Nlsat simplify (#7227)

* dev branch for simplification

Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>

* bug fixes

Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>

* bugfixes

Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>

* fix factorization

Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>

* separate out simplification functionality

* reorder initialization

Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>

* reorder initialization

Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>

* Update README.md

* initial warppers for seq-map/seq-fold

Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>

* expose fold as well

Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>

* add C++ bindings for sequence operations

Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>

* add abs function to API

Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>

* add parameter validation to ternary and 4-ary functions for API #7219

Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>

* add pre-processing and reorder

Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>

* add pre-processing and reorder

Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>

---------

Signed-off-by: Nikolaj Bjorner <nbjorner@microsoft.com>

src/math/lp/lp_settings.h

@@ -128,8 +128,12 @@ struct statistics {
     unsigned m_grobner_conflicts;
     unsigned m_offset_eqs;
     unsigned m_fixed_eqs;
+    ::statistics m_st;
     statistics() { reset(); }
-    void reset() { memset(this, 0, sizeof(*this)); }
+    void reset() {
+        memset(this, 0, sizeof(*this));
+        m_st.reset();
+    }
     void collect_statistics(::statistics& st) const {
         st.update("arith-factorizations", m_num_factorizations);
         st.update("arith-make-feasible", m_make_feasible);
@@ -157,7 +161,7 @@ struct statistics {
         st.update("arith-nla-lemmas", m_nla_lemmas);
         st.update("arith-nra-calls", m_nra_calls);   
         st.update("arith-bounds-improvements", m_nla_bounds_improvements);
-
+        st.copy(m_st);
     }
 };
 

src/math/lp/nra_solver.cpp

@@ -34,7 +34,7 @@ struct solver::imp {
     scoped_ptr<scoped_anum_vector>   m_values; // values provided by LRA solver
     scoped_ptr<scoped_anum> m_tmp1, m_tmp2;
     nla::core&                m_nla_core;
-
+    
     imp(lp::lar_solver& s, reslimit& lim, params_ref const& p, nla::core& nla_core): 
         lra(s), 
         m_limit(lim),
@@ -180,6 +180,7 @@ struct solver::imp {
         }
 
         lbool r = l_undef;
+        statistics& st = m_nla_core.lp_settings().stats().m_st;
         try {
             r = m_nlsat->check();
         }
@@ -188,9 +189,11 @@ struct solver::imp {
                 r = l_undef;
             }
             else {
+                m_nlsat->collect_statistics(st);
                 throw;
             }
         }
+        m_nlsat->collect_statistics(st);
         TRACE("nra",
               m_nlsat->display(tout << r << "\n");
               display(tout);
@@ -234,6 +237,7 @@ struct solver::imp {
         return r;
     }   
 
+
     void add_monic_eq_bound(mon_eq const& m) {
         if (!lra.column_has_lower_bound(m.var()) && 
             !lra.column_has_upper_bound(m.var()))
@@ -374,6 +378,7 @@ struct solver::imp {
         }
         
         lbool r = l_undef;
+        statistics& st = m_nla_core.lp_settings().stats().m_st;
         try {
             r = m_nlsat->check();
         }
@@ -382,9 +387,11 @@ struct solver::imp {
                 r = l_undef;
             }
             else {
+                m_nlsat->collect_statistics(st);
                 throw;
             }
         }
+        m_nlsat->collect_statistics(st);
         
         switch (r) {
         case l_true:
@@ -665,7 +672,7 @@ nlsat::anum_manager& solver::am() {
 scoped_anum& solver::tmp1() { return m_imp->tmp1(); }
 
 scoped_anum& solver::tmp2() { return m_imp->tmp2(); }
-
+ 
 
 void solver::updt_params(params_ref& p) {
     m_imp->updt_params(p);

src/math/polynomial/algebraic_numbers.cpp

@@ -2594,27 +2594,28 @@ namespace algebraic_numbers {
 
         void int_lt(numeral const & a, numeral & b) {
             scoped_mpz v(qm());
+            if (!a.is_basic())
+                refine_until_prec(const_cast<numeral&>(a), 1);
             if (a.is_basic()) {
                 qm().floor(basic_value(a), v);
                 qm().dec(v);
             }
-            else {
+            else                 
-                refine_until_prec(const_cast<numeral&>(a), 1);
+                bqm().floor(qm(), lower(a.to_algebraic()), v);            
-                bqm().floor(qm(), lower(a.to_algebraic()), v);
-            }
             m_wrapper.set(b, v);
         }
 
         void int_gt(numeral const & a, numeral & b) {
             scoped_mpz v(qm());
+            if (!a.is_basic()) 
+                refine_until_prec(const_cast<numeral&>(a), 1);
             if (a.is_basic()) {
                 qm().ceil(basic_value(a), v);
                 qm().inc(v);
             }
-            else {
+            else                
-                refine_until_prec(const_cast<numeral&>(a), 1);
                 bqm().ceil(qm(), upper(a.to_algebraic()), v);
-            }
+            
             m_wrapper.set(b, v);
         }
 

src/math/polynomial/polynomial.cpp

@@ -2504,30 +2504,139 @@ namespace polynomial {
             return p;
         }
 
-        void gcd_simplify(polynomial * p) {
+        void gcd_simplify(polynomial_ref& p, manager::ineq_type t) {
-            if (m_manager.finite()) return;
             auto& m = m_manager.m();
             unsigned sz = p->size();
             if (sz == 0) 
                 return;
             unsigned g = 0;
-            for (unsigned i = 0; i < sz; i++) {
+            for (unsigned i = 0; i < sz; i++) {                
                 if (!m.is_int(p->a(i))) {
+                    gcd_simplify_slow(p, t);
                     return;
                 }
+                if (t != EQ && is_unit(p->m(i)))
+                    continue;
                 int j = m.get_int(p->a(i));
-                if (j == INT_MIN || j == 1 || j == -1)
+                if (j == INT_MIN) {
+                    gcd_simplify_slow(p, t);
+                    return;
+                }
+                if (j == 1 || j == -1)
                     return;
                 g = u_gcd(abs(j), g);
                 if (g == 1) 
                     return;
             }
-            scoped_mpz r(m), gg(m);
+            scoped_mpz gg(m);
             m.set(gg, g);
-            for (unsigned i = 0; i < sz; ++i) {
+            apply_gcd_simplify(gg, p, t);            
-                m.div_gcd(p->a(i), gg, r);
+        }
-                m.set(p->a(i), r);
+
+        void apply_gcd_simplify(mpz & g, polynomial_ref& p, manager::ineq_type t) {
+           
+            auto& m = m_manager.m();
+
+#if 0
+            m.display(verbose_stream() << "gcd ", g);
+            p->display(verbose_stream() << "\n", m_manager, false);
+            char const* tt = "";
+            switch (t) {
+            case ineq_type::GT: tt = ">"; break;
+            case ineq_type::LT: tt = "<"; break;
+            case ineq_type::EQ: tt = "="; break;
             }
+            verbose_stream() << " " << tt << " 0\n ->\n";
+#endif
+            scoped_mpz r(m);
+            unsigned sz = p->size();
+            m_som_buffer.reset();
+            for (unsigned i = 0; i < sz; ++i) {
+                if (t != EQ && is_unit(p->m(i))) {
+                    scoped_mpz one(m);
+                    m.set(one, 1);
+                    if (t == GT) {
+                        // p - 2 - 1 >= 0
+                        // p div 2 + floor((-2 - 1 ) / 2) >= 0
+                        // p div 2 + floor(-3 / 2) >= 0
+                        // p div 2 - 2 >= 0
+                        // p div 2 - 1 > 0
+                        // 
+                        // p + k > 0
+                        // p + k - 1 >= 0
+                        // p div g + (k - 1) div g >= 0
+                        // p div g + (k - 1) div g + 1 > 0
+                        m.sub(p->a(i), one, r);
+                        bool is_neg = m.is_neg(r);
+                        if (is_neg) {
+                            m.neg(r);
+                            m.add(r, g, r);
+                            m.sub(r, one, r);
+                            m.div_gcd(r, g, r);
+                            m.neg(r);
+                        }
+                        else {
+                            m.div_gcd(r, g, r);
+                        }
+                        m.add(r, one, r);
+                    }
+                    else {
+                        // p + k < 0 
+                        // p + k + 1 <= 0
+                        // p div g + (k + 1 + g - 1) div g <= 0
+                        // p div g + (k + 1 + g - 1) div g - 1 < 0
+
+                        m.add(p->a(i), one, r);
+                        bool is_neg = m.is_neg(r);
+
+                        if (is_neg) {
+                            // p - k <= 0
+                            // p <= k
+                            // p div g <= k div g
+                            // p div g - k div g <= 0
+                            // p div g - k div g - 1 < 0
+                            m.neg(r);
+                            m.div_gcd(r, g, r);
+                            m.neg(r);
+                            m.sub(r, one, r);
+                        }
+                        else {
+                            m.div_gcd(p->a(i), g, r);
+                            m.add(p->a(i), g, r);
+                            m.div_gcd(r, g, r);
+                            m.sub(r, one, r);
+                        }
+
+                    }
+                }
+                else {
+                    m.div_gcd(p->a(i), g, r);                    
+                }
+                if (!m.is_zero(r))
+                    m_som_buffer.add(r, p->m(i));
+            }
+            p = m_som_buffer.mk();
+            
+            // p->display(verbose_stream(), m_manager, false);
+            // verbose_stream() << " " << tt << " 0\n";
+        }
+
+        void gcd_simplify_slow(polynomial_ref& p, manager::ineq_type t) {
+            auto& m = m_manager.m();
+            unsigned sz = p->size();
+            scoped_mpz g(m);
+            m.set(g, 0);
+            for (unsigned i = 0; i < sz; i++) {
+                auto const& a = p->a(i);
+                if (m.is_one(a) || m.is_minus_one(a))
+                    return;
+                if (t != EQ && is_unit(p->m(i)))
+                    continue;
+                m.gcd(a, g, g);
+                if (m.is_one(g))
+                    return;
+            }
+            apply_gcd_simplify(g, p, t);
         }
 
         polynomial * mk_zero() {
@@ -6087,9 +6196,11 @@ namespace polynomial {
                     }
                     return false;
                 }
-                if (!m_manager.ge(a1, a2))
+                if (m_manager.eq(a1, a2) || (m1->is_square() && m_manager.ge(a1, a2))) {
-                    return false;
+                    ++i, ++j;
-                ++i, ++j;               
+                    continue;
+                }
+                return false;               
             }
             return i == sz1 && j == sz2;           
         }
@@ -6971,8 +7082,8 @@ namespace polynomial {
         return m_imp->hash(p);
     }
 
-    void manager::gcd_simplify(polynomial * p) {
+    void manager::gcd_simplify(polynomial_ref& p, ineq_type t) {
-        m_imp->gcd_simplify(p);
+        m_imp->gcd_simplify(p, t);
     }
 
     polynomial * manager::coeff(polynomial const * p, var x, unsigned k) {

src/math/polynomial/polynomial.h

@@ -285,7 +285,8 @@ namespace polynomial {
         /**
            \brief Normalize coefficients by dividing by their gcd
         */
-        void gcd_simplify(polynomial* p);
+        enum ineq_type { EQ, LT, GT };
+        void gcd_simplify(polynomial_ref& p, ineq_type t);
 
         /**
            \brief Return true if \c m is the unit monomial.

src/math/polynomial/polynomial_cache.cpp

@@ -117,20 +117,14 @@ namespace polynomial {
         }
 
         void reset_psc_chain_cache() {
-            psc_chain_cache::iterator it  = m_psc_chain_cache.begin();
+            for (auto & k : m_psc_chain_cache)
-            psc_chain_cache::iterator end = m_psc_chain_cache.end();
+                del_psc_chain_entry(k);            
-            for (; it != end; ++it) {
-                del_psc_chain_entry(*it);
-            }
             m_psc_chain_cache.reset();
         }
 
         void reset_factor_cache() {
-            factor_cache::iterator it  = m_factor_cache.begin();
+            for (auto & e : m_factor_cache)
-            factor_cache::iterator end = m_factor_cache.end();
+                del_factor_entry(e);            
-            for (; it != end; ++it) {
-                del_factor_entry(*it);
-            }
             m_factor_cache.reset();
         }
 
@@ -139,7 +133,6 @@ namespace polynomial {
         polynomial * mk_unique(polynomial * p) {
             if (m_in_cache.get(pid(p), false))
                 return p;
-            // m.gcd_simplify(p);
             polynomial * p_prime = m_poly_table.insert_if_not_there(p);
             if (p == p_prime) {
                 m_cached_polys.push_back(p_prime); 

src/nlsat/CMakeLists.txt

@@ -4,6 +4,7 @@ z3_add_component(nlsat
     nlsat_evaluator.cpp
     nlsat_explain.cpp
     nlsat_interval_set.cpp
+    nlsat_simplify.cpp
     nlsat_solver.cpp
     nlsat_types.cpp
   COMPONENT_DEPENDENCIES

src/nlsat/nlsat_simplify.cpp

@@ -0,0 +1,824 @@
+#include "nlsat/nlsat_simplify.h"
+#include "nlsat/nlsat_solver.h"
+#include "nlsat/nlsat_scoped_literal_vector.h"
+#include "util/dependency.h"
+#include "util/map.h"
+
+namespace nlsat {
+
+    struct simplify::imp {
+
+        solver& s;
+        atom_vector&  m_atoms;
+        clause_vector& m_clauses, m_learned;
+        pmanager& m_pm;
+        literal_vector m_lemma;
+        vector<ptr_vector<clause>> m_var_occurs;
+
+
+        imp(solver& s, atom_vector& atoms, clause_vector& clauses, clause_vector& learned, pmanager& pm):
+            s(s), 
+            m_atoms(atoms), 
+            m_clauses(clauses), 
+            m_learned(learned),
+            m_pm(pm) {}
+
+        void operator()() {
+
+            // for now just remove all learned clauses.
+            // TODO; check if main clauses are subsumed by learned, 
+            // then promote learned to main.
+            for (auto c : m_learned)
+                s.del_clause(c);
+            m_learned.reset();
+            
+            IF_VERBOSE(3, s.display(verbose_stream() << "before\n"));
+            unsigned sz = m_clauses.size();
+            while (true) {
+
+                subsumption_simplify();
+
+                while (elim_uncnstr())
+                    ;
+
+                simplify_literals();
+
+                while (fm())
+                    ;
+
+                if (m_clauses.size() >= sz)
+                    break;
+
+                split_factors();
+
+                sz = m_clauses.size();
+            }
+
+            IF_VERBOSE(3, s.display(verbose_stream() << "after\n"));
+        }
+
+        //
+        // Apply gcd simplification to literals
+        // 
+        void simplify_literals() {
+            u_map<literal> b2l;
+            scoped_literal_vector lits(s);
+            polynomial_ref p(m_pm);
+            ptr_buffer<poly> ps;
+            buffer<bool> is_even;
+            unsigned num_atoms = m_atoms.size();
+            for (unsigned j = 0; j < num_atoms; ++j) {
+                atom* a1 = m_atoms[j];
+                if (a1 && a1->is_ineq_atom()) {
+                    ineq_atom const& a = *to_ineq_atom(a1);
+                    ps.reset();
+                    is_even.reset();
+                    for (unsigned i = 0; i < a.size(); ++i) {                        
+                        p = a.p(i);
+                        ps.push_back(p);
+                        is_even.push_back(a.is_even(i));
+                    }
+                    literal l = s.mk_ineq_literal(a.get_kind(), ps.size(), ps.data(), is_even.data(), true);
+                    if (l == null_literal)
+                        continue;
+                    lits.push_back(l);
+                    if (a.m_bool_var != l.var()) {
+                        IF_VERBOSE(3, s.display(verbose_stream() << "simplify ", a) << " -> ";
+                        s.display(verbose_stream(), l) << "\n");
+                        b2l.insert(a.m_bool_var, l);
+                    }
+                }
+            }
+            update_clauses(b2l);
+        }
+
+        void update_clauses(u_map<literal> const& b2l) {
+            bool is_sat = true;
+            literal_vector lits;
+            unsigned n = m_clauses.size();
+
+            for (unsigned i = 0; i < n; ++i) {
+                clause* c = m_clauses[i];
+                lits.reset();
+                bool changed = false;
+                bool is_tautology = false;
+                for (literal l : *c) {
+                    literal lit = null_literal;
+                    if (b2l.find(l.var(), lit)) {
+                        lit = l.sign() ? ~lit : lit;
+                        if (lit == true_literal) 
+                            is_tautology = true;                        
+                        else if (lit != false_literal) 
+                            lits.push_back(lit);                        
+                        changed = true;
+                    }
+                    else 
+                        lits.push_back(l);                    
+                }
+                if (changed) {
+                    c->set_removed();
+                    if (is_tautology) 
+                        continue;                    
+                    s.mk_clause(lits.size(), lits.data(), c->is_learned(), c->assumptions());                    
+                }
+            }
+            cleanup_removed();
+        }
+
+        //
+        // Replace unit literals p*q > 0 by clauses.
+        // 
+        void split_factors() {
+            auto sz = m_clauses.size();
+            for (unsigned i = 0; i < sz; ++i) {
+                auto& c = *m_clauses[i];
+                if (c.size() != 1)
+                    continue;
+                auto lit = c[0];
+                auto a1 = m_atoms[lit.var()];
+                if (!a1)
+                    continue;
+                auto& a = *to_ineq_atom(a1);
+                if (a.size() != 2)
+                    continue;
+
+                auto* p = a.p(0);
+                auto* q = a.p(1);
+                auto is_evenp = a.is_even(0);
+                auto is_evenq = a.is_even(1);
+
+                auto asum = c.assumptions();
+                literal lits[2];
+                clause* c1 = nullptr, * c2 = nullptr;
+
+                c.set_removed();
+                s.inc_simplify();
+                switch (a.get_kind()) {
+                case atom::EQ: {
+                    auto l1 = s.mk_ineq_literal(atom::EQ, 1, &p, &is_evenp, false);
+                    auto l2 = s.mk_ineq_literal(atom::EQ, 1, &q, &is_evenq, false);
+                    if (lit.sign()) {
+                        lits[0] = ~l1;
+                        c1 = s.mk_clause(1, lits, false, asum);
+                        lits[0] = ~l2;
+                        c2 = s.mk_clause(1, lits, false, asum);
+                    }
+                    else {
+                        lits[0] = l1;
+                        lits[1] = l2;
+                        c1 = s.mk_clause(2, lits, false, asum);
+                    }
+                    break;
+                }
+                case atom::LT: {
+                    auto pgt = s.mk_ineq_literal(atom::GT, 1, &p, &is_evenp, false);
+                    auto plt = s.mk_ineq_literal(atom::LT, 1, &p, &is_evenp, false);
+                    auto qgt = s.mk_ineq_literal(atom::GT, 1, &q, &is_evenq, false);
+                    auto qlt = s.mk_ineq_literal(atom::LT, 1, &q, &is_evenq, false);
+                    if (lit.sign()) {
+                        // p*q >= 0 <=> (p < 0 => q <= 0) & (q < 0 => p <= 0)
+                        // (!(p < 0) or !(q > 0)) & (!(q < 0) or !(p > 0))
+
+                        lits[0] = ~plt;
+                        lits[1] = ~qgt;
+                        c1 = s.mk_clause(2, lits, false, asum);
+                        lits[0] = ~qlt;
+                        lits[1] = ~pgt;
+                        c2 = s.mk_clause(2, lits, false, asum);
+                    }
+                    else {
+                        // p*q < 0
+                        // (p > 0 & q < 0) or (q > 0 & p < 0)
+                        // (p > 0 or q > 0) and (p < 0 or q < 0)
+
+                        lits[0] = pgt;
+                        lits[1] = qgt;
+                        c1 = s.mk_clause(2, lits, false, asum);
+                        lits[0] = plt;
+                        lits[1] = qlt;
+                        c2 = s.mk_clause(2, lits, false, asum);
+                    }
+                    break;
+                }
+                case atom::GT: {
+                    auto pgt = s.mk_ineq_literal(atom::GT, 1, &p, &is_evenp, false);
+                    auto plt = s.mk_ineq_literal(atom::LT, 1, &p, &is_evenp, false);
+                    auto qgt = s.mk_ineq_literal(atom::GT, 1, &q, &is_evenq, false);
+                    auto qlt = s.mk_ineq_literal(atom::LT, 1, &q, &is_evenq, false);
+                    if (lit.sign()) {
+                        // p*q <= 0
+                        // (p > 0 => q <= 0) & (p < 0 => q >= 0)
+                        // (!(p > 0) or !(q > 0)) & (!(p < 0) or !(q < 0))
+
+                        lits[0] = ~pgt;
+                        lits[1] = ~qgt;
+                        c1 = s.mk_clause(2, lits, false, asum);
+                        lits[0] = ~qlt;
+                        lits[1] = ~plt;
+                        c2 = s.mk_clause(2, lits, false, asum);
+                    }
+                    else {
+                        // p*q > 0
+                        // (p > 0 or q < 0) & (p < 0 or q > 0)
+                        lits[0] = plt;
+                        lits[1] = qgt;
+                        c1 = s.mk_clause(2, lits, false, asum);
+                        lits[0] = qlt;
+                        lits[1] = pgt;
+                        c2 = s.mk_clause(2, lits, false, asum);
+                    }
+                    break;
+                }
+                }
+                IF_VERBOSE(3, 
+                s.display(verbose_stream(), c) << " ->\n";
+                if (c1) s.display(verbose_stream(), *c1) << "\n";
+                if (c2) s.display(verbose_stream(), *c2) << "\n");
+            }
+            cleanup_removed();
+        }
+
+        bool elim_uncnstr() {
+            // compute variable occurrences
+            if (any_of(m_clauses, [&](clause* c) { return s.has_root_atom(*c); }))
+                return false;
+            compute_occurs();
+            // for each variable occurrence, figure out if it is unconstrained.
+            bool has_removed = false;
+            for (unsigned v = m_var_occurs.size(); v-- > 0; ) {
+                auto& clauses = m_var_occurs[v];
+                if (clauses.size() != 1)
+                    continue;
+                auto& c = *clauses[0];
+                if (c.is_removed())
+                    continue;
+                if (!is_unconstrained(v, c))
+                    continue;
+                s.inc_simplify();
+                c.set_removed();
+                has_removed = true;
+            }
+            cleanup_removed();
+
+            return has_removed;
+        }
+
+        bool is_unconstrained(var x, clause& c) {
+            poly* p;
+            polynomial_ref A(m_pm), B(m_pm);
+            for (auto lit : c) {
+                bool_var b = lit.var();
+                if (!m_atoms[b])
+                    continue;
+                auto& a = *to_ineq_atom(m_atoms[b]);
+                if (!is_single_poly(a, p))
+                    continue;
+
+                if (1 != m_pm.degree(p, x))
+                    continue;
+
+                A = m_pm.coeff(p, x, 1, B);
+
+                if (a.is_eq() && !lit.sign()) {
+                    // A*x + B = 0
+                    if (s.is_int(x) && is_unit(A)) {
+                        s.add_bound(bound_constraint(x, A, B, false, nullptr));
+                        return true;
+                    }
+
+                    if (!s.is_int(x) && m_pm.is_const(A)) {
+                        s.add_bound(bound_constraint(x, A, B, false, nullptr));
+                        return true;
+                    }
+                }
+                // TODO: add other cases for LT and GT atoms
+            }
+            return false;
+        }
+
+        void compute_occurs() {
+            m_var_occurs.reset();
+            for (auto c : m_clauses)
+                compute_occurs(*c);
+        }
+
+        void compute_occurs(clause& c) {
+            var_vector vars;
+            m_pm.begin_vars_incremental();
+            for (auto lit : c) {
+                bool_var b = lit.var();
+                atom* a = m_atoms[b];
+                if (!a)
+                    continue;
+                if (a->is_ineq_atom()) {
+                    auto sz = to_ineq_atom(a)->size();
+                    for (unsigned i = 0; i < sz; ++i) {
+                        auto* p = to_ineq_atom(a)->p(i);
+                        m_pm.vars_incremental(p, vars);
+                    }
+                }
+            }
+            m_pm.end_vars_incremental(vars);
+            unsigned h = 0;
+            for (auto v : vars) {
+                m_var_occurs.reserve(v + 1);
+                m_var_occurs[v].push_back(&c);
+                h |= (1ul << (v % 32ul));
+            }
+            c.set_var_hash(h);
+        }
+
+        bool cleanup_removed() {
+            unsigned j = 0, sz = m_clauses.size();
+            for (unsigned i = 0; i < sz; ++i) {
+                auto c = m_clauses[i];
+                if (c->is_removed())
+                    s.del_clause(c);
+                else
+                    m_clauses[j++] = c;
+            }
+            m_clauses.shrink(j);
+            return j < sz;
+        }
+
+        bool unit_subsumption_simplify(clause& src, clause& c) {
+            if (src.size() != 1)
+                return false;
+            auto u = src[0];
+            for (auto lit : c) {
+                if (subsumes(u, ~lit)) {
+                    
+                    literal_vector lits;
+                    for (auto lit2 : c)
+                        if (lit2 != lit)
+                            lits.push_back(lit2);
+                    if (lits.empty())
+                        return false;
+                    auto a = s.join(c.assumptions(), src.assumptions());
+                    auto cls = s.mk_clause(lits.size(), lits.data(), false, a);
+                    if (cls)
+                        compute_occurs(*cls);
+                    return true;
+                }
+            }
+            return false;
+        }
+
+        //
+        // Subsumption simplification
+        // 
+        // Remove D if C subsumes D
+        //                
+        // Unit subsumption resolution
+        // u is a unit literal (lit or C) is a clause
+        // u => ~lit, then simplify (lit or C) to C
+        // 
+        void subsumption_simplify() {
+            compute_occurs();
+            for (unsigned v = m_var_occurs.size(); v-- > 0; ) {
+                auto& clauses = m_var_occurs[v];
+                unsigned sz = clauses.size();
+                for (unsigned i = 0; i < sz; ++i) {
+                    auto c = clauses[i];
+                    if (c->is_marked() || c->is_removed())
+                        continue;
+                    c->mark();
+                    for (unsigned j = 0; j < sz; ++j) {
+                        auto c2 = clauses[j];
+                        if (c == c2 || c2->is_removed())
+                            continue;
+                        if (subsumes(*c, *c2) || unit_subsumption_simplify(*c, *c2)) {
+                            IF_VERBOSE(3, s.display(verbose_stream() << "subsumes ", *c);
+                            s.display(verbose_stream() << " ", *c2) << "\n");
+                            s.inc_simplify();
+                            c2->set_removed();
+                        }
+                    }
+                }
+            }
+            for (auto c : m_clauses)
+                c->unmark();
+
+            cleanup_removed();
+        }
+
+        // does c1 subsume c2?
+        bool subsumes(clause const& c1, clause const& c2) {
+            if (c1.size() > c2.size())
+                return false;
+            if ((c1.var_hash() & c2.var_hash()) != c1.var_hash())
+                return false;
+            for (auto lit1 : c1) {
+                if (!any_of(c2, [&](auto lit2) { return subsumes(lit1, lit2); }))
+                    return false;
+            }
+            return true;
+        }
+
+        bool subsumes(literal lit1, literal lit2) {
+            if (lit1 == lit2)
+                return true;
+
+            atom* a1 = m_atoms[lit1.var()];
+            atom* a2 = m_atoms[lit2.var()];
+            if (!a1 || !a2)
+                return false;
+
+            // use m_pm.ge(p1, p2) 
+            // whenever lit1 = p1 < 0,    lit2 = p2 < 0
+            // or       lit1 = p1 < 0,    lit2 = !(p2 > 0)
+            // or       lit1 = !(p1 > 0), lit2 = !(p2 > 0)
+            // use m_pm.ge(p2, p1)
+            // whenever lit1 = p1 > 0,    lit2 = p2 > 0
+            // or       lit1 = !(p1 < 0), lit2 = !(p2 < 0)
+            // or       lit1 = p1 > 0,    lit2 = !(p2 < 0)
+            // or       lit1 = !(p1 > 0), lit2 = p2 < 0
+            //
+            if (a1->is_ineq_atom() && a2->is_ineq_atom()) {
+                auto& i1 = *to_ineq_atom(a1);
+                auto& i2 = *to_ineq_atom(a2);
+                auto is_lt1 = !lit1.sign() && a1->get_kind() == atom::kind::LT;
+                auto is_le1 = lit1.sign() && a1->get_kind() == atom::kind::GT;
+                auto is_gt1 = !lit1.sign() && a1->get_kind() == atom::kind::GT;
+                auto is_ge1 = lit1.sign() && a1->get_kind() == atom::kind::LT;
+
+                auto is_lt2 = !lit2.sign() && a2->get_kind() == atom::kind::LT;
+                auto is_le2 = lit2.sign() && a2->get_kind() == atom::kind::GT;
+                auto is_gt2 = !lit2.sign() && a2->get_kind() == atom::kind::GT;
+                auto is_ge2 = lit2.sign() && a2->get_kind() == atom::kind::LT;
+
+                auto check_ge = (is_lt1 && (is_lt2 || is_le2)) || (is_le1 && is_le2);
+                auto check_le = (is_gt1 && (is_gt2 || is_ge2)) || (is_ge1 && is_ge2);
+
+                if (i1.size() != i2.size())
+                    ;
+                else if (check_ge) {
+                    for (unsigned i = 0; i < i1.size(); ++i)
+                        if (!m_pm.ge(i1.p(i), i2.p(i)))
+                            return false;
+                    return true;
+                }
+                else if (check_le) {
+                    for (unsigned i = 0; i < i1.size(); ++i)
+                        if (!m_pm.ge(i2.p(i), i1.p(i)))
+                            return false;
+                    return true;
+                }
+            }
+            return false;
+        }
+
+        //
+// Fourier Motzkin elimination
+//
+
+        bool fm() {
+            if (any_of(m_clauses, [&](clause* c) { return s.has_root_atom(*c); }))
+                return false;
+            compute_occurs();
+
+            for (unsigned v = m_var_occurs.size(); v-- > 0; )
+                apply_fm(v, m_var_occurs[v]);
+
+            return cleanup_removed();
+        }
+
+        // progression of possible features:
+        // . Current: unit literals
+        // . Either lower or upper bound is unit coefficient
+        // . single occurrence of x in C
+        // . (x <= t or x <= s or C) == (x <= max(s, t) or C)
+        //
+
+        bool is_invertible(var x, polynomial_ref& A) {
+            if (!m_pm.is_const(A))
+                return false;
+            if (s.is_int(x) && !is_unit(A))
+                return false;
+            return true;
+        }
+
+        bool apply_fm(var x, ptr_vector<clause>& clauses) {
+            polynomial_ref A(m_pm), B(m_pm);
+            vector<bound_constraint> lo, hi;
+            poly* p = nullptr;
+            bool all_solved = true;
+            for (auto c : clauses) {
+                if (c->is_removed())
+                    continue;
+                if (c->size() != 1) {
+                    all_solved = false;
+                    continue;
+                }
+                literal lit = (*c)[0];
+                bool sign = lit.sign();
+                ineq_atom const& a = *to_ineq_atom(m_atoms[lit.var()]);
+                if (sign && a.is_eq()) {
+                    all_solved = false;
+                    continue;
+                }
+                if (!is_single_poly(a, p)) {
+                    all_solved = false;
+                    continue;
+                }
+                if (1 != m_pm.degree(p, x)) {
+                    all_solved = false;
+                    continue;
+                }
+                A = m_pm.coeff(p, x, 1, B);
+                if (!is_invertible(x, A)) {
+                    all_solved = false;
+                    continue;
+                }
+                auto const& A_value = m_pm.coeff(A, 0);
+                bool is_pos = m_pm.m().is_pos(A_value);
+                bool is_strict = false;
+                switch (a.get_kind()) {
+                case atom::LT:
+                    //  !(Ax + B < 0) == Ax + B >= 0
+                    if (sign)
+                        is_strict = false;
+                    else {
+                        // Ax + B < 0 == -Ax - B > 0
+                        A = -A;
+                        B = -B;
+                        is_pos = !is_pos;
+                        if (s.is_int(x)) {
+                            // Ax + B > 0 == Ax + B - |A| >= 0
+                            if (is_pos)
+                                B = m_pm.sub(B, A);
+                            else
+                                B = m_pm.add(B, A);
+                            is_strict = false;
+                        }
+                        else
+                            is_strict = true;
+                    }
+                    break;
+                case atom::GT:
+                    //  !(Ax + B > 0) == -Ax + -B >= 0
+                    if (sign) {
+                        A = -A;
+                        B = -B;
+                        is_pos = !is_pos;
+                        is_strict = false;
+                    }
+                    else {
+                        // Ax + B > 0
+                        if (s.is_int(x)) {
+                            // Ax + B - |A| >= 0
+                            if (is_pos)
+                                B = m_pm.sub(B, A);
+                            else
+                                B = m_pm.add(B, A);
+                            is_strict = false;
+                        }
+                        else
+                            is_strict = true;
+                    }
+                    break;
+                case atom::EQ: {
+                    all_solved = false;
+                    if (sign)
+                        continue;
+                    bound_constraint l(x, A, B, false, c);
+                    A = -A;
+                    B = -B;
+                    bound_constraint h(x, A, B, false, c);
+                    apply_fm_equality(x, clauses, l, h);
+                    return true;
+                }
+                default:
+                    UNREACHABLE();
+                    break;
+                }
+                auto& set = is_pos ? hi : lo;
+                bool found = false;
+                for (auto const& bound : set) {
+                    if (is_strict == bound.is_strict && m_pm.eq(A, bound.A) && m_pm.eq(B, bound.B))
+                        found = true;
+                }
+                if (found)
+                    continue;
+
+                set.push_back(bound_constraint(x, A, B, is_strict, c));
+
+            }
+
+            if (lo.empty() && hi.empty())
+                return false;
+
+            if (apply_fm_equality(x, clauses, lo, hi))
+                return true;
+
+
+            if (!all_solved)
+                return false;
+
+            IF_VERBOSE(3,
+                verbose_stream() << "x" << x << " lo " << lo.size() << " hi " << hi.size() << "\n";
+            for (auto c : clauses)
+                if (!c->is_removed())
+                    s.display(verbose_stream(), *c) << "\n";
+            );
+
+            auto num_lo = lo.size(), num_hi = hi.size();
+            if (num_lo >= 2 && num_hi >= 2 && (num_lo > 2 || num_hi > 2))
+                return false;
+
+            apply_fm_inequality(x, clauses, lo, hi);
+
+            return true;
+        }
+
+        void apply_fm_inequality(
+            var x, ptr_vector<clause>& clauses,
+            vector<bound_constraint>& lo, vector<bound_constraint>& hi) {
+
+            polynomial_ref C(m_pm), D(m_pm);
+            for (auto c : clauses)
+                c->set_removed();
+
+            for (auto const& l : lo) {
+                // l.A * x + l.B, l.is_strict;, l.A < 0
+                for (auto const& h : hi) {
+                    // h.A * x + h.B, h.is_strict; h.A > 0
+                    // (l.A x + l.B)*h.A + (h.A x + h.B)*|l.A| >= 0
+                    C = m_pm.mul(l.B, h.A);
+                    D = m_pm.mul(h.B, l.A);
+                    C = m_pm.sub(C, D);
+                    poly* p = C.get();
+                    bool is_even = false;
+                    m_lemma.reset();
+                    if (l.is_strict || h.is_strict)
+                        m_lemma.push_back(s.mk_ineq_literal(atom::GT, 1, &p, &is_even, true));
+                    else
+                        m_lemma.push_back(~s.mk_ineq_literal(atom::LT, 1, &p, &is_even, true));
+                    if (m_lemma[0] == true_literal)
+                        continue;
+                    auto a = s.join(l.c->assumptions(), h.c->assumptions());
+                    auto cls = s.mk_clause(m_lemma.size(), m_lemma.data(), false, a);
+                    if (cls)
+                        compute_occurs(*cls);
+                    IF_VERBOSE(3, s.display(verbose_stream() << "add resolvent ", *cls) << "\n");
+                }
+            }
+
+            // track updates for model reconstruction
+            for (auto const& l : lo)
+                s.add_bound(l);
+            for (auto const& h : hi)
+                s.add_bound(h);
+        }
+
+        literal substitute_var(var x, poly* p, poly* q, literal lit) {
+            auto b = lit.var();
+            auto a = m_atoms[b];
+            if (!a)
+                return lit;
+            SASSERT(a->is_ineq_atom());
+            auto& a1 = *to_ineq_atom(a);
+            auto r = substitute_var(x, p, q, a1);
+            if (r == null_literal)
+                r = lit;
+            else if (lit.sign())
+                r.neg();
+            return r;
+        }
+
+        literal substitute_var(var x, poly* p, poly* q, ineq_atom const& a) {
+            unsigned sz = a.size();
+            bool_vector even;
+            polynomial_ref pr(m_pm), qq(q, m_pm);
+            qq = -qq;
+            polynomial_ref_vector ps(m_pm);
+            bool change = false;
+            auto k = a.get_kind();
+            for (unsigned i = 0; i < sz; ++i) {
+                poly* po = a.p(i);
+                m_pm.substitute(po, x, qq, p, pr);
+                change |= pr != po;
+                TRACE("nlsat", tout << pr << "\n";);
+                if (m_pm.is_zero(pr)) {
+                    ps.reset();
+                    even.reset();
+                    ps.push_back(pr);
+                    even.push_back(false);
+                    break;
+                }
+                if (m_pm.is_const(pr)) {
+                    if (!a.is_even(i) && m_pm.m().is_neg(m_pm.coeff(pr, 0)))
+                        k = atom::flip(k);
+                    continue;
+                }
+                ps.push_back(pr);
+                even.push_back(a.is_even(i));
+            }
+            if (!change)
+                return null_literal;
+            return s.mk_ineq_literal(k, ps.size(), ps.data(), even.data(), true);
+        }
+
+        bool apply_fm_equality(
+            var x, ptr_vector<clause>& clauses,
+            vector<bound_constraint>& lo, vector<bound_constraint>& hi) {
+            for (auto& l : lo) {
+                if (l.is_strict)
+                    continue;
+                l.A = -l.A;
+                l.B = -l.B;
+                for (auto& h : hi) {
+                    if (h.is_strict)
+                        continue;
+                    if (!m_pm.eq(l.B, h.B))
+                        continue;
+                    if (!m_pm.eq(l.A, h.A))
+                        continue;
+                    l.A = -l.A;
+                    l.B = -l.B;
+                    apply_fm_equality(x, clauses, l, h);
+                    s.inc_simplify();
+                    return true;
+                }
+                l.A = -l.A;
+                l.B = -l.B;
+            }
+            return false;
+        }
+
+        void apply_fm_equality(
+            var x, ptr_vector<clause>& clauses,
+            bound_constraint& l, bound_constraint& h) {
+            auto a1 = s.join(l.c->assumptions(), h.c->assumptions());
+            s.inc_ref(a1);
+
+            polynomial_ref A(l.A), B(l.B);
+
+            if (m_pm.is_neg(l.A)) {
+                A = -A;
+                B = -B;
+            }
+
+            for (auto c : clauses) {
+                if (c->is_removed())
+                    continue;
+                c->set_removed();
+                if (c == l.c || c == h.c)
+                    continue;
+                m_lemma.reset();
+                bool is_tautology = false;
+                for (literal lit : *c) {
+                    lit = substitute_var(x, A, B, lit);
+                    m_lemma.push_back(lit);
+                    if (lit == true_literal)
+                        is_tautology = true;
+                }
+                if (is_tautology)
+                    continue;
+                auto a = s.join(c->assumptions(), a1);
+                auto cls = s.mk_clause(m_lemma.size(), m_lemma.data(), false, a);
+
+                IF_VERBOSE(3,
+                    if (cls) {
+                        s.display_proc()(verbose_stream(), x) << " * " << l.A << " = " << l.B << "\n";
+                        s.display(verbose_stream(), *c) << " -> ";
+                        s.display(verbose_stream(), *cls) << " - ";
+                        s.display(verbose_stream(), *l.c) << " ";
+                        s.display(verbose_stream(), *h.c) << "\n";
+                    });
+                if (cls)
+                    compute_occurs(*cls);
+            }
+            s.dec_ref(a1);
+            // track updates for model reconstruction
+            s.add_bound(l);
+            s.add_bound(h);
+            s.inc_simplify();
+        }
+
+        bool is_single_poly(ineq_atom const& a, poly*& p) {
+            unsigned sz = a.size();
+            return sz == 1 && a.is_odd(0) && (p = a.p(0), true);
+        }
+
+        bool is_unit(polynomial_ref const& p) {
+            if (!m_pm.is_const(p))
+                return false;
+            auto const& c = m_pm.coeff(p, 0);
+            return m_pm.m().is_one(c) || m_pm.m().is_minus_one(c);
+        }
+    };
+
+    simplify::simplify(solver& s, atom_vector& atoms, clause_vector& clauses, clause_vector& learned, pmanager& pm) {
+        m_imp = alloc(imp, s, atoms, clauses, learned, pm);
+    }
+
+    simplify::~simplify() {
+        dealloc(m_imp);
+    }
+
+    void simplify::operator()() {
+        (*m_imp)();
+    }
+
+};

src/nlsat/nlsat_simplify.h

@@ -0,0 +1,16 @@
+#pragma once
+
+#include "nlsat/nlsat_types.h"
+#include "nlsat/nlsat_clause.h"
+
+namespace nlsat {
+    class simplify {
+        struct imp;
+        imp * m_imp;
+    public:
+        simplify(solver& s, atom_vector& atoms, clause_vector& clauses, clause_vector& learned, pmanager & pm);
+        ~simplify();
+
+        void operator()();
+    };
+}

src/nlsat/nlsat_solver.h

@@ -24,6 +24,7 @@ Revision History:
 #include "util/params.h"
 #include "util/statistics.h"
 #include "util/rlimit.h"
+#include "util/dependency.h"
 
 namespace nlsat {
 
@@ -36,6 +37,15 @@ namespace nlsat {
         virtual std::ostream& operator()(std::ostream& out, assumption a) const = 0;
     };
 
+    struct bound_constraint {
+        var x;
+        polynomial_ref A, B;
+        bool is_strict;
+        clause* c;
+        bound_constraint(var x, polynomial_ref& A, polynomial_ref& B, bool is_strict, clause* c) :
+            x(x), A(A), B(B), is_strict(is_strict), c(c) {}
+    };
+
     class solver {
         struct imp;
         struct ctx;
@@ -103,7 +113,7 @@ namespace nlsat {
            e[i] = 1 if is_even[i] is false
            e[i] = 2 if is_even[i] is true
         */
-        literal mk_ineq_literal(atom::kind k, unsigned sz, poly * const * ps, bool const * is_even);
+        literal mk_ineq_literal(atom::kind k, unsigned sz, poly * const * ps, bool const * is_even, bool simplify = false);
 
         /**
            \brief Create an atom of the form: x=root[i](p), x<root[i](p), x>root[i](p)
@@ -114,6 +124,9 @@ namespace nlsat {
         void inc_ref(literal l) { inc_ref(l.var()); }
         void dec_ref(bool_var b);
         void dec_ref(literal l) { dec_ref(l.var()); }
+        void inc_ref(assumption a);
+        void dec_ref(assumption a);
+        
         
         /**
            \brief Create a new clause.
@@ -172,6 +185,17 @@ namespace nlsat {
         void get_bvalues(svector<bool_var> const& bvars, svector<lbool>& vs);
         void set_bvalues(svector<lbool> const& vs);
 
+        /**
+        * \brief Access functions for simplify module.
+        */
+        void del_clause(clause* c);
+        clause* mk_clause(unsigned n, literal const* lits, bool learned, internal_assumption a);
+        bool has_root_atom(clause const& c) const;
+        assumption join(assumption a, assumption b);
+
+        void inc_simplify();
+        void add_bound(bound_constraint const& c);
+
         /**
            \brief reorder variables. 
          */
@@ -244,6 +268,8 @@ namespace nlsat {
 
         std::ostream& display(std::ostream & out, unsigned n, literal const* ls) const;
 
+        std::ostream& display(std::ostream& out, clause const& c) const;
+
         std::ostream& display(std::ostream & out, literal_vector const& ls) const;
 
         std::ostream& display(std::ostream & out, atom const& a) const;
@@ -254,9 +280,10 @@ namespace nlsat {
 
         std::ostream& display_smt2(std::ostream & out, literal_vector const& ls) const;
 
-                std::ostream& display_smt2(std::ostream & out) const;
 
 
+        std::ostream& display_smt2(std::ostream & out) const;
+
         /**
            \brief Display variable
         */

src/nlsat/nlsat_types.h

@@ -32,6 +32,7 @@ namespace nlsat {
 #define NLSAT_VB_LVL 10
     typedef void *                     assumption;
     typedef void *                     assumption_set;
+    typedef void *                     internal_assumption;
 
     typedef sat::bool_var              bool_var;
     typedef sat::bool_var_vector       bool_var_vector;
@@ -74,6 +75,7 @@ namespace nlsat {
         }
     protected:
         friend class solver;
+        friend class simplify;
         kind     m_kind;
         unsigned m_ref_count;
         bool_var m_bool_var;

src/util/small_object_allocator.cpp

@@ -28,6 +28,7 @@ Revision History:
 # include <iostream>
 #endif
 
+
 small_object_allocator::small_object_allocator(char const * id) {
     for (unsigned i = 0; i < NUM_SLOTS; i++) {
         m_chunks[i] = nullptr;
@@ -98,7 +99,9 @@ void small_object_allocator::deallocate(size_t size, void * p) {
 
 
 void * small_object_allocator::allocate(size_t size) {
-    if (size == 0) return nullptr;
+    if (size == 0) 
+        return nullptr;
+
 
 
 #if defined(Z3DEBUG) && !defined(_WINDOWS)
@@ -109,6 +112,8 @@ void * small_object_allocator::allocate(size_t size) {
     if (size >= SMALL_OBJ_SIZE - (1 << PTR_ALIGNMENT)) {
         return memory::allocate(size);
     }
+
+
 #ifdef Z3DEBUG
     size_t osize = size;
 #endif