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z3/lib/arith_simplifier_plugin.cpp
Leonardo de Moura e9eab22e5c Z3 sources 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
2012-10-02 11:35:25 -07:00

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/*++
Copyright (c) 2007 Microsoft Corporation
Module Name:
arith_simplifier_plugin.cpp
Abstract:
Simplifier for the arithmetic family.
Author:
Leonardo (leonardo) 2008-01-08
--*/
#include"arith_simplifier_plugin.h"
#include"ast_pp.h"
#include"ast_ll_pp.h"
#include"ast_smt2_pp.h"
arith_simplifier_plugin::~arith_simplifier_plugin() {
}
arith_simplifier_plugin::arith_simplifier_plugin(ast_manager & m, basic_simplifier_plugin & b, arith_simplifier_params & p):
poly_simplifier_plugin(symbol("arith"), m, OP_ADD, OP_MUL, OP_UMINUS, OP_SUB, OP_NUM),
m_params(p),
m_util(m),
m_bsimp(b),
m_int_zero(m),
m_real_zero(m) {
m_int_zero = m_util.mk_numeral(rational(0), true);
m_real_zero = m_util.mk_numeral(rational(0), false);
}
/**
\brief Return true if the first monomial of t is negative.
*/
bool arith_simplifier_plugin::is_neg_poly(expr * t) const {
if (m_util.is_add(t)) {
t = to_app(t)->get_arg(0);
}
if (m_util.is_mul(t)) {
t = to_app(t)->get_arg(0);
rational r;
bool is_int;
if (m_util.is_numeral(t, r, is_int))
return r.is_neg();
}
return false;
}
void arith_simplifier_plugin::get_monomial_gcd(expr_ref_vector& monomials, numeral& g) {
g = numeral::zero();
numeral n;
for (unsigned i = 0; !g.is_one() && i < monomials.size(); ++i) {
expr* e = monomials[i].get();
if (is_numeral(e, n)) {
g = gcd(abs(n), g);
}
else if (is_mul(e) && is_numeral(to_app(e)->get_arg(0), n)) {
g = gcd(abs(n), g);
}
else {
g = numeral::one();
return;
}
}
if (g.is_zero()) {
g = numeral::one();
}
}
void arith_simplifier_plugin::div_monomial(expr_ref_vector& monomials, numeral const& g) {
numeral n;
for (unsigned i = 0; i < monomials.size(); ++i) {
expr* e = monomials[i].get();
if (is_numeral(e, n)) {
SASSERT((n/g).is_int());
monomials[i] = mk_numeral(n/g);
}
else if (is_mul(e) && is_numeral(to_app(e)->get_arg(0), n)) {
SASSERT((n/g).is_int());
monomials[i] = mk_mul(n/g, to_app(e)->get_arg(1));
}
else {
UNREACHABLE();
}
}
}
void arith_simplifier_plugin::gcd_reduce_monomial(expr_ref_vector& monomials, numeral& k) {
numeral g, n;
get_monomial_gcd(monomials, g);
g = gcd(abs(k), g);
if (g.is_one()) {
return;
}
SASSERT(g.is_pos());
k = k / g;
div_monomial(monomials, g);
}
template<arith_simplifier_plugin::op_kind Kind>
void arith_simplifier_plugin::mk_le_ge_eq_core(expr * arg1, expr * arg2, expr_ref & result) {
set_curr_sort(arg1);
bool is_int = m_curr_sort->get_decl_kind() == INT_SORT;
expr_ref_vector monomials(m_manager);
rational k;
TRACE("arith_eq_bug", tout << mk_ismt2_pp(arg1, m_manager) << "\n" << mk_ismt2_pp(arg2, m_manager) << "\n";);
process_sum_of_monomials(false, arg1, monomials, k);
process_sum_of_monomials(true, arg2, monomials, k);
k.neg();
if (is_int) {
numeral g;
get_monomial_gcd(monomials, g);
if (!g.is_one()) {
div_monomial(monomials, g);
switch(Kind) {
case LE:
//
// g*monmials' <= k
// <=>
// monomials' <= floor(k/g)
//
k = floor(k/g);
break;
case GE:
//
// g*monmials' >= k
// <=>
// monomials' >= ceil(k/g)
//
k = ceil(k/g);
break;
case EQ:
k = k/g;
if (!k.is_int()) {
result = m_manager.mk_false();
return;
}
break;
}
}
}
expr_ref lhs(m_manager);
mk_sum_of_monomials(monomials, lhs);
if (m_util.is_numeral(lhs)) {
SASSERT(lhs == mk_zero());
if (( Kind == LE && numeral::zero() <= k) ||
( Kind == GE && numeral::zero() >= k) ||
( Kind == EQ && numeral::zero() == k))
result = m_manager.mk_true();
else
result = m_manager.mk_false();
}
else {
if (is_neg_poly(lhs)) {
expr_ref neg_lhs(m_manager);
mk_uminus(lhs, neg_lhs);
lhs = neg_lhs;
k.neg();
expr * rhs = m_util.mk_numeral(k, is_int);
switch (Kind) {
case LE:
result = m_util.mk_ge(lhs, rhs);
break;
case GE:
result = m_util.mk_le(lhs, rhs);
break;
case EQ:
result = m_manager.mk_eq(lhs, rhs);
break;
}
}
else {
expr * rhs = m_util.mk_numeral(k, is_int);
switch (Kind) {
case LE:
result = m_util.mk_le(lhs, rhs);
break;
case GE:
result = m_util.mk_ge(lhs, rhs);
break;
case EQ:
result = m_manager.mk_eq(lhs, rhs);
break;
}
}
}
}
void arith_simplifier_plugin::mk_arith_eq(expr * arg1, expr * arg2, expr_ref & result) {
mk_le_ge_eq_core<EQ>(arg1, arg2, result);
}
void arith_simplifier_plugin::mk_le(expr * arg1, expr * arg2, expr_ref & result) {
mk_le_ge_eq_core<LE>(arg1, arg2, result);
}
void arith_simplifier_plugin::mk_ge(expr * arg1, expr * arg2, expr_ref & result) {
mk_le_ge_eq_core<GE>(arg1, arg2, result);
}
void arith_simplifier_plugin::mk_lt(expr * arg1, expr * arg2, expr_ref & result) {
expr_ref tmp(m_manager);
mk_le(arg2, arg1, tmp);
m_bsimp.mk_not(tmp, result);
}
void arith_simplifier_plugin::mk_gt(expr * arg1, expr * arg2, expr_ref & result) {
expr_ref tmp(m_manager);
mk_le(arg1, arg2, tmp);
m_bsimp.mk_not(tmp, result);
}
void arith_simplifier_plugin::gcd_normalize(numeral & coeff, expr_ref& term) {
if (!abs(coeff).is_one()) {
set_curr_sort(term);
SASSERT(m_curr_sort->get_decl_kind() == INT_SORT);
expr_ref_vector monomials(m_manager);
rational k;
monomials.push_back(mk_numeral(numeral(coeff), true));
process_sum_of_monomials(false, term, monomials, k);
gcd_reduce_monomial(monomials, k);
numeral coeff1;
if (!is_numeral(monomials[0].get(), coeff1)) {
UNREACHABLE();
}
if (coeff1 == coeff) {
return;
}
monomials[0] = mk_numeral(k, true);
coeff = coeff1;
mk_sum_of_monomials(monomials, term);
}
}
void arith_simplifier_plugin::mk_div(expr * arg1, expr * arg2, expr_ref & result) {
set_curr_sort(arg1);
numeral v1, v2;
bool is_int;
if (m_util.is_numeral(arg2, v2, is_int) && !v2.is_zero()) {
SASSERT(!is_int);
if (m_util.is_numeral(arg1, v1, is_int))
result = m_util.mk_numeral(v1/v2, false);
else {
numeral k(1);
k /= v2;
expr_ref inv_arg2(m_util.mk_numeral(k, false), m_manager);
mk_mul(inv_arg2, arg1, result);
}
}
else
result = m_util.mk_div(arg1, arg2);
}
void arith_simplifier_plugin::mk_idiv(expr * arg1, expr * arg2, expr_ref & result) {
set_curr_sort(arg1);
numeral v1, v2;
bool is_int;
if (m_util.is_numeral(arg1, v1, is_int) && m_util.is_numeral(arg2, v2, is_int) && !v2.is_zero())
result = m_util.mk_numeral(div(v1, v2), is_int);
else
result = m_util.mk_idiv(arg1, arg2);
}
void arith_simplifier_plugin::prop_mod_const(expr * e, unsigned depth, numeral const& k, expr_ref& result) {
SASSERT(m_util.is_int(e));
SASSERT(k.is_int() && k.is_pos());
numeral n;
bool is_int;
if (depth == 0) {
result = e;
}
else if (is_add(e) || is_mul(e)) {
expr_ref_vector args(m_manager);
expr_ref tmp(m_manager);
app* a = to_app(e);
for (unsigned i = 0; i < a->get_num_args(); ++i) {
prop_mod_const(a->get_arg(i), depth - 1, k, tmp);
args.push_back(tmp);
}
reduce(a->get_decl(), args.size(), args.c_ptr(), result);
}
else if (m_util.is_numeral(e, n, is_int) && is_int) {
result = mk_numeral(mod(n, k), true);
}
else {
result = e;
}
}
void arith_simplifier_plugin::mk_mod(expr * arg1, expr * arg2, expr_ref & result) {
set_curr_sort(arg1);
numeral v1, v2;
bool is_int;
if (m_util.is_numeral(arg1, v1, is_int) && m_util.is_numeral(arg2, v2, is_int) && !v2.is_zero()) {
result = m_util.mk_numeral(mod(v1, v2), is_int);
}
else if (m_util.is_numeral(arg2, v2, is_int) && is_int && v2.is_one()) {
result = m_util.mk_numeral(numeral(0), true);
}
else if (m_util.is_numeral(arg2, v2, is_int) && is_int && v2.is_pos()) {
expr_ref tmp(m_manager);
prop_mod_const(arg1, 5, v2, tmp);
result = m_util.mk_mod(tmp, arg2);
}
else {
result = m_util.mk_mod(arg1, arg2);
}
}
void arith_simplifier_plugin::mk_rem(expr * arg1, expr * arg2, expr_ref & result) {
set_curr_sort(arg1);
numeral v1, v2;
bool is_int;
if (m_util.is_numeral(arg1, v1, is_int) && m_util.is_numeral(arg2, v2, is_int) && !v2.is_zero()) {
numeral m = mod(v1, v2);
//
// rem(v1,v2) = if v2 >= 0 then mod(v1,v2) else -mod(v1,v2)
//
if (v2.is_neg()) {
m.neg();
}
result = m_util.mk_numeral(m, is_int);
}
else if (m_util.is_numeral(arg2, v2, is_int) && is_int && v2.is_one()) {
result = m_util.mk_numeral(numeral(0), true);
}
else if (m_util.is_numeral(arg2, v2, is_int) && is_int && !v2.is_zero()) {
expr_ref tmp(m_manager);
prop_mod_const(arg1, 5, v2, tmp);
result = m_util.mk_mod(tmp, arg2);
if (v2.is_neg()) {
result = m_util.mk_uminus(result);
}
}
else {
result = m_util.mk_rem(arg1, arg2);
}
}
void arith_simplifier_plugin::mk_to_real(expr * arg, expr_ref & result) {
numeral v;
if (m_util.is_numeral(arg, v))
result = m_util.mk_numeral(v, false);
else
result = m_util.mk_to_real(arg);
}
void arith_simplifier_plugin::mk_to_int(expr * arg, expr_ref & result) {
numeral v;
if (m_util.is_numeral(arg, v))
result = m_util.mk_numeral(floor(v), true);
else if (m_util.is_to_real(arg))
result = to_app(arg)->get_arg(0);
else
result = m_util.mk_to_int(arg);
}
void arith_simplifier_plugin::mk_is_int(expr * arg, expr_ref & result) {
numeral v;
if (m_util.is_numeral(arg, v))
result = v.is_int()?m_manager.mk_true():m_manager.mk_false();
else if (m_util.is_to_real(arg))
result = m_manager.mk_true();
else
result = m_util.mk_is_int(arg);
}
bool arith_simplifier_plugin::reduce(func_decl * f, unsigned num_args, expr * const * args, expr_ref & result) {
set_reduce_invoked();
SASSERT(f->get_family_id() == m_fid);
TRACE("arith_simplifier_plugin", tout << mk_pp(f, m_manager) << "\n";
for (unsigned i = 0; i < num_args; i++) tout << mk_pp(args[i], m_manager) << "\n";);
arith_op_kind k = static_cast<arith_op_kind>(f->get_decl_kind());
switch (k) {
case OP_NUM: return false;
case OP_LE: if (m_presimp) return false; SASSERT(num_args == 2); mk_le(args[0], args[1], result); break;
case OP_GE: if (m_presimp) return false; SASSERT(num_args == 2); mk_ge(args[0], args[1], result); break;
case OP_LT: if (m_presimp) return false; SASSERT(num_args == 2); mk_lt(args[0], args[1], result); break;
case OP_GT: if (m_presimp) return false; SASSERT(num_args == 2); mk_gt(args[0], args[1], result); break;
case OP_ADD: mk_add(num_args, args, result); break;
case OP_SUB: mk_sub(num_args, args, result); break;
case OP_UMINUS: SASSERT(num_args == 1); mk_uminus(args[0], result); break;
case OP_MUL:
mk_mul(num_args, args, result);
TRACE("arith_simplifier_plugin", tout << mk_pp(result, m_manager) << "\n";);
break;
case OP_DIV: SASSERT(num_args == 2); mk_div(args[0], args[1], result); break;
case OP_IDIV: SASSERT(num_args == 2); mk_idiv(args[0], args[1], result); break;
case OP_REM: SASSERT(num_args == 2); mk_rem(args[0], args[1], result); break;
case OP_MOD: SASSERT(num_args == 2); mk_mod(args[0], args[1], result); break;
case OP_TO_REAL: SASSERT(num_args == 1); mk_to_real(args[0], result); break;
case OP_TO_INT: SASSERT(num_args == 1); mk_to_int(args[0], result); break;
case OP_IS_INT: SASSERT(num_args == 1); mk_is_int(args[0], result); break;
case OP_POWER: return false;
case OP_IRRATIONAL_ALGEBRAIC_NUM: return false;
default:
UNREACHABLE();
return false;
}
TRACE("arith_simplifier_plugin", tout << mk_pp(result.get(), m_manager) << "\n";);
return true;
}
bool arith_simplifier_plugin::is_arith_term(expr * n) const {
return n->get_kind() == AST_APP && to_app(n)->get_family_id() == m_fid;
}
bool arith_simplifier_plugin::reduce_eq(expr * lhs, expr * rhs, expr_ref & result) {
TRACE("reduce_eq_bug", tout << mk_ismt2_pp(lhs, m_manager) << "\n" << mk_ismt2_pp(rhs, m_manager) << "\n";);
set_reduce_invoked();
if (m_presimp) {
return false;
}
if (m_params.m_arith_expand_eqs) {
expr_ref le(m_manager), ge(m_manager);
mk_le_ge_eq_core<LE>(lhs, rhs, le);
mk_le_ge_eq_core<GE>(lhs, rhs, ge);
m_bsimp.mk_and(le, ge, result);
return true;
}
if (m_params.m_arith_process_all_eqs || is_arith_term(lhs) || is_arith_term(rhs)) {
mk_arith_eq(lhs, rhs, result);
return true;
}
return false;
}
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