/*! @brief From T|- QUANTIFIER (vars): e we get T|-QUANTIFIER(vars') e
* where vars' is obtained from vars by removing all bound variables
* not used in e. If vars' is empty the produced theorem is just T|-e
*/
Theorem QuantTheoremProducer::boundVarElim(const Theorem& t1)
{
const Expr e=t1.getExpr();
const Expr body = e.getBody();
if(CHECK_PROOFS) {
CHECK_SOUND(e.isForall() || e.isExists(),
"bound var elimination: "
+e.toString());
}
ExprMap<bool> boundVars; //a mapping of bound variables in the body to true
ExprMap<bool> visited; //to make sure expressions aren't traversed
//multiple times
recFindBoundVars(body, boundVars, visited);
vector<Expr> quantVars;
const vector<Expr>& origVars = e.getVars();
for(vector<Expr>::const_iterator it = origVars.begin(),
iend=origVars.end(); it!=iend; ++it)
{
if(boundVars.count(*it) > 0)
quantVars.push_back(*it);
}
// If all variables are used, just return the original theorem
if(quantVars.size() == origVars.size())
return t1;
Proof pf;
if(withProof()) {
vector<Expr> es;
vector<Proof> pfs;
es.push_back(e);
es.insert(es.end(), quantVars.begin(), quantVars.end());
pfs.push_back(t1.getProof());
pf= newPf("bound_variable_elimination", es, pfs);
}
if(quantVars.size() == 0)
return(newTheorem(e.getBody(), t1.getAssumptionsRef(), pf));
Expr newQuantExpr;
if(e.isForall())
newQuantExpr = d_theoryQuant->getEM()->newClosureExpr(FORALL, quantVars, body);
else
newQuantExpr = d_theoryQuant->getEM()->newClosureExpr(EXISTS, quantVars, body);
return(newTheorem(newQuantExpr, t1.getAssumptionsRef(), pf));
}
//!adjust the order of bound vars, newBvs begin first
Theorem QuantTheoremProducer::adjustVarUniv(const Theorem& t1, const std::vector<Expr>& newBvs){
const Expr e=t1.getExpr();
const Expr body = e.getBody();
if(CHECK_PROOFS) {
CHECK_SOUND(e.isForall(),
"adjustVarUniv: "
+e.toString());
}
const vector<Expr>& origVars = e.getVars();
ExprMap<bool> oldmap;
for(vector<Expr>::const_iterator it = origVars.begin(),
iend=origVars.end(); it!=iend; ++it) {
oldmap[*it]=true;
}
vector<Expr> quantVars;
for(vector<Expr>::const_iterator it = newBvs.begin(),
iend=newBvs.end(); it!=iend; ++it) {
if(oldmap.count(*it) > 0)
quantVars.push_back(*it);
}
if(quantVars.size() == origVars.size())
return t1;
ExprMap<bool> newmap;
for(vector<Expr>::const_iterator it = newBvs.begin(),
iend=newBvs.end(); it!=iend; ++it) {
newmap[*it]=true;
}
for(vector<Expr>::const_iterator it = origVars.begin(),
iend=origVars.end(); it!=iend; ++it) {
if(newmap.count(*it)<=0){
quantVars.push_back(*it);
};
}
Proof pf;
if(withProof()) {
vector<Expr> es;
vector<Proof> pfs;
es.push_back(e);
es.insert(es.end(), quantVars.begin(), quantVars.end());
pfs.push_back(t1.getProof());
pf= newPf("adjustVarUniv", es, pfs);
}
Expr newQuantExpr;
newQuantExpr = d_theoryQuant->getEM()->newClosureExpr(FORALL, quantVars, body);
return(newTheorem(newQuantExpr, t1.getAssumptionsRef(), pf));
}
Theorem
QuantTheoremProducer::normalizeQuant(const Expr& quant) {
if(CHECK_PROOFS) {
CHECK_SOUND(quant.isForall()||quant.isExists(),
"normalizeQuant: expr must be FORALL or EXISTS\n"
+quant.toString());
}
std::map<Expr,int>::iterator typeIter;
std::string base("_BD");
int counter(0);
vector<Expr> newVars;
const std::vector<Expr>& cur_vars = quant.getVars();
for(size_t j =0; j<cur_vars.size(); j++){
Type t = cur_vars[j].getType();
int typeIndex ;
typeIter = d_typeFound.find(t.getExpr());
if(d_typeFound.end() == typeIter){
typeIndex = d_typeFound.size();
d_typeFound[t.getExpr()] = typeIndex;
}
else{
typeIndex = typeIter->second;
}
counter++;
std::stringstream stringType;
stringType << counter << "TY" << typeIndex ;
std::string out_str = base + stringType.str();
Expr newExpr = d_theoryQuant->getEM()->newBoundVarExpr(out_str, int2string(counter));
newExpr.setType(t);
newVars.push_back(newExpr);
}
vector<vector<Expr> > trigs = quant.getTriggers();
for(size_t i = 0 ; i < trigs.size(); i++){
for(size_t j = 0 ; j < trigs[i].size(); j++){
trigs[i][j] = trigs[i][j].substExpr(cur_vars,newVars);
}
}
Expr normBody = quant.getBody().substExpr(cur_vars,newVars);
Expr normQuant = d_theoryQuant->getEM()->newClosureExpr(quant.isForall()?FORALL:EXISTS, newVars, normBody, trigs);
Proof pf;
if(withProof())
pf = newPf("normalizeQuant", quant, normQuant);
return newRWTheorem(quant, normQuant, Assumptions::emptyAssump(), pf);
}
Expr VCCmd::skolemizeAx(const Expr& e)
{
vector<Expr>vars;
const vector<Expr>boundVars = e.getVars();
for(unsigned int i=0; i<boundVars.size(); i++) {
Expr skolV(e.skolemExpr(i));
vars.push_back(skolV);
}
Expr sub = e.getBody().substExpr(boundVars, vars);
return e.iffExpr(sub);
}
//! find all bound variables in e and maps them to true in boundVars
void QuantTheoremProducer::recFindBoundVars(const Expr& e,
ExprMap<bool> & boundVars, ExprMap<bool> &visited)
{
if(visited.count(e)>0)
return;
else
visited[e] = true;
if(e.getKind() == BOUND_VAR)
boundVars[e] = true;
if(e.getKind() == EXISTS || e.getKind() == FORALL)
recFindBoundVars(e.getBody(), boundVars, visited);
for(Expr::iterator it = e.begin(); it!=e.end(); ++it)
recFindBoundVars(*it, boundVars, visited);
}
void VCCmd::findAxioms(const Expr& e, ExprMap<bool>& skolemAxioms,
ExprMap<bool>& visited) {
if(visited.count(e)>0)
return;
else visited[e] = true;
if(e.isSkolem()) {
skolemAxioms.insert(e.getExistential(), true);
return;
}
if(e.isClosure()) {
findAxioms(e.getBody(), skolemAxioms, visited);
}
if(e.arity()>0) {
Expr::iterator end = e.end();
for(Expr::iterator i = e.begin(); i!=end; ++i)
findAxioms(*i, skolemAxioms, visited);
}
}
//! convert (forall (x) ... forall (y)) into (forall (x y)...)
//! convert (exists (x) ... exists (y)) into (exists (x y)...)
Theorem QuantTheoremProducer::pullVarOut(const Theorem& t1){
const Expr thm_expr=t1.getExpr();
if(CHECK_PROOFS) {
CHECK_SOUND(thm_expr.isForall() || thm_expr.isExists(),
"pullVarOut: "
+thm_expr.toString());
}
const Expr outBody = thm_expr.getBody();
// if(((outBody.isAnd() && outBody[1].isForall()) ||
// (outBody.isImpl() && outBody[1].isForall()) ||
// (outBody.isNot() && outBody[0].isAnd() && outBody[0][1].isExists()) )){
// return t1;
// }
if (thm_expr.isForall()){
if((outBody.isNot() && outBody[0].isAnd() && outBody[0][1].isExists())){
vector<Expr> bVarsOut = thm_expr.getVars();
const Expr innerExists =outBody[0][1];
const Expr innerBody = innerExists.getBody();
vector<Expr> bVarsIn = innerExists.getVars();
for(vector<Expr>::iterator i=bVarsIn.begin(), iend=bVarsIn.end(); i!=iend; i++){
bVarsOut.push_back(*i);
}
Proof pf;
if(withProof()) {
vector<Expr> es;
vector<Proof> pfs;
es.push_back(thm_expr);
es.insert(es.end(), bVarsIn.begin(), bVarsIn.end());
pfs.push_back(t1.getProof());
pf= newPf("pullVarOut", es, pfs);
}
Expr newbody;
newbody=(outBody[0][0].notExpr()).orExpr(innerBody.notExpr());
Expr newQuantExpr;
newQuantExpr = d_theoryQuant->getEM()->newClosureExpr(FORALL, bVarsOut, newbody);
return(newTheorem(newQuantExpr, t1.getAssumptionsRef(), pf));
}
else if ((outBody.isAnd() && outBody[1].isForall()) ||
(outBody.isImpl() && outBody[1].isForall())){
vector<Expr> bVarsOut = thm_expr.getVars();
const Expr innerForall=outBody[1];
const Expr innerBody = innerForall.getBody();
vector<Expr> bVarsIn = innerForall.getVars();
for(vector<Expr>::iterator i=bVarsIn.begin(), iend=bVarsIn.end(); i!=iend; i++){
bVarsOut.push_back(*i);
}
Proof pf;
if(withProof()) {
vector<Expr> es;
vector<Proof> pfs;
es.push_back(thm_expr);
es.insert(es.end(), bVarsIn.begin(), bVarsIn.end());
pfs.push_back(t1.getProof());
pf= newPf("pullVarOut", es, pfs);
}
Expr newbody;
if(outBody.isAnd()){
newbody=outBody[0].andExpr(innerBody);
}
else if(outBody.isImpl()){
newbody=outBody[0].impExpr(innerBody);
}
Expr newQuantExpr;
newQuantExpr = d_theoryQuant->getEM()->newClosureExpr(FORALL, bVarsOut, newbody);
return(newTheorem(newQuantExpr, t1.getAssumptionsRef(), pf));
}
return t1; // case cannot be handled now.
}
else if (thm_expr.isExists()){
if ((outBody.isAnd() && outBody[1].isExists()) ||
(outBody.isImpl() && outBody[1].isExists())){
vector<Expr> bVarsOut = thm_expr.getVars();
const Expr innerExists = outBody[1];
const Expr innerBody = innerExists.getBody();
vector<Expr> bVarsIn = innerExists.getVars();
for(vector<Expr>::iterator i=bVarsIn.begin(), iend=bVarsIn.end(); i!=iend; i++){
bVarsOut.push_back(*i);
}
Proof pf;
if(withProof()) {
vector<Expr> es;
vector<Proof> pfs;
es.push_back(thm_expr);
es.insert(es.end(), bVarsIn.begin(), bVarsIn.end());
pfs.push_back(t1.getProof());
pf= newPf("pullVarOut", es, pfs);
}
Expr newbody;
if(outBody.isAnd()){
newbody=outBody[0].andExpr(innerBody);
}
else if(outBody.isImpl()){
newbody=outBody[0].impExpr(innerBody);
}
Expr newQuantExpr;
newQuantExpr = d_theoryQuant->getEM()->newClosureExpr(EXISTS, bVarsOut, newbody);
return(newTheorem(newQuantExpr, t1.getAssumptionsRef(), pf));
}
}
return t1;
}
Theorem QuantTheoremProducer::partialUniversalInst(const Theorem& t1, const vector<Expr>& terms, int quantLevel){
cout<<"error in partial inst" << endl;
Expr e = t1.getExpr();
const vector<Expr>& boundVars = e.getVars();
if(CHECK_PROOFS) {
CHECK_SOUND(boundVars.size() >= terms.size(),
"Universal instantiation: size of terms array does "
"not match quanitfied variables array size");
CHECK_SOUND(e.isForall(),
"universal instantiation: expr must be FORALL:\n"
+e.toString());
for(unsigned int i=0; i<terms.size(); i++){
CHECK_SOUND(d_theoryQuant->getBaseType(boundVars[i]) ==
d_theoryQuant->getBaseType(terms[i]),
"partial Universal instantiation: type mismatch");
}
}
//build up a conjunction of type predicates for expression
Expr tr = e.getEM()->trueExpr();
Expr typePred = tr;
for(unsigned int i=0; i<terms.size(); i++) {
Expr p = d_theoryQuant->getTypePred(boundVars[i].getType(),terms[i]);
if(p!=tr) {
if(typePred==tr)
typePred = p;
else
typePred = typePred.andExpr(p);
}
}
Proof pf;
if(withProof()) {
vector<Proof> pfs;
vector<Expr> es;
pfs.push_back(t1.getProof());
es.push_back(e);
es.insert(es.end(), terms.begin(), terms.end());
pf= newPf("partial_universal_instantiation", es, pfs);
}
if(terms.size() == boundVars.size()){
Expr inst = e.getBody().substExpr(e.getVars(), terms);
Expr imp;
if(typePred == tr)
imp = inst;
else
imp = typePred.impExpr(inst);
return(newTheorem(imp, t1.getAssumptionsRef(), pf));
}
else{
vector<Expr> newBoundVars;
for(size_t i=0; i<terms.size(); i++) {
newBoundVars.push_back(boundVars[i]);
}
vector<Expr>leftBoundVars;
for(size_t i=terms.size(); i<boundVars.size(); i++) {
leftBoundVars.push_back(boundVars[i]);
}
Expr tempinst = e.getBody().substExpr(newBoundVars, terms);
Expr inst = d_theoryQuant->getEM()->newClosureExpr(FORALL, leftBoundVars, tempinst);
Expr imp;
if(typePred == tr)
imp = inst;
else
imp = typePred.impExpr(inst);
Theorem res = (newTheorem(imp, t1.getAssumptionsRef(), pf));
int thmLevel = t1.getQuantLevel();
if(quantLevel >= thmLevel) {
res.setQuantLevel(quantLevel+1);
}
else{
//k ret.setQuantLevel(thmLevel+1);
res.setQuantLevel(thmLevel);
}
return res;
}
}
Theorem QuantTheoremProducer::universalInst(const Theorem& t1, const vector<Expr>& terms){
Expr e = t1.getExpr();
const vector<Expr>& boundVars = e.getVars();
if(CHECK_PROOFS) {
CHECK_SOUND(boundVars.size() == terms.size(),
"Universal instantiation: size of terms array does "
"not match quanitfied variables array size");
CHECK_SOUND(e.isForall(),
"universal instantiation: expr must be FORALL:\n"
+e.toString());
for(unsigned int i=0; i<terms.size(); i++)
CHECK_SOUND(d_theoryQuant->getBaseType(boundVars[i]) ==
d_theoryQuant->getBaseType(terms[i]),
"Universal instantiation: type mismatch");
}
//build up a conjunction of type predicates for expression
Expr tr = e.getEM()->trueExpr();
Expr typePred = tr;
unsigned qlevel=0, qlevelMax = 0;
for(unsigned int i=0; i<terms.size(); i++) {
Expr p = d_theoryQuant->getTypePred(boundVars[i].getType(),terms[i]);
if(p!=tr) {
if(typePred==tr)
typePred = p;
else
typePred = typePred.andExpr(p);
}
qlevel = d_theoryQuant->theoryCore()->getQuantLevelForTerm(terms[i]);
if (qlevel > qlevelMax) qlevel = qlevelMax;
}
Expr inst = e.getBody().substExpr(e.getVars(), terms);
// Expr inst = e.getBody().substExprQuant(e.getVars(), terms);
// Expr inst = e.getBody().substExpr(e.getVars(), terms);
Proof pf;
if(withProof()) {
vector<Proof> pfs;
vector<Expr> es;
pfs.push_back(t1.getProof());
es.push_back(e);
es.push_back(Expr(RAW_LIST,terms));
// es.insert(es.end(), terms.begin(), terms.end());
es.push_back(inst);
pf= newPf("universal_elimination3", es, pfs);
}
// Expr inst = e.getBody().substExpr(e.getVars(), terms);
Expr imp;
if( typePred == tr ) //just for easy life, yeting, change this asap
imp = inst;
else
imp = typePred.impExpr(inst);
Theorem ret = newTheorem(imp, t1.getAssumptionsRef(), pf);
unsigned thmLevel = t1.getQuantLevel();
if(qlevel >= thmLevel) {
ret.setQuantLevel(qlevel+1);
}
else{
// ret.setQuantLevel(thmLevel+1);
ret.setQuantLevel(thmLevel+1);
}
// ret.setQuantLevel(qlevel+1);
return ret;
}