<?xml version="1.0"?>
<?xml-stylesheet type="text/xsl" href="cpf3HTML.xsl"?><certificationProblem xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="cpf3.xsd">
  <cpfVersion>3.0</cpfVersion>
  <lookupTables>
    <ruleTable>
      <indexToRule>
        <index>1</index>
        <rule>
          <funapp>
            <name>f</name>
            <var>y</var>
            <funapp>
              <name>f</name>
              <funapp>
                <name>g</name>
                <var>x</var>
              </funapp>
              <funapp><name>b</name></funapp>
            </funapp>
          </funapp>
          <funapp>
            <name>g</name>
            <funapp>
              <name>f</name>
              <funapp>
                <name>f</name>
                <funapp>
                  <name>g</name>
                  <funapp>
                    <name>f</name>
                    <funapp><name>a</name></funapp>
                    <funapp><name>b</name></funapp>
                  </funapp>
                </funapp>
                <var>y</var>
              </funapp>
              <var>x</var>
            </funapp>
          </funapp>          
        </rule>
      </indexToRule>
    </ruleTable>
  </lookupTables>
  <input>
    <trsInput>
      <trs>
        <rules>
          <ruleIndex>1</ruleIndex>
        </rules>
      </trs>
    </trsInput>
  </input>
  <property>
    <termination/>
  </property>
  <answer>
    <yes/>
  </answer>
  <proof>
    <trsTerminationProof>
      <ruleRemoval>
        <interpretation>
          <type>
            <coreMatrixInterpretation>
              <domain>
                <integers/>
              </domain>
              <dimension>2</dimension>
              <indices>
                <nat>1</nat>
                <nat>2</nat>
              </indices>
              <mode>
                <M/>
              </mode>
            </coreMatrixInterpretation>
          </type>
          <interpret>
            <name>a</name>
            <arity>0</arity>
            <matrix>
              <vector>
                <integer>0</integer>
                <integer>1</integer>
              </vector>
              <vector>
                <integer>0</integer>
                <integer>1</integer>
              </vector>
            </matrix>
          </interpret>
          <interpret>
            <name>b</name>
            <arity>0</arity>
            <matrix>
              <vector>
                <integer>1</integer>
                <integer>0</integer>
              </vector>
              <vector>
                <integer>5</integer>
                <integer>4</integer>
              </vector>
            </matrix>
          </interpret>
          <interpret>
            <name>g</name>
            <arity>1</arity>
            <sum>
              <matrix>
                <vector>
                  <integer>0</integer>
                  <integer>0</integer>
                </vector>
                <vector>
                  <integer>1</integer>
                  <integer>2</integer>
                </vector>
              </matrix>              
              <product>
                <matrix>
                  <vector>
                    <integer>1</integer>
                    <integer>0</integer>
                  </vector>
                  <vector>
                    <integer>1</integer>
                    <integer>0</integer>
                  </vector>
                </matrix>
                <variable>1</variable>
              </product>              
            </sum>
          </interpret>
          <interpret>
            <name>f</name>
            <arity>2</arity>
            <sum>
              <product>
                <matrix>
                  <vector>
                    <integer>0</integer>
                    <integer>1</integer>
                  </vector>
                  <vector>
                    <integer>1</integer>
                    <integer>1</integer>
                  </vector>
                </matrix>
                <variable>1</variable>
              </product>              
              <product>
                <matrix>
                  <vector>
                    <integer>2</integer>
                    <integer>0</integer>
                  </vector>
                  <vector>
                    <integer>0</integer>
                    <integer>1</integer>
                  </vector>
                </matrix>
                <variable>2</variable>
              </product>              
            </sum>
          </interpret>
        </interpretation>
        <trs>
          <rules>
            <ruleIndex>1</ruleIndex>
          </rules>
        </trs>
        <trsTerminationProof>
          <rIsEmpty/>
        </trsTerminationProof>
      </ruleRemoval>
    </trsTerminationProof>
  </proof>
  <metaInformation>
    <inputInfos>
      <inputInfo>Example 23</inputInfo>      
    </inputInfos>
  </metaInformation>
</certificationProblem>
