SimpleSMT.hs

{-# LANGUAGE Safe #-}
{-# LANGUAGE RecordWildCards #-}
{-# LANGUAGE PatternGuards #-}
-- | A module for interacting with an SMT solver, using SmtLib-2 format.
module SimpleSMT
  (
    -- * Basic Solver Interface
    Solver(..)
  , newSolver
  , newSolverNotify
  , ackCommand
  , simpleCommand
  , simpleCommandMaybe
  , loadFile
  , loadString

    -- ** S-Expressions
  , SExpr(..)
  , showsSExpr, ppSExpr, readSExpr

    -- ** Logging and Debugging
  , Logger(..)
  , newLogger
  , withLogLevel
  , logMessageAt
  , logIndented

    -- * Common SmtLib-2 Commands
  , setLogic, setLogicMaybe
  , setOption, setOptionMaybe
  , produceUnsatCores
  , named
  , push, pushMany
  , pop, popMany
  , inNewScope
  , declare
  , declareFun
  , declareDatatype
  , define
  , defineFun
  , defineFunRec
  , defineFunsRec
  , assert
  , check
  , Result(..)
  , getExprs, getExpr
  , getConsts, getConst
  , getUnsatCore
  , Value(..)
  , sexprToVal

    -- * Convenience Functions for SmtLib-2 Expressions
  , fam
  , fun
  , const
  , app

    -- * Convenience Functions for SmtLib-2 identifiers
  , quoteSymbol
  , symbol
  , keyword
  , as

    -- ** Types
  , tInt
  , tBool
  , tReal
  , tArray
  , tBits

    -- ** Literals
  , int
  , real
  , bool
  , bvBin
  , bvHex
  , value

    -- ** Connectives
  , not
  , and
  , andMany
  , or
  , orMany
  , xor
  , implies

    -- ** If-then-else
  , ite

    -- ** Relational Predicates
  , eq
  , distinct
  , gt
  , lt
  , geq
  , leq
  , bvULt
  , bvULeq
  , bvSLt
  , bvSLeq

    -- ** Arithmetic
  , add
  , addMany
  , sub
  , neg
  , mul
  , abs
  , div
  , mod
  , divisible
  , realDiv
  , toInt
  , toReal

    -- ** Bit Vectors
  , concat
  , extract
  , bvNot
  , bvNeg
  , bvAnd
  , bvXOr
  , bvOr
  , bvAdd
  , bvSub
  , bvMul
  , bvUDiv
  , bvURem
  , bvSDiv
  , bvSRem
  , bvShl
  , bvLShr
  , bvAShr
  , signExtend
  , zeroExtend

    -- ** Arrays
  , select
  , store
  ) where

import Prelude hiding (not, and, or, abs, div, mod, concat, const)
import qualified Prelude as P
import Data.Char(isSpace, isDigit)
import Data.List(unfoldr,intersperse)
import Data.Bits(testBit)
import Data.IORef(newIORef, atomicModifyIORef, modifyIORef', readIORef,
                  writeIORef)
import System.Process(runInteractiveProcess, waitForProcess, terminateProcess)
import System.IO (hFlush, hGetLine, hGetContents, hPutStrLn, stdout, hClose)
import System.Exit(ExitCode)
import qualified Control.Exception as X
import Control.Concurrent(forkIO)
import Control.Monad(forever,when,void)
import Text.Read(readMaybe)
import Data.Ratio((%), numerator, denominator)
import Numeric(showHex, readHex, showFFloat)


-- | Results of checking for satisfiability.
data Result = Sat         -- ^ The assertions are satisfiable
            | Unsat       -- ^ The assertions are unsatisfiable
            | Unknown     -- ^ The result is inconclusive
              deriving (Eq,Show)

-- | Common values returned by SMT solvers.
data Value =  Bool  !Bool           -- ^ Boolean value
            | Int   !Integer        -- ^ Integral value
            | Real  !Rational       -- ^ Rational value
            | Bits  !Int !Integer   -- ^ Bit vector: width, value
            | Other !SExpr          -- ^ Some other value
              deriving (Eq,Show)

-- | S-expressions. These are the basic format for SmtLib-2.
data SExpr  = Atom String
            | List [SExpr]
              deriving (Eq, Ord, Show)

-- | Show an s-expression.
--
-- >>> let Just (e, _) = readSExpr "(assert (= ((_ map (- (Int Int) Int)) a1Cl a1Cm) a1Cv))"
-- >>> putStrLn $ showsSExpr e ""
-- (assert (= ((_ map (- (Int Int) Int)) a1Cl a1Cm) a1Cv))
showsSExpr :: SExpr -> ShowS
showsSExpr ex =
  case ex of
    Atom x  -> showString x
    List [] -> showString "()"
    List (e0 : es) -> showChar '(' . showsSExpr e0 .
                       foldr (\e m -> showChar ' ' . showsSExpr e . m)
                       (showChar ')') es


-- | Show an s-expression in a somewhat readable fashion.
--
-- >>> let Just (e, _) = readSExpr "(assert (= ((_ map (- (Int Int) Int)) a1Cl a1Cm) a1Cv))"
-- >>> putStrLn $ ppSExpr e ""
-- (assert
--    (=
--       (
--         (_
--            map
--            (-
--               (Int Int)
--               Int))
--         a1Cl
--         a1Cm)
--       a1Cv))
ppSExpr :: SExpr -> ShowS
ppSExpr = go 0
  where
  tab n = showString (replicate n ' ')
  many  = foldr (.) id

  new n e = showChar '\n' . tab n . go n e

  small n es =
    case es of
      [] -> Just []
      e : more
        | n <= 0 -> Nothing
        | otherwise -> case e of
                         Atom x -> (showString x :) <$> small (n-1) more
                         _      -> Nothing

  go :: Int -> SExpr -> ShowS
  go n ex =
    case ex of
      Atom x        -> showString x
      List es
        | Just fs <- small 5 es ->
          showChar '(' . many (intersperse (showChar ' ') fs) . showChar ')'

      List (Atom x : es) -> showString "(" . showString x .
                                many (map (new (n+3)) es) . showString ")"

      List es -> showString "(" . many (map (new (n+2)) es) . showString ")"

-- | Parse an s-expression.
--
-- >>> readSExpr "(_ map (- (Int Int) Int)) a1Cl a1Cm)"
-- Just (List [Atom "_",Atom "map",List [Atom "-",List [Atom "Int",Atom "Int"],Atom "Int"]]," a1Cl a1Cm)")
readSExpr :: String -> Maybe (SExpr, String)
readSExpr (c : more) | isSpace c = readSExpr more
readSExpr (';' : more) = readSExpr $ drop 1 $ dropWhile (/= '\n') more
readSExpr ('|' : more) = do (sym, '|' : rest) <- pure (span ((/=) '|') more)
                            Just (Atom ('|' : sym ++ ['|']), rest)
readSExpr ('(' : more) = do (xs,more1) <- list more
                            return (List xs, more1)
  where
  list (c : txt) | isSpace c = list txt
  list (')' : txt) = return ([], txt)
  list txt         = do (v,txt1) <- readSExpr txt
                        (vs,txt2) <- list txt1
                        return (v:vs, txt2)
readSExpr txt     = case break end txt of
                       (as,bs) | P.not (null as) -> Just (Atom as, bs)
                       _ -> Nothing
  where end x = x == ')' || isSpace x


--------------------------------------------------------------------------------

-- | An interactive solver process.
data Solver = Solver
  { command   :: SExpr -> IO SExpr
    -- ^ Send a command to the solver.

  , stop :: IO ExitCode
    -- ^ Wait for the solver to finish and exit gracefully.

  , forceStop :: IO ExitCode
    -- ^ Terminate the solver without waiting for it to finish.
  }


-- | Start a new solver process.
newSolver :: String       {- ^ Executable -}            ->
             [String]     {- ^ Arguments -}             ->
             Maybe Logger {- ^ Optional logging here -} ->
             IO Solver
newSolver n xs l = newSolverNotify n xs l Nothing

newSolverNotify ::
  String        {- ^ Executable -}            ->
  [String]      {- ^ Arguments -}             ->
  Maybe Logger  {- ^ Optional logging here -} ->
  Maybe (ExitCode -> IO ()) {- ^ Do this when the solver exits -} ->
  IO Solver
newSolverNotify exe opts mbLog mbOnExit =
  do (hIn, hOut, hErr, h) <- runInteractiveProcess exe opts Nothing Nothing

     let info a = case mbLog of
                    Nothing -> return ()
                    Just l  -> logMessage l a

     _ <- forkIO $ forever (do errs <- hGetLine hErr
                               info ("[stderr] " ++ errs))
                    `X.catch` \X.SomeException {} -> return ()

     case mbOnExit of
       Nothing -> pure ()
       Just this -> void (forkIO (this =<< waitForProcess h))

     getResponse <-
       do txt <- hGetContents hOut                  -- Read *all* output
          ref <- newIORef (unfoldr readSExpr txt)  -- Parse, and store result
          return $ atomicModifyIORef ref $ \xs ->
                      case xs of
                        []     -> (xs, Nothing)
                        y : ys -> (ys, Just y)

     let cmd c = do let txt = showsSExpr c ""
                    info ("[send->] " ++ txt)
                    hPutStrLn hIn txt
                    hFlush hIn

         command c =
           do cmd c
              mb <- getResponse
              case mb of
                Just res -> do info ("[<-recv] " ++ showsSExpr res "")
                               return res
                Nothing  -> fail "Missing response from solver"

         waitAndCleanup =
           do ec <- waitForProcess h
              X.catch (do hClose hIn
                          hClose hOut
                          hClose hErr)
                      (\ex -> info (show (ex::X.IOException)))
              return ec

         forceStop = terminateProcess h *> waitAndCleanup

         stop =
           do cmd (List [Atom "exit"])
                `X.catch` (\X.SomeException{} -> pure ())
              waitAndCleanup

         solver = Solver { .. }

     setOption solver ":print-success" "true"
     setOption solver ":produce-models" "true"

     return solver


-- | Load the contents of a file.
loadFile :: Solver -> FilePath -> IO ()
loadFile s file = loadString s =<< readFile file

-- | Load a raw SMT string.
loadString :: Solver -> String -> IO ()
loadString s str = go (dropComments str)
  where
  go txt
    | all isSpace txt = return ()
    | otherwise =
      case readSExpr txt of
        Just (e,rest) -> command s e >> go rest
        Nothing       -> fail $ unlines [ "Failed to parse SMT file."
                                        , txt
                                        ]

  dropComments = unlines . map dropComment . lines
  dropComment xs = case break (== ';') xs of
                     (as,_:_) -> as
                     _ -> xs




-- | A command with no interesting result.
ackCommand :: Solver -> SExpr -> IO ()
ackCommand proc c =
  do res <- command proc c
     case res of
       Atom "success" -> return ()
       _  -> fail $ unlines
                      [ "Unexpected result from the SMT solver:"
                      , "  Expected: success"
                      , "  Result: " ++ showsSExpr res ""
                      ]

-- | A command entirely made out of atoms, with no interesting result.
simpleCommand :: Solver -> [String] -> IO ()
simpleCommand proc = ackCommand proc . List . map Atom

-- | Run a command and return True if successful, and False if unsupported.
-- This is useful for setting options that unsupported by some solvers, but used
-- by others.
simpleCommandMaybe :: Solver -> [String] -> IO Bool
simpleCommandMaybe proc c =
  do res <- command proc (List (map Atom c))
     case res of
       Atom "success"     -> return True
       Atom "unsupported" -> return False
       _                  -> fail $ unlines
                                      [ "Unexpected result from the SMT solver:"
                                      , "  Expected: success or unsupported"
                                      , "  Result: " ++ showsSExpr res ""
                                      ]


-- | Set a solver option.
setOption :: Solver -> String -> String -> IO ()
setOption s x y = simpleCommand s [ "set-option", x, y ]

-- | Set a solver option, returning False if the option is unsupported.
setOptionMaybe :: Solver -> String -> String -> IO Bool
setOptionMaybe s x y = simpleCommandMaybe s [ "set-option", x, y ]

-- | Set the solver's logic.  Usually, this should be done first.
setLogic :: Solver -> String -> IO ()
setLogic s x = simpleCommand s [ "set-logic", x ]


-- | Set the solver's logic, returning False if the logic is unsupported.
setLogicMaybe :: Solver -> String -> IO Bool
setLogicMaybe s x = simpleCommandMaybe s [ "set-logic", x ]

-- | Request unsat cores.  Returns if the solver supports them.
produceUnsatCores :: Solver -> IO Bool
produceUnsatCores s = setOptionMaybe s ":produce-unsat-cores" "true"

-- | Checkpoint state.  A special case of 'pushMany'.
push :: Solver -> IO ()
push proc = pushMany proc 1

-- | Restore to last check-point.  A special case of 'popMany'.
pop :: Solver -> IO ()
pop proc = popMany proc 1

-- | Push multiple scopes.
pushMany :: Solver -> Integer -> IO ()
pushMany proc n = simpleCommand proc [ "push", show n ]

-- | Pop multiple scopes.
popMany :: Solver -> Integer -> IO ()
popMany proc n = simpleCommand proc [ "pop", show n ]

-- | Execute the IO action in a new solver scope (push before, pop after)
inNewScope :: Solver -> IO a -> IO a
inNewScope s m =
  do push s
     m `X.finally` pop s



-- | Declare a constant.  A common abbreviation for 'declareFun'.
-- For convenience, returns an the declared name as a constant expression.
declare :: Solver -> String -> SExpr -> IO SExpr
declare proc f t = declareFun proc f [] t

-- | Declare a function or a constant.
-- For convenience, returns an the declared name as a constant expression.
declareFun :: Solver -> String -> [SExpr] -> SExpr -> IO SExpr
declareFun proc f as r =
  do ackCommand proc $ fun "declare-fun" [ Atom f, List as, r ]
     return (const f)

-- | Declare an ADT using the format introduced in SmtLib 2.6.
declareDatatype ::
  Solver ->
  String {- ^ datatype name -} ->
  [String] {- ^ sort parameters -} ->
  [(String, [(String, SExpr)])] {- ^ constructors -} ->
  IO ()
declareDatatype proc t [] cs =
  ackCommand proc $
    fun "declare-datatype" $
      [ Atom t
      , List [ List (Atom c : [ List [Atom s, argTy] | (s, argTy) <- args]) | (c, args) <- cs ]
      ]
declareDatatype proc t ps cs =
  ackCommand proc $
    fun "declare-datatype" $
      [ Atom t
      , fun "par" $
          [ List (map Atom ps)
          , List [ List (Atom c : [ List [Atom s, argTy] | (s, argTy) <- args]) | (c, args) <- cs ]
          ]
      ]


-- | Declare a constant.  A common abbreviation for 'declareFun'.
-- For convenience, returns the defined name as a constant expression.
define :: Solver ->
          String {- ^ New symbol -} ->
          SExpr  {- ^ Symbol type -} ->
          SExpr  {- ^ Symbol definition -} ->
          IO SExpr
define proc f t e = defineFun proc f [] t e

-- | Define a function or a constant.
-- For convenience, returns an the defined name as a constant expression.
defineFun :: Solver ->
             String           {- ^ New symbol -} ->
             [(String,SExpr)] {- ^ Parameters, with types -} ->
             SExpr            {- ^ Type of result -} ->
             SExpr            {- ^ Definition -} ->
             IO SExpr
defineFun proc f as t e =
  do ackCommand proc $ fun "define-fun"
                     $ [ Atom f, List [ List [const x,a] | (x,a) <- as ], t, e]
     return (const f)

-- | Define a recursive function or a constant.  For convenience,
-- returns an the defined name as a constant expression.  This body
-- takes the function name as an argument.
defineFunRec :: Solver ->
                String           {- ^ New symbol -} ->
                [(String,SExpr)] {- ^ Parameters, with types -} ->
                SExpr            {- ^ Type of result -} ->
                (SExpr -> SExpr) {- ^ Definition -} ->
                IO SExpr
defineFunRec proc f as t e =
  do let fs = const f
     ackCommand proc $ fun "define-fun-rec"
                     $ [ Atom f, List [ List [const x,a] | (x,a) <- as ], t, e fs]
     return fs

-- | Define a recursive function or a constant.  For convenience,
-- returns an the defined name as a constant expression.  This body
-- takes the function name as an argument.
defineFunsRec :: Solver ->
                 [(String, [(String,SExpr)], SExpr, SExpr)] ->
                 IO ()
defineFunsRec proc ds = ackCommand proc $ fun "define-funs-rec" [ decls, bodies ]
  where
    oneArg (f, args, t, _) = List [ Atom f, List [ List [const x,a] | (x,a) <- args ], t]
    decls  = List (map oneArg ds)
    bodies = List (map (\(_, _, _, body) -> body) ds)


-- | Assume a fact.
assert :: Solver -> SExpr -> IO ()
assert proc e = ackCommand proc $ fun "assert" [e]

-- | Check if the current set of assertion is consistent.
check :: Solver -> IO Result
check proc =
  do res <- command proc (List [ Atom "check-sat" ])
     case res of
       Atom "unsat"   -> return Unsat
       Atom "unknown" -> return Unknown
       Atom "sat"     -> return Sat
       _ -> fail $ unlines
              [ "Unexpected result from the SMT solver:"
              , "  Expected: unsat, unknown, or sat"
              , "  Result: " ++ showsSExpr res ""
              ]

-- | Convert an s-expression to a value.
sexprToVal :: SExpr -> Value
sexprToVal expr =
  case expr of
    Atom "true"                    -> Bool True
    Atom "false"                   -> Bool False
    Atom ('#' : 'b' : ds)
      | Just n <- binLit ds         -> Bits (length ds) n
    Atom ('#' : 'x' : ds)
      | [(n,[])] <- readHex ds      -> Bits (4 * length ds) n
    Atom txt
      | Just n <- readMaybe txt     -> Int n
    List [ Atom "-", x ]
      | Int a <- sexprToVal x    -> Int (negate a)
    List [ Atom "/", x, y ]
      | Int a <- sexprToVal x
      , Int b <- sexprToVal y    -> Real (a % b)
    _ -> Other expr

  where
  binLit cs = do ds <- mapM binDigit cs
                 return $ sum $ zipWith (*) (reverse ds) powers2
  powers2   = 1 : map (2 *) powers2
  binDigit '0' = Just 0
  binDigit '1' = Just 1
  binDigit _   = Nothing

-- | Get the values of some s-expressions.
-- Only valid after a 'Sat' result.
getExprs :: Solver -> [SExpr] -> IO [(SExpr, Value)]
getExprs proc vals =
  do res <- command proc $ List [ Atom "get-value", List vals ]
     case res of
       List xs -> mapM getAns xs
       _ -> fail $ unlines
                 [ "Unexpected response from the SMT solver:"
                 , "  Exptected: a list"
                 , "  Result: " ++ showsSExpr res ""
                 ]
  where
  getAns expr =
    case expr of
      List [ e, v ] -> return (e, sexprToVal v)
      _             -> fail $ unlines
                            [ "Unexpected response from the SMT solver:"
                            , "  Expected: (expr val)"
                            , "  Result: " ++ showsSExpr expr ""
                            ]

-- | Get the values of some constants in the current model.
-- A special case of 'getExprs'.
-- Only valid after a 'Sat' result.
getConsts :: Solver -> [String] -> IO [(String, Value)]
getConsts proc xs =
  do ans <- getExprs proc (map Atom xs)
     return [ (x,e) | (Atom x, e) <- ans ]


-- | Get the value of a single expression.
getExpr :: Solver -> SExpr -> IO Value
getExpr proc x =
  do [ (_,v) ] <- getExprs proc [x]
     return v

-- | Get the value of a single constant.
getConst :: Solver -> String -> IO Value
getConst proc x = getExpr proc (Atom x)

-- | Returns the names of the (named) formulas involved in a contradiction.
getUnsatCore :: Solver -> IO [String]
getUnsatCore s =
  do res <- command s $ List [ Atom "get-unsat-core" ]
     case res of
       List xs -> mapM fromAtom xs
       _       -> unexpected "a list of atoms" res
  where
  fromAtom x =
    case x of
      Atom a -> return a
      _      -> unexpected "an atom" x

  unexpected x e =
    fail $ unlines [ "Unexpected response from the SMT Solver:"
                   , "  Expected: " ++ x
                   , "  Result: " ++ showsSExpr e ""
                   ]

--------------------------------------------------------------------------------


-- | A constant, corresponding to a family indexed by some integers.
fam :: String -> [Integer] -> SExpr
fam f is = List (Atom "_" : Atom f : map (Atom . show) is)

-- | An SMT function.
fun :: String -> [SExpr] -> SExpr
fun f [] = Atom f
fun f as = List (Atom f : as)

-- | An SMT constant.  A special case of 'fun'.
const :: String -> SExpr
const f = fun f []

app :: SExpr -> [SExpr] -> SExpr
app f xs = List (f : xs)

-- Identifiers -----------------------------------------------------------------------

-- | Symbols are either simple or quoted (c.f. SMTLIB v2.6 S3.1).
-- This predicate indicates whether a character is allowed in a simple
-- symbol.  Note that only ASCII letters are allowed.
allowedSimpleChar :: Char -> Bool
allowedSimpleChar c =
  isDigit c || c `elem` (['a' .. 'z'] ++ ['A' .. 'Z'] ++ "~!@$%^&*_-+=<>.?/")

isSimpleSymbol :: String -> Bool
isSimpleSymbol s@(c : _) = P.not (isDigit c) && all allowedSimpleChar s
isSimpleSymbol _         = False

quoteSymbol :: String -> String
quoteSymbol s
  | isSimpleSymbol s = s
  | otherwise        = '|' : s ++ "|"

symbol :: String -> SExpr
symbol = Atom . quoteSymbol

keyword :: String -> SExpr
keyword s = Atom (':' : s)

-- | Generate a type annotation for a symbol
as :: SExpr -> SExpr -> SExpr
as s t = fun "as" [s, t]

-- Types -----------------------------------------------------------------------


-- | The type of integers.
tInt :: SExpr
tInt = const "Int"

-- | The type of booleans.
tBool :: SExpr
tBool = const "Bool"

-- | The type of reals.
tReal :: SExpr
tReal = const "Real"

-- | The type of arrays.
tArray :: SExpr {- ^ Type of indexes  -} ->
          SExpr {- ^ Type of elements -} ->
          SExpr
tArray x y = fun "Array" [x,y]

-- | The type of bit vectors.
tBits :: Integer {- ^ Number of bits -} ->
         SExpr
tBits w = fam "BitVec" [w]



-- Literals --------------------------------------------------------------------

-- | Boolean literals.
bool :: Bool -> SExpr
bool b = const (if b then "true" else "false")

-- | Integer literals.
int :: Integer -> SExpr
int x | x < 0     = neg (int (negate x))
      | otherwise = Atom (show x)

-- | Real (well, rational) literals.
real :: Rational -> SExpr
real x
  | toRational y == x = Atom (showFFloat Nothing y "")
  | otherwise = realDiv (int (numerator x)) (int (denominator x))
  where y = fromRational x :: Double

-- | A bit vector represented in binary.
--
--     * If the value does not fit in the bits, then the bits will be increased.
--     * The width should be strictly positive.
bvBin :: Int {- ^ Width, in bits -} -> Integer {- ^ Value -} -> SExpr
bvBin w v = const ("#b" ++ bits)
  where
  bits = reverse [ if testBit v n then '1' else '0' | n <- [ 0 .. w - 1 ] ]

-- | A bit vector represented in hex.
--
--    * If the value does not fit in the bits, the bits will be increased to
--      the next multiple of 4 that will fit the value.
--    * If the width is not a multiple of 4, it will be rounded
--      up so that it is.
--    * The width should be strictly positive.
bvHex :: Int {- ^ Width, in bits -} -> Integer {- ^ Value -} -> SExpr
bvHex w v
  | v >= 0    = const ("#x" ++ padding ++ hex)
  | otherwise = bvHex w (2^w + v)
  where
  hex     = showHex v ""
  padding = replicate (P.div (w + 3) 4 - length hex) '0'


-- | Render a value as an expression.  Bit-vectors are rendered in hex,
-- if their width is a multiple of 4, and in binary otherwise.
value :: Value -> SExpr
value val =
  case val of
    Bool b    -> bool b
    Int n     -> int n
    Real r    -> real r
    Bits w v | P.mod w 4 == 0 -> bvHex w v
             | otherwise      -> bvBin w v
    Other o   -> o

-- Connectives -----------------------------------------------------------------

-- | Logical negation.
not :: SExpr -> SExpr
not p = fun "not" [p]

-- | Conjunction.
and :: SExpr -> SExpr -> SExpr
and p q = fun "and" [p,q]

andMany :: [SExpr] -> SExpr
andMany xs = if null xs then bool True else fun "and" xs

-- | Disjunction.
or :: SExpr -> SExpr -> SExpr
or p q = fun "or" [p,q]

orMany :: [SExpr] -> SExpr
orMany xs = if null xs then bool False else fun "or" xs

-- | Exclusive-or.
xor :: SExpr -> SExpr -> SExpr
xor p q = fun "xor" [p,q]

-- | Implication.
implies :: SExpr -> SExpr -> SExpr
implies p q = fun "=>" [p,q]


-- If-then-else ----------------------------------------------------------------

-- | If-then-else.  This is polymorphic and can be used to construct any term.
ite :: SExpr -> SExpr -> SExpr -> SExpr
ite x y z = fun "ite" [x,y,z]




-- Relations -------------------------------------------------------------------

-- | Equality.
eq :: SExpr -> SExpr -> SExpr
eq x y = fun "=" [x,y]

distinct :: [SExpr] -> SExpr
distinct xs = if null xs then bool True else fun "distinct" xs

-- | Greater-then
gt :: SExpr -> SExpr -> SExpr
gt x y = fun ">" [x,y]

-- | Less-then.
lt :: SExpr -> SExpr -> SExpr
lt x y = fun "<" [x,y]

-- | Greater-than-or-equal-to.
geq :: SExpr -> SExpr -> SExpr
geq x y = fun ">=" [x,y]

-- | Less-than-or-equal-to.
leq :: SExpr -> SExpr -> SExpr
leq x y = fun "<=" [x,y]

-- | Unsigned less-than on bit-vectors.
bvULt :: SExpr -> SExpr -> SExpr
bvULt x y = fun "bvult" [x,y]

-- | Unsigned less-than-or-equal on bit-vectors.
bvULeq :: SExpr -> SExpr -> SExpr
bvULeq x y = fun "bvule" [x,y]

-- | Signed less-than on bit-vectors.
bvSLt :: SExpr -> SExpr -> SExpr
bvSLt x y = fun "bvslt" [x,y]

-- | Signed less-than-or-equal on bit-vectors.
bvSLeq :: SExpr -> SExpr -> SExpr
bvSLeq x y = fun "bvsle" [x,y]




-- | Addition.
-- See also 'bvAdd'
add :: SExpr -> SExpr -> SExpr
add x y = fun "+" [x,y]

addMany :: [SExpr] -> SExpr
addMany xs = if null xs then int 0 else fun "+" xs

-- | Subtraction.
sub :: SExpr -> SExpr -> SExpr
sub x y = fun "-" [x,y]

-- | Arithmetic negation for integers and reals.
-- See also 'bvNeg'.
neg :: SExpr -> SExpr
neg x = fun "-" [x]

-- | Multiplication.
mul :: SExpr -> SExpr -> SExpr
mul x y = fun "*" [x,y]

-- | Absolute value.
abs :: SExpr -> SExpr
abs x = fun "abs" [x]

-- | Integer division.
div :: SExpr -> SExpr -> SExpr
div x y = fun "div" [x,y]

-- | Modulus.
mod :: SExpr -> SExpr -> SExpr
mod x y = fun "mod" [x,y]

-- | Is the number divisible by the given constant.
divisible :: SExpr -> Integer -> SExpr
divisible x n = List [ fam "divisible" [n], x ]

-- | Division of real numbers.
realDiv :: SExpr -> SExpr -> SExpr
realDiv x y = fun "/" [x,y]

-- | Bit vector concatenation.
concat :: SExpr -> SExpr -> SExpr
concat x y = fun "concat" [x,y]

-- | Extend to the signed equivalent bitvector by @i@ bits
signExtend :: Integer -> SExpr -> SExpr
signExtend i x = List [ fam "sign_extend" [i], x ]

-- | Extend with zeros to the unsigned equivalent bitvector
-- by @i@ bits
zeroExtend :: Integer -> SExpr -> SExpr
zeroExtend i x = List [ fam "zero_extend" [i], x ]

-- | Satisfies @toInt x <= x@ (i.e., this is like Haskell's 'floor')
toInt :: SExpr -> SExpr
toInt e = fun "to_int" [e]

-- | Promote an integer to a real
toReal :: SExpr -> SExpr
toReal e = fun "to_real" [e]

-- | Extract a sub-sequence of a bit vector.
extract :: SExpr -> Integer -> Integer -> SExpr
extract x y z = List [ fam "extract" [y,z], x ]

-- | Bitwise negation.
bvNot :: SExpr -> SExpr
bvNot x = fun "bvnot" [x]

-- | Bitwise conjuction.
bvAnd :: SExpr -> SExpr -> SExpr
bvAnd x y = fun "bvand" [x,y]

-- | Bitwise disjunction.
bvOr :: SExpr -> SExpr -> SExpr
bvOr x y = fun "bvor" [x,y]

-- | Bitwise exclusive or.
bvXOr :: SExpr -> SExpr -> SExpr
bvXOr x y = fun "bvxor" [x,y]

-- | Bit vector arithmetic negation.
bvNeg :: SExpr -> SExpr
bvNeg x = fun "bvneg" [x]

-- | Addition of bit vectors.
bvAdd :: SExpr -> SExpr -> SExpr
bvAdd x y = fun "bvadd" [x,y]

-- | Subtraction of bit vectors.
bvSub :: SExpr -> SExpr -> SExpr
bvSub x y = fun "bvsub" [x,y]



-- | Multiplication of bit vectors.
bvMul :: SExpr -> SExpr -> SExpr
bvMul x y = fun "bvmul" [x,y]

-- | Bit vector unsigned division.
bvUDiv :: SExpr -> SExpr -> SExpr
bvUDiv x y = fun "bvudiv" [x,y]

-- | Bit vector unsigned reminder.
bvURem :: SExpr -> SExpr -> SExpr
bvURem x y = fun "bvurem" [x,y]

-- | Bit vector signed division.
bvSDiv :: SExpr -> SExpr -> SExpr
bvSDiv x y = fun "bvsdiv" [x,y]

-- | Bit vector signed reminder.
bvSRem :: SExpr -> SExpr -> SExpr
bvSRem x y = fun "bvsrem" [x,y]




-- | Shift left.
bvShl :: SExpr {- ^ value -} -> SExpr {- ^ shift amount -} -> SExpr
bvShl x y = fun "bvshl" [x,y]

-- | Logical shift right.
bvLShr :: SExpr {- ^ value -} -> SExpr {- ^ shift amount -} -> SExpr
bvLShr x y = fun "bvlshr" [x,y]

-- | Arithemti shift right (copies most significant bit).
bvAShr :: SExpr {- ^ value -} -> SExpr {- ^ shift amount -} -> SExpr
bvAShr x y = fun "bvashr" [x,y]


-- | Get an elemeent of an array.
select :: SExpr {- ^ array -} -> SExpr {- ^ index -} -> SExpr
select x y = fun "select" [x,y]

-- | Update an array
store :: SExpr {- ^ array -}     ->
         SExpr {- ^ index -}     ->
         SExpr {- ^ new value -} ->
         SExpr
store x y z = fun "store" [x,y,z]


--------------------------------------------------------------------------------
-- Attributes

named :: String -> SExpr -> SExpr
named x e = fun "!" [e, Atom ":named", Atom x ]


--------------------------------------------------------------------------------

-- | Log messages with minimal formatting. Mostly for debugging.
data Logger = Logger
  { logMessage :: String -> IO ()
    -- ^ Log a message.

  , logLevel   :: IO Int

  , logSetLevel:: Int -> IO ()

  , logTab     :: IO ()
    -- ^ Increase indentation.

  , logUntab   :: IO ()
    -- ^ Decrease indentation.
  }

-- | Run an IO action with the logger set to a specific level, restoring it when
-- done.
withLogLevel :: Logger -> Int -> IO a -> IO a
withLogLevel Logger { .. } l m =
  do l0 <- logLevel
     X.bracket_ (logSetLevel l) (logSetLevel l0) m

logIndented :: Logger -> IO a -> IO a
logIndented Logger { .. } = X.bracket_ logTab logUntab

-- | Log a message at a specific log level.
logMessageAt :: Logger -> Int -> String -> IO ()
logMessageAt logger l msg = withLogLevel logger l (logMessage logger msg)

-- | A simple stdout logger.  Shows only messages logged at a level that is
-- greater than or equal to the passed level.
newLogger :: Int -> IO Logger
newLogger l =
  do tab <- newIORef 0
     lev <- newIORef 0
     let logLevel    = readIORef lev
         logSetLevel = writeIORef lev

         shouldLog m =
           do cl <- logLevel
              when (cl >= l) m

         logMessage x = shouldLog $
           do let ls = lines x
              t <- readIORef tab
              putStr $ unlines [ replicate t ' ' ++ l | l <- ls ]
              hFlush stdout

         logTab   = shouldLog (modifyIORef' tab (+ 2))
         logUntab = shouldLog (modifyIORef' tab (subtract 2))
     return Logger { .. }