diff --git a/src/math/lp/dioph_eq.cpp b/src/math/lp/dioph_eq.cpp
index efc313ec6..665d97ed0 100644
--- a/src/math/lp/dioph_eq.cpp
+++ b/src/math/lp/dioph_eq.cpp
@@ -2,52 +2,171 @@
 #include "math/lp/int_solver.h"
 #include "math/lp/lar_solver.h"
 #include "math/lp/lp_utils.h"
+#include <list>
 
 namespace lp {
-
+    // This class represents a term with an added constant number c, in form sum {x_i*a_i} + c.
+    
+    class term_o:public lar_term {        
+        mpq m_c;        
+    public:
+        const mpq& c() const { return m_c; }
+        mpq& c() { return m_c; }
+        term_o():m_c(0) {}
+        term_o& operator*=(mpq const& k) {
+            for (auto & t : m_coeffs)
+                t.m_value *= k;
+            return *this;
+        }   
+    };
     class dioph_eq::imp {
     public:
         int_solver& lia;
         lar_solver& lra;
-
-        imp(int_solver& lia, lar_solver& lra): lia(lia), lra(lra) {}
-        vector<lar_term> m_e;
-            
+        unsigned    m_fresh_x_vars_start;
+        unsigned    m_last_fresh_x_var;
+        
+        // set F 
+        std::list<term_o*> m_f;
+        // set S
+        std::list<term_o*> m_s;
+        
+    t    imp(int_solver& lia, lar_solver& lra): lia(lia), lra(lra) {
+            m_fresh_x_vars_start = lra.number_of_vars();
+            m_last_fresh_x_var = -1;
+        }
+        
+        bool belongs_to_list(term_o* t, std::list<term_o*> l) {
+            for (auto& item : l) {
+                if (item == t) {
+                    return true;
+                }
+            }
+            return false;
+        }
+    
         void init() {
             unsigned n_of_rows = static_cast<unsigned>(lra.r_basis().size());
             unsigned skipped = 0;
+            unsigned fn = 0;
             for (unsigned i = 0; i < n_of_rows; i++) {
                 auto & row = lra.get_row(i);
-                lar_term t;
-                bool is_int = true;
-                for (const auto & p : row) {
-                    if (!lia.column_is_int(p.var()))
-                        is_int = false;
+                term_o t;
+                bool all_vars_are_int = true;
+
+                for (const auto & p : row) {                   
+                    if (!lia.column_is_int(p.var())){
+                        all_vars_are_int = false;
+                        break;
+                    }
+                    
                 }
-                if (is_int) {
-                    lar_term t;
+
+                if (all_vars_are_int) {
+                    term_o* t = new term_o();
                     const auto lcm = get_denominators_lcm(row);
                     for (const auto & p: row) {
-                        t.add_monomial(lcm * p.coeff(), p.var());
+                        t->add_monomial(lcm * p.coeff(), p.var());
+                        if(lia.is_fixed(p.var())) {
+                            t->c() += lia.lower_bound(p.var()).x;
+                        }
                     }
-                    m_e.push_back(t);
+                    m_f.push_back(t);
+                }
+            }
+            
+        }
+
+        // returns true if no conflict is found and false otherwise
+        bool normalize_e_by_gcd(term_o& e) {
+            mpq g(0);
+            for (auto & p : e) {
+                if (g.is_zero()) {
+                    g = abs(p.coeff());
                 } else {
-                    skipped ++;
+                    g = gcd(g, p.coeff());
+                }                
+            }
+            if (g.is_zero()) {
+                UNREACHABLE();
+            }
+            if (g.is_one())
+                return true;
+            std::cout << "reached\n";    
+            e.c() *= (1/g);
+            if (!e.c().is_int()) {
+                // conlict: todo - collect the conflict
+                NOT_IMPLEMENTED_YET();
+                return false;                 
+            }
+            return true;
+        }
+        // returns true if no conflict is found and false otherwise
+        bool normalize_by_gcd() {
+            for (auto * e: m_f) {
+                if (!normalize_e_by_gcd(*e)) {
+                    return false;
                 }
             }
-            if (m_e.size() > 0) {
-                std::cout << "collected " << m_e.size() << ", skipped " << skipped << "\n";
-                for (const auto & t: m_e) {
-                    lra.print_term(t, std::cout) << "\n";
-                }
-                std::cout << "________________________\n";
-            }
+            return true;
         }
 
         lia_move check() {
             init();
-            return lia_move::undef;
+            while(m_f.size()) {
+                if (!normalize_by_gcd()) 
+                    return lia_move::unsat;
+                rewrite_eqs();
+            }
+            return lia_move::sat;
         }
+        std::list<term_o*>::iterator pick_eh() {
+            return m_f.begin(); // TODO: make a smarter joice
+        }
+
+        void substitute(unsigned k, term_o* s)  {
+            NOT_IMPLEMENTED_YET();
+        }
+        // this is the step 6 or 7 of the algorithm
+        void rewrite_eqs() {
+            auto eh_it = pick_eh();
+            term_o* eh = *eh_it;
+            // looking for minimal in absolute value coefficient
+            bool first = true;
+            mpq ahk;
+            unsigned k;
+            int k_sign;
+            for (auto& p: *eh) {
+                if (first || abs(p.coeff()) < ahk) {
+                    ahk = abs(p.coeff());
+                    k_sign = p.coeff().is_pos()? 1:-1;
+                    k = p.j();
+                    if (ahk.is_one())
+                        break;
+                    first = false;    
+                }
+            }
+            if (ahk.is_one()) {
+                // step 6
+                term_o *s_term = new term_o();
+                s_term->j() = k; // keep the assigned variable in m_j of the term
+                for (auto& p:*eh) {
+                    if (p.j() == k) continue;
+                    s_term->add_monomial(-k_sign*p.coeff(), p.j());
+                }
+                m_f.erase(eh_it);
+                m_s.push_back(s_term);
+                substitute(k, s_term);
+            } else {
+                // step 7
+                // the fresh variable 
+                unsigned xt = m_last_fresh_x_var == -1? m_fresh_x_vars_start: m_last_fresh_x_var++;
+                
+                
+            }
+                
+        }
+        
     };
 // Constructor definition
     dioph_eq::dioph_eq(int_solver& lia): lia(lia) {
diff --git a/src/math/lp/int_solver.cpp b/src/math/lp/int_solver.cpp
index 88ac2a1bb..fc9572fa7 100644
--- a/src/math/lp/int_solver.cpp
+++ b/src/math/lp/int_solver.cpp
@@ -165,7 +165,8 @@ namespace lp {
         }
 
         lia_move solve_dioph_eq() {
-            //dioph_eq de(lia);
+            dioph_eq de(lia);
+            de.check();
             return lia_move::undef;
         }
 
diff --git a/src/math/lp/lar_term.h b/src/math/lp/lar_term.h
index 8f8dd2f4a..659158b96 100644
--- a/src/math/lp/lar_term.h
+++ b/src/math/lp/lar_term.h
@@ -25,6 +25,7 @@ namespace lp {
 // represents a linear expressieon
 class lar_term {
     typedef unsigned lpvar;
+protected:
     u_map<mpq> m_coeffs;
     // the column index related to the term
     lpvar m_j = -1;