diff --git a/src/nlsat/nlsat_explain.cpp b/src/nlsat/nlsat_explain.cpp
index 2278e53dd..420174506 100644
--- a/src/nlsat/nlsat_explain.cpp
+++ b/src/nlsat/nlsat_explain.cpp
@@ -574,7 +574,7 @@ namespace nlsat {
             if (is_const(p))
                 return;
             if (m_factor) {
-                TRACE("nlsat_explain", tout << "adding factors of\n"; display(tout, p); tout << "\n";);
+                TRACE("nlsat_explain", display(tout << "adding factors of\n", p); tout << "\n";);
                 factor(p, m_factors);
                 polynomial_ref f(m_pm);
                 for (unsigned i = 0; i < m_factors.size(); i++) {
@@ -1457,7 +1457,7 @@ namespace nlsat {
             process(num, ls);
             reset_already_added();
             m_result = nullptr;
-            TRACE("nlsat_explain", tout << "[explain] result\n"; display(tout, result););
+            TRACE("nlsat_explain", display(tout << "[explain] result\n", result););
             CASSERT("nlsat", check_already_added());
         }
 
@@ -1466,7 +1466,12 @@ namespace nlsat {
             
             m_result = &result;
             svector<literal> lits;
-            TRACE("nlsat", tout << "project x" << x << "\n"; m_solver.display(tout););
+            TRACE("nlsat", tout << "project x" << x << "\n"; 
+                  for (unsigned i = 0; i < num; ++i) {
+                      m_solver.display(tout, ls[i]) << " ";
+                  }
+                  tout << "\n";
+                  m_solver.display(tout););
                   
             DEBUG_CODE(
                 for (unsigned i = 0; i < num; ++i) {
@@ -1509,8 +1514,15 @@ namespace nlsat {
                 result.set(i, ~result[i]);
             }
             DEBUG_CODE(
-                for (unsigned i = 0; i < result.size(); ++i) {
-                    SASSERT(l_true == m_solver.value(result[i]));
+                TRACE("nlsat", 
+                      for (literal l : result) {
+                          m_solver.display(tout << " ", l);
+                      }
+                      tout << "\n";
+                      );
+                for (literal l : result) {
+                    CTRACE("nlsat", l_true != m_solver.value(l), m_solver.display(tout, l) << " " << m_solver.value(l) << "\n";);
+                    SASSERT(l_true == m_solver.value(l));
                 });
 
         }
@@ -1621,21 +1633,21 @@ namespace nlsat {
                 roots.reset();
                 m_am.isolate_roots(p, undef_var_assignment(m_assignment, x), roots);
                 bool glb_valid = false, lub_valid = false;
-                for (unsigned j = 0; j < roots.size(); ++j) {
-                    int s = m_am.compare(x_val, roots[j]);
+                for (auto const& r : roots) {
+                    int s = m_am.compare(x_val, r);
                     SASSERT(s != 0);
+
+                    if (s < 0 && (!lub_valid || m_am.lt(r, lub))) {
+                        lub_index = i;
+                        m_am.set(lub, r);
+                    }
+
+                    if (s > 0 && (!glb_valid || m_am.lt(glb, r))) {
+                        glb_index = i;
+                        m_am.set(glb, r);                        
+                    }
                     lub_valid |= s < 0;
                     glb_valid |= s > 0;
-
-                    if (s < 0 && m_am.lt(roots[j], lub)) {
-                        lub_index = i;
-                        m_am.set(lub, roots[j]);
-                    }
-
-                    if (s > 0 && m_am.lt(glb, roots[j])) {
-                        glb_index = i;
-                        m_am.set(glb, roots[j]);                        
-                    }
                 }
                 if (glb_valid) {
                     ++num_glb;
@@ -1701,6 +1713,7 @@ namespace nlsat {
         }
 
         void project_pairs(var x, unsigned idx, polynomial_ref_vector const& ps) {
+            TRACE("nlsat_explain", tout << "project pairs\n";);
             polynomial_ref p(m_pm);
             p = ps.get(idx);
             for (unsigned i = 0; i < ps.size(); ++i) {
diff --git a/src/nlsat/nlsat_scoped_literal_vector.h b/src/nlsat/nlsat_scoped_literal_vector.h
index 61760f06d..dffaa169f 100644
--- a/src/nlsat/nlsat_scoped_literal_vector.h
+++ b/src/nlsat/nlsat_scoped_literal_vector.h
@@ -34,9 +34,8 @@ namespace nlsat {
         bool empty() const { return m_lits.empty(); }
         literal operator[](unsigned i) const { return m_lits[i]; }
         void reset() {
-            unsigned sz = m_lits.size();
-            for (unsigned i = 0; i < sz; i++) {
-                m_solver.dec_ref(m_lits[i]);
+            for (literal l : m_lits) {
+                m_solver.dec_ref(l);
             }
             m_lits.reset();
         }
@@ -50,6 +49,8 @@ namespace nlsat {
             m_lits[i] = l;
         }
         literal const * c_ptr() const { return m_lits.c_ptr(); }
+        literal const * begin() const { return m_lits.begin(); }
+        literal const * end() const { return m_lits.end(); }
         void shrink(unsigned new_sz) {
             SASSERT(new_sz <= m_lits.size());
             unsigned sz = m_lits.size();
diff --git a/src/nlsat/nlsat_solver.cpp b/src/nlsat/nlsat_solver.cpp
index 9767448b1..f430f3a0c 100644
--- a/src/nlsat/nlsat_solver.cpp
+++ b/src/nlsat/nlsat_solver.cpp
@@ -270,7 +270,7 @@ namespace nlsat {
             TRACE("ref", tout << "inc: " << b << "\n";);
             if (b == null_bool_var)
                 return;
-            if (m_atoms[b] == 0)
+            if (m_atoms[b] == nullptr)
                 return;
             m_atoms[b]->inc_ref();
         }
@@ -395,7 +395,7 @@ namespace nlsat {
         bool_var mk_bool_var_core() {
             bool_var b = m_bid_gen.mk();
             m_num_bool_vars++;
-            m_atoms         .setx(b, 0, 0);
+            m_atoms         .setx(b, nullptr, nullptr);
             m_bvalues       .setx(b, l_undef, l_undef);
             m_levels        .setx(b, UINT_MAX, UINT_MAX);
             m_justifications.setx(b, null_justification, null_justification);
@@ -469,7 +469,7 @@ namespace nlsat {
             //SASSERT(m_bvalues[b] == l_undef);
             m_num_bool_vars--;
             m_dead[b]  = true;
-            m_atoms[b] = 0;
+            m_atoms[b] = nullptr;
             m_bid_gen.recycle(b);
         }
 
@@ -494,6 +494,7 @@ namespace nlsat {
         void del(atom * a) {
             if (a == nullptr)
                 return ;
+            TRACE("nlsat", display(tout << "del: b" << a->m_bool_var << " ", *a) << "\n";);
             if (a->is_ineq_atom())
                 del(to_ineq_atom(a));
             else
@@ -555,6 +556,7 @@ namespace nlsat {
                 bool_var b = mk_bool_var_core();
                 m_atoms[b] = atom;
                 atom->m_bool_var = b;
+                TRACE("nlsat", display(tout << "create: b" << atom->m_bool_var << " ", *atom) << "\n";);
                 return b;
             }
         }
@@ -749,7 +751,7 @@ namespace nlsat {
             m_levels[b]  = UINT_MAX;
             del_jst(m_allocator, m_justifications[b]);
             m_justifications[b] = null_justification;
-            if (m_atoms[b] == 0 && b < m_bk)
+            if (m_atoms[b] == nullptr && b < m_bk)
                 m_bk = b;
         }
 
@@ -895,7 +897,7 @@ namespace nlsat {
            \brief Assign literal using the given justification
          */
         void assign(literal l, justification j) {
-            TRACE("nlsat", tout << "assigning literal:\n"; display(tout, l); 
+            TRACE("nlsat", tout << "assigning literal: "; display(tout, l); 
                   tout << "\njustification kind: " << j.get_kind() << "\n";);
             SASSERT(assigned_value(l) == l_undef);
             SASSERT(j != null_justification);
@@ -926,6 +928,7 @@ namespace nlsat {
         */
         lbool value(literal l) {
             lbool val = assigned_value(l);
+            TRACE("nlsat_verbose", display(tout << " assigned value " << val << " for ", l) << "\n";);
             if (val != l_undef) {
                 return val;
             }
@@ -941,7 +944,7 @@ namespace nlsat {
             val = to_lbool(m_evaluator.eval(a, l.sign()));
             TRACE("value_bug", tout << "value of: "; display(tout, l); tout << " := " << val << "\n"; 
                   tout << "xk: " << m_xk << ", a->max_var(): " << a->max_var() << "\n";
-                  display_assignment(tout););
+                  display_assignment(tout););            
             return val;
         }
 
@@ -2011,9 +2014,9 @@ namespace nlsat {
         }
 
         bool can_reorder() const {
-            for (unsigned i = 0; i < m_atoms.size(); ++i) {
-                if (m_atoms[i]) {
-                    if (m_atoms[i]->is_root_atom()) return false;
+            for (atom * a : m_atoms) {
+                if (a) {
+                    if (a->is_root_atom()) return false;
                 }
             }
             return true;
@@ -2100,16 +2103,13 @@ namespace nlsat {
            \brief After variable reordering some lemmas containing root atoms may be ill-formed.
         */
         void del_ill_formed_lemmas() {
-            unsigned sz = m_learned.size();
             unsigned j  = 0;
-            for (unsigned i = 0; i < sz; i++) {
-                clause * c = m_learned[i];
+            for (clause* c : m_learned) {
                 if (ill_formed(*c)) {
                     del_clause(c);
                 }
                 else {
-                    m_learned[j] = c;
-                    j++;
+                    m_learned[j++] = c;
                 }
             }
             m_learned.shrink(j);
@@ -2119,9 +2119,8 @@ namespace nlsat {
             \brief Return true if the clause contains an ill formed root atom
         */
         bool ill_formed(clause const & c) {
-            unsigned sz = c.size();
-            for (unsigned i = 0; i < sz; i++) {
-                bool_var b = c[i].var();
+            for (literal lit : c) {
+                bool_var b = lit.var();
                 atom * a = m_atoms[b];
                 if (a == nullptr)
                     continue;
@@ -2139,19 +2138,16 @@ namespace nlsat {
         void reinit_cache() {
             reinit_cache(m_clauses);
             reinit_cache(m_learned);
-            for (unsigned i = 0; i < m_atoms.size(); ++i) {
-                reinit_cache(m_atoms[i]);
-            }
+            for (atom* a : m_atoms) 
+                reinit_cache(a);            
         }
         void reinit_cache(clause_vector const & cs) {
-            unsigned sz = cs.size();
-            for (unsigned i = 0; i < sz; i++)
-                reinit_cache(*(cs[i]));
+            for (clause* c : cs) 
+                reinit_cache(*c);
         }
         void reinit_cache(clause const & c) {
-            unsigned sz = c.size();
-            for (unsigned i = 0; i < sz; i++)
-                reinit_cache(c[i]);
+            for (literal l : c) 
+                reinit_cache(l);
         }
         void reinit_cache(literal l) {
             bool_var b = l.var();
@@ -2566,9 +2562,12 @@ namespace nlsat {
         std::ostream& display_bool_assignment(std::ostream & out) const {
             unsigned sz = m_atoms.size();
             for (bool_var b = 0; b < sz; b++) {
-                if (m_atoms[b] == 0 && m_bvalues[b] != l_undef) {
+                if (m_atoms[b] == nullptr && m_bvalues[b] != l_undef) {
                     out << "b" << b << " -> " << (m_bvalues[b] == l_true ? "true" : "false") << "\n";
                 }
+                else if (m_atoms[b] != nullptr && m_bvalues[b] != l_undef) {
+                    display(out << "b" << b << " ", *m_atoms[b]) << " -> " << (m_bvalues[b] == l_true ? "true" : "false") << "\n";
+                }
             }
             TRACE("nlsat_bool_assignment",
                   for (bool_var b = 0; b < sz; b++) {
@@ -3035,7 +3034,7 @@ namespace nlsat {
         std::ostream& display_smt2_bool_decls(std::ostream & out) const {
             unsigned sz = m_atoms.size();
             for (unsigned i = 0; i < sz; i++) {
-                if (m_atoms[i] == 0)
+                if (m_atoms[i] == nullptr)
                     out << "(declare-fun b" << i << " () Bool)\n";
             }
             return out;
@@ -3162,13 +3161,24 @@ namespace nlsat {
 
     void solver::get_bvalues(svector<lbool>& vs) {
         vs.reset();
-        vs.append(m_imp->m_bvalues);
+        unsigned sz = m_imp->m_bvalues.size();
+        for (bool_var b = 0; b < sz; ++b) {
+            if (m_imp->m_atoms[b] == nullptr) {
+                vs.push_back(m_imp->m_bvalues[b]);
+            }
+            else {
+                vs.push_back(l_undef); // don't save values from atoms.
+            }
+        }
+        TRACE("nlsat", display(tout););
     }
 
     void solver::set_bvalues(svector<lbool> const& vs) {
+        TRACE("nlsat", display(tout););
         m_imp->m_bvalues.reset();
         m_imp->m_bvalues.append(vs);
         m_imp->m_bvalues.resize(m_imp->m_atoms.size(), l_undef);        
+        TRACE("nlsat", display(tout););
     }
     
     var solver::mk_var(bool is_int) {
@@ -3199,27 +3209,28 @@ namespace nlsat {
         return m_imp->mk_clause(num_lits, lits, a);
     }
 
-    void solver::display(std::ostream & out) const {
-        m_imp->display(out);
+    std::ostream& solver::display(std::ostream & out) const {
+        return m_imp->display(out);
     }
 
-    void solver::display(std::ostream & out, literal l) const {
-        m_imp->display(out, l);
+    std::ostream& solver::display(std::ostream & out, literal l) const {
+        return m_imp->display(out, l);
     }
 
-    void solver::display(std::ostream & out, unsigned n, literal const* ls) const {
+    std::ostream& solver::display(std::ostream & out, unsigned n, literal const* ls) const {
         for (unsigned i = 0; i < n; ++i) {
             display(out, ls[i]);
             out << ";  ";
         }
+        return out;
     }
 
-    void solver::display(std::ostream & out, var x) const {
-        m_imp->m_display_var(out, x);
+    std::ostream& solver::display(std::ostream & out, var x) const {
+        return m_imp->m_display_var(out, x);
     }
 
-    void solver::display(std::ostream & out, atom const& a) const {
-        m_imp->display(out, a, m_imp->m_display_var);
+    std::ostream& solver::display(std::ostream & out, atom const& a) const {
+        return m_imp->display(out, a, m_imp->m_display_var);
     }
 
     display_var_proc const & solver::display_proc() const {
diff --git a/src/nlsat/nlsat_solver.h b/src/nlsat/nlsat_solver.h
index e7b250f64..4ba1225bd 100644
--- a/src/nlsat/nlsat_solver.h
+++ b/src/nlsat/nlsat_solver.h
@@ -225,21 +225,21 @@ namespace nlsat {
         /**
            \brief Display solver's state.
         */
-        void display(std::ostream & out) const;
+        std::ostream& display(std::ostream & out) const;
 
         /**
            \brief Display literal
         */
-        void display(std::ostream & out, literal l) const;
+        std::ostream& display(std::ostream & out, literal l) const;
 
-        void display(std::ostream & out, unsigned n, literal const* ls) const;
+        std::ostream& display(std::ostream & out, unsigned n, literal const* ls) const;
 
-        void display(std::ostream & out, atom const& a) const;
+        std::ostream& display(std::ostream & out, atom const& a) const;
 
         /**
            \brief Display variable
         */
-        void display(std::ostream & out, var x) const;
+        std::ostream& display(std::ostream & out, var x) const;
         
         display_var_proc const & display_proc() const;
     };
diff --git a/src/qe/nlqsat.cpp b/src/qe/nlqsat.cpp
index d8d6705fe..95f6b9b21 100644
--- a/src/qe/nlqsat.cpp
+++ b/src/qe/nlqsat.cpp
@@ -205,8 +205,7 @@ namespace qe {
             nlsat::scoped_literal_vector new_result(m_solver);
             result.reset();
             // project quantified Boolean variables.
-            for (unsigned i = 0; i < m_asms.size(); ++i) {
-                nlsat::literal lit = m_asms[i];
+            for (nlsat::literal lit : m_asms) {
                 if (!m_b2a.contains(lit.var()) || fvars.contains(lit.var())) {
                     result.push_back(lit);
                 }
@@ -215,12 +214,13 @@ namespace qe {
             // project quantified real variables.
             // They are sorted by size, so we project the largest variables first to avoid 
             // renaming variables. 
-            for (unsigned i = vars.size(); i > 0;) {
-                --i;
+            for (unsigned i = vars.size(); i-- > 0;) {
                 new_result.reset();
+                TRACE("qe", m_solver.display(tout << "project: ", vars[i]) << "\n";);
                 ex.project(vars[i], result.size(), result.c_ptr(), new_result);
                 result.swap(new_result);
-                TRACE("qe", m_solver.display(tout, result.size(), result.c_ptr()); tout << "\n";);
+                TRACE("qe", m_solver.display(tout, vars[i]) << ": ";
+                      m_solver.display(tout, result.size(), result.c_ptr()); tout << "\n";);
             }
             negate_clause(result);
         }
@@ -596,6 +596,7 @@ namespace qe {
         }
 
         void display(std::ostream& out) {
+            out << "level " << level() << "\n";
             display_preds(out);
             display_assumptions(out);
             m_solver.display(out << "solver:\n");
@@ -682,7 +683,7 @@ namespace qe {
                     }
                     else if (m_t2x.is_var(v)) {
                         nlsat::var w = m_t2x.to_var(v);
-                        TRACE("qe", tout << mk_pp(v, m) << " |-> " << w << "\n";);
+                        TRACE("qe", tout << mk_pp(v, m) << " |-> x" << w << "\n";);
                         m_bound_rvars.back().push_back(w);
                         m_rvar2level.setx(w, lvl, max_level());
                     }
@@ -724,13 +725,11 @@ namespace qe {
         }
 
         void init_var2expr() {
-            expr2var::iterator it = m_t2x.begin(), end = m_t2x.end();
-            for (; it != end; ++it) {
-                m_x2t.insert(it->m_value, it->m_key);
+            for (auto const& kv : m_t2x) {
+                m_x2t.insert(kv.m_value, kv.m_key);
             }
-            it = m_a2b.begin(), end = m_a2b.end();
-            for (; it != end; ++it) {
-                m_b2a.insert(it->m_value, it->m_key);
+            for (auto const& kv : m_a2b) {
+                m_b2a.insert(kv.m_value, kv.m_key);
             }
         }
 
@@ -741,10 +740,9 @@ namespace qe {
             bool ok = true;
             model_ref md = alloc(model, m);
             arith_util util(m);
-            expr2var::iterator it = m_t2x.begin(), end = m_t2x.end();
-            for (; it != end; ++it) {
-                nlsat::var x = it->m_value;
-                expr * t = it->m_key;
+            for (auto const& kv : m_t2x) {
+                nlsat::var x = kv.m_value;
+                expr * t = kv.m_key;
                 if (!is_uninterp_const(t) || !m_free_vars.contains(t) || m_aux_vars.contains(t))
                     continue;
                 expr * v;
@@ -760,10 +758,9 @@ namespace qe {
                 }
                 md->register_decl(to_app(t)->get_decl(), v);
             }
-            it = m_a2b.begin(), end = m_a2b.end();
-            for (; it != end; ++it) {
-                expr * a = it->m_key;
-                nlsat::bool_var b = it->m_value;
+            for (auto const& kv : m_a2b) {
+                expr * a = kv.m_key;
+                nlsat::bool_var b = kv.m_value;
                 if (a == nullptr || !is_uninterp_const(a) || b == m_is_true.var() || !m_free_vars.contains(a) || m_aux_vars.contains(a))
                     continue;
                 lbool val = m_bmodel0.get(b, l_undef);