Data.hs

-----------------------------------------------------------------------------
-- |
-- Module      :  Data.SBV.BitVectors.Data
-- Copyright   :  (c) Levent Erkok
-- License     :  BSD3
-- Maintainer  :  erkokl@gmail.com
-- Stability   :  experimental
--
-- Internal data-structures for the sbv library
-----------------------------------------------------------------------------

{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE TypeSynonymInstances       #-}
{-# LANGUAGE TypeOperators              #-}
{-# LANGUAGE MultiParamTypeClasses      #-}
{-# LANGUAGE ScopedTypeVariables        #-}
{-# LANGUAGE FlexibleInstances          #-}
{-# LANGUAGE PatternGuards              #-}
{-# LANGUAGE DefaultSignatures          #-}

module Data.SBV.BitVectors.Data
 ( SBool, SWord8, SWord16, SWord32, SWord64
 , SInt8, SInt16, SInt32, SInt64, SInteger, SReal
 , SymWord(..)
 , CW(..), CWVal(..), cwSameType, cwIsBit, cwToBool
 , mkConstCW ,liftCW2, mapCW, mapCW2
 , SW(..), trueSW, falseSW, trueCW, falseCW
 , SBV(..), NodeId(..), mkSymSBV
 , ArrayContext(..), ArrayInfo, SymArray(..), SFunArray(..), mkSFunArray, SArray(..), arrayUIKind
 , sbvToSW, sbvToSymSW
 , SBVExpr(..), newExpr
 , cache, uncache, uncacheAI, HasKind(..)
 , Op(..), NamedSymVar, UnintKind(..), getTableIndex, Pgm, Symbolic, runSymbolic, runSymbolic', State, inProofMode, SBVRunMode(..), Kind(..), Outputtable(..), Result(..)
 , getTraceInfo, getConstraints, addConstraint
 , SBVType(..), newUninterpreted, unintFnUIKind, addAxiom
 , Quantifier(..), needsExistentials
 , SMTLibPgm(..), SMTLibVersion(..)
 ) where

import Control.DeepSeq      (NFData(..))
import Control.Monad        (when)
import Control.Monad.Reader (MonadReader, ReaderT, ask, runReaderT)
import Control.Monad.Trans  (MonadIO, liftIO)
import Data.Char            (isAlpha, isAlphaNum)
import Data.Generics        (Data(..), dataTypeName, dataTypeOf, tyconUQname)
import Data.Int             (Int8, Int16, Int32, Int64)
import Data.Word            (Word8, Word16, Word32, Word64)
import Data.IORef           (IORef, newIORef, modifyIORef, readIORef, writeIORef)
import Data.List            (intercalate, sortBy)
import Data.Maybe           (isJust, fromJust)

import qualified Data.IntMap   as IMap (IntMap, empty, size, toAscList, lookup, insert, insertWith)
import qualified Data.Map      as Map  (Map, empty, toList, size, insert, lookup)
import qualified Data.Foldable as F    (toList)
import qualified Data.Sequence as S    (Seq, empty, (|>))

import System.Mem.StableName
import System.Random

import Data.SBV.BitVectors.AlgReals
import Data.SBV.Utils.Lib

-- | A constant value
data CWVal = CWAlgReal       AlgReal    -- ^ algebraic real
           | CWInteger       Integer    -- ^ bit-vector/unbounded integer
           | CWUninterpreted String     -- ^ value of an uninterpreted kind
           deriving (Eq, Ord)

-- | 'CW' represents a concrete word of a fixed size:
-- Endianness is mostly irrelevant (see the 'FromBits' class).
-- For signed words, the most significant digit is considered to be the sign.
data CW = CW { cwKind   :: !Kind
             , cwVal    :: !CWVal
             }
        deriving (Eq, Ord)

-- | Are two CW's of the same type?
cwSameType :: CW -> CW -> Bool
cwSameType x y = cwKind x == cwKind y

-- | Is this a bit?
cwIsBit :: CW -> Bool
cwIsBit x = case cwKind x of
              KBounded False 1 -> True
              _                -> False

-- | Convert a CW to a Haskell boolean (NB. Assumes input is well-kinded)
cwToBool :: CW -> Bool
cwToBool x = cwVal x /= CWInteger 0

-- | Normalize a CW. 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.)
normCW :: CW -> CW
normCW c@(CW (KBounded signed sz) (CWInteger v)) = c { cwVal = CWInteger 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    = v `mod` (2 ^ sz)
normCW c = c

-- | Kind of symbolic value
data Kind = KBounded Bool Int
          | KUnbounded
          | KReal
          | KUninterpreted String
          deriving (Eq, Ord)

instance Show Kind where
  show (KBounded False 1) = "SBool"
  show (KBounded False n) = "SWord" ++ show n
  show (KBounded True n)  = "SInt"  ++ show n
  show KUnbounded         = "SInteger"
  show KReal              = "SReal"
  show (KUninterpreted s) = s

-- | A symbolic node id
newtype NodeId = NodeId Int deriving (Eq, Ord)

-- | A symbolic word, tracking it's signedness and size.
data SW = SW Kind NodeId deriving (Eq, Ord)

-- | Quantifiers: forall or exists. Note that we allow
-- arbitrary nestings.
data Quantifier = ALL | EX deriving Eq

-- | Are there any existential quantifiers?
needsExistentials :: [Quantifier] -> Bool
needsExistentials = (EX `elem`)

-- | Constant False as a SW. Note that this value always occupies slot -2.
falseSW :: SW
falseSW = SW (KBounded False 1) $ NodeId (-2)

-- | Constant False as a SW. Note that this value always occupies slot -1.
trueSW :: SW
trueSW  = SW (KBounded False 1) $ NodeId (-1)

-- | Constant False as a CW. We represent it using the integer value 0.
falseCW :: CW
falseCW = CW (KBounded False 1) (CWInteger 0)

-- | Constant True as a CW. We represent it using the integer value 1.
trueCW :: CW
trueCW  = CW (KBounded False 1) (CWInteger 1)

-- | A simple type for SBV computations, used mainly for uninterpreted constants.
-- We keep track of the signedness/size of the arguments. A non-function will
-- have just one entry in the list.
newtype SBVType = SBVType [Kind]
             deriving (Eq, Ord)

-- | how many arguments does the type take?
typeArity :: SBVType -> Int
typeArity (SBVType xs) = length xs - 1

instance Show SBVType where
  show (SBVType []) = error "SBV: internal error, empty SBVType"
  show (SBVType xs) = intercalate " -> " $ map show xs

-- | Symbolic operations
data Op = Plus | Times | Minus
        | Quot | Rem
        | Equal | NotEqual
        | LessThan | GreaterThan | LessEq | GreaterEq
        | Ite
        | And | Or  | XOr | Not
        | Shl Int | Shr Int | Rol Int | Ror Int
        | Extract Int Int -- Extract i j: extract bits i to j. Least significant bit is 0 (big-endian)
        | Join  -- Concat two words to form a bigger one, in the order given
        | LkUp (Int, Kind, Kind, Int) !SW !SW   -- (table-index, arg-type, res-type, length of the table) index out-of-bounds-value
        | ArrEq   Int Int
        | ArrRead Int
        | Uninterpreted String
        deriving (Eq, Ord)

-- | A symbolic expression
data SBVExpr = SBVApp !Op ![SW]
             deriving (Eq, Ord)

-- | A class for capturing values that have a sign and a size (finite or infinite)
-- minimal complete definition: kindOf. This class can be automatically derived
-- for data-types that have a 'Data' instance; this is useful for creating uninterpreted
-- sorts.
class HasKind a where
  kindOf          :: a -> Kind
  hasSign         :: a -> Bool
  intSizeOf       :: a -> Int
  isBounded       :: a -> Bool
  isReal          :: a -> Bool
  isInteger       :: a -> Bool
  isUninterpreted :: a -> Bool
  showType        :: a -> String
  -- defaults
  hasSign x = case kindOf x of
                  KBounded b _     -> b
                  KUnbounded       -> True
                  KReal            -> True
                  KUninterpreted{} -> False
  intSizeOf x = case kindOf x of
                  KBounded _ s     -> s
                  KUnbounded       -> error "SBV.HasKind.intSizeOf((S)Integer)"
                  KReal            -> error "SBV.HasKind.intSizeOf((S)Real)"
                  KUninterpreted s -> error $ "SBV.HasKind.intSizeOf: Uninterpreted sort: " ++ s
  isBounded       x | KBounded{}       <- kindOf x = True
                    | True                         = False
  isReal          x | KReal{}          <- kindOf x = True
                    | True                         = False
  isInteger      x  | KUnbounded{}     <- kindOf x = True
                    | True                         = False
  isUninterpreted x | KUninterpreted{} <- kindOf x = True
                    | True                         = False
  showType = show . kindOf

  -- default signature for uninterpreted kinds
  default kindOf :: Data a => a -> Kind
  kindOf = KUninterpreted . tyconUQname . dataTypeName . dataTypeOf

instance HasKind Bool    where kindOf _ = KBounded False 1
instance HasKind Int8    where kindOf _ = KBounded True  8
instance HasKind Word8   where kindOf _ = KBounded False 8
instance HasKind Int16   where kindOf _ = KBounded True  16
instance HasKind Word16  where kindOf _ = KBounded False 16
instance HasKind Int32   where kindOf _ = KBounded True  32
instance HasKind Word32  where kindOf _ = KBounded False 32
instance HasKind Int64   where kindOf _ = KBounded True  64
instance HasKind Word64  where kindOf _ = KBounded False 64
instance HasKind Integer where kindOf _ = KUnbounded
instance HasKind AlgReal where kindOf _ = KReal

-- | Lift a unary function thruough a CW
liftCW :: (AlgReal -> b) -> (Integer -> b) -> (String -> b) -> CW -> b
liftCW f _ _ (CW _ (CWAlgReal v))       = f v
liftCW _ g _ (CW _ (CWInteger v))       = g v
liftCW _ _ h (CW _ (CWUninterpreted v)) = h v

-- | Lift a binary function through a CW
liftCW2 :: (AlgReal -> AlgReal -> b) -> (Integer -> Integer -> b) -> (String -> String -> b) -> CW -> CW -> b
liftCW2 f g h x y = case (cwVal x, cwVal y) of
                      (CWAlgReal a,       CWAlgReal b)       -> f a b
                      (CWInteger a,       CWInteger b)       -> g a b
                      (CWUninterpreted a, CWUninterpreted b) -> h a b
                      _                                      -> error $ "SBV.liftCW2: impossible, incompatible args received: " ++ show (x, y)

-- | Map a unary function through a CW
mapCW :: (AlgReal -> AlgReal) -> (Integer -> Integer) -> (String -> String) -> CW -> CW
mapCW f g h x  = normCW $ CW (cwKind x) $ case cwVal x of
                                            CWAlgReal a       -> CWAlgReal       (f a)
                                            CWInteger a       -> CWInteger       (g a)
                                            CWUninterpreted a -> CWUninterpreted (h a)

-- | Map a binary function through a CW
mapCW2 :: (AlgReal -> AlgReal -> AlgReal) -> (Integer -> Integer -> Integer) -> (String -> String -> String) -> CW -> CW -> CW
mapCW2 f g h x y = case (cwSameType x y, cwVal x, cwVal y) of
                     (True, CWAlgReal a,       CWAlgReal b)       -> normCW $ CW (cwKind x) (CWAlgReal       (f a b))
                     (True, CWInteger a,       CWInteger b)       -> normCW $ CW (cwKind x) (CWInteger       (g a b))
                     (True, CWUninterpreted a, CWUninterpreted b) -> normCW $ CW (cwKind x) (CWUninterpreted (h a b))
                     _                        -> error $ "SBV.mapCW2: impossible, incompatible args received: " ++ show (x, y)

instance HasKind CW where
  kindOf = cwKind

instance HasKind SW where
  kindOf (SW k _) = k

instance Show CW where
  show w | cwIsBit w = show (cwToBool w)
  show w             = liftCW show show id w ++ " :: " ++ showType w

instance Show SW where
  show (SW _ (NodeId n))
    | n < 0 = "s_" ++ show (abs n)
    | True  = 's' : show n

instance Show Op where
  show (Shl i) = "<<"  ++ show i
  show (Shr i) = ">>"  ++ show i
  show (Rol i) = "<<<" ++ show i
  show (Ror i) = ">>>" ++ show i
  show (Extract i j) = "choose [" ++ show i ++ ":" ++ show j ++ "]"
  show (LkUp (ti, at, rt, l) i e)
        = "lookup(" ++ tinfo ++ ", " ++ show i ++ ", " ++ show e ++ ")"
        where tinfo = "table" ++ show ti ++ "(" ++ show at ++ " -> " ++ show rt ++ ", " ++ show l ++ ")"
  show (ArrEq i j)   = "array_" ++ show i ++ " == array_" ++ show j
  show (ArrRead i)   = "select array_" ++ show i
  show (Uninterpreted i) = "[uninterpreted] " ++ i
  show op
    | Just s <- op `lookup` syms = s
    | True                       = error "impossible happened; can't find op!"
    where syms = [ (Plus, "+"), (Times, "*"), (Minus, "-")
                 , (Quot, "quot")
                 , (Rem,  "rem")
                 , (Equal, "=="), (NotEqual, "/=")
                 , (LessThan, "<"), (GreaterThan, ">"), (LessEq, "<"), (GreaterEq, ">")
                 , (Ite, "if_then_else")
                 , (And, "&"), (Or, "|"), (XOr, "^"), (Not, "~")
                 , (Join, "#")
                 ]

-- | To improve hash-consing, take advantage of commutative operators by
-- reordering their arguments.
reorder :: SBVExpr -> SBVExpr
reorder s = case s of
              SBVApp op [a, b] | isCommutative op && a > b -> SBVApp op [b, a]
              _ -> s
  where isCommutative :: Op -> Bool
        isCommutative o = o `elem` [Plus, Times, Equal, NotEqual, And, Or, XOr]

instance Show SBVExpr where
  show (SBVApp Ite [t, a, b]) = unwords ["if", show t, "then", show a, "else", show b]
  show (SBVApp (Shl i) [a])   = unwords [show a, "<<", show i]
  show (SBVApp (Shr i) [a])   = unwords [show a, ">>", show i]
  show (SBVApp (Rol i) [a])   = unwords [show a, "<<<", show i]
  show (SBVApp (Ror i) [a])   = unwords [show a, ">>>", show i]
  show (SBVApp op  [a, b])    = unwords [show a, show op, show b]
  show (SBVApp op  args)      = unwords (show op : map show args)

-- | A program is a sequence of assignments
type Pgm = S.Seq (SW, SBVExpr)

-- | 'NamedSymVar' pairs symbolic words and user given/automatically generated names
type NamedSymVar = (SW, String)

-- | 'UnintKind' pairs array names and uninterpreted constants with their "kinds"
-- used mainly for printing counterexamples
data UnintKind = UFun Int String | UArr Int String      -- in each case, arity and the aliasing name
 deriving Show

-- | Result of running a symbolic computation
data Result = Result (Bool, Bool)                  -- contains unbounded integers/reals
                     [String]                      -- uninterpreted sorts
                     [(String, CW)]                -- quick-check counter-example information (if any)
                     [(String, [String])]          -- uninterpeted code segments
                     [(Quantifier, NamedSymVar)]   -- inputs (possibly existential)
                     [(SW, CW)]                    -- constants
                     [((Int, Kind, Kind), [SW])]   -- tables (automatically constructed) (tableno, index-type, result-type) elts
                     [(Int, ArrayInfo)]            -- arrays (user specified)
                     [(String, SBVType)]           -- uninterpreted constants
                     [(String, [String])]          -- axioms
                     Pgm                           -- assignments
                     [SW]                          -- additional constraints (boolean)
                     [SW]                          -- outputs

-- | Extract the constraints from a result
getConstraints :: Result -> [SW]
getConstraints (Result _ _ _ _ _ _ _ _ _ _ _ cstrs _) = cstrs

-- | Extract the traced-values from a result (quick-check)
getTraceInfo :: Result -> [(String, CW)]
getTraceInfo (Result _ _ tvals _ _ _ _ _ _ _ _ _ _) = tvals

instance Show Result where
  show (Result _ _ _ _ _ cs _ _ [] [] _ [] [r])
    | Just c <- r `lookup` cs
    = show c
  show (Result _ sorts _ cgs is cs ts as uis axs xs cstrs os)  = intercalate "\n" $
                   (if null sorts then [] else "SORTS" : map ("  " ++) sorts)
                ++ ["INPUTS"]
                ++ map shn is
                ++ ["CONSTANTS"]
                ++ map shc cs
                ++ ["TABLES"]
                ++ map sht ts
                ++ ["ARRAYS"]
                ++ map sha as
                ++ ["UNINTERPRETED CONSTANTS"]
                ++ map shui uis
                ++ ["USER GIVEN CODE SEGMENTS"]
                ++ concatMap shcg cgs
                ++ ["AXIOMS"]
                ++ map shax axs
                ++ ["DEFINE"]
                ++ map (\(s, e) -> "  " ++ shs s ++ " = " ++ show e) (F.toList xs)
                ++ ["CONSTRAINTS"]
                ++ map (("  " ++) . show) cstrs
                ++ ["OUTPUTS"]
                ++ map (("  " ++) . show) os
    where shs sw = show sw ++ " :: " ++ showType sw
          sht ((i, at, rt), es)  = "  Table " ++ show i ++ " : " ++ show at ++ "->" ++ show rt ++ " = " ++ show es
          shc (sw, cw) = "  " ++ show sw ++ " = " ++ show cw
          shcg (s, ss) = ("Variable: " ++ s) : map ("  " ++) ss
          shn (q, (sw, nm)) = "  " ++ ni ++ " :: " ++ showType sw ++ ex ++ alias
            where ni = show sw
                  ex | q == ALL = ""
                     | True     = ", existential"
                  alias | ni == nm = ""
                        | True     = ", aliasing " ++ show nm
          sha (i, (nm, (ai, bi), ctx)) = "  " ++ ni ++ " :: " ++ show ai ++ " -> " ++ show bi ++ alias
                                       ++ "\n     Context: "     ++ show ctx
            where ni = "array_" ++ show i
                  alias | ni == nm = ""
                        | True     = ", aliasing " ++ show nm
          shui (nm, t) = "  [uninterpreted] " ++ nm ++ " :: " ++ show t
          shax (nm, ss) = "  -- user defined axiom: " ++ nm ++ "\n  " ++ intercalate "\n  " ss

-- | The context of a symbolic array as created
data ArrayContext = ArrayFree (Maybe SW)     -- ^ A new array, with potential initializer for each cell
                  | ArrayReset Int SW        -- ^ An array created from another array by fixing each element to another value
                  | ArrayMutate Int SW SW    -- ^ An array created by mutating another array at a given cell
                  | ArrayMerge  SW Int Int   -- ^ An array created by symbolically merging two other arrays

instance Show ArrayContext where
  show (ArrayFree Nothing)  = " initialized with random elements"
  show (ArrayFree (Just s)) = " initialized with " ++ show s ++ " :: " ++ showType s
  show (ArrayReset i s)     = " reset array_" ++ show i ++ " with " ++ show s ++ " :: " ++ showType s
  show (ArrayMutate i a b)  = " cloned from array_" ++ show i ++ " with " ++ show a ++ " :: " ++ showType a ++ " |-> " ++ show b ++ " :: " ++ showType b
  show (ArrayMerge s i j)   = " merged arrays " ++ show i ++ " and " ++ show j ++ " on condition " ++ show s

-- | Expression map, used for hash-consing
type ExprMap   = Map.Map SBVExpr SW

-- | Constants are stored in a map, for hash-consing
type CnstMap   = Map.Map CW SW

-- | Tables generated during a symbolic run
type TableMap  = Map.Map [SW] (Int, Kind, Kind)

-- | Representation for symbolic arrays
type ArrayInfo = (String, (Kind, Kind), ArrayContext)

-- | Arrays generated during a symbolic run
type ArrayMap  = IMap.IntMap ArrayInfo

-- | Uninterpreted-constants generated during a symbolic run
type UIMap     = Map.Map String SBVType

-- | Code-segments for Uninterpreted-constants, as given by the user
type CgMap     = Map.Map String [String]

-- | Cached values, implementing sharing
type Cache a   = IMap.IntMap [(StableName (State -> IO a), a)]

-- | Convert an SBV-type to the kind-of uninterpreted value it represents
unintFnUIKind :: (String, SBVType) -> (String, UnintKind)
unintFnUIKind (s, t) = (s, UFun (typeArity t) s)

-- | Convert an array value type to the kind-of uninterpreted value it represents
arrayUIKind :: (Int, ArrayInfo) -> Maybe (String, UnintKind)
arrayUIKind (i, (nm, _, ctx)) 
  | external ctx = Just ("array_" ++ show i, UArr 1 nm) -- arrays are always 1-dimensional in the SMT-land. (Unless encoded explicitly)
  | True         = Nothing
  where external (ArrayFree{})   = True
        external (ArrayReset{})  = False
        external (ArrayMutate{}) = False
        external (ArrayMerge{})  = False

-- | Different means of running a symbolic piece of code
data SBVRunMode = Proof Bool      -- ^ Symbolic simulation mode, for proof purposes. Bool is True if it's a sat instance
                | CodeGen         -- ^ Code generation mode
                | Concrete StdGen -- ^ Concrete simulation mode. The StdGen is for the pConstrain acceptance in cross runs

-- | Is this a concrete run? (i.e., quick-check or test-generation like)
isConcreteMode :: SBVRunMode -> Bool
isConcreteMode (Concrete _) = True
isConcreteMode (Proof{})    = False
isConcreteMode CodeGen      = False

-- | The state of the symbolic interpreter
data State  = State { runMode       :: SBVRunMode
                    , rStdGen       :: IORef StdGen
                    , rCInfo        :: IORef [(String, CW)]
                    , rctr          :: IORef Int
                    , rUnBounded    :: IORef (Bool, Bool)     -- SInteger, SReal
                    , rinps         :: IORef [(Quantifier, NamedSymVar)]
                    , rConstraints  :: IORef [SW]
                    , routs         :: IORef [SW]
                    , rtblMap       :: IORef TableMap
                    , spgm          :: IORef Pgm
                    , rconstMap     :: IORef CnstMap
                    , rexprMap      :: IORef ExprMap
                    , rArrayMap     :: IORef ArrayMap
                    , rUIMap        :: IORef UIMap
                    , rCgMap        :: IORef CgMap
                    , raxioms       :: IORef [(String, [String])]
                    , rSWCache      :: IORef (Cache SW)
                    , rAICache      :: IORef (Cache Int)
                    , rSorts        :: IORef [String]
                    }

-- | Are we running in proof mode?
inProofMode :: State -> Bool
inProofMode s = case runMode s of
                  Proof{}    -> True
                  CodeGen    -> False
                  Concrete{} -> False

-- | The "Symbolic" value. Either a constant (@Left@) or a symbolic
-- value (@Right Cached@). Note that caching is essential for making
-- sure sharing is preserved. The parameter 'a' is phantom, but is
-- extremely important in keeping the user interface strongly typed.
data SBV a = SBV !Kind !(Either CW (Cached SW))

-- | A symbolic boolean/bit
type SBool   = SBV Bool

-- | 8-bit unsigned symbolic value
type SWord8  = SBV Word8

-- | 16-bit unsigned symbolic value
type SWord16 = SBV Word16

-- | 32-bit unsigned symbolic value
type SWord32 = SBV Word32

-- | 64-bit unsigned symbolic value
type SWord64 = SBV Word64

-- | 8-bit signed symbolic value, 2's complement representation
type SInt8   = SBV Int8

-- | 16-bit signed symbolic value, 2's complement representation
type SInt16  = SBV Int16

-- | 32-bit signed symbolic value, 2's complement representation
type SInt32  = SBV Int32

-- | 64-bit signed symbolic value, 2's complement representation
type SInt64  = SBV Int64

-- | Infinite precision signed symbolic value
type SInteger = SBV Integer

-- | Infinite precision symbolic algebraic real value
type SReal = SBV AlgReal

-- Not particularly "desirable", but will do if needed
instance Show (SBV a) where
  show (SBV _ (Left c))  = show c
  show (SBV k (Right _)) = "<symbolic> :: " ++ show k

-- Equality constraint on SBV values. Not desirable since we can't really compare two
-- symbolic values, but will do.
instance Eq (SBV a) where
  SBV _ (Left a) == SBV _ (Left b) = a == b
  a == b = error $ "Comparing symbolic bit-vectors; Use (.==) instead. Received: " ++ show (a, b)
  SBV _ (Left a) /= SBV _ (Left b) = a /= b
  a /= b = error $ "Comparing symbolic bit-vectors; Use (./=) instead. Received: " ++ show (a, b)

instance HasKind a => HasKind (SBV a) where
  kindOf _ = kindOf (undefined :: a)

-- | Increment the variable counter
incCtr :: State -> IO Int
incCtr s = do ctr <- readIORef (rctr s)
              let i = ctr + 1
              i `seq` writeIORef (rctr s) i
              return ctr

-- | Generate a random value, for quick-check and test-gen purposes
throwDice :: State -> IO Double
throwDice st = do g <- readIORef (rStdGen st)
                  let (r, g') = randomR (0, 1) g
                  writeIORef (rStdGen st) g'
                  return r

-- | Create a new uninterpreted symbol, possibly with user given code
newUninterpreted :: State -> String -> SBVType -> Maybe [String] -> IO ()
newUninterpreted st nm t mbCode
  | null nm || not (isAlpha (head nm)) || not (all validChar (tail nm))
  = error $ "Bad uninterpreted constant name: " ++ show nm ++ ". Must be a valid identifier."
  | True = do
        uiMap <- readIORef (rUIMap st)
        case nm `Map.lookup` uiMap of
          Just t' -> if t /= t'
                     then error $  "Uninterpreted constant " ++ show nm ++ " used at incompatible types\n"
                                ++ "      Current type      : " ++ show t ++ "\n"
                                ++ "      Previously used at: " ++ show t'
                     else return ()
          Nothing -> do modifyIORef (rUIMap st) (Map.insert nm t)
                        when (isJust mbCode) $ modifyIORef (rCgMap st) (Map.insert nm (fromJust mbCode))
  where validChar x = isAlphaNum x || x `elem` "_"

-- | Create a new constant; hash-cons as necessary
newConst :: State -> CW -> IO SW
newConst st c = do
  constMap <- readIORef (rconstMap st)
  case c `Map.lookup` constMap of
    Just sw -> return sw
    Nothing -> do ctr <- incCtr st
                  let k = kindOf c
                      sw = SW k (NodeId ctr)
                  () <- case kindOf c of
                         KUnbounded -> modifyIORef (rUnBounded st) (\(_, y) -> (True, y))
                         KReal      -> modifyIORef (rUnBounded st) (\(x, _) -> (x, True))
                         _          -> return ()
                  modifyIORef (rconstMap st) (Map.insert c sw)
                  return sw
{-# INLINE newConst #-}

-- | Create a new table; hash-cons as necessary
getTableIndex :: State -> Kind -> Kind -> [SW] -> IO Int
getTableIndex st at rt elts = do
  tblMap <- readIORef (rtblMap st)
  case elts `Map.lookup` tblMap of
    Just (i, _, _)  -> return i
    Nothing         -> do let i = Map.size tblMap
                          modifyIORef (rtblMap st) (Map.insert elts (i, at, rt))
                          return i

-- | Create a constant word
mkConstCW :: Integral a => Kind -> a -> CW
mkConstCW KReal a = normCW $ CW KReal (CWAlgReal (fromInteger (toInteger a)))
mkConstCW k     a = normCW $ CW k     (CWInteger (toInteger a))

-- | Create a new expression; hash-cons as necessary
newExpr :: State -> Kind -> SBVExpr -> IO SW
newExpr st k app = do
   let e = reorder app
   exprMap <- readIORef (rexprMap st)
   case e `Map.lookup` exprMap of
     Just sw -> return sw
     Nothing -> do ctr <- incCtr st
                   let sw = SW k (NodeId ctr)
                   () <- case k of
                          KUnbounded -> modifyIORef (rUnBounded st) (\(_, y) -> (True, y))
                          KReal      -> modifyIORef (rUnBounded st) (\(x, _) -> (x, True))
                          _          -> return ()
                   modifyIORef (spgm st)     (flip (S.|>) (sw, e))
                   modifyIORef (rexprMap st) (Map.insert e sw)
                   return sw
{-# INLINE newExpr #-}

-- | Convert a symbolic value to a symbolic-word
sbvToSW :: State -> SBV a -> IO SW
sbvToSW st (SBV _ (Left c))  = newConst st c
sbvToSW st (SBV _ (Right f)) = uncache f st

-------------------------------------------------------------------------
-- * Symbolic Computations
-------------------------------------------------------------------------
-- | A Symbolic computation. Represented by a reader monad carrying the
-- state of the computation, layered on top of IO for creating unique
-- references to hold onto intermediate results.
newtype Symbolic a = Symbolic (ReaderT State IO a)
                   deriving (Functor, Monad, MonadIO, MonadReader State)

-- | Create a symbolic value, based on the quantifier we have. If an explicit quantifier is given, we just use that.
-- If not, then we pick existential for SAT calls and universal for everything else.
mkSymSBV :: forall a. (Random a, SymWord a) => Maybe Quantifier -> Kind -> Maybe String -> Symbolic (SBV a)
mkSymSBV mbQ k mbNm = do
        st <- ask
        let q = case (mbQ, runMode st) of
                  (Just x,  _)           -> x   -- user given, just take it
                  (Nothing, Concrete{})  -> ALL -- concrete simulation, pick universal
                  (Nothing, Proof True)  -> EX  -- sat mode, pick existential
                  (Nothing, Proof False) -> ALL -- proof mode, pick universal
                  (Nothing, CodeGen)     -> ALL -- code generation, pick universal
        case runMode st of
          Concrete _ | q == EX -> case mbNm of
                                    Nothing -> error $ "Cannot quick-check in the presence of existential variables, type: " ++ showType (undefined :: SBV a)
                                    Just nm -> error $ "Cannot quick-check in the presence of existential variable " ++ nm ++ " :: " ++ showType (undefined :: SBV a)
          Concrete _           -> do v@(SBV _ (Left cw)) <- liftIO randomIO
                                     liftIO $ modifyIORef (rCInfo st) ((maybe "_" id mbNm, cw):)
                                     return v
          _          -> do ctr <- liftIO $ incCtr st
                           let nm = maybe ('s':show ctr) id mbNm
                               sw = SW k (NodeId ctr)
                           () <- case k of
                                   KUnbounded -> liftIO $ modifyIORef (rUnBounded st) (\(_, y) -> (True, y))
                                   KReal      -> liftIO $ modifyIORef (rUnBounded st) (\(x, _) -> (x, True))
                                   _          -> return ()
                           liftIO $ modifyIORef (rinps st) ((q, (sw, nm)):)
                           return $ SBV k $ Right $ cache (const (return sw))

-- | Convert a symbolic value to an SW, inside the Symbolic monad
sbvToSymSW :: SBV a -> Symbolic SW
sbvToSymSW sbv = do
        st <- ask
        liftIO $ sbvToSW st sbv

-- | A class representing what can be returned from a symbolic computation.
class Outputtable a where
  -- | Mark an interim result as an output. Useful when constructing Symbolic programs
  -- that return multiple values, or when the result is programmatically computed.
  output :: a -> Symbolic a

instance Outputtable (SBV a) where
  output i@(SBV _ (Left c)) = do
          st <- ask
          sw <- liftIO $ newConst st c
          liftIO $ modifyIORef (routs st) (sw:)
          return i
  output i@(SBV _ (Right f)) = do
          st <- ask
          sw <- liftIO $ uncache f st
          liftIO $ modifyIORef (routs st) (sw:)
          return i

instance Outputtable a => Outputtable [a] where
  output = mapM output

instance Outputtable () where
  output = return

instance (Outputtable a, Outputtable b) => Outputtable (a, b) where
  output = mlift2 (,) output output

instance (Outputtable a, Outputtable b, Outputtable c) => Outputtable (a, b, c) where
  output = mlift3 (,,) output output output

instance (Outputtable a, Outputtable b, Outputtable c, Outputtable d) => Outputtable (a, b, c, d) where
  output = mlift4 (,,,) output output output output

instance (Outputtable a, Outputtable b, Outputtable c, Outputtable d, Outputtable e) => Outputtable (a, b, c, d, e) where
  output = mlift5 (,,,,) output output output output output

instance (Outputtable a, Outputtable b, Outputtable c, Outputtable d, Outputtable e, Outputtable f) => Outputtable (a, b, c, d, e, f) where
  output = mlift6 (,,,,,) output output output output output output

instance (Outputtable a, Outputtable b, Outputtable c, Outputtable d, Outputtable e, Outputtable f, Outputtable g) => Outputtable (a, b, c, d, e, f, g) where
  output = mlift7 (,,,,,,) output output output output output output output

instance (Outputtable a, Outputtable b, Outputtable c, Outputtable d, Outputtable e, Outputtable f, Outputtable g, Outputtable h) => Outputtable (a, b, c, d, e, f, g, h) where
  output = mlift8 (,,,,,,,) output output output output output output output output

-- | Add a user specified axiom to the generated SMT-Lib file. The first argument is a mere
-- string, use for commenting purposes. The second argument is intended to hold the multiple-lines
-- of the axiom text as expressed in SMT-Lib notation. Note that we perform no checks on the axiom
-- itself, to see whether it's actually well-formed or is sensical by any means.
-- A separate formalization of SMT-Lib would be very useful here.
addAxiom :: String -> [String] -> Symbolic ()
addAxiom nm ax = do
        st <- ask
        liftIO $ modifyIORef (raxioms st) ((nm, ax) :)

-- | Run a symbolic computation in Proof mode and return a 'Result'. The boolean
-- argument indicates if this is a sat instance or not.
runSymbolic :: Bool -> Symbolic a -> IO Result
runSymbolic b c = snd `fmap` runSymbolic' (Proof b) c

-- | Run a symbolic computation, and return a extra value paired up with the 'Result'
runSymbolic' :: SBVRunMode -> Symbolic a -> IO (a, Result)
runSymbolic' currentRunMode (Symbolic c) = do
   ctr       <- newIORef (-2) -- start from -2; False and True will always occupy the first two elements
   cInfo     <- newIORef []
   pgm       <- newIORef S.empty
   emap      <- newIORef Map.empty
   cmap      <- newIORef Map.empty
   inps      <- newIORef []
   outs      <- newIORef []
   tables    <- newIORef Map.empty
   arrays    <- newIORef IMap.empty
   uis       <- newIORef Map.empty
   cgs       <- newIORef Map.empty
   axioms    <- newIORef []
   swCache   <- newIORef IMap.empty
   aiCache   <- newIORef IMap.empty
   unbounded <- newIORef (False, False)
   cstrs     <- newIORef []
   sorts     <- newIORef []
   rGen      <- case currentRunMode of
                  Concrete g -> newIORef g
                  _          -> newStdGen >>= newIORef
   let st = State { runMode      = currentRunMode
                  , rStdGen      = rGen
                  , rCInfo       = cInfo
                  , rctr         = ctr
                  , rUnBounded   = unbounded
                  , rinps        = inps
                  , routs        = outs
                  , rtblMap      = tables
                  , spgm         = pgm
                  , rconstMap    = cmap
                  , rArrayMap    = arrays
                  , rexprMap     = emap
                  , rUIMap       = uis
                  , rCgMap       = cgs
                  , raxioms      = axioms
                  , rSWCache     = swCache
                  , rAICache     = aiCache
                  , rConstraints = cstrs
                  , rSorts       = sorts
                  }
   _ <- newConst st (mkConstCW (KBounded False 1) (0::Integer)) -- s(-2) == falseSW
   _ <- newConst st (mkConstCW (KBounded False 1) (1::Integer)) -- s(-1) == trueSW
   r <- runReaderT c st
   rpgm  <- readIORef pgm
   inpsO <- reverse `fmap` readIORef inps
   outsO <- reverse `fmap` readIORef outs
   let swap (a, b) = (b, a)
       cmp  (a, _) (b, _) = a `compare` b
   cnsts <- (sortBy cmp . map swap . Map.toList) `fmap` readIORef (rconstMap st)
   tbls  <- (sortBy (\((x, _, _), _) ((y, _, _), _) -> x `compare` y) . map swap . Map.toList) `fmap` readIORef tables
   arrs  <- IMap.toAscList `fmap` readIORef arrays
   unint <- Map.toList `fmap` readIORef uis
   axs   <- reverse `fmap` readIORef axioms
   boundInfo <- readIORef unbounded
   cgMap <- Map.toList `fmap` readIORef cgs
   traceVals <- reverse `fmap` readIORef cInfo
   extraCstrs <- reverse `fmap` readIORef cstrs
   usorts <- reverse `fmap` readIORef sorts
   return $ (r, Result boundInfo usorts traceVals cgMap inpsO cnsts tbls arrs unint axs rpgm extraCstrs outsO)

-------------------------------------------------------------------------------
-- * Symbolic Words
-------------------------------------------------------------------------------
-- | A 'SymWord' is a potential symbolic bitvector that can be created instances of
-- to be fed to a symbolic program. Note that these methods are typically not needed
-- in casual uses with 'prove', 'sat', 'allSat' etc, as default instances automatically
-- provide the necessary bits.
class (HasKind a, Ord a) => SymWord a where
  -- | Create a user named input (universal)
  forall :: String -> Symbolic (SBV a)
  -- | Create an automatically named input
  forall_ :: Symbolic (SBV a)
  -- | Get a bunch of new words
  mkForallVars :: Int -> Symbolic [SBV a]
  -- | Create an existential variable
  exists  :: String -> Symbolic (SBV a)
  -- | Create an automatically named existential variable
  exists_ :: Symbolic (SBV a)
  -- | Create a bunch of existentials
  mkExistVars :: Int -> Symbolic [SBV a]
  -- | Create a free variable, universal in a proof, existential in sat
  free :: String -> Symbolic (SBV a)
  -- | Create an unnamed free variable, universal in proof, existential in sat
  free_ :: Symbolic (SBV a)
  -- | Create a bunch of free vars
  mkFreeVars :: Int -> Symbolic [SBV a]
  -- | Similar to free; Just a more convenient name
  symbolic  :: String -> Symbolic (SBV a)
  -- | Similar to mkFreeVars; but automatically gives names based on the strings
  symbolics :: [String] -> Symbolic [SBV a]
  -- | Turn a literal constant to symbolic
  literal :: a -> SBV a
  -- | Extract a literal, if the value is concrete
  unliteral :: SBV a -> Maybe a
  -- | Extract a literal, from a CW representation
  fromCW :: CW -> a
  -- | Is the symbolic word concrete?
  isConcrete :: SBV a -> Bool
  -- | Is the symbolic word really symbolic?
  isSymbolic :: SBV a -> Bool
  -- | Does it concretely satisfy the given predicate?
  isConcretely :: SBV a -> (a -> Bool) -> Bool
  -- | max/minbounds, if available. Note that we don't want
  -- to impose "Bounded" on our class as Integer is not Bounded but it is a SymWord
  mbMaxBound, mbMinBound :: Maybe a
  -- | One stop allocator
  mkSymWord :: Maybe Quantifier -> Maybe String -> Symbolic (SBV a)

  -- minimal complete definiton, Nothing.
  -- Giving no instances is ok when defining an uninterpreted sort, but otherwise you really
  -- want to define: mbMaxBound, mbMinBound, literal, fromCW, mkSymWord
  forall   = mkSymWord (Just ALL) . Just
  forall_  = mkSymWord (Just ALL)   Nothing
  exists   = mkSymWord (Just EX)  . Just
  exists_  = mkSymWord (Just EX)    Nothing
  free     = mkSymWord Nothing    . Just
  free_    = mkSymWord Nothing      Nothing
  mkForallVars n = mapM (const forall_) [1 .. n]
  mkExistVars n  = mapM (const exists_) [1 .. n]
  mkFreeVars n   = mapM (const free_)   [1 .. n]
  symbolic       = free
  symbolics      = mapM symbolic
  unliteral (SBV _ (Left c))  = Just $ fromCW c
  unliteral _                 = Nothing
  isConcrete (SBV _ (Left _)) = True
  isConcrete _                = False
  isSymbolic = not . isConcrete
  isConcretely s p
    | Just i <- unliteral s = p i
    | True                  = False
  -- Followings, you really want to define them unless the instance is for an uninterpreted sort
  mbMaxBound = Nothing
  mbMinBound = Nothing
  literal x = error $ "Cannot create symbolic literals for kind: " ++ show (kindOf x)
  fromCW cw = error $ "Cannot convert CW " ++ show cw ++ " to kind " ++ show (kindOf (undefined :: a))

  default mkSymWord :: Data a => Maybe Quantifier -> Maybe String -> Symbolic (SBV a)
  mkSymWord mbQ mbNm = do
        let sortName = tyconUQname . dataTypeName . dataTypeOf $ (undefined :: a)
        st <- ask
        let -- TBD: Is this list comprehensive?
            reserved = ["Int", "Real", "List", "Array", "Bool"]
        when (sortName `elem` reserved) $ error $ "SBV.registerSort: " ++ show sortName ++ " is a reserved sort; please use a different name"
        curSorts <- liftIO $ readIORef (rSorts st)
        when (sortName `notElem` curSorts) $ liftIO $ modifyIORef (rSorts st) (sortName :)
        let k = KUninterpreted sortName
            q = case (mbQ, runMode st) of
                  (Just x,  _)           -> x
                  (Nothing, Proof True)  -> EX
                  (Nothing, Proof False) -> ALL
                  (Nothing, Concrete{})  -> error $ "SBV.registerSort: Uninterpreted sort " ++ sortName ++ " can not be used in concrete simulation mode."
                  (Nothing, CodeGen)     -> error $ "SBV.registerSort: Uninterpreted sort " ++ sortName ++ " can not be used in code-generation mode."
        ctr <- liftIO $ incCtr st
        let sw = SW k (NodeId ctr)
            nm = maybe ('s':show ctr) id mbNm
        liftIO $ modifyIORef (rinps st) ((q, (sw, nm)):)
        return $ SBV k $ Right $ cache (const (return sw))

instance (Random a, SymWord a) => Random (SBV a) where
  randomR (l, h) g = case (unliteral l, unliteral h) of
                       (Just lb, Just hb) -> let (v, g') = randomR (lb, hb) g in (literal (v :: a), g')
                       _                  -> error $ "SBV.Random: Cannot generate random values with symbolic bounds"
  random         g = let (v, g') = random g in (literal (v :: a) , g')
---------------------------------------------------------------------------------
-- * Symbolic Arrays
---------------------------------------------------------------------------------

-- | Flat arrays of symbolic values
-- An @array a b@ is an array indexed by the type @'SBV' a@, with elements of type @'SBV' b@
-- If an initial value is not provided in 'newArray_' and 'newArray' methods, then the elements
-- are left unspecified, i.e., the solver is free to choose any value. This is the right thing
-- to do if arrays are used as inputs to functions to be verified, typically. 
--
-- While it's certainly possible for user to create instances of 'SymArray', the
-- 'SArray' and 'SFunArray' instances already provided should cover most use cases
-- in practice. (There are some differences between these models, however, see the corresponding
-- declaration.)
--
--
-- Minimal complete definition: All methods are required, no defaults.
class SymArray array where
  -- | Create a new array, with an optional initial value
  newArray_      :: (HasKind a, HasKind b) => Maybe (SBV b) -> Symbolic (array a b)
  -- | Create a named new array, with an optional initial value
  newArray       :: (HasKind a, HasKind b) => String -> Maybe (SBV b) -> Symbolic (array a b)
  -- | Read the array element at @a@
  readArray      :: array a b -> SBV a -> SBV b
  -- | Reset all the elements of the array to the value @b@
  resetArray     :: SymWord b => array a b -> SBV b -> array a b
  -- | Update the element at @a@ to be @b@
  writeArray     :: SymWord b => array a b -> SBV a -> SBV b -> array a b
  -- | Merge two given arrays on the symbolic condition
  -- Intuitively: @mergeArrays cond a b = if cond then a else b@.
  -- Merging pushes the if-then-else choice down on to elements
  mergeArrays    :: SymWord b => SBV Bool -> array a b -> array a b -> array a b

-- | Arrays implemented in terms of SMT-arrays: <http://goedel.cs.uiowa.edu/smtlib/theories/ArraysEx.smt2>
--
--   * Maps directly to SMT-lib arrays
--
--   * Reading from an unintialized value is OK and yields an uninterpreted result
--
--   * Can check for equality of these arrays
--
--   * Cannot quick-check theorems using @SArray@ values
--
--   * Typically slower as it heavily relies on SMT-solving for the array theory
--
data SArray a b = SArray (Kind, Kind) (Cached ArrayIndex)

-- | An array index is simple an int value
type ArrayIndex = Int

instance (HasKind a, HasKind b) => Show (SArray a b) where
  show (SArray{}) = "SArray<" ++ showType (undefined :: a) ++ ":" ++ showType (undefined :: b) ++ ">"

instance SymArray SArray where
  newArray_  = declNewSArray (\t -> "array_" ++ show t)
  newArray n = declNewSArray (const n)
  readArray (SArray (_, bk) f) a = SBV bk $ Right $ cache r
     where r st = do arr <- uncacheAI f st
                     i   <- sbvToSW st a
                     newExpr st bk (SBVApp (ArrRead arr) [i])
  resetArray (SArray ainfo f) b = SArray ainfo $ cache g
     where g st = do amap <- readIORef (rArrayMap st)
                     val <- sbvToSW st b
                     i <- uncacheAI f st
                     let j = IMap.size amap
                     j `seq` modifyIORef (rArrayMap st) (IMap.insert j ("array_" ++ show j, ainfo, ArrayReset i val))
                     return j
  writeArray (SArray ainfo f) a b = SArray ainfo $ cache g
     where g st = do arr  <- uncacheAI f st
                     addr <- sbvToSW st a
                     val  <- sbvToSW st b
                     amap <- readIORef (rArrayMap st)
                     let j = IMap.size amap
                     j `seq` modifyIORef (rArrayMap st) (IMap.insert j ("array_" ++ show j, ainfo, ArrayMutate arr addr val))
                     return j
  mergeArrays t (SArray ainfo a) (SArray _ b) = SArray ainfo $ cache h
    where h st = do ai <- uncacheAI a st
                    bi <- uncacheAI b st
                    ts <- sbvToSW st t
                    amap <- readIORef (rArrayMap st)
                    let k = IMap.size amap
                    k `seq` modifyIORef (rArrayMap st) (IMap.insert k ("array_" ++ show k, ainfo, ArrayMerge ts ai bi))
                    return k

-- | Declare a new symbolic array, with a potential initial value
declNewSArray :: forall a b. (HasKind a, HasKind b) => (Int -> String) -> Maybe (SBV b) -> Symbolic (SArray a b)
declNewSArray mkNm mbInit = do
   let aknd = kindOf (undefined :: a)
       bknd = kindOf (undefined :: b)
   st <- ask
   amap <- liftIO $ readIORef $ rArrayMap st
   let i = IMap.size amap
       nm = mkNm i
   actx <- liftIO $ case mbInit of
                     Nothing   -> return $ ArrayFree Nothing
                     Just ival -> sbvToSW st ival >>= \sw -> return $ ArrayFree (Just sw)
   liftIO $ modifyIORef (rArrayMap st) (IMap.insert i (nm, (aknd, bknd), actx))
   return $ SArray (aknd, bknd) $ cache $ const $ return i

-- | Arrays implemented internally as functions
--
--    * Internally handled by the library and not mapped to SMT-Lib
--
--    * Reading an uninitialized value is considered an error (will throw exception)
--
--    * Cannot check for equality (internally represented as functions)
--
--    * Can quick-check
--
--    * Typically faster as it gets compiled away during translation
--
data SFunArray a b = SFunArray (SBV a -> SBV b)

instance (HasKind a, HasKind b) => Show (SFunArray a b) where
  show (SFunArray _) = "SFunArray<" ++ showType (undefined :: a) ++ ":" ++ showType (undefined :: b) ++ ">"

-- | Lift a function to an array. Useful for creating arrays in a pure context. (Otherwise use `newArray`.)
mkSFunArray :: (SBV a -> SBV b) -> SFunArray a b
mkSFunArray = SFunArray

-- | Handling constraints
imposeConstraint :: SBool -> Symbolic ()
imposeConstraint c = do st <- ask
                        case runMode st of
                          CodeGen -> error "SBV: constraints are not allowed in code-generation"
                          _       -> do liftIO $ do v <- sbvToSW st c
                                                    modifyIORef (rConstraints st) (v:)

-- | Add a constraint with a given probability
addConstraint :: Maybe Double -> SBool -> SBool -> Symbolic ()
addConstraint Nothing  c _  = imposeConstraint c
addConstraint (Just t) c c'
  | t < 0 || t > 1
  = error $ "SBV: pConstrain: Invalid probability threshold: " ++ show t ++ ", must be in [0, 1]."
  | True
  = do st <- ask
       when (not (isConcreteMode (runMode st))) $ error "SBV: pConstrain only allowed in 'genTest' or 'quickCheck' contexts."
       case () of
         () | t > 0 && t < 1 -> liftIO (throwDice st) >>= \d -> imposeConstraint (if d <= t then c else c')
            | t > 0          -> imposeConstraint c
            | True           -> imposeConstraint c'

---------------------------------------------------------------------------------
-- * Cached values
---------------------------------------------------------------------------------

-- | We implement a peculiar caching mechanism, applicable to the use case in
-- implementation of SBV's.  Whenever we do a state based computation, we do
-- not want to keep on evaluating it in the then-current state. That will
-- produce essentially a semantically equivalent value. Thus, we want to run
-- it only once, and reuse that result, capturing the sharing at the Haskell
-- level. This is similar to the "type-safe observable sharing" work, but also
-- takes into the account of how symbolic simulation executes.
--
-- See Andy Gill's type-safe obervable sharing trick for the inspiration behind
-- this technique: <http://ittc.ku.edu/~andygill/paper.php?label=DSLExtract09>
--
-- Note that this is *not* a general memo utility!
newtype Cached a = Cached (State -> IO a)

-- | Cache a state-based computation
cache :: (State -> IO a) -> Cached a
cache = Cached

-- | Uncache a previously cached computation
uncache :: Cached SW -> State -> IO SW
uncache = uncacheGen rSWCache

-- | Uncache, retrieving array indexes
uncacheAI :: Cached ArrayIndex -> State -> IO ArrayIndex
uncacheAI = uncacheGen rAICache

-- | Generic uncaching. Note that this is entirely safe, since we do it in the IO monad.
uncacheGen :: (State -> IORef (Cache a)) -> Cached a -> State -> IO a
uncacheGen getCache (Cached f) st = do
        let rCache = getCache st
        stored <- readIORef rCache
        sn <- f `seq` makeStableName f
        let h = hashStableName sn
        case maybe Nothing (sn `lookup`) (h `IMap.lookup` stored) of
          Just r  -> return r
          Nothing -> do r <- f st
                        r `seq` modifyIORef rCache (IMap.insertWith (++) h [(sn, r)])
                        return r

-- | Representation of SMTLib Program versions, currently we only know of versions 1 and 2.
-- (NB. Eventually, we should just drop SMTLib1.)
data SMTLibVersion = SMTLib1
                   | SMTLib2
                   deriving Eq

-- | Representation of an SMT-Lib program. In between pre and post goes the refuted models
data SMTLibPgm = SMTLibPgm SMTLibVersion  ( [(String, SW)]          -- alias table
                                          , [String]                -- pre: declarations.
                                          , [String])               -- post: formula
instance NFData SMTLibVersion
instance NFData SMTLibPgm

instance Show SMTLibPgm where
  show (SMTLibPgm _ (_, pre, post)) = intercalate "\n" $ pre ++ post

-- Other Technicalities..
instance NFData CW where
  rnf (CW x y) = x `seq` y `seq` ()

instance NFData Result where
  rnf (Result isInf sorts qcInfo cgs inps consts tbls arrs uis axs pgm cstr outs)
        = rnf isInf `seq` rnf sorts `seq` rnf qcInfo `seq` rnf cgs
                    `seq` rnf inps  `seq` rnf consts `seq` rnf tbls
                    `seq` rnf arrs  `seq` rnf uis    `seq` rnf axs
                    `seq` rnf pgm   `seq` rnf cstr   `seq` rnf outs

instance NFData Kind
instance NFData ArrayContext
instance NFData SW
instance NFData SBVExpr
instance NFData Quantifier
instance NFData SBVType
instance NFData UnintKind
instance NFData a => NFData (Cached a) where
  rnf (Cached f) = f `seq` ()
instance NFData a => NFData (SBV a) where
  rnf (SBV x y) = rnf x `seq` rnf y `seq` ()
instance NFData Pgm