)
-> IO (e (BaseFloatType fpp))
floatIEEEArithBinOp ctor sym x y =
let BaseFloatRepr fpp = exprType x in sbMakeExpr sym $ ctor fpp x y
floatIEEEArithBinOpR
:: (e ~ Expr t)
=> ( FloatPrecisionRepr fpp
-> RoundingMode
-> e (BaseFloatType fpp)
-> e (BaseFloatType fpp)
-> App e (BaseFloatType fpp)
)
-> ExprBuilder t st fs
-> RoundingMode
-> e (BaseFloatType fpp)
-> e (BaseFloatType fpp)
-> IO (e (BaseFloatType fpp))
floatIEEEArithBinOpR ctor sym r x y =
let BaseFloatRepr fpp = exprType x in sbMakeExpr sym $ ctor fpp r x y
floatIEEEArithUnOp
:: (e ~ Expr t)
=> ( FloatPrecisionRepr fpp
-> e (BaseFloatType fpp)
-> App e (BaseFloatType fpp)
)
-> ExprBuilder t st fs
-> e (BaseFloatType fpp)
-> IO (e (BaseFloatType fpp))
floatIEEEArithUnOp ctor sym x =
let BaseFloatRepr fpp = exprType x in sbMakeExpr sym $ ctor fpp x
floatIEEEArithUnOpR
:: (e ~ Expr t)
=> ( FloatPrecisionRepr fpp
-> RoundingMode
-> e (BaseFloatType fpp)
-> App e (BaseFloatType fpp)
)
-> ExprBuilder t st fs
-> RoundingMode
-> e (BaseFloatType fpp)
-> IO (e (BaseFloatType fpp))
floatIEEEArithUnOpR ctor sym r x =
let BaseFloatRepr fpp = exprType x in sbMakeExpr sym $ ctor fpp r x
floatIEEELogicBinOp
:: (e ~ Expr t)
=> (e (BaseFloatType fpp) -> e (BaseFloatType fpp) -> App e BaseBoolType)
-> ExprBuilder t st fs
-> e (BaseFloatType fpp)
-> e (BaseFloatType fpp)
-> IO (e BaseBoolType)
floatIEEELogicBinOp ctor sym x y = sbMakeExpr sym $ ctor x y
floatIEEELogicUnOp
:: (e ~ Expr t)
=> (e (BaseFloatType fpp) -> App e BaseBoolType)
-> ExprBuilder t st fs
-> e (BaseFloatType fpp)
-> IO (e BaseBoolType)
floatIEEELogicUnOp ctor sym x = sbMakeExpr sym $ ctor x
----------------------------------------------------------------------
-- Float interpretations
type instance SymInterpretedFloatType (ExprBuilder t st (Flags FloatReal)) fi =
BaseRealType
instance IsInterpretedFloatExprBuilder (ExprBuilder t st (Flags FloatReal)) where
iFloatPZero sym _ = return $ realZero sym
iFloatNZero sym _ = return $ realZero sym
iFloatNaN _ _ = fail "NaN cannot be represented as a real value."
iFloatPInf _ _ = fail "+Infinity cannot be represented as a real value."
iFloatNInf _ _ = fail "-Infinity cannot be represented as a real value."
iFloatLitRational sym _ = realLit sym
iFloatLitSingle sym = realLit sym . toRational
iFloatLitDouble sym = realLit sym . toRational
iFloatLitLongDouble sym x =
case fp80ToRational x of
Nothing -> fail ("80-bit floating point value does not represent a rational number: " ++ show x)
Just r -> realLit sym r
iFloatNeg = realNeg
iFloatAbs = realAbs
iFloatSqrt sym _ = realSqrt sym
iFloatAdd sym _ = realAdd sym
iFloatSub sym _ = realSub sym
iFloatMul sym _ = realMul sym
iFloatDiv sym _ = realDiv sym
iFloatRem = realMod
iFloatMin sym x y = do
c <- realLe sym x y
realIte sym c x y
iFloatMax sym x y = do
c <- realGe sym x y
realIte sym c x y
iFloatFMA sym _ x y z = do
tmp <- (realMul sym x y)
realAdd sym tmp z
iFloatEq = realEq
iFloatNe = realNe
iFloatFpEq = realEq
iFloatFpApart = realNe
iFloatLe = realLe
iFloatLt = realLt
iFloatGe = realGe
iFloatGt = realGt
iFloatIte = realIte
iFloatIsNaN sym _ = return $ falsePred sym
iFloatIsInf sym _ = return $ falsePred sym
iFloatIsZero sym = realEq sym $ realZero sym
iFloatIsPos sym = realLt sym $ realZero sym
iFloatIsNeg sym = realGt sym $ realZero sym
iFloatIsSubnorm sym _ = return $ falsePred sym
iFloatIsNorm sym = realNe sym $ realZero sym
iFloatCast _ _ _ = return
iFloatRound sym r x =
integerToReal sym =<< case r of
RNA -> realRound sym x
RTP -> realCeil sym x
RTN -> realFloor sym x
RTZ -> do
is_pos <- realLt sym (realZero sym) x
iteM intIte sym is_pos (realFloor sym x) (realCeil sym x)
RNE -> fail "Unsupported rond to nearest even for real values."
iFloatFromBinary sym _ x
| Just (FnApp fn args) <- asNonceApp x
, "uninterpreted_real_to_float_binary" == solverSymbolAsText (symFnName fn)
, UninterpFnInfo param_types (BaseBVRepr _) <- symFnInfo fn
, (Ctx.Empty Ctx.:> BaseRealRepr) <- param_types
, (Ctx.Empty Ctx.:> rval) <- args
= return rval
| otherwise = mkFreshUninterpFnApp sym
"uninterpreted_real_from_float_binary"
(Ctx.Empty Ctx.:> x)
knownRepr
iFloatToBinary sym fi x =
mkFreshUninterpFnApp sym
"uninterpreted_real_to_float_binary"
(Ctx.Empty Ctx.:> x)
(floatInfoToBVTypeRepr fi)
iBVToFloat sym _ _ = uintToReal sym
iSBVToFloat sym _ _ = sbvToReal sym
iRealToFloat _ _ _ = return
iFloatToBV sym w _ x = realToBV sym x w
iFloatToSBV sym w _ x = realToSBV sym x w
iFloatToReal _ = return
iFloatBaseTypeRepr _ _ = knownRepr
type instance SymInterpretedFloatType (ExprBuilder t st (Flags FloatUninterpreted)) fi =
BaseBVType (FloatInfoToBitWidth fi)
instance IsInterpretedFloatExprBuilder (ExprBuilder t st (Flags FloatUninterpreted)) where
iFloatPZero sym =
floatUninterpArithCt "uninterpreted_float_pzero" sym . iFloatBaseTypeRepr sym
iFloatNZero sym =
floatUninterpArithCt "uninterpreted_float_nzero" sym . iFloatBaseTypeRepr sym
iFloatNaN sym =
floatUninterpArithCt "uninterpreted_float_nan" sym . iFloatBaseTypeRepr sym
iFloatPInf sym =
floatUninterpArithCt "uninterpreted_float_pinf" sym . iFloatBaseTypeRepr sym
iFloatNInf sym =
floatUninterpArithCt "uninterpreted_float_ninf" sym . iFloatBaseTypeRepr sym
iFloatLitRational sym fi x = iRealToFloat sym fi RNE =<< realLit sym x
iFloatLitSingle sym x =
iFloatFromBinary sym SingleFloatRepr
=<< (bvLit sym knownNat $ BV.word32 $ IEEE754.floatToWord x)
iFloatLitDouble sym x =
iFloatFromBinary sym DoubleFloatRepr
=<< (bvLit sym knownNat $ BV.word64 $ IEEE754.doubleToWord x)
iFloatLitLongDouble sym x =
iFloatFromBinary sym X86_80FloatRepr
=<< (bvLit sym knownNat $ BV.mkBV knownNat $ fp80ToBits x)
iFloatNeg = floatUninterpArithUnOp "uninterpreted_float_neg"
iFloatAbs = floatUninterpArithUnOp "uninterpreted_float_abs"
iFloatSqrt = floatUninterpArithUnOpR "uninterpreted_float_sqrt"
iFloatAdd = floatUninterpArithBinOpR "uninterpreted_float_add"
iFloatSub = floatUninterpArithBinOpR "uninterpreted_float_sub"
iFloatMul = floatUninterpArithBinOpR "uninterpreted_float_mul"
iFloatDiv = floatUninterpArithBinOpR "uninterpreted_float_div"
iFloatRem = floatUninterpArithBinOp "uninterpreted_float_rem"
iFloatMin = floatUninterpArithBinOp "uninterpreted_float_min"
iFloatMax = floatUninterpArithBinOp "uninterpreted_float_max"
iFloatFMA sym r x y z = do
let ret_type = exprType x
r_arg <- roundingModeToSymInt sym r
mkUninterpFnApp sym
"uninterpreted_float_fma"
(Ctx.empty Ctx.:> r_arg Ctx.:> x Ctx.:> y Ctx.:> z)
ret_type
iFloatEq = isEq
iFloatNe sym x y = notPred sym =<< isEq sym x y
iFloatFpEq = floatUninterpLogicBinOp "uninterpreted_float_fp_eq"
iFloatFpApart = floatUninterpLogicBinOp "uninterpreted_float_fp_apart"
iFloatLe = floatUninterpLogicBinOp "uninterpreted_float_le"
iFloatLt = floatUninterpLogicBinOp "uninterpreted_float_lt"
iFloatGe sym x y = floatUninterpLogicBinOp "uninterpreted_float_le" sym y x
iFloatGt sym x y = floatUninterpLogicBinOp "uninterpreted_float_lt" sym y x
iFloatIte = baseTypeIte
iFloatIsNaN = floatUninterpLogicUnOp "uninterpreted_float_is_nan"
iFloatIsInf = floatUninterpLogicUnOp "uninterpreted_float_is_inf"
iFloatIsZero = floatUninterpLogicUnOp "uninterpreted_float_is_zero"
iFloatIsPos = floatUninterpLogicUnOp "uninterpreted_float_is_pos"
iFloatIsNeg = floatUninterpLogicUnOp "uninterpreted_float_is_neg"
iFloatIsSubnorm = floatUninterpLogicUnOp "uninterpreted_float_is_subnorm"
iFloatIsNorm = floatUninterpLogicUnOp "uninterpreted_float_is_norm"
iFloatCast sym =
floatUninterpCastOp "uninterpreted_float_cast" sym . iFloatBaseTypeRepr sym
iFloatRound = floatUninterpArithUnOpR "uninterpreted_float_round"
iFloatFromBinary _ _ = return
iFloatToBinary _ _ = return
iBVToFloat sym =
floatUninterpCastOp "uninterpreted_bv_to_float" sym . iFloatBaseTypeRepr sym
iSBVToFloat sym =
floatUninterpCastOp "uninterpreted_sbv_to_float" sym . iFloatBaseTypeRepr sym
iRealToFloat sym =
floatUninterpCastOp "uninterpreted_real_to_float" sym . iFloatBaseTypeRepr sym
iFloatToBV sym =
floatUninterpCastOp "uninterpreted_float_to_bv" sym . BaseBVRepr
iFloatToSBV sym =
floatUninterpCastOp "uninterpreted_float_to_sbv" sym . BaseBVRepr
iFloatToReal sym x =
mkUninterpFnApp sym
"uninterpreted_float_to_real"
(Ctx.empty Ctx.:> x)
knownRepr
iFloatBaseTypeRepr _ = floatInfoToBVTypeRepr
floatUninterpArithBinOp
:: (e ~ Expr t) => String -> ExprBuilder t st fs -> e bt -> e bt -> IO (e bt)
floatUninterpArithBinOp fn sym x y =
let ret_type = exprType x
in mkUninterpFnApp sym fn (Ctx.empty Ctx.:> x Ctx.:> y) ret_type
floatUninterpArithBinOpR
:: (e ~ Expr t)
=> String
-> ExprBuilder t st fs
-> RoundingMode
-> e bt
-> e bt
-> IO (e bt)
floatUninterpArithBinOpR fn sym r x y = do
let ret_type = exprType x
r_arg <- roundingModeToSymInt sym r
mkUninterpFnApp sym fn (Ctx.empty Ctx.:> r_arg Ctx.:> x Ctx.:> y) ret_type
floatUninterpArithUnOp
:: (e ~ Expr t) => String -> ExprBuilder t st fs -> e bt -> IO (e bt)
floatUninterpArithUnOp fn sym x =
let ret_type = exprType x
in mkUninterpFnApp sym fn (Ctx.empty Ctx.:> x) ret_type
floatUninterpArithUnOpR
:: (e ~ Expr t)
=> String
-> ExprBuilder t st fs
-> RoundingMode
-> e bt
-> IO (e bt)
floatUninterpArithUnOpR fn sym r x = do
let ret_type = exprType x
r_arg <- roundingModeToSymInt sym r
mkUninterpFnApp sym fn (Ctx.empty Ctx.:> r_arg Ctx.:> x) ret_type
floatUninterpArithCt
:: (e ~ Expr t)
=> String
-> ExprBuilder t st fs
-> BaseTypeRepr bt
-> IO (e bt)
floatUninterpArithCt fn sym ret_type =
mkUninterpFnApp sym fn Ctx.empty ret_type
floatUninterpLogicBinOp
:: (e ~ Expr t)
=> String
-> ExprBuilder t st fs
-> e bt
-> e bt
-> IO (e BaseBoolType)
floatUninterpLogicBinOp fn sym x y =
mkUninterpFnApp sym fn (Ctx.empty Ctx.:> x Ctx.:> y) knownRepr
floatUninterpLogicUnOp
:: (e ~ Expr t)
=> String
-> ExprBuilder t st fs
-> e bt
-> IO (e BaseBoolType)
floatUninterpLogicUnOp fn sym x =
mkUninterpFnApp sym fn (Ctx.empty Ctx.:> x) knownRepr
floatUninterpCastOp
:: (e ~ Expr t)
=> String
-> ExprBuilder t st fs
-> BaseTypeRepr bt
-> RoundingMode
-> e bt'
-> IO (e bt)
floatUninterpCastOp fn sym ret_type r x = do
r_arg <- roundingModeToSymInt sym r
mkUninterpFnApp sym fn (Ctx.empty Ctx.:> r_arg Ctx.:> x) ret_type
roundingModeToSymInt
:: (sym ~ ExprBuilder t st fs) => sym -> RoundingMode -> IO (SymInteger sym)
roundingModeToSymInt sym = intLit sym . toInteger . fromEnum
type instance SymInterpretedFloatType (ExprBuilder t st (Flags FloatIEEE)) fi =
BaseFloatType (FloatInfoToPrecision fi)
instance IsInterpretedFloatExprBuilder (ExprBuilder t st (Flags FloatIEEE)) where
iFloatPZero sym = floatPZero sym . floatInfoToPrecisionRepr
iFloatNZero sym = floatNZero sym . floatInfoToPrecisionRepr
iFloatNaN sym = floatNaN sym . floatInfoToPrecisionRepr
iFloatPInf sym = floatPInf sym . floatInfoToPrecisionRepr
iFloatNInf sym = floatNInf sym . floatInfoToPrecisionRepr
iFloatLitRational sym = floatLitRational sym . floatInfoToPrecisionRepr
iFloatLitSingle sym x =
floatFromBinary sym knownRepr
=<< (bvLit sym knownNat $ BV.word32 $ IEEE754.floatToWord x)
iFloatLitDouble sym x =
floatFromBinary sym knownRepr
=<< (bvLit sym knownNat $ BV.word64 $ IEEE754.doubleToWord x)
iFloatLitLongDouble sym (X86_80Val e s) = do
el <- bvLit sym (knownNat @16) $ BV.word16 e
sl <- bvLit sym (knownNat @64) $ BV.word64 s
fl <- bvConcat sym el sl
floatFromBinary sym knownRepr fl
-- n.b. This may not be valid semantically for operations
-- performed on 80-bit values, but it allows them to be present in
-- formulas.
iFloatNeg = floatNeg
iFloatAbs = floatAbs
iFloatSqrt = floatSqrt
iFloatAdd = floatAdd
iFloatSub = floatSub
iFloatMul = floatMul
iFloatDiv = floatDiv
iFloatRem = floatRem
iFloatMin = floatMin
iFloatMax = floatMax
iFloatFMA = floatFMA
iFloatEq = floatEq
iFloatNe = floatNe
iFloatFpEq = floatFpEq
iFloatFpApart = floatFpApart
iFloatLe = floatLe
iFloatLt = floatLt
iFloatGe = floatGe
iFloatGt = floatGt
iFloatIte = floatIte
iFloatIsNaN = floatIsNaN
iFloatIsInf = floatIsInf
iFloatIsZero = floatIsZero
iFloatIsPos = floatIsPos
iFloatIsNeg = floatIsNeg
iFloatIsSubnorm = floatIsSubnorm
iFloatIsNorm = floatIsNorm
iFloatCast sym = floatCast sym . floatInfoToPrecisionRepr
iFloatRound = floatRound
iFloatFromBinary sym fi x = case fi of
HalfFloatRepr -> floatFromBinary sym knownRepr x
SingleFloatRepr -> floatFromBinary sym knownRepr x
DoubleFloatRepr -> floatFromBinary sym knownRepr x
QuadFloatRepr -> floatFromBinary sym knownRepr x
X86_80FloatRepr -> fail "x86_80 is not an IEEE-754 format."
DoubleDoubleFloatRepr -> fail "double-double is not an IEEE-754 format."
iFloatToBinary sym fi x = case fi of
HalfFloatRepr -> floatToBinary sym x
SingleFloatRepr -> floatToBinary sym x
DoubleFloatRepr -> floatToBinary sym x
QuadFloatRepr -> floatToBinary sym x
X86_80FloatRepr -> fail "x86_80 is not an IEEE-754 format."
DoubleDoubleFloatRepr -> fail "double-double is not an IEEE-754 format."
iBVToFloat sym = bvToFloat sym . floatInfoToPrecisionRepr
iSBVToFloat sym = sbvToFloat sym . floatInfoToPrecisionRepr
iRealToFloat sym = realToFloat sym . floatInfoToPrecisionRepr
iFloatToBV = floatToBV
iFloatToSBV = floatToSBV
iFloatToReal = floatToReal
iFloatBaseTypeRepr _ = BaseFloatRepr . floatInfoToPrecisionRepr
instance IsSymExprBuilder (ExprBuilder t st fs) where
freshConstant sym nm tp = do
v <- sbMakeBoundVar sym nm tp UninterpVarKind Nothing
updateVarBinding sym nm (VarSymbolBinding v)
return $! BoundVarExpr v
freshBoundedBV sym nm w Nothing Nothing = freshConstant sym nm (BaseBVRepr w)
freshBoundedBV sym nm w mlo mhi =
do unless boundsOK (Ex.throwIO (InvalidRange (BaseBVRepr w) (fmap toInteger mlo) (fmap toInteger mhi)))
v <- sbMakeBoundVar sym nm (BaseBVRepr w) UninterpVarKind (Just $! (BVD.range w lo hi))
updateVarBinding sym nm (VarSymbolBinding v)
return $! BoundVarExpr v
where
boundsOK = lo <= hi && minUnsigned w <= lo && hi <= maxUnsigned w
lo = maybe (minUnsigned w) toInteger mlo
hi = maybe (maxUnsigned w) toInteger mhi
freshBoundedSBV sym nm w Nothing Nothing = freshConstant sym nm (BaseBVRepr w)
freshBoundedSBV sym nm w mlo mhi =
do unless boundsOK (Ex.throwIO (InvalidRange (BaseBVRepr w) mlo mhi))
v <- sbMakeBoundVar sym nm (BaseBVRepr w) UninterpVarKind (Just $! (BVD.range w lo hi))
updateVarBinding sym nm (VarSymbolBinding v)
return $! BoundVarExpr v
where
boundsOK = lo <= hi && minSigned w <= lo && hi <= maxSigned w
lo = fromMaybe (minSigned w) mlo
hi = fromMaybe (maxSigned w) mhi
freshBoundedInt sym nm mlo mhi =
do unless (boundsOK mlo mhi) (Ex.throwIO (InvalidRange BaseIntegerRepr mlo mhi))
v <- sbMakeBoundVar sym nm BaseIntegerRepr UninterpVarKind (absVal mlo mhi)
updateVarBinding sym nm (VarSymbolBinding v)
return $! BoundVarExpr v
where
boundsOK (Just lo) (Just hi) = lo <= hi
boundsOK _ _ = True
absVal Nothing Nothing = Nothing
absVal (Just lo) Nothing = Just $! MultiRange (Inclusive lo) Unbounded
absVal Nothing (Just hi) = Just $! MultiRange Unbounded (Inclusive hi)
absVal (Just lo) (Just hi) = Just $! MultiRange (Inclusive lo) (Inclusive hi)
freshBoundedReal sym nm mlo mhi =
do unless (boundsOK mlo mhi) (Ex.throwIO (InvalidRange BaseRealRepr mlo mhi))
v <- sbMakeBoundVar sym nm BaseRealRepr UninterpVarKind (absVal mlo mhi)
updateVarBinding sym nm (VarSymbolBinding v)
return $! BoundVarExpr v
where
boundsOK (Just lo) (Just hi) = lo <= hi
boundsOK _ _ = True
absVal Nothing Nothing = Nothing
absVal (Just lo) Nothing = Just $! RAV (MultiRange (Inclusive lo) Unbounded) Nothing
absVal Nothing (Just hi) = Just $! RAV (MultiRange Unbounded (Inclusive hi)) Nothing
absVal (Just lo) (Just hi) = Just $! RAV (MultiRange (Inclusive lo) (Inclusive hi)) Nothing
freshLatch sym nm tp = do
v <- sbMakeBoundVar sym nm tp LatchVarKind Nothing
updateVarBinding sym nm (VarSymbolBinding v)
return $! BoundVarExpr v
freshBoundVar sym nm tp =
sbMakeBoundVar sym nm tp QuantifierVarKind Nothing
varExpr _ = BoundVarExpr
forallPred sym bv e = sbNonceExpr sym $ Forall bv e
existsPred sym bv e = sbNonceExpr sym $ Exists bv e
----------------------------------------------------------------------
-- SymFn operations.
-- | Create a function defined in terms of previous functions.
definedFn sym fn_name bound_vars result policy = do
l <- curProgramLoc sym
n <- sbFreshSymFnNonce sym
let fn = ExprSymFn { symFnId = n
, symFnName = fn_name
, symFnInfo = DefinedFnInfo bound_vars result policy
, symFnLoc = l
}
updateVarBinding sym fn_name (FnSymbolBinding fn)
return fn
freshTotalUninterpFn sym fn_name arg_types ret_type = do
n <- sbFreshSymFnNonce sym
l <- curProgramLoc sym
let fn = ExprSymFn { symFnId = n
, symFnName = fn_name
, symFnInfo = UninterpFnInfo arg_types ret_type
, symFnLoc = l
}
seq fn $ do
updateVarBinding sym fn_name (FnSymbolBinding fn)
return fn
applySymFn sym fn args = do
case symFnInfo fn of
DefinedFnInfo bound_vars e policy
| shouldUnfold policy args ->
evalBoundVars sym e bound_vars args
MatlabSolverFnInfo f _ _ -> do
evalMatlabSolverFn f sym args
_ -> sbNonceExpr sym $! FnApp fn args
instance IsInterpretedFloatExprBuilder (ExprBuilder t st fs) => IsInterpretedFloatSymExprBuilder (ExprBuilder t st fs)
--------------------------------------------------------------------------------
-- MatlabSymbolicArrayBuilder instance
instance MatlabSymbolicArrayBuilder (ExprBuilder t st fs) where
mkMatlabSolverFn sym fn_id = do
let key = MatlabFnWrapper fn_id
mr <- stToIO $ PH.lookup (sbMatlabFnCache sym) key
case mr of
Just (ExprSymFnWrapper f) -> return f
Nothing -> do
let tps = matlabSolverArgTypes fn_id
vars <- traverseFC (freshBoundVar sym emptySymbol) tps
r <- evalMatlabSolverFn fn_id sym (fmapFC BoundVarExpr vars)
l <- curProgramLoc sym
n <- sbFreshSymFnNonce sym
let f = ExprSymFn { symFnId = n
, symFnName = emptySymbol
, symFnInfo = MatlabSolverFnInfo fn_id vars r
, symFnLoc = l
}
updateVarBinding sym emptySymbol (FnSymbolBinding f)
stToIO $ PH.insert (sbMatlabFnCache sym) key (ExprSymFnWrapper f)
return f
unsafeUserSymbol :: String -> IO SolverSymbol
unsafeUserSymbol s =
case userSymbol s of
Left err -> fail (show err)
Right symbol -> return symbol
cachedUninterpFn
:: (sym ~ ExprBuilder t st fs)
=> sym
-> SolverSymbol
-> Ctx.Assignment BaseTypeRepr args
-> BaseTypeRepr ret
-> ( sym
-> SolverSymbol
-> Ctx.Assignment BaseTypeRepr args
-> BaseTypeRepr ret
-> IO (SymFn sym args ret)
)
-> IO (SymFn sym args ret)
cachedUninterpFn sym fn_name arg_types ret_type handler = do
fn_cache <- readIORef $ sbUninterpFnCache sym
case Map.lookup fn_key fn_cache of
Just (SomeSymFn fn)
| Just Refl <- testEquality (fnArgTypes fn) arg_types
, Just Refl <- testEquality (fnReturnType fn) ret_type
-> return fn
| otherwise
-> fail "Duplicate uninterpreted function declaration."
Nothing -> do
fn <- handler sym fn_name arg_types ret_type
atomicModifyIORef' (sbUninterpFnCache sym) (\m -> (Map.insert fn_key (SomeSymFn fn) m, ()))
return fn
where fn_key = (fn_name, Some (arg_types Ctx.:> ret_type))
mkUninterpFnApp
:: (sym ~ ExprBuilder t st fs)
=> sym
-> String
-> Ctx.Assignment (SymExpr sym) args
-> BaseTypeRepr ret
-> IO (SymExpr sym ret)
mkUninterpFnApp sym str_fn_name args ret_type = do
fn_name <- unsafeUserSymbol str_fn_name
let arg_types = fmapFC exprType args
fn <- cachedUninterpFn sym fn_name arg_types ret_type freshTotalUninterpFn
applySymFn sym fn args
mkFreshUninterpFnApp
:: (sym ~ ExprBuilder t st fs)
=> sym
-> String
-> Ctx.Assignment (SymExpr sym) args
-> BaseTypeRepr ret
-> IO (SymExpr sym ret)
mkFreshUninterpFnApp sym str_fn_name args ret_type = do
fn_name <- unsafeUserSymbol str_fn_name
let arg_types = fmapFC exprType args
fn <- freshTotalUninterpFn sym fn_name arg_types ret_type
applySymFn sym fn args