+/** Check whether two indices form a dummy pair. */
+bool is_dummy_pair(const idx & i1, const idx & i2);
+
+/** Check whether two expressions form a dummy index pair. */
+bool is_dummy_pair(const ex & e1, const ex & e2);
+
+/** Given a vector of indices, split them into two vectors, one containing
+ * the free indices, the other containing the dummy indices (numeric
+ * indices are neither free nor dummy ones).
+ *
+ * @param it Pointer to start of index vector
+ * @param itend Pointer to end of index vector
+ * @param out_free Vector of free indices (returned, sorted)
+ * @param out_dummy Vector of dummy indices (returned, sorted) */
+void find_free_and_dummy(exvector::const_iterator it, exvector::const_iterator itend, exvector & out_free, exvector & out_dummy);
+
+/** Given a vector of indices, split them into two vectors, one containing
+ * the free indices, the other containing the dummy indices (numeric
+ * indices are neither free nor dummy ones).
+ *
+ * @param v Index vector
+ * @param out_free Vector of free indices (returned, sorted)
+ * @param out_dummy Vector of dummy indices (returned, sorted) */
+inline void find_free_and_dummy(const exvector & v, exvector & out_free, exvector & out_dummy)
+{
+ find_free_and_dummy(v.begin(), v.end(), out_free, out_dummy);
+}
+
+/** Given a vector of indices, find the dummy indices.
+ *
+ * @param v Index vector
+ * @param out_dummy Vector of dummy indices (returned, sorted) */
+inline void find_dummy_indices(const exvector & v, exvector & out_dummy)
+{
+ exvector free_indices;
+ find_free_and_dummy(v.begin(), v.end(), free_indices, out_dummy);
+}
+
+/** Count the number of dummy index pairs in an index vector. */
+inline size_t count_dummy_indices(const exvector & v)
+{
+ exvector free_indices, dummy_indices;
+ find_free_and_dummy(v.begin(), v.end(), free_indices, dummy_indices);
+ return dummy_indices.size();
+}
+
+/** Count the number of dummy index pairs in an index vector. */
+inline size_t count_free_indices(const exvector & v)
+{
+ exvector free_indices, dummy_indices;
+ find_free_and_dummy(v.begin(), v.end(), free_indices, dummy_indices);
+ return free_indices.size();
+}