{- |

Module : $Header$

Description : termination proofs for equation systems, using AProVE

Copyright : (c) Mingyi Liu and Till Mossakowski and Uni Bremen 2004-2005

License : GPLv2 or higher

Maintainer : xinga@informatik.uni-bremen.de

Stability : provisional

Portability : portable

Termination proofs for equation systems, using AProVE

-}

module CASL.CCC.TerminationProof where

import CASL.AS_Basic_CASL

import CASL.CCC.TermFormula(term, quanti, isApp, arguOfTerm, varOfAxiom, idStr,

substiF, sameOpsApp)

import CASL.ToDoc()

import Common.Id(Token(tokStr))

import Common.Utils(getEnvDef, nubOrd)

import System.Cmd(system)

import System.Directory(getTemporaryDirectory)

{-

Automatic termination proof

using AProVE, see http://aprove.informatik.rwth-aachen.de/

interface to AProVE system, using system

translate CASL signature to AProVE Input Language,

CASL formulas to AProVE term rewrite systems(TRS),

see http://aprove.informatik.rwth-aachen.de/help/html/in/trs.html

if a equation system is terminal, then it is computable.

-}

terminationProof :: Ord f => [FORMULA f] -> [(TERM f, FORMULA f)]

-> IO (Maybe Bool)

terminationProof fs dms = if null fs then return $ Just True else do

let

allVar = nubOrd . concat

varsStr vars str

| null vars = str

| otherwise = if null str then varsStr (tail vars) (tokStr $ head vars)

else varsStr (tail vars) (str ++ " " ++tokStr (head vars))

axiomTrs axs str

| null axs = str

| otherwise = axiomTrs (tail axs) (str ++ axiom2TRS (head axs) dms ++ "\n")

axhead = "(RULES\neq(t,t) -> true\n" ++

"eq(_,_) -> false\n" ++

"when_else(t1,true,t2) -> t1\n" ++

"when_else(t1,false,t2) -> t2\n"

c_vars = "(VAR t t1 t2 " ++ varsStr (allVar $ map varOfAxiom fs) "" ++ ")\n"

c_axms = axhead ++ axiomTrs fs "" ++ ")\n"

ipath = "/Input.trs"

opath = "/Result.txt"

tmpDir <- getTemporaryDirectory

writeFile (tmpDir ++ ipath) (c_vars ++ c_axms)

aprovePath <- getEnvDef "HETS_APROVE"

"CASL/Termination/AProVE.jar"

system ("java -jar " ++ aprovePath ++ " -u cli -m " ++

"wst -p plain " ++ tmpDir ++

ipath ++ " | head -n 1 > " ++ tmpDir ++ opath)

proof <- readFile (tmpDir ++ opath)

return $ case proof of

"YES\n" -> Just True

"NO\n" -> Just False

_ -> Nothing

-- | get the name of a operation symbol

opSymName :: OP_SYMB -> String

opSymName os = case os of

Op_name on -> idStr on

Qual_op_name on _ _ -> idStr on

-- | get the name of a predicate symbol

predSymName :: PRED_SYMB -> String

predSymName ps = case ps of

Pred_name pn -> idStr pn

Qual_pred_name pn _ _ -> idStr pn

-- | translate a casl term to a term of TRS(Terme Rewrite Systems)

term2TRS :: TERM f -> String

term2TRS t =

case (term t) of

Qual_var var _ _ -> tokStr var

Application (Op_name opn) [] _ -> idStr opn

Application (Qual_op_name opn _ _) ts _ ->

if null ts then idStr opn

else idStr opn ++ "(" ++ termsPA ts ++")"

Sorted_term t' _ _ -> term2TRS t'

Cast t' _ _ -> term2TRS t'

Conditional t1 f t2 _ ->

"when_else(" ++ term2TRS t1 ++ "," ++ t_f_str f ++ "," ++

term2TRS t2 ++ ")"

_ -> error "CASL.CCC.TerminationProof.<term2TRS>"

where t_f_str f = case f of -- condition of a term

Strong_equation t1 t2 _ ->

"eq(" ++ term2TRS t1 ++ "," ++ term2TRS t2 ++ ")"

Predication ps ts _ ->

if null ts then predSymName ps

else predSymName ps ++ "(" ++ termsPA ts ++ ")"

_ -> error "CASL.CCC.TerminationProof.<term2TRS_cond>"

-- | translate a list of casl terms to the patterns of a term in TRS

termsPA :: [TERM f] -> String

termsPA ts =

if null ts then "" else tail $ concatMap ((\ s -> ',':s) . term2TRS) ts

-- | translate a casl axiom to TRS-rule:

-- a rule without condition is represented by "A -> B" in

-- Term Rewrite Systems; if there are some conditions, then

-- follow the conditions after the symbol "|".

-- For example : "A -> B | C -> D, E -> F, ..."

axiom2TRS :: (Eq f) => FORMULA f -> [(TERM f,FORMULA f)] -> String

axiom2TRS f doms =

case (quanti f) of

Quantification _ _ f' _ -> axiom2TRS f' doms

Conjunction fs _ -> conjSub fs doms ++ " -> true"

Disjunction fs _ -> disjSub fs doms ++ " -> true"

Implication f1 f2 _ _ ->

case f2 of

Equivalence f3 f4 _ ->

axiom2TRS f3 doms ++ " | " ++ axiom2TRS f1 doms ++ ", " ++

axiom2TRS f4 doms

_ ->

case f1 of

Definedness t _ ->

let dm = filter (sameOpsApp t . fst) doms

phi = snd $ head dm

te = fst $ head dm

p1 = arguOfTerm te

p2 = arguOfTerm t

st = zip p2 p1

in if not $ null dm

then axiom2TRS f2 doms ++ " | " ++

axiom2TRS (substiF st phi) doms

else axiom2TRS f2 doms ++

" | def(" ++ term2TRS t ++ ") -> open"

_ -> axiom2TRS f2 doms ++ " | " ++ axiom2TRS f1 doms

Equivalence f1 f2 _ -> axiom2TRS f1 doms ++ " | " ++ axiom2TRS f2 doms

Negation f' _ ->

case f' of

Conjunction fs _ -> conjSub fs doms ++ " -> false"

Disjunction fs _ -> disjSub fs doms ++ " -> false"

Predication p_s ts _ ->

if null ts

then predSymName p_s ++ " -> false"

else predSymName p_s ++ "(" ++ termsPA ts ++ ") -> false"

Strong_equation t1 t2 _ ->

"eq(" ++ term2TRS t1 ++ "," ++ term2TRS t2 ++ ") -> false"

Definedness t _ -> term2TRS t ++ " -> undefined"

_ -> error "CASL.CCC.TerminationProof.<axiom2TRS_Negation>"

Predication p_s ts _ ->

if null ts then predSymName p_s ++ " -> true"

else predSymName p_s ++ "(" ++ termsPA ts ++ ") -> true"

Definedness t _ -> "def(" ++ term2TRS t ++ ") -> open"

Strong_equation t1 t2 _ ->

case t2 of

Conditional tt1 ff tt2 _ ->

term2TRS t1 ++ " -> " ++ term2TRS tt1 ++ " | " ++

axiomSub ff doms ++ " -> true\n" ++

term2TRS t1 ++ " -> " ++ term2TRS tt2 ++ " | " ++

axiomSub ff doms ++ " -> false"

_ -> if isApp t1

then term2TRS t1 ++ " -> " ++ term2TRS t2

else "eq(" ++ term2TRS t1 ++ "," ++ term2TRS t2 ++

") -> true"

True_atom _ -> "true -> true"

False_atom _ -> "false -> false"

_ -> error "CASL.CCC.TerminationProof.<axiom2TRS>"

-- | translate a casl axiom(without conditions) to a subrule of TRS,

-- | the handling of an implication in a subrule is yet to be correctly defined.

axiomSub :: (Eq f) => FORMULA f -> [(TERM f,FORMULA f)] -> String

axiomSub f doms =

case (quanti f) of

Quantification _ _ f' _ -> axiomSub f' doms

Conjunction fs _ -> conjSub fs doms

Disjunction fs _ -> disjSub fs doms

Negation f' _ ->

case f' of

Conjunction fs _ -> "not(" ++ conjSub fs doms ++ ")"

Disjunction fs _ -> "not(" ++ disjSub fs doms ++ ")"

Predication p_s ts _ ->

if null ts then "not(" ++ predSymName p_s ++ ")"

else "not(" ++ predSymName p_s ++ "(" ++ termsPA ts ++ "))"

Strong_equation t1 t2 _ ->

"not(eq(" ++ term2TRS t1 ++ "," ++ term2TRS t2 ++ "))"

_ -> error "CASL.CCC.TerminationProof.str_substr.<Negation>"

True_atom _ -> "true"

False_atom _ -> "false"

Predication p_s ts _ ->

if null ts then predSymName p_s

else predSymName p_s ++ "(" ++ termsPA ts ++ ")"

Definedness t _ -> "def(" ++ term2TRS t ++ ")"

Strong_equation t1 t2 _ -> "eq(" ++ term2TRS t1 ++ "," ++ term2TRS t2 ++ ")"

i@(Implication _ _ _ _) -> axiom2TRS i doms

_ -> error "CASL.CCC.TerminationProof.axiomSub.<axiomSub>"

-- | translate a list of conjunctive casl formulas to a subrule of TRS

conjSub :: (Eq f) => [FORMULA f] -> [(TERM f,FORMULA f)] -> String

conjSub fs doms =

if length fs == 2 then "and(" ++ axiomSub (head fs) doms ++ "," ++

axiomSub (last fs) doms ++ ")"

else "and(" ++ axiomSub (head fs) doms ++ "," ++ conjSub (tail fs) doms ++ ")"

-- | translate a list of disjunctive casl formulas to a subrule of TRS

disjSub :: (Eq f) => [FORMULA f] -> [(TERM f,FORMULA f)] -> String

disjSub fs doms =

if length fs == 2 then "or(" ++ axiomSub (head fs) doms ++ "," ++

axiomSub (last fs) doms ++ ")"

else "or(" ++ axiomSub (head fs) doms ++ "," ++ disjSub (tail fs) doms ++ ")"