diff --git a/src/sat/sat_aig_cuts.cpp b/src/sat/sat_aig_cuts.cpp
index 556264524..fba07cc28 100644
--- a/src/sat/sat_aig_cuts.cpp
+++ b/src/sat/sat_aig_cuts.cpp
@@ -24,10 +24,13 @@ namespace sat {
         m_config.m_max_cut_size = std::min(cut().max_cut_size, m_config.m_max_cut_size);
         m_cut_set1.init(m_region, m_config.m_max_cutset_size + 1);
         m_cut_set2.init(m_region, m_config.m_max_cutset_size + 1);
+        m_true = null_bool_var;
     }
 
     vector<cut_set> const& aig_cuts::get_cuts() {
+        flush_roots();
         unsigned_vector node_ids = filter_valid_nodes();
+        TRACE("aig_simplifier", display(tout););
         augment(node_ids);
         TRACE("aig_simplifier", display(tout););
         return m_cuts;
@@ -38,15 +41,17 @@ namespace sat {
             cut_set& cs = m_cuts[id];
             node const& n = m_aig[id];
             SASSERT(n.is_valid());
+            // cs.display(std::cout << "augment " << id << "\nbefore\n");
             augment(n, cs);
             for (node const& n2 : m_aux_aig[id]) {
                 augment(n2, cs);
             }
+            // cs.display(std::cout << "after\n");
         }
     }
 
     void aig_cuts::augment(node const& n, cut_set& cs) {
-        unsigned nc = n.num_children();
+        unsigned nc = n.is_var() ? 0 : n.num_children();
         if (n.is_var()) {
             SASSERT(!n.sign());
         }
@@ -68,6 +73,7 @@ namespace sat {
     }
 
     bool aig_cuts::insert_cut(cut const& c, cut_set& cs) {
+        SASSERT(c.m_size > 0);
         while (cs.size() >= m_config.m_max_cutset_size) {
             // never evict the first entry, it is used for the starting point
             unsigned idx = 1 + (m_rand() % (cs.size() - 1));
@@ -110,15 +116,16 @@ namespace sat {
 
     void aig_cuts::augment_aig0(node const& n, cut_set& cs) {
         SASSERT(n.is_and());
-        cut c;          
+        SASSERT(m_true != null_bool_var);
+        cut c(m_true);          
         cs.reset();
         if (n.sign()) {
-            c.m_table = 0;   // constant false
+            c.m_table = 0x0; // constant false
         }
         else {
             c.m_table = 0x3; // constant true
         }
-        cs.insert(c);
+        cs.push_back(c);
     }
 
     void aig_cuts::augment_aig1(node const& n, cut_set& cs) {
@@ -126,7 +133,7 @@ namespace sat {
         literal lit = child(n, 0);
         unsigned round = 0;
         for (auto const& a : m_cuts[lit.var()]) {
-            cut c;
+            cut c(lit.var());
             c.set_table(a.m_table);
             if (n.sign()) c.negate();
             if (insert_cut(c, cs) && ++round >= m_config.m_max_insertions) 
@@ -219,28 +226,106 @@ namespace sat {
         unsigned offset = m_literals.size();
         node n(head.sign(), op, sz, offset);
         m_literals.append(sz, args);
+        if (op == and_op || op == xor_op) {
+            std::sort(m_literals.c_ptr() + offset, m_literals.c_ptr() + offset + sz);
+        }
         for (unsigned i = 0; i < sz; ++i) {
             if (!m_aig[args[i].var()].is_valid()) {
                 add_var(args[i].var());
             }
         }
-        if (!m_aig[v].is_valid() || m_aig[v].is_var() || (sz == 0)) {
+        if (n.is_const() && m_true == null_bool_var) {
+            m_true = v;
+        }
+        if (!m_aig[v].is_valid() || m_aig[v].is_var() || n.is_const()) {
             m_aig[v] = n;
             init_cut_set(v);
+            if (n.is_const()) {
+                augment_aig0(n, m_cuts[v]);
+            }
         }
-        else if (eq(n, m_aig[v]) || !insert_aux(v, n)) {
+        else if (m_aig[v].is_const() || eq(n, m_aig[v]) || !insert_aux(v, n)) {
             m_literals.shrink(m_literals.size() - sz);
             TRACE("aig_simplifier", tout << "duplicate\n";);
         }
+        for (auto const& c : m_cuts[v]) SASSERT(c.m_size > 0); 
         SASSERT(m_aig[v].is_valid());
     }
 
+    void aig_cuts::set_root(bool_var v, literal r) {
+        IF_VERBOSE(2, verbose_stream() << "set-root " << v << " -> " << r << "\n");
+        m_roots.push_back(std::make_pair(v, r));
+    }
+
+    void aig_cuts::flush_roots() {
+        if (m_roots.empty()) return;
+        literal_vector to_root;
+        for (unsigned i = 0; i < m_aig.size(); ++i) {
+            to_root.push_back(literal(i, false));
+        }
+        for (unsigned i = m_roots.size(); i-- > 0; ) {
+            bool_var v = m_roots[i].first;
+            literal  r = m_roots[i].second;
+            literal rr = to_root[r.var()];
+            to_root[v] = r.sign() ? ~rr : rr;
+        }
+        for (unsigned i = 0; i < m_aig.size(); ++i) {
+            node& n = m_aig[i];
+            // invalidate nodes that have been rooted
+            if (to_root[i] != literal(i, false)) {
+                m_aux_aig[i].reset();
+                m_aig[i] = node();
+                m_cuts[i].reset();
+            }
+            else if (n.is_valid()) {
+                flush_roots(to_root, n);
+                for (node & n2 : m_aux_aig[i]) {
+                    flush_roots(to_root, n2);
+                }
+            }
+        }
+        for (cut_set& cs : m_cuts) {
+            flush_roots(to_root, cs);
+        }
+        m_roots.reset();
+        TRACE("aig_simplifier", display(tout););
+    }
+
+    void aig_cuts::flush_roots(literal_vector const& to_root, node& n) {
+        bool changed = false;
+        for (unsigned i = 0; i < n.num_children(); ++i) {
+            literal& lit = m_literals[n.offset() + i];
+            if (to_root[lit.var()] != lit) {
+                changed = true;
+                lit = lit.sign() ? ~to_root[lit.var()] : to_root[lit.var()];
+            }
+        }
+        if (changed) {
+            std::sort(m_literals.c_ptr() + n.offset(), m_literals.c_ptr() + n.offset() + n.num_children());
+        }
+    }
+
+    void aig_cuts::flush_roots(literal_vector const& to_root, cut_set& cs) {
+        unsigned j = 0;        
+        for (cut& c : cs) {
+            bool has_stale = false;
+            for (unsigned v : c) {
+                has_stale |= (to_root[v] != literal(v, false));
+            }
+            if (!has_stale) {
+                cs[j++] = c;
+            }
+        }
+        cs.shrink(j);
+    }
+
     void aig_cuts::init_cut_set(unsigned id) {
         node const& n = m_aig[id];
         SASSERT(n.is_valid());
         auto& cut_set = m_cuts[id];
         cut_set.init(m_region, m_config.m_max_cutset_size + 1);
-        cut_set.push_back(cut(id));  // TBD: if entry is a constant?
+        cut_set.reset();
+        cut_set.push_back(cut(id));  
         m_aux_aig[id].reset();        
     }
 
@@ -281,8 +366,9 @@ namespace sat {
             out << id << " == ";
             display(out, m_aig[id]) << "\n";
             for (auto const& n : m_aux_aig[id]) {
-                display(out << " ", n) << "\n";
+                display(out << "   ", n) << "\n";
             }
+            m_cuts[id].display(out);
         }
         return out;
     }
@@ -290,10 +376,10 @@ namespace sat {
     std::ostream& aig_cuts::display(std::ostream& out, node const& n) const {
         if (n.sign()) out << "! ";
         switch (n.op()) {
-        case var_op: out << "var "; break;
-        case and_op: out << "and "; break;
-        case xor_op: out << "xor "; break;
-        case ite_op: out << "ite "; break;
+        case var_op: out << "var "; return out;
+        case and_op: out << "& "; break;
+        case xor_op: out << "^ "; break;
+        case ite_op: out << "? "; break;
         default: break;
         }
         for (unsigned i = 0; i < n.num_children(); ++i) {
diff --git a/src/sat/sat_aig_cuts.h b/src/sat/sat_aig_cuts.h
index 3f6bf49d9..41eb0f436 100644
--- a/src/sat/sat_aig_cuts.h
+++ b/src/sat/sat_aig_cuts.h
@@ -31,6 +31,16 @@ namespace sat {
         no_op
     };
 
+    inline std::ostream& operator<<(std::ostream& out, bool_op op) {
+        switch (op) {
+        case var_op: return out << "v";
+        case and_op: return out << "&";
+        case ite_op: return out << "?";
+        case xor_op: return out << "^";
+        default: return out << "";
+        }
+    }
+
     class aig_cuts {
 
         struct config {
@@ -50,14 +60,15 @@ namespace sat {
         public:
             node(): m_sign(false), m_op(no_op), m_num_children(UINT_MAX), m_offset(UINT_MAX) {}
             explicit node(unsigned v): m_sign(false), m_op(var_op), m_num_children(UINT_MAX), m_offset(v) {}
-            explicit node(bool negate, bool_op op, unsigned num_children, unsigned offset): 
-                m_sign(negate), m_op(op), m_num_children(num_children), m_offset(offset) {}
+            explicit node(bool negate, bool_op op, unsigned nc, unsigned o): 
+                m_sign(negate), m_op(op), m_num_children(nc), m_offset(o) {}
             bool is_valid() const { return m_offset != UINT_MAX; }
             bool_op op() const { return m_op; }
             bool is_var() const { return m_op == var_op; }
             bool is_and() const { return m_op == and_op; }
             bool is_xor() const { return m_op == xor_op; }
             bool is_ite() const { return m_op == ite_op; }
+            bool is_const() const { return is_and() && num_children() == 0; }
             unsigned var() const { SASSERT(is_var()); return m_offset; }
             bool sign() const { return m_sign; }
             unsigned num_children() const { SASSERT(!is_var()); return m_num_children; }
@@ -71,6 +82,8 @@ namespace sat {
         region                m_region;
         cut_set               m_cut_set1, m_cut_set2;
         vector<cut_set>       m_cuts;
+        bool_var              m_true;
+        svector<std::pair<bool_var, literal>> m_roots;
 
         void reserve(unsigned v);
         bool insert_aux(unsigned v, node const& n);
@@ -89,6 +102,10 @@ namespace sat {
 
         bool insert_cut(cut const& c, cut_set& cs);
 
+        void flush_roots();
+        void flush_roots(literal_vector const& to_root, node& n);
+        void flush_roots(literal_vector const& to_root, cut_set& cs);
+
         std::ostream& display(std::ostream& out, node const& n) const;
 
         literal child(node const& n, unsigned idx) const { SASSERT(!n.is_var()); SASSERT(idx < n.num_children()); return m_literals[n.offset() + idx]; }
@@ -97,6 +114,7 @@ namespace sat {
         aig_cuts();
         void add_var(unsigned v);
         void add_node(literal head, bool_op op, unsigned sz, literal const* args);
+        void set_root(bool_var v, literal r);
 
         vector<cut_set> const & get_cuts();
 
diff --git a/src/sat/sat_aig_simplifier.cpp b/src/sat/sat_aig_simplifier.cpp
index 296e47d15..6044624f8 100644
--- a/src/sat/sat_aig_simplifier.cpp
+++ b/src/sat/sat_aig_simplifier.cpp
@@ -76,6 +76,10 @@ namespace sat {
         m_stats.m_num_xors++;
     }
 
+    void aig_simplifier::set_root(bool_var v, literal r) {
+        m_aig_cuts.set_root(v, r);
+    }
+
     void aig_simplifier::operator()() {
         report _report(*this);
         TRACE("aig_simplifier", s.display(tout););
@@ -159,6 +163,7 @@ namespace sat {
             for (auto& cut : cuts[i]) {
                 unsigned j = 0;
                 if (cut2id.find(&cut, j)) {
+                    if (i == j) std::cout << "dup: " << i << "\n";
                     VERIFY(i != j);
                     literal v(i, false);
                     literal w(j, false);
diff --git a/src/sat/sat_aig_simplifier.h b/src/sat/sat_aig_simplifier.h
index 58962941d..5d88033f7 100644
--- a/src/sat/sat_aig_simplifier.h
+++ b/src/sat/sat_aig_simplifier.h
@@ -51,6 +51,7 @@ namespace sat {
         void add_xor(literal head, unsigned sz, literal const* args);
         void add_ite(literal head, literal c, literal t, literal e);
         void add_iff(literal head, literal l1, literal l2);
+        void set_root(bool_var v, literal r);
     };
 }
 
diff --git a/src/sat/sat_cutset.cpp b/src/sat/sat_cutset.cpp
index e23c2b866..227638ee0 100644
--- a/src/sat/sat_cutset.cpp
+++ b/src/sat/sat_cutset.cpp
@@ -33,6 +33,7 @@ namespace sat {
     */
     
     bool cut_set::insert(cut const& c) {
+        SASSERT(c.m_size > 0);
         unsigned i = 0, j = 0;
         for (; i < size(); ++i) {
             cut const& a = (*this)[i];
@@ -61,6 +62,14 @@ namespace sat {
         }
         return true;
     }
+
+    std::ostream& cut_set::display(std::ostream& out) const {
+        for (auto const& cut : *this) {
+            cut.display(out) << "\n";
+        }
+        return out;
+    }
+
     
     /**
        \brief shift table 'a' by adding elements from 'c'.
@@ -116,10 +125,12 @@ namespace sat {
     }
     
     std::ostream& cut::display(std::ostream& out) const {
+        out << "{";
         for (unsigned i = 0; i < m_size; ++i) {
-            out << (*this)[i] << " ";
-        }
-        out << "t: ";
+            out << (*this)[i];
+            if (i + 1 < m_size) out << " ";
+        }        
+        out << "} t: ";
         for (unsigned i = 0; i < (1u << m_size); ++i) {
             if (0 != (m_table & (1ull << i))) out << "1"; else out << "0";
         }    
diff --git a/src/sat/sat_cutset.h b/src/sat/sat_cutset.h
index 6da26a7d8..361e0c889 100644
--- a/src/sat/sat_cutset.h
+++ b/src/sat/sat_cutset.h
@@ -111,9 +111,11 @@ namespace sat {
     public:
         cut_set(): m_region(nullptr), m_size(0), m_max_size(0), m_cuts(nullptr) {}
         void init(region& r, unsigned sz) { 
+            m_max_size = sz;
+            SASSERT(!m_region || m_cuts);
+            if (m_region) return;
             m_region = &r; 
             m_cuts = new (r) cut[sz]; 
-            m_max_size = sz; 
         }
         bool insert(cut const& c);
         bool no_duplicates() const;
@@ -121,7 +123,8 @@ namespace sat {
         cut * begin() const { return m_cuts; }
         cut * end() const { return m_cuts + m_size; }
         cut & back() { return m_cuts[m_size-1]; }
-        void push_back(cut const& c) { 
+        void push_back(cut const& c) {
+            SASSERT(c.m_size > 0);
             if (m_size == m_max_size) {
                 m_max_size *= 2;
                 cut* new_cuts = new (*m_region) cut[m_max_size]; 
@@ -135,6 +138,7 @@ namespace sat {
         void shrink(unsigned j) { m_size = j; }
         void swap(cut_set& other) { std::swap(m_size, other.m_size); std::swap(m_cuts, other.m_cuts); std::swap(m_max_size, other.m_max_size); }
         void evict(unsigned idx) {  m_cuts[idx] = m_cuts[--m_size]; }
+        std::ostream& display(std::ostream& out) const;
     };
 
 }
diff --git a/src/sat/sat_elim_eqs.cpp b/src/sat/sat_elim_eqs.cpp
index 908304fb5..06968ec69 100644
--- a/src/sat/sat_elim_eqs.cpp
+++ b/src/sat/sat_elim_eqs.cpp
@@ -230,6 +230,7 @@ namespace sat {
             literal  l(v, false);
             literal r  = roots[v];
             SASSERT(v != r.var());
+            if (m_solver.m_aig_simplifier) m_solver.m_aig_simplifier->set_root(v, r);
             bool set_root = m_solver.set_root(l, r);
             bool root_ok = !m_solver.is_external(v) || set_root;
             if (m_solver.is_assumption(v) || (m_solver.is_external(v) && (m_solver.is_incremental() || !root_ok))) {