Split string-specific operators from TheoryStringsRewriter (#3920)

Organization towards theory of sequences.

The motivation of this PR is to ensure that string-specific operators in the rewriter are in their own file; thus the use of mkConst<String> / getConst<String> is allowable in rewriter_str.cpp.

1 files changed, 5597 insertions, 0 deletions

diff --git a/src/theory/strings/sequences_rewriter.cpp b/src/theory/strings/sequences_rewriter.cpp
new file mode 100644
index 000000000..f4a1cd411
--- /dev/null
+++ b/src/theory/strings/sequences_rewriter.cpp
@@ -0,0 +1,5597 @@
+/*********************                                                        */
+/*! \file sequences_rewriter.cpp
+ ** \verbatim
+ ** Top contributors (to current version):
+ **   Andrew Reynolds, Andres Noetzli, Tianyi Liang
+ ** 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 the theory of strings.
+ **
+ ** Implementation of the theory of strings.
+ **/
+
+#include "theory/strings/sequences_rewriter.h"
+
+#include <stdint.h>
+#include <algorithm>
+
+#include "expr/node_builder.h"
+#include "options/strings_options.h"
+#include "smt/logic_exception.h"
+#include "theory/arith/arith_msum.h"
+#include "theory/strings/regexp_operation.h"
+#include "theory/strings/strings_rewriter.h"
+#include "theory/strings/theory_strings_utils.h"
+#include "theory/strings/word.h"
+#include "theory/theory.h"
+#include "util/integer.h"
+#include "util/rational.h"
+
+using namespace std;
+using namespace CVC4;
+using namespace CVC4::kind;
+using namespace CVC4::theory;
+using namespace CVC4::theory::strings;
+
+Node SequencesRewriter::simpleRegexpConsume(std::vector<Node>& mchildren,
+                                            std::vector<Node>& children,
+                                            int dir)
+{
+  Trace("regexp-ext-rewrite-debug")
+      << "Simple reg exp consume, dir=" << dir << ":" << std::endl;
+  Trace("regexp-ext-rewrite-debug")
+      << "  mchildren : " << mchildren << std::endl;
+  Trace("regexp-ext-rewrite-debug") << "  children : " << children << std::endl;
+  NodeManager* nm = NodeManager::currentNM();
+  unsigned tmin = dir < 0 ? 0 : dir;
+  unsigned tmax = dir < 0 ? 1 : dir;
+  // try to remove off front and back
+  for (unsigned t = 0; t < 2; t++)
+  {
+    if (tmin <= t && t <= tmax)
+    {
+      bool do_next = true;
+      while (!children.empty() && !mchildren.empty() && do_next)
+      {
+        do_next = false;
+        Node xc = mchildren[mchildren.size() - 1];
+        Node rc = children[children.size() - 1];
+        Assert(rc.getKind() != kind::REGEXP_CONCAT);
+        Assert(xc.getKind() != kind::STRING_CONCAT);
+        if (rc.getKind() == kind::STRING_TO_REGEXP)
+        {
+          if (xc == rc[0])
+          {
+            children.pop_back();
+            mchildren.pop_back();
+            do_next = true;
+            Trace("regexp-ext-rewrite-debug") << "...strip equal" << std::endl;
+          }
+          else if (xc.isConst() && rc[0].isConst())
+          {
+            // split the constant
+            int index;
+            Node s = splitConstant(xc, rc[0], index, t == 0);
+            Trace("regexp-ext-rewrite-debug")
+                << "CRE: Regexp const split : " << xc << " " << rc[0] << " -> "
+                << s << " " << index << " " << t << std::endl;
+            if (s.isNull())
+            {
+              Trace("regexp-ext-rewrite-debug")
+                  << "...return false" << std::endl;
+              return NodeManager::currentNM()->mkConst(false);
+            }
+            else
+            {
+              Trace("regexp-ext-rewrite-debug")
+                  << "...strip equal const" << std::endl;
+              children.pop_back();
+              mchildren.pop_back();
+              if (index == 0)
+              {
+                mchildren.push_back(s);
+              }
+              else
+              {
+                children.push_back(nm->mkNode(STRING_TO_REGEXP, s));
+              }
+            }
+            do_next = true;
+          }
+        }
+        else if (xc.isConst())
+        {
+          // check for constants
+          CVC4::String s = xc.getConst<String>();
+          if (Word::isEmpty(xc))
+          {
+            Trace("regexp-ext-rewrite-debug") << "...ignore empty" << std::endl;
+            // ignore and continue
+            mchildren.pop_back();
+            do_next = true;
+          }
+          else if (rc.getKind() == kind::REGEXP_RANGE
+                   || rc.getKind() == kind::REGEXP_SIGMA)
+          {
+            std::vector<unsigned> ssVec;
+            ssVec.push_back(t == 0 ? s.back() : s.front());
+            CVC4::String ss(ssVec);
+            if (testConstStringInRegExp(ss, 0, rc))
+            {
+              // strip off one character
+              mchildren.pop_back();
+              if (s.size() > 1)
+              {
+                if (t == 0)
+                {
+                  mchildren.push_back(NodeManager::currentNM()->mkConst(
+                      s.substr(0, s.size() - 1)));
+                }
+                else
+                {
+                  mchildren.push_back(
+                      NodeManager::currentNM()->mkConst(s.substr(1)));
+                }
+              }
+              children.pop_back();
+              do_next = true;
+            }
+            else
+            {
+              return NodeManager::currentNM()->mkConst(false);
+            }
+          }
+          else if (rc.getKind() == kind::REGEXP_INTER
+                   || rc.getKind() == kind::REGEXP_UNION)
+          {
+            // see if any/each child does not work
+            bool result_valid = true;
+            Node result;
+            Node emp_s = NodeManager::currentNM()->mkConst(::CVC4::String(""));
+            for (unsigned i = 0; i < rc.getNumChildren(); i++)
+            {
+              std::vector<Node> mchildren_s;
+              std::vector<Node> children_s;
+              mchildren_s.push_back(xc);
+              utils::getConcat(rc[i], children_s);
+              Node ret = simpleRegexpConsume(mchildren_s, children_s, t);
+              if (!ret.isNull())
+              {
+                // one conjunct cannot be satisfied, return false
+                if (rc.getKind() == kind::REGEXP_INTER)
+                {
+                  return ret;
+                }
+              }
+              else
+              {
+                if (children_s.empty())
+                {
+                  // if we were able to fully consume, store the result
+                  Assert(mchildren_s.size() <= 1);
+                  if (mchildren_s.empty())
+                  {
+                    mchildren_s.push_back(emp_s);
+                  }
+                  if (result.isNull())
+                  {
+                    result = mchildren_s[0];
+                  }
+                  else if (result != mchildren_s[0])
+                  {
+                    result_valid = false;
+                  }
+                }
+                else
+                {
+                  result_valid = false;
+                }
+              }
+            }
+            if (result_valid)
+            {
+              if (result.isNull())
+              {
+                // all disjuncts cannot be satisfied, return false
+                Assert(rc.getKind() == kind::REGEXP_UNION);
+                return NodeManager::currentNM()->mkConst(false);
+              }
+              else
+              {
+                // all branches led to the same result
+                children.pop_back();
+                mchildren.pop_back();
+                if (result != emp_s)
+                {
+                  mchildren.push_back(result);
+                }
+                do_next = true;
+              }
+            }
+          }
+          else if (rc.getKind() == kind::REGEXP_STAR)
+          {
+            // check if there is no way that this star can be unrolled even once
+            std::vector<Node> mchildren_s;
+            mchildren_s.insert(
+                mchildren_s.end(), mchildren.begin(), mchildren.end());
+            if (t == 1)
+            {
+              std::reverse(mchildren_s.begin(), mchildren_s.end());
+            }
+            std::vector<Node> children_s;
+            utils::getConcat(rc[0], children_s);
+            Trace("regexp-ext-rewrite-debug")
+                << "...recursive call on body of star" << std::endl;
+            Node ret = simpleRegexpConsume(mchildren_s, children_s, t);
+            if (!ret.isNull())
+            {
+              Trace("regexp-ext-rewrite-debug")
+                  << "CRE : regexp star infeasable " << xc << " " << rc
+                  << std::endl;
+              children.pop_back();
+              if (!children.empty())
+              {
+                Trace("regexp-ext-rewrite-debug") << "...continue" << std::endl;
+                do_next = true;
+              }
+            }
+            else
+            {
+              if (children_s.empty())
+              {
+                // check if beyond this, we can't do it or there is nothing
+                // left, if so, repeat
+                bool can_skip = false;
+                if (children.size() > 1)
+                {
+                  std::vector<Node> mchildren_ss;
+                  mchildren_ss.insert(
+                      mchildren_ss.end(), mchildren.begin(), mchildren.end());
+                  std::vector<Node> children_ss;
+                  children_ss.insert(
+                      children_ss.end(), children.begin(), children.end() - 1);
+                  if (t == 1)
+                  {
+                    std::reverse(mchildren_ss.begin(), mchildren_ss.end());
+                    std::reverse(children_ss.begin(), children_ss.end());
+                  }
+                  if (simpleRegexpConsume(mchildren_ss, children_ss, t)
+                          .isNull())
+                  {
+                    can_skip = true;
+                  }
+                }
+                if (!can_skip)
+                {
+                  Trace("regexp-ext-rewrite-debug")
+                      << "...can't skip" << std::endl;
+                  // take the result of fully consuming once
+                  if (t == 1)
+                  {
+                    std::reverse(mchildren_s.begin(), mchildren_s.end());
+                  }
+                  mchildren.clear();
+                  mchildren.insert(
+                      mchildren.end(), mchildren_s.begin(), mchildren_s.end());
+                  do_next = true;
+                }
+                else
+                {
+                  Trace("regexp-ext-rewrite-debug")
+                      << "...can skip " << rc << " from " << xc << std::endl;
+                }
+              }
+            }
+          }
+        }
+        if (!do_next)
+        {
+          Trace("regexp-ext-rewrite")
+              << "Cannot consume : " << xc << " " << rc << std::endl;
+        }
+      }
+    }
+    if (dir != 0)
+    {
+      std::reverse(children.begin(), children.end());
+      std::reverse(mchildren.begin(), mchildren.end());
+    }
+  }
+  return Node::null();
+}
+
+Node SequencesRewriter::rewriteEquality(Node node)
+{
+  Assert(node.getKind() == kind::EQUAL);
+  if (node[0] == node[1])
+  {
+    return NodeManager::currentNM()->mkConst(true);
+  }
+  else if (node[0].isConst() && node[1].isConst())
+  {
+    return NodeManager::currentNM()->mkConst(false);
+  }
+
+  // ( ~contains( s, t ) V ~contains( t, s ) ) => ( s == t ---> false )
+  for (unsigned r = 0; r < 2; r++)
+  {
+    // must call rewrite contains directly to avoid infinite loop
+    // we do a fix point since we may rewrite contains terms to simpler
+    // contains terms.
+    Node ctn = checkEntailContains(node[r], node[1 - r], false);
+    if (!ctn.isNull())
+    {
+      if (!ctn.getConst<bool>())
+      {
+        return returnRewrite(node, ctn, "eq-nctn");
+      }
+      else
+      {
+        // definitely contains but not syntactically equal
+        // We may be able to simplify, e.g.
+        //  str.++( x, "a" ) == "a"  ----> x = ""
+      }
+    }
+  }
+
+  // ( len( s ) != len( t ) ) => ( s == t ---> false )
+  // This covers cases like str.++( x, x ) == "a" ---> false
+  Node len0 = NodeManager::currentNM()->mkNode(kind::STRING_LENGTH, node[0]);
+  Node len1 = NodeManager::currentNM()->mkNode(kind::STRING_LENGTH, node[1]);
+  Node len_eq = len0.eqNode(len1);
+  len_eq = Rewriter::rewrite(len_eq);
+  if (len_eq.isConst() && !len_eq.getConst<bool>())
+  {
+    return returnRewrite(node, len_eq, "eq-len-deq");
+  }
+
+  std::vector<Node> c[2];
+  for (unsigned i = 0; i < 2; i++)
+  {
+    utils::getConcat(node[i], c[i]);
+  }
+
+  // check if the prefix, suffix mismatches
+  //   For example, str.++( x, "a", y ) == str.++( x, "bc", z ) ---> false
+  unsigned minsize = std::min(c[0].size(), c[1].size());
+  for (unsigned r = 0; r < 2; r++)
+  {
+    for (unsigned i = 0; i < minsize; i++)
+    {
+      unsigned index1 = r == 0 ? i : (c[0].size() - 1) - i;
+      unsigned index2 = r == 0 ? i : (c[1].size() - 1) - i;
+      if (c[0][index1].isConst() && c[1][index2].isConst())
+      {
+        CVC4::String s = c[0][index1].getConst<String>();
+        CVC4::String t = c[1][index2].getConst<String>();
+        unsigned len_short = s.size() <= t.size() ? s.size() : t.size();
+        bool isSameFix =
+            r == 1 ? s.rstrncmp(t, len_short) : s.strncmp(t, len_short);
+        if (!isSameFix)
+        {
+          Node ret = NodeManager::currentNM()->mkConst(false);
+          return returnRewrite(node, ret, "eq-nfix");
+        }
+      }
+      if (c[0][index1] != c[1][index2])
+      {
+        break;
+      }
+    }
+  }
+
+  // standard ordering
+  if (node[0] > node[1])
+  {
+    return NodeManager::currentNM()->mkNode(kind::EQUAL, node[1], node[0]);
+  }
+  return node;
+}
+
+Node SequencesRewriter::rewriteEqualityExt(Node node)
+{
+  Assert(node.getKind() == EQUAL);
+  if (node[0].getType().isInteger())
+  {
+    return rewriteArithEqualityExt(node);
+  }
+  if (node[0].getType().isString())
+  {
+    return rewriteStrEqualityExt(node);
+  }
+  return node;
+}
+
+Node SequencesRewriter::rewriteStrEqualityExt(Node node)
+{
+  Assert(node.getKind() == EQUAL && node[0].getType().isString());
+
+  NodeManager* nm = NodeManager::currentNM();
+  std::vector<Node> c[2];
+  Node new_ret;
+  for (unsigned i = 0; i < 2; i++)
+  {
+    utils::getConcat(node[i], c[i]);
+  }
+  // ------- equality unification
+  bool changed = false;
+  for (unsigned i = 0; i < 2; i++)
+  {
+    while (!c[0].empty() && !c[1].empty() && c[0].back() == c[1].back())
+    {
+      c[0].pop_back();
+      c[1].pop_back();
+      changed = true;
+    }
+    // splice constants
+    if (!c[0].empty() && !c[1].empty() && c[0].back().isConst()
+        && c[1].back().isConst())
+    {
+      Node cs[2];
+      size_t csl[2];
+      for (unsigned j = 0; j < 2; j++)
+      {
+        cs[j] = c[j].back();
+        csl[j] = Word::getLength(cs[j]);
+      }
+      size_t larger = csl[0] > csl[1] ? 0 : 1;
+      size_t smallerSize = csl[1 - larger];
+      if (cs[1 - larger]
+          == (i == 0 ? Word::suffix(cs[larger], smallerSize)
+                     : Word::prefix(cs[larger], smallerSize)))
+      {
+        size_t sizeDiff = csl[larger] - smallerSize;
+        c[larger][c[larger].size() - 1] =
+            i == 0 ? Word::prefix(cs[larger], sizeDiff)
+                   : Word::suffix(cs[larger], sizeDiff);
+        c[1 - larger].pop_back();
+        changed = true;
+      }
+    }
+    for (unsigned j = 0; j < 2; j++)
+    {
+      std::reverse(c[j].begin(), c[j].end());
+    }
+  }
+  if (changed)
+  {
+    // e.g. x++y = x++z ---> y = z, "AB" ++ x = "A" ++ y --> "B" ++ x = y
+    Node s1 = utils::mkConcat(STRING_CONCAT, c[0]);
+    Node s2 = utils::mkConcat(STRING_CONCAT, c[1]);
+    new_ret = s1.eqNode(s2);
+    node = returnRewrite(node, new_ret, "str-eq-unify");
+  }
+
+  // ------- homogeneous constants
+  for (unsigned i = 0; i < 2; i++)
+  {
+    Node cn = checkEntailHomogeneousString(node[i]);
+    if (!cn.isNull() && !Word::isEmpty(cn))
+    {
+      Assert(cn.isConst());
+      Assert(Word::getLength(cn) == 1);
+      unsigned hchar = cn.getConst<String>().front();
+
+      // The operands of the concat on each side of the equality without
+      // constant strings
+      std::vector<Node> trimmed[2];
+      // Counts the number of `hchar`s on each side
+      size_t numHChars[2] = {0, 0};
+      for (size_t j = 0; j < 2; j++)
+      {
+        // Sort the operands of the concats on both sides of the equality
+        // (since both sides may only contain one char, the order does not
+        // matter)
+        std::sort(c[j].begin(), c[j].end());
+        for (const Node& cc : c[j])
+        {
+          if (cc.isConst())
+          {
+            // Count the number of `hchar`s in the string constant and make
+            // sure that all chars are `hchar`s
+            std::vector<unsigned> veccc = cc.getConst<String>().getVec();
+            for (size_t k = 0, size = veccc.size(); k < size; k++)
+            {
+              if (veccc[k] != hchar)
+              {
+                // This conflict case should mostly should be taken care of by
+                // multiset reasoning in the strings rewriter, but we recognize
+                // this conflict just in case.
+                new_ret = nm->mkConst(false);
+                return returnRewrite(
+                    node, new_ret, "string-eq-const-conflict-non-homog");
+              }
+              numHChars[j]++;
+            }
+          }
+          else
+          {
+            trimmed[j].push_back(cc);
+          }
+        }
+      }
+
+      // We have to remove the same number of `hchar`s from both sides, so the
+      // side with less `hchar`s determines how many we can remove
+      size_t trimmedConst = std::min(numHChars[0], numHChars[1]);
+      for (size_t j = 0; j < 2; j++)
+      {
+        size_t diff = numHChars[j] - trimmedConst;
+        if (diff != 0)
+        {
+          // Add a constant string to the side with more `hchar`s to restore
+          // the difference in number of `hchar`s
+          std::vector<unsigned> vec(diff, hchar);
+          trimmed[j].push_back(nm->mkConst(String(vec)));
+        }
+      }
+
+      Node lhs = utils::mkConcat(STRING_CONCAT, trimmed[i]);
+      Node ss = utils::mkConcat(STRING_CONCAT, trimmed[1 - i]);
+      if (lhs != node[i] || ss != node[1 - i])
+      {
+        // e.g.
+        //  "AA" = y ++ x ---> "AA" = x ++ y if x < y
+        //  "AAA" = y ++ "A" ++ z ---> "AA" = y ++ z
+        new_ret = lhs.eqNode(ss);
+        node = returnRewrite(node, new_ret, "str-eq-homog-const");
+      }
+    }
+  }
+
+  // ------- rewrites for (= "" _)
+  Node empty = nm->mkConst(::CVC4::String(""));
+  for (size_t i = 0; i < 2; i++)
+  {
+    if (node[i] == empty)
+    {
+      Node ne = node[1 - i];
+      if (ne.getKind() == STRING_STRREPL)
+      {
+        // (= "" (str.replace x y x)) ---> (= x "")
+        if (ne[0] == ne[2])
+        {
+          Node ret = nm->mkNode(EQUAL, ne[0], empty);
+          return returnRewrite(node, ret, "str-emp-repl-x-y-x");
+        }
+
+        // (= "" (str.replace x y "A")) ---> (and (= x "") (not (= y "")))
+        if (checkEntailNonEmpty(ne[2]))
+        {
+          Node ret =
+              nm->mkNode(AND,
+                         nm->mkNode(EQUAL, ne[0], empty),
+                         nm->mkNode(NOT, nm->mkNode(EQUAL, ne[1], empty)));
+          return returnRewrite(node, ret, "str-emp-repl-emp");
+        }
+
+        // (= "" (str.replace x "A" "")) ---> (str.prefix x "A")
+        if (checkEntailLengthOne(ne[1]) && ne[2] == empty)
+        {
+          Node ret = nm->mkNode(STRING_PREFIX, ne[0], ne[1]);
+          return returnRewrite(node, ret, "str-emp-repl-emp");
+        }
+      }
+      else if (ne.getKind() == STRING_SUBSTR)
+      {
+        Node zero = nm->mkConst(Rational(0));
+
+        if (checkEntailArith(ne[1], false) && checkEntailArith(ne[2], true))
+        {
+          // (= "" (str.substr x 0 m)) ---> (= "" x) if m > 0
+          if (ne[1] == zero)
+          {
+            Node ret = nm->mkNode(EQUAL, ne[0], empty);
+            return returnRewrite(node, ret, "str-emp-substr-leq-len");
+          }
+
+          // (= "" (str.substr x n m)) ---> (<= (str.len x) n)
+          // if n >= 0 and m > 0
+          Node ret = nm->mkNode(LEQ, nm->mkNode(STRING_LENGTH, ne[0]), ne[1]);
+          return returnRewrite(node, ret, "str-emp-substr-leq-len");
+        }
+
+        // (= "" (str.substr "A" 0 z)) ---> (<= z 0)
+        if (checkEntailNonEmpty(ne[0]) && ne[1] == zero)
+        {
+          Node ret = nm->mkNode(LEQ, ne[2], zero);
+          return returnRewrite(node, ret, "str-emp-substr-leq-z");
+        }
+      }
+    }
+  }
+
+  // ------- rewrites for (= (str.replace _ _ _) _)
+  for (size_t i = 0; i < 2; i++)
+  {
+    if (node[i].getKind() == STRING_STRREPL)
+    {
+      Node repl = node[i];
+      Node x = node[1 - i];
+
+      // (= "A" (str.replace "" x y)) ---> (= "" (str.replace "A" y x))
+      if (checkEntailNonEmpty(x) && repl[0] == empty)
+      {
+        Node ret = nm->mkNode(
+            EQUAL, empty, nm->mkNode(STRING_STRREPL, x, repl[2], repl[1]));
+        return returnRewrite(node, ret, "str-eq-repl-emp");
+      }
+
+      // (= x (str.replace y x y)) ---> (= x y)
+      if (repl[0] == repl[2] && x == repl[1])
+      {
+        Node ret = nm->mkNode(EQUAL, x, repl[0]);
+        return returnRewrite(node, ret, "str-eq-repl-to-eq");
+      }
+
+      // (= x (str.replace x "A" "B")) ---> (not (str.contains x "A"))
+      if (x == repl[0])
+      {
+        Node eq = Rewriter::rewrite(nm->mkNode(EQUAL, repl[1], repl[2]));
+        if (eq.isConst() && !eq.getConst<bool>())
+        {
+          Node ret = nm->mkNode(NOT, nm->mkNode(STRING_STRCTN, x, repl[1]));
+          return returnRewrite(node, ret, "str-eq-repl-not-ctn");
+        }
+      }
+
+      // (= (str.replace x y z) z) --> (or (= x y) (= x z))
+      // if (str.len y) = (str.len z)
+      if (repl[2] == x)
+      {
+        Node lenY = nm->mkNode(STRING_LENGTH, repl[1]);
+        Node lenZ = nm->mkNode(STRING_LENGTH, repl[2]);
+        if (checkEntailArithEq(lenY, lenZ))
+        {
+          Node ret = nm->mkNode(OR,
+                                nm->mkNode(EQUAL, repl[0], repl[1]),
+                                nm->mkNode(EQUAL, repl[0], repl[2]));
+          return returnRewrite(node, ret, "str-eq-repl-to-dis");
+        }
+      }
+    }
+  }
+
+  // Try to rewrite (= x y) into a conjunction of equalities based on length
+  // entailment.
+  //
+  // (<= (str.len x) (str.++ y1 ... yn)) AND (= x (str.++ y1 ... yn)) --->
+  //  (and (= x (str.++ y1' ... ym')) (= y1'' "") ... (= yk'' ""))
+  //
+  // where yi' and yi'' correspond to some yj and
+  //   (<= (str.len x) (str.++ y1' ... ym'))
+  for (unsigned i = 0; i < 2; i++)
+  {
+    if (node[1 - i].getKind() == STRING_CONCAT)
+    {
+      new_ret = inferEqsFromContains(node[i], node[1 - i]);
+      if (!new_ret.isNull())
+      {
+        return returnRewrite(node, new_ret, "str-eq-conj-len-entail");
+      }
+    }
+  }
+
+  if (node[0].getKind() == STRING_CONCAT && node[1].getKind() == STRING_CONCAT)
+  {
+    // (= (str.++ x_1 ... x_i x_{i + 1} ... x_n)
+    //    (str.++ y_1 ... y_j y_{j + 1} ... y_m)) --->
+    //  (and (= (str.++ x_1 ... x_i) (str.++ y_1 ... y_j))
+    //       (= (str.++ x_{i + 1} ... x_n) (str.++ y_{j + 1} ... y_m)))
+    //
+    // if (str.len (str.++ x_1 ... x_i)) = (str.len (str.++ y_1 ... y_j))
+    //
+    // This rewrite performs length-based equality splitting: If we can show
+    // that two prefixes have the same length, we can split an equality into
+    // two equalities, one over the prefixes and another over the suffixes.
+    std::vector<Node> v0, v1;
+    utils::getConcat(node[0], v0);
+    utils::getConcat(node[1], v1);
+    size_t startRhs = 0;
+    for (size_t i = 0, size0 = v0.size(); i <= size0; i++)
+    {
+      std::vector<Node> pfxv0(v0.begin(), v0.begin() + i);
+      Node pfx0 = utils::mkConcat(STRING_CONCAT, pfxv0);
+      for (size_t j = startRhs, size1 = v1.size(); j <= size1; j++)
+      {
+        if (!(i == 0 && j == 0) && !(i == v0.size() && j == v1.size()))
+        {
+          std::vector<Node> pfxv1(v1.begin(), v1.begin() + j);
+          Node pfx1 = utils::mkConcat(STRING_CONCAT, pfxv1);
+          Node lenPfx0 = nm->mkNode(STRING_LENGTH, pfx0);
+          Node lenPfx1 = nm->mkNode(STRING_LENGTH, pfx1);
+
+          if (checkEntailArithEq(lenPfx0, lenPfx1))
+          {
+            std::vector<Node> sfxv0(v0.begin() + i, v0.end());
+            std::vector<Node> sfxv1(v1.begin() + j, v1.end());
+            Node ret =
+                nm->mkNode(kind::AND,
+                           pfx0.eqNode(pfx1),
+                           utils::mkConcat(STRING_CONCAT, sfxv0)
+                               .eqNode(utils::mkConcat(STRING_CONCAT, sfxv1)));
+            return returnRewrite(node, ret, "split-eq");
+          }
+          else if (checkEntailArith(lenPfx1, lenPfx0, true))
+          {
+            // The prefix on the right-hand side is strictly longer than the
+            // prefix on the left-hand side, so we try to strip the right-hand
+            // prefix by the length of the left-hand prefix
+            //
+            // Example:
+            // (= (str.++ "A" x y) (str.++ x "AB" z)) --->
+            //   (and (= (str.++ "A" x) (str.++ x "A")) (= y (str.++ "B" z)))
+            std::vector<Node> rpfxv1;
+            if (stripSymbolicLength(pfxv1, rpfxv1, 1, lenPfx0))
+            {
+              std::vector<Node> sfxv0(v0.begin() + i, v0.end());
+              pfxv1.insert(pfxv1.end(), v1.begin() + j, v1.end());
+              Node ret = nm->mkNode(
+                  kind::AND,
+                  pfx0.eqNode(utils::mkConcat(STRING_CONCAT, rpfxv1)),
+                  utils::mkConcat(STRING_CONCAT, sfxv0)
+                      .eqNode(utils::mkConcat(STRING_CONCAT, pfxv1)));
+              return returnRewrite(node, ret, "split-eq-strip-r");
+            }
+
+            // If the prefix of the right-hand side is (strictly) longer than
+            // the prefix of the left-hand side, we can advance the left-hand
+            // side (since the length of the right-hand side is only increasing
+            // in the inner loop)
+            break;
+          }
+          else if (checkEntailArith(lenPfx0, lenPfx1, true))
+          {
+            // The prefix on the left-hand side is strictly longer than the
+            // prefix on the right-hand side, so we try to strip the left-hand
+            // prefix by the length of the right-hand prefix
+            //
+            // Example:
+            // (= (str.++ x "AB" z) (str.++ "A" x y)) --->
+            //   (and (= (str.++ x "A") (str.++ "A" x)) (= (str.++ "B" z) y))
+            std::vector<Node> rpfxv0;
+            if (stripSymbolicLength(pfxv0, rpfxv0, 1, lenPfx1))
+            {
+              pfxv0.insert(pfxv0.end(), v0.begin() + i, v0.end());
+              std::vector<Node> sfxv1(v1.begin() + j, v1.end());
+              Node ret = nm->mkNode(
+                  kind::AND,
+                  utils::mkConcat(STRING_CONCAT, rpfxv0).eqNode(pfx1),
+                  utils::mkConcat(STRING_CONCAT, pfxv0)
+                      .eqNode(utils::mkConcat(STRING_CONCAT, sfxv1)));
+              return returnRewrite(node, ret, "split-eq-strip-l");
+            }
+
+            // If the prefix of the left-hand side is (strictly) longer than
+            // the prefix of the right-hand side, then we don't need to check
+            // that right-hand prefix for future left-hand prefixes anymore
+            // (since they are increasing in length)
+            startRhs = j + 1;
+          }
+        }
+      }
+    }
+  }
+
+  return node;
+}
+
+Node SequencesRewriter::rewriteArithEqualityExt(Node node)
+{
+  Assert(node.getKind() == EQUAL && node[0].getType().isInteger());
+
+  // cases where we can solve the equality
+
+  // notice we cannot rewrite str.to.int(x)=n to x="n" due to leading zeroes.
+
+  return node;
+}
+
+// TODO (#1180) add rewrite
+//  str.++( str.substr( x, n1, n2 ), str.substr( x, n1+n2, n3 ) ) --->
+//  str.substr( x, n1, n2+n3 )
+Node SequencesRewriter::rewriteConcat(Node node)
+{
+  Assert(node.getKind() == kind::STRING_CONCAT);
+  Trace("strings-rewrite-debug")
+      << "Strings::rewriteConcat start " << node << std::endl;
+  NodeManager* nm = NodeManager::currentNM();
+  Node retNode = node;
+  std::vector<Node> node_vec;
+  Node preNode = Node::null();
+  for (Node tmpNode : node)
+  {
+    if (tmpNode.getKind() == STRING_CONCAT)
+    {
+      unsigned j = 0;
+      // combine the first term with the previous constant if applicable
+      if (!preNode.isNull())
+      {
+        if (tmpNode[0].isConst())
+        {
+          preNode = nm->mkConst(
+              preNode.getConst<String>().concat(tmpNode[0].getConst<String>()));
+          node_vec.push_back(preNode);
+        }
+        else
+        {
+          node_vec.push_back(preNode);
+          node_vec.push_back(tmpNode[0]);
+        }
+        preNode = Node::null();
+        ++j;
+      }
+      // insert the middle terms to node_vec
+      if (j <= tmpNode.getNumChildren() - 1)
+      {
+        node_vec.insert(node_vec.end(), tmpNode.begin() + j, tmpNode.end() - 1);
+      }
+      // take the last term as the current
+      tmpNode = tmpNode[tmpNode.getNumChildren() - 1];
+    }
+    if (!tmpNode.isConst())
+    {
+      if (!preNode.isNull())
+      {
+        if (preNode.isConst() && !Word::isEmpty(preNode))
+        {
+          node_vec.push_back(preNode);
+        }
+        preNode = Node::null();
+      }
+      node_vec.push_back(tmpNode);
+    }
+    else
+    {
+      if (preNode.isNull())
+      {
+        preNode = tmpNode;
+      }
+      else
+      {
+        std::vector<Node> vec;
+        vec.push_back(preNode);
+        vec.push_back(tmpNode);
+        preNode = Word::mkWord(vec);
+      }
+    }
+  }
+  if (!preNode.isNull() && (!preNode.isConst() || !Word::isEmpty(preNode)))
+  {
+    node_vec.push_back(preNode);
+  }
+
+  // Sort adjacent operands in str.++ that all result in the same string or the
+  // empty string.
+  //
+  // E.g.: (str.++ ... (str.replace "A" x "") "A" (str.substr "A" 0 z) ...) -->
+  // (str.++ ... [sort those 3 arguments] ... )
+  size_t lastIdx = 0;
+  Node lastX;
+  for (size_t i = 0, nsize = node_vec.size(); i < nsize; i++)
+  {
+    Node s = getStringOrEmpty(node_vec[i]);
+    bool nextX = false;
+    if (s != lastX)
+    {
+      nextX = true;
+    }
+
+    if (nextX)
+    {
+      std::sort(node_vec.begin() + lastIdx, node_vec.begin() + i);
+      lastX = s;
+      lastIdx = i;
+    }
+  }
+  std::sort(node_vec.begin() + lastIdx, node_vec.end());
+
+  retNode = utils::mkConcat(STRING_CONCAT, node_vec);
+  Trace("strings-rewrite-debug")
+      << "Strings::rewriteConcat end " << retNode << std::endl;
+  return retNode;
+}
+
+Node SequencesRewriter::rewriteConcatRegExp(TNode node)
+{
+  Assert(node.getKind() == kind::REGEXP_CONCAT);
+  NodeManager* nm = NodeManager::currentNM();
+  Trace("strings-rewrite-debug")
+      << "Strings::rewriteConcatRegExp flatten " << node << std::endl;
+  Node retNode = node;
+  std::vector<Node> vec;
+  bool changed = false;
+  Node emptyRe;
+  for (const Node& c : node)
+  {
+    if (c.getKind() == REGEXP_CONCAT)
+    {
+      changed = true;
+      for (const Node& cc : c)
+      {
+        vec.push_back(cc);
+      }
+    }
+    else if (c.getKind() == STRING_TO_REGEXP && c[0].isConst()
+             && Word::isEmpty(c[0]))
+    {
+      changed = true;
+      emptyRe = c;
+    }
+    else if (c.getKind() == REGEXP_EMPTY)
+    {
+      // re.++( ..., empty, ... ) ---> empty
+      std::vector<Node> nvec;
+      return nm->mkNode(REGEXP_EMPTY, nvec);
+    }
+    else
+    {
+      vec.push_back(c);
+    }
+  }
+  if (changed)
+  {
+    // flatten
+    // this handles nested re.++ and elimination or str.to.re(""), e.g.:
+    // re.++( re.++( R1, R2 ), str.to.re(""), R3 ) ---> re.++( R1, R2, R3 )
+    if (vec.empty())
+    {
+      Assert(!emptyRe.isNull());
+      retNode = emptyRe;
+    }
+    else
+    {
+      retNode = vec.size() == 1 ? vec[0] : nm->mkNode(REGEXP_CONCAT, vec);
+    }
+    return returnRewrite(node, retNode, "re.concat-flatten");
+  }
+  Trace("strings-rewrite-debug")
+      << "Strings::rewriteConcatRegExp start " << node << std::endl;
+  std::vector<Node> cvec;
+  // the current accumulation of constant strings
+  std::vector<Node> preReStr;
+  // whether the last component was (_)*
+  bool lastAllStar = false;
+  String emptyStr = String("");
+  // this loop checks to see if components can be combined or dropped
+  for (unsigned i = 0, size = vec.size(); i <= size; i++)
+  {
+    Node curr;
+    if (i < size)
+    {
+      curr = vec[i];
+      Assert(curr.getKind() != REGEXP_CONCAT);
+    }
+    // update preReStr
+    if (!curr.isNull() && curr.getKind() == STRING_TO_REGEXP)
+    {
+      lastAllStar = false;
+      preReStr.push_back(curr[0]);
+      curr = Node::null();
+    }
+    else if (!preReStr.empty())
+    {
+      Assert(!lastAllStar);
+      // this groups consecutive strings a++b ---> ab
+      Node acc = nm->mkNode(STRING_TO_REGEXP,
+                            utils::mkConcat(STRING_CONCAT, preReStr));
+      cvec.push_back(acc);
+      preReStr.clear();
+    }
+    else if (!curr.isNull() && lastAllStar)
+    {
+      // if empty, drop it
+      // e.g. this ensures we rewrite (_)* ++ (a)* ---> (_)*
+      if (isConstRegExp(curr) && testConstStringInRegExp(emptyStr, 0, curr))
+      {
+        curr = Node::null();
+      }
+    }
+    if (!curr.isNull())
+    {
+      lastAllStar = false;
+      if (curr.getKind() == REGEXP_STAR)
+      {
+        // we can group stars (a)* ++ (a)* ---> (a)*
+        if (!cvec.empty() && cvec.back() == curr)
+        {
+          curr = Node::null();
+        }
+        else if (curr[0].getKind() == REGEXP_SIGMA)
+        {
+          Assert(!lastAllStar);
+          lastAllStar = true;
+          // go back and remove empty ones from back of cvec
+          // e.g. this ensures we rewrite (a)* ++ (_)* ---> (_)*
+          while (!cvec.empty() && isConstRegExp(cvec.back())
+                 && testConstStringInRegExp(emptyStr, 0, cvec.back()))
+          {
+            cvec.pop_back();
+          }
+        }
+      }
+    }
+    if (!curr.isNull())
+    {
+      cvec.push_back(curr);
+    }
+  }
+  Assert(!cvec.empty());
+  retNode = utils::mkConcat(REGEXP_CONCAT, cvec);
+  if (retNode != node)
+  {
+    // handles all cases where consecutive re constants are combined or dropped
+    // as described in the loop above.
+    return returnRewrite(node, retNode, "re.concat");
+  }
+
+  // flipping adjacent star arguments
+  changed = false;
+  for (size_t i = 0, size = cvec.size() - 1; i < size; i++)
+  {
+    if (cvec[i].getKind() == REGEXP_STAR && cvec[i][0] == cvec[i + 1])
+    {
+      // by convention, flip the order (a*)++a ---> a++(a*)
+      std::swap(cvec[i], cvec[i + 1]);
+      changed = true;
+    }
+  }
+  if (changed)
+  {
+    retNode = utils::mkConcat(REGEXP_CONCAT, cvec);
+    return returnRewrite(node, retNode, "re.concat.opt");
+  }
+  return node;
+}
+
+Node SequencesRewriter::rewriteStarRegExp(TNode node)
+{
+  Assert(node.getKind() == REGEXP_STAR);
+  NodeManager* nm = NodeManager::currentNM();
+  Node retNode = node;
+  if (node[0].getKind() == REGEXP_STAR)
+  {
+    // ((R)*)* ---> R*
+    return returnRewrite(node, node[0], "re-star-nested-star");
+  }
+  else if (node[0].getKind() == STRING_TO_REGEXP && node[0][0].isConst()
+           && Word::isEmpty(node[0][0]))
+  {
+    // ("")* ---> ""
+    return returnRewrite(node, node[0], "re-star-empty-string");
+  }
+  else if (node[0].getKind() == REGEXP_EMPTY)
+  {
+    // (empty)* ---> ""
+    retNode = nm->mkNode(STRING_TO_REGEXP, nm->mkConst(String("")));
+    return returnRewrite(node, retNode, "re-star-empty");
+  }
+  else if (node[0].getKind() == REGEXP_UNION)
+  {
+    // simplification of unions under star
+    if (hasEpsilonNode(node[0]))
+    {
+      bool changed = false;
+      std::vector<Node> node_vec;
+      for (const Node& nc : node[0])
+      {
+        if (nc.getKind() == STRING_TO_REGEXP && nc[0].isConst()
+            && Word::isEmpty(nc[0]))
+        {
+          // can be removed
+          changed = true;
+        }
+        else
+        {
+          node_vec.push_back(nc);
+        }
+      }
+      if (changed)
+      {
+        retNode = node_vec.size() == 1 ? node_vec[0]
+                                       : nm->mkNode(REGEXP_UNION, node_vec);
+        retNode = nm->mkNode(REGEXP_STAR, retNode);
+        // simplification of union beneath star based on loop above
+        // for example, ( "" | "a" )* ---> ("a")*
+        return returnRewrite(node, retNode, "re-star-union");
+      }
+    }
+  }
+  return node;
+}
+
+Node SequencesRewriter::rewriteAndOrRegExp(TNode node)
+{
+  Kind nk = node.getKind();
+  Assert(nk == REGEXP_UNION || nk == REGEXP_INTER);
+  Trace("strings-rewrite-debug")
+      << "Strings::rewriteAndOrRegExp start " << node << std::endl;
+  std::vector<Node> node_vec;
+  for (const Node& ni : node)
+  {
+    if (ni.getKind() == nk)
+    {
+      for (const Node& nic : ni)
+      {
+        if (std::find(node_vec.begin(), node_vec.end(), nic) == node_vec.end())
+        {
+          node_vec.push_back(nic);
+        }
+      }
+    }
+    else if (ni.getKind() == REGEXP_EMPTY)
+    {
+      if (nk == REGEXP_INTER)
+      {
+        return returnRewrite(node, ni, "re.and-empty");
+      }
+      // otherwise, can ignore
+    }
+    else if (ni.getKind() == REGEXP_STAR && ni[0].getKind() == REGEXP_SIGMA)
+    {
+      if (nk == REGEXP_UNION)
+      {
+        return returnRewrite(node, ni, "re.or-all");
+      }
+      // otherwise, can ignore
+    }
+    else if (std::find(node_vec.begin(), node_vec.end(), ni) == node_vec.end())
+    {
+      node_vec.push_back(ni);
+    }
+  }
+  NodeManager* nm = NodeManager::currentNM();
+  std::vector<Node> nvec;
+  Node retNode;
+  if (node_vec.empty())
+  {
+    if (nk == REGEXP_INTER)
+    {
+      retNode = nm->mkNode(REGEXP_STAR, nm->mkNode(REGEXP_SIGMA, nvec));
+    }
+    else
+    {
+      retNode = nm->mkNode(kind::REGEXP_EMPTY, nvec);
+    }
+  }
+  else
+  {
+    retNode = node_vec.size() == 1 ? node_vec[0] : nm->mkNode(nk, node_vec);
+  }
+  if (retNode != node)
+  {
+    // flattening and removing children, based on loop above
+    return returnRewrite(node, retNode, "re.andor-flatten");
+  }
+  return node;
+}
+
+Node SequencesRewriter::rewriteLoopRegExp(TNode node)
+{
+  Assert(node.getKind() == REGEXP_LOOP);
+  Node retNode = node;
+  Node r = node[0];
+  if (r.getKind() == REGEXP_STAR)
+  {
+    return returnRewrite(node, r, "re.loop-star");
+  }
+  TNode n1 = node[1];
+  NodeManager* nm = NodeManager::currentNM();
+  CVC4::Rational rMaxInt(String::maxSize());
+  AlwaysAssert(n1.isConst()) << "re.loop contains non-constant integer (1).";
+  AlwaysAssert(n1.getConst<Rational>().sgn() >= 0)
+      << "Negative integer in string REGEXP_LOOP (1)";
+  Assert(n1.getConst<Rational>() <= rMaxInt)
+      << "Exceeded UINT32_MAX in string REGEXP_LOOP (1)";
+  uint32_t l = n1.getConst<Rational>().getNumerator().toUnsignedInt();
+  std::vector<Node> vec_nodes;
+  for (unsigned i = 0; i < l; i++)
+  {
+    vec_nodes.push_back(r);
+  }
+  if (node.getNumChildren() == 3)
+  {
+    TNode n2 = Rewriter::rewrite(node[2]);
+    Node n =
+        vec_nodes.size() == 0
+            ? nm->mkNode(STRING_TO_REGEXP, nm->mkConst(String("")))
+            : vec_nodes.size() == 1 ? r : nm->mkNode(REGEXP_CONCAT, vec_nodes);
+    AlwaysAssert(n2.isConst()) << "re.loop contains non-constant integer (2).";
+    AlwaysAssert(n2.getConst<Rational>().sgn() >= 0)
+        << "Negative integer in string REGEXP_LOOP (2)";
+    Assert(n2.getConst<Rational>() <= rMaxInt)
+        << "Exceeded UINT32_MAX in string REGEXP_LOOP (2)";
+    uint32_t u = n2.getConst<Rational>().getNumerator().toUnsignedInt();
+    if (u <= l)
+    {
+      retNode = n;
+    }
+    else
+    {
+      std::vector<Node> vec2;
+      vec2.push_back(n);
+      for (unsigned j = l; j < u; j++)
+      {
+        vec_nodes.push_back(r);
+        n = utils::mkConcat(REGEXP_CONCAT, vec_nodes);
+        vec2.push_back(n);
+      }
+      retNode = nm->mkNode(REGEXP_UNION, vec2);
+    }
+  }
+  else
+  {
+    Node rest = nm->mkNode(REGEXP_STAR, r);
+    retNode = vec_nodes.size() == 0
+                  ? rest
+                  : vec_nodes.size() == 1
+                        ? nm->mkNode(REGEXP_CONCAT, r, rest)
+                        : nm->mkNode(REGEXP_CONCAT,
+                                     nm->mkNode(REGEXP_CONCAT, vec_nodes),
+                                     rest);
+  }
+  Trace("strings-lp") << "Strings::lp " << node << " => " << retNode
+                      << std::endl;
+  if (retNode != node)
+  {
+    return returnRewrite(node, retNode, "re.loop");
+  }
+  return node;
+}
+
+bool SequencesRewriter::isConstRegExp(TNode t)
+{
+  if (t.getKind() == kind::STRING_TO_REGEXP)
+  {
+    return t[0].isConst();
+  }
+  else if (t.isVar())
+  {
+    return false;
+  }
+  else
+  {
+    for (unsigned i = 0; i < t.getNumChildren(); ++i)
+    {
+      if (!isConstRegExp(t[i]))
+      {
+        return false;
+      }
+    }
+    return true;
+  }
+}
+
+bool SequencesRewriter::testConstStringInRegExp(CVC4::String& s,
+                                                unsigned int index_start,
+                                                TNode r)
+{
+  Assert(index_start <= s.size());
+  Trace("regexp-debug") << "Checking " << s << " in " << r << ", starting at "
+                        << index_start << std::endl;
+  Assert(!r.isVar());
+  Kind k = r.getKind();
+  switch (k)
+  {
+    case kind::STRING_TO_REGEXP:
+    {
+      CVC4::String s2 = s.substr(index_start, s.size() - index_start);
+      if (r[0].isConst())
+      {
+        return (s2 == r[0].getConst<String>());
+      }
+      else
+      {
+        Assert(false) << "RegExp contains variables";
+        return false;
+      }
+    }
+    case kind::REGEXP_CONCAT:
+    {
+      if (s.size() != index_start)
+      {
+        std::vector<int> vec_k(r.getNumChildren(), -1);
+        int start = 0;
+        int left = (int)s.size() - index_start;
+        int i = 0;
+        while (i < (int)r.getNumChildren())
+        {
+          bool flag = true;
+          if (i == (int)r.getNumChildren() - 1)
+          {
+            if (testConstStringInRegExp(s, index_start + start, r[i]))
+            {
+              return true;
+            }
+          }
+          else if (i == -1)
+          {
+            return false;
+          }
+          else
+          {
+            for (vec_k[i] = vec_k[i] + 1; vec_k[i] <= left; ++vec_k[i])
+            {
+              CVC4::String t = s.substr(index_start + start, vec_k[i]);
+              if (testConstStringInRegExp(t, 0, r[i]))
+              {
+                start += vec_k[i];
+                left -= vec_k[i];
+                flag = false;
+                ++i;
+                vec_k[i] = -1;
+                break;
+              }
+            }
+          }
+
+          if (flag)
+          {
+            --i;
+            if (i >= 0)
+            {
+              start -= vec_k[i];
+              left += vec_k[i];
+            }
+          }
+        }
+        return false;
+      }
+      else
+      {
+        for (unsigned i = 0; i < r.getNumChildren(); ++i)
+        {
+          if (!testConstStringInRegExp(s, index_start, r[i])) return false;
+        }
+        return true;
+      }
+    }
+    case kind::REGEXP_UNION:
+    {
+      for (unsigned i = 0; i < r.getNumChildren(); ++i)
+      {
+        if (testConstStringInRegExp(s, index_start, r[i])) return true;
+      }
+      return false;
+    }
+    case kind::REGEXP_INTER:
+    {
+      for (unsigned i = 0; i < r.getNumChildren(); ++i)
+      {
+        if (!testConstStringInRegExp(s, index_start, r[i])) return false;
+      }
+      return true;
+    }
+    case kind::REGEXP_STAR:
+    {
+      if (s.size() != index_start)
+      {
+        for (unsigned i = s.size() - index_start; i > 0; --i)
+        {
+          CVC4::String t = s.substr(index_start, i);
+          if (testConstStringInRegExp(t, 0, r[0]))
+          {
+            if (index_start + i == s.size()
+                || testConstStringInRegExp(s, index_start + i, r))
+            {
+              return true;
+            }
+          }
+        }
+        return false;
+      }
+      else
+      {
+        return true;
+      }
+    }
+    case kind::REGEXP_EMPTY: { return false;
+    }
+    case kind::REGEXP_SIGMA:
+    {
+      if (s.size() == index_start + 1)
+      {
+        return true;
+      }
+      else
+      {
+        return false;
+      }
+    }
+    case kind::REGEXP_RANGE:
+    {
+      if (s.size() == index_start + 1)
+      {
+        unsigned a = r[0].getConst<String>().front();
+        a = String::convertUnsignedIntToCode(a);
+        unsigned b = r[1].getConst<String>().front();
+        b = String::convertUnsignedIntToCode(b);
+        unsigned c = s.back();
+        c = String::convertUnsignedIntToCode(c);
+        return (a <= c && c <= b);
+      }
+      else
+      {
+        return false;
+      }
+    }
+    case kind::REGEXP_LOOP:
+    {
+      uint32_t l = r[1].getConst<Rational>().getNumerator().toUnsignedInt();
+      if (s.size() == index_start)
+      {
+        return l == 0 ? true : testConstStringInRegExp(s, index_start, r[0]);
+      }
+      else if (l == 0 && r[1] == r[2])
+      {
+        return false;
+      }
+      else
+      {
+        Assert(r.getNumChildren() == 3)
+            << "String rewriter error: LOOP has 2 children";
+        if (l == 0)
+        {
+          // R{0,u}
+          uint32_t u = r[2].getConst<Rational>().getNumerator().toUnsignedInt();
+          for (unsigned len = s.size() - index_start; len >= 1; len--)
+          {
+            CVC4::String t = s.substr(index_start, len);
+            if (testConstStringInRegExp(t, 0, r[0]))
+            {
+              if (len + index_start == s.size())
+              {
+                return true;
+              }
+              else
+              {
+                Node num2 =
+                    NodeManager::currentNM()->mkConst(CVC4::Rational(u - 1));
+                Node r2 = NodeManager::currentNM()->mkNode(
+                    kind::REGEXP_LOOP, r[0], r[1], num2);
+                if (testConstStringInRegExp(s, index_start + len, r2))
+                {
+                  return true;
+                }
+              }
+            }
+          }
+          return false;
+        }
+        else
+        {
+          // R{l,l}
+          Assert(r[1] == r[2])
+              << "String rewriter error: LOOP nums are not equal";
+          if (l > s.size() - index_start)
+          {
+            if (testConstStringInRegExp(s, s.size(), r[0]))
+            {
+              l = s.size() - index_start;
+            }
+            else
+            {
+              return false;
+            }
+          }
+          for (unsigned len = 1; len <= s.size() - index_start; len++)
+          {
+            CVC4::String t = s.substr(index_start, len);
+            if (testConstStringInRegExp(t, 0, r[0]))
+            {
+              Node num2 =
+                  NodeManager::currentNM()->mkConst(CVC4::Rational(l - 1));
+              Node r2 = NodeManager::currentNM()->mkNode(
+                  kind::REGEXP_LOOP, r[0], num2, num2);
+              if (testConstStringInRegExp(s, index_start + len, r2))
+              {
+                return true;
+              }
+            }
+          }
+          return false;
+        }
+      }
+    }
+    case REGEXP_COMPLEMENT:
+    {
+      return !testConstStringInRegExp(s, index_start, r[0]);
+      break;
+    }
+    default:
+    {
+      Assert(!RegExpOpr::isRegExpKind(k));
+      return false;
+    }
+  }
+}
+
+Node SequencesRewriter::rewriteMembership(TNode node)
+{
+  NodeManager* nm = NodeManager::currentNM();
+  Node retNode = node;
+  Node x = node[0];
+  Node r = node[1];
+
+  if (r.getKind() == kind::REGEXP_EMPTY)
+  {
+    retNode = NodeManager::currentNM()->mkConst(false);
+  }
+  else if (x.isConst() && isConstRegExp(r))
+  {
+    // test whether x in node[1]
+    CVC4::String s = x.getConst<String>();
+    retNode =
+        NodeManager::currentNM()->mkConst(testConstStringInRegExp(s, 0, r));
+  }
+  else if (r.getKind() == kind::REGEXP_SIGMA)
+  {
+    Node one = nm->mkConst(Rational(1));
+    retNode = one.eqNode(nm->mkNode(STRING_LENGTH, x));
+  }
+  else if (r.getKind() == kind::REGEXP_STAR)
+  {
+    if (x.isConst())
+    {
+      String s = x.getConst<String>();
+      if (s.size() == 0)
+      {
+        retNode = nm->mkConst(true);
+        // e.g. (str.in.re "" (re.* (str.to.re x))) ----> true
+        return returnRewrite(node, retNode, "re-empty-in-str-star");
+      }
+      else if (s.size() == 1)
+      {
+        if (r[0].getKind() == STRING_TO_REGEXP)
+        {
+          retNode = r[0][0].eqNode(x);
+          // e.g. (str.in.re "A" (re.* (str.to.re x))) ----> "A" = x
+          return returnRewrite(node, retNode, "re-char-in-str-star");
+        }
+      }
+    }
+    else if (x.getKind() == STRING_CONCAT)
+    {
+      // (str.in.re (str.++ x1 ... xn) (re.* R)) -->
+      //   (str.in.re x1 (re.* R)) AND ... AND (str.in.re xn (re.* R))
+      //     if the length of all strings in R is one.
+      Node flr = getFixedLengthForRegexp(r[0]);
+      if (!flr.isNull())
+      {
+        Node one = nm->mkConst(Rational(1));
+        if (flr == one)
+        {
+          NodeBuilder<> nb(AND);
+          for (const Node& xc : x)
+          {
+            nb << nm->mkNode(STRING_IN_REGEXP, xc, r);
+          }
+          return returnRewrite(
+              node, nb.constructNode(), "re-in-dist-char-star");
+        }
+      }
+    }
+    if (r[0].getKind() == kind::REGEXP_SIGMA)
+    {
+      retNode = NodeManager::currentNM()->mkConst(true);
+      return returnRewrite(node, retNode, "re-in-sigma-star");
+    }
+  }
+  else if (r.getKind() == kind::REGEXP_CONCAT)
+  {
+    bool allSigma = true;
+    bool allSigmaStrict = true;
+    unsigned allSigmaMinSize = 0;
+    Node constStr;
+    size_t constIdx = 0;
+    size_t nchildren = r.getNumChildren();
+    for (size_t i = 0; i < nchildren; i++)
+    {
+      Node rc = r[i];
+      Assert(rc.getKind() != kind::REGEXP_EMPTY);
+      if (rc.getKind() == kind::REGEXP_SIGMA)
+      {
+        allSigmaMinSize++;
+      }
+      else if (rc.getKind() == REGEXP_STAR && rc[0].getKind() == REGEXP_SIGMA)
+      {
+        allSigmaStrict = false;
+      }
+      else if (rc.getKind() == STRING_TO_REGEXP)
+      {
+        if (constStr.isNull())
+        {
+          constStr = rc[0];
+          constIdx = i;
+        }
+        else
+        {
+          allSigma = false;
+          break;
+        }
+      }
+      else
+      {
+        allSigma = false;
+        break;
+      }
+    }
+    if (allSigma)
+    {
+      if (constStr.isNull())
+      {
+        // x in re.++(_*, _, _) ---> str.len(x) >= 2
+        Node num = nm->mkConst(Rational(allSigmaMinSize));
+        Node lenx = nm->mkNode(STRING_LENGTH, x);
+        retNode = nm->mkNode(allSigmaStrict ? EQUAL : GEQ, lenx, num);
+        return returnRewrite(node, retNode, "re-concat-pure-allchar");
+      }
+      else if (allSigmaMinSize == 0 && nchildren >= 3 && constIdx != 0
+               && constIdx != nchildren - 1)
+      {
+        // x in re.++(_*, "abc", _*) ---> str.contains(x, "abc")
+        retNode = nm->mkNode(STRING_STRCTN, x, constStr);
+        return returnRewrite(node, retNode, "re-concat-to-contains");
+      }
+    }
+  }
+  else if (r.getKind() == kind::REGEXP_INTER
+           || r.getKind() == kind::REGEXP_UNION)
+  {
+    std::vector<Node> mvec;
+    for (unsigned i = 0; i < r.getNumChildren(); i++)
+    {
+      mvec.push_back(
+          NodeManager::currentNM()->mkNode(kind::STRING_IN_REGEXP, x, r[i]));
+    }
+    retNode = NodeManager::currentNM()->mkNode(
+        r.getKind() == kind::REGEXP_INTER ? kind::AND : kind::OR, mvec);
+  }
+  else if (r.getKind() == kind::STRING_TO_REGEXP)
+  {
+    retNode = x.eqNode(r[0]);
+  }
+  else if (r.getKind() == REGEXP_RANGE)
+  {
+    // x in re.range( char_i, char_j ) ---> i <= str.code(x) <= j
+    Node xcode = nm->mkNode(STRING_TO_CODE, x);
+    retNode =
+        nm->mkNode(AND,
+                   nm->mkNode(LEQ, nm->mkNode(STRING_TO_CODE, r[0]), xcode),
+                   nm->mkNode(LEQ, xcode, nm->mkNode(STRING_TO_CODE, r[1])));
+  }
+  else if (r.getKind() == REGEXP_COMPLEMENT)
+  {
+    retNode = nm->mkNode(STRING_IN_REGEXP, x, r[0]).negate();
+  }
+  else if (x != node[0] || r != node[1])
+  {
+    retNode = NodeManager::currentNM()->mkNode(kind::STRING_IN_REGEXP, x, r);
+  }
+
+  // do simple consumes
+  if (retNode == node)
+  {
+    if (r.getKind() == kind::REGEXP_STAR)
+    {
+      for (unsigned dir = 0; dir <= 1; dir++)
+      {
+        std::vector<Node> mchildren;
+        utils::getConcat(x, mchildren);
+        bool success = true;
+        while (success)
+        {
+          success = false;
+          std::vector<Node> children;
+          utils::getConcat(r[0], children);
+          Node scn = simpleRegexpConsume(mchildren, children, dir);
+          if (!scn.isNull())
+          {
+            Trace("regexp-ext-rewrite")
+                << "Regexp star : const conflict : " << node << std::endl;
+            return scn;
+          }
+          else if (children.empty())
+          {
+            // fully consumed one copy of the STAR
+            if (mchildren.empty())
+            {
+              Trace("regexp-ext-rewrite")
+                  << "Regexp star : full consume : " << node << std::endl;
+              return NodeManager::currentNM()->mkConst(true);
+            }
+            else
+            {
+              retNode = nm->mkNode(STRING_IN_REGEXP,
+                                   utils::mkConcat(STRING_CONCAT, mchildren),
+                                   r);
+              success = true;
+            }
+          }
+        }
+        if (retNode != node)
+        {
+          Trace("regexp-ext-rewrite") << "Regexp star : rewrite " << node
+                                      << " -> " << retNode << std::endl;
+          break;
+        }
+      }
+    }
+    else
+    {
+      std::vector<Node> children;
+      utils::getConcat(r, children);
+      std::vector<Node> mchildren;
+      utils::getConcat(x, mchildren);
+      unsigned prevSize = children.size() + mchildren.size();
+      Node scn = simpleRegexpConsume(mchildren, children);
+      if (!scn.isNull())
+      {
+        Trace("regexp-ext-rewrite")
+            << "Regexp : const conflict : " << node << std::endl;
+        return scn;
+      }
+      else
+      {
+        if ((children.size() + mchildren.size()) != prevSize)
+        {
+          // Given a membership (str.++ x1 ... xn) in (re.++ r1 ... rm),
+          // above, we strip components to construct an equivalent membership:
+          // (str.++ xi .. xj) in (re.++ rk ... rl).
+          Node xn = utils::mkConcat(STRING_CONCAT, mchildren);
+          Node emptyStr = nm->mkConst(String(""));
+          if (children.empty())
+          {
+            // If we stripped all components on the right, then the left is
+            // equal to the empty string.
+            // e.g. (str.++ "a" x) in (re.++ (str.to.re "a")) ---> (= x "")
+            retNode = xn.eqNode(emptyStr);
+          }
+          else
+          {
+            // otherwise, construct the updated regular expression
+            retNode = nm->mkNode(
+                STRING_IN_REGEXP, xn, utils::mkConcat(REGEXP_CONCAT, children));
+          }
+          Trace("regexp-ext-rewrite") << "Regexp : rewrite : " << node << " -> "
+                                      << retNode << std::endl;
+          return returnRewrite(node, retNode, "re-simple-consume");
+        }
+      }
+    }
+  }
+  return retNode;
+}
+
+RewriteResponse SequencesRewriter::postRewrite(TNode node)
+{
+  Trace("strings-postrewrite")
+      << "Strings::postRewrite start " << node << std::endl;
+  NodeManager* nm = NodeManager::currentNM();
+  Node retNode = node;
+  Node orig = retNode;
+  Kind nk = node.getKind();
+  if (nk == kind::STRING_CONCAT)
+  {
+    retNode = rewriteConcat(node);
+  }
+  else if (nk == kind::EQUAL)
+  {
+    retNode = rewriteEquality(node);
+  }
+  else if (nk == kind::STRING_LENGTH)
+  {
+    Kind nk0 = node[0].getKind();
+    if (node[0].isConst())
+    {
+      retNode = nm->mkConst(Rational(Word::getLength(node[0])));
+    }
+    else if (nk0 == kind::STRING_CONCAT)
+    {
+      Node tmpNode = node[0];
+      if (tmpNode.isConst())
+      {
+        retNode = nm->mkConst(Rational(Word::getLength(tmpNode)));
+      }
+      else if (tmpNode.getKind() == kind::STRING_CONCAT)
+      {
+        std::vector<Node> node_vec;
+        for (unsigned int i = 0; i < tmpNode.getNumChildren(); ++i)
+        {
+          if (tmpNode[i].isConst())
+          {
+            node_vec.push_back(
+                nm->mkConst(Rational(Word::getLength(tmpNode[i]))));
+          }
+          else
+          {
+            node_vec.push_back(NodeManager::currentNM()->mkNode(
+                kind::STRING_LENGTH, tmpNode[i]));
+          }
+        }
+        retNode = NodeManager::currentNM()->mkNode(kind::PLUS, node_vec);
+      }
+    }
+    else if (nk0 == STRING_STRREPL || nk0 == STRING_STRREPLALL)
+    {
+      Node len1 = Rewriter::rewrite(nm->mkNode(STRING_LENGTH, node[0][1]));
+      Node len2 = Rewriter::rewrite(nm->mkNode(STRING_LENGTH, node[0][2]));
+      if (len1 == len2)
+      {
+        // len( y ) == len( z ) => len( str.replace( x, y, z ) ) ---> len( x )
+        retNode = nm->mkNode(STRING_LENGTH, node[0][0]);
+      }
+    }
+    else if (nk0 == STRING_TOLOWER || nk0 == STRING_TOUPPER
+             || nk0 == STRING_REV)
+    {
+      // len( f( x ) ) == len( x ) where f is tolower, toupper, or rev.
+      retNode = nm->mkNode(STRING_LENGTH, node[0][0]);
+    }
+  }
+  else if (nk == kind::STRING_CHARAT)
+  {
+    Node one = NodeManager::currentNM()->mkConst(Rational(1));
+    retNode = NodeManager::currentNM()->mkNode(
+        kind::STRING_SUBSTR, node[0], node[1], one);
+  }
+  else if (nk == kind::STRING_SUBSTR)
+  {
+    retNode = rewriteSubstr(node);
+  }
+  else if (nk == kind::STRING_STRCTN)
+  {
+    retNode = rewriteContains(node);
+  }
+  else if (nk == kind::STRING_LT)
+  {
+    // eliminate s < t ---> s != t AND s <= t
+    retNode = nm->mkNode(AND,
+                         node[0].eqNode(node[1]).negate(),
+                         nm->mkNode(STRING_LEQ, node[0], node[1]));
+  }
+  else if (nk == kind::STRING_LEQ)
+  {
+    retNode = StringsRewriter::rewriteStringLeq(node);
+  }
+  else if (nk == kind::STRING_STRIDOF)
+  {
+    retNode = rewriteIndexof(node);
+  }
+  else if (nk == kind::STRING_STRREPL)
+  {
+    retNode = rewriteReplace(node);
+  }
+  else if (nk == kind::STRING_STRREPLALL)
+  {
+    retNode = rewriteReplaceAll(node);
+  }
+  else if (nk == STRING_TOLOWER || nk == STRING_TOUPPER)
+  {
+    retNode = StringsRewriter::rewriteStrConvert(node);
+  }
+  else if (nk == STRING_REV)
+  {
+    retNode = rewriteStrReverse(node);
+  }
+  else if (nk == kind::STRING_PREFIX || nk == kind::STRING_SUFFIX)
+  {
+    retNode = rewritePrefixSuffix(node);
+  }
+  else if (nk == STRING_IS_DIGIT)
+  {
+    // eliminate str.is_digit(s) ----> 48 <= str.to_code(s) <= 57
+    Node t = nm->mkNode(STRING_TO_CODE, node[0]);
+    retNode = nm->mkNode(AND,
+                         nm->mkNode(LEQ, nm->mkConst(Rational(48)), t),
+                         nm->mkNode(LEQ, t, nm->mkConst(Rational(57))));
+  }
+  else if (nk == kind::STRING_ITOS)
+  {
+    retNode = StringsRewriter::rewriteIntToStr(node);
+  }
+  else if (nk == kind::STRING_STOI)
+  {
+    retNode = StringsRewriter::rewriteStrToInt(node);
+  }
+  else if (nk == kind::STRING_IN_REGEXP)
+  {
+    retNode = rewriteMembership(node);
+  }
+  else if (nk == STRING_TO_CODE)
+  {
+    retNode = StringsRewriter::rewriteStringToCode(node);
+  }
+  else if (nk == STRING_FROM_CODE)
+  {
+    retNode = StringsRewriter::rewriteStringFromCode(node);
+  }
+  else if (nk == REGEXP_CONCAT)
+  {
+    retNode = rewriteConcatRegExp(node);
+  }
+  else if (nk == REGEXP_UNION || nk == REGEXP_INTER)
+  {
+    retNode = rewriteAndOrRegExp(node);
+  }
+  else if (nk == REGEXP_DIFF)
+  {
+    retNode = nm->mkNode(
+        REGEXP_INTER, node[0], nm->mkNode(REGEXP_COMPLEMENT, node[1]));
+  }
+  else if (nk == REGEXP_STAR)
+  {
+    retNode = rewriteStarRegExp(node);
+  }
+  else if (nk == REGEXP_PLUS)
+  {
+    retNode =
+        nm->mkNode(REGEXP_CONCAT, node[0], nm->mkNode(REGEXP_STAR, node[0]));
+  }
+  else if (nk == REGEXP_OPT)
+  {
+    retNode = nm->mkNode(REGEXP_UNION,
+                         nm->mkNode(STRING_TO_REGEXP, nm->mkConst(String(""))),
+                         node[0]);
+  }
+  else if (nk == REGEXP_RANGE)
+  {
+    if (node[0] == node[1])
+    {
+      retNode = nm->mkNode(STRING_TO_REGEXP, node[0]);
+    }
+  }
+  else if (nk == REGEXP_LOOP)
+  {
+    retNode = rewriteLoopRegExp(node);
+  }
+
+  Trace("strings-postrewrite")
+      << "Strings::postRewrite returning " << retNode << std::endl;
+  if (orig != retNode)
+  {
+    Trace("strings-rewrite-debug")
+        << "Strings: post-rewrite " << orig << " to " << retNode << std::endl;
+  }
+  return RewriteResponse(orig == retNode ? REWRITE_DONE : REWRITE_AGAIN_FULL,
+                         retNode);
+}
+
+bool SequencesRewriter::hasEpsilonNode(TNode node)
+{
+  for (unsigned int i = 0; i < node.getNumChildren(); i++)
+  {
+    if (node[i].getKind() == kind::STRING_TO_REGEXP && node[i][0].isConst()
+        && Word::isEmpty(node[i][0]))
+    {
+      return true;
+    }
+  }
+  return false;
+}
+
+RewriteResponse SequencesRewriter::preRewrite(TNode node)
+{
+  return RewriteResponse(REWRITE_DONE, node);
+}
+
+Node SequencesRewriter::rewriteSubstr(Node node)
+{
+  Assert(node.getKind() == kind::STRING_SUBSTR);
+
+  NodeManager* nm = NodeManager::currentNM();
+  if (node[0].isConst())
+  {
+    if (Word::isEmpty(node[0]))
+    {
+      Node ret = node[0];
+      return returnRewrite(node, ret, "ss-emptystr");
+    }
+    // rewriting for constant arguments
+    if (node[1].isConst() && node[2].isConst())
+    {
+      Node s = node[0];
+      CVC4::Rational rMaxInt(String::maxSize());
+      uint32_t start;
+      if (node[1].getConst<Rational>() > rMaxInt)
+      {
+        // start beyond the maximum size of strings
+        // thus, it must be beyond the end point of this string
+        Node ret = Word::mkEmptyWord(node.getType());
+        return returnRewrite(node, ret, "ss-const-start-max-oob");
+      }
+      else if (node[1].getConst<Rational>().sgn() < 0)
+      {
+        // start before the beginning of the string
+        Node ret = Word::mkEmptyWord(node.getType());
+        return returnRewrite(node, ret, "ss-const-start-neg");
+      }
+      else
+      {
+        start = node[1].getConst<Rational>().getNumerator().toUnsignedInt();
+        if (start >= Word::getLength(node[0]))
+        {
+          // start beyond the end of the string
+          Node ret = Word::mkEmptyWord(node.getType());
+          return returnRewrite(node, ret, "ss-const-start-oob");
+        }
+      }
+      if (node[2].getConst<Rational>() > rMaxInt)
+      {
+        // take up to the end of the string
+        size_t lenS = Word::getLength(s);
+        Node ret = Word::suffix(s, lenS - start);
+        return returnRewrite(node, ret, "ss-const-len-max-oob");
+      }
+      else if (node[2].getConst<Rational>().sgn() <= 0)
+      {
+        Node ret = Word::mkEmptyWord(node.getType());
+        return returnRewrite(node, ret, "ss-const-len-non-pos");
+      }
+      else
+      {
+        uint32_t len =
+            node[2].getConst<Rational>().getNumerator().toUnsignedInt();
+        if (start + len > Word::getLength(node[0]))
+        {
+          // take up to the end of the string
+          size_t lenS = Word::getLength(s);
+          Node ret = Word::suffix(s, lenS - start);
+          return returnRewrite(node, ret, "ss-const-end-oob");
+        }
+        else
+        {
+          // compute the substr using the constant string
+          Node ret = Word::substr(s, start, len);
+          return returnRewrite(node, ret, "ss-const-ss");
+        }
+      }
+    }
+  }
+  Node zero = nm->mkConst(CVC4::Rational(0));
+
+  // if entailed non-positive length or negative start point
+  if (checkEntailArith(zero, node[1], true))
+  {
+    Node ret = Word::mkEmptyWord(node.getType());
+    return returnRewrite(node, ret, "ss-start-neg");
+  }
+  else if (checkEntailArith(zero, node[2]))
+  {
+    Node ret = Word::mkEmptyWord(node.getType());
+    return returnRewrite(node, ret, "ss-len-non-pos");
+  }
+
+  if (node[0].getKind() == STRING_SUBSTR)
+  {
+    // (str.substr (str.substr x a b) c d) ---> "" if c >= b
+    //
+    // Note that this rewrite can be generalized to:
+    //
+    // (str.substr x a b) ---> "" if a >= (str.len x)
+    //
+    // This can be done when we generalize our entailment methods to
+    // accept an optional context. Then we could conjecture that
+    // (str.substr x a b) rewrites to "" and do a case analysis:
+    //
+    // - a < 0 or b < 0 (the result is trivially empty in these cases)
+    // - a >= (str.len x) assuming that { a >= 0, b >= 0 }
+    //
+    // For example, for (str.substr (str.substr x a a) a a), we could
+    // then deduce that under those assumptions, "a" is an
+    // over-approximation of the length of (str.substr x a a), which
+    // then allows us to reason that the result of the whole term must
+    // be empty.
+    if (checkEntailArith(node[1], node[0][2]))
+    {
+      Node ret = Word::mkEmptyWord(node.getType());
+      return returnRewrite(node, ret, "ss-start-geq-len");
+    }
+  }
+  else if (node[0].getKind() == STRING_STRREPL)
+  {
+    // (str.substr (str.replace x y z) 0 n)
+    // 	 ---> (str.replace (str.substr x 0 n) y z)
+    // if (str.len y) = 1 and (str.len z) = 1
+    if (node[1] == zero)
+    {
+      if (checkEntailLengthOne(node[0][1], true)
+          && checkEntailLengthOne(node[0][2], true))
+      {
+        Node ret = nm->mkNode(
+            kind::STRING_STRREPL,
+            nm->mkNode(kind::STRING_SUBSTR, node[0][0], node[1], node[2]),
+            node[0][1],
+            node[0][2]);
+        return returnRewrite(node, ret, "substr-repl-swap");
+      }
+    }
+  }
+
+  std::vector<Node> n1;
+  utils::getConcat(node[0], n1);
+
+  // definite inclusion
+  if (node[1] == zero)
+  {
+    Node curr = node[2];
+    std::vector<Node> childrenr;
+    if (stripSymbolicLength(n1, childrenr, 1, curr))
+    {
+      if (curr != zero && !n1.empty())
+      {
+        childrenr.push_back(nm->mkNode(kind::STRING_SUBSTR,
+                                       utils::mkConcat(STRING_CONCAT, n1),
+                                       node[1],
+                                       curr));
+      }
+      Node ret = utils::mkConcat(STRING_CONCAT, childrenr);
+      return returnRewrite(node, ret, "ss-len-include");
+    }
+  }
+
+  // symbolic length analysis
+  for (unsigned r = 0; r < 2; r++)
+  {
+    // the amount of characters we can strip
+    Node curr;
+    if (r == 0)
+    {
+      if (node[1] != zero)
+      {
+        // strip up to start point off the start of the string
+        curr = node[1];
+      }
+    }
+    else if (r == 1)
+    {
+      Node tot_len =
+          Rewriter::rewrite(nm->mkNode(kind::STRING_LENGTH, node[0]));
+      Node end_pt = Rewriter::rewrite(nm->mkNode(kind::PLUS, node[1], node[2]));
+      if (node[2] != tot_len)
+      {
+        if (checkEntailArith(node[2], tot_len))
+        {
+          // end point beyond end point of string, map to tot_len
+          Node ret = nm->mkNode(kind::STRING_SUBSTR, node[0], node[1], tot_len);
+          return returnRewrite(node, ret, "ss-end-pt-norm");
+        }
+        else
+        {
+          // strip up to ( str.len(node[0]) - end_pt ) off the end of the string
+          curr = Rewriter::rewrite(nm->mkNode(kind::MINUS, tot_len, end_pt));
+        }
+      }
+
+      // (str.substr s x y) --> "" if x < len(s) |= 0 >= y
+      Node n1_lt_tot_len =
+          Rewriter::rewrite(nm->mkNode(kind::LT, node[1], tot_len));
+      if (checkEntailArithWithAssumption(n1_lt_tot_len, zero, node[2], false))
+      {
+        Node ret = Word::mkEmptyWord(node.getType());
+        return returnRewrite(node, ret, "ss-start-entails-zero-len");
+      }
+
+      // (str.substr s x y) --> "" if 0 < y |= x >= str.len(s)
+      Node non_zero_len =
+          Rewriter::rewrite(nm->mkNode(kind::LT, zero, node[2]));
+      if (checkEntailArithWithAssumption(non_zero_len, node[1], tot_len, false))
+      {
+        Node ret = Word::mkEmptyWord(node.getType());
+        return returnRewrite(node, ret, "ss-non-zero-len-entails-oob");
+      }
+
+      // (str.substr s x y) --> "" if x >= 0 |= 0 >= str.len(s)
+      Node geq_zero_start =
+          Rewriter::rewrite(nm->mkNode(kind::GEQ, node[1], zero));
+      if (checkEntailArithWithAssumption(geq_zero_start, zero, tot_len, false))
+      {
+        Node ret = Word::mkEmptyWord(node.getType());
+        return returnRewrite(node, ret, "ss-geq-zero-start-entails-emp-s");
+      }
+
+      // (str.substr s x x) ---> "" if (str.len s) <= 1
+      if (node[1] == node[2] && checkEntailLengthOne(node[0]))
+      {
+        Node ret = Word::mkEmptyWord(node.getType());
+        return returnRewrite(node, ret, "ss-len-one-z-z");
+      }
+    }
+    if (!curr.isNull())
+    {
+      // strip off components while quantity is entailed positive
+      int dir = r == 0 ? 1 : -1;
+      std::vector<Node> childrenr;
+      if (stripSymbolicLength(n1, childrenr, dir, curr))
+      {
+        if (r == 0)
+        {
+          Node ret = nm->mkNode(kind::STRING_SUBSTR,
+                                utils::mkConcat(STRING_CONCAT, n1),
+                                curr,
+                                node[2]);
+          return returnRewrite(node, ret, "ss-strip-start-pt");
+        }
+        else
+        {
+          Node ret = nm->mkNode(kind::STRING_SUBSTR,
+                                utils::mkConcat(STRING_CONCAT, n1),
+                                node[1],
+                                node[2]);
+          return returnRewrite(node, ret, "ss-strip-end-pt");
+        }
+      }
+    }
+  }
+  // combine substr
+  if (node[0].getKind() == kind::STRING_SUBSTR)
+  {
+    Node start_inner = node[0][1];
+    Node start_outer = node[1];
+    if (checkEntailArith(start_outer) && checkEntailArith(start_inner))
+    {
+      // both are positive
+      // thus, start point is definitely start_inner+start_outer.
+      // We can rewrite if it for certain what the length is
+
+      // the length of a string from the inner substr subtracts the start point
+      // of the outer substr
+      Node len_from_inner =
+          Rewriter::rewrite(nm->mkNode(kind::MINUS, node[0][2], start_outer));
+      Node len_from_outer = node[2];
+      Node new_len;
+      // take quantity that is for sure smaller than the other
+      if (len_from_inner == len_from_outer)
+      {
+        new_len = len_from_inner;
+      }
+      else if (checkEntailArith(len_from_inner, len_from_outer))
+      {
+        new_len = len_from_outer;
+      }
+      else if (checkEntailArith(len_from_outer, len_from_inner))
+      {
+        new_len = len_from_inner;
+      }
+      if (!new_len.isNull())
+      {
+        Node new_start = nm->mkNode(kind::PLUS, start_inner, start_outer);
+        Node ret =
+            nm->mkNode(kind::STRING_SUBSTR, node[0][0], new_start, new_len);
+        return returnRewrite(node, ret, "ss-combine");
+      }
+    }
+  }
+  Trace("strings-rewrite-nf") << "No rewrites for : " << node << std::endl;
+  return node;
+}
+
+Node SequencesRewriter::rewriteContains(Node node)
+{
+  Assert(node.getKind() == kind::STRING_STRCTN);
+  NodeManager* nm = NodeManager::currentNM();
+
+  if (node[0] == node[1])
+  {
+    Node ret = NodeManager::currentNM()->mkConst(true);
+    return returnRewrite(node, ret, "ctn-eq");
+  }
+  if (node[0].isConst())
+  {
+    CVC4::String s = node[0].getConst<String>();
+    if (node[1].isConst())
+    {
+      Node ret = nm->mkConst(Word::find(node[0], node[1]) != std::string::npos);
+      return returnRewrite(node, ret, "ctn-const");
+    }
+    else
+    {
+      Node t = node[1];
+      if (Word::isEmpty(node[0]))
+      {
+        Node len1 =
+            NodeManager::currentNM()->mkNode(kind::STRING_LENGTH, node[1]);
+        if (checkEntailArith(len1, true))
+        {
+          // we handle the false case here since the rewrite for equality
+          // uses this function, hence we want to conclude false if possible.
+          // len(x)>0 => contains( "", x ) ---> false
+          Node ret = NodeManager::currentNM()->mkConst(false);
+          return returnRewrite(node, ret, "ctn-lhs-emptystr");
+        }
+      }
+      else if (checkEntailLengthOne(t))
+      {
+        const std::vector<unsigned>& vec = s.getVec();
+
+        NodeBuilder<> nb(OR);
+        nb << nm->mkConst(String("")).eqNode(t);
+        for (unsigned c : vec)
+        {
+          std::vector<unsigned> sv = {c};
+          nb << nm->mkConst(String(sv)).eqNode(t);
+        }
+
+        // str.contains("ABCabc", t) --->
+        // t = "" v t = "A" v t = "B" v t = "C" v t = "a" v t = "b" v t = "c"
+        // if len(t) <= 1
+        Node ret = nb;
+        return returnRewrite(node, ret, "ctn-split");
+      }
+      else if (node[1].getKind() == kind::STRING_CONCAT)
+      {
+        int firstc, lastc;
+        if (!canConstantContainConcat(node[0], node[1], firstc, lastc))
+        {
+          Node ret = NodeManager::currentNM()->mkConst(false);
+          return returnRewrite(node, ret, "ctn-nconst-ctn-concat");
+        }
+      }
+    }
+  }
+  if (node[1].isConst())
+  {
+    size_t len = Word::getLength(node[1]);
+    if (len == 0)
+    {
+      // contains( x, "" ) ---> true
+      Node ret = NodeManager::currentNM()->mkConst(true);
+      return returnRewrite(node, ret, "ctn-rhs-emptystr");
+    }
+    else if (len == 1)
+    {
+      // The following rewrites are specific to a single character second
+      // argument of contains, where we can reason that this character is
+      // not split over multiple components in the first argument.
+      if (node[0].getKind() == STRING_CONCAT)
+      {
+        std::vector<Node> nc1;
+        utils::getConcat(node[0], nc1);
+        NodeBuilder<> nb(OR);
+        for (const Node& ncc : nc1)
+        {
+          nb << nm->mkNode(STRING_STRCTN, ncc, node[1]);
+        }
+        Node ret = nb.constructNode();
+        // str.contains( x ++ y, "A" ) --->
+        //   str.contains( x, "A" ) OR str.contains( y, "A" )
+        return returnRewrite(node, ret, "ctn-concat-char");
+      }
+      else if (node[0].getKind() == STRING_STRREPL)
+      {
+        Node rplDomain = checkEntailContains(node[0][1], node[1]);
+        if (!rplDomain.isNull() && !rplDomain.getConst<bool>())
+        {
+          Node d1 = nm->mkNode(STRING_STRCTN, node[0][0], node[1]);
+          Node d2 =
+              nm->mkNode(AND,
+                         nm->mkNode(STRING_STRCTN, node[0][0], node[0][1]),
+                         nm->mkNode(STRING_STRCTN, node[0][2], node[1]));
+          Node ret = nm->mkNode(OR, d1, d2);
+          // If str.contains( y, "A" ) ---> false, then:
+          // str.contains( str.replace( x, y, z ), "A" ) --->
+          //   str.contains( x, "A" ) OR
+          //   ( str.contains( x, y ) AND str.contains( z, "A" ) )
+          return returnRewrite(node, ret, "ctn-repl-char");
+        }
+      }
+    }
+  }
+  std::vector<Node> nc1;
+  utils::getConcat(node[0], nc1);
+  std::vector<Node> nc2;
+  utils::getConcat(node[1], nc2);
+
+  // component-wise containment
+  std::vector<Node> nc1rb;
+  std::vector<Node> nc1re;
+  if (componentContains(nc1, nc2, nc1rb, nc1re) != -1)
+  {
+    Node ret = NodeManager::currentNM()->mkConst(true);
+    return returnRewrite(node, ret, "ctn-component");
+  }
+
+  // strip endpoints
+  std::vector<Node> nb;
+  std::vector<Node> ne;
+  if (stripConstantEndpoints(nc1, nc2, nb, ne))
+  {
+    Node ret = NodeManager::currentNM()->mkNode(
+        kind::STRING_STRCTN, utils::mkConcat(STRING_CONCAT, nc1), node[1]);
+    return returnRewrite(node, ret, "ctn-strip-endpt");
+  }
+
+  for (const Node& n : nc2)
+  {
+    if (n.getKind() == kind::STRING_STRREPL)
+    {
+      // (str.contains x (str.replace y z w)) --> false
+      // if (str.contains x y) = false and (str.contains x w) = false
+      //
+      // Reasoning: (str.contains x y) checks that x does not contain y if the
+      // replacement does not change y. (str.contains x w) checks that if the
+      // replacement changes anything in y, the w makes it impossible for it to
+      // occur in x.
+      Node ctnConst = checkEntailContains(node[0], n[0]);
+      if (!ctnConst.isNull() && !ctnConst.getConst<bool>())
+      {
+        Node ctnConst2 = checkEntailContains(node[0], n[2]);
+        if (!ctnConst2.isNull() && !ctnConst2.getConst<bool>())
+        {
+          Node res = nm->mkConst(false);
+          return returnRewrite(node, res, "ctn-rpl-non-ctn");
+        }
+      }
+
+      // (str.contains x (str.++ w (str.replace x y x) z)) --->
+      //   (and (= w "") (= x (str.replace x y x)) (= z ""))
+      //
+      // TODO: Remove with under-/over-approximation
+      if (node[0] == n[0] && node[0] == n[2])
+      {
+        Node ret;
+        if (nc2.size() > 1)
+        {
+          Node emp = nm->mkConst(CVC4::String(""));
+          NodeBuilder<> nb2(kind::AND);
+          for (const Node& n2 : nc2)
+          {
+            if (n2 == n)
+            {
+              nb2 << nm->mkNode(kind::EQUAL, node[0], node[1]);
+            }
+            else
+            {
+              nb2 << nm->mkNode(kind::EQUAL, emp, n2);
+            }
+          }
+          ret = nb2.constructNode();
+        }
+        else
+        {
+          ret = nm->mkNode(kind::EQUAL, node[0], node[1]);
+        }
+        return returnRewrite(node, ret, "ctn-repl-self");
+      }
+    }
+  }
+
+  // length entailment
+  Node len_n1 = NodeManager::currentNM()->mkNode(kind::STRING_LENGTH, node[0]);
+  Node len_n2 = NodeManager::currentNM()->mkNode(kind::STRING_LENGTH, node[1]);
+  if (checkEntailArith(len_n2, len_n1, true))
+  {
+    // len( n2 ) > len( n1 ) => contains( n1, n2 ) ---> false
+    Node ret = NodeManager::currentNM()->mkConst(false);
+    return returnRewrite(node, ret, "ctn-len-ineq");
+  }
+
+  // multi-set reasoning
+  //   For example, contains( str.++( x, "b" ), str.++( "a", x ) ) ---> false
+  //   since the number of a's in the second argument is greater than the number
+  //   of a's in the first argument
+  if (checkEntailMultisetSubset(node[0], node[1]))
+  {
+    Node ret = nm->mkConst(false);
+    return returnRewrite(node, ret, "ctn-mset-nss");
+  }
+
+  if (checkEntailArith(len_n2, len_n1, false))
+  {
+    // len( n2 ) >= len( n1 ) => contains( n1, n2 ) ---> n1 = n2
+    Node ret = node[0].eqNode(node[1]);
+    return returnRewrite(node, ret, "ctn-len-ineq-nstrict");
+  }
+
+  // splitting
+  if (node[0].getKind() == kind::STRING_CONCAT)
+  {
+    if (node[1].isConst())
+    {
+      CVC4::String t = node[1].getConst<String>();
+      // Below, we are looking for a constant component of node[0]
+      // has no overlap with node[1], which means we can split.
+      // Notice that if the first or last components had no
+      // overlap, these would have been removed by strip
+      // constant endpoints above.
+      // Hence, we consider only the inner children.
+      for (unsigned i = 1; i < (node[0].getNumChildren() - 1); i++)
+      {
+        // constant contains
+        if (node[0][i].isConst())
+        {
+          CVC4::String s = node[0][i].getConst<String>();
+          // if no overlap, we can split into disjunction
+          if (s.noOverlapWith(t))
+          {
+            std::vector<Node> nc0;
+            utils::getConcat(node[0], nc0);
+            std::vector<Node> spl[2];
+            spl[0].insert(spl[0].end(), nc0.begin(), nc0.begin() + i);
+            Assert(i < nc0.size() - 1);
+            spl[1].insert(spl[1].end(), nc0.begin() + i + 1, nc0.end());
+            Node ret = NodeManager::currentNM()->mkNode(
+                kind::OR,
+                NodeManager::currentNM()->mkNode(
+                    kind::STRING_STRCTN,
+                    utils::mkConcat(STRING_CONCAT, spl[0]),
+                    node[1]),
+                NodeManager::currentNM()->mkNode(
+                    kind::STRING_STRCTN,
+                    utils::mkConcat(STRING_CONCAT, spl[1]),
+                    node[1]));
+            return returnRewrite(node, ret, "ctn-split");
+          }
+        }
+      }
+    }
+  }
+  else if (node[0].getKind() == kind::STRING_SUBSTR)
+  {
+    // (str.contains (str.substr x n (str.len y)) y) --->
+    //   (= (str.substr x n (str.len y)) y)
+    //
+    // TODO: Remove with under-/over-approximation
+    if (node[0][2] == nm->mkNode(kind::STRING_LENGTH, node[1]))
+    {
+      Node ret = nm->mkNode(kind::EQUAL, node[0], node[1]);
+      return returnRewrite(node, ret, "ctn-substr");
+    }
+  }
+  else if (node[0].getKind() == kind::STRING_STRREPL)
+  {
+    if (node[1].isConst() && node[0][1].isConst() && node[0][2].isConst())
+    {
+      if (Word::noOverlapWith(node[1], node[0][1])
+          && Word::noOverlapWith(node[1], node[0][2]))
+      {
+        // (str.contains (str.replace x c1 c2) c3) ---> (str.contains x c3)
+        // if there is no overlap between c1 and c3 and none between c2 and c3
+        Node ret = nm->mkNode(STRING_STRCTN, node[0][0], node[1]);
+        return returnRewrite(node, ret, "ctn-repl-cnsts-to-ctn");
+      }
+    }
+
+    if (node[0][0] == node[0][2])
+    {
+      // (str.contains (str.replace x y x) y) ---> (str.contains x y)
+      if (node[0][1] == node[1])
+      {
+        Node ret = nm->mkNode(kind::STRING_STRCTN, node[0][0], node[1]);
+        return returnRewrite(node, ret, "ctn-repl-to-ctn");
+      }
+
+      // (str.contains (str.replace x y x) z) ---> (str.contains x z)
+      // if (str.len z) <= 1
+      if (checkEntailLengthOne(node[1]))
+      {
+        Node ret = nm->mkNode(kind::STRING_STRCTN, node[0][0], node[1]);
+        return returnRewrite(node, ret, "ctn-repl-len-one-to-ctn");
+      }
+    }
+
+    // (str.contains (str.replace x y z) z) --->
+    //   (or (str.contains x y) (str.contains x z))
+    if (node[0][2] == node[1])
+    {
+      Node ret = nm->mkNode(OR,
+                            nm->mkNode(STRING_STRCTN, node[0][0], node[0][1]),
+                            nm->mkNode(STRING_STRCTN, node[0][0], node[0][2]));
+      return returnRewrite(node, ret, "ctn-repl-to-ctn-disj");
+    }
+
+    // (str.contains (str.replace x y z) w) --->
+    //   (str.contains (str.replace x y "") w)
+    // if (str.contains z w) ---> false and (str.len w) = 1
+    if (checkEntailLengthOne(node[1]))
+    {
+      Node ctn = checkEntailContains(node[1], node[0][2]);
+      if (!ctn.isNull() && !ctn.getConst<bool>())
+      {
+        Node empty = nm->mkConst(String(""));
+        Node ret = nm->mkNode(
+            kind::STRING_STRCTN,
+            nm->mkNode(kind::STRING_STRREPL, node[0][0], node[0][1], empty),
+            node[1]);
+        return returnRewrite(node, ret, "ctn-repl-simp-repl");
+      }
+    }
+  }
+
+  if (node[1].getKind() == kind::STRING_STRREPL)
+  {
+    // (str.contains x (str.replace y x y)) --->
+    //   (str.contains x y)
+    if (node[0] == node[1][1] && node[1][0] == node[1][2])
+    {
+      Node ret = nm->mkNode(kind::STRING_STRCTN, node[0], node[1][0]);
+      return returnRewrite(node, ret, "ctn-repl");
+    }
+
+    // (str.contains x (str.replace "" x y)) --->
+    //   (= "" (str.replace "" x y))
+    //
+    // Note: Length-based reasoning is not sufficient to get this rewrite. We
+    // can neither show that str.len(str.replace("", x, y)) - str.len(x) >= 0
+    // nor str.len(x) - str.len(str.replace("", x, y)) >= 0
+    Node emp = nm->mkConst(CVC4::String(""));
+    if (node[0] == node[1][1] && node[1][0] == emp)
+    {
+      Node ret = nm->mkNode(kind::EQUAL, emp, node[1]);
+      return returnRewrite(node, ret, "ctn-repl-empty");
+    }
+  }
+
+  Trace("strings-rewrite-nf") << "No rewrites for : " << node << std::endl;
+  return node;
+}
+
+Node SequencesRewriter::rewriteIndexof(Node node)
+{
+  Assert(node.getKind() == kind::STRING_STRIDOF);
+  NodeManager* nm = NodeManager::currentNM();
+
+  if (node[2].isConst() && node[2].getConst<Rational>().sgn() < 0)
+  {
+    // z<0  implies  str.indexof( x, y, z ) --> -1
+    Node negone = nm->mkConst(Rational(-1));
+    return returnRewrite(node, negone, "idof-neg");
+  }
+
+  // evaluation and simple cases
+  std::vector<Node> children0;
+  utils::getConcat(node[0], children0);
+  if (children0[0].isConst() && node[1].isConst() && node[2].isConst())
+  {
+    CVC4::Rational rMaxInt(CVC4::String::maxSize());
+    if (node[2].getConst<Rational>() > rMaxInt)
+    {
+      // We know that, due to limitations on the size of string constants
+      // in our implementation, that accessing a position greater than
+      // rMaxInt is guaranteed to be out of bounds.
+      Node negone = nm->mkConst(Rational(-1));
+      return returnRewrite(node, negone, "idof-max");
+    }
+    Assert(node[2].getConst<Rational>().sgn() >= 0);
+    Node s = children0[0];
+    Node t = node[1];
+    uint32_t start =
+        node[2].getConst<Rational>().getNumerator().toUnsignedInt();
+    std::size_t ret = Word::find(s, t, start);
+    if (ret != std::string::npos)
+    {
+      Node retv = nm->mkConst(Rational(static_cast<unsigned>(ret)));
+      return returnRewrite(node, retv, "idof-find");
+    }
+    else if (children0.size() == 1)
+    {
+      Node negone = nm->mkConst(Rational(-1));
+      return returnRewrite(node, negone, "idof-nfind");
+    }
+  }
+
+  if (node[0] == node[1])
+  {
+    if (node[2].isConst())
+    {
+      if (node[2].getConst<Rational>().sgn() == 0)
+      {
+        // indexof( x, x, 0 ) --> 0
+        Node zero = nm->mkConst(Rational(0));
+        return returnRewrite(node, zero, "idof-eq-cst-start");
+      }
+    }
+    if (checkEntailArith(node[2], true))
+    {
+      // y>0  implies  indexof( x, x, y ) --> -1
+      Node negone = nm->mkConst(Rational(-1));
+      return returnRewrite(node, negone, "idof-eq-nstart");
+    }
+    Node emp = nm->mkConst(CVC4::String(""));
+    if (node[0] != emp)
+    {
+      // indexof( x, x, z ) ---> indexof( "", "", z )
+      Node ret = nm->mkNode(STRING_STRIDOF, emp, emp, node[2]);
+      return returnRewrite(node, ret, "idof-eq-norm");
+    }
+  }
+
+  Node len0 = nm->mkNode(STRING_LENGTH, node[0]);
+  Node len1 = nm->mkNode(STRING_LENGTH, node[1]);
+  Node len0m2 = nm->mkNode(MINUS, len0, node[2]);
+
+  if (node[1].isConst())
+  {
+    if (Word::isEmpty(node[1]))
+    {
+      if (checkEntailArith(len0, node[2]) && checkEntailArith(node[2]))
+      {
+        // len(x)>=z ^ z >=0 implies indexof( x, "", z ) ---> z
+        return returnRewrite(node, node[2], "idof-emp-idof");
+      }
+    }
+  }
+
+  if (checkEntailArith(len1, len0m2, true))
+  {
+    // len(x)-z < len(y)  implies  indexof( x, y, z ) ----> -1
+    Node negone = nm->mkConst(Rational(-1));
+    return returnRewrite(node, negone, "idof-len");
+  }
+
+  Node fstr = node[0];
+  if (!node[2].isConst() || node[2].getConst<Rational>().sgn() != 0)
+  {
+    fstr = nm->mkNode(kind::STRING_SUBSTR, node[0], node[2], len0);
+    fstr = Rewriter::rewrite(fstr);
+  }
+
+  Node cmp_conr = checkEntailContains(fstr, node[1]);
+  Trace("strings-rewrite-debug") << "For " << node << ", check contains("
+                                 << fstr << ", " << node[1] << ")" << std::endl;
+  Trace("strings-rewrite-debug") << "...got " << cmp_conr << std::endl;
+  std::vector<Node> children1;
+  utils::getConcat(node[1], children1);
+  if (!cmp_conr.isNull())
+  {
+    if (cmp_conr.getConst<bool>())
+    {
+      if (node[2].isConst() && node[2].getConst<Rational>().sgn() == 0)
+      {
+        // past the first position in node[0] that contains node[1], we can drop
+        std::vector<Node> nb;
+        std::vector<Node> ne;
+        int cc = componentContains(children0, children1, nb, ne, true, 1);
+        if (cc != -1 && !ne.empty())
+        {
+          // For example:
+          // str.indexof(str.++(x,y,z),y,0) ---> str.indexof(str.++(x,y),y,0)
+          Node nn = utils::mkConcat(STRING_CONCAT, children0);
+          Node ret = nm->mkNode(kind::STRING_STRIDOF, nn, node[1], node[2]);
+          return returnRewrite(node, ret, "idof-def-ctn");
+        }
+
+        // Strip components from the beginning that are guaranteed not to match
+        if (stripConstantEndpoints(children0, children1, nb, ne, 1))
+        {
+          // str.indexof(str.++("AB", x, "C"), "C", 0) --->
+          // 2 + str.indexof(str.++(x, "C"), "C", 0)
+          Node ret =
+              nm->mkNode(kind::PLUS,
+                         nm->mkNode(kind::STRING_LENGTH,
+                                    utils::mkConcat(STRING_CONCAT, nb)),
+                         nm->mkNode(kind::STRING_STRIDOF,
+                                    utils::mkConcat(STRING_CONCAT, children0),
+                                    node[1],
+                                    node[2]));
+          return returnRewrite(node, ret, "idof-strip-cnst-endpts");
+        }
+      }
+
+      // strip symbolic length
+      Node new_len = node[2];
+      std::vector<Node> nr;
+      if (stripSymbolicLength(children0, nr, 1, new_len))
+      {
+        // For example:
+        // z>str.len( x1 ) and str.contains( x2, y )-->true
+        // implies
+        // str.indexof( str.++( x1, x2 ), y, z ) --->
+        // str.len( x1 ) + str.indexof( x2, y, z-str.len(x1) )
+        Node nn = utils::mkConcat(STRING_CONCAT, children0);
+        Node ret =
+            nm->mkNode(kind::PLUS,
+                       nm->mkNode(kind::MINUS, node[2], new_len),
+                       nm->mkNode(kind::STRING_STRIDOF, nn, node[1], new_len));
+        return returnRewrite(node, ret, "idof-strip-sym-len");
+      }
+    }
+    else
+    {
+      // str.contains( x, y ) --> false  implies  str.indexof(x,y,z) --> -1
+      Node negone = nm->mkConst(Rational(-1));
+      return returnRewrite(node, negone, "idof-nctn");
+    }
+  }
+  else
+  {
+    Node new_len = node[2];
+    std::vector<Node> nr;
+    if (stripSymbolicLength(children0, nr, 1, new_len))
+    {
+      // Normalize the string before the start index.
+      //
+      // For example:
+      // str.indexof(str.++("ABCD", x), y, 3) --->
+      // str.indexof(str.++("AAAD", x), y, 3)
+      Node nodeNr = utils::mkConcat(STRING_CONCAT, nr);
+      Node normNr = lengthPreserveRewrite(nodeNr);
+      if (normNr != nodeNr)
+      {
+        std::vector<Node> normNrChildren;
+        utils::getConcat(normNr, normNrChildren);
+        std::vector<Node> children(normNrChildren);
+        children.insert(children.end(), children0.begin(), children0.end());
+        Node nn = utils::mkConcat(STRING_CONCAT, children);
+        Node res = nm->mkNode(kind::STRING_STRIDOF, nn, node[1], node[2]);
+        return returnRewrite(node, res, "idof-norm-prefix");
+      }
+    }
+  }
+
+  if (node[2].isConst() && node[2].getConst<Rational>().sgn() == 0)
+  {
+    std::vector<Node> cb;
+    std::vector<Node> ce;
+    if (stripConstantEndpoints(children0, children1, cb, ce, -1))
+    {
+      Node ret = utils::mkConcat(STRING_CONCAT, children0);
+      ret = nm->mkNode(STRING_STRIDOF, ret, node[1], node[2]);
+      // For example:
+      // str.indexof( str.++( x, "A" ), "B", 0 ) ---> str.indexof( x, "B", 0 )
+      return returnRewrite(node, ret, "rpl-pull-endpt");
+    }
+  }
+
+  Trace("strings-rewrite-nf") << "No rewrites for : " << node << std::endl;
+  return node;
+}
+
+Node SequencesRewriter::rewriteReplace(Node node)
+{
+  Assert(node.getKind() == kind::STRING_STRREPL);
+  NodeManager* nm = NodeManager::currentNM();
+
+  if (node[1].isConst() && Word::isEmpty(node[1]))
+  {
+    Node ret = nm->mkNode(STRING_CONCAT, node[2], node[0]);
+    return returnRewrite(node, ret, "rpl-rpl-empty");
+  }
+
+  std::vector<Node> children0;
+  utils::getConcat(node[0], children0);
+
+  if (node[1].isConst() && children0[0].isConst())
+  {
+    Node s = children0[0];
+    Node t = node[1];
+    std::size_t p = Word::find(s, t);
+    if (p == std::string::npos)
+    {
+      if (children0.size() == 1)
+      {
+        return returnRewrite(node, node[0], "rpl-const-nfind");
+      }
+    }
+    else
+    {
+      Node s1 = Word::substr(s, 0, p);
+      Node s3 = Word::substr(s, p + Word::getLength(t));
+      std::vector<Node> children;
+      if (!Word::isEmpty(s1))
+      {
+        children.push_back(s1);
+      }
+      children.push_back(node[2]);
+      if (!Word::isEmpty(s3))
+      {
+        children.push_back(s3);
+      }
+      children.insert(children.end(), children0.begin() + 1, children0.end());
+      Node ret = utils::mkConcat(STRING_CONCAT, children);
+      return returnRewrite(node, ret, "rpl-const-find");
+    }
+  }
+
+  // rewrites that apply to both replace and replaceall
+  Node rri = rewriteReplaceInternal(node);
+  if (!rri.isNull())
+  {
+    // printing of the rewrite managed by the call above
+    return rri;
+  }
+
+  if (node[0] == node[2])
+  {
+    // ( len( y )>=len(x) ) => str.replace( x, y, x ) ---> x
+    Node l0 = NodeManager::currentNM()->mkNode(kind::STRING_LENGTH, node[0]);
+    Node l1 = NodeManager::currentNM()->mkNode(kind::STRING_LENGTH, node[1]);
+    if (checkEntailArith(l1, l0))
+    {
+      return returnRewrite(node, node[0], "rpl-rpl-len-id");
+    }
+
+    // (str.replace x y x) ---> (str.replace x (str.++ y1 ... yn) x)
+    // if 1 >= (str.len x) and (= y "") ---> (= y1 "") ... (= yn "")
+    if (checkEntailLengthOne(node[0]))
+    {
+      Node empty = nm->mkConst(String(""));
+      Node rn1 = Rewriter::rewrite(
+          rewriteEqualityExt(nm->mkNode(EQUAL, node[1], empty)));
+      if (rn1 != node[1])
+      {
+        std::vector<Node> emptyNodes;
+        bool allEmptyEqs;
+        std::tie(allEmptyEqs, emptyNodes) = collectEmptyEqs(rn1);
+
+        if (allEmptyEqs)
+        {
+          Node nn1 = utils::mkConcat(STRING_CONCAT, emptyNodes);
+          if (node[1] != nn1)
+          {
+            Node ret = nm->mkNode(STRING_STRREPL, node[0], nn1, node[2]);
+            return returnRewrite(node, ret, "rpl-x-y-x-simp");
+          }
+        }
+      }
+    }
+  }
+
+  std::vector<Node> children1;
+  utils::getConcat(node[1], children1);
+
+  // check if contains definitely does (or does not) hold
+  Node cmp_con = nm->mkNode(kind::STRING_STRCTN, node[0], node[1]);
+  Node cmp_conr = Rewriter::rewrite(cmp_con);
+  if (!checkEntailContains(node[0], node[1]).isNull())
+  {
+    if (cmp_conr.getConst<bool>())
+    {
+      // component-wise containment
+      std::vector<Node> cb;
+      std::vector<Node> ce;
+      int cc = componentContains(children0, children1, cb, ce, true, 1);
+      if (cc != -1)
+      {
+        if (cc == 0 && children0[0] == children1[0])
+        {
+          // definitely a prefix, can do the replace
+          // for example,
+          //   str.replace( str.++( x, "ab" ), str.++( x, "a" ), y )  --->
+          //   str.++( y, "b" )
+          std::vector<Node> cres;
+          cres.push_back(node[2]);
+          cres.insert(cres.end(), ce.begin(), ce.end());
+          Node ret = utils::mkConcat(STRING_CONCAT, cres);
+          return returnRewrite(node, ret, "rpl-cctn-rpl");
+        }
+        else if (!ce.empty())
+        {
+          // we can pull remainder past first definite containment
+          // for example,
+          //   str.replace( str.++( x, "ab" ), "a", y ) --->
+          //   str.++( str.replace( str.++( x, "a" ), "a", y ), "b" )
+          // this is independent of whether the second argument may be empty
+          std::vector<Node> scc;
+          scc.push_back(NodeManager::currentNM()->mkNode(
+              kind::STRING_STRREPL,
+              utils::mkConcat(STRING_CONCAT, children0),
+              node[1],
+              node[2]));
+          scc.insert(scc.end(), ce.begin(), ce.end());
+          Node ret = utils::mkConcat(STRING_CONCAT, scc);
+          return returnRewrite(node, ret, "rpl-cctn");
+        }
+      }
+    }
+    else
+    {
+      // ~contains( t, s ) => ( replace( t, s, r ) ----> t )
+      return returnRewrite(node, node[0], "rpl-nctn");
+    }
+  }
+  else if (cmp_conr.getKind() == kind::EQUAL || cmp_conr.getKind() == kind::AND)
+  {
+    // Rewriting the str.contains may return equalities of the form (= x "").
+    // In that case, we can substitute the variables appearing in those
+    // equalities with the empty string in the third argument of the
+    // str.replace. For example:
+    //
+    // (str.replace x (str.++ x y) y) --> (str.replace x (str.++ x y) "")
+    //
+    // This can be done because str.replace changes x iff (str.++ x y) is in x
+    // but that means that y must be empty in that case. Thus, we can
+    // substitute y with "" in the third argument. Note that the third argument
+    // does not matter when the str.replace does not apply.
+    //
+    Node empty = nm->mkConst(::CVC4::String(""));
+
+    std::vector<Node> emptyNodes;
+    bool allEmptyEqs;
+    std::tie(allEmptyEqs, emptyNodes) = collectEmptyEqs(cmp_conr);
+
+    if (emptyNodes.size() > 0)
+    {
+      // Perform the substitutions
+      std::vector<TNode> substs(emptyNodes.size(), TNode(empty));
+      Node nn2 = node[2].substitute(
+          emptyNodes.begin(), emptyNodes.end(), substs.begin(), substs.end());
+
+      // If the contains rewrites to a conjunction of empty-string equalities
+      // and we are doing the replacement in an empty string, we can rewrite
+      // the string-to-replace with a concatenation of all the terms that must
+      // be empty:
+      //
+      // (str.replace "" y z) ---> (str.replace "" (str.++ y1 ... yn)  z)
+      // if (str.contains "" y) ---> (and (= y1 "") ... (= yn ""))
+      if (node[0] == empty && allEmptyEqs)
+      {
+        std::vector<Node> emptyNodesList(emptyNodes.begin(), emptyNodes.end());
+        Node nn1 = utils::mkConcat(STRING_CONCAT, emptyNodesList);
+        if (nn1 != node[1] || nn2 != node[2])
+        {
+          Node res = nm->mkNode(kind::STRING_STRREPL, node[0], nn1, nn2);
+          return returnRewrite(node, res, "rpl-emp-cnts-substs");
+        }
+      }
+
+      if (nn2 != node[2])
+      {
+        Node res = nm->mkNode(kind::STRING_STRREPL, node[0], node[1], nn2);
+        return returnRewrite(node, res, "rpl-cnts-substs");
+      }
+    }
+  }
+
+  if (cmp_conr != cmp_con)
+  {
+    if (checkEntailNonEmpty(node[1]))
+    {
+      // pull endpoints that can be stripped
+      // for example,
+      //   str.replace( str.++( "b", x, "b" ), "a", y ) --->
+      //   str.++( "b", str.replace( x, "a", y ), "b" )
+      std::vector<Node> cb;
+      std::vector<Node> ce;
+      if (stripConstantEndpoints(children0, children1, cb, ce))
+      {
+        std::vector<Node> cc;
+        cc.insert(cc.end(), cb.begin(), cb.end());
+        cc.push_back(NodeManager::currentNM()->mkNode(
+            kind::STRING_STRREPL,
+            utils::mkConcat(STRING_CONCAT, children0),
+            node[1],
+            node[2]));
+        cc.insert(cc.end(), ce.begin(), ce.end());
+        Node ret = utils::mkConcat(STRING_CONCAT, cc);
+        return returnRewrite(node, ret, "rpl-pull-endpt");
+      }
+    }
+  }
+
+  children1.clear();
+  utils::getConcat(node[1], children1);
+  Node lastChild1 = children1[children1.size() - 1];
+  if (lastChild1.getKind() == kind::STRING_SUBSTR)
+  {
+    // (str.replace x (str.++ t (str.substr y i j)) z) --->
+    // (str.replace x (str.++ t
+    //                  (str.substr y i (+ (str.len x) 1 (- (str.len t))))) z)
+    // if j > len(x)
+    //
+    // Reasoning: If the string to be replaced is longer than x, then it does
+    // not matter how much longer it is, the result is always x. Thus, it is
+    // fine to only look at the prefix of length len(x) + 1 - len(t).
+
+    children1.pop_back();
+    // Length of the non-substr components in the second argument
+    Node partLen1 = nm->mkNode(kind::STRING_LENGTH,
+                               utils::mkConcat(STRING_CONCAT, children1));
+    Node maxLen1 = nm->mkNode(kind::PLUS, partLen1, lastChild1[2]);
+
+    Node zero = nm->mkConst(Rational(0));
+    Node one = nm->mkConst(Rational(1));
+    Node len0 = nm->mkNode(kind::STRING_LENGTH, node[0]);
+    Node len0_1 = nm->mkNode(kind::PLUS, len0, one);
+    // Check len(t) + j > len(x) + 1
+    if (checkEntailArith(maxLen1, len0_1, true))
+    {
+      children1.push_back(nm->mkNode(
+          kind::STRING_SUBSTR,
+          lastChild1[0],
+          lastChild1[1],
+          nm->mkNode(
+              kind::PLUS, len0, one, nm->mkNode(kind::UMINUS, partLen1))));
+      Node res = nm->mkNode(kind::STRING_STRREPL,
+                            node[0],
+                            utils::mkConcat(STRING_CONCAT, children1),
+                            node[2]);
+      return returnRewrite(node, res, "repl-subst-idx");
+    }
+  }
+
+  if (node[0].getKind() == STRING_STRREPL)
+  {
+    Node x = node[0];
+    Node y = node[1];
+    Node z = node[2];
+    if (x[0] == x[2] && x[0] == y)
+    {
+      // (str.replace (str.replace y w y) y z) -->
+      //   (str.replace (str.replace y w z) y z)
+      // if (str.len w) >= (str.len z) and w != z
+      //
+      // Reasoning: There are two cases: (1) w does not appear in y and (2) w
+      // does appear in y.
+      //
+      // Case (1): In this case, the reasoning is trivial. The
+      // inner replace does not do anything, so we can just replace its third
+      // argument with any string.
+      //
+      // Case (2): After the inner replace, we are guaranteed to have a string
+      // that contains y at the index of w in the original string y. The outer
+      // replace then replaces that y with z, so we can short-circuit that
+      // replace by directly replacing w with z in the inner replace. We can
+      // only do that if the result of the new inner replace does not contain
+      // y, otherwise we end up doing two replaces that are different from the
+      // original expression. We enforce that by requiring that the length of w
+      // has to be greater or equal to the length of z and that w and z have to
+      // be different. This makes sure that an inner replace changes a string
+      // to a string that is shorter than y, making it impossible for the outer
+      // replace to match.
+      Node w = x[1];
+
+      // (str.len w) >= (str.len z)
+      Node wlen = nm->mkNode(kind::STRING_LENGTH, w);
+      Node zlen = nm->mkNode(kind::STRING_LENGTH, z);
+      if (checkEntailArith(wlen, zlen))
+      {
+        // w != z
+        Node wEqZ = Rewriter::rewrite(nm->mkNode(kind::EQUAL, w, z));
+        if (wEqZ.isConst() && !wEqZ.getConst<bool>())
+        {
+          Node ret = nm->mkNode(kind::STRING_STRREPL,
+                                nm->mkNode(kind::STRING_STRREPL, y, w, z),
+                                y,
+                                z);
+          return returnRewrite(node, ret, "repl-repl-short-circuit");
+        }
+      }
+    }
+  }
+
+  if (node[1].getKind() == STRING_STRREPL)
+  {
+    if (node[1][0] == node[0])
+    {
+      if (node[1][0] == node[1][2] && node[1][0] == node[2])
+      {
+        // str.replace( x, str.replace( x, y, x ), x ) ---> x
+        return returnRewrite(node, node[0], "repl-repl2-inv-id");
+      }
+      bool dualReplIteSuccess = false;
+      Node cmp_con2 = checkEntailContains(node[1][0], node[1][2]);
+      if (!cmp_con2.isNull() && !cmp_con2.getConst<bool>())
+      {
+        // str.contains( x, z ) ---> false
+        //   implies
+        // str.replace( x, str.replace( x, y, z ), w ) --->
+        // ite( str.contains( x, y ), x, w )
+        dualReplIteSuccess = true;
+      }
+      else
+      {
+        // str.contains( y, z ) ---> false and str.contains( z, y ) ---> false
+        //   implies
+        // str.replace( x, str.replace( x, y, z ), w ) --->
+        // ite( str.contains( x, y ), x, w )
+        cmp_con2 = checkEntailContains(node[1][1], node[1][2]);
+        if (!cmp_con2.isNull() && !cmp_con2.getConst<bool>())
+        {
+          cmp_con2 = checkEntailContains(node[1][2], node[1][1]);
+          if (!cmp_con2.isNull() && !cmp_con2.getConst<bool>())
+          {
+            dualReplIteSuccess = true;
+          }
+        }
+      }
+      if (dualReplIteSuccess)
+      {
+        Node res = nm->mkNode(ITE,
+                              nm->mkNode(STRING_STRCTN, node[0], node[1][1]),
+                              node[0],
+                              node[2]);
+        return returnRewrite(node, res, "repl-dual-repl-ite");
+      }
+    }
+
+    bool invSuccess = false;
+    if (node[1][1] == node[0])
+    {
+      if (node[1][0] == node[1][2])
+      {
+        // str.replace(x, str.replace(y, x, y), w) ---> str.replace(x, y, w)
+        invSuccess = true;
+      }
+      else if (node[1][1] == node[2] || node[1][0] == node[2])
+      {
+        // str.contains(y, z) ----> false and ( y == w or x == w ) implies
+        //   implies
+        // str.replace(x, str.replace(y, x, z), w) ---> str.replace(x, y, w)
+        Node cmp_con2 = checkEntailContains(node[1][0], node[1][2]);
+        invSuccess = !cmp_con2.isNull() && !cmp_con2.getConst<bool>();
+      }
+    }
+    else
+    {
+      // str.contains(x, z) ----> false and str.contains(x, w) ----> false
+      //   implies
+      // str.replace(x, str.replace(y, z, w), u) ---> str.replace(x, y, u)
+      Node cmp_con2 = checkEntailContains(node[0], node[1][1]);
+      if (!cmp_con2.isNull() && !cmp_con2.getConst<bool>())
+      {
+        cmp_con2 = checkEntailContains(node[0], node[1][2]);
+        invSuccess = !cmp_con2.isNull() && !cmp_con2.getConst<bool>();
+      }
+    }
+    if (invSuccess)
+    {
+      Node res = nm->mkNode(kind::STRING_STRREPL, node[0], node[1][0], node[2]);
+      return returnRewrite(node, res, "repl-repl2-inv");
+    }
+  }
+  if (node[2].getKind() == STRING_STRREPL)
+  {
+    if (node[2][1] == node[0])
+    {
+      // str.contains( z, w ) ----> false implies
+      // str.replace( x, w, str.replace( z, x, y ) ) ---> str.replace( x, w, z )
+      Node cmp_con2 = checkEntailContains(node[1], node[2][0]);
+      if (!cmp_con2.isNull() && !cmp_con2.getConst<bool>())
+      {
+        Node res =
+            nm->mkNode(kind::STRING_STRREPL, node[0], node[1], node[2][0]);
+        return returnRewrite(node, res, "repl-repl3-inv");
+      }
+    }
+    if (node[2][0] == node[1])
+    {
+      bool success = false;
+      if (node[2][0] == node[2][2] && node[2][1] == node[0])
+      {
+        // str.replace( x, y, str.replace( y, x, y ) ) ---> x
+        success = true;
+      }
+      else
+      {
+        // str.contains( x, z ) ----> false implies
+        // str.replace( x, y, str.replace( y, z, w ) ) ---> x
+        cmp_con = checkEntailContains(node[0], node[2][1]);
+        success = !cmp_con.isNull() && !cmp_con.getConst<bool>();
+      }
+      if (success)
+      {
+        return returnRewrite(node, node[0], "repl-repl3-inv-id");
+      }
+    }
+  }
+  // miniscope based on components that do not contribute to contains
+  // for example,
+  //   str.replace( x ++ y ++ x ++ y, "A", z ) -->
+  //   str.replace( x ++ y, "A", z ) ++ x ++ y
+  // since if "A" occurs in x ++ y ++ x ++ y, then it must occur in x ++ y.
+  if (checkEntailLengthOne(node[1]))
+  {
+    Node lastLhs;
+    unsigned lastCheckIndex = 0;
+    for (unsigned i = 1, iend = children0.size(); i < iend; i++)
+    {
+      unsigned checkIndex = children0.size() - i;
+      std::vector<Node> checkLhs;
+      checkLhs.insert(
+          checkLhs.end(), children0.begin(), children0.begin() + checkIndex);
+      Node lhs = utils::mkConcat(STRING_CONCAT, checkLhs);
+      Node rhs = children0[checkIndex];
+      Node ctn = checkEntailContains(lhs, rhs);
+      if (!ctn.isNull() && ctn.getConst<bool>())
+      {
+        lastLhs = lhs;
+        lastCheckIndex = checkIndex;
+      }
+      else
+      {
+        break;
+      }
+    }
+    if (!lastLhs.isNull())
+    {
+      std::vector<Node> remc(children0.begin() + lastCheckIndex,
+                             children0.end());
+      Node rem = utils::mkConcat(STRING_CONCAT, remc);
+      Node ret =
+          nm->mkNode(STRING_CONCAT,
+                     nm->mkNode(STRING_STRREPL, lastLhs, node[1], node[2]),
+                     rem);
+      // for example:
+      //   str.replace( x ++ x, "A", y ) ---> str.replace( x, "A", y ) ++ x
+      // Since we know that the first occurrence of "A" cannot be in the
+      // second occurrence of x. Notice this is specific to single characters
+      // due to complications with finds that span multiple components for
+      // non-characters.
+      return returnRewrite(node, ret, "repl-char-ncontrib-find");
+    }
+  }
+
+  // TODO (#1180) incorporate these?
+  // contains( t, s ) =>
+  //   replace( replace( x, t, s ), s, r ) ----> replace( x, t, r )
+  // contains( t, s ) =>
+  //   contains( replace( t, s, r ), r ) ----> true
+
+  Trace("strings-rewrite-nf") << "No rewrites for : " << node << std::endl;
+  return node;
+}
+
+Node SequencesRewriter::rewriteReplaceAll(Node node)
+{
+  Assert(node.getKind() == STRING_STRREPLALL);
+
+  if (node[0].isConst() && node[1].isConst())
+  {
+    std::vector<Node> children;
+    Node s = node[0];
+    Node t = node[1];
+    if (Word::isEmpty(s) || Word::isEmpty(t))
+    {
+      return returnRewrite(node, node[0], "replall-empty-find");
+    }
+    std::size_t sizeS = Word::getLength(s);
+    std::size_t sizeT = Word::getLength(t);
+    std::size_t index = 0;
+    std::size_t curr = 0;
+    do
+    {
+      curr = Word::find(s, t, index);
+      if (curr != std::string::npos)
+      {
+        if (curr > index)
+        {
+          children.push_back(Word::substr(s, index, curr - index));
+        }
+        children.push_back(node[2]);
+        index = curr + sizeT;
+      }
+      else
+      {
+        children.push_back(Word::substr(s, index, sizeS - index));
+      }
+    } while (curr != std::string::npos && curr < sizeS);
+    // constant evaluation
+    Node res = utils::mkConcat(STRING_CONCAT, children);
+    return returnRewrite(node, res, "replall-const");
+  }
+
+  // rewrites that apply to both replace and replaceall
+  Node rri = rewriteReplaceInternal(node);
+  if (!rri.isNull())
+  {
+    // printing of the rewrite managed by the call above
+    return rri;
+  }
+
+  Trace("strings-rewrite-nf") << "No rewrites for : " << node << std::endl;
+  return node;
+}
+
+Node SequencesRewriter::rewriteReplaceInternal(Node node)
+{
+  Kind nk = node.getKind();
+  Assert(nk == STRING_STRREPL || nk == STRING_STRREPLALL);
+
+  if (node[1] == node[2])
+  {
+    return returnRewrite(node, node[0], "rpl-id");
+  }
+
+  if (node[0] == node[1])
+  {
+    // only holds for replaceall if non-empty
+    if (nk == STRING_STRREPL || checkEntailNonEmpty(node[1]))
+    {
+      return returnRewrite(node, node[2], "rpl-replace");
+    }
+  }
+
+  return Node::null();
+}
+
+Node SequencesRewriter::rewriteStrReverse(Node node)
+{
+  Assert(node.getKind() == STRING_REV);
+  NodeManager* nm = NodeManager::currentNM();
+  Node x = node[0];
+  if (x.isConst())
+  {
+    std::vector<unsigned> nvec = node[0].getConst<String>().getVec();
+    std::reverse(nvec.begin(), nvec.end());
+    Node retNode = nm->mkConst(String(nvec));
+    return returnRewrite(node, retNode, "str-conv-const");
+  }
+  else if (x.getKind() == STRING_CONCAT)
+  {
+    std::vector<Node> children;
+    for (const Node& nc : x)
+    {
+      children.push_back(nm->mkNode(STRING_REV, nc));
+    }
+    std::reverse(children.begin(), children.end());
+    // rev( x1 ++ x2 ) --> rev( x2 ) ++ rev( x1 )
+    Node retNode = nm->mkNode(STRING_CONCAT, children);
+    return returnRewrite(node, retNode, "str-rev-minscope-concat");
+  }
+  else if (x.getKind() == STRING_REV)
+  {
+    // rev( rev( x ) ) --> x
+    Node retNode = x[0];
+    return returnRewrite(node, retNode, "str-rev-idem");
+  }
+  return node;
+}
+
+Node SequencesRewriter::rewritePrefixSuffix(Node n)
+{
+  Assert(n.getKind() == kind::STRING_PREFIX
+         || n.getKind() == kind::STRING_SUFFIX);
+  bool isPrefix = n.getKind() == kind::STRING_PREFIX;
+  if (n[0] == n[1])
+  {
+    Node ret = NodeManager::currentNM()->mkConst(true);
+    return returnRewrite(n, ret, "suf/prefix-eq");
+  }
+  if (n[0].isConst())
+  {
+    CVC4::String t = n[0].getConst<String>();
+    if (t.isEmptyString())
+    {
+      Node ret = NodeManager::currentNM()->mkConst(true);
+      return returnRewrite(n, ret, "suf/prefix-empty-const");
+    }
+  }
+  if (n[1].isConst())
+  {
+    Node s = n[1];
+    size_t lenS = Word::getLength(s);
+    if (n[0].isConst())
+    {
+      Node ret = NodeManager::currentNM()->mkConst(false);
+      Node t = n[0];
+      size_t lenT = Word::getLength(t);
+      if (lenS >= lenT)
+      {
+        if ((isPrefix && t == Word::prefix(s, lenT))
+            || (!isPrefix && t == Word::suffix(s, lenT)))
+        {
+          ret = NodeManager::currentNM()->mkConst(true);
+        }
+      }
+      return returnRewrite(n, ret, "suf/prefix-const");
+    }
+    else if (lenS == 0)
+    {
+      Node ret = n[0].eqNode(n[1]);
+      return returnRewrite(n, ret, "suf/prefix-empty");
+    }
+    else if (lenS == 1)
+    {
+      // (str.prefix x "A") and (str.suffix x "A") are equivalent to
+      // (str.contains "A" x )
+      Node ret =
+          NodeManager::currentNM()->mkNode(kind::STRING_STRCTN, n[1], n[0]);
+      return returnRewrite(n, ret, "suf/prefix-ctn");
+    }
+  }
+  Node lens = NodeManager::currentNM()->mkNode(kind::STRING_LENGTH, n[0]);
+  Node lent = NodeManager::currentNM()->mkNode(kind::STRING_LENGTH, n[1]);
+  Node val;
+  if (isPrefix)
+  {
+    val = NodeManager::currentNM()->mkConst(::CVC4::Rational(0));
+  }
+  else
+  {
+    val = NodeManager::currentNM()->mkNode(kind::MINUS, lent, lens);
+  }
+
+  // Check if we can turn the prefix/suffix into equalities by showing that the
+  // prefix/suffix is at least as long as the string
+  Node eqs = inferEqsFromContains(n[1], n[0]);
+  if (!eqs.isNull())
+  {
+    return returnRewrite(n, eqs, "suf/prefix-to-eqs");
+  }
+
+  // general reduction to equality + substr
+  Node retNode = n[0].eqNode(
+      NodeManager::currentNM()->mkNode(kind::STRING_SUBSTR, n[1], val, lens));
+
+  return retNode;
+}
+
+Node SequencesRewriter::splitConstant(Node a, Node b, int& index, bool isRev)
+{
+  Assert(a.isConst() && b.isConst());
+  size_t lenA = Word::getLength(a);
+  size_t lenB = Word::getLength(b);
+  index = lenA <= lenB ? 1 : 0;
+  size_t len_short = index == 1 ? lenA : lenB;
+  bool cmp =
+      isRev ? a.getConst<String>().rstrncmp(b.getConst<String>(), len_short)
+            : a.getConst<String>().strncmp(b.getConst<String>(), len_short);
+  if (cmp)
+  {
+    Node l = index == 0 ? a : b;
+    if (isRev)
+    {
+      int new_len = l.getConst<String>().size() - len_short;
+      return Word::substr(l, 0, new_len);
+    }
+    else
+    {
+      return Word::substr(l, len_short);
+    }
+  }
+  // not the same prefix/suffix
+  return Node::null();
+}
+
+bool SequencesRewriter::canConstantContainConcat(Node c,
+                                                 Node n,
+                                                 int& firstc,
+                                                 int& lastc)
+{
+  Assert(c.isConst());
+  CVC4::String t = c.getConst<String>();
+  const std::vector<unsigned>& tvec = t.getVec();
+  Assert(n.getKind() == kind::STRING_CONCAT);
+  // must find constant components in order
+  size_t pos = 0;
+  firstc = -1;
+  lastc = -1;
+  for (unsigned i = 0; i < n.getNumChildren(); i++)
+  {
+    if (n[i].isConst())
+    {
+      firstc = firstc == -1 ? i : firstc;
+      lastc = i;
+      CVC4::String s = n[i].getConst<String>();
+      size_t new_pos = t.find(s, pos);
+      if (new_pos == std::string::npos)
+      {
+        return false;
+      }
+      else
+      {
+        pos = new_pos + s.size();
+      }
+    }
+    else if (n[i].getKind() == kind::STRING_ITOS && checkEntailArith(n[i][0]))
+    {
+      // find the first occurrence of a digit starting at pos
+      while (pos < tvec.size() && !String::isDigit(tvec[pos]))
+      {
+        pos++;
+      }
+      if (pos == tvec.size())
+      {
+        return false;
+      }
+      // must consume at least one digit here
+      pos++;
+    }
+  }
+  return true;
+}
+
+bool SequencesRewriter::canConstantContainList(Node c,
+                                               std::vector<Node>& l,
+                                               int& firstc,
+                                               int& lastc)
+{
+  Assert(c.isConst());
+  // must find constant components in order
+  size_t pos = 0;
+  firstc = -1;
+  lastc = -1;
+  for (unsigned i = 0; i < l.size(); i++)
+  {
+    if (l[i].isConst())
+    {
+      firstc = firstc == -1 ? i : firstc;
+      lastc = i;
+      size_t new_pos = Word::find(c, l[i], pos);
+      if (new_pos == std::string::npos)
+      {
+        return false;
+      }
+      else
+      {
+        pos = new_pos + Word::getLength(l[i]);
+      }
+    }
+  }
+  return true;
+}
+
+bool SequencesRewriter::stripSymbolicLength(std::vector<Node>& n1,
+                                            std::vector<Node>& nr,
+                                            int dir,
+                                            Node& curr)
+{
+  Assert(dir == 1 || dir == -1);
+  Assert(nr.empty());
+  Node zero = NodeManager::currentNM()->mkConst(CVC4::Rational(0));
+  bool ret = false;
+  bool success;
+  unsigned sindex = 0;
+  do
+  {
+    Assert(!curr.isNull());
+    success = false;
+    if (curr != zero && sindex < n1.size())
+    {
+      unsigned sindex_use = dir == 1 ? sindex : ((n1.size() - 1) - sindex);
+      if (n1[sindex_use].isConst())
+      {
+        // could strip part of a constant
+        Node lowerBound = getConstantArithBound(Rewriter::rewrite(curr));
+        if (!lowerBound.isNull())
+        {
+          Assert(lowerBound.isConst());
+          Rational lbr = lowerBound.getConst<Rational>();
+          if (lbr.sgn() > 0)
+          {
+            Assert(checkEntailArith(curr, true));
+            CVC4::String s = n1[sindex_use].getConst<String>();
+            Node ncl =
+                NodeManager::currentNM()->mkConst(CVC4::Rational(s.size()));
+            Node next_s =
+                NodeManager::currentNM()->mkNode(kind::MINUS, lowerBound, ncl);
+            next_s = Rewriter::rewrite(next_s);
+            Assert(next_s.isConst());
+            // we can remove the entire constant
+            if (next_s.getConst<Rational>().sgn() >= 0)
+            {
+              curr = Rewriter::rewrite(
+                  NodeManager::currentNM()->mkNode(kind::MINUS, curr, ncl));
+              success = true;
+              sindex++;
+            }
+            else
+            {
+              // we can remove part of the constant
+              // lower bound minus the length of a concrete string is negative,
+              // hence lowerBound cannot be larger than long max
+              Assert(lbr < Rational(String::maxSize()));
+              curr = Rewriter::rewrite(NodeManager::currentNM()->mkNode(
+                  kind::MINUS, curr, lowerBound));
+              uint32_t lbsize = lbr.getNumerator().toUnsignedInt();
+              Assert(lbsize < s.size());
+              if (dir == 1)
+              {
+                // strip partially from the front
+                nr.push_back(
+                    NodeManager::currentNM()->mkConst(s.prefix(lbsize)));
+                n1[sindex_use] = NodeManager::currentNM()->mkConst(
+                    s.suffix(s.size() - lbsize));
+              }
+              else
+              {
+                // strip partially from the back
+                nr.push_back(
+                    NodeManager::currentNM()->mkConst(s.suffix(lbsize)));
+                n1[sindex_use] = NodeManager::currentNM()->mkConst(
+                    s.prefix(s.size() - lbsize));
+              }
+              ret = true;
+            }
+            Assert(checkEntailArith(curr));
+          }
+          else
+          {
+            // we cannot remove the constant
+          }
+        }
+      }
+      else
+      {
+        Node next_s = NodeManager::currentNM()->mkNode(
+            kind::MINUS,
+            curr,
+            NodeManager::currentNM()->mkNode(kind::STRING_LENGTH,
+                                             n1[sindex_use]));
+        next_s = Rewriter::rewrite(next_s);
+        if (checkEntailArith(next_s))
+        {
+          success = true;
+          curr = next_s;
+          sindex++;
+        }
+      }
+    }
+  } while (success);
+  if (sindex > 0)
+  {
+    if (dir == 1)
+    {
+      nr.insert(nr.begin(), n1.begin(), n1.begin() + sindex);
+      n1.erase(n1.begin(), n1.begin() + sindex);
+    }
+    else
+    {
+      nr.insert(nr.end(), n1.end() - sindex, n1.end());
+      n1.erase(n1.end() - sindex, n1.end());
+    }
+    ret = true;
+  }
+  return ret;
+}
+
+int SequencesRewriter::componentContains(std::vector<Node>& n1,
+                                         std::vector<Node>& n2,
+                                         std::vector<Node>& nb,
+                                         std::vector<Node>& ne,
+                                         bool computeRemainder,
+                                         int remainderDir)
+{
+  Assert(nb.empty());
+  Assert(ne.empty());
+  // if n2 is a singleton, we can do optimized version here
+  if (n2.size() == 1)
+  {
+    for (unsigned i = 0; i < n1.size(); i++)
+    {
+      Node n1rb;
+      Node n1re;
+      if (componentContainsBase(n1[i], n2[0], n1rb, n1re, 0, computeRemainder))
+      {
+        if (computeRemainder)
+        {
+          n1[i] = n2[0];
+          if (remainderDir != -1)
+          {
+            if (!n1re.isNull())
+            {
+              ne.push_back(n1re);
+            }
+            ne.insert(ne.end(), n1.begin() + i + 1, n1.end());
+            n1.erase(n1.begin() + i + 1, n1.end());
+          }
+          else if (!n1re.isNull())
+          {
+            n1[i] = Rewriter::rewrite(NodeManager::currentNM()->mkNode(
+                kind::STRING_CONCAT, n1[i], n1re));
+          }
+          if (remainderDir != 1)
+          {
+            nb.insert(nb.end(), n1.begin(), n1.begin() + i);
+            n1.erase(n1.begin(), n1.begin() + i);
+            if (!n1rb.isNull())
+            {
+              nb.push_back(n1rb);
+            }
+          }
+          else if (!n1rb.isNull())
+          {
+            n1[i] = Rewriter::rewrite(NodeManager::currentNM()->mkNode(
+                kind::STRING_CONCAT, n1rb, n1[i]));
+          }
+        }
+        return i;
+      }
+    }
+  }
+  else if (n1.size() >= n2.size())
+  {
+    unsigned diff = n1.size() - n2.size();
+    for (unsigned i = 0; i <= diff; i++)
+    {
+      Node n1rb_first;
+      Node n1re_first;
+      // first component of n2 must be a suffix
+      if (componentContainsBase(n1[i],
+                                n2[0],
+                                n1rb_first,
+                                n1re_first,
+                                1,
+                                computeRemainder && remainderDir != 1))
+      {
+        Assert(n1re_first.isNull());
+        for (unsigned j = 1; j < n2.size(); j++)
+        {
+          // are we in the last component?
+          if (j + 1 == n2.size())
+          {
+            Node n1rb_last;
+            Node n1re_last;
+            // last component of n2 must be a prefix
+            if (componentContainsBase(n1[i + j],
+                                      n2[j],
+                                      n1rb_last,
+                                      n1re_last,
+                                      -1,
+                                      computeRemainder && remainderDir != -1))
+            {
+              Assert(n1rb_last.isNull());
+              if (computeRemainder)
+              {
+                if (remainderDir != -1)
+                {
+                  if (!n1re_last.isNull())
+                  {
+                    ne.push_back(n1re_last);
+                  }
+                  ne.insert(ne.end(), n1.begin() + i + j + 1, n1.end());
+                  n1.erase(n1.begin() + i + j + 1, n1.end());
+                  n1[i + j] = n2[j];
+                }
+                if (remainderDir != 1)
+                {
+                  n1[i] = n2[0];
+                  nb.insert(nb.end(), n1.begin(), n1.begin() + i);
+                  n1.erase(n1.begin(), n1.begin() + i);
+                  if (!n1rb_first.isNull())
+                  {
+                    nb.push_back(n1rb_first);
+                  }
+                }
+              }
+              return i;
+            }
+            else
+            {
+              break;
+            }
+          }
+          else if (n1[i + j] != n2[j])
+          {
+            break;
+          }
+        }
+      }
+    }
+  }
+  return -1;
+}
+
+bool SequencesRewriter::componentContainsBase(
+    Node n1, Node n2, Node& n1rb, Node& n1re, int dir, bool computeRemainder)
+{
+  Assert(n1rb.isNull());
+  Assert(n1re.isNull());
+
+  NodeManager* nm = NodeManager::currentNM();
+
+  if (n1 == n2)
+  {
+    return true;
+  }
+  else
+  {
+    if (n1.isConst() && n2.isConst())
+    {
+      size_t len1 = Word::getLength(n1);
+      size_t len2 = Word::getLength(n2);
+      if (len2 < len1)
+      {
+        if (dir == 1)
+        {
+          if (Word::suffix(n1, len2) == n2)
+          {
+            if (computeRemainder)
+            {
+              n1rb = Word::prefix(n1, len1 - len2);
+            }
+            return true;
+          }
+        }
+        else if (dir == -1)
+        {
+          if (Word::prefix(n1, len2) == n2)
+          {
+            if (computeRemainder)
+            {
+              n1re = Word::suffix(n1, len1 - len2);
+            }
+            return true;
+          }
+        }
+        else
+        {
+          size_t f = Word::find(n1, n2);
+          if (f != std::string::npos)
+          {
+            if (computeRemainder)
+            {
+              if (f > 0)
+              {
+                n1rb = Word::prefix(n1, f);
+              }
+              if (len1 > f + len2)
+              {
+                n1re = Word::suffix(n1, len1 - (f + len2));
+              }
+            }
+            return true;
+          }
+        }
+      }
+    }
+    else
+    {
+      // cases for:
+      //   n1 = x   containing   n2 = substr( x, n2[1], n2[2] )
+      if (n2.getKind() == kind::STRING_SUBSTR)
+      {
+        if (n2[0] == n1)
+        {
+          bool success = true;
+          Node start_pos = n2[1];
+          Node end_pos = nm->mkNode(kind::PLUS, n2[1], n2[2]);
+          Node len_n2s = nm->mkNode(kind::STRING_LENGTH, n2[0]);
+          if (dir == 1)
+          {
+            // To be a suffix, start + length must be greater than
+            // or equal to the length of the string.
+            success = checkEntailArith(end_pos, len_n2s);
+          }
+          else if (dir == -1)
+          {
+            // To be a prefix, must literally start at 0, since
+            //   if we knew it started at <0, it should be rewritten to "",
+            //   if we knew it started at 0, then n2[1] should be rewritten to
+            //   0.
+            success = start_pos.isConst()
+                      && start_pos.getConst<Rational>().sgn() == 0;
+          }
+          if (success)
+          {
+            if (computeRemainder)
+            {
+              // we can only compute the remainder if start_pos and end_pos
+              // are known to be non-negative.
+              if (!checkEntailArith(start_pos) || !checkEntailArith(end_pos))
+              {
+                return false;
+              }
+              if (dir != 1)
+              {
+                n1rb = nm->mkNode(kind::STRING_SUBSTR,
+                                  n2[0],
+                                  nm->mkConst(Rational(0)),
+                                  start_pos);
+              }
+              if (dir != -1)
+              {
+                n1re = nm->mkNode(kind::STRING_SUBSTR, n2[0], end_pos, len_n2s);
+              }
+            }
+            return true;
+          }
+        }
+      }
+
+      if (!computeRemainder && dir == 0)
+      {
+        if (n1.getKind() == STRING_STRREPL)
+        {
+          // (str.contains (str.replace x y z) w) ---> true
+          // if (str.contains x w) --> true and (str.contains z w) ---> true
+          Node xCtnW = checkEntailContains(n1[0], n2);
+          if (!xCtnW.isNull() && xCtnW.getConst<bool>())
+          {
+            Node zCtnW = checkEntailContains(n1[2], n2);
+            if (!zCtnW.isNull() && zCtnW.getConst<bool>())
+            {
+              return true;
+            }
+          }
+        }
+      }
+    }
+  }
+  return false;
+}
+
+bool SequencesRewriter::stripConstantEndpoints(std::vector<Node>& n1,
+                                               std::vector<Node>& n2,
+                                               std::vector<Node>& nb,
+                                               std::vector<Node>& ne,
+                                               int dir)
+{
+  Assert(nb.empty());
+  Assert(ne.empty());
+
+  NodeManager* nm = NodeManager::currentNM();
+  bool changed = false;
+  // for ( forwards, backwards ) direction
+  for (unsigned r = 0; r < 2; r++)
+  {
+    if (dir == 0 || (r == 0 && dir == 1) || (r == 1 && dir == -1))
+    {
+      unsigned index0 = r == 0 ? 0 : n1.size() - 1;
+      unsigned index1 = r == 0 ? 0 : n2.size() - 1;
+      bool removeComponent = false;
+      Node n1cmp = n1[index0];
+
+      if (n1cmp.isConst() && n1cmp.getConst<String>().size() == 0)
+      {
+        return false;
+      }
+
+      std::vector<Node> sss;
+      std::vector<Node> sls;
+      n1cmp = decomposeSubstrChain(n1cmp, sss, sls);
+      Trace("strings-rewrite-debug2")
+          << "stripConstantEndpoints : Compare " << n1cmp << " " << n2[index1]
+          << ", dir = " << dir << std::endl;
+      if (n1cmp.isConst())
+      {
+        CVC4::String s = n1cmp.getConst<String>();
+        // overlap is an overapproximation of the number of characters
+        // n2[index1] can match in s
+        unsigned overlap = s.size();
+        if (n2[index1].isConst())
+        {
+          CVC4::String t = n2[index1].getConst<String>();
+          std::size_t ret = r == 0 ? s.find(t) : s.rfind(t);
+          if (ret == std::string::npos)
+          {
+            if (n1.size() == 1)
+            {
+              // can remove everything
+              //   e.g. str.contains( "abc", str.++( "ba", x ) ) -->
+              //   str.contains( "", str.++( "ba", x ) )
+              removeComponent = true;
+            }
+            else if (sss.empty())  // only if not substr
+            {
+              // check how much overlap there is
+              // This is used to partially strip off the endpoint
+              // e.g. str.contains( str.++( "abc", x ), str.++( "cd", y ) ) -->
+              // str.contains( str.++( "c", x ), str.++( "cd", y ) )
+              overlap = r == 0 ? s.overlap(t) : t.overlap(s);
+            }
+            else
+            {
+              // if we are looking at a substring, we can remove the component
+              // if there is no overlap
+              //   e.g. str.contains( str.++( str.substr( "c", i, j ), x), "a" )
+              //        --> str.contains( x, "a" )
+              removeComponent = ((r == 0 ? s.overlap(t) : t.overlap(s)) == 0);
+            }
+          }
+          else if (sss.empty())  // only if not substr
+          {
+            Assert(ret < s.size());
+            // can strip off up to the find position, e.g.
+            // str.contains( str.++( "abc", x ), str.++( "b", y ) ) -->
+            // str.contains( str.++( "bc", x ), str.++( "b", y ) ),
+            // and
+            // str.contains( str.++( x, "abbd" ), str.++( y, "b" ) ) -->
+            // str.contains( str.++( x, "abb" ), str.++( y, "b" ) )
+            overlap = s.size() - ret;
+          }
+        }
+        else
+        {
+          // inconclusive
+        }
+        // process the overlap
+        if (overlap < s.size())
+        {
+          changed = true;
+          if (overlap == 0)
+          {
+            removeComponent = true;
+          }
+          else
+          {
+            // can drop the prefix (resp. suffix) from the first (resp. last)
+            // component
+            if (r == 0)
+            {
+              nb.push_back(nm->mkConst(s.prefix(s.size() - overlap)));
+              n1[index0] = nm->mkConst(s.suffix(overlap));
+            }
+            else
+            {
+              ne.push_back(nm->mkConst(s.suffix(s.size() - overlap)));
+              n1[index0] = nm->mkConst(s.prefix(overlap));
+            }
+          }
+        }
+      }
+      else if (n1cmp.getKind() == kind::STRING_ITOS)
+      {
+        if (n2[index1].isConst())
+        {
+          CVC4::String t = n2[index1].getConst<String>();
+
+          if (n1.size() == 1)
+          {
+            // if n1.size()==1, then if n2[index1] is not a number, we can drop
+            // the entire component
+            //    e.g. str.contains( int.to.str(x), "123a45") --> false
+            if (!t.isNumber())
+            {
+              removeComponent = true;
+            }
+          }
+          else
+          {
+            const std::vector<unsigned>& tvec = t.getVec();
+            Assert(tvec.size() > 0);
+
+            // if n1.size()>1, then if the first (resp. last) character of
+            // n2[index1]
+            //  is not a digit, we can drop the entire component, e.g.:
+            //    str.contains( str.++( int.to.str(x), y ), "a12") -->
+            //    str.contains( y, "a12" )
+            //    str.contains( str.++( y, int.to.str(x) ), "a0b") -->
+            //    str.contains( y, "a0b" )
+            unsigned i = r == 0 ? 0 : (tvec.size() - 1);
+            if (!String::isDigit(tvec[i]))
+            {
+              removeComponent = true;
+            }
+          }
+        }
+      }
+      if (removeComponent)
+      {
+        // can drop entire first (resp. last) component
+        if (r == 0)
+        {
+          nb.push_back(n1[index0]);
+          n1.erase(n1.begin(), n1.begin() + 1);
+        }
+        else
+        {
+          ne.push_back(n1[index0]);
+          n1.pop_back();
+        }
+        if (n1.empty())
+        {
+          // if we've removed everything, just return (we will rewrite to false)
+          return true;
+        }
+        else
+        {
+          changed = true;
+        }
+      }
+    }
+  }
+  // TODO (#1180) : computing the maximal overlap in this function may be
+  // important.
+  // str.contains( str.++( str.to.int(x), str.substr(y,0,3) ), "2aaaa" ) --->
+  // false
+  //   ...since str.to.int(x) can contain at most 1 character from "2aaaa",
+  //   leaving 4 characters
+  //      which is larger that the upper bound for length of str.substr(y,0,3),
+  //      which is 3.
+  return changed;
+}
+
+Node SequencesRewriter::canonicalStrForSymbolicLength(Node len)
+{
+  NodeManager* nm = NodeManager::currentNM();
+
+  Node res;
+  if (len.getKind() == kind::CONST_RATIONAL)
+  {
+    // c -> "A" repeated c times
+    Rational ratLen = len.getConst<Rational>();
+    Assert(ratLen.getDenominator() == 1);
+    Integer intLen = ratLen.getNumerator();
+    res = nm->mkConst(String(std::string(intLen.getUnsignedInt(), 'A')));
+  }
+  else if (len.getKind() == kind::PLUS)
+  {
+    // x + y -> norm(x) + norm(y)
+    NodeBuilder<> concatBuilder(kind::STRING_CONCAT);
+    for (const auto& n : len)
+    {
+      Node sn = canonicalStrForSymbolicLength(n);
+      if (sn.isNull())
+      {
+        return Node::null();
+      }
+      std::vector<Node> snChildren;
+      utils::getConcat(sn, snChildren);
+      concatBuilder.append(snChildren);
+    }
+    res = concatBuilder.constructNode();
+  }
+  else if (len.getKind() == kind::MULT && len.getNumChildren() == 2
+           && len[0].isConst())
+  {
+    // c * x -> norm(x) repeated c times
+    Rational ratReps = len[0].getConst<Rational>();
+    Assert(ratReps.getDenominator() == 1);
+    Integer intReps = ratReps.getNumerator();
+
+    Node nRep = canonicalStrForSymbolicLength(len[1]);
+    std::vector<Node> nRepChildren;
+    utils::getConcat(nRep, nRepChildren);
+    NodeBuilder<> concatBuilder(kind::STRING_CONCAT);
+    for (size_t i = 0, reps = intReps.getUnsignedInt(); i < reps; i++)
+    {
+      concatBuilder.append(nRepChildren);
+    }
+    res = concatBuilder.constructNode();
+  }
+  else if (len.getKind() == kind::STRING_LENGTH)
+  {
+    // len(x) -> x
+    res = len[0];
+  }
+  return res;
+}
+
+Node SequencesRewriter::lengthPreserveRewrite(Node n)
+{
+  NodeManager* nm = NodeManager::currentNM();
+  Node len = Rewriter::rewrite(nm->mkNode(kind::STRING_LENGTH, n));
+  Node res = canonicalStrForSymbolicLength(len);
+  return res.isNull() ? n : res;
+}
+
+Node SequencesRewriter::checkEntailContains(Node a, Node b, bool fullRewriter)
+{
+  NodeManager* nm = NodeManager::currentNM();
+  Node ctn = nm->mkNode(kind::STRING_STRCTN, a, b);
+
+  if (fullRewriter)
+  {
+    ctn = Rewriter::rewrite(ctn);
+  }
+  else
+  {
+    Node prev;
+    do
+    {
+      prev = ctn;
+      ctn = rewriteContains(ctn);
+    } while (prev != ctn && ctn.getKind() == kind::STRING_STRCTN);
+  }
+
+  Assert(ctn.getType().isBoolean());
+  return ctn.isConst() ? ctn : Node::null();
+}
+
+bool SequencesRewriter::checkEntailNonEmpty(Node a)
+{
+  Node len = NodeManager::currentNM()->mkNode(STRING_LENGTH, a);
+  len = Rewriter::rewrite(len);
+  return checkEntailArith(len, true);
+}
+
+bool SequencesRewriter::checkEntailLengthOne(Node s, bool strict)
+{
+  NodeManager* nm = NodeManager::currentNM();
+  Node one = nm->mkConst(Rational(1));
+  Node len = nm->mkNode(STRING_LENGTH, s);
+  len = Rewriter::rewrite(len);
+  return checkEntailArith(one, len) && (!strict || checkEntailArith(len, true));
+}
+
+bool SequencesRewriter::checkEntailArithEq(Node a, Node b)
+{
+  if (a == b)
+  {
+    return true;
+  }
+  else
+  {
+    Node ar = Rewriter::rewrite(a);
+    Node br = Rewriter::rewrite(b);
+    return ar == br;
+  }
+}
+
+bool SequencesRewriter::checkEntailArith(Node a, Node b, bool strict)
+{
+  if (a == b)
+  {
+    return !strict;
+  }
+  else
+  {
+    Node diff = NodeManager::currentNM()->mkNode(kind::MINUS, a, b);
+    return checkEntailArith(diff, strict);
+  }
+}
+
+struct StrCheckEntailArithTag
+{
+};
+struct StrCheckEntailArithComputedTag
+{
+};
+/** Attribute true for expressions for which checkEntailArith returned true */
+typedef expr::Attribute<StrCheckEntailArithTag, bool> StrCheckEntailArithAttr;
+typedef expr::Attribute<StrCheckEntailArithComputedTag, bool>
+    StrCheckEntailArithComputedAttr;
+
+bool SequencesRewriter::checkEntailArith(Node a, bool strict)
+{
+  if (a.isConst())
+  {
+    return a.getConst<Rational>().sgn() >= (strict ? 1 : 0);
+  }
+
+  Node ar =
+      strict
+          ? NodeManager::currentNM()->mkNode(
+                kind::MINUS, a, NodeManager::currentNM()->mkConst(Rational(1)))
+          : a;
+  ar = Rewriter::rewrite(ar);
+
+  if (ar.getAttribute(StrCheckEntailArithComputedAttr()))
+  {
+    return ar.getAttribute(StrCheckEntailArithAttr());
+  }
+
+  bool ret = checkEntailArithInternal(ar);
+  if (!ret)
+  {
+    // try with approximations
+    ret = checkEntailArithApprox(ar);
+  }
+  // cache the result
+  ar.setAttribute(StrCheckEntailArithAttr(), ret);
+  ar.setAttribute(StrCheckEntailArithComputedAttr(), true);
+  return ret;
+}
+
+bool SequencesRewriter::checkEntailArithApprox(Node ar)
+{
+  Assert(Rewriter::rewrite(ar) == ar);
+  NodeManager* nm = NodeManager::currentNM();
+  std::map<Node, Node> msum;
+  Trace("strings-ent-approx-debug")
+      << "Setup arithmetic approximations for " << ar << std::endl;
+  if (!ArithMSum::getMonomialSum(ar, msum))
+  {
+    Trace("strings-ent-approx-debug")
+        << "...failed to get monomial sum!" << std::endl;
+    return false;
+  }
+  // for each monomial v*c, mApprox[v] a list of
+  // possibilities for how the term can be soundly approximated, that is,
+  // if mApprox[v] contains av, then v*c > av*c. Notice that if c
+  // is positive, then v > av, otherwise if c is negative, then v < av.
+  // In other words, av is an under-approximation if c is positive, and an
+  // over-approximation if c is negative.
+  bool changed = false;
+  std::map<Node, std::vector<Node> > mApprox;
+  // map from approximations to their monomial sums
+  std::map<Node, std::map<Node, Node> > approxMsums;
+  // aarSum stores each monomial that does not have multiple approximations
+  std::vector<Node> aarSum;
+  for (std::pair<const Node, Node>& m : msum)
+  {
+    Node v = m.first;
+    Node c = m.second;
+    Trace("strings-ent-approx-debug")
+        << "Get approximations " << v << "..." << std::endl;
+    if (v.isNull())
+    {
+      Node mn = c.isNull() ? nm->mkConst(Rational(1)) : c;
+      aarSum.push_back(mn);
+    }
+    else
+    {
+      // c.isNull() means c = 1
+      bool isOverApprox = !c.isNull() && c.getConst<Rational>().sgn() == -1;
+      std::vector<Node>& approx = mApprox[v];
+      std::unordered_set<Node, NodeHashFunction> visited;
+      std::vector<Node> toProcess;
+      toProcess.push_back(v);
+      do
+      {
+        Node curr = toProcess.back();
+        Trace("strings-ent-approx-debug") << "  process " << curr << std::endl;
+        curr = Rewriter::rewrite(curr);
+        toProcess.pop_back();
+        if (visited.find(curr) == visited.end())
+        {
+          visited.insert(curr);
+          std::vector<Node> currApprox;
+          getArithApproximations(curr, currApprox, isOverApprox);
+          if (currApprox.empty())
+          {
+            Trace("strings-ent-approx-debug")
+                << "...approximation: " << curr << std::endl;
+            // no approximations, thus curr is a possibility
+            approx.push_back(curr);
+          }
+          else
+          {
+            toProcess.insert(
+                toProcess.end(), currApprox.begin(), currApprox.end());
+          }
+        }
+      } while (!toProcess.empty());
+      Assert(!approx.empty());
+      // if we have only one approximation, move it to final
+      if (approx.size() == 1)
+      {
+        changed = v != approx[0];
+        Node mn = ArithMSum::mkCoeffTerm(c, approx[0]);
+        aarSum.push_back(mn);
+        mApprox.erase(v);
+      }
+      else
+      {
+        // compute monomial sum form for each approximation, used below
+        for (const Node& aa : approx)
+        {
+          if (approxMsums.find(aa) == approxMsums.end())
+          {
+            CVC4_UNUSED bool ret =
+                ArithMSum::getMonomialSum(aa, approxMsums[aa]);
+            Assert(ret);
+          }
+        }
+        changed = true;
+      }
+    }
+  }
+  if (!changed)
+  {
+    // approximations had no effect, return
+    Trace("strings-ent-approx-debug") << "...no approximations" << std::endl;
+    return false;
+  }
+  // get the current "fixed" sum for the abstraction of ar
+  Node aar = aarSum.empty()
+                 ? nm->mkConst(Rational(0))
+                 : (aarSum.size() == 1 ? aarSum[0] : nm->mkNode(PLUS, aarSum));
+  aar = Rewriter::rewrite(aar);
+  Trace("strings-ent-approx-debug")
+      << "...processed fixed sum " << aar << " with " << mApprox.size()
+      << " approximated monomials." << std::endl;
+  // if we have a choice of how to approximate
+  if (!mApprox.empty())
+  {
+    // convert aar back to monomial sum
+    std::map<Node, Node> msumAar;
+    if (!ArithMSum::getMonomialSum(aar, msumAar))
+    {
+      return false;
+    }
+    if (Trace.isOn("strings-ent-approx"))
+    {
+      Trace("strings-ent-approx")
+          << "---- Check arithmetic entailment by under-approximation " << ar
+          << " >= 0" << std::endl;
+      Trace("strings-ent-approx") << "FIXED:" << std::endl;
+      ArithMSum::debugPrintMonomialSum(msumAar, "strings-ent-approx");
+      Trace("strings-ent-approx") << "APPROX:" << std::endl;
+      for (std::pair<const Node, std::vector<Node> >& a : mApprox)
+      {
+        Node c = msum[a.first];
+        Trace("strings-ent-approx") << "  ";
+        if (!c.isNull())
+        {
+          Trace("strings-ent-approx") << c << " * ";
+        }
+        Trace("strings-ent-approx")
+            << a.second << " ...from " << a.first << std::endl;
+      }
+      Trace("strings-ent-approx") << std::endl;
+    }
+    Rational one(1);
+    // incorporate monomials one at a time that have a choice of approximations
+    while (!mApprox.empty())
+    {
+      Node v;
+      Node vapprox;
+      int maxScore = -1;
+      // Look at each approximation, take the one with the best score.
+      // Notice that we are in the process of trying to prove
+      // ( c1*t1 + .. + cn*tn ) + ( approx_1 | ... | approx_m ) >= 0,
+      // where c1*t1 + .. + cn*tn is the "fixed" component of our sum (aar)
+      // and approx_1 ... approx_m are possible approximations. The
+      // intution here is that we want coefficients c1...cn to be positive.
+      // This is because arithmetic string terms t1...tn (which may be
+      // applications of len, indexof, str.to.int) are never entailed to be
+      // negative. Hence, we add the approx_i that contributes the "most"
+      // towards making all constants c1...cn positive and cancelling negative
+      // monomials in approx_i itself.
+      for (std::pair<const Node, std::vector<Node> >& nam : mApprox)
+      {
+        Node cr = msum[nam.first];
+        for (const Node& aa : nam.second)
+        {
+          unsigned helpsCancelCount = 0;
+          unsigned addsObligationCount = 0;
+          std::map<Node, Node>::iterator it;
+          // we are processing an approximation cr*( c1*t1 + ... + cn*tn )
+          for (std::pair<const Node, Node>& aam : approxMsums[aa])
+          {
+            // Say aar is of the form t + c*ti, and aam is the monomial ci*ti
+            // where ci != 0. We say aam:
+            // (1) helps cancel if c != 0 and c>0 != ci>0
+            // (2) adds obligation if c>=0 and c+ci<0
+            Node ti = aam.first;
+            Node ci = aam.second;
+            if (!cr.isNull())
+            {
+              ci = ci.isNull() ? cr
+                               : Rewriter::rewrite(nm->mkNode(MULT, ci, cr));
+            }
+            Trace("strings-ent-approx-debug") << ci << "*" << ti << " ";
+            int ciSgn = ci.isNull() ? 1 : ci.getConst<Rational>().sgn();
+            it = msumAar.find(ti);
+            if (it != msumAar.end())
+            {
+              Node c = it->second;
+              int cSgn = c.isNull() ? 1 : c.getConst<Rational>().sgn();
+              if (cSgn == 0)
+              {
+                addsObligationCount += (ciSgn == -1 ? 1 : 0);
+              }
+              else if (cSgn != ciSgn)
+              {
+                helpsCancelCount++;
+                Rational r1 = c.isNull() ? one : c.getConst<Rational>();
+                Rational r2 = ci.isNull() ? one : ci.getConst<Rational>();
+                Rational r12 = r1 + r2;
+                if (r12.sgn() == -1)
+                {
+                  addsObligationCount++;
+                }
+              }
+            }
+            else
+            {
+              addsObligationCount += (ciSgn == -1 ? 1 : 0);
+            }
+          }
+          Trace("strings-ent-approx-debug")
+              << "counts=" << helpsCancelCount << "," << addsObligationCount
+              << " for " << aa << " into " << aar << std::endl;
+          int score = (addsObligationCount > 0 ? 0 : 2)
+                      + (helpsCancelCount > 0 ? 1 : 0);
+          // if its the best, update v and vapprox
+          if (v.isNull() || score > maxScore)
+          {
+            v = nam.first;
+            vapprox = aa;
+            maxScore = score;
+          }
+        }
+        if (!v.isNull())
+        {
+          break;
+        }
+      }
+      Trace("strings-ent-approx")
+          << "- Decide " << v << " = " << vapprox << std::endl;
+      // we incorporate v approximated by vapprox into the overall approximation
+      // for ar
+      Assert(!v.isNull() && !vapprox.isNull());
+      Assert(msum.find(v) != msum.end());
+      Node mn = ArithMSum::mkCoeffTerm(msum[v], vapprox);
+      aar = nm->mkNode(PLUS, aar, mn);
+      // update the msumAar map
+      aar = Rewriter::rewrite(aar);
+      msumAar.clear();
+      if (!ArithMSum::getMonomialSum(aar, msumAar))
+      {
+        Assert(false);
+        Trace("strings-ent-approx")
+            << "...failed to get monomial sum!" << std::endl;
+        return false;
+      }
+      // we have processed the approximation for v
+      mApprox.erase(v);
+    }
+    Trace("strings-ent-approx") << "-----------------" << std::endl;
+  }
+  if (aar == ar)
+  {
+    Trace("strings-ent-approx-debug")
+        << "...approximation had no effect" << std::endl;
+    // this should never happen, but we avoid the infinite loop for sanity here
+    Assert(false);
+    return false;
+  }
+  // Check entailment on the approximation of ar.
+  // Notice that this may trigger further reasoning by approximation. For
+  // example, len( replace( x ++ y, substr( x, 0, n ), z ) ) may be
+  // under-approximated as len( x ) + len( y ) - len( substr( x, 0, n ) ) on
+  // this call, where in the recursive call we may over-approximate
+  // len( substr( x, 0, n ) ) as len( x ). In this example, we can infer
+  // that len( replace( x ++ y, substr( x, 0, n ), z ) ) >= len( y ) in two
+  // steps.
+  if (checkEntailArith(aar))
+  {
+    Trace("strings-ent-approx")
+        << "*** StrArithApprox: showed " << ar
+        << " >= 0 using under-approximation!" << std::endl;
+    Trace("strings-ent-approx")
+        << "*** StrArithApprox: under-approximation was " << aar << std::endl;
+    return true;
+  }
+  return false;
+}
+
+void SequencesRewriter::getArithApproximations(Node a,
+                                               std::vector<Node>& approx,
+                                               bool isOverApprox)
+{
+  NodeManager* nm = NodeManager::currentNM();
+  // We do not handle PLUS here since this leads to exponential behavior.
+  // Instead, this is managed, e.g. during checkEntailArithApprox, where
+  // PLUS terms are expanded "on-demand" during the reasoning.
+  Trace("strings-ent-approx-debug")
+      << "Get arith approximations " << a << std::endl;
+  Kind ak = a.getKind();
+  if (ak == MULT)
+  {
+    Node c;
+    Node v;
+    if (ArithMSum::getMonomial(a, c, v))
+    {
+      bool isNeg = c.getConst<Rational>().sgn() == -1;
+      getArithApproximations(v, approx, isNeg ? !isOverApprox : isOverApprox);
+      for (unsigned i = 0, size = approx.size(); i < size; i++)
+      {
+        approx[i] = nm->mkNode(MULT, c, approx[i]);
+      }
+    }
+  }
+  else if (ak == STRING_LENGTH)
+  {
+    Kind aak = a[0].getKind();
+    if (aak == STRING_SUBSTR)
+    {
+      // over,under-approximations for len( substr( x, n, m ) )
+      Node lenx = nm->mkNode(STRING_LENGTH, a[0][0]);
+      if (isOverApprox)
+      {
+        // m >= 0 implies
+        //  m >= len( substr( x, n, m ) )
+        if (checkEntailArith(a[0][2]))
+        {
+          approx.push_back(a[0][2]);
+        }
+        if (checkEntailArith(lenx, a[0][1]))
+        {
+          // n <= len( x ) implies
+          //   len( x ) - n >= len( substr( x, n, m ) )
+          approx.push_back(nm->mkNode(MINUS, lenx, a[0][1]));
+        }
+        else
+        {
+          // len( x ) >= len( substr( x, n, m ) )
+          approx.push_back(lenx);
+        }
+      }
+      else
+      {
+        // 0 <= n and n+m <= len( x ) implies
+        //   m <= len( substr( x, n, m ) )
+        Node npm = nm->mkNode(PLUS, a[0][1], a[0][2]);
+        if (checkEntailArith(a[0][1]) && checkEntailArith(lenx, npm))
+        {
+          approx.push_back(a[0][2]);
+        }
+        // 0 <= n and n+m >= len( x ) implies
+        //   len(x)-n <= len( substr( x, n, m ) )
+        if (checkEntailArith(a[0][1]) && checkEntailArith(npm, lenx))
+        {
+          approx.push_back(nm->mkNode(MINUS, lenx, a[0][1]));
+        }
+      }
+    }
+    else if (aak == STRING_STRREPL)
+    {
+      // over,under-approximations for len( replace( x, y, z ) )
+      // notice this is either len( x ) or ( len( x ) + len( z ) - len( y ) )
+      Node lenx = nm->mkNode(STRING_LENGTH, a[0][0]);
+      Node leny = nm->mkNode(STRING_LENGTH, a[0][1]);
+      Node lenz = nm->mkNode(STRING_LENGTH, a[0][2]);
+      if (isOverApprox)
+      {
+        if (checkEntailArith(leny, lenz))
+        {
+          // len( y ) >= len( z ) implies
+          //   len( x ) >= len( replace( x, y, z ) )
+          approx.push_back(lenx);
+        }
+        else
+        {
+          // len( x ) + len( z ) >= len( replace( x, y, z ) )
+          approx.push_back(nm->mkNode(PLUS, lenx, lenz));
+        }
+      }
+      else
+      {
+        if (checkEntailArith(lenz, leny) || checkEntailArith(lenz, lenx))
+        {
+          // len( y ) <= len( z ) or len( x ) <= len( z ) implies
+          //   len( x ) <= len( replace( x, y, z ) )
+          approx.push_back(lenx);
+        }
+        else
+        {
+          // len( x ) - len( y ) <= len( replace( x, y, z ) )
+          approx.push_back(nm->mkNode(MINUS, lenx, leny));
+        }
+      }
+    }
+    else if (aak == STRING_ITOS)
+    {
+      // over,under-approximations for len( int.to.str( x ) )
+      if (isOverApprox)
+      {
+        if (checkEntailArith(a[0][0], false))
+        {
+          if (checkEntailArith(a[0][0], true))
+          {
+            // x > 0 implies
+            //   x >= len( int.to.str( x ) )
+            approx.push_back(a[0][0]);
+          }
+          else
+          {
+            // x >= 0 implies
+            //   x+1 >= len( int.to.str( x ) )
+            approx.push_back(
+                nm->mkNode(PLUS, nm->mkConst(Rational(1)), a[0][0]));
+          }
+        }
+      }
+      else
+      {
+        if (checkEntailArith(a[0][0]))
+        {
+          // x >= 0 implies
+          //   len( int.to.str( x ) ) >= 1
+          approx.push_back(nm->mkConst(Rational(1)));
+        }
+        // other crazy things are possible here, e.g.
+        // len( int.to.str( len( y ) + 10 ) ) >= 2
+      }
+    }
+  }
+  else if (ak == STRING_STRIDOF)
+  {
+    // over,under-approximations for indexof( x, y, n )
+    if (isOverApprox)
+    {
+      Node lenx = nm->mkNode(STRING_LENGTH, a[0]);
+      Node leny = nm->mkNode(STRING_LENGTH, a[1]);
+      if (checkEntailArith(lenx, leny))
+      {
+        // len( x ) >= len( y ) implies
+        //   len( x ) - len( y ) >= indexof( x, y, n )
+        approx.push_back(nm->mkNode(MINUS, lenx, leny));
+      }
+      else
+      {
+        // len( x ) >= indexof( x, y, n )
+        approx.push_back(lenx);
+      }
+    }
+    else
+    {
+      // TODO?:
+      // contains( substr( x, n, len( x ) ), y ) implies
+      //   n <= indexof( x, y, n )
+      // ...hard to test, runs risk of non-termination
+
+      // -1 <= indexof( x, y, n )
+      approx.push_back(nm->mkConst(Rational(-1)));
+    }
+  }
+  else if (ak == STRING_STOI)
+  {
+    // over,under-approximations for str.to.int( x )
+    if (isOverApprox)
+    {
+      // TODO?:
+      // y >= 0 implies
+      //   y >= str.to.int( int.to.str( y ) )
+    }
+    else
+    {
+      // -1 <= str.to.int( x )
+      approx.push_back(nm->mkConst(Rational(-1)));
+    }
+  }
+  Trace("strings-ent-approx-debug") << "Return " << approx.size() << std::endl;
+}
+
+bool SequencesRewriter::checkEntailMultisetSubset(Node a, Node b)
+{
+  NodeManager* nm = NodeManager::currentNM();
+
+  std::vector<Node> avec;
+  utils::getConcat(getMultisetApproximation(a), avec);
+  std::vector<Node> bvec;
+  utils::getConcat(b, bvec);
+
+  std::map<Node, unsigned> num_nconst[2];
+  std::map<Node, unsigned> num_const[2];
+  for (unsigned j = 0; j < 2; j++)
+  {
+    std::vector<Node>& jvec = j == 0 ? avec : bvec;
+    for (const Node& cc : jvec)
+    {
+      if (cc.isConst())
+      {
+        num_const[j][cc]++;
+      }
+      else
+      {
+        num_nconst[j][cc]++;
+      }
+    }
+  }
+  bool ms_success = true;
+  for (std::pair<const Node, unsigned>& nncp : num_nconst[0])
+  {
+    if (nncp.second > num_nconst[1][nncp.first])
+    {
+      ms_success = false;
+      break;
+    }
+  }
+  if (ms_success)
+  {
+    // count the number of constant characters in the first argument
+    std::map<Node, unsigned> count_const[2];
+    std::vector<Node> chars;
+    for (unsigned j = 0; j < 2; j++)
+    {
+      for (std::pair<const Node, unsigned>& ncp : num_const[j])
+      {
+        Node cn = ncp.first;
+        Assert(cn.isConst());
+        std::vector<unsigned> cc_vec;
+        const std::vector<unsigned>& cvec = cn.getConst<String>().getVec();
+        for (unsigned i = 0, size = cvec.size(); i < size; i++)
+        {
+          // make the character
+          cc_vec.clear();
+          cc_vec.insert(cc_vec.end(), cvec.begin() + i, cvec.begin() + i + 1);
+          Node ch = nm->mkConst(String(cc_vec));
+          count_const[j][ch] += ncp.second;
+          if (std::find(chars.begin(), chars.end(), ch) == chars.end())
+          {
+            chars.push_back(ch);
+          }
+        }
+      }
+    }
+    Trace("strings-entail-ms-ss")
+        << "For " << a << " and " << b << " : " << std::endl;
+    for (const Node& ch : chars)
+    {
+      Trace("strings-entail-ms-ss") << "  # occurrences of substring ";
+      Trace("strings-entail-ms-ss") << ch << " in arguments is ";
+      Trace("strings-entail-ms-ss")
+          << count_const[0][ch] << " / " << count_const[1][ch] << std::endl;
+      if (count_const[0][ch] < count_const[1][ch])
+      {
+        return true;
+      }
+    }
+
+    // TODO (#1180): count the number of 2,3,4,.. character substrings
+    // for example:
+    // str.contains( str.++( x, "cbabc" ), str.++( "cabbc", x ) ) ---> false
+    // since the second argument contains more occurrences of "bb".
+    // note this is orthogonal reasoning to inductive reasoning
+    // via regular membership reduction in Liang et al CAV 2015.
+  }
+  return false;
+}
+
+Node SequencesRewriter::checkEntailHomogeneousString(Node a)
+{
+  NodeManager* nm = NodeManager::currentNM();
+
+  std::vector<Node> avec;
+  utils::getConcat(getMultisetApproximation(a), avec);
+
+  bool cValid = false;
+  unsigned c = 0;
+  for (const Node& ac : avec)
+  {
+    if (ac.isConst())
+    {
+      std::vector<unsigned> acv = ac.getConst<String>().getVec();
+      for (unsigned cc : acv)
+      {
+        if (!cValid)
+        {
+          cValid = true;
+          c = cc;
+        }
+        else if (c != cc)
+        {
+          // Found a different character
+          return Node::null();
+        }
+      }
+    }
+    else
+    {
+      // Could produce a different character
+      return Node::null();
+    }
+  }
+
+  if (!cValid)
+  {
+    return nm->mkConst(String(""));
+  }
+
+  std::vector<unsigned> cv = {c};
+  return nm->mkConst(String(cv));
+}
+
+Node SequencesRewriter::getMultisetApproximation(Node a)
+{
+  NodeManager* nm = NodeManager::currentNM();
+  if (a.getKind() == STRING_SUBSTR)
+  {
+    return a[0];
+  }
+  else if (a.getKind() == STRING_STRREPL)
+  {
+    return getMultisetApproximation(nm->mkNode(STRING_CONCAT, a[0], a[2]));
+  }
+  else if (a.getKind() == STRING_CONCAT)
+  {
+    NodeBuilder<> nb(STRING_CONCAT);
+    for (const Node& ac : a)
+    {
+      nb << getMultisetApproximation(ac);
+    }
+    return nb.constructNode();
+  }
+  else
+  {
+    return a;
+  }
+}
+
+bool SequencesRewriter::checkEntailArithWithEqAssumption(Node assumption,
+                                                         Node a,
+                                                         bool strict)
+{
+  Assert(assumption.getKind() == kind::EQUAL);
+  Assert(Rewriter::rewrite(assumption) == assumption);
+
+  // Find candidates variables to compute substitutions for
+  std::unordered_set<Node, NodeHashFunction> candVars;
+  std::vector<Node> toVisit = {assumption};
+  while (!toVisit.empty())
+  {
+    Node curr = toVisit.back();
+    toVisit.pop_back();
+
+    if (curr.getKind() == kind::PLUS || curr.getKind() == kind::MULT
+        || curr.getKind() == kind::MINUS || curr.getKind() == kind::EQUAL)
+    {
+      for (const auto& currChild : curr)
+      {
+        toVisit.push_back(currChild);
+      }
+    }
+    else if (curr.isVar() && Theory::theoryOf(curr) == THEORY_ARITH)
+    {
+      candVars.insert(curr);
+    }
+    else if (curr.getKind() == kind::STRING_LENGTH)
+    {
+      candVars.insert(curr);
+    }
+  }
+
+  // Check if any of the candidate variables are in n
+  Node v;
+  Assert(toVisit.empty());
+  toVisit.push_back(a);
+  while (!toVisit.empty())
+  {
+    Node curr = toVisit.back();
+    toVisit.pop_back();
+
+    for (const auto& currChild : curr)
+    {
+      toVisit.push_back(currChild);
+    }
+
+    if (candVars.find(curr) != candVars.end())
+    {
+      v = curr;
+      break;
+    }
+  }
+
+  if (v.isNull())
+  {
+    // No suitable candidate found
+    return false;
+  }
+
+  Node solution = ArithMSum::solveEqualityFor(assumption, v);
+  if (solution.isNull())
+  {
+    // Could not solve for v
+    return false;
+  }
+
+  a = a.substitute(TNode(v), TNode(solution));
+  return checkEntailArith(a, strict);
+}
+
+bool SequencesRewriter::checkEntailArithWithAssumption(Node assumption,
+                                                       Node a,
+                                                       Node b,
+                                                       bool strict)
+{
+  Assert(Rewriter::rewrite(assumption) == assumption);
+
+  NodeManager* nm = NodeManager::currentNM();
+
+  if (!assumption.isConst() && assumption.getKind() != kind::EQUAL)
+  {
+    // We rewrite inequality assumptions from x <= y to x + (str.len s) = y
+    // where s is some fresh string variable. We use (str.len s) because
+    // (str.len s) must be non-negative for the equation to hold.
+    Node x, y;
+    if (assumption.getKind() == kind::GEQ)
+    {
+      x = assumption[0];
+      y = assumption[1];
+    }
+    else
+    {
+      // (not (>= s t)) --> (>= (t - 1) s)
+      Assert(assumption.getKind() == kind::NOT
+             && assumption[0].getKind() == kind::GEQ);
+      x = nm->mkNode(kind::MINUS, assumption[0][1], nm->mkConst(Rational(1)));
+      y = assumption[0][0];
+    }
+
+    Node s = nm->mkBoundVar("slackVal", nm->stringType());
+    Node slen = nm->mkNode(kind::STRING_LENGTH, s);
+    assumption = Rewriter::rewrite(
+        nm->mkNode(kind::EQUAL, x, nm->mkNode(kind::PLUS, y, slen)));
+  }
+
+  Node diff = nm->mkNode(kind::MINUS, a, b);
+  bool res = false;
+  if (assumption.isConst())
+  {
+    bool assumptionBool = assumption.getConst<bool>();
+    if (assumptionBool)
+    {
+      res = checkEntailArith(diff, strict);
+    }
+    else
+    {
+      res = true;
+    }
+  }
+  else
+  {
+    res = checkEntailArithWithEqAssumption(assumption, diff, strict);
+  }
+  return res;
+}
+
+bool SequencesRewriter::checkEntailArithWithAssumptions(
+    std::vector<Node> assumptions, Node a, Node b, bool strict)
+{
+  // TODO: We currently try to show the entailment with each assumption
+  // independently. In the future, we should make better use of multiple
+  // assumptions.
+  bool res = false;
+  for (const auto& assumption : assumptions)
+  {
+    Assert(Rewriter::rewrite(assumption) == assumption);
+
+    if (checkEntailArithWithAssumption(assumption, a, b, strict))
+    {
+      res = true;
+      break;
+    }
+  }
+  return res;
+}
+
+Node SequencesRewriter::getConstantArithBound(Node a, bool isLower)
+{
+  Assert(Rewriter::rewrite(a) == a);
+  Node ret;
+  if (a.isConst())
+  {
+    ret = a;
+  }
+  else if (a.getKind() == kind::STRING_LENGTH)
+  {
+    if (isLower)
+    {
+      ret = NodeManager::currentNM()->mkConst(Rational(0));
+    }
+  }
+  else if (a.getKind() == kind::PLUS || a.getKind() == kind::MULT)
+  {
+    std::vector<Node> children;
+    bool success = true;
+    for (unsigned i = 0; i < a.getNumChildren(); i++)
+    {
+      Node ac = getConstantArithBound(a[i], isLower);
+      if (ac.isNull())
+      {
+        ret = ac;
+        success = false;
+        break;
+      }
+      else
+      {
+        if (ac.getConst<Rational>().sgn() == 0)
+        {
+          if (a.getKind() == kind::MULT)
+          {
+            ret = ac;
+            success = false;
+            break;
+          }
+        }
+        else
+        {
+          if (a.getKind() == kind::MULT)
+          {
+            if ((ac.getConst<Rational>().sgn() > 0) != isLower)
+            {
+              ret = Node::null();
+              success = false;
+              break;
+            }
+          }
+          children.push_back(ac);
+        }
+      }
+    }
+    if (success)
+    {
+      if (children.empty())
+      {
+        ret = NodeManager::currentNM()->mkConst(Rational(0));
+      }
+      else if (children.size() == 1)
+      {
+        ret = children[0];
+      }
+      else
+      {
+        ret = NodeManager::currentNM()->mkNode(a.getKind(), children);
+        ret = Rewriter::rewrite(ret);
+      }
+    }
+  }
+  Trace("strings-rewrite-cbound")
+      << "Constant " << (isLower ? "lower" : "upper") << " bound for " << a
+      << " is " << ret << std::endl;
+  Assert(ret.isNull() || ret.isConst());
+  // entailment check should be at least as powerful as computing a lower bound
+  Assert(!isLower || ret.isNull() || ret.getConst<Rational>().sgn() < 0
+         || checkEntailArith(a, false));
+  Assert(!isLower || ret.isNull() || ret.getConst<Rational>().sgn() <= 0
+         || checkEntailArith(a, true));
+  return ret;
+}
+
+Node SequencesRewriter::getFixedLengthForRegexp(Node n)
+{
+  NodeManager* nm = NodeManager::currentNM();
+  if (n.getKind() == STRING_TO_REGEXP)
+  {
+    Node ret = nm->mkNode(STRING_LENGTH, n[0]);
+    ret = Rewriter::rewrite(ret);
+    if (ret.isConst())
+    {
+      return ret;
+    }
+  }
+  else if (n.getKind() == REGEXP_SIGMA || n.getKind() == REGEXP_RANGE)
+  {
+    return nm->mkConst(Rational(1));
+  }
+  else if (n.getKind() == REGEXP_UNION || n.getKind() == REGEXP_INTER)
+  {
+    Node ret;
+    for (const Node& nc : n)
+    {
+      Node flc = getFixedLengthForRegexp(nc);
+      if (flc.isNull() || (!ret.isNull() && ret != flc))
+      {
+        return Node::null();
+      }
+      else if (ret.isNull())
+      {
+        // first time
+        ret = flc;
+      }
+    }
+    return ret;
+  }
+  else if (n.getKind() == REGEXP_CONCAT)
+  {
+    NodeBuilder<> nb(PLUS);
+    for (const Node& nc : n)
+    {
+      Node flc = getFixedLengthForRegexp(nc);
+      if (flc.isNull())
+      {
+        return flc;
+      }
+      nb << flc;
+    }
+    Node ret = nb.constructNode();
+    ret = Rewriter::rewrite(ret);
+    return ret;
+  }
+  return Node::null();
+}
+
+bool SequencesRewriter::checkEntailArithInternal(Node a)
+{
+  Assert(Rewriter::rewrite(a) == a);
+  // check whether a >= 0
+  if (a.isConst())
+  {
+    return a.getConst<Rational>().sgn() >= 0;
+  }
+  else if (a.getKind() == kind::STRING_LENGTH)
+  {
+    // str.len( t ) >= 0
+    return true;
+  }
+  else if (a.getKind() == kind::PLUS || a.getKind() == kind::MULT)
+  {
+    for (unsigned i = 0; i < a.getNumChildren(); i++)
+    {
+      if (!checkEntailArithInternal(a[i]))
+      {
+        return false;
+      }
+    }
+    // t1 >= 0 ^ ... ^ tn >= 0 => t1 op ... op tn >= 0
+    return true;
+  }
+
+  return false;
+}
+
+Node SequencesRewriter::decomposeSubstrChain(Node s,
+                                             std::vector<Node>& ss,
+                                             std::vector<Node>& ls)
+{
+  Assert(ss.empty());
+  Assert(ls.empty());
+  while (s.getKind() == STRING_SUBSTR)
+  {
+    ss.push_back(s[1]);
+    ls.push_back(s[2]);
+    s = s[0];
+  }
+  std::reverse(ss.begin(), ss.end());
+  std::reverse(ls.begin(), ls.end());
+  return s;
+}
+
+Node SequencesRewriter::mkSubstrChain(Node base,
+                                      const std::vector<Node>& ss,
+                                      const std::vector<Node>& ls)
+{
+  NodeManager* nm = NodeManager::currentNM();
+  for (unsigned i = 0, size = ss.size(); i < size; i++)
+  {
+    base = nm->mkNode(STRING_SUBSTR, base, ss[i], ls[i]);
+  }
+  return base;
+}
+
+Node SequencesRewriter::getStringOrEmpty(Node n)
+{
+  NodeManager* nm = NodeManager::currentNM();
+  Node res;
+  while (res.isNull())
+  {
+    switch (n.getKind())
+    {
+      case kind::STRING_STRREPL:
+      {
+        Node empty = nm->mkConst(::CVC4::String(""));
+        if (n[0] == empty)
+        {
+          // (str.replace "" x y) --> y
+          n = n[2];
+          break;
+        }
+
+        if (checkEntailLengthOne(n[0]) && n[2] == empty)
+        {
+          // (str.replace "A" x "") --> "A"
+          res = n[0];
+          break;
+        }
+
+        res = n;
+        break;
+      }
+      case kind::STRING_SUBSTR:
+      {
+        if (checkEntailLengthOne(n[0]))
+        {
+          // (str.substr "A" x y) --> "A"
+          res = n[0];
+          break;
+        }
+        res = n;
+        break;
+      }
+      default:
+      {
+        res = n;
+        break;
+      }
+    }
+  }
+  return res;
+}
+
+bool SequencesRewriter::inferZerosInSumGeq(Node x,
+                                           std::vector<Node>& ys,
+                                           std::vector<Node>& zeroYs)
+{
+  Assert(zeroYs.empty());
+
+  NodeManager* nm = NodeManager::currentNM();
+
+  // Check if we can show that y1 + ... + yn >= x
+  Node sum = (ys.size() > 1) ? nm->mkNode(PLUS, ys) : ys[0];
+  if (!checkEntailArith(sum, x))
+  {
+    return false;
+  }
+
+  // Try to remove yi one-by-one and check if we can still show:
+  //
+  // y1 + ... + yi-1 +  yi+1 + ... + yn >= x
+  //
+  // If that's the case, we know that yi can be zero and the inequality still
+  // holds.
+  size_t i = 0;
+  while (i < ys.size())
+  {
+    Node yi = ys[i];
+    std::vector<Node>::iterator pos = ys.erase(ys.begin() + i);
+    if (ys.size() > 1)
+    {
+      sum = nm->mkNode(PLUS, ys);
+    }
+    else
+    {
+      sum = ys.size() == 1 ? ys[0] : nm->mkConst(Rational(0));
+    }
+
+    if (checkEntailArith(sum, x))
+    {
+      zeroYs.push_back(yi);
+    }
+    else
+    {
+      ys.insert(pos, yi);
+      i++;
+    }
+  }
+  return true;
+}
+
+Node SequencesRewriter::inferEqsFromContains(Node x, Node y)
+{
+  NodeManager* nm = NodeManager::currentNM();
+  Node emp = nm->mkConst(String(""));
+
+  Node xLen = nm->mkNode(STRING_LENGTH, x);
+  std::vector<Node> yLens;
+  if (y.getKind() != STRING_CONCAT)
+  {
+    yLens.push_back(nm->mkNode(STRING_LENGTH, y));
+  }
+  else
+  {
+    for (const Node& yi : y)
+    {
+      yLens.push_back(nm->mkNode(STRING_LENGTH, yi));
+    }
+  }
+
+  std::vector<Node> zeroLens;
+  if (x == emp)
+  {
+    // If x is the empty string, then all ys must be empty, too, and we can
+    // skip the expensive checks. Note that this is just a performance
+    // optimization.
+    zeroLens.swap(yLens);
+  }
+  else
+  {
+    // Check if we can infer that str.len(x) <= str.len(y). If that is the
+    // case, try to minimize the sum in str.len(x) <= str.len(y1) + ... +
+    // str.len(yn) (where y = y1 ++ ... ++ yn) while keeping the inequality
+    // true. The terms that can have length zero without making the inequality
+    // false must be all be empty if (str.contains x y) is true.
+    if (!inferZerosInSumGeq(xLen, yLens, zeroLens))
+    {
+      // We could not prove that the inequality holds
+      return Node::null();
+    }
+    else if (yLens.size() == y.getNumChildren())
+    {
+      // We could only prove that the inequality holds but not that any of the
+      // ys must be empty
+      return nm->mkNode(EQUAL, x, y);
+    }
+  }
+
+  if (y.getKind() != STRING_CONCAT)
+  {
+    if (zeroLens.size() == 1)
+    {
+      // y is not a concatenation and we found that it must be empty, so just
+      // return (= y "")
+      Assert(zeroLens[0][0] == y);
+      return nm->mkNode(EQUAL, y, emp);
+    }
+    else
+    {
+      Assert(yLens.size() == 1 && yLens[0][0] == y);
+      return nm->mkNode(EQUAL, x, y);
+    }
+  }
+
+  std::vector<Node> cs;
+  for (const Node& yiLen : yLens)
+  {
+    Assert(std::find(y.begin(), y.end(), yiLen[0]) != y.end());
+    cs.push_back(yiLen[0]);
+  }
+
+  NodeBuilder<> nb(AND);
+  // (= x (str.++ y1' ... ym'))
+  if (!cs.empty())
+  {
+    nb << nm->mkNode(EQUAL, x, utils::mkConcat(STRING_CONCAT, cs));
+  }
+  // (= y1'' "") ... (= yk'' "")
+  for (const Node& zeroLen : zeroLens)
+  {
+    Assert(std::find(y.begin(), y.end(), zeroLen[0]) != y.end());
+    nb << nm->mkNode(EQUAL, zeroLen[0], emp);
+  }
+
+  // (and (= x (str.++ y1' ... ym')) (= y1'' "") ... (= yk'' ""))
+  return nb.constructNode();
+}
+
+std::pair<bool, std::vector<Node> > SequencesRewriter::collectEmptyEqs(Node x)
+{
+  NodeManager* nm = NodeManager::currentNM();
+  Node empty = nm->mkConst(::CVC4::String(""));
+
+  // Collect the equalities of the form (= x "") (sorted)
+  std::set<TNode> emptyNodes;
+  bool allEmptyEqs = true;
+  if (x.getKind() == kind::EQUAL)
+  {
+    if (x[0] == empty)
+    {
+      emptyNodes.insert(x[1]);
+    }
+    else if (x[1] == empty)
+    {
+      emptyNodes.insert(x[0]);
+    }
+    else
+    {
+      allEmptyEqs = false;
+    }
+  }
+  else if (x.getKind() == kind::AND)
+  {
+    for (const Node& c : x)
+    {
+      if (c.getKind() == kind::EQUAL)
+      {
+        if (c[0] == empty)
+        {
+          emptyNodes.insert(c[1]);
+        }
+        else if (c[1] == empty)
+        {
+          emptyNodes.insert(c[0]);
+        }
+      }
+      else
+      {
+        allEmptyEqs = false;
+      }
+    }
+  }
+
+  if (emptyNodes.size() == 0)
+  {
+    allEmptyEqs = false;
+  }
+
+  return std::make_pair(
+      allEmptyEqs, std::vector<Node>(emptyNodes.begin(), emptyNodes.end()));
+}
+
+Node SequencesRewriter::returnRewrite(Node node, Node ret, const char* c)
+{
+  Trace("strings-rewrite") << "Rewrite " << node << " to " << ret << " by " << c
+                           << "." << std::endl;
+
+  NodeManager* nm = NodeManager::currentNM();
+
+  // standard post-processing
+  // We rewrite (string) equalities immediately here. This allows us to forego
+  // the standard invariant on equality rewrites (that s=t must rewrite to one
+  // of { s=t, t=s, true, false } ).
+  Kind retk = ret.getKind();
+  if (retk == OR || retk == AND)
+  {
+    std::vector<Node> children;
+    bool childChanged = false;
+    for (const Node& cret : ret)
+    {
+      Node creter = cret;
+      if (cret.getKind() == EQUAL)
+      {
+        creter = rewriteEqualityExt(cret);
+      }
+      else if (cret.getKind() == NOT && cret[0].getKind() == EQUAL)
+      {
+        creter = nm->mkNode(NOT, rewriteEqualityExt(cret[0]));
+      }
+      childChanged = childChanged || cret != creter;
+      children.push_back(creter);
+    }
+    if (childChanged)
+    {
+      ret = nm->mkNode(retk, children);
+    }
+  }
+  else if (retk == NOT && ret[0].getKind() == EQUAL)
+  {
+    ret = nm->mkNode(NOT, rewriteEqualityExt(ret[0]));
+  }
+  else if (retk == EQUAL && node.getKind() != EQUAL)
+  {
+    Trace("strings-rewrite")
+        << "Apply extended equality rewrite on " << ret << std::endl;
+    ret = rewriteEqualityExt(ret);
+  }
+  return ret;
+}