sbv-11.3: Data/SBV/Core/Concrete.hs
-----------------------------------------------------------------------------
-- |
-- Module : Data.SBV.Core.Concrete
-- Copyright : (c) Levent Erkok
-- License : BSD3
-- Maintainer: erkokl@gmail.com
-- Stability : experimental
--
-- Operations on concrete values
-----------------------------------------------------------------------------
{-# LANGUAGE DeriveAnyClass #-}
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE DeriveDataTypeable #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE Rank2Types #-}
{-# OPTIONS_GHC -Wall -Werror #-}
module Data.SBV.Core.Concrete where
import Control.Monad (replicateM)
import Control.DeepSeq (NFData)
import Data.Bits
import System.Random (randomIO, randomRIO)
import Data.Char (chr, isSpace)
import Data.List (intercalate)
import Data.SBV.Core.Kind
import Data.SBV.Core.AlgReals
import Data.SBV.Core.SizedFloats
import Data.Proxy
import Data.SBV.Utils.Numeric (fpIsEqualObjectH, fpCompareObjectH)
import Data.Set (Set)
import qualified Data.Set as Set
import qualified Data.Generics as G
import GHC.Generics
-- | A 'RCSet' is either a regular set or a set given by its complement from the corresponding universal set.
data RCSet a = RegularSet (Set a)
| ComplementSet (Set a)
deriving (NFData, G.Data, Generic)
-- | Show instance. Regular sets are shown as usual.
-- Complements are shown "U -" notation.
instance Show a => Show (RCSet a) where
show rcs = case rcs of
ComplementSet s | Set.null s -> "U"
| True -> "U - " ++ sh (Set.toAscList s)
RegularSet s -> sh (Set.toAscList s)
where sh xs = '{' : intercalate "," (map show xs) ++ "}"
-- | Structural equality for 'RCSet'. We need Eq/Ord instances for 'RCSet' because we want to put them in maps/tables. But
-- we don't want to derive these, nor make it an instance! Why? Because the same set can have multiple representations if the underlying
-- type is finite. For instance, @{True} = U - {False}@ for boolean sets! Instead, we use the following two functions,
-- which are equivalent to Eq/Ord instances and work for our purposes, but we do not export these to the user.
eqRCSet :: Eq a => RCSet a -> RCSet a -> Bool
eqRCSet (RegularSet a) (RegularSet b) = a == b
eqRCSet (ComplementSet a) (ComplementSet b) = a == b
eqRCSet _ _ = False
-- | Comparing 'RCSet' values. See comments for 'eqRCSet' on why we don't define the 'Ord' instance.
compareRCSet :: Ord a => RCSet a -> RCSet a -> Ordering
compareRCSet (RegularSet a) (RegularSet b) = a `compare` b
compareRCSet (RegularSet _) (ComplementSet _) = LT
compareRCSet (ComplementSet _) (RegularSet _) = GT
compareRCSet (ComplementSet a) (ComplementSet b) = a `compare` b
instance HasKind a => HasKind (RCSet a) where
kindOf _ = KSet (kindOf (Proxy @a))
-- | Underlying type for SMTLib arrays, as a list of key-value pairs, with a default for unmapped
-- elements. Note that this type matches the typical models returned by SMT-solvers.
-- When we store the array, we do not bother removing earlier writes, so there might be duplicates.
-- That is, we store the history of the writes. The earlier a pair is in the list, the "later" it
-- is done, i.e., it takes precedence over the latter entries.
data ArrayModel a b = ArrayModel [(a, b)] b
deriving (G.Data, Generic, NFData)
-- | The kind of an ArrayModel
instance (HasKind a, HasKind b) => HasKind (ArrayModel a b) where
kindOf _ = KArray (kindOf (Proxy @a)) (kindOf (Proxy @b))
-- | A constant value.
-- Note: If you add a new constructor here, make sure you add the
-- corresponding equality in the instance "Eq CVal" and "Ord CVal"!
data CVal = CAlgReal !AlgReal -- ^ Algebraic real
| CInteger !Integer -- ^ Bit-vector/unbounded integer
| CFloat !Float -- ^ Float
| CDouble !Double -- ^ Double
| CFP !FP -- ^ Arbitrary float
| CRational Rational -- ^ Rational
| CChar !Char -- ^ Character
| CString !String -- ^ String
| CList ![CVal] -- ^ List
| CSet !(RCSet CVal) -- ^ Set. Can be regular or complemented.
| CUserSort !(Maybe Int, String) -- ^ Value of an uninterpreted/user kind. The Maybe Int shows index position for enumerations
| CTuple ![CVal] -- ^ Tuple
| CMaybe !(Maybe CVal) -- ^ Maybe
| CEither !(Either CVal CVal) -- ^ Disjoint union
| CArray !(ArrayModel CVal CVal) -- ^ Arrays are backed by look-up tables concretely
deriving (G.Data, Generic, NFData)
-- | Assign a rank to constant values, this is structural and helps with ordering
cvRank :: CVal -> Int
cvRank CAlgReal {} = 0
cvRank CInteger {} = 1
cvRank CFloat {} = 2
cvRank CDouble {} = 3
cvRank CFP {} = 4
cvRank CRational {} = 5
cvRank CChar {} = 6
cvRank CString {} = 7
cvRank CList {} = 8
cvRank CSet {} = 9
cvRank CUserSort {} = 10
cvRank CTuple {} = 11
cvRank CMaybe {} = 12
cvRank CEither {} = 13
cvRank CArray {} = 14
-- | Eq instance for CVal. Note that we cannot simply derive Eq/Ord, since CVAlgReal doesn't have proper
-- instances for these when values are infinitely precise reals. However, we do
-- need a structural eq/ord for Map indexes; so define custom ones here:
instance Eq CVal where
CAlgReal a == CAlgReal b = a `algRealStructuralEqual` b
CInteger a == CInteger b = a == b
CFloat a == CFloat b = a `fpIsEqualObjectH` b -- We don't want +0/-0 to be confused; and also we want NaN = NaN here!
CDouble a == CDouble b = a `fpIsEqualObjectH` b -- ditto
CRational a == CRational b = a == b
CFP a == CFP b = a `arbFPIsEqualObjectH` b
CChar a == CChar b = a == b
CString a == CString b = a == b
CList a == CList b = a == b
CSet a == CSet b = a `eqRCSet` b
CUserSort a == CUserSort b = a == b
CTuple a == CTuple b = a == b
CMaybe a == CMaybe b = a == b
CEither a == CEither b = a == b
-- This is legit since we don't use this equality for actual semantic" equality, but rather as an index into maps
CArray (ArrayModel a1 d1) == CArray (ArrayModel a2 d2) = (a1, d1) == (a2, d2)
a == b = if cvRank a == cvRank b
then error $ unlines [ ""
, "*** Data.SBV.Eq.CVal: Impossible happened: same rank in comparison fallthru"
, "***"
, "*** Received: " ++ show (cvRank a, cvRank b)
, "***"
, "*** Please report this as a bug!"
]
else False
-- | Ord instance for CVal. Same comments as the 'Eq' instance why this cannot be derived.
instance Ord CVal where
CAlgReal a `compare` CAlgReal b = a `algRealStructuralCompare` b
CInteger a `compare` CInteger b = a `compare` b
CFloat a `compare` CFloat b = a `fpCompareObjectH` b
CDouble a `compare` CDouble b = a `fpCompareObjectH` b
CRational a `compare` CRational b = a `compare` b
CFP a `compare` CFP b = a `arbFPCompareObjectH` b
CChar a `compare` CChar b = a `compare` b
CString a `compare` CString b = a `compare` b
CList a `compare` CList b = a `compare` b
CSet a `compare` CSet b = a `compareRCSet` b
CUserSort a `compare` CUserSort b = a `compare` b
CTuple a `compare` CTuple b = a `compare` b
CMaybe a `compare` CMaybe b = a `compare` b
CEither a `compare` CEither b = a `compare` b
-- This is legit since we don't use this equality for actual semantic order, but rather as an index into maps
CArray (ArrayModel a1 d1) `compare` CArray (ArrayModel a2 d2) = (a1, d1) `compare` (a2, d2)
a `compare` b = let ra = cvRank a
rb = cvRank b
in if ra == rb
then error $ unlines [ ""
, "*** Data.SBV.Ord.CVal: Impossible happened: same rank in comparison fallthru"
, "***"
, "*** Received: " ++ show (ra, rb)
, "***"
, "*** Please report this as a bug!"
]
else cvRank a `compare` cvRank b
-- | A t'CV' represents a concrete word of a fixed size:
-- For signed words, the most significant digit is considered to be the sign.
data CV = CV { _cvKind :: !Kind
, cvVal :: !CVal
}
deriving (Eq, Ord, G.Data, NFData, Generic)
-- | A generalized CV allows for expressions involving infinite and epsilon values/intervals Used in optimization problems.
data GeneralizedCV = ExtendedCV ExtCV
| RegularCV CV
-- | A simple expression type over extended values, covering infinity, epsilon and intervals.
data ExtCV = Infinite Kind -- infinity
| Epsilon Kind -- epsilon
| Interval ExtCV ExtCV -- closed interval
| BoundedCV CV -- a bounded value (i.e., neither infinity, nor epsilon). Note that this cannot appear at top, but can appear as a sub-expr.
| AddExtCV ExtCV ExtCV -- addition
| MulExtCV ExtCV ExtCV -- multiplication
-- | Kind instance for Extended CV
instance HasKind ExtCV where
kindOf (Infinite k) = k
kindOf (Epsilon k) = k
kindOf (Interval l _) = kindOf l
kindOf (BoundedCV c) = kindOf c
kindOf (AddExtCV l _) = kindOf l
kindOf (MulExtCV l _) = kindOf l
-- | Show instance, shows with the kind
instance Show ExtCV where
show = showExtCV True
-- | Show an extended CV, with kind if required
showExtCV :: Bool -> ExtCV -> String
showExtCV = go False
where go parens shk extCV = case extCV of
Infinite{} -> withKind False "oo"
Epsilon{} -> withKind False "epsilon"
Interval l u -> withKind True $ '[' : showExtCV False l ++ " .. " ++ showExtCV False u ++ "]"
BoundedCV c -> showCV shk c
AddExtCV l r -> par $ withKind False $ add (go True False l) (go True False r)
-- a few niceties here to grok -oo and -epsilon
MulExtCV (BoundedCV (CV KUnbounded (CInteger (-1)))) Infinite{} -> withKind False "-oo"
MulExtCV (BoundedCV (CV KReal (CAlgReal (-1)))) Infinite{} -> withKind False "-oo"
MulExtCV (BoundedCV (CV KUnbounded (CInteger (-1)))) Epsilon{} -> withKind False "-epsilon"
MulExtCV (BoundedCV (CV KReal (CAlgReal (-1)))) Epsilon{} -> withKind False "-epsilon"
MulExtCV l r -> par $ withKind False $ mul (go True False l) (go True False r)
where par v | parens = '(' : v ++ ")"
| True = v
withKind isInterval v | not shk = v
| isInterval = v ++ " :: [" ++ showBaseKind (kindOf extCV) ++ "]"
| True = v ++ " :: " ++ showBaseKind (kindOf extCV)
add :: String -> String -> String
add n ('-':v) = n ++ " - " ++ v
add n v = n ++ " + " ++ v
mul :: String -> String -> String
mul n v = n ++ " * " ++ v
-- | Is this a regular CV?
isRegularCV :: GeneralizedCV -> Bool
isRegularCV RegularCV{} = True
isRegularCV ExtendedCV{} = False
-- | 'Kind' instance for CV
instance HasKind CV where
kindOf (CV k _) = k
-- | 'Kind' instance for generalized CV
instance HasKind GeneralizedCV where
kindOf (ExtendedCV e) = kindOf e
kindOf (RegularCV c) = kindOf c
-- | Are two CV's of the same type?
cvSameType :: CV -> CV -> Bool
cvSameType x y = kindOf x == kindOf y
-- | Convert a CV to a Haskell boolean (NB. Assumes input is well-kinded)
cvToBool :: CV -> Bool
cvToBool x = cvVal x /= CInteger 0
-- | Normalize a CV. Essentially performs modular arithmetic to make sure the
-- value can fit in the given bit-size. Note that this is rather tricky for
-- negative values, due to asymmetry. (i.e., an 8-bit negative number represents
-- values in the range -128 to 127; thus we have to be careful on the negative side.)
normCV :: CV -> CV
normCV c@(CV (KBounded signed sz) (CInteger v)) = c { cvVal = CInteger norm }
where norm | sz == 0 = 0
| signed = let rg = 2 ^ (sz - 1)
in case divMod v rg of
(a, b) | even a -> b
(_, b) -> b - rg
| True = {- We really want to do:
v `mod` (2 ^ sz)
Below is equivalent, and hopefully faster!
-}
v .&. (((1 :: Integer) `shiftL` sz) - 1)
normCV c@(CV KBool (CInteger v)) = c { cvVal = CInteger (v .&. 1) }
normCV c = c
{-# INLINE normCV #-}
-- | Constant False as a t'CV'. We represent it using the integer value 0.
falseCV :: CV
falseCV = CV KBool (CInteger 0)
-- | Constant True as a t'CV'. We represent it using the integer value 1.
trueCV :: CV
trueCV = CV KBool (CInteger 1)
-- | Map a unary function through a t'CV'.
mapCV :: (AlgReal -> AlgReal)
-> (Integer -> Integer)
-> (Float -> Float)
-> (Double -> Double)
-> (FP -> FP)
-> (Rational -> Rational)
-> CV -> CV
mapCV r i f d af ra x = normCV $ CV (kindOf x) $ case cvVal x of
CAlgReal a -> CAlgReal (r a)
CInteger a -> CInteger (i a)
CFloat a -> CFloat (f a)
CDouble a -> CDouble (d a)
CFP a -> CFP (af a)
CRational a -> CRational (ra a)
CChar{} -> error "Data.SBV.mapCV: Unexpected call through mapCV with chars!"
CString{} -> error "Data.SBV.mapCV: Unexpected call through mapCV with strings!"
CUserSort{} -> error "Data.SBV.mapCV: Unexpected call through mapCV with uninterpreted sorts!"
CList{} -> error "Data.SBV.mapCV: Unexpected call through mapCV with lists!"
CSet{} -> error "Data.SBV.mapCV: Unexpected call through mapCV with sets!"
CTuple{} -> error "Data.SBV.mapCV: Unexpected call through mapCV with tuples!"
CMaybe{} -> error "Data.SBV.mapCV: Unexpected call through mapCV with maybe!"
CEither{} -> error "Data.SBV.mapCV: Unexpected call through mapCV with either!"
CArray{} -> error "Data.SBV.mapCV: Unexpected call through mapCV with arrays!"
-- | Map a binary function through a t'CV'.
mapCV2 :: (AlgReal -> AlgReal -> AlgReal)
-> (Integer -> Integer -> Integer)
-> (Float -> Float -> Float)
-> (Double -> Double -> Double)
-> (FP -> FP -> FP)
-> (Rational -> Rational -> Rational)
-> CV -> CV -> CV
mapCV2 r i f d af ra x y = case (cvSameType x y, cvVal x, cvVal y) of
(True, CAlgReal a, CAlgReal b) -> normCV $ CV (kindOf x) (CAlgReal (r a b))
(True, CInteger a, CInteger b) -> normCV $ CV (kindOf x) (CInteger (i a b))
(True, CFloat a, CFloat b) -> normCV $ CV (kindOf x) (CFloat (f a b))
(True, CDouble a, CDouble b) -> normCV $ CV (kindOf x) (CDouble (d a b))
(True, CFP a, CFP b) -> normCV $ CV (kindOf x) (CFP (af a b))
(True, CRational a, CRational b) -> normCV $ CV (kindOf x) (CRational (ra a b))
(True, CChar{}, CChar{}) -> unexpected "chars!"
(True, CString{}, CString{}) -> unexpected "strings!"
(True, CUserSort{}, CUserSort{}) -> unexpected "uninterpreted constants!"
(True, CList{}, CList{}) -> unexpected "lists!"
(True, CTuple{}, CTuple{}) -> unexpected "tuples!"
(True, CMaybe{}, CMaybe{}) -> unexpected "maybes!"
(True, CEither{}, CEither{}) -> unexpected "eithers!"
_ -> unexpected $ "incompatible args: " ++ show (x, y)
where unexpected w = error $ unlines [ ""
, "*** Data.SBV.mapCV2: Unexpected call through mapCV2 with " ++ w
, "*** Please report this as a bug!"
]
-- | Show instance for t'CV'.
instance Show CV where
show = showCV True
-- | Show instance for Generalized t'CV'
instance Show GeneralizedCV where
show (ExtendedCV k) = showExtCV True k
show (RegularCV c) = showCV True c
-- | Show a CV, with kind info if bool is True
showCV :: Bool -> CV -> String
showCV shk w | isBoolean w = show (cvToBool w) ++ (if shk then " :: Bool" else "")
showCV shk w = sh (cvVal w) ++ kInfo
where kInfo | shk = " :: " ++ showBaseKind wk
| True = ""
wk = kindOf w
sh (CAlgReal v) = show v
sh (CInteger v) = show v
sh (CFloat v) = show v
sh (CDouble v) = show v
sh (CFP v) = show v
sh (CRational v) = show v
sh (CChar v) = show v
sh (CString v) = show v
sh (CUserSort v) = snd v
sh (CList v) = shL v
sh (CSet v) = shS v
sh (CTuple v) = shT v
sh (CMaybe v) = shM v
sh (CEither v) = shE v
sh (CArray v) = shA v
shL xs = "[" ++ intercalate "," (map (showCV False . CV ke) xs) ++ "]"
where ke = case wk of
KList k -> k
_ -> error $ "Data.SBV.showCV: Impossible happened, expected list, got: " ++ show wk
-- we represent complements as @U - set@. This might be confusing, but is utterly cute!
shS :: RCSet CVal -> String
shS eru = case eru of
RegularSet e -> set e
ComplementSet e | Set.null e -> "U"
| True -> "U - " ++ set e
where set xs = "{" ++ intercalate "," (map (showCV False . CV ke) (Set.toList xs)) ++ "}"
ke = case wk of
KSet k -> k
_ -> error $ "Data.SBV.showCV: Impossible happened, expected set, got: " ++ show wk
shT :: [CVal] -> String
shT xs = "(" ++ intercalate "," xs' ++ ")"
where xs' = case wk of
KTuple ks | length ks == length xs -> zipWith (\k x -> showCV False (CV k x)) ks xs
_ -> error $ "Data.SBV.showCV: Impossible happened, expected tuple (of length " ++ show (length xs) ++ "), got: " ++ show wk
shM :: Maybe CVal -> String
shM c = case (c, wk) of
(Nothing, KMaybe{}) -> "Nothing"
(Just x, KMaybe k) -> "Just " ++ paren (showCV False (CV k x))
_ -> error $ "Data.SBV.showCV: Impossible happened, expected maybe, got: " ++ show wk
shE :: Either CVal CVal -> String
shE val
| KEither k1 k2 <- wk = case val of
Left x -> "Left " ++ paren (showCV False (CV k1 x))
Right y -> "Right " ++ paren (showCV False (CV k2 y))
| True = error $ "Data.SBV.showCV: Impossible happened, expected sum, got: " ++ show wk
shA :: ArrayModel CVal CVal -> String
shA (ArrayModel assocs def)
| KArray k1 k2 <- wk = "([" ++ intercalate "," [showCV False (CV (KTuple [k1, k2]) (CTuple [a, b])) | (a, b) <- assocs] ++ "], " ++ showCV False (CV k2 def) ++ ")"
| True = error $ "Data.SBV.showCV: Impossible happened, expected array, got: " ++ show wk
-- kind of crude, but works ok
paren v
| needsParen = '(' : v ++ ")"
| True = v
where needsParen = case dropWhile isSpace v of
[] -> False
rest@(x:_) -> x == '-' || (any isSpace rest && x `notElem` "{[(")
-- | Create a constant word from an integral.
mkConstCV :: Integral a => Kind -> a -> CV
mkConstCV KBool a = normCV $ CV KBool (CInteger (toInteger a))
mkConstCV k@KBounded{} a = normCV $ CV k (CInteger (toInteger a))
mkConstCV KUnbounded a = normCV $ CV KUnbounded (CInteger (toInteger a))
mkConstCV KReal a = normCV $ CV KReal (CAlgReal (fromInteger (toInteger a)))
mkConstCV KFloat a = normCV $ CV KFloat (CFloat (fromInteger (toInteger a)))
mkConstCV KDouble a = normCV $ CV KDouble (CDouble (fromInteger (toInteger a)))
mkConstCV k@(KFP eb sb) a = normCV $ CV k (CFP (fpFromInteger eb sb (toInteger a)))
mkConstCV KRational a = normCV $ CV KRational (CRational (fromInteger (toInteger a)))
mkConstCV KChar a = error $ "Unexpected call to mkConstCV (Char) with value: " ++ show (toInteger a)
mkConstCV KString a = error $ "Unexpected call to mkConstCV (String) with value: " ++ show (toInteger a)
mkConstCV (KUserSort s _) a = error $ "Unexpected call to mkConstCV with user kind: " ++ s ++ " with value: " ++ show (toInteger a)
mkConstCV k@KList{} a = error $ "Unexpected call to mkConstCV (" ++ show k ++ ") with value: " ++ show (toInteger a)
mkConstCV k@KSet{} a = error $ "Unexpected call to mkConstCV (" ++ show k ++ ") with value: " ++ show (toInteger a)
mkConstCV k@KTuple{} a = error $ "Unexpected call to mkConstCV (" ++ show k ++ ") with value: " ++ show (toInteger a)
mkConstCV k@KMaybe{} a = error $ "Unexpected call to mkConstCV (" ++ show k ++ ") with value: " ++ show (toInteger a)
mkConstCV k@KEither{} a = error $ "Unexpected call to mkConstCV (" ++ show k ++ ") with value: " ++ show (toInteger a)
mkConstCV k@KArray{} a = error $ "Unexpected call to mkConstCV (" ++ show k ++ ") with value: " ++ show (toInteger a)
-- | Generate a random constant value ('CVal') of the correct kind. We error out for a completely uninterpreted type.
randomCVal :: Kind -> IO CVal
randomCVal k =
case k of
KBool -> CInteger <$> randomRIO (0, 1)
KBounded s w -> CInteger <$> randomRIO (bounds s w)
KUnbounded -> CInteger <$> randomIO
KReal -> CAlgReal <$> randomIO
KFloat -> CFloat <$> randomIO
KDouble -> CDouble <$> randomIO
KRational -> CRational <$> randomIO
-- Rather bad, but OK
KFP eb sb -> do sgn <- randomRIO (0 :: Integer, 1)
let sign = sgn == 1
e <- randomRIO (0 :: Integer, 2^eb-1)
s <- randomRIO (0 :: Integer, 2^sb-1)
pure $ CFP $ fpFromRawRep sign (e, eb) (s, sb)
-- TODO: KString/KChar currently only go for 0..255; include unicode?
KString -> do l <- randomRIO (0, 100)
CString <$> replicateM l (chr <$> randomRIO (0, 255))
KChar -> CChar . chr <$> randomRIO (0, 255)
KUserSort s es -> case es of
Just vs@(_:_) -> do i <- randomRIO (0, length vs - 1)
pure $ CUserSort (Just i, vs !! i)
_ -> error $ "randomCVal: Not supported for completely uninterpreted type: " ++ s
KList ek -> do l <- randomRIO (0, 100)
CList <$> replicateM l (randomCVal ek)
KSet ek -> do i <- randomIO -- regular or complement
l <- randomRIO (0, 100) -- some set upto 100 elements
vals <- Set.fromList <$> replicateM l (randomCVal ek)
return $ CSet $ if i then RegularSet vals else ComplementSet vals
KTuple ks -> CTuple <$> traverse randomCVal ks
KMaybe ke -> do i <- randomIO
if i
then return $ CMaybe Nothing
else CMaybe . Just <$> randomCVal ke
KEither k1 k2 -> do i <- randomIO
if i
then CEither . Left <$> randomCVal k1
else CEither . Right <$> randomCVal k2
KArray k1 k2 -> do l <- randomRIO (0, 100)
ks <- replicateM l (randomCVal k1)
vs <- replicateM l (randomCVal k2)
def <- randomCVal k2
return $ CArray $ ArrayModel (zip ks vs) def
where
bounds :: Bool -> Int -> (Integer, Integer)
bounds False w = (0, 2^w - 1)
bounds True w = (-x, x-1) where x = 2^(w-1)
-- | Generate a random constant value (i.e., t'CV') of the correct kind.
randomCV :: Kind -> IO CV
randomCV k = CV k <$> randomCVal k
{- HLint ignore module "Redundant if" -}