{- |

Module : $Header$

Description : consistency checking of free types

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 : non-portable (imports Logic.Logic)

Consistency checking of free types

-}

module CASL.CCC.FreeTypes (checkFreeType) where

import CASL.Sign -- Sign, OpType

import CASL.Morphism

import CASL.AS_Basic_CASL -- FORMULA, OP_{NAME,SYMB}, TERM, SORT, VAR

import qualified Data.Map as Map

import qualified Data.Set as Set

import qualified Common.Lib.Rel as Rel

import CASL.CCC.TermFormula

import CASL.CCC.TerminationProof

import Common.AS_Annotation

import Common.Consistency

import Common.Result

import Common.Id

import Data.List (nub,intersect,delete)

inhabited :: [SORT] -> [Constraint] -> [SORT]

inhabited sorts constrs = iterateInhabited sorts

where (_,ops,_)=recover_Sort_gen_ax constrs

argsRes=concat $

map (\os-> case os of

Op_name _->[]

Qual_op_name _ ot _->

case ot of

Op_type _ args res _ -> [(args,res)]

) ops

iterateInhabited l =

if l==newL then newL else iterateInhabited newL

where newL =foldr (\(ags,rs) l'->

if (all (\s->elem s l') ags)

&& (not (elem rs l'))

then rs:l'

else l') l argsRes

{------------------------------------------------------------------------

function checkFreeType:

- check if leading symbols are new (not in the image of morphism),

if not, return them as obligations

- generated datatype is free

- if new sort is not etype or esort, it can not be empty.

- the leading terms consist of variables and constructors only,

if not, return Nothing

- split function leading_Symb into

leading_Term_Predication

and

extract_leading_symb

- collect all operation symbols from recover_Sort_gen_ax fconstrs

(= constructors)

- no variable occurs twice in a leading term, if not, return Nothing

- check that patterns do not overlap, if not, return obligations

This means:

in each group of the grouped axioms:

all patterns of leading terms/formulas are disjoint

this means: either leading symbol is a variable,

and there is just one axiom

otherwise, group axioms according to leading symbol

no symbol may be a variable

check recursively the arguments of

constructor in each group

- sufficient completeness

- termination proof

-------------------------------------------------------------------------}

-- | free datatypes and recursive equations are consistent

checkFreeType :: (Sign () (),[Named (FORMULA ())]) -> Morphism () () ()

-> [Named (FORMULA ())]

-> Result (Maybe (ConsistencyStatus,[FORMULA ()]))

checkFreeType (osig,osens) m fsn

| null fsn && null nSorts = return (Just (Conservative,[]))

| not $ null notFreeSorts =

let (Id ts _ pos) = head notFreeSorts

sname = concat $ map tokStr ts

in warning Nothing (sname ++ " is not freely generated") pos

| not $ null nefsorts =

let (Id ts _ pos) = head nefsorts

sname = concat $ map tokStr ts

in warning (Just (Inconsistent,[])) (sname ++ " is not inhabited") pos

| elem Nothing l_Syms =

let pos = snd $ head $ filter (\f'-> (fst f') == Nothing) $

map leadingSymPos _axioms

in warning Nothing "axiom is not definitional" pos

| not $ null $ un_p_axioms =

let pos = getRange $ (take 1 un_p_axioms)

in warning Nothing "partial axiom is not definitional" pos

| (length dom_l) /= (length $ nub $ dom_l) =

let pos = getRange $ (take 1 dualDom)

dualOS = head $ filter (\o-> elem o $ delete o dom_l) dom_l

dualDom = filter (\f-> domain_os f dualOS) p_axioms

in warning Nothing "partial axiom is not definitional" pos

| not $ null $ pcheck =

let pos = getRange $ (take 1 pcheck)

in warning Nothing "partial axiom is not definitional" pos

| not $ and $ map (checkTerms tsig constructors) $

map arguOfTerm leadingTerms=

let (Application os _ _) = tt

tt= head $ filter (\t->not $ checkTerms tsig constructors $

arguOfTerm t) $ leadingTerms

pos = axiomRangeforTerm _axioms tt

in warning Nothing ("a leading term of " ++ (opSymName os) ++

" consists of not only variables and constructors") pos

| not $ and $ map (checkTerms tsig constructors) $

map arguOfPred leadingPreds=

let (Predication ps _ pos) = quanti pf

pf= head $ filter (\p->not $ checkTerms tsig constructors $

arguOfPred p) $ leadingPreds

in warning Nothing ("a leading predicate of " ++ (predSymName ps) ++

" consists of not only variables and constructors") pos

| not $ and $ map checkVar_App leadingTerms =

let (Application os _ _) = tt

tt = head $ filter (\t->not $ checkVar_App t) leadingTerms

pos = axiomRangeforTerm _axioms tt

in warning Nothing ("a variable occurs twice in a leading term of " ++

opSymName os) pos

| not $ and $ map checkVar_Pred leadingPreds =

let (Predication ps _ pos) = quanti pf

pf = head $ filter (\p->not $ checkVar_Pred p) leadingPreds

in warning Nothing ("a variable occurs twice in a leading " ++

"predicate of " ++ predSymName ps) pos

| not $ null notcomplete =

let symb_p = leadingSymPos $ head $ head notcomplete

pos = snd $ symb_p

sname = case (fst symb_p) of

Just (Left opS) -> opSymName opS

Just (Right pS) -> predSymName pS

_ -> error "CASL.CCC.FreeTypes.<Symb_Name>"

in warning (Just (Conservative,obligations)) ("the definition of " ++

sname ++ " is not complete") pos

| (not $ null fs_terminalProof) && (proof /= Just True)=

if proof == Just False

then warning (Just (Conservative,obligations))

"not terminating" nullRange

else warning (Just (Conservative,obligations))

"cannot prove termination" nullRange

| not $ null obligations =

return (Just (conStatus,obligations))

| otherwise = return (Just (conStatus,[]))

{-

call the symbols in the image of the signature morphism "new"

- each new sort must be a free type,

i.e. it must occur in a sort generation constraint that is marked as free

(Sort_gen_ax constrs True)

such that the sort is in srts,

where (srts,ops,_)=recover_Sort_gen_ax constrs

if not, output "don't know"

and there must be one term of that sort (inhabited)

if not, output "no"

- group the axioms according to their leading operation/predicate symbol,

i.e. the f resp. the p in

forall x_1:s_n .... x_n:s_n . f(t_1,...,t_m)=t

forall x_1:s_n .... x_n:s_n . phi => f(t_1,...,t_m)=t

Implication Application Strong_equation

forall x_1:s_n .... x_n:s_n . p(t_1,...,t_m)<=>phi

forall x_1:s_n .... x_n:s_n . phi1 => p(t_1,...,t_m)<=>phi

Implication Predication Equivalence

if there are axioms not being of this form, output "don't know"

-}

where

fs = map sentence (filter is_user_or_sort_gen fsn)

fs_terminalProof = filter (\f->(not $ isSortGen f) &&

(not $ is_Membership f) &&

(not $ is_ex_quanti f) &&

(not $ isDomain f)) fs

ofs = map sentence (filter is_user_or_sort_gen osens)

sig = imageOfMorphism m

tsig = mtarget m

oldSorts = (sortSet osig)

oSorts = Set.toList oldSorts

allSorts = sortSet $ tsig

esorts = Set.toList $ emptySortSet tsig

newSorts = Set.filter (\s-> not $ Set.member s oldSorts) allSorts

nSorts = Set.toList newSorts

sR = Rel.toList $ sortRel tsig

subs = map fst sR

axOfS = filter (\f-> (isSortGen f) ||

(is_Membership f)) fs

notFreeSorts = filter (\s->(is_free_gen_sort s axOfS) == Just False) nSorts

oldOpMap = opMap sig

oldPredMap = predMap sig

fconstrs = concat $ map constraintOfAxiom (ofs ++ fs)

(srts,constructors_o,_) = recover_Sort_gen_ax fconstrs

constructors = constructorOverload tsig op_map constructors_o

f_Inhabited = inhabited oSorts fconstrs

gens = intersect nSorts srts

fsorts = filter (\s-> not $ elem s esorts) $ intersect nSorts srts

nefsorts = filter (\s->not $ elem s f_Inhabited) fsorts

op_map = opMap $ tsig

axioms = filter (\f-> (not $ isSortGen f) &&

(not $ is_Membership f) &&

(not $ is_ex_quanti f)) fs

memberships = filter (\f-> is_Membership f) fs

info_subsort = concat $ map (infoSubsort esorts) memberships

dataStatus = if null nSorts then Definitional

else if not $ null $ filter (\s-> (not $ elem s subs) &&

(not $ elem s gens)) nSorts

then Conservative

else Monomorphic

_axioms = map quanti axioms

l_Syms = map leadingSym axioms -- leading_Symbol

{-

check the definitional form of the partial axioms

-}

p_axioms = filter partialAxiom _axioms -- all partial axioms

pax_with_def = filter containDef p_axioms

ax_without_dom = filter (not.isDomain) _axioms

pax_without_def = filter (not.containDef) p_axioms

un_p_axioms = filter (not.correctDef) pax_with_def

dom_l = domainOpSymbs p_axioms

pcheck = filter (\f-> case leadingSym f of

Just (Left opS) ->

elem opS dom_l

_ -> False) pax_without_def

domains = domainList fs

{-

check if leading symbols are new (not in the image of morphism),

if not, return it as proof obligation

-}

op_fs = filter (\f-> case leadingSym f of

Just (Left _) -> True

_ -> False) axioms

pred_fs = filter (\f-> case leadingSym f of

Just (Right _) -> True

_ -> False) axioms

find_ot (ident,ot) = case Map.lookup ident oldOpMap of

Nothing -> False

Just ots -> Set.member ot ots

find_pt (ident,pt) = case Map.lookup ident oldPredMap of

Nothing -> False

Just pts -> Set.member pt pts

oOps = filter (\a-> find_ot $ head $ filterOp $ leadingSym a) op_fs

oPreds = filter (\a-> find_pt $ head $ filterPred $ leadingSym a) pred_fs

{-

the leading terms consist of variables and constructors only

-}

ltp = map leading_Term_Predication _axioms -- leading_term_pred

leadingTerms = concat $ map (\tp->case tp of

Just (Left t)->[t]

_ -> []) $ ltp -- leading Term

leadingPreds = concat $ map (\tp->case tp of

Just (Right f)->[f]

_ -> []) $ ltp

{-

check that patterns do not overlap, if not, return proof obligation.

-}

axGroup = groupAxioms ax_without_dom

axPairs =

case axGroup of

Just sym_fs -> concat $ map pairs $ map snd sym_fs

Nothing -> error "CASL.CCC.FreeTypes.<axPairs>"

olPairs = filter (\a-> checkPatterns $

(patternsOfAxiom $ fst a,

patternsOfAxiom $ snd a)) axPairs

subst (f1,f2) = ((f1,sb1),(f2,sb2))

where p1= patternsOfAxiom f1

p2= patternsOfAxiom f2

(sb1,sb2) = st ((p1,[]),(p2,[]))

st ((pa1,s1),(pa2,s2))

| null pa1 =(s1,s2)

| head pa1 == head pa2 = st ((tail pa1,s1),(tail pa2,s2))

| isVar $ head pa1 = st ((tail pa1,s1++[(head pa2,head pa1)]),

(tail pa2,s2))

| isVar $ head pa2 = st ((tail pa1,s1),

(tail pa2,s2++[(head pa1,head pa2)]))

| otherwise = st (((patternsOfTerm $ head pa1)++(tail pa1),s1),

((patternsOfTerm $ head pa2)++(tail pa2),s2))

olPairsWithS = map subst olPairs

overlap_qu = map overlapQuery olPairsWithS

overlap_query = map (\f-> Quantification Universal

(varDeclOfF f)

f

nullRange) overlap_qu

{-

check the sufficient completeness

-}

axGroups =

case axGroup of

Just sym_fs -> map snd sym_fs

Nothing -> error "CASL.CCC.FreeTypes.<axGroups>"

notcomplete = filter (\f'-> not $ completePatterns constructors $

map patternsOfAxiom f') axGroups

ex_axioms = filter is_ex_quanti $ fs

proof = terminationProof fs_terminalProof domains

obligations = oOps ++ oPreds ++ ex_axioms ++ info_subsort ++ overlap_query

defStatus = if null $ (oOps ++ oPreds ++ ex_axioms ++ overlap_query)

then Definitional

else Conservative

conStatus = min dataStatus defStatus

-- | group the axioms according to their leading symbol,

-- output Nothing if there is some axiom in incorrect form

groupAxioms :: [FORMULA f] -> Maybe [(Either OP_SYMB PRED_SYMB,[FORMULA f])]

groupAxioms phis = do

symbs <- mapM leadingSym phis

return (filterA (zip symbs phis) [])

where filterA [] _=[]

filterA (p:ps) symb=

let fp=fst p

p'= if elem fp symb then []

else [(fp,snd $ unzip $ filter (\x->(fst x)==fp) (p:ps))]

symb'= if not $ (elem fp symb) then fp:symb

else symb

in p'++(filterA ps symb')

filterOp :: Maybe (Either OP_SYMB PRED_SYMB) -> [(OP_NAME,OpType)]

filterOp symb = case symb of

Just (Left (Qual_op_name ident (Op_type k ags rs _) _))->

[(ident, OpType {opKind=k, opArgs=ags, opRes=rs})]

_ -> []

filterPred :: Maybe (Either OP_SYMB PRED_SYMB) -> [(OP_NAME,PredType)]

filterPred symb = case symb of

Just (Right (Qual_pred_name ident (Pred_type s _) _)) ->

[(ident, PredType {predArgs=s})]

_ -> []

-- | a leading term and a predication consist of

-- | variables and constructors only

checkTerms :: Sign f e -> [OP_SYMB] -> [TERM f] -> Bool

checkTerms sig cons ts = all checkT ts

where checkT (Sorted_term tt _ _) = checkT tt

checkT (Qual_var _ _ _) = True

checkT (Application subop subts _) =

(isCons sig cons subop) &&

(all checkT subts)

checkT _ = False

-- | no variable occurs twice in a leading term

checkVar_App :: (Eq f) => TERM f -> Bool

checkVar_App (Application _ ts _) = noDuplation $ concat $ map varOfTerm ts

checkVar_App _ = error "CASL.CCC.FreeTypes<checkVar_App>"

-- | no variable occurs twice in a leading predication

checkVar_Pred :: (Eq f) => FORMULA f -> Bool

checkVar_Pred (Predication _ ts _) = noDuplation $ concat $ map varOfTerm ts

checkVar_Pred _ = error "CASL.CCC.FreeTypes<checkVar_Pred>"

-- check whether all element are identic

allIdentic :: (Eq a) => [a] -> Bool

allIdentic ts = if (length $ nub ts) <= 1 then True

else False

-- | there are no duplicate items in a list

noDuplation :: (Eq a) => [a] -> Bool

noDuplation l = if (length $ nub l) == (length l) then True

else False

-- | create all possible pairs from a list

pairs :: [a] -> [(a,a)]

pairs ps= if length ps <=1 then []

else (map (\x'->((head ps),x')) $ tail ps) ++

(pairs $ tail ps)

-- | get the patterns of a term

patternsOfTerm :: TERM f -> [TERM f]

patternsOfTerm t = case t of

Qual_var _ _ _ -> [t]

Application (Qual_op_name _ _ _) ts _-> ts

Sorted_term t' _ _ -> patternsOfTerm t'

Cast t' _ _ -> patternsOfTerm t'

_ -> []

-- | get the patterns of a axiom

patternsOfAxiom :: FORMULA f -> [TERM f]

patternsOfAxiom f = case f of

Quantification _ _ f' _ -> patternsOfAxiom f'

Negation f' _ -> patternsOfAxiom f'

Implication _ f' _ _ -> patternsOfAxiom f'

Equivalence f' _ _ -> patternsOfAxiom f'

Predication _ ts _ -> ts

Existl_equation t _ _ -> patternsOfTerm t

Strong_equation t _ _ -> patternsOfTerm t

_ -> []

-- | check whether two patterns are overlapped

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

checkPatterns (ps1,ps2)

| ps1 == ps2 = True

| (isVar $ head ps1) ||

(isVar $ head ps2) = checkPatterns (tail ps1,tail ps2)

| otherwise = if sameOps_App (head ps1) (head ps2)

then checkPatterns $

((patternsOfTerm $ head ps1)++(tail ps1),

(patternsOfTerm $ head ps2)++(tail ps2))

else False

-- | get the axiom from left hand side of a implication,

-- | if there is no implication, then return atomic formula true

conditionAxiom :: FORMULA f -> [FORMULA f]

conditionAxiom f = case f of

Quantification _ _ f' _ -> conditionAxiom f'

Implication f' _ _ _ -> [f']

_ -> [True_atom nullRange]

-- | get the axiom from right hand side of a equivalence,

-- | if there is no equivalence, then return atomic formula true

resultAxiom :: FORMULA f -> [FORMULA f]

resultAxiom f = case f of

Quantification _ _ f' _ -> resultAxiom f'

Implication _ f' _ _ -> resultAxiom f'

Equivalence _ f' _ -> [f']

_ -> [True_atom nullRange]

-- | get the term from left hand side of a equation in a formula,

-- | if there is no equation, then return a simple id

resultTerm :: FORMULA f -> [TERM f]

resultTerm f = case f of

Quantification _ _ f' _ -> resultTerm f'

Implication _ f' _ _ -> resultTerm f'

Negation (Definedness _ _) _ ->

[varOrConst (mkSimpleId "undefined")]

Strong_equation _ t _ -> [t]

Existl_equation _ t _ -> [t]

_ -> [varOrConst (mkSimpleId "unknown")]

-- | create the proof obligation for a pair of overlapped formulas

overlapQuery :: Eq f => ((FORMULA f,[(TERM f,TERM f)]),

(FORMULA f,[(TERM f,TERM f)])) -> FORMULA f

overlapQuery ((a1,s1),(a2,s2)) =

case leadingSym a1 of

Just (Left _)

| (containNeg a1) &&

(not $ containNeg a2) ->

Implication (Conjunction [con1,con2] nullRange)

(Negation (Definedness resT2 nullRange)

nullRange)

True

nullRange

| (containNeg a2) &&

(not $ containNeg a1)->

Implication (Conjunction [con1,con2] nullRange)

(Negation (Definedness resT1 nullRange)

nullRange)

True

nullRange

| (containNeg a1) &&

(containNeg a2) ->

True_atom nullRange

| otherwise ->

Implication (Conjunction [con1,con2] nullRange)

(Strong_equation resT1 resT2 nullRange)

True

nullRange

Just (Right _)

| (containNeg a1) &&

(not $ containNeg a2) ->

Implication (Conjunction [con1,con2] nullRange)

(Negation resA2 nullRange)

True

nullRange

| (containNeg a2) &&

(not $ containNeg a1) ->

Implication (Conjunction [con1,con2] nullRange)

(Negation resA1 nullRange)

True

nullRange

| (containNeg a1) &&

(containNeg a2) ->

True_atom nullRange

| otherwise ->

Implication (Conjunction [con1,con2] nullRange)

(Conjunction [resA1,resA2] nullRange)

True

nullRange

_ -> error "CASL.CCC.FreeTypes.<overlapQuery>"

where cond = concat $ map conditionAxiom [a1,a2]

resT = concat $ map resultTerm [a1,a2]

resA = concat $ map resultAxiom [a1,a2]

con1 = substiF s1 $ head cond

con2 = substiF s2 $ last cond

resT1 = substitute s1 $ head resT

resT2 = substitute s2 $ last resT

resA1 = substiF s1 $ head resA

resA2 = substiF s2 $ last resA

-- | check whether the patterns of a function or predicate are complete

completePatterns :: (Eq f) => [OP_SYMB] -> ([[TERM f]]) -> Bool

completePatterns cons pas

| all null pas = True

| all isVar $ map head pas = completePatterns cons (map tail pas)

| otherwise = if elem (con_ts $ map head pas) s_cons

then all id $ map (completePatterns cons) $ pa_group pas

else False

where s_op_os c = case c of

Op_name _ -> []

Qual_op_name on ot _ -> [(res_OP_TYPE ot,on)]

s_sum sns = map (\s->(s, map snd $ filter (\c-> (fst c)==s) sns)) $

nub $ map fst sns

s_cons = s_sum $ concat $ map s_op_os cons

s_op_t t = case t of

Application os _ _ -> s_op_os os

_ -> []

con_ts ts = head $ s_sum $ concat $ map s_op_t ts

opN t = case t of

Application os _ _ -> opSymbName os

_ -> genName "unknown"

pa_group p = map p_g $ map (\o->

filter (\a-> (isVar $ head a) ||

((opN $ head $ a) == o)) p) $

snd $ head $

filter (\sc-> fst sc == (sortOfTerm $

head $ head p)) s_cons

p_g p = map (\p'-> if isVar $ head p'

then (take (maximum $ map (length.arguOfTerm.head)

p) $ repeat $ head p') ++ tail p'

else (arguOfTerm $ head p') ++ tail p') p