diff --git a/scripts/update_api.py b/scripts/update_api.py
index fa6111482..3d8a5dfaf 100644
--- a/scripts/update_api.py
+++ b/scripts/update_api.py
@@ -949,6 +949,11 @@ def def_API(name, result, params):
                 log_c.write(" }\n")
                 log_c.write("  Au(a%s);\n" % sz)
                 exe_c.write("in.get_uint_array(%s)" % i)
+            elif ty == INT:
+                log_c.write("U(a%s[i]);" % i)
+                log_c.write(" }\n")
+                log_c.write("  Au(a%s);\n" % sz)
+                exe_c.write("in.get_int_array(%s)" % i)
             else:
                 error ("unsupported parameter for %s, %s" % (name, p))
         elif kind == OUT_ARRAY:
diff --git a/src/api/z3_api.h b/src/api/z3_api.h
index 2b8c74e65..57ab9a1b7 100644
--- a/src/api/z3_api.h
+++ b/src/api/z3_api.h
@@ -874,6 +874,12 @@ typedef enum
 
       - Z3_OP_DT_ACCESSOR: datatype accessor.
 
+      - Z3_OP_PB_AT_MOST: Cardinality constraint. 
+              E.g., x + y + z <= 2
+      
+      - Z3_OP_PB_LE: Generalized Pseudo-Boolean cardinality constraint.
+              Example  2*x + 3*y <= 4
+
       - Z3_OP_UNINTERPRETED: kind used for uninterpreted symbols.
 */
 typedef enum {
@@ -1054,6 +1060,10 @@ typedef enum {
     Z3_OP_DT_RECOGNISER,
     Z3_OP_DT_ACCESSOR,
 
+    // Pseudo Booleans
+    Z3_OP_PB_AT_MOST=0x900,
+    Z3_OP_PB_LE,
+
     Z3_OP_UNINTERPRETED         
 } Z3_decl_kind;
 
@@ -3758,6 +3768,29 @@ END_MLAPI_EXCLUDE
     Z3_sort Z3_API Z3_get_relation_column(__in Z3_context c, __in Z3_sort s, unsigned col);
 
 
+    /**
+       \brief Pseudo-Boolean relations.
+
+       Encode p1 + p2 + ... + pn <= k
+
+       def_API('Z3_mk_atmost', AST, (_in(CONTEXT), _in(UINT), _in_array(1,AST), _in(UINT)))
+    */
+
+    Z3_ast Z3_API Z3_mk_atmost(__in Z3_context c, __in unsigned num_args, 
+                               __in_ecount(num_args) Z3_ast const args[], __in unsigned k);
+
+    /**
+       \brief Pseudo-Boolean relations.
+
+       Encode k1*p1 + k2*p2 + ... + kn*pn <= k
+
+       def_API('Z3_mk_pble', AST, (_in(CONTEXT), _in(UINT), _in_array(1,AST), _in_array(1,INT), _in(INT)))
+    */
+
+    Z3_ast Z3_API Z3_mk_pble(__in Z3_context c, __in unsigned num_args, 
+                             __in_ecount(num_args) Z3_ast const args[], __in_ecount(num_args) int coeffs[],
+                             __in int k);
+
     /**
        \mlonly {3 {L Function Declarations}} \endmlonly
     */
diff --git a/src/api/z3_replayer.cpp b/src/api/z3_replayer.cpp
index acdb10bf6..f13b2cbb8 100644
--- a/src/api/z3_replayer.cpp
+++ b/src/api/z3_replayer.cpp
@@ -44,7 +44,7 @@ struct z3_replayer::imp {
     size_t_map<void *>       m_heap;
     svector<z3_replayer_cmd> m_cmds;
 
-    enum value_kind { INT64, UINT64, DOUBLE, STRING, SYMBOL, OBJECT, UINT_ARRAY, SYMBOL_ARRAY, OBJECT_ARRAY };
+    enum value_kind { INT64, UINT64, DOUBLE, STRING, SYMBOL, OBJECT, UINT_ARRAY, INT_ARRAY, SYMBOL_ARRAY, OBJECT_ARRAY };
 
     struct value { 
         value_kind m_kind;
@@ -68,6 +68,7 @@ struct z3_replayer::imp {
     vector<ptr_vector<void> >   m_obj_arrays;
     vector<svector<Z3_symbol> > m_sym_arrays;
     vector<unsigned_vector>     m_unsigned_arrays;
+    vector<svector<int> >       m_int_arrays;
 
     imp(z3_replayer & o, std::istream & in):
         m_owner(o),
@@ -295,6 +296,15 @@ struct z3_replayer::imp {
                 v.push_back(static_cast<unsigned>(m_args[i].m_uint));
             }
         }
+        if (k == INT64) {
+            aidx = m_int_arrays.size();
+            nk   = INT_ARRAY;
+            m_int_arrays.push_back(svector<int>());
+            svector<int> & v = m_int_arrays.back();
+            for (unsigned i = asz - sz; i < asz; i++) {
+                v.push_back(static_cast<int>(m_args[i].m_int));
+            }
+        }
         else if (k == SYMBOL) {
             aidx = m_sym_arrays.size();
             nk   = SYMBOL_ARRAY;
@@ -547,6 +557,13 @@ struct z3_replayer::imp {
         return m_unsigned_arrays[idx].c_ptr();
     }
 
+    int * get_int_array(unsigned pos) const {
+        if (pos >= m_args.size() || m_args[pos].m_kind != INT_ARRAY)
+            throw_invalid_reference();
+        unsigned idx = static_cast<unsigned>(m_args[pos].m_uint);
+        return m_int_arrays[idx].c_ptr();
+    }
+
     Z3_symbol * get_symbol_array(unsigned pos) const {
         if (pos >= m_args.size() || m_args[pos].m_kind != SYMBOL_ARRAY)
             throw_invalid_reference();
@@ -615,6 +632,7 @@ struct z3_replayer::imp {
         m_obj_arrays.reset();
         m_sym_arrays.reset();
         m_unsigned_arrays.reset();
+        m_int_arrays.reset();
     }
     
   
@@ -673,6 +691,10 @@ unsigned * z3_replayer::get_uint_array(unsigned pos) const {
     return m_imp->get_uint_array(pos);
 }
 
+int * z3_replayer::get_int_array(unsigned pos) const {
+    return m_imp->get_int_array(pos);
+}
+
 Z3_symbol * z3_replayer::get_symbol_array(unsigned pos) const {
     return m_imp->get_symbol_array(pos);
 }
diff --git a/src/api/z3_replayer.h b/src/api/z3_replayer.h
index 6de4bdb39..412032eb7 100644
--- a/src/api/z3_replayer.h
+++ b/src/api/z3_replayer.h
@@ -49,6 +49,7 @@ public:
     void * get_obj(unsigned pos) const;
 
     unsigned * get_uint_array(unsigned pos) const;
+    int * get_int_array(unsigned pos) const;
     Z3_symbol * get_symbol_array(unsigned pos) const;
     void ** get_obj_array(unsigned pos) const;
 
diff --git a/src/ast/card_decl_plugin.cpp b/src/ast/card_decl_plugin.cpp
index d32a2f5f2..6a15792e5 100644
--- a/src/ast/card_decl_plugin.cpp
+++ b/src/ast/card_decl_plugin.cpp
@@ -72,6 +72,16 @@ app * card_util::mk_at_most_k(unsigned num_args, expr * const * args, unsigned k
     return m.mk_app(m_fid, OP_AT_MOST_K, 1, &param, num_args, args, m.mk_bool_sort());
 }
 
+app * card_util::mk_le(unsigned num_args, int const * coeffs, expr * const * args, int k) {
+    vector<parameter> params;
+    params.push_back(parameter(k));
+    for (unsigned i = 0; i < num_args; ++i) {
+        params.push_back(parameter(coeffs[i]));
+    }
+    return m.mk_app(m_fid, OP_PB_LE, params.size(), params.c_ptr(), num_args, args, m.mk_bool_sort());
+}
+
+
 bool card_util::is_at_most_k(app *a) const {
     return is_app_of(a, m_fid, OP_AT_MOST_K);
 }
diff --git a/src/ast/card_decl_plugin.h b/src/ast/card_decl_plugin.h
index 28a22b0be..3ac681284 100644
--- a/src/ast/card_decl_plugin.h
+++ b/src/ast/card_decl_plugin.h
@@ -71,8 +71,9 @@ class card_util {
 public:
     card_util(ast_manager& m):m(m), m_fid(m.mk_family_id("card")) {}
     ast_manager & get_manager() const { return m; }
+    family_id get_family_id() const { return m_fid; }
     app * mk_at_most_k(unsigned num_args, expr * const * args, unsigned k);
-    app * mk_le(unsigned num_args, int * const* coeffs, expr * const * args, int k);
+    app * mk_le(unsigned num_args, int const * coeffs, expr * const * args, int k);
     bool is_at_most_k(app *a) const;
     bool is_at_most_k(app *a, unsigned& k) const;
     int  get_k(app *a) const;
diff --git a/src/opt/core_maxsat.cpp b/src/opt/core_maxsat.cpp
index e131526cf..b4472971e 100644
--- a/src/opt/core_maxsat.cpp
+++ b/src/opt/core_maxsat.cpp
@@ -65,6 +65,7 @@ namespace opt {
                       for (unsigned i = 0; i < ans.size(); ++i) {
                           tout << mk_pp(ans[i].get(), m) << "\n";
                       });
+                IF_VERBOSE(1, verbose_stream() << "(maxsat.core sat: " << ans.size() << "\n";);
                 if (ans.size() > m_answer.size()) {
                     m_answer.reset();
                     m_answer.append(ans);
@@ -91,6 +92,7 @@ namespace opt {
                     core_vars.insert(get_not(core[i]));
                     block_vars.remove(core[i]);
                 }
+                IF_VERBOSE(1, verbose_stream() << "(maxsat.core unsat (core size = " << core.size() << ")\n";);
                 if (core.empty()) {
                     m_upper = m_answer.size();
                     return l_true;
diff --git a/src/opt/fu_malik.cpp b/src/opt/fu_malik.cpp
index 5c5bace02..6ba3bba39 100644
--- a/src/opt/fu_malik.cpp
+++ b/src/opt/fu_malik.cpp
@@ -73,7 +73,7 @@ namespace opt {
         */
 
         lbool step() {
-            IF_VERBOSE(1, verbose_stream() << "(opt.max_sat step)\n";);
+            IF_VERBOSE(1, verbose_stream() << "(opt.max_sat step)\n";); // add some count, add some information of # of soft constraints still possibly sat.
             expr_ref_vector assumptions(m), block_vars(m);
             for (unsigned i = 0; i < m_soft.size(); ++i) {
                 assumptions.push_back(m.mk_not(m_aux[i].get()));