<?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">
  <input>
    <trsInput>
      <trs>
        <rules>
          <rule>
            <lhs>
              <funapp>
                <name>id</name>
                <arg>
                  <var>x</var>
                </arg>
              </funapp>
            </lhs>
            <rhs>
              <var>x</var>
            </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>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>
          <rule>
            <lhs>
              <funapp>
                <name>div</name>
                <arg>
                  <funapp>
                    <name>0</name>
                  </funapp>
                </arg>
                <arg>
                  <funapp>
                    <name>s</name>
                    <arg>
                      <var>y</var>
                    </arg>
                  </funapp>
                </arg>
              </funapp>
            </lhs>
            <rhs>
              <funapp>
                <name>0</name>
              </funapp>
            </rhs>
          </rule>
          <rule>
            <lhs>
              <funapp>
                <name>div</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>s</name>
                <arg>
                  <funapp>
                    <name>div</name>
                    <arg>
                      <funapp>
                        <name>minus</name>
                        <arg>
                          <var>x</var>
                        </arg>
                        <arg>
                          <var>y</var>
                        </arg>
                      </funapp>
                    </arg>
                    <arg>
                      <funapp>
                        <name>id</name>
                        <arg>
                          <funapp>
                            <name>s</name>
                            <arg>
                              <var>y</var>
                            </arg>
                          </funapp>
                        </arg>
                      </funapp>
                    </arg>
                  </funapp>
                </arg>
              </funapp>
            </rhs>
          </rule>
        </rules>
      </trs>
    </trsInput>
  </input>
  <cpfVersion>2.0</cpfVersion>
  <proof>
    <trsTerminationProof>
      <dpTrans>
        <dps>
          <rules>
            <rule>
              <lhs>
                <funapp>
                  <sharp>
                    <name>div</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>div</name>
                  </sharp>
                  <arg>
                    <funapp>
                      <name>minus</name>
                      <arg>
                        <var>x</var>
                      </arg>
                      <arg>
                        <var>y</var>
                      </arg>
                    </funapp>
                  </arg>
                  <arg>
                    <funapp>
                      <name>id</name>
                      <arg>
                        <funapp>
                          <name>s</name>
                          <arg>
                            <var>y</var>
                          </arg>
                        </funapp>
                      </arg>
                    </funapp>
                  </arg>
                </funapp>
              </rhs>
            </rule>
            <rule>
              <lhs>
                <funapp>
                  <sharp>
                    <name>div</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>div</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>id</name>
                  </sharp>
                  <arg>
                    <funapp>
                      <name>s</name>
                      <arg>
                        <var>y</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>
        <markedSymbols>true</markedSymbols>
        <dpProof>
          <depGraphProc>
            <component>
              <dps>
                <rules>
                  <rule>
                    <lhs>
                      <funapp>
                        <sharp>
                          <name>div</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>div</name>
                        </sharp>
                        <arg>
                          <funapp>
                            <name>minus</name>
                            <arg>
                              <var>x</var>
                            </arg>
                            <arg>
                              <var>y</var>
                            </arg>
                          </funapp>
                        </arg>
                        <arg>
                          <funapp>
                            <name>id</name>
                            <arg>
                              <funapp>
                                <name>s</name>
                                <arg>
                                  <var>y</var>
                                </arg>
                              </funapp>
                            </arg>
                          </funapp>
                        </arg>
                      </funapp>
                    </rhs>
                  </rule>
                </rules>
              </dps>
              <realScc>true</realScc>
              <dpProof>
                <redPairUrProc>
                  <orderingConstraintProof>
                    <redPair>
                      <interpretation>
                        <type>
                          <polynomial>
                            <domain>
                              <naturals/>
                            </domain>
                            <degree>1</degree>
                          </polynomial>
                        </type>
                        <interpret>
                          <sharp>
                            <name>div</name>
                          </sharp>
                          <arity>2</arity>
                          <polynomial>
                            <sum>
                              <polynomial>
                                <coefficient>
                                  <integer>1</integer>
                                </coefficient>
                              </polynomial>
                              <polynomial>
                                <product>
                                  <polynomial>
                                    <coefficient>
                                      <integer>2</integer>
                                    </coefficient>
                                  </polynomial>
                                  <polynomial>
                                    <variable>1</variable>
                                  </polynomial>
                                </product>
                              </polynomial>
                            </sum>
                          </polynomial>
                        </interpret>
                        <interpret>
                          <name>minus</name>
                          <arity>2</arity>
                          <polynomial>
                            <variable>1</variable>
                          </polynomial>
                        </interpret>
                        <interpret>
                          <name>s</name>
                          <arity>1</arity>
                          <polynomial>
                            <sum>
                              <polynomial>
                                <coefficient>
                                  <integer>1</integer>
                                </coefficient>
                              </polynomial>
                              <polynomial>
                                <variable>1</variable>
                              </polynomial>
                            </sum>
                          </polynomial>
                        </interpret>
                      </interpretation>
                    </redPair>
                  </orderingConstraintProof>
                  <dps>
                    <rules/>
                  </dps>
                  <usableRules>
                    <rules>
                      <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>
                      <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>
                    </rules>
                  </usableRules>
                  <dpProof>
                    <pIsEmpty/>
                  </dpProof>
                </redPairUrProc>
              </dpProof>
            </component>
            <component>
              <dps>
                <rules>
                  <rule>
                    <lhs>
                      <funapp>
                        <sharp>
                          <name>div</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>div</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>id</name>
                        </sharp>
                        <arg>
                          <funapp>
                            <name>s</name>
                            <arg>
                              <var>y</var>
                            </arg>
                          </funapp>
                        </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>
                </rules>
              </dps>
              <realScc>true</realScc>
              <dpProof>
                <monoRedPairUrProc>
                  <orderingConstraintProof>
                    <redPair>
                      <interpretation>
                        <type>
                          <polynomial>
                            <domain>
                              <naturals/>
                            </domain>
                            <degree>1</degree>
                          </polynomial>
                        </type>
                        <interpret>
                          <sharp>
                            <name>minus</name>
                          </sharp>
                          <arity>2</arity>
                          <polynomial>
                            <sum>
                              <polynomial>
                                <variable>1</variable>
                              </polynomial>
                              <polynomial>
                                <variable>2</variable>
                              </polynomial>
                            </sum>
                          </polynomial>
                        </interpret>
                        <interpret>
                          <name>minus</name>
                          <arity>2</arity>
                          <polynomial>
                            <sum>
                              <polynomial>
                                <variable>1</variable>
                              </polynomial>
                              <polynomial>
                                <variable>2</variable>
                              </polynomial>
                            </sum>
                          </polynomial>
                        </interpret>
                        <interpret>
                          <name>s</name>
                          <arity>1</arity>
                          <polynomial>
                            <sum>
                              <polynomial>
                                <coefficient>
                                  <integer>1</integer>
                                </coefficient>
                              </polynomial>
                              <polynomial>
                                <variable>1</variable>
                              </polynomial>
                            </sum>
                          </polynomial>
                        </interpret>
                      </interpretation>
                    </redPair>
                  </orderingConstraintProof>
                  <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>
                    </rules>
                  </dps>
                  <trs>
                    <rules/>
                  </trs>
                  <usableRules>
                    <rules/>
                  </usableRules>
                  <dpProof>
                    <redPairProc>
                      <orderingConstraintProof>
                        <redPair>
                          <interpretation>
                            <type>
                              <polynomial>
                                <domain>
                                  <naturals/>
                                </domain>
                                <degree>1</degree>
                              </polynomial>
                            </type>
                            <interpret>
                              <sharp>
                                <name>minus</name>
                              </sharp>
                              <arity>2</arity>
                              <polynomial>
                                <sum>
                                  <polynomial>
                                    <variable>1</variable>
                                  </polynomial>
                                  <polynomial>
                                    <variable>2</variable>
                                  </polynomial>
                                </sum>
                              </polynomial>
                            </interpret>
                            <interpret>
                              <name>s</name>
                              <arity>1</arity>
                              <polynomial>
                                <sum>
                                  <polynomial>
                                    <coefficient>
                                      <integer>1</integer>
                                    </coefficient>
                                  </polynomial>
                                  <polynomial>
                                    <variable>1</variable>
                                  </polynomial>
                                </sum>
                              </polynomial>
                            </interpret>
                          </interpretation>
                        </redPair>
                      </orderingConstraintProof>
                      <dps>
                        <rules/>
                      </dps>
                      <dpProof>
                        <pIsEmpty/>
                      </dpProof>
                    </redPairProc>
                  </dpProof>
                </monoRedPairUrProc>
              </dpProof>
            </component>
          </depGraphProc>
        </dpProof>
      </dpTrans>
    </trsTerminationProof>
  </proof>
</certificationProblem>

