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diff --git a/src/theory/arith/nl/nl_lemma_utils.cpp b/src/theory/arith/nl/nl_lemma_utils.cpp
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+++ b/src/theory/arith/nl/nl_lemma_utils.cpp
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+/*********************                                                        */
+/*! \file nl_lemma_utils.cpp
+ ** \verbatim
+ ** Top contributors (to current version):
+ **   Andrew Reynolds
+ ** This file is part of the CVC4 project.
+ ** Copyright (c) 2009-2019 by the authors listed in the file AUTHORS
+ ** in the top-level source directory) and their institutional affiliations.
+ ** All rights reserved.  See the file COPYING in the top-level source
+ ** directory for licensing information.\endverbatim
+ **
+ ** \brief Implementation of utilities for the non-linear solver
+ **/
+
+#include "theory/arith/nl/nl_lemma_utils.h"
+
+#include "theory/arith/nl/nl_model.h"
+
+namespace CVC4 {
+namespace theory {
+namespace arith {
+namespace nl {
+
+bool SortNlModel::operator()(Node i, Node j)
+{
+  int cv = d_nlm->compare(i, j, d_isConcrete, d_isAbsolute);
+  if (cv == 0)
+  {
+    return i < j;
+  }
+  return d_reverse_order ? cv < 0 : cv > 0;
+}
+
+bool SortNonlinearDegree::operator()(Node i, Node j)
+{
+  unsigned i_count = getDegree(i);
+  unsigned j_count = getDegree(j);
+  return i_count == j_count ? (i < j) : (i_count < j_count ? true : false);
+}
+
+unsigned SortNonlinearDegree::getDegree(Node n) const
+{
+  std::map<Node, unsigned>::const_iterator it = d_mdegree.find(n);
+  Assert(it != d_mdegree.end());
+  return it->second;
+}
+
+Node ArgTrie::add(Node d, const std::vector<Node>& args)
+{
+  ArgTrie* at = this;
+  for (const Node& a : args)
+  {
+    at = &(at->d_children[a]);
+  }
+  if (at->d_data.isNull())
+  {
+    at->d_data = d;
+  }
+  return at->d_data;
+}
+
+}  // namespace nl
+}  // namespace arith
+}  // namespace theory
+}  // namespace CVC4