{- |

Module : $Header$

Description : termination proofs for equation systems, using AProVE

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

License : similar to LGPL, see HetCATS/LICENSE.txt or LIZENZ.txt

Maintainer : xinga@informatik.uni-bremen.de

Stability : provisional

Portability : portable

Termination proofs for equation systems, using AProVE

-}

module CASL.CCC.TerminationProof where

import CASL.ToDoc ()

import CASL.AS_Basic_CASL

-- import Common.DocUtils

import CASL.CCC.TermFormula

import Common.Id

import Common.Utils (getEnvDef)

import System.Cmd

import System.IO.Unsafe

import System.Directory

-- import Debug.Trace

import Data.List (nub)

{-

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 :: Eq f => [FORMULA f] -> [(TERM f,FORMULA f)] -> Maybe Bool

terminationProof fs dms

| null fs = Just True

| proof == "YES\n" = Just True

| proof == "MAYBE\n" = Nothing

| proof == "NO\n" = Just False

| otherwise = error "CASL.CCC.TerminationProof.<terminationProof>"

where

allVar vs = nub $ concat vs

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 ++ (axiom_trs (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"

proof = unsafePerformIO (do

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)

res <- readFile (tmpDir ++ opath)

return res)

-- | 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)

term_trs :: TERM f -> String

term_trs t =

case (term t) of

Simple_id simid -> tokStr simid

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) ++ "(" ++ terms_pa ts ++")")

Sorted_term t' _ _ ->

term_trs t'

Cast t' _ _ ->

term_trs t'

Conditional t1 f t2 _ ->

("when_else("++(term_trs t1)++","++

(t_f_str f)++","++(term_trs t2)++")")

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

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

Strong_equation t1 t2 _ ->

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

Predication ps ts _ ->

if null ts then predSymName ps

else ((predSymName ps) ++ "(" ++

(terms_pa ts) ++ ")")

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

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

terms_pa :: [TERM f] -> String

terms_pa ts =

if null ts then ""

else tail $ concat $ map (\s->","++s) $ map term_trs 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, ..."

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

axiom_trs f doms =

case (quanti f) of

Quantification _ _ f' _ -> axiom_trs f' doms

Conjunction fs _ ->

conj_sub fs ++ " -> true"

Disjunction fs _ ->

disj_sub fs ++ " -> true"

Implication f1 f2 _ _ ->

case f2 of

Equivalence f3 f4 _ ->

(axiom_trs f3 doms) ++ " | " ++

(axiom_trs f1 doms) ++ ", " ++

(axiom_trs f4 doms)

_ ->

case f1 of

Definedness t _ ->

let dm = filter (\d -> sameOps_App t (fst d)) 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 (axiom_trs f2 doms) ++ " | " ++

(axiom_trs (substiF st phi) doms)

else (axiom_trs f2 doms) ++

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

_ -> (axiom_trs f2 doms) ++ " | " ++

(axiom_trs f1 doms)

Equivalence f1 f2 _ ->

(axiom_trs f1 doms) ++ " | " ++

(axiom_trs f2 doms)

Negation f' _ ->

case f' of

Conjunction fs _ ->

conj_sub fs ++ " -> false"

Disjunction fs _ ->

disj_sub fs ++ " -> false"

Predication p_s ts _ ->

if null ts

then (predSymName p_s) ++ " -> false"

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

Strong_equation t1 t2 _ ->

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

Definedness t _ ->

(term_trs t) ++ " -> undefined"

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

Predication p_s ts _ ->

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

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

Definedness t _ ->

"def(" ++ (term_trs t) ++ ") -> open"

Strong_equation t1 t2 _ ->

case t2 of

Conditional tt1 ff tt2 _ ->

(term_trs t1) ++ " -> " ++ (term_trs tt1) ++ " | " ++

(axiom_sub ff) ++ " -> true\n" ++

(term_trs t1) ++ " -> " ++ (term_trs tt2) ++ " | " ++

(axiom_sub ff) ++ " -> false"

_ -> if isApp t1

then (term_trs t1) ++ " -> " ++ (term_trs t2)

else "eq(" ++ (term_trs t1) ++ "," ++ (term_trs t2) ++

") -> true"

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

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

-- | the arrow " -> " occurs not in a subrule.

axiom_sub :: FORMULA f -> String

axiom_sub f =

case (quanti f) of

Quantification _ _ f' _ -> axiom_sub f'

Conjunction fs _ ->

conj_sub fs

Disjunction fs _ ->

disj_sub fs

Negation f' _ ->

case f' of

Conjunction fs _ ->

"not(" ++ conj_sub fs ++ ")"

Disjunction fs _ ->

"not(" ++ disj_sub fs ++ ")"

Predication p_s ts _ ->

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

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

Strong_equation t1 t2 _ ->

"not(eq(" ++ (term_trs t1) ++ "," ++ (term_trs 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) ++ "(" ++ (terms_pa ts) ++ ")"

Definedness t _ ->

"def(" ++ (term_trs t) ++ ")"

Strong_equation t1 t2 _ ->

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

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

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

conj_sub :: [FORMULA f] -> String

conj_sub fs =

if length fs == 2 then ("and(" ++ (axiom_sub $ head fs) ++ "," ++

(axiom_sub $ last fs) ++ ")")

else ("and(" ++ (axiom_sub $ head fs) ++ "," ++

(conj_sub $ tail fs) ++ ")")

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

disj_sub :: [FORMULA f] -> String

disj_sub fs =

if length fs == 2 then ("or(" ++ (axiom_sub $ head fs) ++ "," ++

(axiom_sub $ last fs) ++ ")")

else ("or(" ++ (axiom_sub $ head fs) ++ "," ++

(disj_sub $ tail fs) ++ ")")