diff --git a/src/util/lp/emonomials.cpp b/src/util/lp/emonomials.cpp
index 90735d23e..4d4c6e1f3 100644
--- a/src/util/lp/emonomials.cpp
+++ b/src/util/lp/emonomials.cpp
@@ -280,7 +280,6 @@ void emonomials::do_canonize(monomial & m) const {
 
 bool emonomials::is_canonized(const monomial & m) const {
     monomial mm(m);
-    do_canonize(mm);
     return mm.rvars() == m.rvars();
 }
 
diff --git a/src/util/lp/monomial.h b/src/util/lp/monomial.h
index 9ce13de8a..ead38b4f6 100644
--- a/src/util/lp/monomial.h
+++ b/src/util/lp/monomial.h
@@ -61,8 +61,7 @@ public:
     unsigned visited() const { return m_visited; }
     unsigned& visited() { return m_visited; }
     svector<lpvar> const& rvars() const { return m_rvars; }
-    bool sign() const { return m_rsign; }
-    rational rsign() const { return rational(m_rsign ? -1 : 1); }
+    bool rsign() const { return m_rsign; }
     void reset_rfields() { m_rsign = false; m_rvars.reset(); SASSERT(m_rvars.size() == 0); }
     void push_rvar(signed_var sv) { m_rsign ^= sv.sign(); m_rvars.push_back(sv.var()); }
     void sort_rvars() {
diff --git a/src/util/lp/nla_basics_lemmas.cpp b/src/util/lp/nla_basics_lemmas.cpp
index 1a3d9fda8..57b5badf6 100644
--- a/src/util/lp/nla_basics_lemmas.cpp
+++ b/src/util/lp/nla_basics_lemmas.cpp
@@ -27,7 +27,7 @@ basics::basics(core * c) : common(c) {}
 // Monomials m and n vars have the same values, up to "sign"
 // Generate a lemma if values of m.var() and n.var() are not the same up to sign
 bool basics::basic_sign_lemma_on_two_monomials(const monomial& m, const monomial& n) {
-    const rational& sign = m.rsign() * n.rsign();
+    const rational sign = sign_to_rat(m.rsign() ^ n.rsign());
      if (val(m) == val(n) * sign)
         return false;
      TRACE("nla_solver", tout << "sign contradiction:\nm = " << pp_mon(c(), m) << "n= " << pp_mon(c(), n) << "sign: " << sign << "\n";);
@@ -478,14 +478,25 @@ void basics::generate_pl(const monomial& rm, const factorization& fc, int factor
         explain(rm);
     }
     TRACE("nla_solver", c().print_lemma(tout); );
-}   
+}
+
+bool basics::is_separated_from_zero(const factorization& f) const {
+    for (const factor& fc: f) {
+        lpvar j = var(fc);
+        if (!(c().var_has_positive_lower_bound(j) || c().var_has_negative_upper_bound(j))) {
+            return false;
+        }
+    }
+    return true;
+}
+
+
 // here we use the fact xy = 0 -> x = 0 or y = 0
 void basics::basic_lemma_for_mon_zero_model_based(const monomial& rm, const factorization& f) {        
     TRACE("nla_solver",  c().trace_print_monomial_and_factorization(rm, f, tout););
     SASSERT(val(rm).is_zero()&& ! c().rm_check(rm));
     add_empty_lemma();
-    int sign =  c().get_derived_sign(rm, f);
-    if (sign == 0) {
+    if (is_separated_from_zero(f)) {
         c().mk_ineq(var(rm), llc::NE);        
         for (auto j : f) {
             c().mk_ineq(var(j), llc::EQ);
@@ -496,7 +507,6 @@ void basics::basic_lemma_for_mon_zero_model_based(const monomial& rm, const fact
             c().explain_separation_from_zero(var(j));
         }            
     }
-    explain(rm);
     explain(f);
     TRACE("nla_solver", c().print_lemma(tout););
 }
@@ -692,7 +702,7 @@ void basics::basic_lemma_for_mon_neutral_model_based(const monomial& rm, const f
 // use the fact
 // 1 * 1 ... * 1 * x * 1 ... * 1 = x
 bool basics::basic_lemma_for_mon_neutral_from_factors_to_monomial_model_based(const monomial& rm, const factorization& f) {
-    rational sign = rm.rsign();
+    rational sign = sign_to_rat(rm.rsign());
     TRACE("nla_solver_bl", tout << "f = "; c().print_factorization(f, tout); tout << ", sign = " << sign << '\n'; );
     lpvar not_one = -1;
     for (auto j : f){
diff --git a/src/util/lp/nla_basics_lemmas.h b/src/util/lp/nla_basics_lemmas.h
index 12e32153a..5fc63270a 100644
--- a/src/util/lp/nla_basics_lemmas.h
+++ b/src/util/lp/nla_basics_lemmas.h
@@ -103,5 +103,6 @@ struct basics: common {
     // none of the factors is zero and the product is not zero
     // -> |fc[factor_index]| <= |rm|
     void generate_pl(const monomial& rm, const factorization& fc, int factor_index);   
+    bool is_separated_from_zero(const factorization&) const;
 };
 }
diff --git a/src/util/lp/nla_common.cpp b/src/util/lp/nla_common.cpp
index d35b684b6..fe156bdb6 100644
--- a/src/util/lp/nla_common.cpp
+++ b/src/util/lp/nla_common.cpp
@@ -40,15 +40,14 @@ template <typename T> lpvar common::var(T const& t) const { return c().var(t); }
 template lpvar common::var<factor>(factor const& t) const;
 template lpvar common::var<monomial>(monomial const& t) const;
 void common::add_empty_lemma() { c().add_empty_lemma(); }
-template <typename T> rational common::canonize_sign(const T& t) const {
+template <typename T> bool common::canonize_sign(const T& t) const {
     return c().canonize_sign(t);
 }
 
-template rational common::canonize_sign<monomial>(const monomial& t) const;
-template rational common::canonize_sign<factor>(const factor& t) const;
-rational common::canonize_sign(lpvar j) const {
-    return c().canonize_sign_of_var(j);
-}
+template bool common::canonize_sign<monomial>(const monomial&) const;
+template bool common::canonize_sign<factor>(const factor&) const;
+template bool common::canonize_sign<lpvar>(const lpvar&) const;
+
 void common::mk_ineq(lp::lar_term& t, llc cmp, const rational& rs){
     c().mk_ineq(t, cmp, rs);
 }
@@ -68,6 +67,10 @@ void common::mk_ineq(const rational& a, lpvar j, const rational& b, lpvar k, llc
     c().mk_ineq(a, j, b, k, cmp);
 }
 
+void common::mk_ineq(bool a, lpvar j, bool b, lpvar k, llc cmp) {
+    c().mk_ineq(sign_to_rat(a), j, sign_to_rat(b), k, cmp);
+}
+
 void common::mk_ineq(const rational& a ,lpvar j, lpvar k, llc cmp, const rational& rs) {
     c().mk_ineq(a, j, k, cmp, rs);
 }
diff --git a/src/util/lp/nla_common.h b/src/util/lp/nla_common.h
index 1606ae20b..73a6282ef 100644
--- a/src/util/lp/nla_common.h
+++ b/src/util/lp/nla_common.h
@@ -57,8 +57,7 @@ struct common {
     template <typename T> void explain(const T&);
     void explain(lpvar);
     void add_empty_lemma();
-    template <typename T> rational canonize_sign(const T&) const;
-    rational canonize_sign(lpvar) const;
+    template <typename T> bool canonize_sign(const T&) const;
     void mk_ineq(lp::lar_term& t, llc cmp, const rational& rs);
     void mk_ineq(const rational& a, lpvar j, const rational& b, lpvar k, llc cmp, const rational& rs);
 
@@ -67,6 +66,7 @@ struct common {
     void mk_ineq(lpvar j, const rational& b, lpvar k, llc cmp);
 
     void mk_ineq(const rational& a, lpvar j, const rational& b, lpvar k, llc cmp);
+    void mk_ineq(bool a, lpvar j, bool b, lpvar k, llc cmp);
 
     void mk_ineq(const rational& a ,lpvar j, lpvar k, llc cmp, const rational& rs);
 
diff --git a/src/util/lp/nla_core.cpp b/src/util/lp/nla_core.cpp
index b7259b25b..a8b8af41b 100644
--- a/src/util/lp/nla_core.cpp
+++ b/src/util/lp/nla_core.cpp
@@ -92,23 +92,23 @@ svector<lpvar> core::sorted_rvars(const factor& f) const {
 
 // the value of the factor is equal to the value of the variable multiplied
 // by the canonize_sign
-rational core::canonize_sign(const factor& f) const {
-    return f.rat_sign() * (f.is_var()? canonize_sign_of_var(f.var()) : m_emons[f.var()].rsign());
+bool core::canonize_sign(const factor& f) const {
+    return f.sign() ^ (f.is_var()? canonize_sign(f.var()) : m_emons[f.var()].rsign());
 }
 
-rational core::canonize_sign_of_var(lpvar j) const {
-    return m_evars.find(j).rsign();        
+bool core::canonize_sign(lpvar j) const {
+    return m_evars.find(j).sign();        
 }
 
 bool core::canonize_sign_is_correct(const monomial& m) const {
-    rational r(1);
+    bool r = false;
     for (lpvar j : m.vars()) {
-        r *= canonize_sign_of_var(j);
+        r ^= canonize_sign(j);
     }
     return r == m.rsign();
 }
 
-rational core::canonize_sign(const monomial& m) const {
+bool core::canonize_sign(const monomial& m) const {
     SASSERT(canonize_sign_is_correct(m));
     return m.rsign();
 }
@@ -811,18 +811,6 @@ void core::explain_separation_from_zero(lpvar j) {
         explain_existing_upper_bound(j);
 }
 
-int core::get_derived_sign(const monomial& rm, const factorization& f) const {
-    rational sign = rm.rsign(); // this is the flip sign of the variable var(rm)
-    SASSERT(!sign.is_zero());
-    for (const factor& fc: f) {
-        lpvar j = var(fc);
-        if (!(var_has_positive_lower_bound(j) || var_has_negative_upper_bound(j))) {
-            return 0;
-        }
-        sign *= m_evars.find(j).rsign();
-    }
-    return nla::rat_sign(sign);
-}
 void core::trace_print_monomial_and_factorization(const monomial& rm, const factorization& f, std::ostream& out) const {
     out << "rooted vars: ";
     print_product(rm.rvars(), out);
@@ -1390,11 +1378,7 @@ std::unordered_set<lpvar> core::collect_vars(const lemma& l) const {
 // divides bc by c, so bc = b*c
 bool core::divide(const monomial& bc, const factor& c, factor & b) const {
     svector<lpvar> c_rvars = sorted_rvars(c);
-    TRACE("nla_solver",
-          tout << "c_rvars = ";
-          print_product(c_rvars, tout);
-          tout << "\nbc_rvars = ";
-          print_product(bc.rvars(), tout););
+    TRACE("nla_solver_div", tout << "c_rvars = "; print_product(c_rvars, tout); tout << "\nbc_rvars = "; print_product(bc.rvars(), tout););
     if (!lp::is_proper_factor(c_rvars, bc.rvars()))
         return false;
             
@@ -1412,12 +1396,10 @@ bool core::divide(const monomial& bc, const factor& c, factor & b) const {
         b = factor(sv->var(), factor_type::MON);
     }
     SASSERT(!b.sign());
-    // we should have bc.vars()*canonize_sign(bc) = bc.rvar() = b.rvars()*c.rvars() =
-    // = canonize_sign(b)*b.vars()* canonize_sign(c)*c.vars().
-    // so bc.vars()*canonize_sign(bc) = canonize_sign(b)*b.vars()* canonize_sign(c)*c.vars().
-    // but canonize_sign(b) now is canonize_sign_of_var(b.m_var) or canonize_sign(m_monomials[b.m_var])
-    // so we are adjusting it
-    b.sign() = (canonize_sign(b) * canonize_sign(c) * canonize_sign(bc) == rational(1))? false: true; 
+    // We have bc = canonize_sign(bc)*bc.rvars() = canonize_sign(b)*b.rvars()*canonize_sign(c)*c.rvars().
+    // Dividing by bc.rvars() we get canonize_sign(bc) = canonize_sign(b)*canonize_sign(c)
+    // Currently, canonize_sign(b) is 1, we might need to adjust it
+    b.sign() = canonize_sign(b) ^ canonize_sign(c) ^ canonize_sign(bc); 
     TRACE("nla_solver", tout << "success div:"; print_factor(b, tout););
     return true;
 }
@@ -1438,8 +1420,8 @@ void core::negate_factor_equality(const factor& c,
 }
     
 void core::negate_factor_relation(const rational& a_sign, const factor& a, const rational& b_sign, const factor& b) {
-    rational a_fs = canonize_sign(a);
-    rational b_fs = canonize_sign(b);
+    rational a_fs = sign_to_rat(canonize_sign(a));
+    rational b_fs = sign_to_rat(canonize_sign(b));
     llc cmp = a_sign*val(a) < b_sign*val(b)? llc::GE : llc::LE;
     mk_ineq(a_fs*a_sign, var(a), - b_fs*b_sign, var(b), cmp);
 }
diff --git a/src/util/lp/nla_core.h b/src/util/lp/nla_core.h
index c39016c1e..babec870b 100644
--- a/src/util/lp/nla_core.h
+++ b/src/util/lp/nla_core.h
@@ -116,13 +116,13 @@ public:
     void add_empty_lemma();
     // the value of the factor is equal to the value of the variable multiplied
     // by the canonize_sign
-    rational canonize_sign(const factor& f) const;
+    bool canonize_sign(const factor& f) const;
 
-    rational canonize_sign_of_var(lpvar j) const;
+    bool canonize_sign(lpvar j) const;
     
     // the value of the rooted monomias is equal to the value of the m.var() variable multiplied
     // by the canonize_sign
-    rational canonize_sign(const monomial& m) const;
+    bool canonize_sign(const monomial& m) const;
     
 
     void deregister_monomial_from_monomialomials (const monomial & m, unsigned i);
@@ -186,6 +186,8 @@ public:
     
     void mk_ineq(lp::lar_term& t, llc cmp, const rational& rs);
     void mk_ineq(const rational& a, lpvar j, const rational& b, lpvar k, llc cmp, const rational& rs);
+    void mk_ineq(bool a, lpvar j, bool b, lpvar k, llc cmp, const rational& rs);
+    void mk_ineq(bool a, lpvar j, bool b, lpvar k, llc cmp);
     void mk_ineq(lpvar j, const rational& b, lpvar k, llc cmp, const rational& rs);
     void mk_ineq(lpvar j, const rational& b, lpvar k, llc cmp);
     void mk_ineq(const rational& a, lpvar j, const rational& b, lpvar k, llc cmp);
@@ -240,20 +242,13 @@ public:
 
     const monomial* find_monomial_of_vars(const svector<lpvar>& vars) const;
 
-
-    int get_derived_sign(const monomial& rm, const factorization& f) const;
-
-
     bool var_has_positive_lower_bound(lpvar j) const;
 
     bool var_has_negative_upper_bound(lpvar j) const;
     
     bool var_is_separated_from_zero(lpvar j) const;
-    
 
-    bool vars_are_equiv(lpvar a, lpvar b) const;
-
-    
+    bool vars_are_equiv(lpvar a, lpvar b) const;    
     void explain_equiv_vars(lpvar a, lpvar b);
     void explain(const factorization& f, lp::explanation& exp);
     bool explain_upper_bound(const lp::lar_term& t, const rational& rs, lp::explanation& e) const;
@@ -353,6 +348,7 @@ struct pp_rmon {
     pp_rmon(core const& c, monomial const& m): c(c), m(m) {}
     pp_rmon(core const& c, lpvar v): c(c), m(c.m_emons[v]) {}
 };
+inline rational sign_to_rat(bool s) { return rational(s? -1 : 1); }
 inline std::ostream& operator<<(std::ostream& out, pp_mon const& p) { return p.c.print_monomial(p.m, out); }
 inline std::ostream& operator<<(std::ostream& out, pp_rmon const& p) { return p.c.print_monomial_with_vars(p.m, out); }
 
diff --git a/src/util/lp/nla_order_lemmas.cpp b/src/util/lp/nla_order_lemmas.cpp
index b474c490e..0fd458290 100644
--- a/src/util/lp/nla_order_lemmas.cpp
+++ b/src/util/lp/nla_order_lemmas.cpp
@@ -209,12 +209,14 @@ bool order::order_lemma_on_ac_and_bc(const monomial& rm_ac,
         order_lemma_on_ac_and_bc_and_factors(rm_ac, ac_f[!k], ac_f[k], rm_bd, b);
 }
 
-// TBD: document what lemma is created here.
 
+// Here ab is a binary factorization of m.
+// We try to find a monomial n = cd, such that |b| = |d| 
+// and get a lemma m R n & |b| = |d| => ab/|b| R cd /|d|, where R is a relation
 void order::order_lemma_on_factorization(const monomial& m, const factorization& ab) {
-    rational sign = m.rsign();
+    bool sign = m.rsign();
     for (factor f: ab)
-        sign *= _().canonize_sign(f);
+        sign ^= _().canonize_sign(f);
     const rational fv = val(ab[0]) * val(ab[1]);
     const rational mv = sign * val(m);
     TRACE("nla_solver",
@@ -225,7 +227,7 @@ void order::order_lemma_on_factorization(const monomial& m, const factorization&
     bool gt = mv > fv;
     TRACE("nla_solver", tout << "m="; _().print_monomial_with_vars(m, tout); tout << "\nfactorization="; _().print_factorization(ab, tout););
     for (unsigned j = 0, k = 1; j < 2; j++, k--) {
-        order_lemma_on_ab(m, sign, var(ab[k]), var(ab[j]), gt);
+        order_lemma_on_ab(m, sign_to_rat(sign), var(ab[k]), var(ab[j]), gt);
         explain(ab); explain(m);
         TRACE("nla_solver", _().print_lemma(tout););
         order_lemma_on_ac_explore(m, ab, j == 1);
@@ -265,7 +267,7 @@ void order::generate_ol(const monomial& ac,
     add_empty_lemma();
     rational rc_sign = rational(c_sign);
     mk_ineq(rc_sign * canonize_sign(c), var(c), llc::LE);
-    mk_ineq(canonize_sign(ac), var(ac), -canonize_sign(bc), var(bc), ab_cmp);
+    mk_ineq(canonize_sign(ac), var(ac), !canonize_sign(bc), var(bc), ab_cmp);
     mk_ineq(canonize_sign(a)*rc_sign, var(a), - canonize_sign(b)*rc_sign, var(b), negate(ab_cmp));
     explain(ac);
     explain(a);