YES
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ient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><coefficient><integer>1</integer></coefficient></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret><interpret><name>if_min</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>-2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>le</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>true</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>false</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>if_replace</name><arity>4</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial><polynomial><variable>4</variable></polynomial></sum></polynomial></interpret><interpret><name>eq</name><arity>2</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules/></dps><usableRules><rules><rule><lhs><funapp><name>replace</name><arg><var>n</var></arg><arg><var>m</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>nil</name></funapp></rhs></rule><rule><lhs><funapp><name>replace</name><arg><var>n</var></arg><arg><var>m</var></arg><arg><funapp><name>cons</name><arg><var>k</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>if_replace</name><arg><funapp><name>eq</name><arg><var>n</var></arg><arg><var>k</var></arg></funapp></arg><arg><var>n</var></arg><arg><var>m</var></arg><arg><funapp><name>cons</name><arg><var>k</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_replace</name><arg><funapp><name>true</name></funapp></arg><arg><var>n</var></arg><arg><var>m</var></arg><arg><funapp><name>cons</name><arg><var>k</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_replace</name><arg><funapp><name>false</name></funapp></arg><arg><var>n</var></arg><arg><var>m</var></arg><arg><funapp><name>cons</name><arg><var>k</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>k</var></arg><arg><funapp><name>replace</name><arg><var>n</var></arg><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><pIsEmpty/></dpProof></redPairUrProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>sort</name></sharp><arg><funapp><name>cons</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>replace</name></sharp><arg><funapp><name>min</name><arg><funapp><name>cons</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>sort</name></sharp><arg><funapp><name>cons</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>min</name></sharp><arg><funapp><name>cons</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>replace</name></sharp><arg><var>n</var></arg><arg><var>m</var></arg><arg><funapp><name>cons</name><arg><var>k</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_replace</name></sharp><arg><funapp><name>eq</name><arg><var>n</var></arg><arg><var>k</var></arg></funapp></arg><arg><var>n</var></arg><arg><var>m</var></arg><arg><funapp><name>cons</name><arg><var>k</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_replace</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>n</var></arg><arg><var>m</var></arg><arg><funapp><name>cons</name><arg><var>k</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>replace</name></sharp><arg><var>n</var></arg><arg><var>m</var></arg><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>eq</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>eq</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>m</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule><rule><lhs><funapp><name>eq</name><arg><funapp><name>s</name><arg><var>n</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule><rule><lhs><funapp><name>eq</name><arg><funapp><name>s</name><arg><var>n</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>m</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>eq</name><arg><var>n</var></arg><arg><var>m</var></arg></funapp></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>eq</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>eq</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp><funapp><name>eq</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>eq</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x1</var></arg></funapp></arg></funapp></innermostLhss><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>if_replace</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>n</var></arg><arg><var>m</var></arg><arg><funapp><name>cons</name><arg><var>k</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>replace</name></sharp><arg><var>n</var></arg><arg><var>m</var></arg><arg><var>x</var></arg></funapp></rhs></rule><edge><position>2</position><strict>false</strict><position>1</position></edge><edge><position>3</position><strict>false</strict><position>2</position></edge><edge><position>4</position><strict>true</strict><position>3</position></edge></sizeChangeGraph><sizeChangeGraph><rule><lhs><funapp><sharp><name>replace</name></sharp><arg><var>n</var></arg><arg><var>m</var></arg><arg><funapp><name>cons</name><arg><var>k</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_replace</name></sharp><arg><funapp><name>eq</name><arg><var>n</var></arg><arg><var>k</var></arg></funapp></arg><arg><var>n</var></arg><arg><var>m</var></arg><arg><funapp><name>cons</name><arg><var>k</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><edge><position>1</position><strict>false</strict><position>2</position></edge><edge><position>2</position><strict>false</strict><position>3</position></edge><edge><position>3</position><strict>false</strict><position>4</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>min</name></sharp><arg><funapp><name>cons</name><arg><var>n</var></arg><arg><funapp><name>cons</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>le</name><arg><var>n</var></arg><arg><var>m</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>n</var></arg><arg><funapp><name>cons</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>cons</name><arg><var>n</var></arg><arg><funapp><name>cons</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>min</name></sharp><arg><funapp><name>cons</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>n</var></arg><arg><funapp><name>cons</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>min</name></sharp><arg><funapp><name>cons</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>le</name><arg><funapp><name>0</name></funapp></arg><arg><var>m</var></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>le</name><arg><funapp><name>s</name><arg><var>n</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule><rule><lhs><funapp><name>le</name><arg><funapp><name>s</name><arg><var>n</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>m</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>le</name><arg><var>n</var></arg><arg><var>m</var></arg></funapp></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>le</name><arg><funapp><name>0</name></funapp></arg><arg><var>x0</var></arg></funapp><funapp><name>le</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>le</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x1</var></arg></funapp></arg></funapp></innermostLhss><dpProof><redPairUrProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>if_min</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>le</name><arity>2</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>true</name><arity>0</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>false</name><arity>0</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret><interpret><sharp><name>min</name></sharp><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>cons</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>min</name></sharp><arg><funapp><name>cons</name><arg><var>n</var></arg><arg><funapp><name>cons</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>le</name><arg><var>n</var></arg><arg><var>m</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>n</var></arg><arg><funapp><name>cons</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><usableRules><rules/></usableRules><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>min</name></sharp><arg><funapp><name>cons</name><arg><var>n</var></arg><arg><funapp><name>cons</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>le</name><arg><var>n</var></arg><arg><var>m</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>n</var></arg><arg><funapp><name>cons</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><edge><position>1</position><strict>false</strict><position>2</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></redPairUrProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>replace</name></sharp><arg><var>n</var></arg><arg><var>m</var></arg><arg><funapp><name>cons</name><arg><var>k</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>eq</name></sharp><arg><var>n</var></arg><arg><var>k</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>min</name></sharp><arg><funapp><name>cons</name><arg><var>n</var></arg><arg><funapp><name>cons</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>le</name></sharp><arg><var>n</var></arg><arg><var>m</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>eq</name></sharp><arg><funapp><name>s</name><arg><var>n</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>m</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>eq</name></sharp><arg><var>n</var></arg><arg><var>m</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules/></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss/><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>eq</name></sharp><arg><funapp><name>s</name><arg><var>n</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>m</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>eq</name></sharp><arg><var>n</var></arg><arg><var>m</var></arg></funapp></rhs></rule><edge><position>1</position><strict>true</strict><position>1</position></edge><edge><position>2</position><strict>true</strict><position>2</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>le</name></sharp><arg><funapp><name>s</name><arg><var>n</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>m</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>le</name></sharp><arg><var>n</var></arg><arg><var>m</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules/></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss/><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>le</name></sharp><arg><funapp><name>s</name><arg><var>n</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>m</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>le</name></sharp><arg><var>n</var></arg><arg><var>m</var></arg></funapp></rhs></rule><edge><position>1</position><strict>true</strict><position>1</position></edge><edge><position>2</position><strict>true</strict><position>2</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component></depGraphProc></dpProof></dpTrans></trsTerminationProof></proof><origin><proofOrigin><tool><name>AProVE</name><version>AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
</version><strategy>Statistics for single proof: 100.00 % (19 real / 0 unknown / 0 assumptions / 19 total proof steps)</strategy><url>http://aprove.informatik.rwth-aachen.de</url></tool><toolUser><firstName>John</firstName><lastName>Doe</lastName></toolUser></proofOrigin><inputOrigin/></origin></certificationProblem>