sbv-10.3: Data/SBV/SMT/SMTLib2.hs
-----------------------------------------------------------------------------
-- |
-- Module : Data.SBV.SMT.SMTLib2
-- Copyright : (c) Levent Erkok
-- License : BSD3
-- Maintainer: erkokl@gmail.com
-- Stability : experimental
--
-- Conversion of symbolic programs to SMTLib format, Using v2 of the standard
-----------------------------------------------------------------------------
{-# LANGUAGE NamedFieldPuns #-}
{-# LANGUAGE PatternGuards #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE ViewPatterns #-}
{-# OPTIONS_GHC -Wall -Werror #-}
module Data.SBV.SMT.SMTLib2(cvt, cvtExp, cvtCV, cvtInc, declUserFuns, constructTables) where
import Data.Bits (bit)
import Data.List (intercalate, partition, nub, elemIndex)
import Data.Maybe (listToMaybe, catMaybes)
import qualified Data.Foldable as F (toList)
import qualified Data.Map.Strict as M
import qualified Data.IntMap.Strict as IM
import Data.Set (Set)
import qualified Data.Set as Set
import Data.SBV.Core.Data
import Data.SBV.Core.Kind (smtType, needsFlattening)
import Data.SBV.SMT.Utils
import Data.SBV.Control.Types
import Data.SBV.Core.Symbolic ( QueryContext(..), SetOp(..), CnstMap, getUserName', getSV, regExpToSMTString
, SMTDef(..), ResultInp(..), ProgInfo(..), SpecialRelOp(..)
)
import Data.SBV.Utils.PrettyNum (smtRoundingMode, cvToSMTLib)
import qualified Data.Generics.Uniplate.Data as G
import qualified Data.Graph as DG
tbd :: String -> a
tbd e = error $ "SBV.SMTLib2: Not-yet-supported: " ++ e
-- | Translate a problem into an SMTLib2 script
cvt :: SMTLibConverter ([String], [String])
cvt ctx curProgInfo kindInfo isSat comments allInputs (allConsts, consts) tbls arrs uis defs (SBVPgm asgnsSeq) cstrs out cfg = (pgm, exportedDefs)
where hasInteger = KUnbounded `Set.member` kindInfo
hasReal = KReal `Set.member` kindInfo
hasFP = not (null [() | KFP{} <- Set.toList kindInfo])
|| KFloat `Set.member` kindInfo
|| KDouble `Set.member` kindInfo
hasString = KString `Set.member` kindInfo
hasRegExp = (not . null) [() | (_ :: RegExOp) <- G.universeBi asgnsSeq]
hasChar = KChar `Set.member` kindInfo
hasRounding = not $ null [s | (s, _) <- usorts, s == "RoundingMode"]
hasBVs = not (null [() | KBounded{} <- Set.toList kindInfo])
usorts = [(s, dt) | KUserSort s dt <- Set.toList kindInfo]
trueUSorts = [s | (s, _) <- usorts, s /= "RoundingMode"]
tupleArities = findTupleArities kindInfo
hasNonBVArrays = (not . null) [() | (_, (_, (k1, k2), _)) <- arrs, not (isBounded k1 && isBounded k2)]
hasArrayInits = (not . null) $ [() | (_, (_, _, ArrayFree (Left (Just _)))) <- arrs]
++ [() | (_, (_, _, ArrayFree (Right _ ))) <- arrs]
hasOverflows = (not . null) [() | (_ :: OvOp) <- G.universeBi asgnsSeq]
hasQuantBools = (not . null) [() | QuantifiedBool _ <- G.universeBi asgnsSeq]
hasList = any isList kindInfo
hasSets = any isSet kindInfo
hasTuples = not . null $ tupleArities
hasEither = any isEither kindInfo
hasMaybe = any isMaybe kindInfo
hasRational = any isRational kindInfo
rm = roundingMode cfg
solverCaps = capabilities (solver cfg)
hasFoldMap = let isFoldMap SeqMap{} = True
isFoldMap SeqMapI{} = True
isFoldMap SeqFoldLeft{} = True
isFoldMap SeqFoldLeftI{} = True
isFoldMap _ = False
in (not . null) [ () | o :: SeqOp <- G.universeBi asgnsSeq, isFoldMap o]
(needsQuantifiers, needsSpecialRels) = case curProgInfo of
ProgInfo hasQ srs tcs -> (hasQ, not (null srs && null tcs))
-- Is there a reason why we can't handle this problem?
-- NB. There's probably a lot more checking we can do here, but this is a start:
doesntHandle = listToMaybe [nope w | (w, have, need) <- checks, need && not (have solverCaps)]
where checks = [ ("data types", supportsDataTypes, hasTuples || hasEither || hasMaybe)
, ("folds and maps", supportsFoldAndMap, hasFoldMap)
, ("set operations", supportsSets, hasSets)
, ("bit vectors", supportsBitVectors, hasBVs)
, ("special relations", supportsSpecialRels, needsSpecialRels)
, ("needs quantifiers", supportsQuantifiers, needsQuantifiers)
, ("unbounded integers", supportsUnboundedInts, hasInteger)
, ("algebraic reals", supportsReals, hasReal)
, ("floating-point numbers", supportsIEEE754, hasFP)
, ("uninterpreted sorts", supportsUninterpretedSorts, not (null trueUSorts))
]
nope w = [ "*** Given problem requires support for " ++ w
, "*** But the chosen solver (" ++ show (name (solver cfg)) ++ ") doesn't support this feature."
]
-- Some cases require all, some require none. Sigh..
setAll reason = ["(set-logic ALL) ; " ++ reason ++ ", using catch-all."]
-- Determining the logic is surprisingly tricky!
logic
-- user told us what to do: so just take it:
| Just l <- case [l | SetLogic l <- solverSetOptions cfg] of
[] -> Nothing
[l] -> Just l
ls -> error $ unlines [ ""
, "*** Only one setOption call to 'setLogic' is allowed, found: " ++ show (length ls)
, "*** " ++ unwords (map show ls)
]
= case l of
Logic_NONE -> ["; NB. Not setting the logic per user request of Logic_NONE"]
_ -> ["(set-logic " ++ show l ++ ") ; NB. User specified."]
-- There's a reason why we can't handle this problem:
| Just cantDo <- doesntHandle
= error $ unlines $ [ ""
, "*** SBV is unable to choose a proper solver configuration:"
, "***"
]
++ cantDo
++ [ "***"
, "*** Please report this as a feature request, either for SBV or the backend solver."
]
-- Otherwise, we try to determine the most suitable logic.
-- NB. This isn't really fool proof!
-- we never set QF_S (ALL seems to work better in all cases)
| needsSpecialRels = ["; has special relations, no logic set."]
-- Things that require ALL
| hasInteger = setAll "has unbounded values"
| hasRational = setAll "has rational values"
| hasReal = setAll "has algebraic reals"
| not (null trueUSorts) = setAll "has user-defined sorts"
| hasNonBVArrays = setAll "has non-bitvector arrays"
| hasTuples = setAll "has tuples"
| hasEither = setAll "has either type"
| hasMaybe = setAll "has maybe type"
| hasSets = setAll "has sets"
| hasList = setAll "has lists"
| hasChar = setAll "has chars"
| hasString = setAll "has strings"
| hasRegExp = setAll "has regular expressions"
| hasArrayInits = setAll "has array initializers"
| hasOverflows = setAll "has overflow checks"
| hasQuantBools = setAll "has quantified booleans"
| hasFP || hasRounding
= if needsQuantifiers
then ["(set-logic ALL)"]
else if hasBVs
then ["(set-logic QF_FPBV)"]
else ["(set-logic QF_FP)"]
-- If we're in a user query context, we'll pick ALL, otherwise
-- we'll stick to some bit-vector logic based on what we see in the problem.
-- This is controversial, but seems to work well in practice.
| True
= case ctx of
QueryExternal -> ["(set-logic ALL) ; external query, using all logics."]
QueryInternal -> if supportsBitVectors solverCaps
then ["(set-logic " ++ qs ++ as ++ ufs ++ "BV)"]
else ["(set-logic ALL)"] -- fall-thru
where qs | not needsQuantifiers = "QF_"
| True = ""
as | null arrs = ""
| True = "A"
ufs | null uis && null tbls = "" -- we represent tables as UFs
| True = "UF"
-- SBV always requires the production of models!
getModels = "(set-option :produce-models true)"
: concat [flattenConfig | any needsFlattening kindInfo, Just flattenConfig <- [supportsFlattenedModels solverCaps]]
-- process all other settings we're given. If an option cannot be repeated, we only take the last one.
userSettings = map setSMTOption $ filter (not . isLogic) $ foldr comb [] $ solverSetOptions cfg
where -- Logic is already processed, so drop it:
isLogic SetLogic{} = True
isLogic _ = False
-- SBV sets diagnostic-output channel on some solvers. If the user also gives it, let's just
-- take it by only taking the last one
isDiagOutput DiagnosticOutputChannel{} = True
isDiagOutput _ = False
comb o rest
| isDiagOutput o && any isDiagOutput rest = rest
| True = o : rest
settings = userSettings -- NB. Make sure this comes first!
++ getModels
++ logic
(inputs, trackerVars)
= case allInputs of
ResultTopInps ists -> ists
ResultLamInps ps -> error $ unlines [ ""
, "*** Data.SBV.smtLib2: Unexpected lambda inputs in conversion"
, "***"
, "*** Saw: " ++ show ps
]
pgm = map ("; " ++) comments
++ settings
++ [ "; --- uninterpreted sorts ---" ]
++ concatMap declSort usorts
++ [ "; --- tuples ---" ]
++ concatMap declTuple tupleArities
++ [ "; --- sums ---" ]
++ (if containsSum kindInfo then declSum else [])
++ (if containsMaybe kindInfo then declMaybe else [])
++ (if containsRationals kindInfo then declRationals else [])
++ [ "; --- literal constants ---" ]
++ concatMap (declConst cfg) consts
++ [ "; --- top level inputs ---"]
++ concat [declareFun s (SBVType [kindOf s]) (userName s) | var <- inputs, let s = getSV var]
++ [ "; --- optimization tracker variables ---" | not (null trackerVars) ]
++ concat [declareFun s (SBVType [kindOf s]) (Just ("tracks " <> nm)) | var <- trackerVars, let s = getSV var, let nm = getUserName' var]
++ [ "; --- constant tables ---" ]
++ concatMap (uncurry (:) . mkTable) constTables
++ [ "; --- non-constant tables ---" ]
++ map nonConstTable nonConstTables
++ [ "; --- arrays ---" ]
++ concat arrayConstants
++ [ "; --- uninterpreted constants ---" ]
++ concatMap (declUI curProgInfo) uis
++ [ "; --- SBV Function definitions" | not (null funcMap) ]
++ concat [ declSBVFunc op nm | (op, nm) <- M.toAscList funcMap ]
++ [ "; --- user defined functions ---"]
++ userDefs
++ [ "; --- assignments ---" ]
++ concatMap (declDef curProgInfo cfg tableMap funcMap) asgns
++ [ "; --- arrayDelayeds ---" ]
++ concat arrayDelayeds
++ [ "; --- arraySetups ---" ]
++ concat arraySetups
++ [ "; --- delayedEqualities ---" ]
++ map (\s -> "(assert " ++ s ++ ")") delayedEqualities
++ [ "; --- formula ---" ]
++ finalAssert
userDefs = declUserFuns defs
exportedDefs
| null userDefs
= ["; No calls to 'smtFunction' found."]
| True
= "; Automatically generated by SBV. Do not modify!" : userDefs
(tableMap, constTables, nonConstTables) = constructTables rm consts tbls
(arrayConstants, arrayDelayeds, arraySetups) = unzip3 $ map (declArray cfg allConsts) arrs
delayedEqualities = concatMap snd nonConstTables
finalAssert
| noConstraints = []
| True = map (\(attr, v) -> "(assert " ++ addAnnotations attr (mkLiteral v) ++ ")") hardAsserts
++ map (\(attr, v) -> "(assert-soft " ++ addAnnotations attr (mkLiteral v) ++ ")") softAsserts
where mkLiteral (Left v) = cvtSV v
mkLiteral (Right v) = "(not " ++ cvtSV v ++ ")"
(noConstraints, assertions) = finalAssertions
hardAsserts, softAsserts :: [([(String, String)], Either SV SV)]
hardAsserts = [(attr, v) | (False, attr, v) <- assertions]
softAsserts = [(attr, v) | (True, attr, v) <- assertions]
finalAssertions :: (Bool, [(Bool, [(String, String)], Either SV SV)]) -- If Left: positive, Right: negative
finalAssertions
| null finals = (True, [(False, [], Left trueSV)])
| True = (False, finals)
where finals = cstrs' ++ maybe [] (\r -> [(False, [], r)]) mbO
cstrs' = [(isSoft, attrs, c') | (isSoft, attrs, c) <- F.toList cstrs, Just c' <- [pos c]]
mbO | isSat = pos out
| True = neg out
neg s
| s == falseSV = Nothing
| s == trueSV = Just $ Left falseSV
| True = Just $ Right s
pos s
| s == trueSV = Nothing
| s == falseSV = Just $ Left falseSV
| True = Just $ Left s
-- SBV only functions.
funcMap = M.fromList reverses
where reverses = zip (nub [op | op@(SeqOp SBVReverse{}) <- G.universeBi asgnsSeq])
["sbv.reverse_" ++ show i | i <- [(0::Int)..]]
asgns = F.toList asgnsSeq
userNameMap = M.fromList $ map (\nSymVar -> (getSV nSymVar, getUserName' nSymVar)) inputs
userName s = case M.lookup s userNameMap of
Just u | show s /= u -> Just $ "tracks user variable " ++ show u
_ -> Nothing
-- Declare "known" SBV functions here
declSBVFunc :: Op -> String -> [String]
declSBVFunc op nm = case op of
SeqOp (SBVReverse KString) -> mkStringRev
SeqOp (SBVReverse (KList k)) -> mkSeqRev (KList k)
_ -> error $ "Data.SBV.declSBVFunc: Unexpected internal function: " ++ show (op, nm)
where mkStringRev = [ "(define-fun-rec " ++ nm ++ " ((str String)) String"
, " (ite (= str \"\")"
, " \"\""
, " (str.++ (" ++ nm ++ " (str.substr str 1 (- (str.len str) 1)))"
, " (str.substr str 0 1))))"
]
mkSeqRev k = [ "(define-fun-rec " ++ nm ++ " ((lst " ++ t ++ ")) " ++ t
, " (ite (= lst (as seq.empty " ++ t ++ "))"
, " (as seq.empty " ++ t ++ ")"
, " (seq.++ (" ++ nm ++ " (seq.extract lst 1 (- (seq.len lst) 1))) (seq.unit (seq.nth lst 0)))))"
]
where t = smtType k
-- | Declare new sorts
declSort :: (String, Maybe [String]) -> [String]
declSort (s, _)
| s == "RoundingMode" -- built-in-sort; so don't declare.
= []
declSort (s, Nothing) = [ "(declare-sort " ++ s ++ " 0) ; N.B. Uninterpreted sort."
, "(declare-fun " ++ s ++ "_witness () " ++ s ++ ")"
]
declSort (s, Just fs) = [ "(declare-datatypes ((" ++ s ++ " 0)) ((" ++ unwords (map (\c -> "(" ++ c ++ ")") fs) ++ ")))"
, "(define-fun " ++ s ++ "_constrIndex ((x " ++ s ++ ")) Int"
] ++ [" " ++ body fs (0::Int)] ++ [")"]
where body [] _ = ""
body [_] i = show i
body (c:cs) i = "(ite (= x " ++ c ++ ") " ++ show i ++ " " ++ body cs (i+1) ++ ")"
-- | Declare tuple datatypes
--
-- eg:
--
-- @
-- (declare-datatypes ((SBVTuple2 2)) ((par (T1 T2)
-- ((mkSBVTuple2 (proj_1_SBVTuple2 T1)
-- (proj_2_SBVTuple2 T2))))))
-- @
declTuple :: Int -> [String]
declTuple arity
| arity == 0 = ["(declare-datatypes ((SBVTuple0 0)) (((mkSBVTuple0))))"]
| arity == 1 = error "Data.SBV.declTuple: Unexpected one-tuple"
| True = (l1 ++ "(par (" ++ unwords [param i | i <- [1..arity]] ++ ")")
: [pre i ++ proj i ++ post i | i <- [1..arity]]
where l1 = "(declare-datatypes ((SBVTuple" ++ show arity ++ " " ++ show arity ++ ")) ("
l2 = replicate (length l1) ' ' ++ "((mkSBVTuple" ++ show arity ++ " "
tab = replicate (length l2) ' '
pre 1 = l2
pre _ = tab
proj i = "(proj_" ++ show i ++ "_SBVTuple" ++ show arity ++ " " ++ param i ++ ")"
post i = if i == arity then ")))))" else ""
param i = "T" ++ show i
-- | Find the set of tuple sizes to declare, eg (2-tuple, 5-tuple).
-- NB. We do *not* need to recursively go into list/tuple kinds here,
-- because register-kind function automatically registers all subcomponent
-- kinds, thus everything we need is available at the top-level.
findTupleArities :: Set Kind -> [Int]
findTupleArities ks = Set.toAscList
$ Set.map length
$ Set.fromList [ tupKs | KTuple tupKs <- Set.toList ks ]
-- | Is @Either@ being used?
containsSum :: Set Kind -> Bool
containsSum = not . Set.null . Set.filter isEither
-- | Is @Maybe@ being used?
containsMaybe :: Set Kind -> Bool
containsMaybe = not . Set.null . Set.filter isMaybe
-- | Is @Rational@ being used?
containsRationals :: Set Kind -> Bool
containsRationals = not . Set.null . Set.filter isRational
declSum :: [String]
declSum = [ "(declare-datatypes ((SBVEither 2)) ((par (T1 T2)"
, " ((left_SBVEither (get_left_SBVEither T1))"
, " (right_SBVEither (get_right_SBVEither T2))))))"
]
declMaybe :: [String]
declMaybe = [ "(declare-datatypes ((SBVMaybe 1)) ((par (T)"
, " ((nothing_SBVMaybe)"
, " (just_SBVMaybe (get_just_SBVMaybe T))))))"
]
-- Internally, we do *not* keep the rationals in reduced form! So, the boolean operators explicitly do the math
-- to make sure equivalent values are treated correctly.
declRationals :: [String]
declRationals = [ "(declare-datatype SBVRational ((SBV.Rational (sbv.rat.numerator Int) (sbv.rat.denominator Int))))"
, ""
, "(define-fun sbv.rat.eq ((x SBVRational) (y SBVRational)) Bool"
, " (= (* (sbv.rat.numerator x) (sbv.rat.denominator y))"
, " (* (sbv.rat.denominator x) (sbv.rat.numerator y)))"
, ")"
, ""
, "(define-fun sbv.rat.notEq ((x SBVRational) (y SBVRational)) Bool"
, " (not (sbv.rat.eq x y))"
, ")"
, ""
, "(define-fun sbv.rat.lt ((x SBVRational) (y SBVRational)) Bool"
, " (< (* (sbv.rat.numerator x) (sbv.rat.denominator y))"
, " (* (sbv.rat.denominator x) (sbv.rat.numerator y)))"
, ")"
, ""
, "(define-fun sbv.rat.leq ((x SBVRational) (y SBVRational)) Bool"
, " (<= (* (sbv.rat.numerator x) (sbv.rat.denominator y))"
, " (* (sbv.rat.denominator x) (sbv.rat.numerator y)))"
, ")"
, ""
, "(define-fun sbv.rat.plus ((x SBVRational) (y SBVRational)) SBVRational"
, " (SBV.Rational (+ (* (sbv.rat.numerator x) (sbv.rat.denominator y))"
, " (* (sbv.rat.denominator x) (sbv.rat.numerator y)))"
, " (* (sbv.rat.denominator x) (sbv.rat.denominator y)))"
, ")"
, ""
, "(define-fun sbv.rat.minus ((x SBVRational) (y SBVRational)) SBVRational"
, " (SBV.Rational (- (* (sbv.rat.numerator x) (sbv.rat.denominator y))"
, " (* (sbv.rat.denominator x) (sbv.rat.numerator y)))"
, " (* (sbv.rat.denominator x) (sbv.rat.denominator y)))"
, ")"
, ""
, "(define-fun sbv.rat.times ((x SBVRational) (y SBVRational)) SBVRational"
, " (SBV.Rational (* (sbv.rat.numerator x) (sbv.rat.numerator y))"
, " (* (sbv.rat.denominator x) (sbv.rat.denominator y)))"
, ")"
, ""
, "(define-fun sbv.rat.uneg ((x SBVRational)) SBVRational"
, " (SBV.Rational (* (- 1) (sbv.rat.numerator x)) (sbv.rat.denominator x))"
, ")"
, ""
, "(define-fun sbv.rat.abs ((x SBVRational)) SBVRational"
, " (SBV.Rational (abs (sbv.rat.numerator x)) (sbv.rat.denominator x))"
, ")"
]
-- | Convert in a query context.
-- NB. We do not store everything in @newKs@ below, but only what we need
-- to do as an extra in the incremental context. See `Data.SBV.Core.Symbolic.registerKind`
-- for a list of what we include, in case something doesn't show up
-- and you need it!
cvtInc :: SMTLibIncConverter [String]
cvtInc curProgInfo inps newKs (allConsts, consts) arrs tbls uis (SBVPgm asgnsSeq) cstrs cfg =
-- any new settings?
settings
-- sorts
++ concatMap declSort [(s, dt) | KUserSort s dt <- newKinds]
-- tuples. NB. Only declare the new sizes, old sizes persist.
++ concatMap declTuple (findTupleArities newKs)
-- sums
++ (if containsSum newKs then declSum else [])
++ (if containsMaybe newKs then declMaybe else [])
-- constants
++ concatMap (declConst cfg) consts
-- inputs
++ concatMap declInp inps
-- arrays
++ concat arrayConstants
-- uninterpreteds
++ concatMap (declUI curProgInfo) uis
-- table declarations
++ tableDecls
-- expressions
++ concatMap (declDef curProgInfo cfg tableMap funcMap) (F.toList asgnsSeq)
-- delayed equalities
++ concat arrayDelayeds
-- table setups
++ concat tableAssigns
-- array setups
++ concat arraySetups
-- extra constraints
++ map (\(isSoft, attr, v) -> "(assert" ++ (if isSoft then "-soft " else " ") ++ addAnnotations attr (cvtSV v) ++ ")") (F.toList cstrs)
where -- The following is not really kosher; if it happens that a "new" variant of a function is used only incrementally.
-- But we'll punt on this for now, as it should be rare and can be "worked-around" if necessary.
funcMap = M.empty
rm = roundingMode cfg
newKinds = Set.toList newKs
declInp (getSV -> s) = declareFun s (SBVType [kindOf s]) Nothing
(arrayConstants, arrayDelayeds, arraySetups) = unzip3 $ map (declArray cfg allConsts) arrs
(tableMap, allTables) = (tm, ct ++ nct)
where (tm, ct, nct) = constructTables rm consts tbls
(tableDecls, tableAssigns) = unzip $ map mkTable allTables
-- If we need flattening in models, do emit the required lines if preset
settings
| any needsFlattening newKinds
= concat (catMaybes [supportsFlattenedModels solverCaps])
| True
= []
where solverCaps = capabilities (solver cfg)
declDef :: ProgInfo -> SMTConfig -> TableMap -> FunctionMap -> (SV, SBVExpr) -> [String]
declDef curProgInfo cfg tableMap funcMap (s, expr) =
case expr of
SBVApp (Label m) [e] -> defineFun cfg (s, cvtSV e) (Just m)
e -> defineFun cfg (s, cvtExp curProgInfo caps rm tableMap funcMap e) Nothing
where caps = capabilities (solver cfg)
rm = roundingMode cfg
defineFun :: SMTConfig -> (SV, String) -> Maybe String -> [String]
defineFun cfg (s, def) mbComment
| hasDefFun = ["(define-fun " ++ varT ++ " " ++ def ++ ")" ++ cmnt]
| True = [ "(declare-fun " ++ varT ++ ")" ++ cmnt
, "(assert (= " ++ var ++ " " ++ def ++ "))"
]
where var = show s
varT = var ++ " " ++ svFunType [] s
cmnt = maybe "" (" ; " ++) mbComment
hasDefFun = supportsDefineFun $ capabilities (solver cfg)
-- Declare constants. NB. We don't declare true/false; but just inline those as necessary
declConst :: SMTConfig -> (SV, CV) -> [String]
declConst cfg (s, c)
| s == falseSV || s == trueSV
= []
| True
= defineFun cfg (s, cvtCV (roundingMode cfg) c) Nothing
-- Make a function equality of nm against the internal function fun
mkRelEq :: String -> (String, String) -> Kind -> String
mkRelEq nm (fun, order) ak = res
where lhs = "(" ++ nm ++ " x y)"
rhs = "((_ " ++ fun ++ " " ++ order ++ ") x y)"
tk = smtType ak
res = "(forall ((x " ++ tk ++ ") (y " ++ tk ++ ")) (= " ++ lhs ++ " " ++ rhs ++ "))"
declUI :: ProgInfo -> (String, (Bool, Maybe [String], SBVType)) -> [String]
declUI ProgInfo{progTransClosures} (i, (_, _, t)) = declareName i t Nothing ++ declClosure
where declClosure | Just external <- lookup i progTransClosures
= declareName external t Nothing
++ ["(assert " ++ mkRelEq external ("transitive-closure", i) (argKind t) ++ ")"]
| True
= []
argKind (SBVType [ka, _, KBool]) = ka
argKind _ = error $ "declUI: Unexpected type for name: " ++ show (i, t)
-- Note that even though we get all user defined-functions here (i.e., lambda and axiom), we can only have defined-functions
-- and axioms. We spit axioms as is; and topologically sort the definitions.
declUserFuns :: [(SMTDef, SBVType)] -> [String]
declUserFuns ds
| not (null lambdas)
= error $ unlines $ "Data.SBV.declUserFuns: Unexpected anonymous functions in declDef:" : map show lambdas
| True
= declFuncs others
where (lambdas, others) = partition (isLam . fst) ds
isLam SMTLam{} = True
isLam SMTDef{} = False
-- We need to topologically sort the user given definitions and axioms and put them in the proper order and construct.
-- Note that there are no anonymous functions at this level.
declFuncs :: [(SMTDef, SBVType)] -> [String]
declFuncs ds = map declGroup sorted
where mkNode d = (d, getKey d, getDeps d)
getKey (d, _) = case d of
SMTDef n _ _ _ _ -> n
SMTLam{} -> error $ "Data.SBV.declFuns: Unexpected definition kind: " ++ show d
getDeps (SMTDef _ _ d _ _, _) = d
getDeps (l@SMTLam{}, t) = error $ "Data.SBV.declFuns: Unexpected definition: " ++ show (l, t)
mkDecl Nothing rt = "() " ++ rt
mkDecl (Just p) rt = p ++ " " ++ rt
sorted = DG.stronglyConnComp (map mkNode ds)
declGroup (DG.AcyclicSCC b) = declUserDef False b
declGroup (DG.CyclicSCC bs) = case bs of
[] -> error "Data.SBV.declFuns: Impossible happened: an empty cyclic group was returned!"
[x] -> declUserDef True x
xs -> declUserDefMulti xs
declUserDef _ d@(SMTLam{}, _) = error $ "Data.SBV.declFuns: Unexpected anonymous lambda in user-defined functions: " ++ show d
declUserDef isRec (SMTDef nm fk deps param body, ty) = ("; " ++ nm ++ " :: " ++ show ty ++ recursive ++ frees ++ "\n") ++ s
where (recursive, definer) | isRec = (" [Recursive]", "define-fun-rec")
| True = ("", "define-fun")
otherDeps = filter (/= nm) deps
frees | null otherDeps = ""
| True = " [Refers to: " ++ intercalate ", " otherDeps ++ "]"
decl = mkDecl param (smtType fk)
s = "(" ++ definer ++ " " ++ nm ++ " " ++ decl ++ "\n" ++ body 2 ++ ")"
-- declare a bunch of mutually-recursive functions
declUserDefMulti bs = render $ map collect bs
where collect d@(SMTLam{}, _) = error $ "Data.SBV.declFuns: Unexpected lambda in user-defined mutual-recursion group: " ++ show d
collect (SMTDef nm fk deps param body, ty) = (deps, nm, ty, '(' : nm ++ " " ++ decl ++ ")", body 3)
where decl = mkDecl param (smtType fk)
render defs = intercalate "\n" $
[ "; " ++ intercalate ", " [n ++ " :: " ++ show ty | (_, n, ty, _, _) <- defs]
, "(define-funs-rec"
]
++ [ open i ++ param d ++ close1 i | (i, d) <- zip [1..] defs]
++ [ open i ++ dump d ++ close2 i | (i, d) <- zip [1..] defs]
where open 1 = " ("
open _ = " "
param (_deps, _nm, _ty, p, _body) = p
dump (deps, nm, ty, _, body) = "; Definition of: " ++ nm ++ " :: " ++ show ty ++ ". [Refers to: " ++ intercalate ", " deps ++ "]"
++ "\n" ++ body
ld = length defs
close1 n = if n == ld then ")" else ""
close2 n = if n == ld then "))" else ""
mkTable :: (((Int, Kind, Kind), [SV]), [String]) -> (String, [String])
mkTable (((i, ak, rk), _elts), is) = (decl, zipWith wrap [(0::Int)..] is ++ setup)
where t = "table" ++ show i
decl = "(declare-fun " ++ t ++ " (" ++ smtType ak ++ ") " ++ smtType rk ++ ")"
-- Arrange for initializers
mkInit idx = "table" ++ show i ++ "_initializer_" ++ show (idx :: Int)
initializer = "table" ++ show i ++ "_initializer"
wrap index s = "(define-fun " ++ mkInit index ++ " () Bool " ++ s ++ ")"
lis = length is
setup
| lis == 0 = [ "(define-fun " ++ initializer ++ " () Bool true) ; no initialization needed"
]
| lis == 1 = [ "(define-fun " ++ initializer ++ " () Bool " ++ mkInit 0 ++ ")"
, "(assert " ++ initializer ++ ")"
]
| True = [ "(define-fun " ++ initializer ++ " () Bool (and " ++ unwords (map mkInit [0..lis - 1]) ++ "))"
, "(assert " ++ initializer ++ ")"
]
nonConstTable :: (((Int, Kind, Kind), [SV]), [String]) -> String
nonConstTable (((i, ak, rk), _elts), _) = decl
where t = "table" ++ show i
decl = "(declare-fun " ++ t ++ " (" ++ smtType ak ++ ") " ++ smtType rk ++ ")"
constructTables :: RoundingMode -> [(SV, CV)] -> [((Int, Kind, Kind), [SV])]
-> ( IM.IntMap String -- table enumeration
, [(((Int, Kind, Kind), [SV]), [String])] -- constant tables
, [(((Int, Kind, Kind), [SV]), [String])] -- non-constant tables
)
constructTables rm consts tbls = (tableMap, constTables, nonConstTables)
where allTables = [(t, genTableData rm (map fst consts) t) | t <- tbls]
constTables = [(t, d) | (t, Left d) <- allTables]
nonConstTables = [(t, d) | (t, Right d) <- allTables]
tableMap = IM.fromList $ map grab allTables
grab (((t, _, _), _), _) = (t, "table" ++ show t)
-- Left if all constants, Right if otherwise
genTableData :: RoundingMode -> [SV] -> ((Int, Kind, Kind), [SV]) -> Either [String] [String]
genTableData rm consts ((i, aknd, _), elts)
| null post = Left (map (mkEntry . snd) pre)
| True = Right (map (mkEntry . snd) (pre ++ post))
where (pre, post) = partition fst (zipWith mkElt elts [(0::Int)..])
t = "table" ++ show i
mkElt x k = (isReady, (idx, cvtSV x))
where idx = cvtCV rm (mkConstCV aknd k)
isReady = x `Set.member` constsSet
mkEntry (idx, v) = "(= (" ++ t ++ " " ++ idx ++ ") " ++ v ++ ")"
constsSet = Set.fromList consts
-- Declare arrays
declArray :: SMTConfig -> CnstMap -> (Int, ArrayInfo) -> ([String], [String], [String])
declArray cfg consts (i, (_, (aKnd, bKnd), ctx)) = ( adecl : zipWith wrap [(0::Int)..] (map snd pre)
, zipWith wrap [lpre..] (map snd post)
, setup
)
where constMapping = M.fromList [(s, c) | (c, s) <- M.assocs consts]
constNames = M.keys constMapping
(pre, post) = partition fst ctxInfo
nm = "array_" ++ show i
atyp = "(Array " ++ smtType aKnd ++ " " ++ smtType bKnd ++ ")"
adecl = case ctx of
ArrayFree (Right lam) -> "(define-fun " ++ nm ++ " () " ++ atyp ++ " " ++ lam ++ ")"
ArrayFree (Left (Just v)) -> "(define-fun " ++ nm ++ " () " ++ atyp ++ " ((as const " ++ atyp ++ ") " ++ constInit v ++ "))"
ArrayFree (Left Nothing)
| bKnd == KChar -> -- Can't support yet, because we need to make sure all the elements are length-1 strings. So, punt for now.
tbd "Free array declarations containing SChars"
_ -> "(declare-fun " ++ nm ++ " () " ++ atyp ++ ")"
-- CVC4 chokes if the initializer is not a constant. (Z3 is ok with it.) So, print it as
-- a constant if we have it in the constants; otherwise, we merely print it and hope for the best.
constInit v = case v `M.lookup` constMapping of
Nothing -> cvtSV v -- Z3 will work, CVC4 will choke. Others don't even support this.
Just c -> cvtCV (roundingMode cfg) c -- Z3 and CVC4 will work. Other's don't support this.
ctxInfo = case ctx of
ArrayFree _ -> []
ArrayMutate j a b -> [(all (`elem` constNames) [a, b], "(= " ++ nm ++ " (store array_" ++ show j ++ " " ++ cvtSV a ++ " " ++ cvtSV b ++ "))")]
ArrayMerge t j k -> [(t `elem` constNames, "(= " ++ nm ++ " (ite " ++ cvtSV t ++ " array_" ++ show j ++ " array_" ++ show k ++ "))")]
-- Arrange for initializers
mkInit idx = "array_" ++ show i ++ "_initializer_" ++ show (idx :: Int)
initializer = "array_" ++ show i ++ "_initializer"
wrap index s = "(define-fun " ++ mkInit index ++ " () Bool " ++ s ++ ")"
lpre = length pre
lAll = lpre + length post
setup
| lAll == 0 = [ "(define-fun " ++ initializer ++ " () Bool true) ; no initialization needed" ]
| lAll == 1 = [ "(define-fun " ++ initializer ++ " () Bool " ++ mkInit 0 ++ ")"
, "(assert " ++ initializer ++ ")"
]
| True = [ "(define-fun " ++ initializer ++ " () Bool (and " ++ unwords (map mkInit [0..lAll - 1]) ++ "))"
, "(assert " ++ initializer ++ ")"
]
svType :: SV -> String
svType s = smtType (kindOf s)
svFunType :: [SV] -> SV -> String
svFunType ss s = "(" ++ unwords (map svType ss) ++ ") " ++ svType s
cvtType :: SBVType -> String
cvtType (SBVType []) = error "SBV.SMT.SMTLib2.cvtType: internal: received an empty type!"
cvtType (SBVType xs) = "(" ++ unwords (map smtType body) ++ ") " ++ smtType ret
where (body, ret) = (init xs, last xs)
type TableMap = IM.IntMap String
type FunctionMap = M.Map Op String
-- Present an SV, simply show
cvtSV :: SV -> String
cvtSV = show
cvtCV :: RoundingMode -> CV -> String
cvtCV = cvToSMTLib
getTable :: TableMap -> Int -> String
getTable m i
| Just tn <- i `IM.lookup` m = tn
| True = "table" ++ show i -- constant tables are always named this way
cvtExp :: ProgInfo -> SolverCapabilities -> RoundingMode -> TableMap -> FunctionMap -> SBVExpr -> String
cvtExp curProgInfo caps rm tableMap functionMap expr@(SBVApp _ arguments) = sh expr
where hasPB = supportsPseudoBooleans caps
hasInt2bv = supportsInt2bv caps
hasDistinct = supportsDistinct caps
specialRels = progSpecialRels curProgInfo
bvOp = all isBounded arguments
intOp = any isUnbounded arguments
ratOp = any isRational arguments
realOp = any isReal arguments
fpOp = any (\a -> isDouble a || isFloat a || isFP a) arguments
boolOp = all isBoolean arguments
charOp = any isChar arguments
stringOp = any isString arguments
listOp = any isList arguments
bad | intOp = error $ "SBV.SMTLib2: Unsupported operation on unbounded integers: " ++ show expr
| True = error $ "SBV.SMTLib2: Unsupported operation on real values: " ++ show expr
ensureBVOrBool = bvOp || boolOp || bad
ensureBV = bvOp || bad
addRM s = s ++ " " ++ smtRoundingMode rm
hd _ (a:_) = a
hd w [] = error $ "Impossible: " ++ w ++ ": Received empty list of args!"
-- lift a binary op
lift2 o _ [x, y] = "(" ++ o ++ " " ++ x ++ " " ++ y ++ ")"
lift2 o _ sbvs = error $ "SBV.SMTLib2.sh.lift2: Unexpected arguments: " ++ show (o, sbvs)
-- lift an arbitrary arity operator
liftN o _ xs = "(" ++ o ++ " " ++ unwords xs ++ ")"
-- lift a binary operation with rounding-mode added; used for floating-point arithmetic
lift2WM o fo | fpOp = lift2 (addRM fo)
| True = lift2 o
lift1FP o fo | fpOp = lift1 fo
| True = lift1 o
liftAbs sgned args | fpOp = lift1 "fp.abs" sgned args
| intOp = lift1 "abs" sgned args
| bvOp, sgned = mkAbs fArg "bvslt" "bvneg"
| bvOp = fArg
| True = mkAbs fArg "<" "-"
where fArg = hd "liftAbs" args
mkAbs x cmp neg = "(ite " ++ ltz ++ " " ++ nx ++ " " ++ x ++ ")"
where ltz = "(" ++ cmp ++ " " ++ x ++ " " ++ z ++ ")"
nx = "(" ++ neg ++ " " ++ x ++ ")"
z = cvtCV rm (mkConstCV (kindOf (hd "liftAbs.arguments" arguments)) (0::Integer))
lift2B bOp vOp
| boolOp = lift2 bOp
| True = lift2 vOp
lift1B bOp vOp
| boolOp = lift1 bOp
| True = lift1 vOp
eqBV = lift2 "="
neqBV = liftN "distinct"
equal sgn sbvs
| fpOp = lift2 "fp.eq" sgn sbvs
| True = lift2 "=" sgn sbvs
-- Do not use distinct on floats; because +0/-0, and NaNs mess
-- up the meaning. Just go with reqular equals.
notEqual sgn sbvs
| fpOp || not hasDistinct = liftP sbvs
| True = liftN "distinct" sgn sbvs
where liftP xs@[_, _] = "(not " ++ equal sgn xs ++ ")"
liftP args = "(and " ++ unwords (walk args) ++ ")"
walk [] = []
walk (e:es) = map (\e' -> liftP [e, e']) es ++ walk es
lift2S oU oS sgn = lift2 (if sgn then oS else oU) sgn
liftNS oU oS sgn = liftN (if sgn then oS else oU) sgn
lift2Cmp o fo | fpOp = lift2 fo
| True = lift2 o
unintComp o [a, b]
| KUserSort s (Just _) <- kindOf (hd "unintComp" arguments)
= let idx v = "(" ++ s ++ "_constrIndex " ++ v ++ ")" in "(" ++ o ++ " " ++ idx a ++ " " ++ idx b ++ ")"
unintComp o sbvs = error $ "SBV.SMT.SMTLib2.sh.unintComp: Unexpected arguments: " ++ show (o, sbvs, map kindOf arguments)
stringOrChar KString = True
stringOrChar KChar = True
stringOrChar _ = False
stringCmp swap o [a, b]
| stringOrChar (kindOf (hd "stringCmp" arguments))
= let (a1, a2) | swap = (b, a)
| True = (a, b)
in "(" ++ o ++ " " ++ a1 ++ " " ++ a2 ++ ")"
stringCmp _ o sbvs = error $ "SBV.SMT.SMTLib2.sh.stringCmp: Unexpected arguments: " ++ show (o, sbvs)
-- NB. Likewise for sequences
seqCmp swap o [a, b]
| KList{} <- kindOf (hd "seqCmp" arguments)
= let (a1, a2) | swap = (b, a)
| True = (a, b)
in "(" ++ o ++ " " ++ a1 ++ " " ++ a2 ++ ")"
seqCmp _ o sbvs = error $ "SBV.SMT.SMTLib2.sh.seqCmp: Unexpected arguments: " ++ show (o, sbvs)
lift1 o _ [x] = "(" ++ o ++ " " ++ x ++ ")"
lift1 o _ sbvs = error $ "SBV.SMT.SMTLib2.sh.lift1: Unexpected arguments: " ++ show (o, sbvs)
-- We fully qualify the constructor with their types to work around type checking issues
-- Note that this is rather bizarre, as we're tagging the constructor with its *result* type,
-- not its full function type as one would expect. But this is per the spec: Pg. 27 of SMTLib 2.6 spec
-- says:
--
-- To simplify sort checking, a function symbol in a term can be annotated with one of its result sorts sigma.
--
-- I wish it was the full type here not just the result, but we go with the spec. Also see: <http://github.com/Z3Prover/z3/issues/2135>
-- and in particular <http://github.com/Z3Prover/z3/issues/2135#issuecomment-477636435>
dtConstructor fld args res = "((as " ++ fld ++ " " ++ smtType res ++ ") " ++ unwords (map cvtSV args) ++ ")"
-- Similarly, we fully qualify the accessors with their types to work around type checking issues
-- Unfortunately, z3 and CVC4 are behaving differently, so we tie this ascription to a solver capability.
dtAccessor fld params res
| supportsDirectAccessors caps = dResult
| True = aResult
where dResult = "(_ is " ++ fld ++ ")"
ps = " (" ++ unwords (map smtType params) ++ ") "
aResult = "(_ is (" ++ fld ++ ps ++ smtType res ++ "))"
sh (SBVApp Ite [a, b, c]) = "(ite " ++ cvtSV a ++ " " ++ cvtSV b ++ " " ++ cvtSV c ++ ")"
sh (SBVApp (LkUp (t, aKnd, _, l) i e) [])
| needsCheck = "(ite " ++ cond ++ cvtSV e ++ " " ++ lkUp ++ ")"
| True = lkUp
where needsCheck = case aKnd of
KBool -> (2::Integer) > fromIntegral l
KBounded _ n -> (2::Integer)^n > fromIntegral l
KUnbounded -> True
KReal -> error "SBV.SMT.SMTLib2.cvtExp: unexpected real valued index"
KFloat -> error "SBV.SMT.SMTLib2.cvtExp: unexpected float valued index"
KDouble -> error "SBV.SMT.SMTLib2.cvtExp: unexpected double valued index"
KFP{} -> error "SBV.SMT.SMTLib2.cvtExp: unexpected arbitrary float valued index"
KRational{} -> error "SBV.SMT.SMTLib2.cvtExp: unexpected rational valued index"
KChar -> error "SBV.SMT.SMTLib2.cvtExp: unexpected char valued index"
KString -> error "SBV.SMT.SMTLib2.cvtExp: unexpected string valued index"
KList k -> error $ "SBV.SMT.SMTLib2.cvtExp: unexpected list valued: " ++ show k
KSet k -> error $ "SBV.SMT.SMTLib2.cvtExp: unexpected set valued: " ++ show k
KTuple k -> error $ "SBV.SMT.SMTLib2.cvtExp: unexpected tuple valued: " ++ show k
KMaybe k -> error $ "SBV.SMT.SMTLib2.cvtExp: unexpected maybe valued: " ++ show k
KEither k1 k2 -> error $ "SBV.SMT.SMTLib2.cvtExp: unexpected sum valued: " ++ show (k1, k2)
KUserSort s _ -> error $ "SBV.SMT.SMTLib2.cvtExp: unexpected uninterpreted valued index: " ++ s
lkUp = "(" ++ getTable tableMap t ++ " " ++ cvtSV i ++ ")"
cond
| hasSign i = "(or " ++ le0 ++ " " ++ gtl ++ ") "
| True = gtl ++ " "
(less, leq) = case aKnd of
KBool -> error "SBV.SMT.SMTLib2.cvtExp: unexpected boolean valued index"
KBounded{} -> if hasSign i then ("bvslt", "bvsle") else ("bvult", "bvule")
KUnbounded -> ("<", "<=")
KReal -> ("<", "<=")
KFloat -> ("fp.lt", "fp.leq")
KDouble -> ("fp.lt", "fp.leq")
KRational -> ("sbv.rat.lt", "sbv.rat.leq")
KFP{} -> ("fp.lt", "fp.leq")
KChar -> error "SBV.SMT.SMTLib2.cvtExp: unexpected string valued index"
KString -> error "SBV.SMT.SMTLib2.cvtExp: unexpected string valued index"
KList k -> error $ "SBV.SMT.SMTLib2.cvtExp: unexpected sequence valued index: " ++ show k
KSet k -> error $ "SBV.SMT.SMTLib2.cvtExp: unexpected set valued index: " ++ show k
KTuple k -> error $ "SBV.SMT.SMTLib2.cvtExp: unexpected tuple valued index: " ++ show k
KMaybe k -> error $ "SBV.SMT.SMTLib2.cvtExp: unexpected maybe valued index: " ++ show k
KEither k1 k2 -> error $ "SBV.SMT.SMTLib2.cvtExp: unexpected sum valued index: " ++ show (k1, k2)
KUserSort s _ -> error $ "SBV.SMT.SMTLib2.cvtExp: unexpected uninterpreted valued index: " ++ s
mkCnst = cvtCV rm . mkConstCV (kindOf i)
le0 = "(" ++ less ++ " " ++ cvtSV i ++ " " ++ mkCnst 0 ++ ")"
gtl = "(" ++ leq ++ " " ++ mkCnst l ++ " " ++ cvtSV i ++ ")"
sh (SBVApp (KindCast f t) [a]) = handleKindCast hasInt2bv f t (cvtSV a)
sh (SBVApp (ArrEq i j) []) = "(= array_" ++ show i ++ " array_" ++ show j ++")"
sh (SBVApp (ArrRead i) [a]) = "(select array_" ++ show i ++ " " ++ cvtSV a ++ ")"
sh (SBVApp (Uninterpreted nm) []) = nm
sh (SBVApp (Uninterpreted nm) args) = "(" ++ nm ++ " " ++ unwords (map cvtSV args) ++ ")"
sh (SBVApp (QuantifiedBool i) []) = i
sh (SBVApp (QuantifiedBool i) args) = error $ "SBV.SMT.SMTLib2.cvtExp: unexpected arguments to quantified boolean: " ++ show (i, args)
sh a@(SBVApp (SpecialRelOp k o) args)
| not (null args)
= error $ "SBV.SMT.SMTLib2.cvtExp: unexpected arguments to special op: " ++ show a
| True
= let order = case o `elemIndex` specialRels of
Just i -> i
Nothing -> error $ unlines [ "SBV.SMT.SMTLib2.cvtExp: Cannot find " ++ show o ++ " in the special-relations list."
, "Known relations: " ++ intercalate ", " (map show specialRels)
]
asrt nm fun = mkRelEq nm (fun, show order) k
in case o of
IsPartialOrder nm -> asrt nm "partial-order"
IsLinearOrder nm -> asrt nm "linear-order"
IsTreeOrder nm -> asrt nm "tree-order"
IsPiecewiseLinearOrder nm -> asrt nm "piecewise-linear-order"
sh (SBVApp (Extract i j) [a]) | ensureBV = "((_ extract " ++ show i ++ " " ++ show j ++ ") " ++ cvtSV a ++ ")"
sh (SBVApp (Rol i) [a])
| bvOp = rot "rotate_left" i a
| True = bad
sh (SBVApp (Ror i) [a])
| bvOp = rot "rotate_right" i a
| True = bad
sh (SBVApp Shl [a, i])
| bvOp = shft "bvshl" "bvshl" a i
| True = bad
sh (SBVApp Shr [a, i])
| bvOp = shft "bvlshr" "bvashr" a i
| True = bad
sh (SBVApp (ZeroExtend i) [a])
| bvOp = "((_ zero_extend " ++ show i ++ ") " ++ cvtSV a ++ ")"
| True = bad
sh (SBVApp (SignExtend i) [a])
| bvOp = "((_ sign_extend " ++ show i ++ ") " ++ cvtSV a ++ ")"
| True = bad
sh (SBVApp op args)
| Just f <- lookup op smtBVOpTable, ensureBVOrBool
= f (any hasSign args) (map cvtSV args)
where -- The first 4 operators below do make sense for Integer's in Haskell, but there's
-- no obvious counterpart for them in the SMTLib translation.
-- TODO: provide support for these.
smtBVOpTable = [ (And, lift2B "and" "bvand")
, (Or, lift2B "or" "bvor")
, (XOr, lift2B "xor" "bvxor")
, (Not, lift1B "not" "bvnot")
, (Join, lift2 "concat")
]
sh (SBVApp (Label _) [a]) = cvtSV a -- This won't be reached; but just in case!
sh (SBVApp (IEEEFP (FP_Cast kFrom kTo m)) args) = handleFPCast kFrom kTo (cvtSV m) (unwords (map cvtSV args))
sh (SBVApp (IEEEFP w ) args) = "(" ++ show w ++ " " ++ unwords (map cvtSV args) ++ ")"
sh (SBVApp (NonLinear w) args) = "(" ++ show w ++ " " ++ unwords (map cvtSV args) ++ ")"
sh (SBVApp (PseudoBoolean pb) args)
| hasPB = handlePB pb args'
| True = reducePB pb args'
where args' = map cvtSV args
sh (SBVApp (OverflowOp op) args) = "(" ++ show op ++ " " ++ unwords (map cvtSV args) ++ ")"
-- Note the unfortunate reversal in StrInRe..
sh (SBVApp (StrOp (StrInRe r)) args) = "(str.in.re " ++ unwords (map cvtSV args) ++ " " ++ regExpToSMTString r ++ ")"
-- StrUnit is no-op, since a character in SMTLib is the same as a string
sh (SBVApp (StrOp StrUnit) [a]) = cvtSV a
sh (SBVApp (StrOp op) args) = "(" ++ show op ++ " " ++ unwords (map cvtSV args) ++ ")"
sh (SBVApp (RegExOp o@RegExEq{}) []) = show o
sh (SBVApp (RegExOp o@RegExNEq{}) []) = show o
-- Reverse is special, since we need to generate call to the internally generated function
sh inp@(SBVApp op@(SeqOp SBVReverse{}) args) = "(" ++ ops ++ " " ++ unwords (map cvtSV args) ++ ")"
where ops = case op `M.lookup` functionMap of
Just s -> s
Nothing -> error $ "SBV.SMT.SMTLib2.cvtExp.sh: impossible happened; can't translate: " ++ show inp
sh (SBVApp (SeqOp op) args) = "(" ++ show op ++ " " ++ unwords (map cvtSV args) ++ ")"
sh (SBVApp (SetOp SetEqual) args) = "(= " ++ unwords (map cvtSV args) ++ ")"
sh (SBVApp (SetOp SetMember) [e, s]) = "(select " ++ cvtSV s ++ " " ++ cvtSV e ++ ")"
sh (SBVApp (SetOp SetInsert) [e, s]) = "(store " ++ cvtSV s ++ " " ++ cvtSV e ++ " true)"
sh (SBVApp (SetOp SetDelete) [e, s]) = "(store " ++ cvtSV s ++ " " ++ cvtSV e ++ " false)"
sh (SBVApp (SetOp SetIntersect) args) = "(intersection " ++ unwords (map cvtSV args) ++ ")"
sh (SBVApp (SetOp SetUnion) args) = "(union " ++ unwords (map cvtSV args) ++ ")"
sh (SBVApp (SetOp SetSubset) args) = "(subset " ++ unwords (map cvtSV args) ++ ")"
sh (SBVApp (SetOp SetDifference) args) = "(setminus " ++ unwords (map cvtSV args) ++ ")"
sh (SBVApp (SetOp SetComplement) args) = "(complement " ++ unwords (map cvtSV args) ++ ")"
sh (SBVApp (TupleConstructor 0) []) = "mkSBVTuple0"
sh (SBVApp (TupleConstructor n) args) = "((as mkSBVTuple" ++ show n ++ " " ++ smtType (KTuple (map kindOf args)) ++ ") " ++ unwords (map cvtSV args) ++ ")"
sh (SBVApp (TupleAccess i n) [tup]) = "(proj_" ++ show i ++ "_SBVTuple" ++ show n ++ " " ++ cvtSV tup ++ ")"
sh (SBVApp (EitherConstructor k1 k2 False) [arg]) = dtConstructor "left_SBVEither" [arg] (KEither k1 k2)
sh (SBVApp (EitherConstructor k1 k2 True ) [arg]) = dtConstructor "right_SBVEither" [arg] (KEither k1 k2)
sh (SBVApp (EitherIs k1 k2 False) [arg]) = '(' : dtAccessor "left_SBVEither" [k1] (KEither k1 k2) ++ " " ++ cvtSV arg ++ ")"
sh (SBVApp (EitherIs k1 k2 True ) [arg]) = '(' : dtAccessor "right_SBVEither" [k2] (KEither k1 k2) ++ " " ++ cvtSV arg ++ ")"
sh (SBVApp (EitherAccess False) [arg]) = "(get_left_SBVEither " ++ cvtSV arg ++ ")"
sh (SBVApp (EitherAccess True ) [arg]) = "(get_right_SBVEither " ++ cvtSV arg ++ ")"
sh (SBVApp RationalConstructor [t, b]) = "(SBV.Rational " ++ cvtSV t ++ " " ++ cvtSV b ++ ")"
sh (SBVApp (MaybeConstructor k False) []) = dtConstructor "nothing_SBVMaybe" [] (KMaybe k)
sh (SBVApp (MaybeConstructor k True) [arg]) = dtConstructor "just_SBVMaybe" [arg] (KMaybe k)
sh (SBVApp (MaybeIs k False) [arg]) = '(' : dtAccessor "nothing_SBVMaybe" [] (KMaybe k) ++ " " ++ cvtSV arg ++ ")"
sh (SBVApp (MaybeIs k True ) [arg]) = '(' : dtAccessor "just_SBVMaybe" [k] (KMaybe k) ++ " " ++ cvtSV arg ++ ")"
sh (SBVApp MaybeAccess [arg]) = "(get_just_SBVMaybe " ++ cvtSV arg ++ ")"
sh (SBVApp Implies [a, b]) = "(=> " ++ cvtSV a ++ " " ++ cvtSV b ++ ")"
sh inp@(SBVApp op args)
| intOp, Just f <- lookup op smtOpIntTable
= f True (map cvtSV args)
| boolOp, Just f <- lookup op boolComps
= f (map cvtSV args)
| bvOp, Just f <- lookup op smtOpBVTable
= f (any hasSign args) (map cvtSV args)
| realOp, Just f <- lookup op smtOpRealTable
= f (any hasSign args) (map cvtSV args)
| ratOp, Just f <- lookup op ratOpTable
= f (map cvtSV args)
| fpOp, Just f <- lookup op smtOpFloatDoubleTable
= f (any hasSign args) (map cvtSV args)
| charOp || stringOp, Just f <- lookup op smtStringTable
= f (map cvtSV args)
| listOp, Just f <- lookup op smtListTable
= f (map cvtSV args)
| Just f <- lookup op uninterpretedTable
= f (map cvtSV args)
| True
= if not (null args) && isUserSort (hd "isUserSort" args)
then error $ unlines [ ""
, "*** Cannot translate operator : " ++ show op
, "*** When applied to arguments of kind: " ++ intercalate ", " (nub (map (show . kindOf) args))
, "*** Found as part of the expression : " ++ show inp
, "***"
, "*** Note that uninterpreted kinds only support equality."
, "*** If you believe this is in error, please report!"
]
else error $ "SBV.SMT.SMTLib2.cvtExp.sh: impossible happened; can't translate: " ++ show inp
where smtOpBVTable = [ (Plus, lift2 "bvadd")
, (Minus, lift2 "bvsub")
, (Times, lift2 "bvmul")
, (UNeg, lift1B "not" "bvneg")
, (Abs, liftAbs)
, (Quot, lift2S "bvudiv" "bvsdiv")
, (Rem, lift2S "bvurem" "bvsrem")
, (Equal, eqBV)
, (NotEqual, neqBV)
, (LessThan, lift2S "bvult" "bvslt")
, (GreaterThan, lift2S "bvugt" "bvsgt")
, (LessEq, lift2S "bvule" "bvsle")
, (GreaterEq, lift2S "bvuge" "bvsge")
]
-- Boolean comparisons.. SMTLib's bool type doesn't do comparisons, but Haskell does.. Sigh
boolComps = [ (LessThan, blt)
, (GreaterThan, blt . swp)
, (LessEq, blq)
, (GreaterEq, blq . swp)
]
where blt [x, y] = "(and (not " ++ x ++ ") " ++ y ++ ")"
blt xs = error $ "SBV.SMT.SMTLib2.boolComps.blt: Impossible happened, incorrect arity (expected 2): " ++ show xs
blq [x, y] = "(or (not " ++ x ++ ") " ++ y ++ ")"
blq xs = error $ "SBV.SMT.SMTLib2.boolComps.blq: Impossible happened, incorrect arity (expected 2): " ++ show xs
swp [x, y] = [y, x]
swp xs = error $ "SBV.SMT.SMTLib2.boolComps.swp: Impossible happened, incorrect arity (expected 2): " ++ show xs
smtOpRealTable = smtIntRealShared
++ [ (Quot, lift2WM "/" "fp.div")
]
smtOpIntTable = smtIntRealShared
++ [ (Quot, lift2 "div")
, (Rem, lift2 "mod")
]
smtOpFloatDoubleTable = smtIntRealShared
++ [(Quot, lift2WM "/" "fp.div")]
smtIntRealShared = [ (Plus, lift2WM "+" "fp.add")
, (Minus, lift2WM "-" "fp.sub")
, (Times, lift2WM "*" "fp.mul")
, (UNeg, lift1FP "-" "fp.neg")
, (Abs, liftAbs)
, (Equal, equal)
, (NotEqual, notEqual)
, (LessThan, lift2Cmp "<" "fp.lt")
, (GreaterThan, lift2Cmp ">" "fp.gt")
, (LessEq, lift2Cmp "<=" "fp.leq")
, (GreaterEq, lift2Cmp ">=" "fp.geq")
]
ratOpTable = [ (Plus, lift2Rat "sbv.rat.plus")
, (Minus, lift2Rat "sbv.rat.minus")
, (Times, lift2Rat "sbv.rat.times")
, (UNeg, liftRat "sbv.rat.uneg")
, (Abs, liftRat "sbv.rat.abs")
, (Equal, lift2Rat "sbv.rat.eq")
, (NotEqual, lift2Rat "sbv.rat.notEq")
, (LessThan, lift2Rat "sbv.rat.lt")
, (GreaterThan, lift2Rat "sbv.rat.lt" . swap)
, (LessEq, lift2Rat "sbv.rat.leq")
, (GreaterEq, lift2Rat "sbv.rat.leq" . swap)
]
where lift2Rat o [x, y] = "(" ++ o ++ " " ++ x ++ " " ++ y ++ ")"
lift2Rat o sbvs = error $ "SBV.SMTLib2.sh.lift2Rat: Unexpected arguments: " ++ show (o, sbvs)
liftRat o [x] = "(" ++ o ++ " " ++ x ++ ")"
liftRat o sbvs = error $ "SBV.SMTLib2.sh.lift2Rat: Unexpected arguments: " ++ show (o, sbvs)
swap [x, y] = [y, x]
swap sbvs = error $ "SBV.SMTLib2.sh.swap: Unexpected arguments: " ++ show sbvs
-- equality and comparisons are the only thing that works on uninterpreted sorts and pretty much everything else
uninterpretedTable = [ (Equal, lift2S "=" "=" True)
, (NotEqual, liftNS "distinct" "distinct" True)
, (LessThan, unintComp "<")
, (GreaterThan, unintComp ">")
, (LessEq, unintComp "<=")
, (GreaterEq, unintComp ">=")
]
-- For strings, equality and comparisons are the only operators
smtStringTable = [ (Equal, lift2S "=" "=" True)
, (NotEqual, liftNS "distinct" "distinct" True)
, (LessThan, stringCmp False "str.<")
, (GreaterThan, stringCmp True "str.<")
, (LessEq, stringCmp False "str.<=")
, (GreaterEq, stringCmp True "str.<=")
]
-- For lists, equality is really the only operator
-- Likewise here, things might change for comparisons
smtListTable = [ (Equal, lift2S "=" "=" True)
, (NotEqual, liftNS "distinct" "distinct" True)
, (LessThan, seqCmp False "seq.<")
, (GreaterThan, seqCmp True "seq.<")
, (LessEq, seqCmp False "seq.<=")
, (GreaterEq, seqCmp True "seq.<=")
]
declareFun :: SV -> SBVType -> Maybe String -> [String]
declareFun = declareName . show
-- If we have a char, we have to make sure it's and SMTLib string of length exactly one
-- If we have a rational, we have to make sure the denominator is > 0
-- Otherwise, we just declare the name
declareName :: String -> SBVType -> Maybe String -> [String]
declareName s t@(SBVType inputKS) mbCmnt = decl : restrict
where decl = "(declare-fun " ++ s ++ " " ++ cvtType t ++ ")" ++ maybe "" (" ; " ++) mbCmnt
(args, result) = case inputKS of
[] -> error $ "SBV.declareName: Unexpected empty type for: " ++ show s
_ -> (init inputKS, last inputKS)
-- Does the kind KChar and KRational *not* occur in the kind anywhere?
charRatFree k = null $ [() | KChar <- G.universe k] ++ [() | KRational <- G.universe k]
noCharOrRat = charRatFree result
needsQuant = not $ null args
resultVar | needsQuant = "result"
| True = s
argList = ["a" ++ show i | (i, _) <- zip [1::Int ..] args]
argTList = ["(" ++ a ++ " " ++ smtType k ++ ")" | (a, k) <- zip argList args]
resultExp = "(" ++ s ++ " " ++ unwords argList ++ ")"
restrict | noCharOrRat = []
| needsQuant = [ "(assert (forall (" ++ unwords argTList ++ ")"
, " (let ((" ++ resultVar ++ " " ++ resultExp ++ "))"
]
++ (case constraints of
[] -> [ " true"]
[x] -> [ " " ++ x]
(x:xs) -> ( " (and " ++ x)
: [ " " ++ c | c <- xs]
++ [ " )"])
++ [ " )))"]
| True = case constraints of
[] -> []
[x] -> ["(assert " ++ x ++ ")"]
(x:xs) -> ( "(assert (and " ++ x)
: [ " " ++ c | c <- xs]
++ [ " ))"]
constraints = walk 0 resultVar cstr result
where cstr KChar nm = ["(= 1 (str.len " ++ nm ++ "))"]
cstr KRational nm = ["(< 0 (sbv.rat.denominator " ++ nm ++ "))"]
cstr _ _ = []
mkAnd [] = "true"
mkAnd [c] = c
mkAnd cs = "(and " ++ unwords cs ++ ")"
walk :: Int -> String -> (Kind -> String -> [String]) -> Kind -> [String]
walk _d nm f k@KBool {} = f k nm
walk _d nm f k@KBounded {} = f k nm
walk _d nm f k@KUnbounded{} = f k nm
walk _d nm f k@KReal {} = f k nm
walk _d nm f k@KUserSort {} = f k nm
walk _d nm f k@KFloat {} = f k nm
walk _d nm f k@KDouble {} = f k nm
walk _d nm f k@KRational {} = f k nm
walk _d nm f k@KFP {} = f k nm
walk _d nm f k@KChar {} = f k nm
walk _d nm f k@KString {} = f k nm
walk d nm f (KList k)
| charRatFree k = []
| True = let fnm = "seq" ++ show d
cstrs = walk (d+1) ("(seq.nth " ++ nm ++ " " ++ fnm ++ ")") f k
in ["(forall ((" ++ fnm ++ " " ++ smtType KUnbounded ++ ")) " ++ "(=> (and (>= " ++ fnm ++ " 0) (< " ++ fnm ++ " (seq.len " ++ nm ++ "))) " ++ mkAnd cstrs ++ "))"]
walk d nm f (KSet k)
| charRatFree k = []
| True = let fnm = "set" ++ show d
cstrs = walk (d+1) nm (\sk snm -> ["(=> (select " ++ snm ++ " " ++ fnm ++ ") " ++ c ++ ")" | c <- f sk fnm]) k
in ["(forall ((" ++ fnm ++ " " ++ smtType k ++ ")) " ++ mkAnd cstrs ++ ")"]
walk d nm f (KTuple ks) = let tt = "SBVTuple" ++ show (length ks)
project i = "(proj_" ++ show i ++ "_" ++ tt ++ " " ++ nm ++ ")"
nmks = [(project i, k) | (i, k) <- zip [1::Int ..] ks]
in concatMap (\(n, k) -> walk (d+1) n f k) nmks
walk d nm f km@(KMaybe k) = let n = "(get_just_SBVMaybe " ++ nm ++ ")"
in ["(=> " ++ "((_ is (just_SBVMaybe (" ++ smtType k ++ ") " ++ smtType km ++ ")) " ++ nm ++ ") " ++ c ++ ")" | c <- walk (d+1) n f k]
walk d nm f ke@(KEither k1 k2) = let n1 = "(get_left_SBVEither " ++ nm ++ ")"
n2 = "(get_right_SBVEither " ++ nm ++ ")"
c1 = ["(=> " ++ "((_ is (left_SBVEither (" ++ smtType k1 ++ ") " ++ smtType ke ++ ")) " ++ nm ++ ") " ++ c ++ ")" | c <- walk (d+1) n1 f k1]
c2 = ["(=> " ++ "((_ is (right_SBVEither (" ++ smtType k2 ++ ") " ++ smtType ke ++ ")) " ++ nm ++ ") " ++ c ++ ")" | c <- walk (d+1) n2 f k2]
in c1 ++ c2
-----------------------------------------------------------------------------------------------
-- Casts supported by SMTLib. (From: <http://smtlib.cs.uiowa.edu/theories-FloatingPoint.shtml>)
-- ; from another floating point sort
-- ((_ to_fp eb sb) RoundingMode (_ FloatingPoint mb nb) (_ FloatingPoint eb sb))
--
-- ; from real
-- ((_ to_fp eb sb) RoundingMode Real (_ FloatingPoint eb sb))
--
-- ; from signed machine integer, represented as a 2's complement bit vector
-- ((_ to_fp eb sb) RoundingMode (_ BitVec m) (_ FloatingPoint eb sb))
--
-- ; from unsigned machine integer, represented as bit vector
-- ((_ to_fp_unsigned eb sb) RoundingMode (_ BitVec m) (_ FloatingPoint eb sb))
--
-- ; to unsigned machine integer, represented as a bit vector
-- ((_ fp.to_ubv m) RoundingMode (_ FloatingPoint eb sb) (_ BitVec m))
--
-- ; to signed machine integer, represented as a 2's complement bit vector
-- ((_ fp.to_sbv m) RoundingMode (_ FloatingPoint eb sb) (_ BitVec m))
--
-- ; to real
-- (fp.to_real (_ FloatingPoint eb sb) Real)
-----------------------------------------------------------------------------------------------
handleFPCast :: Kind -> Kind -> String -> String -> String
handleFPCast kFromIn kToIn rm input
| kFrom == kTo
= input
| True
= "(" ++ cast kFrom kTo input ++ ")"
where addRM a s = s ++ " " ++ rm ++ " " ++ a
kFrom = simplify kFromIn
kTo = simplify kToIn
simplify KFloat = KFP 8 24
simplify KDouble = KFP 11 53
simplify k = k
size (eb, sb) = show eb ++ " " ++ show sb
-- To go and back from Ints, we detour through reals
cast KUnbounded (KFP eb sb) a = "(_ to_fp " ++ size (eb, sb) ++ ") " ++ rm ++ " (to_real " ++ a ++ ")"
cast KFP{} KUnbounded a = "to_int (fp.to_real " ++ a ++ ")"
-- To floats
cast (KBounded False _) (KFP eb sb) a = addRM a $ "(_ to_fp_unsigned " ++ size (eb, sb) ++ ")"
cast (KBounded True _) (KFP eb sb) a = addRM a $ "(_ to_fp " ++ size (eb, sb) ++ ")"
cast KReal (KFP eb sb) a = addRM a $ "(_ to_fp " ++ size (eb, sb) ++ ")"
cast KFP{} (KFP eb sb) a = addRM a $ "(_ to_fp " ++ size (eb, sb) ++ ")"
-- From float/double
cast KFP{} (KBounded False m) a = addRM a $ "(_ fp.to_ubv " ++ show m ++ ")"
cast KFP{} (KBounded True m) a = addRM a $ "(_ fp.to_sbv " ++ show m ++ ")"
-- To real
cast KFP{} KReal a = "fp.to_real" ++ " " ++ a
-- Nothing else should come up:
cast f d _ = error $ "SBV.SMTLib2: Unexpected FPCast from: " ++ show f ++ " to " ++ show d
rot :: String -> Int -> SV -> String
rot o c x = "((_ " ++ o ++ " " ++ show c ++ ") " ++ cvtSV x ++ ")"
shft :: String -> String -> SV -> SV -> String
shft oW oS x c = "(" ++ o ++ " " ++ cvtSV x ++ " " ++ cvtSV c ++ ")"
where o = if hasSign x then oS else oW
-- Various casts
handleKindCast :: Bool -> Kind -> Kind -> String -> String
handleKindCast hasInt2bv kFrom kTo a
| kFrom == kTo
= a
| True
= case kFrom of
KBounded s m -> case kTo of
KBounded _ n -> fromBV (if s then signExtend else zeroExtend) m n
KUnbounded -> b2i s m
_ -> tryFPCast
KUnbounded -> case kTo of
KReal -> "(to_real " ++ a ++ ")"
KBounded _ n -> i2b n
_ -> tryFPCast
KReal -> case kTo of
KUnbounded -> "(to_int " ++ a ++ ")"
_ -> tryFPCast
_ -> tryFPCast
where -- See if we can push this down to a float-cast, using sRNE. This happens if one of the kinds is a float/double.
-- Otherwise complain
tryFPCast
| any (\k -> isFloat k || isDouble k) [kFrom, kTo]
= handleFPCast kFrom kTo (smtRoundingMode RoundNearestTiesToEven) a
| True
= error $ "SBV.SMTLib2: Unexpected cast from: " ++ show kFrom ++ " to " ++ show kTo
fromBV upConv m n
| n > m = upConv (n - m)
| m == n = a
| True = extract (n - 1)
b2i False _ = "(bv2nat " ++ a ++ ")"
b2i True 1 = "(ite (= " ++ a ++ " #b0) 0 (- 1))"
b2i True m = "(ite (= " ++ msb ++ " #b0" ++ ") " ++ ifPos ++ " " ++ ifNeg ++ ")"
where offset :: Integer
offset = 2^(m-1)
rest = extract (m - 2)
msb = let top = show (m-1) in "((_ extract " ++ top ++ " " ++ top ++ ") " ++ a ++ ")"
ifPos = "(bv2nat " ++ rest ++")"
ifNeg = "(- " ++ ifPos ++ " " ++ show offset ++ ")"
signExtend i = "((_ sign_extend " ++ show i ++ ") " ++ a ++ ")"
zeroExtend i = "((_ zero_extend " ++ show i ++ ") " ++ a ++ ")"
extract i = "((_ extract " ++ show i ++ " 0) " ++ a ++ ")"
-- Some solvers support int2bv, but not all. So, we use a capability to determine.
--
-- NB. The "manual" implementation works regardless n < 0 or not, because the first thing we
-- do is to compute "reduced" to bring it down to the correct range. It also works
-- regardless were mapping to signed or unsigned bit-vector; because the representation
-- is the same.
i2b n
| hasInt2bv
= "((_ int2bv " ++ show n ++ ") " ++ a ++ ")"
| True
= "(let (" ++ reduced ++ ") (let (" ++ defs ++ ") " ++ body ++ "))"
where b i = show (bit i :: Integer)
reduced = "(__a (mod " ++ a ++ " " ++ b n ++ "))"
mkBit 0 = "(__a0 (ite (= (mod __a 2) 0) #b0 #b1))"
mkBit i = "(__a" ++ show i ++ " (ite (= (mod (div __a " ++ b i ++ ") 2) 0) #b0 #b1))"
defs = unwords (map mkBit [0 .. n - 1])
body = foldr1 (\c r -> "(concat " ++ c ++ " " ++ r ++ ")") ["__a" ++ show i | i <- [n-1, n-2 .. 0]]
-- Translation of pseudo-booleans, in case the solver supports them
handlePB :: PBOp -> [String] -> String
handlePB (PB_AtMost k) args = "((_ at-most " ++ show k ++ ") " ++ unwords args ++ ")"
handlePB (PB_AtLeast k) args = "((_ at-least " ++ show k ++ ") " ++ unwords args ++ ")"
handlePB (PB_Exactly k) args = "((_ pbeq " ++ unwords (map show (k : replicate (length args) 1)) ++ ") " ++ unwords args ++ ")"
handlePB (PB_Eq cs k) args = "((_ pbeq " ++ unwords (map show (k : cs)) ++ ") " ++ unwords args ++ ")"
handlePB (PB_Le cs k) args = "((_ pble " ++ unwords (map show (k : cs)) ++ ") " ++ unwords args ++ ")"
handlePB (PB_Ge cs k) args = "((_ pbge " ++ unwords (map show (k : cs)) ++ ") " ++ unwords args ++ ")"
-- Translation of pseudo-booleans, in case the solver does *not* support them
reducePB :: PBOp -> [String] -> String
reducePB op args = case op of
PB_AtMost k -> "(<= " ++ addIf (repeat 1) ++ " " ++ show k ++ ")"
PB_AtLeast k -> "(>= " ++ addIf (repeat 1) ++ " " ++ show k ++ ")"
PB_Exactly k -> "(= " ++ addIf (repeat 1) ++ " " ++ show k ++ ")"
PB_Le cs k -> "(<= " ++ addIf cs ++ " " ++ show k ++ ")"
PB_Ge cs k -> "(>= " ++ addIf cs ++ " " ++ show k ++ ")"
PB_Eq cs k -> "(= " ++ addIf cs ++ " " ++ show k ++ ")"
where addIf :: [Int] -> String
addIf cs = "(+ " ++ unwords ["(ite " ++ a ++ " " ++ show c ++ " 0)" | (a, c) <- zip args cs] ++ ")"