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/******************************************************************************
* Top contributors (to current version):
* Gereon Kremer
*
* This file is part of the cvc5 project.
*
* Copyright (c) 2009-2021 by the authors listed in the file AUTHORS
* in the top-level source directory and their institutional affiliations.
* All rights reserved. See the file COPYING in the top-level source
* directory for licensing information.
* ****************************************************************************
*
* Implementation of CAD proof generator.
*/
#include "theory/arith/nl/cad/proof_generator.h"
#ifdef CVC5_POLY_IMP
#include "proof/lazy_tree_proof_generator.h"
#include "theory/arith/nl/poly_conversion.h"
namespace cvc5 {
namespace theory {
namespace arith {
namespace nl {
namespace cad {
namespace {
/**
* Retrieves the root indices of the sign-invariant region of v.
*
* We assume that roots holds a sorted list of roots from one polynomial.
* If v is equal to one of these roots, we return (id,id) where id is the index
* of this root within roots. Otherwise, we return the id of the largest root
* below v and the id of the smallest root above v. To make sure a smaller root
* and a larger root always exist, we implicitly extend the roots by -infty and
* infty.
*
* ATTENTION: if we return id, the corresponding root is:
* - id = 0: -infty
* - 0 < id <= roots.size(): roots[id-1]
* - id = roots.size() + 1: infty
*/
std::pair<std::size_t, std::size_t> getRootIDs(
const std::vector<poly::Value>& roots, const poly::Value& v)
{
for (std::size_t i = 0; i < roots.size(); ++i)
{
if (roots[i] == v)
{
return {i + 1, i + 1};
}
if (roots[i] > v)
{
return {i, i + 1};
}
}
return {roots.size(), roots.size() + 1};
}
/**
* Constructs an IndexedRootExpression:
* var ~rel~ root_k(poly)
* where root_k(poly) is "the k'th root of the polynomial".
*
* @param var The variable that is bounded
* @param rel The relation for this constraint
* @param zero A node representing Rational(0)
* @param k The index of the root (starting with 1)
* @param poly The polynomial whose root shall be considered
* @param vm A variable mapper from cvc5 to libpoly variables
*/
Node mkIRP(const Node& var,
Kind rel,
const Node& zero,
std::size_t k,
const poly::Polynomial& poly,
VariableMapper& vm)
{
auto* nm = NodeManager::currentNM();
auto op = nm->mkConst<IndexedRootPredicate>(IndexedRootPredicate(k));
return nm->mkNode(Kind::INDEXED_ROOT_PREDICATE,
op,
nm->mkNode(rel, var, zero),
as_cvc_polynomial(poly, vm));
}
} // namespace
CADProofGenerator::CADProofGenerator(context::Context* ctx,
ProofNodeManager* pnm)
: d_pnm(pnm), d_proofs(pnm, ctx), d_current(nullptr)
{
d_false = NodeManager::currentNM()->mkConst<bool>(false);
d_zero = NodeManager::currentNM()->mkConst<Rational>(0);
}
void CADProofGenerator::startNewProof()
{
d_current = d_proofs.allocateProof();
}
void CADProofGenerator::startRecursive() { d_current->openChild(); }
void CADProofGenerator::endRecursive()
{
d_current->setCurrent(PfRule::ARITH_NL_CAD_RECURSIVE, {}, {d_false}, d_false);
d_current->closeChild();
}
void CADProofGenerator::startScope()
{
d_current->openChild();
d_current->getCurrent().d_rule = PfRule::SCOPE;
}
void CADProofGenerator::endScope(const std::vector<Node>& args)
{
d_current->setCurrent(PfRule::SCOPE, {}, args, d_false);
d_current->closeChild();
}
ProofGenerator* CADProofGenerator::getProofGenerator() const
{
return d_current;
}
void CADProofGenerator::addDirect(Node var,
VariableMapper& vm,
const poly::Polynomial& poly,
const poly::Assignment& a,
poly::SignCondition& sc,
const poly::Interval& interval,
Node constraint)
{
if (is_minus_infinity(get_lower(interval))
&& is_plus_infinity(get_upper(interval)))
{
// "Full conflict", constraint excludes (-inf,inf)
d_current->openChild();
d_current->setCurrent(
PfRule::ARITH_NL_CAD_DIRECT, {constraint}, {d_false}, d_false);
d_current->closeChild();
return;
}
std::vector<Node> res;
auto roots = poly::isolate_real_roots(poly, a);
if (get_lower(interval) == get_upper(interval))
{
// Excludes a single point only
auto ids = getRootIDs(roots, get_lower(interval));
Assert(ids.first == ids.second);
res.emplace_back(mkIRP(var, Kind::EQUAL, d_zero, ids.first, poly, vm));
}
else
{
// Excludes an open interval
if (!is_minus_infinity(get_lower(interval)))
{
// Interval has lower bound that is not -inf
auto ids = getRootIDs(roots, get_lower(interval));
Assert(ids.first == ids.second);
Kind rel = poly::get_lower_open(interval) ? Kind::GT : Kind::GEQ;
res.emplace_back(mkIRP(var, rel, d_zero, ids.first, poly, vm));
}
if (!is_plus_infinity(get_upper(interval)))
{
// Interval has upper bound that is not inf
auto ids = getRootIDs(roots, get_upper(interval));
Assert(ids.first == ids.second);
Kind rel = poly::get_upper_open(interval) ? Kind::LT : Kind::LEQ;
res.emplace_back(mkIRP(var, rel, d_zero, ids.first, poly, vm));
}
}
// Add to proof manager
startScope();
d_current->openChild();
d_current->setCurrent(
PfRule::ARITH_NL_CAD_DIRECT, {constraint}, {d_false}, d_false);
d_current->closeChild();
endScope(res);
}
std::vector<Node> CADProofGenerator::constructCell(Node var,
const CACInterval& i,
const poly::Assignment& a,
const poly::Value& s,
VariableMapper& vm)
{
if (is_minus_infinity(get_lower(i.d_interval))
&& is_plus_infinity(get_upper(i.d_interval)))
{
// "Full conflict", constraint excludes (-inf,inf)
return {};
}
std::vector<Node> res;
// Just use bounds for all polynomials
for (const auto& poly : i.d_mainPolys)
{
auto roots = poly::isolate_real_roots(poly, a);
auto ids = getRootIDs(roots, s);
if (ids.first == ids.second)
{
// Excludes a single point only
res.emplace_back(mkIRP(var, Kind::EQUAL, d_zero, ids.first, poly, vm));
}
else
{
// Excludes an open interval
if (ids.first > 0)
{
// Interval has lower bound that is not -inf
res.emplace_back(mkIRP(var, Kind::GT, d_zero, ids.first, poly, vm));
}
if (ids.second <= roots.size())
{
// Interval has upper bound that is not inf
res.emplace_back(mkIRP(var, Kind::LT, d_zero, ids.first, poly, vm));
}
}
}
return res;
}
std::ostream& operator<<(std::ostream& os, const CADProofGenerator& proof)
{
return os << *proof.d_current;
}
} // namespace cad
} // namespace nl
} // namespace arith
} // namespace theory
} // namespace cvc5
#endif
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