module Cryptol.Symbolic.Prims where
import Data.Bits ()
import Data.List (genericDrop, genericReplicate, genericSplitAt, genericTake, sortBy, transpose)
import Data.Ord (comparing)
import Cryptol.Eval.Value (BitWord(..))
import Cryptol.Prims.Eval (binary, unary, tlamN)
import Cryptol.Prims.Syntax (ECon(..))
import Cryptol.Symbolic.Value
import Cryptol.TypeCheck.AST (Name)
import Cryptol.TypeCheck.Solver.InfNat(Nat'(..), nMul)
import Cryptol.Utils.Panic
import qualified Data.SBV.Dynamic as SBV
#if __GLASGOW_HASKELL__ < 710
import Control.Applicative
#endif
traverseSnd :: Functor f => (a -> f b) -> (t, a) -> f (t, b)
traverseSnd f (x, y) = (,) x <$> f y
evalECon :: ECon -> Value
evalECon econ =
case econ of
ECTrue -> VBit SBV.svTrue
ECFalse -> VBit SBV.svFalse
ECDemote -> ecDemoteV
ECPlus -> binary (arithBinary SBV.svPlus)
ECMinus -> binary (arithBinary SBV.svMinus)
ECMul -> binary (arithBinary SBV.svTimes)
ECDiv -> binary (arithBinary SBV.svQuot)
ECMod -> binary (arithBinary SBV.svRem)
ECExp -> binary (arithBinary sExp)
ECLg2 -> unary (arithUnary sLg2)
ECNeg -> unary (arithUnary SBV.svUNeg)
ECLt -> binary (cmpBinary cmpLt cmpLt SBV.svFalse)
ECGt -> binary (cmpBinary cmpGt cmpGt SBV.svFalse)
ECLtEq -> binary (cmpBinary cmpLtEq cmpLtEq SBV.svTrue)
ECGtEq -> binary (cmpBinary cmpGtEq cmpGtEq SBV.svTrue)
ECEq -> binary (cmpBinary cmpEq cmpEq SBV.svTrue)
ECNotEq -> binary (cmpBinary cmpNotEq cmpNotEq SBV.svFalse)
ECFunEq ->
tlam $ \_ ->
tlam $ \b ->
VFun $ \f ->
VFun $ \g ->
VFun $ \x -> cmpBinary cmpEq cmpEq SBV.svTrue b (fromVFun f x) (fromVFun g x)
ECFunNotEq ->
tlam $ \_ ->
tlam $ \b ->
VFun $ \f ->
VFun $ \g ->
VFun $ \x -> cmpBinary cmpNotEq cmpNotEq SBV.svFalse b (fromVFun f x) (fromVFun g x)
ECMin ->
binary $ \a x y ->
let c = cmpBinary cmpLtEq cmpLtEq SBV.svFalse a x y
in iteValue (fromVBit c) x y
ECMax ->
binary $ \a x y ->
let c = cmpBinary cmpLtEq cmpLtEq SBV.svFalse a y x
in iteValue (fromVBit c) x y
ECAnd -> binary (logicBinary SBV.svAnd SBV.svAnd)
ECOr -> binary (logicBinary SBV.svOr SBV.svOr)
ECXor -> binary (logicBinary SBV.svXOr SBV.svXOr)
ECCompl -> unary (logicUnary SBV.svNot SBV.svNot)
ECZero -> VPoly zeroV
ECShiftL ->
tlam $ \m ->
tlam $ \_ ->
tlam $ \a ->
VFun $ \xs ->
VFun $ \y ->
case xs of
VWord x -> VWord (SBV.svShiftLeft x (fromVWord y))
_ -> selectV shl y
where
shl :: Integer -> Value
shl i =
case numTValue m of
Inf -> dropV i xs
Nat j | i >= j -> replicateV j a (zeroV a)
| otherwise -> catV (dropV i xs) (replicateV i a (zeroV a))
ECShiftR ->
tlam $ \m ->
tlam $ \_ ->
tlam $ \a ->
VFun $ \xs ->
VFun $ \y ->
case xs of
VWord x -> VWord (SBV.svShiftRight x (fromVWord y))
_ -> selectV shr y
where
shr :: Integer -> Value
shr i =
case numTValue m of
Inf -> catV (replicateV i a (zeroV a)) xs
Nat j | i >= j -> replicateV j a (zeroV a)
| otherwise -> catV (replicateV i a (zeroV a)) (takeV (j i) xs)
ECRotL ->
tlam $ \m ->
tlam $ \_ ->
tlam $ \_ ->
VFun $ \xs ->
VFun $ \y ->
case xs of
VWord x -> VWord (SBV.svRotateLeft x (fromVWord y))
_ -> selectV rol y
where
rol :: Integer -> Value
rol i = catV (dropV k xs) (takeV k xs)
where k = i `mod` finTValue m
ECRotR ->
tlam $ \m ->
tlam $ \_ ->
tlam $ \_ ->
VFun $ \xs ->
VFun $ \y ->
case xs of
VWord x -> VWord (SBV.svRotateRight x (fromVWord y))
_ -> selectV ror y
where
ror :: Integer -> Value
ror i = catV (dropV k xs) (takeV k xs)
where k = ( i) `mod` finTValue m
ECCat ->
tlam $ \_ ->
tlam $ \_ ->
tlam $ \_ ->
VFun $ \v1 ->
VFun $ \v2 -> catV v1 v2
ECSplitAt ->
tlam $ \(finTValue -> a) ->
tlam $ \_ ->
tlam $ \_ ->
VFun $ \v -> VTuple [takeV a v, dropV a v]
ECJoin -> tlam $ \ parts ->
tlam $ \ each ->
tlam $ \ a -> lam (joinV parts each a)
ECSplit -> ecSplitV
ECReverse ->
tlam $ \a ->
tlam $ \b ->
lam $ \(fromSeq -> xs) -> toSeq a b (reverse xs)
ECTranspose ->
tlam $ \a ->
tlam $ \b ->
tlam $ \c ->
lam $ \((map fromSeq . fromSeq) -> xs) ->
case numTValue a of
Nat 0 ->
let v = toSeq a c []
in case numTValue b of
Nat n -> toSeq b (tvSeq a c) $ genericReplicate n v
Inf -> VStream $ repeat v
_ -> toSeq b (tvSeq a c) $ map (toSeq a c) $ transpose xs
ECAt ->
tlam $ \_ ->
tlam $ \a ->
tlam $ \_ ->
VFun $ \xs ->
VFun $ \y ->
let err = zeroV a
in selectV (\i -> nthV err xs i) y
ECAtRange ->
tlam $ \_ ->
tlam $ \a ->
tlam $ \_ ->
tlam $ \_ ->
VFun $ \xs ->
VFun $ \ys ->
let err = zeroV a
in mapV (isTBit a) (selectV (\i -> nthV err xs i)) ys
ECAtBack ->
tlam $ \(finTValue -> n) ->
tlam $ \a ->
tlam $ \_ ->
VFun $ \xs ->
VFun $ \y ->
let err = zeroV a
in selectV (\i -> nthV err xs (n 1 i)) y
ECAtRangeBack ->
tlam $ \(finTValue -> n) ->
tlam $ \a ->
tlam $ \_ ->
tlam $ \_ ->
VFun $ \xs ->
VFun $ \ys ->
let err = zeroV a
in mapV (isTBit a) (selectV (\i -> nthV err xs (n 1 i))) ys
ECFromThen -> fromThenV
ECFromTo -> fromToV
ECFromThenTo -> fromThenToV
ECInfFrom ->
tlam $ \(finTValue -> bits) ->
lam $ \(fromVWord -> first) ->
toStream [ VWord (SBV.svPlus first (literalSWord (fromInteger bits) i)) | i <- [0 ..] ]
ECInfFromThen ->
tlam $ \_ ->
lam $ \(fromVWord -> first) ->
lam $ \(fromVWord -> next) ->
toStream (map VWord (iterate (SBV.svPlus (SBV.svMinus next first)) first))
ECError ->
tlam $ \at ->
tlam $ \(finTValue -> _len) ->
VFun $ \_msg -> zeroV at
ECPMul ->
tlam $ \(finTValue -> i) ->
tlam $ \(finTValue -> j) ->
VFun $ \v1 ->
VFun $ \v2 ->
let k = max 1 (i + j) 1
mul _ [] ps = ps
mul as (b:bs) ps = mul (SBV.svFalse : as) bs (ites b (as `addPoly` ps) ps)
xs = map fromVBit (fromSeq v1)
ys = map fromVBit (fromSeq v2)
zs = take (fromInteger k) (mul xs ys [] ++ repeat SBV.svFalse)
in VSeq True (map VBit zs)
ECPDiv ->
tlam $ \(finTValue -> i) ->
tlam $ \_ ->
VFun $ \v1 ->
VFun $ \v2 ->
let xs = map fromVBit (fromSeq v1)
ys = map fromVBit (fromSeq v2)
zs = take (fromInteger i) (fst (mdp (reverse xs) (reverse ys)) ++ repeat SBV.svFalse)
in VSeq True (map VBit (reverse zs))
ECPMod ->
tlam $ \_ ->
tlam $ \(finTValue -> j) ->
VFun $ \v1 ->
VFun $ \v2 ->
let xs = map fromVBit (fromSeq v1)
ys = map fromVBit (fromSeq v2)
zs = take (fromInteger j) (snd (mdp (reverse xs) (reverse ys)) ++ repeat SBV.svFalse)
in VSeq True (map VBit (reverse zs))
ECRandom -> panic "Cryptol.Symbolic.Prims.evalECon"
[ "can't symbolically evaluae ECRandom" ]
selectV :: (Integer -> Value) -> Value -> Value
selectV f v = sel 0 bits
where
bits = map fromVBit (fromSeq v)
sel :: Integer -> [SBool] -> Value
sel offset [] = f offset
sel offset (b : bs) = iteValue b m1 m2
where m1 = sel (offset + 2 ^ length bs) bs
m2 = sel offset bs
replicateV :: Integer
-> TValue
-> Value
-> Value
replicateV n (toTypeVal -> TVBit) x = VSeq True (genericReplicate n x)
replicateV n _ x = VSeq False (genericReplicate n x)
nth :: a -> [a] -> Int -> a
nth def [] _ = def
nth def (x : xs) n
| n == 0 = x
| otherwise = nth def xs (n 1)
nthV :: Value -> Value -> Integer -> Value
nthV err v n =
case v of
VStream xs -> nth err xs (fromInteger n)
VSeq _ xs -> nth err xs (fromInteger n)
VWord x -> let i = SBV.svBitSize x 1 fromInteger n
in if i < 0 then err else
VBit (SBV.svTestBit x i)
_ -> err
mapV :: Bool -> (Value -> Value) -> Value -> Value
mapV isBit f v =
case v of
VSeq _ xs -> VSeq isBit (map f xs)
VStream xs -> VStream (map f xs)
_ -> panic "Cryptol.Symbolic.Prims.mapV" [ "non-mappable value" ]
catV :: Value -> Value -> Value
catV xs (VStream ys) = VStream (fromSeq xs ++ ys)
catV (VWord x) ys = VWord (SBV.svJoin x (fromVWord ys))
catV xs (VWord y) = VWord (SBV.svJoin (fromVWord xs) y)
catV (VSeq b xs) (VSeq _ ys) = VSeq b (xs ++ ys)
catV _ _ = panic "Cryptol.Symbolic.Prims.catV" [ "non-concatenable value" ]
dropV :: Integer -> Value -> Value
dropV 0 xs = xs
dropV n xs =
case xs of
VSeq b xs' -> VSeq b (genericDrop n xs')
VStream xs' -> VStream (genericDrop n xs')
VWord w -> VWord $ SBV.svExtract (SBV.svBitSize w 1 fromInteger n) 0 w
_ -> panic "Cryptol.Symbolic.Prims.dropV" [ "non-droppable value" ]
takeV :: Integer -> Value -> Value
takeV n xs =
case xs of
VWord w -> VWord $ SBV.svExtract (SBV.svBitSize w 1) (SBV.svBitSize w fromInteger n) w
VSeq b xs' -> VSeq b (genericTake n xs')
VStream xs' -> VSeq b (genericTake n xs')
where b = case xs' of VBit _ : _ -> True
_ -> False
_ -> panic "Cryptol.Symbolic.Prims.takeV" [ "non-takeable value" ]
ecDemoteV :: Value
ecDemoteV = tlam $ \valT ->
tlam $ \bitT ->
case (numTValue valT, numTValue bitT) of
(Nat v, Nat bs) -> VWord (literalSWord (fromInteger bs) v)
_ -> evalPanic "Cryptol.Prove.evalECon"
["Unexpected Inf in constant."
, show valT
, show bitT
]
data TypeVal
= TVBit
| TVSeq Int TypeVal
| TVStream TypeVal
| TVTuple [TypeVal]
| TVRecord [(Name, TypeVal)]
| TVFun TypeVal TypeVal
toTypeVal :: TValue -> TypeVal
toTypeVal ty
| isTBit ty = TVBit
| Just (n, ety) <- isTSeq ty = case numTValue n of
Nat w -> TVSeq (fromInteger w) (toTypeVal ety)
Inf -> TVStream (toTypeVal ety)
| Just (aty, bty) <- isTFun ty = TVFun (toTypeVal aty) (toTypeVal bty)
| Just (_, tys) <- isTTuple ty = TVTuple (map toTypeVal tys)
| Just fields <- isTRec ty = TVRecord [ (n, toTypeVal aty) | (n, aty) <- fields ]
| otherwise = panic "Cryptol.Symbolic.Prims.toTypeVal" [ "bad TValue" ]
type Binary = TValue -> Value -> Value -> Value
type Unary = TValue -> Value -> Value
arithBinary :: (SWord -> SWord -> SWord) -> Binary
arithBinary op = loop . toTypeVal
where
loop ty l r =
case ty of
TVBit -> evalPanic "arithBinop" ["Invalid arguments"]
TVSeq _ TVBit -> VWord (op (fromVWord l) (fromVWord r))
TVSeq _ t -> VSeq False (zipWith (loop t) (fromSeq l) (fromSeq r))
TVStream t -> VStream (zipWith (loop t) (fromSeq l) (fromSeq r))
TVTuple ts -> VTuple (zipWith3 loop ts (fromVTuple l) (fromVTuple r))
TVRecord fs -> VRecord [ (f, loop t (lookupRecord f l) (lookupRecord f r)) | (f, t) <- fs ]
TVFun _ t -> VFun (\x -> loop t (fromVFun l x) (fromVFun r x))
arithUnary :: (SWord -> SWord) -> Unary
arithUnary op = loop . toTypeVal
where
loop ty v =
case ty of
TVBit -> evalPanic "arithUnary" ["Invalid arguments"]
TVSeq _ TVBit -> VWord (op (fromVWord v))
TVSeq _ t -> VSeq False (map (loop t) (fromSeq v))
TVStream t -> VStream (map (loop t) (fromSeq v))
TVTuple ts -> VTuple (zipWith loop ts (fromVTuple v))
TVRecord fs -> VRecord [ (f, loop t (lookupRecord f v)) | (f, t) <- fs ]
TVFun _ t -> VFun (\x -> loop t (fromVFun v x))
sExp :: SWord -> SWord -> SWord
sExp x y = go (reverse (unpackWord y))
where go [] = literalSWord (SBV.svBitSize x) 1
go (b : bs) = SBV.svIte b (SBV.svTimes x s) s
where a = go bs
s = SBV.svTimes a a
sLg2 :: SWord -> SWord
sLg2 x = go 0
where
lit n = literalSWord (SBV.svBitSize x) n
go i | i < SBV.svBitSize x = SBV.svIte (SBV.svLessEq x (lit (2^i))) (lit (toInteger i)) (go (i + 1))
| otherwise = lit (toInteger i)
cmpValue :: (SBool -> SBool -> a -> a)
-> (SWord -> SWord -> a -> a)
-> (Value -> Value -> a -> a)
cmpValue fb fw = cmp
where
cmp v1 v2 k =
case (v1, v2) of
(VRecord fs1, VRecord fs2) -> let vals = map snd . sortBy (comparing fst)
in cmpValues (vals fs1) (vals fs2) k
(VTuple vs1 , VTuple vs2 ) -> cmpValues vs1 vs2 k
(VBit b1 , VBit b2 ) -> fb b1 b2 k
(VWord w1 , VWord w2 ) -> fw w1 w2 k
(VSeq _ vs1 , VSeq _ vs2 ) -> cmpValues vs1 vs2 k
(VStream {} , VStream {} ) -> panic "Cryptol.Symbolic.Prims.cmpValue"
[ "Infinite streams are not comparable" ]
(VFun {} , VFun {} ) -> panic "Cryptol.Symbolic.Prims.cmpValue"
[ "Functions are not comparable" ]
(VPoly {} , VPoly {} ) -> panic "Cryptol.Symbolic.Prims.cmpValue"
[ "Polymorphic values are not comparable" ]
(VWord w1 , _ ) -> fw w1 (fromVWord v2) k
(_ , VWord w2 ) -> fw (fromVWord v1) w2 k
(_ , _ ) -> panic "Cryptol.Symbolic.Prims.cmpValue"
[ "type mismatch" ]
cmpValues (x1 : xs1) (x2 : xs2) k = cmp x1 x2 (cmpValues xs1 xs2 k)
cmpValues _ _ k = k
cmpEq :: SWord -> SWord -> SBool -> SBool
cmpEq x y k = SBV.svAnd (SBV.svEqual x y) k
cmpNotEq :: SWord -> SWord -> SBool -> SBool
cmpNotEq x y k = SBV.svOr (SBV.svNotEqual x y) k
cmpLt, cmpGt :: SWord -> SWord -> SBool -> SBool
cmpLt x y k = SBV.svOr (SBV.svLessThan x y) (cmpEq x y k)
cmpGt x y k = SBV.svOr (SBV.svGreaterThan x y) (cmpEq x y k)
cmpLtEq, cmpGtEq :: SWord -> SWord -> SBool -> SBool
cmpLtEq x y k = SBV.svAnd (SBV.svLessEq x y) (cmpNotEq x y k)
cmpGtEq x y k = SBV.svAnd (SBV.svGreaterEq x y) (cmpNotEq x y k)
cmpBinary :: (SBool -> SBool -> SBool -> SBool)
-> (SWord -> SWord -> SBool -> SBool)
-> SBool -> Binary
cmpBinary fb fw b _ty v1 v2 = VBit (cmpValue fb fw v1 v2 b)
errorV :: String -> TValue -> Value
errorV msg = go . toTypeVal
where
go ty =
case ty of
TVBit -> VBit (error msg)
TVSeq n t -> VSeq False (replicate n (go t))
TVStream t -> VStream (repeat (go t))
TVTuple ts -> VTuple [ go t | t <- ts ]
TVRecord fs -> VRecord [ (n, go t) | (n, t) <- fs ]
TVFun _ t -> VFun (const (go t))
zeroV :: TValue -> Value
zeroV = go . toTypeVal
where
go ty =
case ty of
TVBit -> VBit SBV.svFalse
TVSeq n TVBit -> VWord (literalSWord n 0)
TVSeq n t -> VSeq False (replicate n (go t))
TVStream t -> VStream (repeat (go t))
TVTuple ts -> VTuple [ go t | t <- ts ]
TVRecord fs -> VRecord [ (n, go t) | (n, t) <- fs ]
TVFun _ t -> VFun (const (go t))
joinV :: TValue -> TValue -> TValue -> Value -> Value
joinV parts each a v =
let len = toNumTValue (numTValue parts `nMul` numTValue each)
in toSeq len a (concatMap fromSeq (fromSeq v))
ecSplitV :: Value
ecSplitV =
tlam $ \ parts ->
tlam $ \ each ->
tlam $ \ a ->
lam $ \ v ->
let mkChunks f = map (toFinSeq a) $ f $ fromSeq v
in case (numTValue parts, numTValue each) of
(Nat p, Nat e) -> VSeq False $ mkChunks (finChunksOf p e)
(Inf , Nat e) -> toStream $ mkChunks (infChunksOf e)
_ -> evalPanic "splitV" ["invalid type arguments to split"]
infChunksOf :: Integer -> [a] -> [[a]]
infChunksOf each xs = let (as,bs) = genericSplitAt each xs
in as : infChunksOf each bs
finChunksOf :: Integer -> Integer -> [a] -> [[a]]
finChunksOf 0 _ _ = []
finChunksOf parts each xs = let (as,bs) = genericSplitAt each xs
in as : finChunksOf (parts 1) each bs
logicBinary :: (SBool -> SBool -> SBool) -> (SWord -> SWord -> SWord) -> Binary
logicBinary bop op = loop . toTypeVal
where
loop ty l r =
case ty of
TVBit -> VBit (bop (fromVBit l) (fromVBit r))
TVSeq _ TVBit -> VWord (op (fromVWord l) (fromVWord r))
TVSeq _ t -> VSeq False (zipWith (loop t) (fromSeq l) (fromSeq r))
TVStream t -> VStream (zipWith (loop t) (fromSeq l) (fromSeq r))
TVTuple ts -> VTuple (zipWith3 loop ts (fromVTuple l) (fromVTuple r))
TVRecord fs -> VRecord [ (f, loop t (lookupRecord f l) (lookupRecord f r)) | (f, t) <- fs ]
TVFun _ t -> VFun (\x -> loop t (fromVFun l x) (fromVFun r x))
logicUnary :: (SBool -> SBool) -> (SWord -> SWord) -> Unary
logicUnary bop op = loop . toTypeVal
where
loop ty v =
case ty of
TVBit -> VBit (bop (fromVBit v))
TVSeq _ TVBit -> VWord (op (fromVWord v))
TVSeq _ t -> VSeq False (map (loop t) (fromSeq v))
TVStream t -> VStream (map (loop t) (fromSeq v))
TVTuple ts -> VTuple (zipWith loop ts (fromVTuple v))
TVRecord fs -> VRecord [ (f, loop t (lookupRecord f v)) | (f, t) <- fs ]
TVFun _ t -> VFun (\x -> loop t (fromVFun v x))
fromThenV :: Value
fromThenV =
tlamN $ \ first ->
tlamN $ \ next ->
tlamN $ \ bits ->
tlamN $ \ len ->
case (first, next, len, bits) of
(Nat first', Nat next', Nat len', Nat bits') ->
let nums = enumFromThen first' next'
lit i = VWord (literalSWord (fromInteger bits') i)
in VSeq False (genericTake len' (map lit nums))
_ -> evalPanic "fromThenV" ["invalid arguments"]
fromToV :: Value
fromToV =
tlamN $ \ first ->
tlamN $ \ lst ->
tlamN $ \ bits ->
case (first, lst, bits) of
(Nat first', Nat lst', Nat bits') ->
let nums = enumFromThenTo first' (first' + 1) lst'
len = 1 + (lst' first')
lit i = VWord (literalSWord (fromInteger bits') i)
in VSeq False (genericTake len (map lit nums))
_ -> evalPanic "fromThenV" ["invalid arguments"]
fromThenToV :: Value
fromThenToV =
tlamN $ \ first ->
tlamN $ \ next ->
tlamN $ \ lst ->
tlamN $ \ bits ->
tlamN $ \ len ->
case (first, next, lst, len, bits) of
(Nat first', Nat next', Nat lst', Nat len', Nat bits') ->
let nums = enumFromThenTo first' next' lst'
lit i = VWord (literalSWord (fromInteger bits') i)
in VSeq False (genericTake len' (map lit nums))
_ -> evalPanic "fromThenV" ["invalid arguments"]
addPoly :: [SBool] -> [SBool] -> [SBool]
addPoly xs [] = xs
addPoly [] ys = ys
addPoly (x:xs) (y:ys) = SBV.svXOr x y : addPoly xs ys
ites :: SBool -> [SBool] -> [SBool] -> [SBool]
ites s xs ys
| Just t <- SBV.svAsBool s
= if t then xs else ys
| True
= go xs ys
where go [] [] = []
go [] (b:bs) = SBV.svIte s SBV.svFalse b : go [] bs
go (a:as) [] = SBV.svIte s a SBV.svFalse : go as []
go (a:as) (b:bs) = SBV.svIte s a b : go as bs
degree :: [SBool] -> Int
degree xs = walk (length xs 1) $ reverse xs
where walk n [] = n
walk n (b:bs)
| Just t <- SBV.svAsBool b
= if t then n else walk (n1) bs
| True
= n
mdp :: [SBool] -> [SBool] -> ([SBool], [SBool])
mdp xs ys = go (length ys 1) (reverse ys)
where degTop = degree xs
go _ [] = error "SBV.Polynomial.mdp: Impossible happened; exhausted ys before hitting 0"
go n (b:bs)
| n == 0 = (reverse qs, rs)
| True = let (rqs, rrs) = go (n1) bs
in (ites b (reverse qs) rqs, ites b rs rrs)
where degQuot = degTop n
ys' = replicate degQuot SBV.svFalse ++ ys
(qs, rs) = divx (degQuot+1) degTop xs ys'
idx :: [SBool] -> Int -> SBool
idx [] _ = SBV.svFalse
idx (x:_) 0 = x
idx (_:xs) i = idx xs (i1)
divx :: Int -> Int -> [SBool] -> [SBool] -> ([SBool], [SBool])
divx n _ xs _ | n <= 0 = ([], xs)
divx n i xs ys' = (q:qs, rs)
where q = xs `idx` i
xs' = ites q (xs `addPoly` ys') xs
(qs, rs) = divx (n1) (i1) xs' (tail ys')