YES
<?xml version="1.0"?>
<?xml-stylesheet type="text/xsl" href="cpfHTML.xsl"?>
<certificationProblem xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="cpf.xsd">
 <cpfVersion>2.2</cpfVersion>
 <proof>
  <acTerminationProof>
   <acDependencyPairs>
    <equations>
     <rules>
      <rule>
       <lhs>
        <funapp>
         <name>plus</name>
         <arg>
          <var>x</var>
         </arg>
         <arg>
          <var>y</var>
         </arg>
        </funapp>
       </lhs>
       <rhs>
        <funapp>
         <name>plus</name>
         <arg>
          <var>y</var>
         </arg>
         <arg>
          <var>x</var>
         </arg>
        </funapp>
       </rhs>
      </rule>
      <rule>
       <lhs>
        <funapp>
         <name>plus</name>
         <arg>
          <var>x</var>
         </arg>
         <arg>
          <funapp>
           <name>plus</name>
           <arg>
            <var>y</var>
           </arg>
           <arg>
            <var>z</var>
           </arg>
          </funapp>
         </arg>
        </funapp>
       </lhs>
       <rhs>
        <funapp>
         <name>plus</name>
         <arg>
          <funapp>
           <name>plus</name>
           <arg>
            <var>x</var>
           </arg>
           <arg>
            <var>y</var>
           </arg>
          </funapp>
         </arg>
         <arg>
          <var>z</var>
         </arg>
        </funapp>
       </rhs>
      </rule>
     </rules>
    </equations>
    <dpEquations>
     <rules>
      <rule>
       <lhs>
        <funapp>
         <sharp>
         <name>plus</name></sharp>
         <arg>
          <var>x</var>
         </arg>
         <arg>
          <var>y</var>
         </arg>
        </funapp>
       </lhs>
       <rhs>
        <funapp>
         <sharp>
         <name>plus</name></sharp>
         <arg>
          <var>y</var>
         </arg>
         <arg>
          <var>x</var>
         </arg>
        </funapp>
       </rhs>
      </rule>
      <rule>
       <lhs>
        <funapp>
         <sharp>
         <name>plus</name></sharp>
         <arg>
          <var>x</var>
         </arg>
         <arg>
          <funapp>
           <name>plus</name>
           <arg>
            <var>y</var>
           </arg>
           <arg>
            <var>z</var>
           </arg>
          </funapp>
         </arg>
        </funapp>
       </lhs>
       <rhs>
        <funapp>
         <sharp>
         <name>plus</name></sharp>
         <arg>
          <var>x</var>
         </arg>
         <arg>
          <var>y</var>
         </arg>
        </funapp>
       </rhs>
      </rule>
      <rule>
       <lhs>
        <funapp>
         <sharp>
         <name>plus</name></sharp>
         <arg>
          <var>x</var>
         </arg>
         <arg>
          <funapp>
           <name>plus</name>
           <arg>
            <var>y</var>
           </arg>
           <arg>
            <var>z</var>
           </arg>
          </funapp>
         </arg>
        </funapp>
       </lhs>
       <rhs>
        <funapp>
         <sharp>
         <name>plus</name></sharp>
         <arg>
          <funapp>
           <name>plus</name>
           <arg>
            <var>x</var>
           </arg>
           <arg>
            <var>y</var>
           </arg>
          </funapp>
         </arg>
         <arg>
          <var>z</var>
         </arg>
        </funapp>
       </rhs>
      </rule>
     </rules>
    </dpEquations>
    <dps>
     <rules>
      <rule>
       <lhs>
        <funapp>
         <sharp>
         <name>minus</name></sharp>
         <arg>
          <funapp>
           <name>minus</name>
           <arg>
            <var>x</var>
           </arg>
           <arg>
            <var>y</var>
           </arg>
          </funapp>
         </arg>
         <arg>
          <var>z</var>
         </arg>
        </funapp>
       </lhs>
       <rhs>
        <funapp>
         <sharp>
         <name>minus</name></sharp>
         <arg>
          <var>x</var>
         </arg>
         <arg>
          <funapp>
           <name>plus</name>
           <arg>
            <var>y</var>
           </arg>
           <arg>
            <var>z</var>
           </arg>
          </funapp>
         </arg>
        </funapp>
       </rhs>
      </rule>
      <rule>
       <lhs>
        <funapp>
         <sharp>
         <name>minus</name></sharp>
         <arg>
          <funapp>
           <name>minus</name>
           <arg>
            <var>x</var>
           </arg>
           <arg>
            <var>y</var>
           </arg>
          </funapp>
         </arg>
         <arg>
          <var>z</var>
         </arg>
        </funapp>
       </lhs>
       <rhs>
        <funapp>
         <sharp>
         <name>plus</name></sharp>
         <arg>
          <var>y</var>
         </arg>
         <arg>
          <var>z</var>
         </arg>
        </funapp>
       </rhs>
      </rule>
      <rule>
       <lhs>
        <funapp>
         <sharp>
         <name>quot</name></sharp>
         <arg>
          <funapp>
           <name>s</name>
           <arg>
            <var>x</var>
           </arg>
          </funapp>
         </arg>
         <arg>
          <funapp>
           <name>s</name>
           <arg>
            <var>y</var>
           </arg>
          </funapp>
         </arg>
        </funapp>
       </lhs>
       <rhs>
        <funapp>
         <sharp>
         <name>minus</name></sharp>
         <arg>
          <var>x</var>
         </arg>
         <arg>
          <var>y</var>
         </arg>
        </funapp>
       </rhs>
      </rule>
      <rule>
       <lhs>
        <funapp>
         <sharp>
         <name>plus</name></sharp>
         <arg>
          <funapp>
           <name>s</name>
           <arg>
            <var>x</var>
           </arg>
          </funapp>
         </arg>
         <arg>
          <var>y</var>
         </arg>
        </funapp>
       </lhs>
       <rhs>
        <funapp>
         <sharp>
         <name>plus</name></sharp>
         <arg>
          <var>x</var>
         </arg>
         <arg>
          <var>y</var>
         </arg>
        </funapp>
       </rhs>
      </rule>
      <rule>
       <lhs>
        <funapp>
         <sharp>
         <name>minus</name></sharp>
         <arg>
          <funapp>
           <name>s</name>
           <arg>
            <var>x</var>
           </arg>
          </funapp>
         </arg>
         <arg>
          <funapp>
           <name>s</name>
           <arg>
            <var>y</var>
           </arg>
          </funapp>
         </arg>
        </funapp>
       </lhs>
       <rhs>
        <funapp>
         <sharp>
         <name>minus</name></sharp>
         <arg>
          <var>x</var>
         </arg>
         <arg>
          <var>y</var>
         </arg>
        </funapp>
       </rhs>
      </rule>
      <rule>
       <lhs>
        <funapp>
         <sharp>
         <name>quot</name></sharp>
         <arg>
          <funapp>
           <name>s</name>
           <arg>
            <var>x</var>
           </arg>
          </funapp>
         </arg>
         <arg>
          <funapp>
           <name>s</name>
           <arg>
            <var>y</var>
           </arg>
          </funapp>
         </arg>
        </funapp>
       </lhs>
       <rhs>
        <funapp>
         <sharp>
         <name>quot</name></sharp>
         <arg>
          <funapp>
           <name>minus</name>
           <arg>
            <var>x</var>
           </arg>
           <arg>
            <var>y</var>
           </arg>
          </funapp>
         </arg>
         <arg>
          <funapp>
           <name>s</name>
           <arg>
            <var>y</var>
           </arg>
          </funapp>
         </arg>
        </funapp>
       </rhs>
      </rule>
     </rules>
    </dps>
    <extensions>
     <rules>
      <rule>
       <lhs>
        <funapp>
         <name>plus</name>
         <arg>
          <funapp>
           <name>plus</name>
           <arg>
            <funapp>
             <name>0</name>
            </funapp>
           </arg>
           <arg>
            <var>y</var>
           </arg>
          </funapp>
         </arg>
         <arg>
          <var>_1</var>
         </arg>
        </funapp>
       </lhs>
       <rhs>
        <funapp>
         <name>plus</name>
         <arg>
          <var>y</var>
         </arg>
         <arg>
          <var>_1</var>
         </arg>
        </funapp>
       </rhs>
      </rule>
      <rule>
       <lhs>
        <funapp>
         <name>plus</name>
         <arg>
          <funapp>
           <name>plus</name>
           <arg>
            <funapp>
             <name>s</name>
             <arg>
              <var>x</var>
             </arg>
            </funapp>
           </arg>
           <arg>
            <var>y</var>
           </arg>
          </funapp>
         </arg>
         <arg>
          <var>_1</var>
         </arg>
        </funapp>
       </lhs>
       <rhs>
        <funapp>
         <name>plus</name>
         <arg>
          <funapp>
           <name>s</name>
           <arg>
            <funapp>
             <name>plus</name>
             <arg>
              <var>x</var>
             </arg>
             <arg>
              <var>y</var>
             </arg>
            </funapp>
           </arg>
          </funapp>
         </arg>
         <arg>
          <var>_1</var>
         </arg>
        </funapp>
       </rhs>
      </rule>
     </rules>
    </extensions>
    <acDPTerminationProof>
     <acDepGraphProc>
      <component>
       <dps>
        <rules>
         <rule>
          <lhs>
           <funapp>
            <sharp>
            <name>quot</name></sharp>
            <arg>
             <funapp>
              <name>s</name>
              <arg>
               <var>x</var>
              </arg>
             </funapp>
            </arg>
            <arg>
             <funapp>
              <name>s</name>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </arg>
           </funapp>
          </lhs>
          <rhs>
           <funapp>
            <sharp>
            <name>quot</name></sharp>
            <arg>
             <funapp>
              <name>minus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </arg>
            <arg>
             <funapp>
              <name>s</name>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </arg>
           </funapp>
          </rhs>
         </rule>
        </rules>
       </dps>
       <realScc>true</realScc>
       <acDPTerminationProof>
        <acRedPairProc>
         <orderingConstraintProof>
          <redPair>
           <interpretation>
            <type>
             <polynomial>
              <domain><naturals/></domain>
              <degree>1</degree>
             </polynomial>
            </type>
            <interpret>
             <name>s</name>
             <arity>1</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>2</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <name>minus</name>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>1</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <sharp>
             <name>plus</name></sharp>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <name>0</name>
             <arity>0</arity>
             <polynomial>
              <sum>
               <polynomial>
                <coefficient>
                <integer>1</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <name>quot</name>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <sharp>
             <name>minus</name></sharp>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <name>plus</name>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <variable>2</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>1</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <sharp>
             <name>quot</name></sharp>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
           </interpretation>
          </redPair>
         </orderingConstraintProof>
         <dps>
          <rules>
           <rule>
            <lhs>
             <funapp>
              <sharp>
              <name>quot</name></sharp>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>x</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <sharp>
              <name>quot</name></sharp>
              <arg>
               <funapp>
                <name>minus</name>
                <arg>
                 <var>x</var>
                </arg>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </rhs>
           </rule>
          </rules>
         </dps>
         <usableRules>
          <rules>
           <rule>
            <lhs>
             <funapp>
              <name>plus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>y</var>
                </arg>
                <arg>
                 <var>z</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>plus</name>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>x</var>
                </arg>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <var>z</var>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>minus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <funapp>
                <name>0</name>
               </funapp>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <var>x</var>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>minus</name>
              <arg>
               <funapp>
                <name>minus</name>
                <arg>
                 <var>x</var>
                </arg>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <var>z</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>minus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>y</var>
                </arg>
                <arg>
                 <var>z</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>plus</name>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>x</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>s</name>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>x</var>
                </arg>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>plus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>plus</name>
              <arg>
               <var>y</var>
              </arg>
              <arg>
               <var>x</var>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>plus</name>
              <arg>
               <funapp>
                <name>0</name>
               </funapp>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <var>y</var>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>minus</name>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>x</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>minus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </rhs>
           </rule>
          </rules>
         </usableRules>
         <acDPTerminationProof>
          <acDepGraphProc>
          </acDepGraphProc>
         </acDPTerminationProof>
        </acRedPairProc>
       </acDPTerminationProof>
      </component>
      <component>
       <dps>
        <rules>
         <rule>
          <lhs>
           <funapp>
            <sharp>
            <name>quot</name></sharp>
            <arg>
             <funapp>
              <name>s</name>
              <arg>
               <var>x</var>
              </arg>
             </funapp>
            </arg>
            <arg>
             <funapp>
              <name>s</name>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </arg>
           </funapp>
          </lhs>
          <rhs>
           <funapp>
            <sharp>
            <name>minus</name></sharp>
            <arg>
             <var>x</var>
            </arg>
            <arg>
             <var>y</var>
            </arg>
           </funapp>
          </rhs>
         </rule>
        </rules>
       </dps>
       <realScc>false</realScc>
      </component>
      <component>
       <dps>
        <rules>
         <rule>
          <lhs>
           <funapp>
            <sharp>
            <name>minus</name></sharp>
            <arg>
             <funapp>
              <name>s</name>
              <arg>
               <var>x</var>
              </arg>
             </funapp>
            </arg>
            <arg>
             <funapp>
              <name>s</name>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </arg>
           </funapp>
          </lhs>
          <rhs>
           <funapp>
            <sharp>
            <name>minus</name></sharp>
            <arg>
             <var>x</var>
            </arg>
            <arg>
             <var>y</var>
            </arg>
           </funapp>
          </rhs>
         </rule>
         <rule>
          <lhs>
           <funapp>
            <sharp>
            <name>minus</name></sharp>
            <arg>
             <funapp>
              <name>minus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </arg>
            <arg>
             <var>z</var>
            </arg>
           </funapp>
          </lhs>
          <rhs>
           <funapp>
            <sharp>
            <name>minus</name></sharp>
            <arg>
             <var>x</var>
            </arg>
            <arg>
             <funapp>
              <name>plus</name>
              <arg>
               <var>y</var>
              </arg>
              <arg>
               <var>z</var>
              </arg>
             </funapp>
            </arg>
           </funapp>
          </rhs>
         </rule>
        </rules>
       </dps>
       <realScc>true</realScc>
       <acDPTerminationProof>
        <acRedPairProc>
         <orderingConstraintProof>
          <redPair>
           <interpretation>
            <type>
             <polynomial>
              <domain><naturals/></domain>
              <degree>1</degree>
             </polynomial>
            </type>
            <interpret>
             <name>s</name>
             <arity>1</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>2</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <name>minus</name>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <variable>2</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>2</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <sharp>
             <name>plus</name></sharp>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <name>0</name>
             <arity>0</arity>
             <polynomial>
              <sum>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <name>quot</name>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <sharp>
             <name>minus</name></sharp>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <variable>2</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <name>plus</name>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <variable>2</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>1</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <sharp>
             <name>quot</name></sharp>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
           </interpretation>
          </redPair>
         </orderingConstraintProof>
         <dps>
          <rules>
           <rule>
            <lhs>
             <funapp>
              <sharp>
              <name>minus</name></sharp>
              <arg>
               <funapp>
                <name>minus</name>
                <arg>
                 <var>x</var>
                </arg>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <var>z</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <sharp>
              <name>minus</name></sharp>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>y</var>
                </arg>
                <arg>
                 <var>z</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <sharp>
              <name>minus</name></sharp>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>x</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <sharp>
              <name>minus</name></sharp>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </rhs>
           </rule>
          </rules>
         </dps>
         <usableRules>
          <rules>
           <rule>
            <lhs>
             <funapp>
              <name>plus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>y</var>
                </arg>
                <arg>
                 <var>z</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>plus</name>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>x</var>
                </arg>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <var>z</var>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>minus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <funapp>
                <name>0</name>
               </funapp>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <var>x</var>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>minus</name>
              <arg>
               <funapp>
                <name>minus</name>
                <arg>
                 <var>x</var>
                </arg>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <var>z</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>minus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>y</var>
                </arg>
                <arg>
                 <var>z</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>plus</name>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>x</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>s</name>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>x</var>
                </arg>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>plus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>plus</name>
              <arg>
               <var>y</var>
              </arg>
              <arg>
               <var>x</var>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>plus</name>
              <arg>
               <funapp>
                <name>0</name>
               </funapp>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <var>y</var>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>minus</name>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>x</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>minus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </rhs>
           </rule>
          </rules>
         </usableRules>
         <acDPTerminationProof>
          <acDepGraphProc>
          </acDepGraphProc>
         </acDPTerminationProof>
        </acRedPairProc>
       </acDPTerminationProof>
      </component>
      <component>
       <dps>
        <rules>
         <rule>
          <lhs>
           <funapp>
            <sharp>
            <name>minus</name></sharp>
            <arg>
             <funapp>
              <name>minus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </arg>
            <arg>
             <var>z</var>
            </arg>
           </funapp>
          </lhs>
          <rhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <var>y</var>
            </arg>
            <arg>
             <var>z</var>
            </arg>
           </funapp>
          </rhs>
         </rule>
        </rules>
       </dps>
       <realScc>false</realScc>
      </component>
      <component>
       <dps>
        <rules>
         <rule>
          <lhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <var>x</var>
            </arg>
            <arg>
             <funapp>
              <name>plus</name>
              <arg>
               <var>y</var>
              </arg>
              <arg>
               <var>z</var>
              </arg>
             </funapp>
            </arg>
           </funapp>
          </lhs>
          <rhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <funapp>
              <name>plus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </arg>
            <arg>
             <var>z</var>
            </arg>
           </funapp>
          </rhs>
         </rule>
         <rule>
          <lhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <funapp>
              <name>s</name>
              <arg>
               <var>x</var>
              </arg>
             </funapp>
            </arg>
            <arg>
             <var>y</var>
            </arg>
           </funapp>
          </lhs>
          <rhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <var>x</var>
            </arg>
            <arg>
             <var>y</var>
            </arg>
           </funapp>
          </rhs>
         </rule>
         <rule>
          <lhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <var>x</var>
            </arg>
            <arg>
             <funapp>
              <name>plus</name>
              <arg>
               <var>y</var>
              </arg>
              <arg>
               <var>z</var>
              </arg>
             </funapp>
            </arg>
           </funapp>
          </lhs>
          <rhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <var>x</var>
            </arg>
            <arg>
             <var>y</var>
            </arg>
           </funapp>
          </rhs>
         </rule>
         <rule>
          <lhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <var>x</var>
            </arg>
            <arg>
             <var>y</var>
            </arg>
           </funapp>
          </lhs>
          <rhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <var>y</var>
            </arg>
            <arg>
             <var>x</var>
            </arg>
           </funapp>
          </rhs>
         </rule>
        </rules>
       </dps>
       <realScc>true</realScc>
       <acDPTerminationProof>
        <acRedPairProc>
         <orderingConstraintProof>
          <redPair>
           <interpretation>
            <type>
             <polynomial>
              <domain><naturals/></domain>
              <degree>1</degree>
             </polynomial>
            </type>
            <interpret>
             <name>s</name>
             <arity>1</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>2</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <name>minus</name>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <variable>2</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>2</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <sharp>
             <name>plus</name></sharp>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <variable>2</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <name>0</name>
             <arity>0</arity>
             <polynomial>
              <sum>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <name>quot</name>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <sharp>
             <name>minus</name></sharp>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <name>plus</name>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <variable>2</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>1</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <sharp>
             <name>quot</name></sharp>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
           </interpretation>
          </redPair>
         </orderingConstraintProof>
         <dps>
          <rules>
           <rule>
            <lhs>
             <funapp>
              <sharp>
              <name>plus</name></sharp>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>y</var>
                </arg>
                <arg>
                 <var>z</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <sharp>
              <name>plus</name></sharp>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <sharp>
              <name>plus</name></sharp>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>x</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <sharp>
              <name>plus</name></sharp>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </rhs>
           </rule>
          </rules>
         </dps>
         <usableRules>
          <rules>
           <rule>
            <lhs>
             <funapp>
              <name>plus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>y</var>
                </arg>
                <arg>
                 <var>z</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>plus</name>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>x</var>
                </arg>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <var>z</var>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>minus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <funapp>
                <name>0</name>
               </funapp>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <var>x</var>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>minus</name>
              <arg>
               <funapp>
                <name>minus</name>
                <arg>
                 <var>x</var>
                </arg>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <var>z</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>minus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>y</var>
                </arg>
                <arg>
                 <var>z</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>plus</name>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>x</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>s</name>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>x</var>
                </arg>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>plus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>plus</name>
              <arg>
               <var>y</var>
              </arg>
              <arg>
               <var>x</var>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>plus</name>
              <arg>
               <funapp>
                <name>0</name>
               </funapp>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <var>y</var>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>minus</name>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>x</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>minus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </rhs>
           </rule>
          </rules>
         </usableRules>
         <acDPTerminationProof>
          <acDepGraphProc>
           <component>
            <dps>
             <rules>
              <rule>
               <lhs>
                <funapp>
                 <sharp>
                 <name>plus</name></sharp>
                 <arg>
                  <var>x</var>
                 </arg>
                 <arg>
                  <funapp>
                   <name>plus</name>
                   <arg>
                    <var>y</var>
                   </arg>
                   <arg>
                    <var>z</var>
                   </arg>
                  </funapp>
                 </arg>
                </funapp>
               </lhs>
               <rhs>
                <funapp>
                 <sharp>
                 <name>plus</name></sharp>
                 <arg>
                  <funapp>
                   <name>plus</name>
                   <arg>
                    <var>x</var>
                   </arg>
                   <arg>
                    <var>y</var>
                   </arg>
                  </funapp>
                 </arg>
                 <arg>
                  <var>z</var>
                 </arg>
                </funapp>
               </rhs>
              </rule>
              <rule>
               <lhs>
                <funapp>
                 <sharp>
                 <name>plus</name></sharp>
                 <arg>
                  <var>x</var>
                 </arg>
                 <arg>
                  <var>y</var>
                 </arg>
                </funapp>
               </lhs>
               <rhs>
                <funapp>
                 <sharp>
                 <name>plus</name></sharp>
                 <arg>
                  <var>y</var>
                 </arg>
                 <arg>
                  <var>x</var>
                 </arg>
                </funapp>
               </rhs>
              </rule>
             </rules>
            </dps>
            <realScc>true</realScc>
            <acDPTerminationProof><acTrivialProc/>
            </acDPTerminationProof>
           </component>
          </acDepGraphProc>
         </acDPTerminationProof>
        </acRedPairProc>
       </acDPTerminationProof>
      </component>
     </acDepGraphProc>
    </acDPTerminationProof>
    <acDPTerminationProof>
     <acDepGraphProc>
      <component>
       <dps>
        <rules>
         <rule>
          <lhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <var>x</var>
            </arg>
            <arg>
             <funapp>
              <name>plus</name>
              <arg>
               <var>y</var>
              </arg>
              <arg>
               <var>z</var>
              </arg>
             </funapp>
            </arg>
           </funapp>
          </lhs>
          <rhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <var>x</var>
            </arg>
            <arg>
             <var>y</var>
            </arg>
           </funapp>
          </rhs>
         </rule>
         <rule>
          <lhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <funapp>
              <name>plus</name>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>x</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </arg>
            <arg>
             <var>_1</var>
            </arg>
           </funapp>
          </lhs>
          <rhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <funapp>
              <name>s</name>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>x</var>
                </arg>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </arg>
            <arg>
             <var>_1</var>
            </arg>
           </funapp>
          </rhs>
         </rule>
         <rule>
          <lhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <var>x</var>
            </arg>
            <arg>
             <var>y</var>
            </arg>
           </funapp>
          </lhs>
          <rhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <var>y</var>
            </arg>
            <arg>
             <var>x</var>
            </arg>
           </funapp>
          </rhs>
         </rule>
         <rule>
          <lhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <var>x</var>
            </arg>
            <arg>
             <funapp>
              <name>plus</name>
              <arg>
               <var>y</var>
              </arg>
              <arg>
               <var>z</var>
              </arg>
             </funapp>
            </arg>
           </funapp>
          </lhs>
          <rhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <funapp>
              <name>plus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </arg>
            <arg>
             <var>z</var>
            </arg>
           </funapp>
          </rhs>
         </rule>
         <rule>
          <lhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <funapp>
              <name>plus</name>
              <arg>
               <funapp>
                <name>0</name>
               </funapp>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </arg>
            <arg>
             <var>_1</var>
            </arg>
           </funapp>
          </lhs>
          <rhs>
           <funapp>
            <sharp>
            <name>plus</name></sharp>
            <arg>
             <var>y</var>
            </arg>
            <arg>
             <var>_1</var>
            </arg>
           </funapp>
          </rhs>
         </rule>
        </rules>
       </dps>
       <realScc>true</realScc>
       <acDPTerminationProof>
        <acRedPairProc>
         <orderingConstraintProof>
          <redPair>
           <interpretation>
            <type>
             <polynomial>
              <domain><naturals/></domain>
              <degree>1</degree>
             </polynomial>
            </type>
            <interpret>
             <name>s</name>
             <arity>1</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>2</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <name>minus</name>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <variable>2</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>2</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <sharp>
             <name>plus</name></sharp>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <variable>2</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <name>0</name>
             <arity>0</arity>
             <polynomial>
              <sum>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <name>quot</name>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <sharp>
             <name>minus</name></sharp>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <name>plus</name>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <variable>2</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>1</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
            <interpret>
             <sharp>
             <name>quot</name></sharp>
             <arity>2</arity>
             <polynomial>
              <sum>
               <polynomial>
                <variable>1</variable>
               </polynomial>
               <polynomial>
                <coefficient>
                <integer>0</integer></coefficient>
               </polynomial>
              </sum>
             </polynomial>
            </interpret>
           </interpretation>
          </redPair>
         </orderingConstraintProof>
         <dps>
          <rules>
           <rule>
            <lhs>
             <funapp>
              <sharp>
              <name>plus</name></sharp>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <funapp>
                  <name>0</name>
                 </funapp>
                </arg>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <var>_1</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <sharp>
              <name>plus</name></sharp>
              <arg>
               <var>y</var>
              </arg>
              <arg>
               <var>_1</var>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <sharp>
              <name>plus</name></sharp>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>y</var>
                </arg>
                <arg>
                 <var>z</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <sharp>
              <name>plus</name></sharp>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </rhs>
           </rule>
          </rules>
         </dps>
         <usableRules>
          <rules>
           <rule>
            <lhs>
             <funapp>
              <name>plus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>y</var>
                </arg>
                <arg>
                 <var>z</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>plus</name>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>x</var>
                </arg>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <var>z</var>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>minus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <funapp>
                <name>0</name>
               </funapp>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <var>x</var>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>minus</name>
              <arg>
               <funapp>
                <name>minus</name>
                <arg>
                 <var>x</var>
                </arg>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <var>z</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>minus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>y</var>
                </arg>
                <arg>
                 <var>z</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>plus</name>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>x</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>s</name>
              <arg>
               <funapp>
                <name>plus</name>
                <arg>
                 <var>x</var>
                </arg>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>plus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>plus</name>
              <arg>
               <var>y</var>
              </arg>
              <arg>
               <var>x</var>
              </arg>
             </funapp>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>plus</name>
              <arg>
               <funapp>
                <name>0</name>
               </funapp>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <var>y</var>
            </rhs>
           </rule>
           <rule>
            <lhs>
             <funapp>
              <name>minus</name>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>x</var>
                </arg>
               </funapp>
              </arg>
              <arg>
               <funapp>
                <name>s</name>
                <arg>
                 <var>y</var>
                </arg>
               </funapp>
              </arg>
             </funapp>
            </lhs>
            <rhs>
             <funapp>
              <name>minus</name>
              <arg>
               <var>x</var>
              </arg>
              <arg>
               <var>y</var>
              </arg>
             </funapp>
            </rhs>
           </rule>
          </rules>
         </usableRules>
         <acDPTerminationProof>
          <acDepGraphProc>
           <component>
            <dps>
             <rules>
              <rule>
               <lhs>
                <funapp>
                 <sharp>
                 <name>plus</name></sharp>
                 <arg>
                  <var>x</var>
                 </arg>
                 <arg>
                  <funapp>
                   <name>plus</name>
                   <arg>
                    <var>y</var>
                   </arg>
                   <arg>
                    <var>z</var>
                   </arg>
                  </funapp>
                 </arg>
                </funapp>
               </lhs>
               <rhs>
                <funapp>
                 <sharp>
                 <name>plus</name></sharp>
                 <arg>
                  <funapp>
                   <name>plus</name>
                   <arg>
                    <var>x</var>
                   </arg>
                   <arg>
                    <var>y</var>
                   </arg>
                  </funapp>
                 </arg>
                 <arg>
                  <var>z</var>
                 </arg>
                </funapp>
               </rhs>
              </rule>
              <rule>
               <lhs>
                <funapp>
                 <sharp>
                 <name>plus</name></sharp>
                 <arg>
                  <funapp>
                   <name>plus</name>
                   <arg>
                    <funapp>
                     <name>s</name>
                     <arg>
                      <var>x</var>
                     </arg>
                    </funapp>
                   </arg>
                   <arg>
                    <var>y</var>
                   </arg>
                  </funapp>
                 </arg>
                 <arg>
                  <var>_1</var>
                 </arg>
                </funapp>
               </lhs>
               <rhs>
                <funapp>
                 <sharp>
                 <name>plus</name></sharp>
                 <arg>
                  <funapp>
                   <name>s</name>
                   <arg>
                    <funapp>
                     <name>plus</name>
                     <arg>
                      <var>x</var>
                     </arg>
                     <arg>
                      <var>y</var>
                     </arg>
                    </funapp>
                   </arg>
                  </funapp>
                 </arg>
                 <arg>
                  <var>_1</var>
                 </arg>
                </funapp>
               </rhs>
              </rule>
              <rule>
               <lhs>
                <funapp>
                 <sharp>
                 <name>plus</name></sharp>
                 <arg>
                  <var>x</var>
                 </arg>
                 <arg>
                  <var>y</var>
                 </arg>
                </funapp>
               </lhs>
               <rhs>
                <funapp>
                 <sharp>
                 <name>plus</name></sharp>
                 <arg>
                  <var>y</var>
                 </arg>
                 <arg>
                  <var>x</var>
                 </arg>
                </funapp>
               </rhs>
              </rule>
             </rules>
            </dps>
            <realScc>true</realScc>
            <acDPTerminationProof>
             <acRedPairProc>
              <orderingConstraintProof>
               <redPair>
                <interpretation>
                 <type>
                  <polynomial>
                   <domain><naturals/></domain>
                   <degree>1</degree>
                  </polynomial>
                 </type>
                 <interpret>
                  <name>s</name>
                  <arity>1</arity>
                  <polynomial>
                   <sum>
                    <polynomial>
                     <coefficient>
                     <integer>2</integer></coefficient>
                    </polynomial>
                   </sum>
                  </polynomial>
                 </interpret>
                 <interpret>
                  <name>minus</name>
                  <arity>2</arity>
                  <polynomial>
                   <sum>
                    <polynomial>
                     <variable>1</variable>
                    </polynomial>
                    <polynomial>
                     <variable>2</variable>
                    </polynomial>
                    <polynomial>
                     <coefficient>
                     <integer>2</integer></coefficient>
                    </polynomial>
                   </sum>
                  </polynomial>
                 </interpret>
                 <interpret>
                  <sharp>
                  <name>plus</name></sharp>
                  <arity>2</arity>
                  <polynomial>
                   <sum>
                    <polynomial>
                     <variable>1</variable>
                    </polynomial>
                    <polynomial>
                     <variable>2</variable>
                    </polynomial>
                    <polynomial>
                     <coefficient>
                     <integer>0</integer></coefficient>
                    </polynomial>
                   </sum>
                  </polynomial>
                 </interpret>
                 <interpret>
                  <name>0</name>
                  <arity>0</arity>
                  <polynomial>
                   <sum>
                    <polynomial>
                     <coefficient>
                     <integer>0</integer></coefficient>
                    </polynomial>
                   </sum>
                  </polynomial>
                 </interpret>
                 <interpret>
                  <name>quot</name>
                  <arity>2</arity>
                  <polynomial>
                   <sum>
                    <polynomial>
                     <coefficient>
                     <integer>0</integer></coefficient>
                    </polynomial>
                   </sum>
                  </polynomial>
                 </interpret>
                 <interpret>
                  <sharp>
                  <name>minus</name></sharp>
                  <arity>2</arity>
                  <polynomial>
                   <sum>
                    <polynomial>
                     <coefficient>
                     <integer>0</integer></coefficient>
                    </polynomial>
                   </sum>
                  </polynomial>
                 </interpret>
                 <interpret>
                  <name>plus</name>
                  <arity>2</arity>
                  <polynomial>
                   <sum>
                    <polynomial>
                     <variable>1</variable>
                    </polynomial>
                    <polynomial>
                     <variable>2</variable>
                    </polynomial>
                    <polynomial>
                     <coefficient>
                     <integer>1</integer></coefficient>
                    </polynomial>
                   </sum>
                  </polynomial>
                 </interpret>
                 <interpret>
                  <sharp>
                  <name>quot</name></sharp>
                  <arity>2</arity>
                  <polynomial>
                   <sum>
                    <polynomial>
                     <coefficient>
                     <integer>0</integer></coefficient>
                    </polynomial>
                   </sum>
                  </polynomial>
                 </interpret>
                </interpretation>
               </redPair>
              </orderingConstraintProof>
              <dps>
               <rules>
                <rule>
                 <lhs>
                  <funapp>
                   <sharp>
                   <name>plus</name></sharp>
                   <arg>
                    <funapp>
                     <name>plus</name>
                     <arg>
                      <funapp>
                       <name>s</name>
                       <arg>
                        <var>x</var>
                       </arg>
                      </funapp>
                     </arg>
                     <arg>
                      <var>y</var>
                     </arg>
                    </funapp>
                   </arg>
                   <arg>
                    <var>_1</var>
                   </arg>
                  </funapp>
                 </lhs>
                 <rhs>
                  <funapp>
                   <sharp>
                   <name>plus</name></sharp>
                   <arg>
                    <funapp>
                     <name>s</name>
                     <arg>
                      <funapp>
                       <name>plus</name>
                       <arg>
                        <var>x</var>
                       </arg>
                       <arg>
                        <var>y</var>
                       </arg>
                      </funapp>
                     </arg>
                    </funapp>
                   </arg>
                   <arg>
                    <var>_1</var>
                   </arg>
                  </funapp>
                 </rhs>
                </rule>
               </rules>
              </dps>
              <usableRules>
               <rules>
                <rule>
                 <lhs>
                  <funapp>
                   <name>plus</name>
                   <arg>
                    <var>x</var>
                   </arg>
                   <arg>
                    <funapp>
                     <name>plus</name>
                     <arg>
                      <var>y</var>
                     </arg>
                     <arg>
                      <var>z</var>
                     </arg>
                    </funapp>
                   </arg>
                  </funapp>
                 </lhs>
                 <rhs>
                  <funapp>
                   <name>plus</name>
                   <arg>
                    <funapp>
                     <name>plus</name>
                     <arg>
                      <var>x</var>
                     </arg>
                     <arg>
                      <var>y</var>
                     </arg>
                    </funapp>
                   </arg>
                   <arg>
                    <var>z</var>
                   </arg>
                  </funapp>
                 </rhs>
                </rule>
                <rule>
                 <lhs>
                  <funapp>
                   <name>plus</name>
                   <arg>
                    <funapp>
                     <name>s</name>
                     <arg>
                      <var>x</var>
                     </arg>
                    </funapp>
                   </arg>
                   <arg>
                    <var>y</var>
                   </arg>
                  </funapp>
                 </lhs>
                 <rhs>
                  <funapp>
                   <name>s</name>
                   <arg>
                    <funapp>
                     <name>plus</name>
                     <arg>
                      <var>x</var>
                     </arg>
                     <arg>
                      <var>y</var>
                     </arg>
                    </funapp>
                   </arg>
                  </funapp>
                 </rhs>
                </rule>
                <rule>
                 <lhs>
                  <funapp>
                   <name>plus</name>
                   <arg>
                    <var>x</var>
                   </arg>
                   <arg>
                    <var>y</var>
                   </arg>
                  </funapp>
                 </lhs>
                 <rhs>
                  <funapp>
                   <name>plus</name>
                   <arg>
                    <var>y</var>
                   </arg>
                   <arg>
                    <var>x</var>
                   </arg>
                  </funapp>
                 </rhs>
                </rule>
                <rule>
                 <lhs>
                  <funapp>
                   <name>plus</name>
                   <arg>
                    <funapp>
                     <name>0</name>
                    </funapp>
                   </arg>
                   <arg>
                    <var>y</var>
                   </arg>
                  </funapp>
                 </lhs>
                 <rhs>
                  <var>y</var>
                 </rhs>
                </rule>
               </rules>
              </usableRules>
              <acDPTerminationProof>
               <acDepGraphProc>
                <component>
                 <dps>
                  <rules>
                   <rule>
                    <lhs>
                     <funapp>
                      <sharp>
                      <name>plus</name></sharp>
                      <arg>
                       <var>x</var>
                      </arg>
                      <arg>
                       <funapp>
                        <name>plus</name>
                        <arg>
                         <var>y</var>
                        </arg>
                        <arg>
                         <var>z</var>
                        </arg>
                       </funapp>
                      </arg>
                     </funapp>
                    </lhs>
                    <rhs>
                     <funapp>
                      <sharp>
                      <name>plus</name></sharp>
                      <arg>
                       <funapp>
                        <name>plus</name>
                        <arg>
                         <var>x</var>
                        </arg>
                        <arg>
                         <var>y</var>
                        </arg>
                       </funapp>
                      </arg>
                      <arg>
                       <var>z</var>
                      </arg>
                     </funapp>
                    </rhs>
                   </rule>
                   <rule>
                    <lhs>
                     <funapp>
                      <sharp>
                      <name>plus</name></sharp>
                      <arg>
                       <var>x</var>
                      </arg>
                      <arg>
                       <var>y</var>
                      </arg>
                     </funapp>
                    </lhs>
                    <rhs>
                     <funapp>
                      <sharp>
                      <name>plus</name></sharp>
                      <arg>
                       <var>y</var>
                      </arg>
                      <arg>
                       <var>x</var>
                      </arg>
                     </funapp>
                    </rhs>
                   </rule>
                  </rules>
                 </dps>
                 <realScc>true</realScc>
                 <acDPTerminationProof><acTrivialProc/>
                 </acDPTerminationProof>
                </component>
               </acDepGraphProc>
              </acDPTerminationProof>
             </acRedPairProc>
            </acDPTerminationProof>
           </component>
          </acDepGraphProc>
         </acDPTerminationProof>
        </acRedPairProc>
       </acDPTerminationProof>
      </component>
     </acDepGraphProc>
    </acDPTerminationProof>
   </acDependencyPairs>
  </acTerminationProof>
 </proof>
 <origin>
  <proofOrigin>
   <tool>
    <name>NaTT</name>
    <version>1.7</version>
   </tool>
  </proofOrigin>
 </origin>
</certificationProblem>