diff --git a/src/util/lp/emonomials.h b/src/util/lp/emonomials.h
index 5ad7ea70a..79bfb8cbc 100644
--- a/src/util/lp/emonomials.h
+++ b/src/util/lp/emonomials.h
@@ -184,6 +184,8 @@ namespace nla {
 
         monomial const& operator[](lpvar v) const { return var2monomial(v); }
 
+        monomial const& operator[](smon const& m) const { return var2monomial(m.var()); }
+
         bool is_monomial_var(lpvar v) const { return m_var2index.get(v, UINT_MAX) != UINT_MAX; }
 
         /**
diff --git a/src/util/lp/nla_common.h b/src/util/lp/nla_common.h
index 668acd6a5..e363df736 100644
--- a/src/util/lp/nla_common.h
+++ b/src/util/lp/nla_common.h
@@ -46,6 +46,9 @@ struct common {
     common(core* c): m_core(c) {}
     core& c() { return *m_core; }
     const core& c() const { return *m_core; }
+    core& _() { return *m_core; }
+    const core& _() const { return *m_core; }
+
     template <typename T> rational vvr(T const& t) const;
     rational vvr(lpvar) const;
     template <typename T> lpvar var(T const& t) const;
diff --git a/src/util/lp/nla_core.h b/src/util/lp/nla_core.h
index 69e5f1c01..4eaeee5dd 100644
--- a/src/util/lp/nla_core.h
+++ b/src/util/lp/nla_core.h
@@ -351,5 +351,21 @@ public:
 
     inline std::ostream& operator<<(std::ostream& out, pp_mon const& p) { return p.c.print_monomial(p.m, out); }
 
+    struct pp_fac {
+        core const& c;
+        factor const& f;
+        pp_fac(core const& c, factor const& f): c(c), f(f) {}
+    };
+
+    inline std::ostream& operator<<(std::ostream& out, pp_fac const& f) { return f.c.print_factor(f.f, out); }
+
+    struct pp_var {
+        core const& c;
+        lpvar v;
+        pp_var(core const& c, lpvar v): c(c), v(v) {}
+    };
+
+    inline std::ostream& operator<<(std::ostream& out, pp_var const& v) { return v.c.print_var(v.v, out); }
+
 } // end of namespace nla
 
diff --git a/src/util/lp/nla_monotone_lemmas.cpp b/src/util/lp/nla_monotone_lemmas.cpp
index 0e93582bf..36b0014b7 100644
--- a/src/util/lp/nla_monotone_lemmas.cpp
+++ b/src/util/lp/nla_monotone_lemmas.cpp
@@ -23,8 +23,19 @@
 namespace nla {
 
 monotone::monotone(core * c) : common(c) {}
-void monotone::print_monotone_array(const vector<std::pair<std::vector<rational>, unsigned>>& lex_sorted,
-                                std::ostream& out) const {
+
+    
+void monotone::monotonicity_lemma() {
+    unsigned shift = random();
+    unsigned size = c().m_to_refine.size();
+    for(unsigned i = 0; i < size && !done(); i++) { 
+        lpvar v = c().m_to_refine[(i + shift) % size];
+        monotonicity_lemma(c().m_emons[v]);
+    }
+}
+
+void monotone::print_monotone_array(const monotone_array_t& lex_sorted,
+                                    std::ostream& out) const {
     out << "Monotone array :\n";
     for (const auto & t : lex_sorted ){
         out << "(";
@@ -33,7 +44,8 @@ void monotone::print_monotone_array(const vector<std::pair<std::vector<rational>
     }
     out <<  "}";
 }
-bool monotone::monotonicity_lemma_on_lex_sorted_rm_upper(const vector<std::pair<std::vector<rational>, unsigned>>& lex_sorted, unsigned i, const smon& rm) {
+
+bool monotone::monotonicity_lemma_on_lex_sorted_rm_upper(const monotone_array_t& lex_sorted, unsigned i, const smon& rm) {
     const rational v = abs(vvr(rm));
     const auto& key = lex_sorted[i].first;
     TRACE("nla_solver", tout << "rm = " << rm << "i = " << i << std::endl;);
@@ -49,11 +61,10 @@ bool monotone::monotonicity_lemma_on_lex_sorted_rm_upper(const vector<std::pair<
             if (static_cast<int>(strict) != -1 && !has_zero(key)) {
                 generate_monl_strict(rm, rmk, strict);
                 return true;
-            } else {
-                if (vk < v) {
-                    generate_monl(rm, rmk);
-                    return true;
-                }
+            } 
+            else if (vk < v) {
+                generate_monl(rm, rmk);
+                return true;
             }
         }
             
@@ -61,7 +72,7 @@ bool monotone::monotonicity_lemma_on_lex_sorted_rm_upper(const vector<std::pair<
     return false;
 }
 
-bool monotone::monotonicity_lemma_on_lex_sorted_rm_lower(const vector<std::pair<std::vector<rational>, unsigned>>& lex_sorted, unsigned i, const smon& rm) {
+bool monotone::monotonicity_lemma_on_lex_sorted_rm_lower(const monotone_array_t& lex_sorted, unsigned i, const smon& rm) {
     const rational v = abs(vvr(rm));
     const auto& key = lex_sorted[i].first;
     TRACE("nla_solver", tout << "rm = " << rm << "i = " << i << std::endl;);
@@ -92,11 +103,11 @@ bool monotone::monotonicity_lemma_on_lex_sorted_rm_lower(const vector<std::pair<
     return false;
 }
 
-bool monotone::monotonicity_lemma_on_lex_sorted_rm(const vector<std::pair<std::vector<rational>, unsigned>>& lex_sorted, unsigned i, const smon& rm) {
+bool monotone::monotonicity_lemma_on_lex_sorted_rm(const monotone_array_t& lex_sorted, unsigned i, const smon& rm) {
     return monotonicity_lemma_on_lex_sorted_rm_upper(lex_sorted, i, rm)
         || monotonicity_lemma_on_lex_sorted_rm_lower(lex_sorted, i, rm);
 }
-bool monotone::monotonicity_lemma_on_lex_sorted(const vector<std::pair<std::vector<rational>, unsigned>>& lex_sorted) {
+bool monotone::monotonicity_lemma_on_lex_sorted(const monotone_array_t& lex_sorted) {
     for (unsigned i = 0; i < lex_sorted.size(); i++) {
         unsigned rmi = lex_sorted[i].second;
         const smon& rm = c().m_emons.canonical[rmi];
@@ -121,6 +132,7 @@ vector<std::pair<rational, lpvar>> monotone::get_sorted_key_with_vars(const smon
                                   });
     return r;
 } 
+
 void monotone::negate_abs_a_le_abs_b(lpvar a, lpvar b, bool strict) {
     rational av = vvr(a);
     rational as = rational(nla::rat_sign(av));
@@ -138,8 +150,8 @@ void monotone::negate_abs_a_le_abs_b(lpvar a, lpvar b, bool strict) {
 
 // strict version
 void monotone::generate_monl_strict(const smon& a,
-                                const smon& b,
-                                unsigned strict) {
+                                    const smon& b,
+                                    unsigned strict) {
     add_empty_lemma();
     auto akey = get_sorted_key_with_vars(a);
     auto bkey = get_sorted_key_with_vars(b);
@@ -189,7 +201,7 @@ void monotone::negate_abs_a_lt_abs_b(lpvar a, lpvar b) {
 
 // not a strict version
 void monotone::generate_monl(const smon& a,
-                         const smon& b) {
+                             const smon& b) {
     TRACE("nla_solver", 
           tout << "a = " << a << "\n:";
           tout << "b = " << b << "\n:";);
@@ -216,7 +228,7 @@ std::vector<rational> monotone::get_sorted_key(const smon& rm) const {
 }
     
 bool monotone::monotonicity_lemma_on_rms_of_same_arity(const unsigned_vector& rms) { 
-    vector<std::pair<std::vector<rational>, unsigned>> lex_sorted;
+    monotone_array_t lex_sorted;
     for (unsigned i : rms) {
         lex_sorted.push_back(std::make_pair(get_sorted_key(c().m_emons.canonical[i]), i));
     }
@@ -228,28 +240,6 @@ bool monotone::monotonicity_lemma_on_rms_of_same_arity(const unsigned_vector& rm
     TRACE("nla_solver", print_monotone_array(lex_sorted, tout););
     return monotonicity_lemma_on_lex_sorted(lex_sorted);
 }
-    
-void monotone::monotonicity_lemma() {
-    unsigned shift = random();
-    unsigned size = c().m_to_refine.size();
-    for(unsigned i = 0; i < size && !done(); i++) { 
-        lpvar v = c().m_to_refine[(i + shift) % size];
-        monotonicity_lemma(c().m_emons[v]);
-    }
-}
-
-#if 0
-void monotone::monotonicity_lemma() {
-    auto const& vars = m_rm_table.m_to_refine 
-        unsigned sz = vars.size();
-    unsigned start = random();
-    for (unsigned j = 0; !done() && j < sz; ++j) {
-        unsigned i = (start + j) % sz;
-        monotonicity_lemma(*m_emons.var2monomial(vars[i]));
-    }
-}
-
-#endif
 
 void monotone::monotonicity_lemma(monomial const& m) {
     SASSERT(!check_monomial(m));
@@ -262,6 +252,7 @@ void monotone::monotonicity_lemma(monomial const& m) {
     else if (m_val > prod_val)
         monotonicity_lemma_gt(m, prod_val);
 }
+
 void monotone::monotonicity_lemma_gt(const monomial& m, const rational& prod_val) {
     add_empty_lemma();
     for (lpvar j : m) {
diff --git a/src/util/lp/nla_monotone_lemmas.h b/src/util/lp/nla_monotone_lemmas.h
index c46ac8b81..c480460c6 100644
--- a/src/util/lp/nla_monotone_lemmas.h
+++ b/src/util/lp/nla_monotone_lemmas.h
@@ -20,16 +20,18 @@
 #pragma once
 namespace nla {
 class core;
-struct monotone: common {    
+class monotone : common {    
+public:
     monotone(core *core);
-    void print_monotone_array(const vector<std::pair<std::vector<rational>, unsigned>>& lex_sorted,
-                                  std::ostream& out) const;
-    bool monotonicity_lemma_on_lex_sorted_rm_upper(const vector<std::pair<std::vector<rational>, unsigned>>& lex_sorted, unsigned i, const smon& rm);
-    bool monotonicity_lemma_on_lex_sorted_rm_lower(const vector<std::pair<std::vector<rational>, unsigned>>& lex_sorted, unsigned i, const smon& rm);
-    bool monotonicity_lemma_on_lex_sorted_rm(const vector<std::pair<std::vector<rational>, unsigned>>& lex_sorted, unsigned i, const smon& rm);
-    bool monotonicity_lemma_on_lex_sorted(const vector<std::pair<std::vector<rational>, unsigned>>& lex_sorted);
-    bool  monotonicity_lemma_on_rms_of_same_arity(const unsigned_vector& rms);
     void monotonicity_lemma();
+private:
+    typedef vector<std::pair<std::vector<rational>, unsigned>> monotone_array_t;
+    void print_monotone_array(const monotone_array_t& lex_sorted, std::ostream& out) const;
+    bool monotonicity_lemma_on_lex_sorted_rm_upper(const monotone_array_t& lex_sorted, unsigned i, const smon& rm);
+    bool monotonicity_lemma_on_lex_sorted_rm_lower(const monotone_array_t& lex_sorted, unsigned i, const smon& rm);
+    bool monotonicity_lemma_on_lex_sorted_rm(const monotone_array_t& lex_sorted, unsigned i, const smon& rm);
+    bool monotonicity_lemma_on_lex_sorted(const monotone_array_t& lex_sorted);
+    bool  monotonicity_lemma_on_rms_of_same_arity(const unsigned_vector& rms);
     void monotonicity_lemma(monomial const& m);
     void monotonicity_lemma_gt(const monomial& m, const rational& prod_val);    
     void monotonicity_lemma_lt(const monomial& m, const rational& prod_val);    
diff --git a/src/util/lp/nla_order_lemmas.cpp b/src/util/lp/nla_order_lemmas.cpp
index ae1c7bce7..05640f0cf 100644
--- a/src/util/lp/nla_order_lemmas.cpp
+++ b/src/util/lp/nla_order_lemmas.cpp
@@ -15,7 +15,6 @@
 
   Revision History:
 
-
   --*/
 
 #include "util/lp/nla_order_lemmas.h"
@@ -25,13 +24,144 @@
 
 namespace nla {
 
-// a >< b && c > 0  => ac >< bc
-// a >< b && c < 0  => ac <> bc
+void order::order_lemma() {
+    TRACE("nla_solver", );
+    const auto& rm_ref = c().m_to_refine;
+    unsigned start = random();
+    unsigned sz = rm_ref.size();
+    for (unsigned i = 0; i < sz && !done(); ++i) {        
+        const smon& rm = c().m_emons.canonical[rm_ref[(i + start) % sz]];
+        order_lemma_on_rmonomial(rm);
+    }
+}
+
+void order::order_lemma_on_rmonomial(const smon& rm) {
+    TRACE("nla_solver_details",
+          tout << "rm = " << rm << ", orig = " << c().m_emons[rm];);
+
+    for (auto ac : factorization_factory_imp(rm, c())) {
+        if (ac.size() != 2)
+            continue;
+        if (ac.is_mon())
+            order_lemma_on_binomial(*ac.mon());
+        else
+            order_lemma_on_factorization(rm, ac);
+        if (done())
+            break;
+    }
+}
+
+void order::order_lemma_on_binomial(const monomial& ac) {
+    TRACE("nla_solver", tout << pp_mon(c(), ac););
+    SASSERT(!check_monomial(ac) && ac.size() == 2);
+    const rational & mult_val = vvr(ac[0]) * vvr(ac[1]);
+    const rational acv = vvr(ac);
+    bool gt = acv > mult_val;
+    for (unsigned k = 0; k < 2; k++) {
+        order_lemma_on_binomial_k(ac, k == 1, gt);
+        order_lemma_on_factor_binomial_explore(ac, k == 1);
+    }
+}
+
+void order::order_lemma_on_binomial_k(const monomial& m, bool k, bool gt) {
+    SASSERT(gt == (vvr(m) > vvr(m[0]) * vvr(m[1])));
+    order_lemma_on_binomial_sign(m, m[k], m[!k], gt ? 1: -1);
+}
+
+/**
+\brief
+ sign = the sign of vvr(xy) - vvr(x) * vvr(y) != 0
+ y <= 0 or x < a or xy >= ay
+ y <= 0 or x > a or xy <= ay
+ y >= 0 or x < a or xy <= ay
+ y >= 0 or x > a or xy >= ay
+  
+*/
+void order::order_lemma_on_binomial_sign(const monomial& xy, lpvar x, lpvar y, int sign) {
+    SASSERT(!_().mon_has_zero(xy));
+    int sy = rat_sign(vvr(y));
+    add_empty_lemma();
+    mk_ineq(y,                     sy == 1      ? llc::LE : llc::GE); // negate sy
+    mk_ineq(x,                     sy*sign == 1 ? llc::GT : llc::LT , vvr(x)); 
+    mk_ineq(xy.var(), - vvr(x), y, sign == 1    ? llc::LE : llc::GE);
+    TRACE("nla_solver", print_lemma(tout););
+}
+
+void order::order_lemma_on_factor_binomial_explore(const monomial& m1, bool k) {
+    SASSERT(m1.size() == 2);
+    lpvar c = m1[k];
+    for (monomial const& m2 : _().m_emons.get_factors_of(c)) {
+        order_lemma_on_factor_binomial_rm(m1, k, m2);
+        if (done()) {
+            break;
+        }
+    }
+}
+
+void order::order_lemma_on_factor_binomial_rm(const monomial& ac, bool k, const monomial& bd) {
+    smon const& rm_bd = _().m_emons.canonical[bd];
+    factor d(_().m_evars.find(ac[k]).var(), factor_type::VAR);
+    factor b;
+    if (c().divide(rm_bd, d, b)) {
+        order_lemma_on_binomial_ac_bd(ac, k, rm_bd, b, d.var());
+    }
+}
+
+void order::order_lemma_on_binomial_ac_bd(const monomial& ac, bool k, const smon& bd, const factor& b, lpvar d) {
+    TRACE("nla_solver",  
+          tout << "ac=" << pp_mon(c(), ac) << "\nrm= " << bd << ", b= " << pp_fac(c(), b) << ", d= " << pp_var(c(), d) << "\n";);
+    bool p = !k;
+    lpvar a = ac[p];
+    lpvar c = ac[k];
+    SASSERT(_().m_evars.find(c).var() == d);
+    rational acv = vvr(ac);
+    rational av = vvr(a);
+    rational c_sign = rrat_sign(vvr(c));
+    rational d_sign = rrat_sign(vvr(d));
+    rational bdv = vvr(bd);
+    rational bv = vvr(b);
+    auto av_c_s = av*c_sign; auto bv_d_s = bv*d_sign;
+
+    // suppose ac >= bd, then ac/|c| >= bd/|d|.
+    // Notice that ac/|c| = a*c_sign , and bd/|d| = b*d_sign
+    if (acv >= bdv && av_c_s < bv_d_s)
+        generate_mon_ol(ac, a, c_sign, c, bd, b, d_sign, d, llc::LT);
+    else if (acv <= bdv && av_c_s > bv_d_s)
+        generate_mon_ol(ac, a, c_sign, c, bd, b, d_sign, d, llc::GT);
+        
+}
+
+// TBD: document what lemma is created here.
+
+void order::generate_mon_ol(const monomial& ac,                     
+                           lpvar a,
+                           const rational& c_sign,
+                           lpvar c,
+                           const smon& bd,
+                           const factor& b,
+                           const rational& d_sign,
+                           lpvar d,
+                           llc ab_cmp) {
+    add_empty_lemma();
+    mk_ineq(c_sign, c, llc::LE);
+    explain(c); // this explains c == +- d
+    negate_var_factor_relation(c_sign, a, d_sign, b);
+    mk_ineq(ac.var(), -canonize_sign(bd), var(bd), ab_cmp);
+    explain(bd);
+    explain(b);
+    explain(d);
+    TRACE("nla_solver", print_lemma(tout););
+}
+
+
+// a > b  && c > 0 => ac > bc
+// a >< b && c > 0 => ac >< bc
+// a >< b && c < 0 => ac <> bc
 // ac[k] plays the role of c   
 
 bool order::order_lemma_on_ac_and_bc(const smon& rm_ac,
                                      const factorization& ac_f,
-                                     unsigned k,
+                                     bool k,
                                      const smon& rm_bd) {
     TRACE("nla_solver", 
           tout << "rm_ac = " << rm_ac << "\n";
@@ -41,10 +171,36 @@ bool order::order_lemma_on_ac_and_bc(const smon& rm_ac,
     factor b;
     return 
         c().divide(rm_bd, ac_f[k], b) && 
-        order_lemma_on_ac_and_bc_and_factors(rm_ac, ac_f[(k + 1) % 2], ac_f[k], rm_bd, b);
+        order_lemma_on_ac_and_bc_and_factors(rm_ac, ac_f[!k], ac_f[k], rm_bd, b);
 }
 
-bool order::order_lemma_on_ac_explore(const smon& rm, const factorization& ac, unsigned k) {
+// TBD: document what lemma is created here.
+
+void order::order_lemma_on_factorization(const smon& rm, const factorization& ab) {
+    const monomial& m = _().m_emons[rm];
+    TRACE("nla_solver", tout << "orig_sign = " << _().m_emons.orig_sign(rm) << "\n";);
+    rational sign = _().m_emons.orig_sign(rm);
+    for (factor f: ab)
+        sign *= _().canonize_sign(f);
+    const rational & fv = vvr(ab[0]) * vvr(ab[1]);
+    const rational mv = sign * vvr(m);
+    TRACE("nla_solver",
+          tout << "ab.size()=" << ab.size() << "\n";
+          tout << "we should have sign*vvr(m):" << mv << "=(" << sign << ")*(" << vvr(m) <<") to be equal to " << " vvr(ab[0])*vvr(ab[1]):" << fv << "\n";);
+    if (mv == fv)
+        return;
+    bool gt = mv > fv;
+    TRACE("nla_solver_f", 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);
+        explain(ab); explain(m);
+        explain(rm);
+        TRACE("nla_solver", _().print_lemma(tout););
+        order_lemma_on_ac_explore(rm, ab, j == 1);
+    }
+}
+
+bool order::order_lemma_on_ac_explore(const smon& rm, const factorization& ac, bool k) {
     const factor c = ac[k];
     TRACE("nla_solver", tout << "c = "; _().print_factor_with_vars(c, tout); );
     if (c.is_var()) {
@@ -65,29 +221,6 @@ bool order::order_lemma_on_ac_explore(const smon& rm, const factorization& ac, u
     return false;
 }
 
-void order::order_lemma_on_factorization(const smon& rm, const factorization& ab) {
-    const monomial& m = _().m_emons[rm.var()];
-    TRACE("nla_solver", tout << "orig_sign = " << _().m_emons.orig_sign(rm) << "\n";);
-    rational sign = _().m_emons.orig_sign(rm);
-    for (factor f: ab)
-        sign *= _().canonize_sign(f);
-    const rational & fv = vvr(ab[0]) * vvr(ab[1]);
-    const rational mv = sign * vvr(m);
-    TRACE("nla_solver",
-          tout << "ab.size()=" << ab.size() << "\n";
-          tout << "we should have sign*vvr(m):" << mv << "=(" << sign << ")*(" << vvr(m) <<") to be equal to " << " vvr(ab[0])*vvr(ab[1]):" << fv << "\n";);
-    if (mv == fv)
-        return;
-    bool gt = mv > fv;
-    TRACE("nla_solver_f", 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);
-        explain(ab); explain(m);
-        explain(rm);
-        TRACE("nla_solver", _().print_lemma(tout););
-        order_lemma_on_ac_explore(rm, ab, j);
-    }
-}
 // |c_sign| = 1, and c*c_sign > 0
 // ac > bc => ac/|c| > bc/|c| => a*c_sign > b*c_sign
 void order::generate_ol(const smon& ac,                     
@@ -110,25 +243,6 @@ void order::generate_ol(const smon& ac,
     TRACE("nla_solver", _().print_lemma(tout););
 }
 
-void order::generate_mon_ol(const monomial& ac,                     
-                           lpvar a,
-                           const rational& c_sign,
-                           lpvar c,
-                           const smon& bd,
-                           const factor& b,
-                           const rational& d_sign,
-                           lpvar d,
-                           llc ab_cmp) {
-    add_empty_lemma();
-    mk_ineq(c_sign, c, llc::LE);
-    explain(c); // this explains c == +- d
-    negate_var_factor_relation(c_sign, a, d_sign, b);
-    mk_ineq(ac.var(), -canonize_sign(bd), var(bd), ab_cmp);
-    explain(bd);
-    explain(b);
-    explain(d);
-    TRACE("nla_solver", print_lemma(tout););
-}
 
 void order::negate_var_factor_relation(const rational& a_sign, lpvar a, const rational& b_sign, const factor& b) {
     rational b_fs = canonize_sign(b);
@@ -136,16 +250,6 @@ void order::negate_var_factor_relation(const rational& a_sign, lpvar a, const ra
     mk_ineq(a_sign, a, - b_fs*b_sign, var(b), cmp);
 }
 
-void order::order_lemma() {
-    TRACE("nla_solver", );
-    const auto& rm_ref = c().m_to_refine;
-    unsigned start = random();
-    unsigned sz = rm_ref.size();
-    for (unsigned i = sz; i-- > 0 && !done(); ) {        
-        const smon& rm = c().m_emons.canonical[rm_ref[(i + start) % sz]];
-        order_lemma_on_rmonomial(rm);
-    }
-}
 
 bool order::order_lemma_on_ac_and_bc_and_factors(const smon& ac,
                                                  const factor& a,
@@ -231,95 +335,5 @@ void order::order_lemma_on_ab(const monomial& m, const rational& sign, lpvar a,
     else 
         order_lemma_on_ab_lt(m, sign, a, b);
 }
-// a > b && c > 0 => ac > bc
-void order::order_lemma_on_rmonomial(const smon& rm) {
-    TRACE("nla_solver_details",
-          tout << "rm = "; print_product(rm, tout);
-          tout << ", orig = " << pp_mon(c(), c().m_emons[rm.var()]);
-          );
-    for (auto ac : factorization_factory_imp(rm, c())) {
-        if (ac.size() != 2)
-            continue;
-        if (ac.is_mon())
-            order_lemma_on_binomial(*ac.mon());
-        else
-            order_lemma_on_factorization(rm, ac);
-        if (done())
-            break;
-    }
-}
-void order::order_lemma_on_binomial_k(const monomial& m, lpvar k, bool gt) {
-    SASSERT(gt == (vvr(m) > vvr(m[0]) * vvr(m[1])));
-    unsigned p = (k + 1) % 2;
-    order_lemma_on_binomial_sign(m, m[k], m[p], gt? 1: -1);
-}
-// sign it the sign of vvr(m) - vvr(m[0]) * vvr(m[1])
-// m = xy
-// and val(m) != val(x)*val(y)
-// y > 0 and x = a, then xy >= ay
-void order::order_lemma_on_binomial_sign(const monomial& ac, lpvar x, lpvar y, int sign) {
-    SASSERT(!_().mon_has_zero(ac));
-    int sy = rat_sign(vvr(y));
-    add_empty_lemma();
-    mk_ineq(y, sy == 1? llc::LE : llc::GE); // negate sy
-    mk_ineq(x, sy*sign == 1? llc::GT:llc::LT , vvr(x)); // assert x <= vvr(x) if x > 0
-    mk_ineq(ac.var(), - vvr(x), y, sign == 1?llc::LE:llc::GE);
-    TRACE("nla_solver", print_lemma(tout););
-}
-void order::order_lemma_on_factor_binomial_rm(const monomial& ac, unsigned k, const monomial& bd) {
-    smon const& rm_bd = _().m_emons.canonical[bd];
-    factor d(_().m_evars.find(ac[k]).var(), factor_type::VAR);
-    factor b;
-    if (_().divide(rm_bd, d, b)) {
-        order_lemma_on_binomial_ac_bd(ac, k, rm_bd, b, d.var());
-    }
-}
 
-void order::order_lemma_on_binomial_ac_bd(const monomial& ac, unsigned k, const smon& bd, const factor& b, lpvar d) {
-    TRACE("nla_solver",  tout << "ac=" << pp_mon(c(), ac);
-          tout << "\nrm=" << bd;
-          print_factor(b, tout << ", b="); print_var(d, tout << ", d=") << "\n";);
-    int p = (k + 1) % 2;
-    lpvar a = ac[p];
-    lpvar c = ac[k];
-    SASSERT(_().m_evars.find(c).var() == d);
-    rational acv = vvr(ac);
-    rational av = vvr(a);
-    rational c_sign = rrat_sign(vvr(c));
-    rational d_sign = rrat_sign(vvr(d));
-    rational bdv = vvr(bd);
-    rational bv = vvr(b);
-    auto av_c_s = av*c_sign; auto bv_d_s = bv*d_sign;
-
-    // suppose ac >= bd, then ac/|c| >= bd/|d|.
-    // Notice that ac/|c| = a*c_sign , and bd/|d| = b*d_sign
-    if (acv >= bdv && av_c_s < bv_d_s)
-        generate_mon_ol(ac, a, c_sign, c, bd, b, d_sign, d, llc::LT);
-    else if (acv <= bdv && av_c_s > bv_d_s)
-        generate_mon_ol(ac, a, c_sign, c, bd, b, d_sign, d, llc::GT);
-        
 }
-
-void order::order_lemma_on_factor_binomial_explore(const monomial& m, unsigned k) {
-    SASSERT(m.size() == 2);
-    lpvar c = m[k];
-    for (monomial const& m2 : _().m_emons.get_factors_of(c)) {
-        order_lemma_on_factor_binomial_rm(m, k, m2);
-        if (done()) {
-            break;
-        }
-    }
-}
-
-void order::order_lemma_on_binomial(const monomial& ac) {
-    TRACE("nla_solver", tout << pp_mon(c(), ac););
-    SASSERT(!check_monomial(ac) && ac.size() == 2);
-    const rational & mult_val = vvr(ac[0]) * vvr(ac[1]);
-    const rational acv = vvr(ac);
-    bool gt = acv > mult_val;
-    for (unsigned k = 0; k < 2; k++) {
-        order_lemma_on_binomial_k(ac, k, gt);
-        order_lemma_on_factor_binomial_explore(ac, k);
-    }
-}
-} // end of namespace nla
diff --git a/src/util/lp/nla_order_lemmas.h b/src/util/lp/nla_order_lemmas.h
index 1270f5c1e..b32478230 100644
--- a/src/util/lp/nla_order_lemmas.h
+++ b/src/util/lp/nla_order_lemmas.h
@@ -24,11 +24,13 @@
 
 namespace nla {
 class core;
-struct order: common {
-    core& _() { return *m_core; }
-    const core& _() const { return *m_core; }
-    //constructor
+class order: common {
+public:
     order(core *c) : common(c) {}
+    void order_lemma();
+    
+private:
+
     bool order_lemma_on_ac_and_bc_and_factors(const smon& ac,
                                               const factor& a,
                                               const factor& c,
@@ -40,10 +42,10 @@ struct order: common {
     // ac[k] plays the role of c   
     bool order_lemma_on_ac_and_bc(const smon& rm_ac,
                                   const factorization& ac_f,
-                                  unsigned k,
+                                  bool k,
                                   const smon& rm_bd);
     
-    bool order_lemma_on_ac_explore(const smon& rm, const factorization& ac, unsigned k);
+    bool order_lemma_on_ac_explore(const smon& rm, const factorization& ac, bool k);
 
     void order_lemma_on_factorization(const smon& rm, const factorization& ab);
     
@@ -61,14 +63,13 @@ struct order: common {
     */
     void order_lemma_on_ab_lt(const monomial& m, const rational& sign, lpvar a, lpvar b);
     void order_lemma_on_ab(const monomial& m, const rational& sign, lpvar a, lpvar b, bool gt);
-    void order_lemma_on_factor_binomial_explore(const monomial& m, unsigned k);
-    void order_lemma_on_factor_binomial_rm(const monomial& ac, unsigned k, const monomial& bd);
-    void order_lemma_on_binomial_ac_bd(const monomial& ac, unsigned k, const smon& bd, const factor& b, lpvar d);
-    void order_lemma_on_binomial_k(const monomial& m, lpvar k, bool gt);
+    void order_lemma_on_factor_binomial_explore(const monomial& m, bool k);
+    void order_lemma_on_factor_binomial_rm(const monomial& ac, bool k, const monomial& bd);
+    void order_lemma_on_binomial_ac_bd(const monomial& ac, bool k, const smon& bd, const factor& b, lpvar d);
+    void order_lemma_on_binomial_k(const monomial& m, bool k, bool gt);
     void order_lemma_on_binomial_sign(const monomial& ac, lpvar x, lpvar y, int sign);
     void order_lemma_on_binomial(const monomial& ac);
     void order_lemma_on_rmonomial(const smon& rm);
-    void order_lemma();
     // |c_sign| = 1, and c*c_sign > 0
     // ac > bc => ac/|c| > bc/|c| => a*c_sign > b*c_sign
     void generate_ol(const smon& ac,