diff --git a/src/math/polysat/conflict.cpp b/src/math/polysat/conflict.cpp
index f8e47289e..949474ac7 100644
--- a/src/math/polysat/conflict.cpp
+++ b/src/math/polysat/conflict.cpp
@@ -62,11 +62,13 @@ namespace polysat {
     class conflict_resolver {
         inf_saturate m_saturate;
         ex_polynomial_superposition m_poly_sup;
+        free_variable_elimination m_free_variable_elimination;
 
     public:
         conflict_resolver(solver& s)
             : m_saturate(s)
             , m_poly_sup(s)
+            , m_free_variable_elimination(s)
         {}
 
         bool try_resolve_value(pvar v, conflict& core) {
@@ -76,6 +78,11 @@ namespace polysat {
                 return true;
             return false;
         }
+
+        // Analyse current conflict core to extract additional lemmas
+        void find_extra_lemmas(conflict& core) {
+            m_free_variable_elimination.find_lemma(core);
+        }
     };
 
     class header_with_var : public displayable {
@@ -524,6 +531,10 @@ namespace polysat {
     clause_ref conflict::build_lemma() {
         LOG_H3("Build lemma from core");
         LOG("core: " << *this);
+
+        // TODO: do this after each step?
+        m_resolver->find_extra_lemmas(*this);
+
         clause_builder lemma(s);
 
 #if 0
diff --git a/src/math/polysat/constraint.h b/src/math/polysat/constraint.h
index 4e596b7e6..3c8df311b 100644
--- a/src/math/polysat/constraint.h
+++ b/src/math/polysat/constraint.h
@@ -258,6 +258,7 @@ namespace polysat {
 
         void add_to_univariate_solver(solver& s, univariate_solver& us, unsigned dep) const { get()->add_to_univariate_solver(s, us, dep, is_positive()); }
 
+        unsigned_vector const& vars() const { return m_constraint->vars(); }
         unsigned var(unsigned idx) const { return m_constraint->var(idx); }
         bool contains_var(pvar v) const { return m_constraint->contains_var(v); }
 
diff --git a/src/math/polysat/solver.h b/src/math/polysat/solver.h
index 442fbd969..e4c871725 100644
--- a/src/math/polysat/solver.h
+++ b/src/math/polysat/solver.h
@@ -85,6 +85,7 @@ namespace polysat {
         friend class assignment_pp;
         friend class assignments_pp;
         friend class ex_polynomial_superposition;
+        friend class free_variable_elimination;
         friend class inf_saturate;
         friend class constraint_manager;
         friend class scoped_solverv;
diff --git a/src/math/polysat/variable_elimination.cpp b/src/math/polysat/variable_elimination.cpp
index 47f759f38..7cdbad5d1 100644
--- a/src/math/polysat/variable_elimination.cpp
+++ b/src/math/polysat/variable_elimination.cpp
@@ -8,26 +8,134 @@ Module Name:
 Author:
 
     Nikolaj Bjorner (nbjorner) 2021-03-19
-    Jakob Rath 2021-04-6
+    Jakob Rath 2021-04-06
 
 --*/
 #include "math/polysat/variable_elimination.h"
+#include "math/polysat/conflict.h"
+#include "math/polysat/clause_builder.h"
 #include "math/polysat/solver.h"
 #include <algorithm>
 
 namespace polysat {
 
-    bool ve_reduction::perform(solver& s, pvar v, conflict& core) {
-        // without any further hints, we just do core reduction with the stronger premise "C contains a c' that evaluates to false",
-        // and kick out all other constraints.
+    void free_variable_elimination::find_lemma(conflict& core) {
+        LOG_H1("Free Variable Elimination");
+        LOG("core: " << core);
+        LOG("Free variables: " << s.m_free_pvars);
+        for (pvar v : core.vars_occurring_in_constraints())
+            if (!s.is_assigned(v))  // TODO: too restrictive. should also consider variables that will be unassigned only after backjumping (can update this after assignment handling in search state is refactored.)
+                find_lemma(v, core);
+    }
+
+    void free_variable_elimination::find_lemma(pvar v, conflict& core) {
+        LOG_H2("Free Variable Elimination for v" << v);
+        // find constraint that allows computing v from other variables
+        // (currently, consider only equations that contain v with degree 1)
         for (signed_constraint c : core) {
-            if (!c->contains_var(v) && c.is_currently_false(s)) {
-                core.set(c);
-                core.logger().log("ve_reduction");
-                return true;
-            }                
+            if (!c.is_eq())
+                continue;
+            if (c.eq().degree(v) != 1)
+                continue;
+            find_lemma(v, c, core);
         }
-        return false;
+    }
+
+    void free_variable_elimination::find_lemma(pvar v, signed_constraint c, conflict& core) {
+        LOG_H3("Free Variable Elimination for v" << v << " using equation " << c);
+        pdd const& p = c.eq();
+        SASSERT_EQ(p.degree(v), 1);
+        auto& m = p.manager();
+        pdd lc = m.zero();
+        pdd rest = m.zero();
+        p.factor(v, 1, lc, rest);
+        if (rest.is_val())
+            return;
+        // lc * v + rest == p == 0
+        // v == -1 * rest * lc^-1
+
+        SASSERT(!lc.free_vars().contains(v));
+        SASSERT(!rest.free_vars().contains(v));
+
+        LOG("lc: " << lc);
+        LOG("rest: " << rest);
+
+        assignment_t a;
+        pdd const lcs = eval(lc, core, a);
+        LOG("lcs: " << lcs);
+        pdd lci = m.zero();
+        if (!inv(lcs, lci))
+            return;
+
+        pdd const rs = s.subst(a, rest);
+        pdd const vs = -rs * lci;  // this is the polynomial that computes v
+        LOG("vs: " << vs);
+        SASSERT(!vs.free_vars().contains(v));
+
+        // Find another constraint where we want to substitute v
+        for (signed_constraint c_target : core) {
+            if (c == c_target)
+                continue;
+            if (c_target.vars().size() <= 1)
+                continue;
+            if (!c_target.contains_var(v))
+                continue;
+
+            // TODO: helper method constraint::subst(pvar v, pdd const& p)
+            //       (or rather, add it on constraint_manager since we need to allocate/dedup the new constraint)
+            //  For now, just restrict to ule_constraint.
+            if (!c_target->is_ule())
+                continue;
+            // TODO: maybe apply assignment a here as well
+            pdd const lhs = c_target->to_ule().lhs().subst_pdd(v, vs);
+            pdd const rhs = c_target->to_ule().rhs().subst_pdd(v, vs);
+            signed_constraint c_new = s.ule(lhs, rhs);
+            if (c_target.is_negative())
+                c_new.negate();
+            LOG("c_target: " << lit_pp(s, c_target));
+            LOG("c_new:    " << lit_pp(s, c_new));
+
+            clause_builder cb(s);
+            for (auto [w, wv] : a)
+                cb.push(~s.eq(s.var(w), wv));
+            cb.push(~c);
+            cb.push(~c_target);
+            cb.push(c_new);
+            core.add_lemma(cb.build());
+        }
+    }
+
+    // Evaluate p under assignments in the core.
+    pdd free_variable_elimination::eval(pdd const& p, conflict& core, assignment_t& out_assignment) {
+        // TODO: this should probably be a helper method on conflict.
+        // TODO: recognize constraints of the form "v1 == 27" to be used in the assignment?
+        //       (but maybe useful evaluations are always part of core.vars() anyway?)
+
+        assignment_t& a = out_assignment;
+        SASSERT(a.empty());
+
+        for (auto v : p.free_vars())
+            if (core.contains_pvar(v))
+                a.push_back({v, s.get_value(v)});
+
+        pdd q = s.subst(a, p);
+
+        // TODO: like in the old conflict::minimize_vars, we can now try to remove unnecessary variables from a.
+
+        return q;
+    }
+
+    // Compute the multiplicative inverse of p.
+    bool free_variable_elimination::inv(pdd const& p, pdd& out_p_inv) {
+        // TODO: in the non-val case, we could introduce an additional variable to represent the inverse
+        //       (and a constraint p * p_inv == 1)
+        if (!p.is_val())
+            return false;
+        rational iv;
+        if (!p.val().mult_inverse(p.power_of_2(), iv))
+            return false;
+        out_p_inv = p.manager().mk_val(iv);
+        return true;
     }
 
 }
diff --git a/src/math/polysat/variable_elimination.h b/src/math/polysat/variable_elimination.h
index 160580a66..3e9a99c30 100644
--- a/src/math/polysat/variable_elimination.h
+++ b/src/math/polysat/variable_elimination.h
@@ -8,25 +8,27 @@ Module Name:
 Author:
 
     Nikolaj Bjorner (nbjorner) 2021-03-19
-    Jakob Rath 2021-04-6
+    Jakob Rath 2021-04-06
 
 --*/
 #pragma once
-#include "math/polysat/conflict.h"
+
+#include "math/polysat/types.h"
+#include "math/polysat/constraint.h"
 
 namespace polysat {
 
-    class solver;
+    class conflict;
 
-    class variable_elimination_engine {
+    class free_variable_elimination {
+        solver& s;
+        void find_lemma(pvar v, conflict& core);
+        void find_lemma(pvar v, signed_constraint c, conflict& core);
+        pdd eval(pdd const& p, conflict& core, assignment_t& out_assignment);
+        bool inv(pdd const& p, pdd& out_p_inv);
     public:
-        virtual ~variable_elimination_engine() {}
-        virtual bool perform(solver& s, pvar v, conflict& core) = 0;
-    };
-
-    class ve_reduction : public variable_elimination_engine {
-    public:
-        bool perform(solver& s, pvar v, conflict& core) override;
+        free_variable_elimination(solver& s): s(s) {}
+        void find_lemma(conflict& core);
     };
 
 }