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sourcecode:
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module Curry2SMT where
import Data.IORef
import Data.List ( intercalate, isPrefixOf, nub, union )
import Data.Maybe ( catMaybes, fromJust, fromMaybe )
import Numeric ( readHex )
-- Imports from dependencies:
import FlatCurry.Annotated.Goodies ( argTypes, resultType )
import FlatCurry.Types ( showQName )
-- Imports from package modules:
import ESMT
import FlatCurry.Typed.Read ( getAllFunctions )
import FlatCurry.Typed.Goodies
import FlatCurry.Typed.Names
import FlatCurry.Typed.Types
import VerifierState
--- Translates a list of operations specified by their qualified name
--- (together with all operations on which these operation depend on)
--- into an SMT string that axiomatizes their semantics.
funcs2SMT :: IORef VState -> [QName] -> IO Command
funcs2SMT vstref qns = do
funs <- getAllFunctions vstref [] (nub qns)
return $ DefineSigsRec (map fun2SMT funs)
-- Translate a function declaration into a (possibly polymorphic)
-- SMT function declaration.
fun2SMT :: TAFuncDecl -> ([Ident],FunSig,Term)
fun2SMT (AFunc qn _ _ texp rule) =
let fsig = FunSig (transOpName qn)
(map polytype2psort (argTypes texp))
(polytype2psort (resultType texp))
srule = rule2SMT rule
tpars = union (typeParamsOfFunSig fsig) (typeParamsOfTerm srule)
in (tpars, fsig, srule)
where
rule2SMT (AExternal _ s) =
tEqu (tComb (transOpName qn) []) (tComb ("External:" ++ s) [])
rule2SMT (ARule _ vs exp) =
Forall (map (\ (v,t) -> SV v (polytype2psort t)) vs)
(if ndExpr exp then exp2SMT (Just lhs) exp
else tEqu lhs (exp2SMT Nothing exp))
where
lhs = tComb (transOpName qn) (map (TSVar . fst) vs)
-- Translates a typed FlatCurry expression into an SMT expression.
-- If the first argument is an SMT expression, an equation between
-- this expression and the translated result expression is generated.
-- This is useful to axiomatize non-deterministic operations.
exp2SMT :: Maybe Term -> TAExpr -> Term
exp2SMT lhs exp = case exp of
AVar _ i -> makeRHS (TSVar i)
ALit _ l -> makeRHS (lit2SMT l)
AComb _ ct (qn,ftype) args ->
-- TODO: reason about condition `not (null args)`
makeRHS (TComb (cons2SMT (ct /= ConsCall || not (null args)) True qn ftype)
(map (exp2SMT Nothing) args))
ACase _ _ e brs -> let be = exp2SMT Nothing e
in branches2SMT be brs
ALet _ bs e -> Let (map (\ ((v,_),be) -> (v, exp2SMT Nothing be)) bs)
(exp2SMT lhs e)
ATyped _ e _ -> exp2SMT lhs e
AFree _ fvs e -> Forall (map (\ (v,t) -> SV v (polytype2psort t)) fvs)
(exp2SMT lhs e)
AOr _ e1 e2 -> tDisj [exp2SMT lhs e1, exp2SMT lhs e2]
where
makeRHS rhs = maybe rhs (\l -> tEqu l rhs) lhs
branches2SMT _ [] = error "branches2SMT: empty branch list"
branches2SMT be [ABranch p e] = branch2SMT be p e
branches2SMT be (ABranch p e : brs@(_:_)) =
tComb "ite" [patternTest p be, --tEqu be (pat2smt p),
branch2SMT be p e,
branches2SMT be brs]
branch2SMT _ (ALPattern _ _) e = exp2SMT lhs e
branch2SMT be (APattern _ (qf,_) ps) e = case ps of
[] -> exp2SMT lhs e
_ -> Let (map (\ (s,v) -> (v, tComb s [be]))
(zip (selectors qf) (map fst ps)))
(exp2SMT lhs e)
patternTest :: TAPattern -> Term -> Term
patternTest (ALPattern _ l) be = tEqu be (lit2SMT l)
patternTest (APattern ty (qf,_) _) be = constructorTest True qf be ty
--- Translates a constructor name and a term into an SMT formula
--- implementing a test on the term for this constructor.
--- If the first argument is true, parametric sorts are used
--- (i.e., we translate a polymorphic function), otherwise
--- type variables are translated into the sort `TVar`.
constructorTest :: Bool -> QName -> Term -> TypeExpr -> Term
constructorTest withpoly qn be vartype
| qn == pre "[]"
= tEqu be (sortedConst "nil"
((if withpoly then polytype2psort else polytype2sort) vartype))
| qn `elem` map pre ["[]","True","False","LT","EQ","GT","Nothing"]
= tEqu be (tComb (transOpName qn) [])
| qn `elem` map pre ["Just","Left","Right"]
= tComb ("is-" ++ snd qn) [be]
| otherwise
= tComb ("is-" ++ transOpName qn) [be]
--- Computes the SMT selector names for a given constructor.
selectors :: QName -> [String]
selectors qf | qf == ("Prelude",":") = ["head","tail"]
| qf == ("Prelude","Left") = ["left"]
| qf == ("Prelude","Right") = ["right"]
| qf == ("Prelude","Just") = ["just"]
| otherwise = map (genSelName qf) [1..]
--- Translates a FlatCurry type expression into a corresponding SMT sort.
--- Polymorphic type variables are translated into the sort `TVar`.
--- The types `TVar` and `Func` are defined in the SMT prelude
--- which is always loaded.
polytype2sort :: TypeExpr -> Sort
polytype2sort = type2sort [] False False
--- Translates a FlatCurry type expression into a parametric SMT sort.
--- Thus, a polymorphic type variable `i` is translated into the sort `TVari`.
--- These type variables are later processed by the SMT translator.
polytype2psort :: TypeExpr -> Sort
polytype2psort = type2sort [] True False
--- Translates a FlatCurry type expression into a corresponding SMT sort.
--- If the first argument is null, then type variables are
--- translated into the sort `TVar`. If the second argument is true,
--- the type variable index is appended (`TVari`) in order to generate
--- a polymorphic sort (which will later be translated by the
--- SMT translator).
--- If the first argument is not null, we are in the translation
--- of the types of selector operations and the first argument
--- contains the currently defined data types. In this case, type variables
--- are translated into `Ti`, but further nesting of the defined data types
--- are not allowed (since this is not supported by SMT).
--- The types `TVar` and `Func` are defined in the SMT prelude
--- which is always loaded.
type2sort :: [QName] -> Bool -> Bool -> TypeExpr -> Sort
type2sort tdcl poly _ (TVar i) =
SComb (if null tdcl then "TVar" ++ (if poly then show i else "")
else 'T' : show i) []
type2sort tdcl poly _ (FuncType dom ran) =
SComb "Func" (map (type2sort tdcl poly True) [dom,ran])
type2sort tdcl poly nested (TCons qc@(mn,tc) targs)
| null tdcl
= SComb (tcons2SMT qc) argtypes
| otherwise -- we are in the selector definition of a datatype
= if qc `elem` tdcl
then if nested
then error $ "Type '" ++ showQName qc ++
"': nested recursive types not supported by SMT!"
else SComb (tcons2SMT qc) [] -- TODO: check whether arguments
-- are directly recursive, otherwise emit error
else SComb (tcons2SMT (mn,tc)) argtypes
where
argtypes = map (type2sort tdcl poly True) targs
type2sort _ _ _ (ForallType _ _) =
error "Curry2SMT.type2SMT: cannot translate ForallType"
--- Translates a FlatCurry type constructor name into SMT.
tcons2SMT :: QName -> String
tcons2SMT (mn,tc)
| "_Dict#" `isPrefixOf` tc
= "Dict" -- since we do not yet consider class dictionaries...
| mn == "Prelude" && take 3 tc == "(,,"
= "Tuple" ++ show (length tc - 1)
| mn == "Prelude"
= maybe (encodeSpecialChars tc) id (lookup tc transPrimTCons)
| otherwise
= mn ++ "_" ++ encodeSpecialChars tc
----------------------------------------------------------------------------
--- Translates a type declaration into an SMT datatype declaration.
tdecl2SMT :: TypeDecl -> Command
tdecl2SMT (TypeSyn tc _ _ _) =
error $ "Cannot translate type synonym '" ++ showQName tc ++ "' into SMT!"
tdecl2SMT (TypeNew tc _ _ _) =
error $ "Cannot translate newtype '" ++ showQName tc ++ "' into SMT!"
tdecl2SMT (Type tc _ tvars consdecls) =
DeclareDatatypes
[(tcons2SMT tc, length tvars,
DT (map (\ (v,_) -> 'T' : show v) tvars) (map tconsdecl consdecls))]
where
tconsdecl (Cons qn _ _ texps) =
let cname = transOpName qn
in DCons cname (map (texp2sel qn) (zip [1..] texps))
texp2sel cname (i,texp) = (genSelName cname i,
type2sort [tc] False False texp)
--- Generates the name of the i-th selector for a given constructor.
genSelName :: QName -> Int -> String
genSelName qc@(mn,fn) i
| mn == "Prelude" && take 3 fn == "(,,"
= transOpName qc ++ "_" ++ show i
| otherwise
= "sel" ++ show i ++ '-' : transOpName qc
--- Translates a prelude type into an SMT datatype declaration,
--- if necessary.
preludeType2SMT :: String -> [Command]
preludeType2SMT tn
| take 3 tn == "(,,"
= let arity = length tn -1
in [DeclareDatatypes
[(tcons2SMT (pre tn), arity,
DT (map (\v -> 'T':show v) [1 .. arity])
[DCons (transOpName (pre tn)) (map texp2sel [1 .. arity])])]]
| otherwise
= []
where
texp2sel i = (genSelName (pre tn) i, SComb ('T' : show i) [])
---------------------------------------------------------------------------
--- Translates a qualifed name with given result type into an SMT identifier.
--- If the first argument is true and the result type is not a base type,
--- the type is attached via `(as ...)` to resolve overloading problems in SMT.
--- If the second argument is true, parametric sorts are used
--- (i.e., we translate a polymorphic function), otherwise
--- type variables are translated into the sort `TVar`.
cons2SMT :: Bool -> Bool -> QName -> TypeExpr -> QIdent
cons2SMT withas withpoly qf rtype =
if withas && not (isBaseType rtype)
then As (transOpName qf)
((if withpoly then polytype2psort else polytype2sort) rtype)
else Id (transOpName qf)
--- Translates a pattern into an SMT expression.
pat2SMT :: TAPattern -> Term
pat2SMT (ALPattern _ l) = lit2SMT l
pat2SMT (APattern ty (qf,_) ps)
| qf == pre "[]" && null ps
= sortedConst "nil" (polytype2sort ty)
| otherwise
= tComb (transOpName qf) (map (TSVar . fst) ps)
--- Translates a literal into an SMT expression.
--- We represent character as ints.
lit2SMT :: Literal -> Term
lit2SMT (Intc i) = TConst (TInt i)
lit2SMT (Floatc f) = TConst (TFloat f)
lit2SMT (Charc c) = TConst (TInt (ord c))
--- Translates a qualified FlatCurry name into an SMT string.
transOpName :: QName -> String
transOpName (mn,fn)
| mn=="Prelude" = fromMaybe tname (lookup fn (transPrimCons ++ preludePrimOps))
| otherwise = tname
where
tname = mn ++ "_" ++ encodeSpecialChars fn
--- Encode special characters in strings
encodeSpecialChars :: String -> String
encodeSpecialChars = concatMap encChar
where
encChar c | c `elem` "#$%[]()!,"
= let oc = ord c
in ['%', int2hex (oc `div` 16), int2hex(oc `mod` 16)]
| otherwise = [c]
int2hex i = if i<10 then chr (ord '0' + i)
else chr (ord 'A' + i - 10)
--- Translates urlencoded string into equivalent ASCII string.
decodeSpecialChars :: String -> String
decodeSpecialChars [] = []
decodeSpecialChars (c:cs)
| c == '%' = let n = case readHex (take 2 cs) of
[(h,"")] -> h
_ -> 0
in chr n : decodeSpecialChars (drop 2 cs)
| otherwise = c : decodeSpecialChars cs
--- Translates a (translated) SMT string back into qualified FlatCurry name.
--- Returns Nothing if it was not a qualified name.
untransOpName :: String -> Maybe QName
untransOpName s
| "is-" `isPrefixOf` s
= Nothing -- selectors are not a qualified name
| otherwise
= let (mn,ufn) = break (=='_') s
in if null ufn
then Nothing
else Just (mn, decodeSpecialChars (tail ufn))
----------------------------------------------------------------------------
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