diff --git a/src/cmd_context/extra_cmds/dbg_cmds.cpp b/src/cmd_context/extra_cmds/dbg_cmds.cpp
index 7c1383d62..d9e575664 100644
--- a/src/cmd_context/extra_cmds/dbg_cmds.cpp
+++ b/src/cmd_context/extra_cmds/dbg_cmds.cpp
@@ -519,6 +519,7 @@ public:
         model_ref mdl;
         s->get_model(mdl);
         qe::euf_arith_mbi_plugin plugin(s.get(), se.get());
+        plugin.set_shared(vars);
         plugin.project(mdl, lits);
         ctx.regular_stream() << lits << "\n";
     }
diff --git a/src/model/model_evaluator.cpp b/src/model/model_evaluator.cpp
index 1991fc4d2..1bd274316 100644
--- a/src/model/model_evaluator.cpp
+++ b/src/model/model_evaluator.cpp
@@ -673,6 +673,15 @@ bool model_evaluator::is_true(expr_ref_vector const& ts) {
     return true;
 }
 
+bool model_evaluator::are_equal(expr* s, expr* t) {
+    if (m().are_equal(s, t)) return true;
+    if (m().are_distinct(s, t)) return false;
+    expr_ref t1(m()), t2(m());
+    eval(t, t1, true);
+    eval(s, t2, true);
+    return m().are_equal(t1, t2);
+}
+
 bool model_evaluator::eval(expr* t, expr_ref& r, bool model_completion) {
     set_model_completion(model_completion);
     try {
diff --git a/src/model/model_evaluator.h b/src/model/model_evaluator.h
index 2ffcf2e0b..9f2320a66 100644
--- a/src/model/model_evaluator.h
+++ b/src/model/model_evaluator.h
@@ -55,6 +55,8 @@ public:
     bool is_true(expr * t);
     bool is_false(expr * t);
     bool is_true(expr_ref_vector const& ts);
+    bool are_equal(expr* s, expr* t);
+
 
     void set_solver(expr_solver* solver);
     bool has_solver();
diff --git a/src/qe/qe_arith.cpp b/src/qe/qe_arith.cpp
index a0aea08da..38027efbc 100644
--- a/src/qe/qe_arith.cpp
+++ b/src/qe/qe_arith.cpp
@@ -604,6 +604,11 @@ namespace qe {
         return m_imp->maximize(fmls, mdl, t, ge, gt);
     }
 
+    void arith_project_plugin::saturate(model& model, func_decl_ref_vector const& shared, expr_ref_vector& lits) {
+        UNREACHABLE();
+    }
+
+
     bool arith_project(model& model, app* var, expr_ref_vector& lits) {
         ast_manager& m = lits.get_manager();
         arith_project_plugin ap(m);
diff --git a/src/qe/qe_arith.h b/src/qe/qe_arith.h
index b55e63fcf..acff8e13b 100644
--- a/src/qe/qe_arith.h
+++ b/src/qe/qe_arith.h
@@ -31,6 +31,7 @@ namespace qe {
         family_id get_family_id() override;
         void operator()(model& model, app_ref_vector& vars, expr_ref_vector& lits) override;
         vector<def> project(model& model, app_ref_vector& vars, expr_ref_vector& lits) override;
+        void saturate(model& model, func_decl_ref_vector const& shared, expr_ref_vector& lits) override;
 
         opt::inf_eps maximize(expr_ref_vector const& fmls, model& mdl, app* t, expr_ref& ge, expr_ref& gt);
 
diff --git a/src/qe/qe_arrays.cpp b/src/qe/qe_arrays.cpp
index 4bead6684..460f9ea37 100644
--- a/src/qe/qe_arrays.cpp
+++ b/src/qe/qe_arrays.cpp
@@ -21,12 +21,14 @@ Revision History:
 #include "util/lbool.h"
 #include "ast/rewriter/rewriter_def.h"
 #include "ast/expr_functors.h"
+#include "ast/for_each_expr.h"
 #include "ast/rewriter/expr_safe_replace.h"
 #include "ast/rewriter/th_rewriter.h"
 #include "ast/ast_util.h"
 #include "ast/ast_pp.h"
 #include "model/model_evaluator.h"
 #include "qe/qe_arrays.h"
+#include "qe/qe_term_graph.h"
 
 
 namespace {
@@ -357,7 +359,7 @@ namespace qe {
                 ptr_vector<expr> sel_args;
                 sel_args.push_back (arr);
                 sel_args.append(I[i].size(), I[i].c_ptr());
-                expr_ref val_term (m_arr_u.mk_select (sel_args.size (), sel_args.c_ptr ()), m);
+                expr_ref val_term (m_arr_u.mk_select (sel_args), m);
                 // evaluate and assign to ith diff_val_const
                 val = (*m_mev)(val_term);
                 M->register_decl (diff_val_consts.get (i)->get_decl (), val);
@@ -451,7 +453,7 @@ namespace qe {
                         ptr_vector<expr> sel_args;
                         sel_args.push_back (arr1);
                         sel_args.append(idxs.size(), idxs.c_ptr());
-                        expr_ref arr1_idx (m_arr_u.mk_select (sel_args.size (), sel_args.c_ptr ()), m);
+                        expr_ref arr1_idx (m_arr_u.mk_select (sel_args), m);
                         expr_ref eq (m.mk_eq (arr1_idx, x), m);
                         m_aux_lits_v.push_back (eq);
 
@@ -821,7 +823,7 @@ namespace qe {
             ptr_vector<expr> args;
             args.push_back(array);
             args.append(arity, js);
-            expr* r = m_arr_u.mk_select (args.size(), args.c_ptr());
+            expr* r = m_arr_u.mk_select (args);
             m_pinned.push_back (r);
             return r;
         }
@@ -1191,7 +1193,7 @@ namespace qe {
         array_util   a;
         scoped_ptr<contains_app> m_var;
 
-        imp(ast_manager& m): m(m), a(m) {}
+        imp(ast_manager& m): m(m), a(m), m_stores(m) {}
         ~imp() {}
 
         bool solve(model& model, app_ref_vector& vars, expr_ref_vector& lits) {
@@ -1267,7 +1269,7 @@ namespace qe {
 
                     args.push_back (s);
                     args.append(idxs[i].m_values.size(), idxs[i].m_vars);
-                    sel = a.mk_select (args.size (), args.c_ptr ());
+                    sel = a.mk_select (args);
                     val = model(sel);
                     model.register_decl (var->get_decl (), val);
 
@@ -1306,7 +1308,7 @@ namespace qe {
                     }
                     args.push_back(t);
                     args.append(n, s->get_args()+1);
-                    lits.push_back(m.mk_eq(a.mk_select(args.size(), args.c_ptr()), s->get_arg(n+1)));
+                    lits.push_back(m.mk_eq(a.mk_select(args), s->get_arg(n+1)));
                     idxs.push_back(idx);
                     return solve(model, to_app(s->get_arg(0)), t, idxs, vars, lits);
                 case l_undef:
@@ -1357,6 +1359,217 @@ namespace qe {
             }
             return l_undef;
         }
+
+        void saturate(model& model, func_decl_ref_vector const& shared, expr_ref_vector& lits) {
+            term_graph tg(m);
+            tg.set_vars(shared, false);
+            tg.add_model_based_terms(model, lits);
+
+            // need tg to take term and map it to optional rep over the
+            // shared vocabulary if it exists.
+            
+            // . collect shared store expressions, index sorts 
+            // . collect shared index expressions
+            // . assert extensionality (add shared index expressions)
+            // . assert store axioms for collected expressions
+
+            collect_store_expressions(tg, lits);
+            collect_index_expressions(tg, lits);
+
+            TRACE("qe",
+                  tout << "indices\n";
+                  for (auto& kv : m_indices) {
+                      tout << sort_ref(kv.m_key, m) << " |-> " << *kv.m_value << "\n";
+                  }
+                  tout << "stores " << m_stores << "\n";
+                  tout << "arrays\n";
+                  for (auto& kv : m_arrays) {
+                      tout << sort_ref(kv.m_key, m) << " |-> " << *kv.m_value << "\n";
+                  });
+
+            assert_extensionality(model, tg, lits);
+            assert_store_select(model, tg, lits);
+
+            TRACE("qe", tout << lits << "\n";);
+
+            for (auto& kv : m_indices) {
+                dealloc(kv.m_value);
+            }
+            for (auto& kv : m_arrays) {
+                dealloc(kv.m_value);
+            }
+            m_stores.reset();
+            m_indices.reset();
+            m_arrays.reset();
+
+            TRACE("qe", tout << "done: " << lits << "\n";);
+
+        }
+
+        app_ref_vector                  m_stores;
+        obj_map<sort, app_ref_vector*> m_indices;
+        obj_map<sort, app_ref_vector*> m_arrays;
+
+        void add_index_sort(expr* n) {
+            sort* s = m.get_sort(n);
+            if (!m_indices.contains(s)) {
+                m_indices.insert(s, alloc(app_ref_vector, m));
+            }
+        }
+
+        void add_array(app* n) {
+            sort* s = m.get_sort(n);
+            app_ref_vector* vs = nullptr;
+            if (!m_arrays.find(s, vs)) {
+                vs = alloc(app_ref_vector, m);
+                m_arrays.insert(s, vs);
+            }
+            vs->push_back(n);
+        }
+
+        app_ref_vector* is_index(expr* n) {
+            app_ref_vector* result = nullptr;
+            m_indices.find(m.get_sort(n), result);
+            return result;
+        }
+
+        struct for_each_store_proc {
+            imp&               m_imp;
+            term_graph&        tg;
+            for_each_store_proc(imp& i, term_graph& tg) : m_imp(i), tg(tg) {}
+
+            void operator()(app* n) {
+                if (m_imp.a.is_array(n) && tg.get_model_based_rep(n)) {
+                    m_imp.add_array(n);
+                }
+
+                if (m_imp.a.is_store(n) &&
+                    (tg.get_model_based_rep(n->get_arg(0)) ||
+                     tg.get_model_based_rep(n->get_arg(n->get_num_args() - 1)))) {
+                    m_imp.m_stores.push_back(n);
+                    for (unsigned i = 1; i + 1 < n->get_num_args(); ++i) {
+                        m_imp.add_index_sort(n->get_arg(i));
+                    }
+                }
+            }
+            
+            void operator()(expr* e) {}
+        };
+
+        struct for_each_index_proc {
+            imp&               m_imp;
+            term_graph&        tg;
+            for_each_index_proc(imp& i, term_graph& tg) : m_imp(i), tg(tg) {}
+
+            void operator()(app* n) {
+                auto* v = m_imp.is_index(n);
+                if (v && tg.get_model_based_rep(n)) {
+                    v->push_back(n);
+                }
+            }
+            
+            void operator()(expr* e) {}
+
+        };
+
+        void collect_store_expressions(term_graph& tg, expr_ref_vector const& terms) {
+            for_each_store_proc proc(*this, tg);
+            for_each_expr<for_each_store_proc>(proc, terms);
+        }
+
+        void collect_index_expressions(term_graph& tg, expr_ref_vector const& terms) {
+            for_each_index_proc proc(*this, tg);
+            for_each_expr<for_each_index_proc>(proc, terms);
+        }
+
+        bool are_equal(model& mdl, expr* s, expr* t) {
+            return mdl.are_equal(s, t);
+        }
+
+        void assert_extensionality(model & mdl, term_graph& tg, expr_ref_vector& lits) {
+            for (auto& kv : m_arrays) {
+                app_ref_vector const& vs = *kv.m_value;
+                if (vs.size() <= 1) continue;
+                func_decl_ref_vector ext(m);
+                sort* s = kv.m_key;
+                unsigned arity = get_array_arity(s);
+                for (unsigned i = 0; i < arity; ++i) {
+                    ext.push_back(a.mk_array_ext(s, i));
+                } 
+                expr_ref_vector args(m);
+                args.resize(arity + 1);
+                for (unsigned i = 0; i < vs.size(); ++i) {
+                    expr* s = vs[i];
+                    for (unsigned j = i + 1; j < vs.size(); ++j) {
+                        expr* t = vs[j];
+                        if (are_equal(mdl, s, t)) {
+                            lits.push_back(m.mk_eq(s, t));
+                        }
+                        else {
+                            for (unsigned k = 0; k < arity; ++k) {
+                                args[k+1] = m.mk_app(ext.get(k), s, t);
+                            }
+                            args[0] = t;
+                            expr* t1 = a.mk_select(args);
+                            args[0] = s;
+                            expr* s1 = a.mk_select(args);
+                            lits.push_back(m.mk_not(m.mk_eq(t1, s1)));
+                        }
+                    }
+                }
+            }
+        }
+
+        void assert_store_select(model & mdl, term_graph& tg, expr_ref_vector& lits) {
+            for (auto& store : m_stores) {
+                assert_store_select(store, mdl, tg, lits);                
+            }
+        }
+
+        void assert_store_select(app* store, model & mdl, term_graph& tg, expr_ref_vector& lits) {
+            SASSERT(a.is_store(store));
+            ptr_vector<app> indices;
+            for (unsigned i = 1; i + 1 < store->get_num_args(); ++i) {
+                SASSERT(indices.empty());
+                assert_store_select(indices, store, mdl, tg, lits);
+            }
+        }
+
+        void assert_store_select(ptr_vector<app>& indices, app* store, model & mdl, term_graph& tg, expr_ref_vector& lits) {
+            unsigned sz = store->get_num_args();
+            if (indices.size() + 2 == sz) {
+                ptr_vector<expr> args;
+                args.push_back(store);
+                for (expr* idx : indices) args.push_back(idx);
+                for (unsigned i = 1; i + 1 < sz; ++i) {
+                    expr* idx1 = store->get_arg(i);
+                    expr* idx2 = indices[i - 1];
+                    if (!are_equal(mdl, idx1, idx2)) {
+                        lits.push_back(m.mk_not(m.mk_eq(idx1, idx2)));
+                        lits.push_back(m.mk_eq(store->get_arg(sz-1), a.mk_select(args)));
+                        return;
+                    }
+                }
+                for (unsigned i = 1; i + 1 < sz; ++i) {
+                    expr* idx1 = store->get_arg(i);
+                    expr* idx2 = indices[i - 1];
+                    lits.push_back(m.mk_eq(idx1, idx2));
+                }
+                expr* a1 = a.mk_select(args);
+                args[0] = store->get_arg(0);
+                expr* a2 = a.mk_select(args);
+                lits.push_back(m.mk_eq(a1, a2));
+            }
+            else {
+                sort* s = m.get_sort(store->get_arg(indices.size() + 1));
+                for (app* idx : *m_indices[s]) {
+                    indices.push_back(idx);
+                    assert_store_select(indices, store, mdl, tg, lits);
+                    indices.pop_back();
+                }
+            }
+        }
+
     };
 
 
@@ -1417,4 +1630,9 @@ namespace qe {
         return vector<def>();
     }
 
+    void array_project_plugin::saturate(model& model, func_decl_ref_vector const& shared, expr_ref_vector& lits) {
+        m_imp->saturate(model, shared, lits);
+    }
+
+
 };
diff --git a/src/qe/qe_arrays.h b/src/qe/qe_arrays.h
index 3bb90335d..84e318426 100644
--- a/src/qe/qe_arrays.h
+++ b/src/qe/qe_arrays.h
@@ -37,6 +37,8 @@ namespace qe {
         void operator()(model& model, app_ref_vector& vars, expr_ref& fml, app_ref_vector& aux_vars, bool reduce_all_selects);
         family_id get_family_id() override;
         vector<def> project(model& model, app_ref_vector& vars, expr_ref_vector& lits) override;
+        void saturate(model& model, func_decl_ref_vector const& shared, expr_ref_vector& lits) override;
+
     };
 
 };
diff --git a/src/qe/qe_datatypes.cpp b/src/qe/qe_datatypes.cpp
index 4109d7fd9..85bb14640 100644
--- a/src/qe/qe_datatypes.cpp
+++ b/src/qe/qe_datatypes.cpp
@@ -303,6 +303,11 @@ namespace qe {
     vector<def> datatype_project_plugin::project(model& model, app_ref_vector& vars, expr_ref_vector& lits) {
         return vector<def>();
     }
+
+    void datatype_project_plugin::saturate(model& model, func_decl_ref_vector const& shared, expr_ref_vector& lits) {
+        NOT_IMPLEMENTED_YET();
+    }
+
     
     family_id datatype_project_plugin::get_family_id() {
         return m_imp->dt.get_family_id();
diff --git a/src/qe/qe_datatypes.h b/src/qe/qe_datatypes.h
index 0483f4cce..50a3930e9 100644
--- a/src/qe/qe_datatypes.h
+++ b/src/qe/qe_datatypes.h
@@ -36,6 +36,8 @@ namespace qe {
         bool solve(model& model, app_ref_vector& vars, expr_ref_vector& lits) override;
         family_id get_family_id() override;
         vector<def> project(model& model, app_ref_vector& vars, expr_ref_vector& lits) override;
+        void saturate(model& model, func_decl_ref_vector const& shared, expr_ref_vector& lits) override;
+
     };
 
 };
diff --git a/src/qe/qe_mbi.cpp b/src/qe/qe_mbi.cpp
index 750dedbce..71a783c4f 100644
--- a/src/qe/qe_mbi.cpp
+++ b/src/qe/qe_mbi.cpp
@@ -38,7 +38,7 @@ Notes:
 #include "qe/qe_mbi.h"
 #include "qe/qe_term_graph.h"
 #include "qe/qe_arith.h"
-// include "opt/opt_context.h"
+#include "qe/qe_arrays.h"
 
 
 namespace qe {
@@ -263,6 +263,9 @@ namespace qe {
         TRACE("qe", tout << lits << "\n" << *mdl << "\n";);
         TRACE("qe", tout << m_solver->get_assertions() << "\n";);
 
+        // 0. saturation
+        array_project_plugin arp(m);
+        arp.saturate(*mdl, m_shared, lits);
 
         // . arithmetical variables - atomic and in purified positions
         app_ref_vector proxies(m);
diff --git a/src/qe/qe_mbp.h b/src/qe/qe_mbp.h
index 0bb8ba00f..1b7dedbd4 100644
--- a/src/qe/qe_mbp.h
+++ b/src/qe/qe_mbp.h
@@ -57,6 +57,13 @@ namespace qe {
         */
         virtual vector<def> project(model& model, app_ref_vector& vars, expr_ref_vector& lits) = 0;
 
+        /**
+           \brief model based saturation. Saturates theory axioms to equi-satisfiable literals over EUF,
+           such that 'shared' are not retained for EUF.
+         */
+        virtual void saturate(model& model, func_decl_ref_vector const& shared, expr_ref_vector& lits) = 0;
+
+
         static expr_ref pick_equality(ast_manager& m, model& model, expr* t);
         static void erase(expr_ref_vector& lits, unsigned& i);
         static void push_back(expr_ref_vector& lits, expr* lit);
diff --git a/src/qe/qe_term_graph.cpp b/src/qe/qe_term_graph.cpp
index 274c25293..166b18bc9 100644
--- a/src/qe/qe_term_graph.cpp
+++ b/src/qe/qe_term_graph.cpp
@@ -217,6 +217,7 @@ namespace qe {
     bool term_graph::is_variable_proc::operator()(const expr * e) const {
         if (!is_app(e)) return false;
         const app *a = ::to_app(e);
+        TRACE("qe", tout << a->get_family_id() << " " << m_solved.contains(a->get_decl()) << " " << m_decls.contains(a->get_decl()) << "\n";);
         return
             a->get_family_id() == null_family_id &&
             !m_solved.contains(a->get_decl()) &&
@@ -242,12 +243,13 @@ namespace qe {
 
     bool term_graph::term_eq::operator()(term const* a, term const* b) const { return term::cg_eq(a, b); }
 
-    term_graph::term_graph(ast_manager &man) : m(man), m_lits(m), m_pinned(m) {
+    term_graph::term_graph(ast_manager &man) : m(man), m_lits(m), m_pinned(m), m_projector(nullptr) {
         m_plugins.register_plugin(mk_basic_solve_plugin(m, m_is_var));
         m_plugins.register_plugin(mk_arith_solve_plugin(m, m_is_var));
     }
 
     term_graph::~term_graph() {
+        dealloc(m_projector);
         reset();
     }
 
@@ -582,12 +584,14 @@ namespace qe {
         u_map<expr*> m_term2app;
         u_map<expr*> m_root2rep;
 
+
         model_ref m_model;
         expr_ref_vector m_pinned;  // tracks expr in the maps
 
         expr* mk_pure(term const& t) {
+            TRACE("qe", t.display(tout););
             expr* e = nullptr;
-            if (m_term2app.find(t.get_id(), e)) return e;
+            if (find_term2app(t, e)) return e;
             e = t.get_expr();
             if (!is_app(e)) return nullptr;
             app* a = ::to_app(e);
@@ -595,17 +599,20 @@ namespace qe {
             for (term* ch : term::children(t)) {
                 // prefer a node that resembles current child, 
                 // otherwise, pick a root representative, if present.
-                if (m_term2app.find(ch->get_id(), e))
-                kids.push_back(e);
-                else if (m_root2rep.find(ch->get_root().get_id(), e)) 
+                if (find_term2app(*ch, e)) {
                     kids.push_back(e);
-                else 
+                }
+                else if (m_root2rep.find(ch->get_root().get_id(), e)) {
+                    kids.push_back(e);
+                }
+                else {
                     return nullptr;
+                }
                 TRACE("qe_verbose", tout << *ch << " -> " << mk_pp(e, m) << "\n";);
             }
             expr* pure = m.mk_app(a->get_decl(), kids.size(), kids.c_ptr());
             m_pinned.push_back(pure);
-            m_term2app.insert(t.get_id(), pure);
+            add_term2app(t, pure);
             return pure;
         }
 
@@ -621,69 +628,15 @@ namespace qe {
             }
         };           
 
-        void purify() {
-            // - propagate representatives up over parents.
-            //   use work-list + marking to propagate.
-            // - produce equalities over represented classes.
-            // - produce other literals over represented classes
-            //   (walk disequalities in m_lits and represent
-            //   lhs/rhs over decls or excluding decls)
-
-            ptr_vector<term> worklist;
-            for (term * t : m_tg.m_terms) {
-                worklist.push_back(t);
-                t->set_mark(true);
-            }
-            // traverse worklist in order of depth.
-            term_depth td;
-            std::sort(worklist.begin(), worklist.end(), td);
-
-            for (unsigned i = 0; i < worklist.size(); ++i) {
-                term* t = worklist[i];
-                t->set_mark(false);
-                if (m_term2app.contains(t->get_id())) 
-                    continue;
-                if (!t->is_theory() && is_projected(*t))
-                    continue;
-
-                expr* pure = mk_pure(*t);
-                if (!pure) continue;
-
-                m_term2app.insert(t->get_id(), pure);
-                TRACE("qe_verbose", tout << "purified " << *t << " " << mk_pp(pure, m) << "\n";);
-                expr* rep = nullptr;                // ensure that the root has a representative
-                m_root2rep.find(t->get_root().get_id(), rep);
-
-                // update rep with pure if it is better
-                if (pure != rep && is_better_rep(pure, rep)) {
-                    m_root2rep.insert(t->get_root().get_id(), pure);
-                    for (term * p : term::parents(t->get_root())) {
-                        m_term2app.remove(p->get_id());
-                        if (!p->is_marked()) {
-                            p->set_mark(true);
-                            worklist.push_back(p);
-                        }
-                    }
-                }
-            }
-
-            // Here we could also walk equivalence classes that
-            // contain interpreted values by sort and extract
-            // disequalities between non-unique value
-            // representatives.  these disequalities are implied
-            // and can be mined using other means, such as theory
-            // aware core minimization
-            m_tg.reset_marks();
-            TRACE("qe", display(tout << "after purify\n"););
-        }
 
         void solve_core() {
             ptr_vector<term> worklist;
             for (term * t : m_tg.m_terms) {
                 // skip pure terms
-                if (m_term2app.contains(t->get_id())) continue;
-                worklist.push_back(t);
-                t->set_mark(true);
+                if (!in_term2app(*t)) {
+                    worklist.push_back(t);
+                    t->set_mark(true);
+                }
             }
             term_depth td;
             std::sort(worklist.begin(), worklist.end(), td);
@@ -691,13 +644,14 @@ namespace qe {
             for (unsigned i = 0; i < worklist.size(); ++i) {
                 term* t = worklist[i];
                 t->set_mark(false);
-                if (m_term2app.contains(t->get_id()))
+                if (in_term2app(*t))
                     continue;
 
                 expr* pure = mk_pure(*t);
-                if (!pure) continue;
+                if (!pure) 
+                    continue;
 
-                m_term2app.insert(t->get_id(), pure);
+                add_term2app(*t, pure);
                 expr* rep = nullptr;
                 // ensure that the root has a representative
                 m_root2rep.find(t->get_root().get_id(), rep);
@@ -705,7 +659,7 @@ namespace qe {
                 if (!rep) {
                     m_root2rep.insert(t->get_root().get_id(), pure);
                     for (term * p : term::parents(t->get_root())) {
-                        SASSERT(!m_term2app.contains(p->get_id()));
+                        SASSERT(!in_term2app(*p));
                         if (!p->is_marked()) {
                             p->set_mark(true);
                             worklist.push_back(p);
@@ -718,14 +672,14 @@ namespace qe {
 
         bool find_app(term &t, expr *&res) {
             return 
-                m_term2app.find(t.get_id(), res) ||
+                find_term2app(t, res) ||
                 m_root2rep.find(t.get_root().get_id(), res);
         }
 
         bool find_app(expr *lit, expr *&res) {
             term const* t = m_tg.get_term(lit);
             return 
-                m_term2app.find(t->get_id(), res) ||
+                find_term2app(*t, res) ||
                 m_root2rep.find(t->get_root().get_id(), res);
         }
 
@@ -856,7 +810,7 @@ namespace qe {
             term const * r = &t;
             do {
                 expr* member = nullptr;
-                if (m_term2app.find(r->get_id(), member) && !members.contains(member)) {
+                if (find_term2app(*r, member) && !members.contains(member)) {
                     res.push_back (m.mk_eq (rep, member));
                     members.insert(member);
                 }
@@ -865,7 +819,9 @@ namespace qe {
             while (r != &t);
         }
 
-        bool is_projected(const term &t) {return m_tg.m_is_var(t);}
+        bool is_projected(const term &t) {
+            return m_tg.m_is_var(t);
+        }
 
         void mk_unpure_equalities(const term &t, expr_ref_vector &res) {
             expr *rep = nullptr;
@@ -981,6 +937,28 @@ namespace qe {
     public:
         projector(term_graph &tg) : m_tg(tg), m(m_tg.m), m_pinned(m) {}
 
+        void add_term2app(term const& t, expr* a) {
+            m_term2app.insert(t.get_id(), a);
+        }
+
+        void del_term2app(term const& t) {
+            m_term2app.remove(t.get_id());
+        }
+
+        bool find_term2app(term const& t, expr*& r) {
+            return m_term2app.find(t.get_id(), r);
+        }
+
+        expr* find_term2app(term const& t) {
+            expr* r = nullptr;
+            find_term2app(t, r);
+            return r;
+        }
+
+        bool in_term2app(term const& t) {
+            return m_term2app.contains(t.get_id());
+        }
+
         void set_model(model &mdl) { m_model = &mdl; }
 
         void reset() {
@@ -1025,7 +1003,7 @@ namespace qe {
             return res;
         }
 
-        vector<expr_ref_vector> get_partition(model& mdl) {
+        vector<expr_ref_vector> get_partition(model& mdl, bool include_bool) {
             vector<expr_ref_vector> result;
             expr_ref_vector pinned(m);
             obj_map<expr, unsigned> pid;
@@ -1033,7 +1011,7 @@ namespace qe {
             for (term *t : m_tg.m_terms) {
                 expr* a = t->get_expr();
                 if (!is_app(a)) continue;
-                if (m.is_bool(a)) continue;
+                if (m.is_bool(a) && !include_bool) continue;
                 expr_ref val = mdl(a);
                 unsigned p = 0;
                 // NB. works for simple domains Integers, Rationals, 
@@ -1065,6 +1043,63 @@ namespace qe {
             }
             return result;
         }
+
+        void purify() {
+            // - propagate representatives up over parents.
+            //   use work-list + marking to propagate.
+            // - produce equalities over represented classes.
+            // - produce other literals over represented classes
+            //   (walk disequalities in m_lits and represent
+            //   lhs/rhs over decls or excluding decls)
+
+            ptr_vector<term> worklist;
+            for (term * t : m_tg.m_terms) {
+                worklist.push_back(t);
+                t->set_mark(true);
+            }
+            // traverse worklist in order of depth.
+            term_depth td;
+            std::sort(worklist.begin(), worklist.end(), td);
+
+            for (unsigned i = 0; i < worklist.size(); ++i) {
+                term* t = worklist[i];
+                t->set_mark(false);
+                if (in_term2app(*t)) 
+                    continue;
+                if (!t->is_theory() && is_projected(*t))
+                    continue;
+
+                expr* pure = mk_pure(*t);
+                if (!pure) continue;
+
+                add_term2app(*t, pure);
+                TRACE("qe_verbose", tout << "purified " << *t << " " << mk_pp(pure, m) << "\n";);
+                expr* rep = nullptr;                // ensure that the root has a representative
+                m_root2rep.find(t->get_root().get_id(), rep);
+
+                // update rep with pure if it is better
+                if (pure != rep && is_better_rep(pure, rep)) {
+                    m_root2rep.insert(t->get_root().get_id(), pure);
+                    for (term * p : term::parents(t->get_root())) {
+                        del_term2app(*p);
+                        if (!p->is_marked()) {
+                            p->set_mark(true);
+                            worklist.push_back(p);
+                        }
+                    }
+                }
+            }
+
+            // Here we could also walk equivalence classes that
+            // contain interpreted values by sort and extract
+            // disequalities between non-unique value
+            // representatives.  these disequalities are implied
+            // and can be mined using other means, such as theory
+            // aware core minimization
+            m_tg.reset_marks();
+            TRACE("qe", display(tout << "after purify\n"););
+        }
+
     };
 
     void term_graph::set_vars(func_decl_ref_vector const& decls, bool exclude) {
@@ -1094,13 +1129,15 @@ namespace qe {
 
     expr_ref_vector term_graph::get_ackerman_disequalities() {
         m_is_var.reset_solved();
-        term_graph::projector p(*this);
-        return p.get_ackerman_disequalities();
+        dealloc(m_projector);
+        m_projector = alloc(term_graph::projector, *this);
+        return m_projector->get_ackerman_disequalities();
     }
 
     vector<expr_ref_vector> term_graph::get_partition(model& mdl) {
-        term_graph::projector p(*this);
-        return p.get_partition(mdl);
+        dealloc(m_projector);
+        m_projector = alloc(term_graph::projector, *this);
+        return m_projector->get_partition(mdl, false);
     }
 
     expr_ref_vector term_graph::shared_occurrences(family_id fid) {
@@ -1108,4 +1145,42 @@ namespace qe {
         return p.shared_occurrences(fid);        
     }
 
+    void term_graph::add_model_based_terms(model& mdl, expr_ref_vector const& terms) {
+        for (expr* t : terms) {
+            internalize_term(t);
+        }
+        m_is_var.reset_solved();
+        
+        SASSERT(!m_projector);
+        m_projector = alloc(term_graph::projector, *this);        
+
+        // retrieve partition of terms
+        vector<expr_ref_vector> equivs = m_projector->get_partition(mdl, true);
+
+        // merge term graph on equal terms.
+        for (auto const& cs : equivs) {
+            term* t0 = get_term(cs[0]);
+            for (unsigned i = 1; i < cs.size(); ++i) {
+                merge(*t0, *get_term(cs[i]));
+            }
+        }
+        TRACE("qe", 
+              for (auto & es : equivs) {
+                  tout << "equiv: ";
+                  for (expr* t : es) tout << expr_ref(t, m) << " ";
+                  tout << "\n";
+              }
+              display(tout););
+        // create representatives for shared/projected variables.
+        m_projector->set_model(mdl);
+        m_projector->purify();
+
+    }
+
+    expr* term_graph::get_model_based_rep(expr* e) {
+        SASSERT(m_projector);
+        term* t = get_term(e);
+        SASSERT(t && "only get representatives");
+        return m_projector->find_term2app(*t);
+    }
 }
diff --git a/src/qe/qe_term_graph.h b/src/qe/qe_term_graph.h
index 855a0f2bc..ef12ab683 100644
--- a/src/qe/qe_term_graph.h
+++ b/src/qe/qe_term_graph.h
@@ -52,6 +52,7 @@ namespace qe {
         expr_ref_vector    m_lits; // NSB: expr_ref_vector?
         u_map<term* >     m_app2term;
         ast_ref_vector    m_pinned;
+        projector*        m_projector;
         u_map<expr*>      m_term2app;
         plugin_manager<qe::solve_plugin> m_plugins;
         ptr_hashtable<term, term_hash, term_eq> m_cg_table;
@@ -135,6 +136,12 @@ namespace qe {
          */
         expr_ref_vector shared_occurrences(family_id fid);
 
+        /**
+         * Map expression that occurs in added literals into representative if it exists.
+         */
+        void  add_model_based_terms(model& mdl, expr_ref_vector const& terms);
+        expr* get_model_based_rep(expr* e);
+
     };
 
 }