diff --git a/src/api/api_solver.cpp b/src/api/api_solver.cpp
index b93fb42af..2ca54a599 100644
--- a/src/api/api_solver.cpp
+++ b/src/api/api_solver.cpp
@@ -903,6 +903,26 @@ extern "C" {
         Z3_CATCH_RETURN(nullptr);
     }
 
+    Z3_ast Z3_API Z3_solver_congruence_root(Z3_context c, Z3_solver s, Z3_ast a) {
+        Z3_TRY;
+        LOG_Z3_solver_congruence_root(c, s, a);
+        RESET_ERROR_CODE();
+        init_solver(c, s);
+        expr* r = to_solver_ref(s)->congruence_root(to_expr(a));
+        RETURN_Z3(of_expr(r));
+        Z3_CATCH_RETURN(nullptr);
+    }
+
+    Z3_ast Z3_API Z3_solver_congruence_next(Z3_context c, Z3_solver s, Z3_ast a) {
+        Z3_TRY;
+        LOG_Z3_solver_congruence_next(c, s, a);
+        RESET_ERROR_CODE();
+        init_solver(c, s);
+        expr* sib = to_solver_ref(s)->congruence_next(to_expr(a));
+        RETURN_Z3(of_expr(sib));
+        Z3_CATCH_RETURN(nullptr);
+    }
+
     class api_context_obj : public user_propagator::context_obj {
         api::context* c;
     public:
diff --git a/src/api/python/z3/z3.py b/src/api/python/z3/z3.py
index 1f24222ad..695bb939f 100644
--- a/src/api/python/z3/z3.py
+++ b/src/api/python/z3/z3.py
@@ -7241,6 +7241,22 @@ class Solver(Z3PPObject):
         cube are likely more useful to cube on."""
         return self.cube_vs
 
+    def root(self, t):
+        t = _py2expr(t, self.ctx)
+        """Retrieve congruence closure root of the term t relative to the current search state
+        The function primarily works for SimpleSolver. Terms and variables that are
+        eliminated during pre-processing are not visible to the congruence closure.
+        """
+        return _to_expr_ref(Z3_solver_congruence_root(self.ctx.ref(), self.solver, t.ast), self.ctx)
+
+    def next(self, t):
+        t = _py2expr(t, self.ctx)
+        """Retrieve congruence closure sibling of the term t relative to the current search state
+        The function primarily works for SimpleSolver. Terms and variables that are
+        eliminated during pre-processing are not visible to the congruence closure.
+        """
+        return _to_expr_ref(Z3_solver_congruence_next(self.ctx.ref(), self.solver, t.ast), self.ctx)
+
     def proof(self):
         """Return a proof for the last `check()`. Proof construction must be enabled."""
         return _to_expr_ref(Z3_solver_get_proof(self.ctx.ref(), self.solver), self.ctx)
diff --git a/src/api/z3_api.h b/src/api/z3_api.h
index 7a0b47da0..ffa0d8665 100644
--- a/src/api/z3_api.h
+++ b/src/api/z3_api.h
@@ -6882,6 +6882,26 @@ extern "C" {
     */
     void Z3_API Z3_solver_get_levels(Z3_context c, Z3_solver s, Z3_ast_vector literals, unsigned sz,  unsigned levels[]);
 
+    /**
+       \brief retrieve the congruence closure root of an expression.
+       The root is retrieved relative to the state where the solver was in when it completed.
+       If it completed during a set of case splits, the congruence roots are relative to these case splits.
+       That is, the congruences are not consequences but they are true under the current state.
+
+       def_API('Z3_solver_congruence_root', AST, (_in(CONTEXT), _in(SOLVER), _in(AST)))
+    */
+    Z3_ast Z3_API Z3_solver_congruence_root(Z3_context c, Z3_solver s, Z3_ast a);
+
+
+    /**
+       \brief retrieve the next expression in the congruence class. The set of congruent siblings form a cyclic list.
+       Repeated calls on the siblings will result in returning to the original expression.
+
+       def_API('Z3_solver_congruence_next', AST, (_in(CONTEXT), _in(SOLVER), _in(AST)))
+    */
+    Z3_ast Z3_API Z3_solver_congruence_next(Z3_context c, Z3_solver s, Z3_ast a);
+
+
     /**
        \brief register a callback to that retrieves assumed, inferred and deleted clauses during search.
        
diff --git a/src/ast/converters/expr_inverter.cpp b/src/ast/converters/expr_inverter.cpp
index 5553420ad..41a60ccd5 100644
--- a/src/ast/converters/expr_inverter.cpp
+++ b/src/ast/converters/expr_inverter.cpp
@@ -81,7 +81,7 @@ public:
      *
      */
 
-    bool operator()(func_decl* f, unsigned num, expr* const* args, expr_ref& r, expr_ref& side_cond) override {
+    bool operator()(func_decl* f, unsigned num, expr* const* args, expr_ref& r, proof_ref& pr) override {
         SASSERT(f->get_family_id() == m.get_basic_family_id());
         switch (f->get_decl_kind()) {
         case OP_ITE:
@@ -233,7 +233,7 @@ public:
     }
 
 
-    bool operator()(func_decl* f, unsigned num, expr* const* args, expr_ref& r, expr_ref& side_cond) override {
+    bool operator()(func_decl* f, unsigned num, expr* const* args, expr_ref& r, proof_ref& pr) override {
         SASSERT(f->get_family_id() == a.get_family_id());
         switch (f->get_decl_kind()) {
         case OP_ADD:
@@ -531,7 +531,7 @@ class bv_expr_inverter : public iexpr_inverter {
      * y := 0
      *
      */
-    bool operator()(func_decl* f, unsigned num, expr* const* args, expr_ref& r, expr_ref& side_cond) override {
+    bool operator()(func_decl* f, unsigned num, expr* const* args, expr_ref& r, proof_ref& pr) override {
         SASSERT(f->get_family_id() == bv.get_family_id());
         switch (f->get_decl_kind()) {
         case OP_BADD:
@@ -611,7 +611,7 @@ public:
 
     family_id get_fid() const override { return a.get_family_id(); }
 
-    bool operator()(func_decl* f, unsigned num, expr* const* args, expr_ref& r, expr_ref& side_cond) override {
+    bool operator()(func_decl* f, unsigned num, expr* const* args, expr_ref& r, proof_ref& pr) override {
         SASSERT(f->get_family_id() == a.get_family_id());
         switch (f->get_decl_kind()) {
         case OP_SELECT:
@@ -679,7 +679,7 @@ public:
      *   head(x) -> fresh
      *   x := cons(fresh, arb)
      */
-    bool operator()(func_decl* f, unsigned num, expr* const* args, expr_ref& r, expr_ref& side_cond) override {
+    bool operator()(func_decl* f, unsigned num, expr* const* args, expr_ref& r, proof_ref& pr) override {
         if (dt.is_accessor(f)) {
             SASSERT(num == 1);
             if (uncnstr(args[0])) {
@@ -799,7 +799,7 @@ expr_inverter::expr_inverter(ast_manager& m): iexpr_inverter(m) {
 }
 
 
-bool expr_inverter::operator()(func_decl* f, unsigned num, expr* const* args, expr_ref& new_expr, expr_ref& side_cond) {
+bool expr_inverter::operator()(func_decl* f, unsigned num, expr* const* args, expr_ref& new_expr, proof_ref& pr) {
     if (num == 0)
         return false;
             
@@ -812,7 +812,7 @@ bool expr_inverter::operator()(func_decl* f, unsigned num, expr* const* args, ex
         return false;
 
     auto* p = m_inverters.get(fid, nullptr);
-    return p && (*p)(f, num, args, new_expr, side_cond);       
+    return p && (*p)(f, num, args, new_expr, pr);       
 }
 
 bool expr_inverter::mk_diff(expr* t, expr_ref& r) {
@@ -849,3 +849,10 @@ void expr_inverter::set_model_converter(generic_model_converter* mc) {
         if (p)
             p->set_model_converter(mc);
 }
+
+void expr_inverter::set_produce_proofs(bool pr) {
+    m_produce_proofs = pr;
+    for (auto* p : m_inverters)
+        if (p)
+            p->set_produce_proofs(pr);
+}
diff --git a/src/ast/converters/expr_inverter.h b/src/ast/converters/expr_inverter.h
index 5b7965478..60540aff3 100644
--- a/src/ast/converters/expr_inverter.h
+++ b/src/ast/converters/expr_inverter.h
@@ -24,6 +24,7 @@ protected:
     ast_manager&                m;
     std::function<bool(expr*)>  m_is_var;
     generic_model_converter_ref m_mc;
+    bool                        m_produce_proofs = false;
 
     bool uncnstr(expr* e) const { return m_is_var(e); }
     bool uncnstr(unsigned num, expr * const * args) const;
@@ -37,8 +38,9 @@ public:
     virtual ~iexpr_inverter() {}
     virtual void set_is_var(std::function<bool(expr*)>& is_var) { m_is_var = is_var; }
     virtual void set_model_converter(generic_model_converter* mc) { m_mc = mc; }
+    virtual void set_produce_proofs(bool p) { m_produce_proofs = true; }
     
-    virtual bool operator()(func_decl* f, unsigned n, expr* const* args, expr_ref& new_expr, expr_ref& side_cond) = 0;
+    virtual bool operator()(func_decl* f, unsigned n, expr* const* args, expr_ref& new_expr, proof_ref& pr) = 0;
     virtual bool mk_diff(expr* t, expr_ref& r) = 0;
     virtual family_id get_fid() const = 0;
 };
@@ -49,9 +51,10 @@ class expr_inverter : public iexpr_inverter {
 public:
     expr_inverter(ast_manager& m);
     ~expr_inverter() override;
-    bool operator()(func_decl* f, unsigned n, expr* const* args, expr_ref& new_expr, expr_ref& side_cond) override;
+    bool operator()(func_decl* f, unsigned n, expr* const* args, expr_ref& new_expr, proof_ref& pr) override;
     bool mk_diff(expr* t, expr_ref& r) override;
     void set_is_var(std::function<bool(expr*)>& is_var) override; 
     void set_model_converter(generic_model_converter* mc) override;
+    void set_produce_proofs(bool p) override;
     family_id get_fid() const override { return null_family_id;  }
 };
diff --git a/src/ast/simplifiers/elim_unconstrained.cpp b/src/ast/simplifiers/elim_unconstrained.cpp
index df6f92251..41a905dea 100644
--- a/src/ast/simplifiers/elim_unconstrained.cpp
+++ b/src/ast/simplifiers/elim_unconstrained.cpp
@@ -62,7 +62,8 @@ bool elim_unconstrained::is_var_lt(int v1, int v2) const {
 void elim_unconstrained::eliminate() {
 
     while (!m_heap.empty()) {
-        expr_ref r(m), side_cond(m);
+        expr_ref r(m);
+        proof_ref pr(m);
         int v = m_heap.erase_min();
         node& n = get_node(v);
         if (n.m_refcount == 0)
@@ -84,7 +85,7 @@ void elim_unconstrained::eliminate() {
         unsigned sz = m_args.size();
         for (expr* arg : *to_app(t))
             m_args.push_back(reconstruct_term(get_node(arg)));
-        bool inverted = m_inverter(t->get_decl(), to_app(t)->get_num_args(), m_args.data() + sz, r, side_cond);
+        bool inverted = m_inverter(t->get_decl(), to_app(t)->get_num_args(), m_args.data() + sz, r, pr);
         n.m_refcount = 0;
         m_args.shrink(sz);
         if (!inverted) {
@@ -113,7 +114,7 @@ void elim_unconstrained::eliminate() {
 
         IF_VERBOSE(11, verbose_stream() << mk_bounded_pp(get_node(v).m_orig, m) << " " << mk_bounded_pp(t, m) << " -> " << r << " " << get_node(e).m_refcount << "\n";);
 
-        SASSERT(!side_cond && "not implemented to add side conditions\n");
+        SASSERT(!pr && "not implemented to add proofs\n");
     }
 }
 
diff --git a/src/muz/spacer/spacer_iuc_solver.h b/src/muz/spacer/spacer_iuc_solver.h
index 0d4712215..e201a1fe1 100644
--- a/src/muz/spacer/spacer_iuc_solver.h
+++ b/src/muz/spacer/spacer_iuc_solver.h
@@ -122,6 +122,8 @@ public:
     void set_phase(phase* p) override { m_solver.set_phase(p); }
     void move_to_front(expr* e) override { m_solver.move_to_front(e); }
     expr_ref_vector cube(expr_ref_vector&, unsigned) override { return expr_ref_vector(m); }
+    expr* congruence_root(expr* e) override { return e; }
+    expr* congruence_next(expr* e) override { return e; }
     void get_levels(ptr_vector<expr> const& vars, unsigned_vector& depth) override { m_solver.get_levels(vars, depth); }
     expr_ref_vector get_trail(unsigned max_level) override { return m_solver.get_trail(max_level); }
 
diff --git a/src/opt/opt_solver.h b/src/opt/opt_solver.h
index 84d31ed0f..2682fca09 100644
--- a/src/opt/opt_solver.h
+++ b/src/opt/opt_solver.h
@@ -110,6 +110,8 @@ namespace opt {
         void get_levels(ptr_vector<expr> const& vars, unsigned_vector& depth) override; 
         expr_ref_vector get_trail(unsigned max_level) override { return m_context.get_trail(max_level); }
         expr_ref_vector cube(expr_ref_vector&, unsigned) override { return expr_ref_vector(m); }
+        expr* congruence_root(expr* e) override { return e; }
+        expr* congruence_next(expr* e) override { return e; }
         void set_phase(expr* e) override { m_context.set_phase(e); }
         phase* get_phase() override { return m_context.get_phase(); }
         void set_phase(phase* p) override { m_context.set_phase(p); }
diff --git a/src/sat/sat_solver/inc_sat_solver.cpp b/src/sat/sat_solver/inc_sat_solver.cpp
index 41b8e609c..0362d8d3e 100644
--- a/src/sat/sat_solver/inc_sat_solver.cpp
+++ b/src/sat/sat_solver/inc_sat_solver.cpp
@@ -463,6 +463,10 @@ public:
         }
         return fmls;
     }
+
+    expr* congruence_next(expr* e) override { return e; }
+    expr* congruence_root(expr* e) override { return e; }
+
     
     lbool get_consequences_core(expr_ref_vector const& assumptions, expr_ref_vector const& vars, expr_ref_vector& conseq) override {
         init_preprocess();
diff --git a/src/sat/sat_solver/sat_smt_solver.cpp b/src/sat/sat_solver/sat_smt_solver.cpp
index e37d513a0..f5872b05a 100644
--- a/src/sat/sat_solver/sat_smt_solver.cpp
+++ b/src/sat/sat_solver/sat_smt_solver.cpp
@@ -476,6 +476,9 @@ public:
             set_reason_unknown(m_solver.get_reason_unknown());
         return fmls;
     }
+
+    expr* congruence_next(expr* e) override { return e; }
+    expr* congruence_root(expr* e) override { return e; }
     
 
     lbool find_mutexes(expr_ref_vector const& vars, vector<expr_ref_vector>& mutexes) override {
diff --git a/src/smt/smt_kernel.cpp b/src/smt/smt_kernel.cpp
index ae3338f52..c4ecf6787 100644
--- a/src/smt/smt_kernel.cpp
+++ b/src/smt/smt_kernel.cpp
@@ -213,6 +213,20 @@ namespace smt {
         return out;
     }
 
+    expr* kernel::congruence_root(expr * e) {
+        smt::enode* n = m_imp->m_kernel.find_enode(e);
+        if (!n)
+            return e;
+        return n->get_root()->get_expr();
+    }
+
+    expr* kernel::congruence_next(expr * e) {
+        smt::enode* n = m_imp->m_kernel.find_enode(e);
+        if (!n)
+            return e;
+        return n->get_next()->get_expr();
+    }
+
     void kernel::collect_statistics(::statistics & st) const {
         m_imp->m_kernel.collect_statistics(st);
     }
diff --git a/src/smt/smt_kernel.h b/src/smt/smt_kernel.h
index fa4a48406..ccea5caf8 100644
--- a/src/smt/smt_kernel.h
+++ b/src/smt/smt_kernel.h
@@ -239,6 +239,13 @@ namespace smt {
         */
         expr_ref_vector cubes(unsigned depth);
 
+        /**
+           \brief access congruence closure
+        */
+        expr* congruence_next(expr* e);
+
+        expr* congruence_root(expr* e);
+
 
         /**
            \brief retrieve depth of variables from decision stack.
diff --git a/src/smt/smt_solver.cpp b/src/smt/smt_solver.cpp
index 61c7fdda7..4be78b20a 100644
--- a/src/smt/smt_solver.cpp
+++ b/src/smt/smt_solver.cpp
@@ -330,6 +330,10 @@ namespace {
             m_context.get_units(units);
         }
 
+        expr* congruence_next(expr* e) override { return m_context.congruence_next(e); }
+        expr* congruence_root(expr* e) override { return m_context.congruence_root(e); }
+
+
         expr_ref_vector cube(expr_ref_vector& vars, unsigned cutoff) override {
             ast_manager& m = get_manager();
             if (!m_cuber) {
diff --git a/src/solver/combined_solver.cpp b/src/solver/combined_solver.cpp
index 7b1449637..53aa56753 100644
--- a/src/solver/combined_solver.cpp
+++ b/src/solver/combined_solver.cpp
@@ -275,6 +275,10 @@ public:
         return m_solver2->cube(vars, backtrack_level);
     }
 
+    expr* congruence_next(expr* e) override { switch_inc_mode(); return m_solver2->congruence_next(e); }
+    expr* congruence_root(expr* e) override { switch_inc_mode(); return m_solver2->congruence_root(e); }
+
+
     expr * get_assumption(unsigned idx) const override {
         unsigned c1 = m_solver1->get_num_assumptions();
         if (idx < c1) return m_solver1->get_assumption(idx);
diff --git a/src/solver/solver.h b/src/solver/solver.h
index 957cb7c8e..7d7a3eec2 100644
--- a/src/solver/solver.h
+++ b/src/solver/solver.h
@@ -238,6 +238,15 @@ public:
 
     virtual expr_ref_vector cube(expr_ref_vector& vars, unsigned backtrack_level) = 0;
 
+    /**
+       \brief retrieve congruence closure root.
+    */
+    virtual expr* congruence_root(expr* e) = 0;
+
+    /**
+       \brief retrieve congruence closure sibling
+    */
+    virtual expr* congruence_next(expr* e) = 0;
 
     /**
        \brief Display the content of this solver.
diff --git a/src/solver/solver_pool.cpp b/src/solver/solver_pool.cpp
index f5760bde3..411634162 100644
--- a/src/solver/solver_pool.cpp
+++ b/src/solver/solver_pool.cpp
@@ -262,6 +262,9 @@ public:
 
     expr_ref_vector cube(expr_ref_vector& vars, unsigned ) override { return expr_ref_vector(m); }
 
+    expr* congruence_next(expr* e) override { return e; }
+    expr* congruence_root(expr* e) override { return e; }
+
     ast_manager& get_manager() const override { return m_base->get_manager(); }
 
     void refresh(solver* new_base) {
diff --git a/src/solver/tactic2solver.cpp b/src/solver/tactic2solver.cpp
index b65ffde57..cc3ac9336 100644
--- a/src/solver/tactic2solver.cpp
+++ b/src/solver/tactic2solver.cpp
@@ -136,6 +136,9 @@ public:
         return expr_ref_vector(get_manager());
     }
 
+    expr* congruence_next(expr* e) override { return e; }
+    expr* congruence_root(expr* e) override { return e; }
+
     model_converter_ref get_model_converter() const override { return m_mc; }
 
     void get_levels(ptr_vector<expr> const& vars, unsigned_vector& depth) override {
diff --git a/src/tactic/fd_solver/bounded_int2bv_solver.cpp b/src/tactic/fd_solver/bounded_int2bv_solver.cpp
index 7b7ca630e..4ac82c0c2 100644
--- a/src/tactic/fd_solver/bounded_int2bv_solver.cpp
+++ b/src/tactic/fd_solver/bounded_int2bv_solver.cpp
@@ -210,6 +210,8 @@ public:
     void set_reason_unknown(char const* msg) override { m_solver->set_reason_unknown(msg); }
     void get_labels(svector<symbol> & r) override { m_solver->get_labels(r); }
     ast_manager& get_manager() const override { return m;  }
+    expr* congruence_next(expr* e) override { return m_solver->congruence_next(e); }
+    expr* congruence_root(expr* e) override { return m_solver->congruence_root(e); }
     expr_ref_vector cube(expr_ref_vector& vars, unsigned backtrack_level) override { flush_assertions(); return m_solver->cube(vars, backtrack_level); }
     lbool find_mutexes(expr_ref_vector const& vars, vector<expr_ref_vector>& mutexes) override { return m_solver->find_mutexes(vars, mutexes); }
     lbool get_consequences_core(expr_ref_vector const& asms, expr_ref_vector const& vars, expr_ref_vector& consequences) override {
diff --git a/src/tactic/fd_solver/enum2bv_solver.cpp b/src/tactic/fd_solver/enum2bv_solver.cpp
index 7ec5243e7..2690e7033 100644
--- a/src/tactic/fd_solver/enum2bv_solver.cpp
+++ b/src/tactic/fd_solver/enum2bv_solver.cpp
@@ -131,6 +131,9 @@ public:
     expr_ref_vector cube(expr_ref_vector& vars, unsigned backtrack_level) override { 
         return m_solver->cube(vars, backtrack_level); 
     }
+    expr* congruence_next(expr* e) override { return m_solver->congruence_next(e); }
+    expr* congruence_root(expr* e) override { return m_solver->congruence_root(e); }
+
     
     lbool get_consequences_core(expr_ref_vector const& asms, expr_ref_vector const& vars, expr_ref_vector& consequences) override {
         datatype_util dt(m);
diff --git a/src/tactic/fd_solver/pb2bv_solver.cpp b/src/tactic/fd_solver/pb2bv_solver.cpp
index 1a5f7d16a..19f2630f2 100644
--- a/src/tactic/fd_solver/pb2bv_solver.cpp
+++ b/src/tactic/fd_solver/pb2bv_solver.cpp
@@ -122,6 +122,8 @@ public:
     void get_labels(svector<symbol> & r) override { m_solver->get_labels(r); }
     ast_manager& get_manager() const override { return m;  }
     expr_ref_vector cube(expr_ref_vector& vars, unsigned backtrack_level) override { flush_assertions(); return m_solver->cube(vars, backtrack_level); }
+    expr* congruence_next(expr* e) override { return m_solver->congruence_next(e); }
+    expr* congruence_root(expr* e) override { return m_solver->congruence_root(e); }
     lbool find_mutexes(expr_ref_vector const& vars, vector<expr_ref_vector>& mutexes) override { return m_solver->find_mutexes(vars, mutexes); }    
     lbool get_consequences_core(expr_ref_vector const& asms, expr_ref_vector const& vars, expr_ref_vector& consequences) override {
         flush_assertions(); 
diff --git a/src/tactic/fd_solver/smtfd_solver.cpp b/src/tactic/fd_solver/smtfd_solver.cpp
index 3729a2ad1..5676c6ee8 100644
--- a/src/tactic/fd_solver/smtfd_solver.cpp
+++ b/src/tactic/fd_solver/smtfd_solver.cpp
@@ -2086,6 +2086,10 @@ namespace smtfd {
         expr_ref_vector cube(expr_ref_vector& vars, unsigned backtrack_level) override { 
             return expr_ref_vector(m);
         }
+
+        expr* congruence_root(expr* e) override { return e; }
+
+        expr* congruence_next(expr* e) override { return e; }
         
         lbool get_consequences_core(expr_ref_vector const& asms, expr_ref_vector const& vars, expr_ref_vector& consequences) override {
             return l_undef;