YES
<?xml version="1.0" encoding="UTF-8" standalone="no"?><?xml-stylesheet type="text/xsl" href="cpfHTML.xsl"?><certificationProblem xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="cpf.xsd"><input><acRewriteSystem><trs><rules><rule><lhs><funapp><name>union</name><arg><funapp><name>empty</name></funapp></arg><arg><var>X</var></arg></funapp></lhs><rhs><var>X</var></rhs></rule><rule><lhs><funapp><name>max</name><arg><funapp><name>singl</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>max</name><arg><funapp><name>union</name><arg><funapp><name>singl</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>singl</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>max</name><arg><funapp><name>union</name><arg><funapp><name>singl</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><funapp><name>singl</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>max</name><arg><funapp><name>union</name><arg><funapp><name>singl</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>singl</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>max</name><arg><funapp><name>union</name><arg><funapp><name>singl</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>union</name><arg><var>Y</var></arg><arg><var>Z</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>max</name><arg><funapp><name>union</name><arg><funapp><name>singl</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>singl</name><arg><funapp><name>max</name><arg><funapp><name>union</name><arg><var>Y</var></arg><arg><var>Z</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></trs><Asymbols><name>union</name></Asymbols><Csymbols><name>union</name></Csymbols></acRewriteSystem></input><cpfVersion>2.1</cpfVersion><proof><acTerminationProof><acRuleRemoval><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>union</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>empty</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>max</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>singl</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></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>s</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><trs><rules><rule><lhs><funapp><name>union</name><arg><funapp><name>empty</name></funapp></arg><arg><var>X</var></arg></funapp></lhs><rhs><var>X</var></rhs></rule></rules></trs><acTerminationProof><acRuleRemoval><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>union</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>max</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>singl</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>3</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><trs><rules><rule><lhs><funapp><name>max</name><arg><funapp><name>union</name><arg><funapp><name>singl</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>singl</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule></rules></trs><acTerminationProof><acRuleRemoval><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>2</degree></polynomial></type><interpret><name>union</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><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>max</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>singl</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><trs><rules><rule><lhs><funapp><name>max</name><arg><funapp><name>union</name><arg><funapp><name>singl</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><funapp><name>singl</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>max</name><arg><funapp><name>union</name><arg><funapp><name>singl</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>singl</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></trs><acTerminationProof><acDependencyPairs><equations><rules><rule><lhs><funapp><name>union</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>union</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>union</name><arg><funapp><name>union</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>union</name><arg><var>x</var></arg><arg><funapp><name>union</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule></rules></equations><dpEquations><rules/></dpEquations><dps><rules><rule><lhs><funapp><sharp><name>max</name></sharp><arg><funapp><name>union</name><arg><funapp><name>singl</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>union</name><arg><var>Y</var></arg><arg><var>Z</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>max</name></sharp><arg><funapp><name>union</name><arg><funapp><name>singl</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>singl</name><arg><funapp><name>max</name><arg><funapp><name>union</name><arg><var>Y</var></arg><arg><var>Z</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>max</name></sharp><arg><funapp><name>union</name><arg><funapp><name>singl</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>union</name><arg><var>Y</var></arg><arg><var>Z</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>max</name></sharp><arg><funapp><name>union</name><arg><var>Y</var></arg><arg><var>Z</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><extensions><rules/></extensions><acDPTerminationProof><acMonoRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>union</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>max</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>singl</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>max</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>max</name></sharp><arg><funapp><name>union</name><arg><funapp><name>singl</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>union</name><arg><var>Y</var></arg><arg><var>Z</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>max</name></sharp><arg><funapp><name>union</name><arg><var>Y</var></arg><arg><var>Z</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><trs><rules/></trs><usableRules><rules><rule><lhs><funapp><name>max</name><arg><funapp><name>singl</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>max</name><arg><funapp><name>union</name><arg><funapp><name>singl</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>union</name><arg><var>Y</var></arg><arg><var>Z</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>max</name><arg><funapp><name>union</name><arg><funapp><name>singl</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>singl</name><arg><funapp><name>max</name><arg><funapp><name>union</name><arg><var>Y</var></arg><arg><var>Z</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>union</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>union</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>union</name><arg><funapp><name>union</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>union</name><arg><var>x</var></arg><arg><funapp><name>union</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><acDPTerminationProof><acRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>max</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>union</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>singl</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>max</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>max</name></sharp><arg><funapp><name>union</name><arg><funapp><name>singl</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>union</name><arg><var>Y</var></arg><arg><var>Z</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>max</name></sharp><arg><funapp><name>union</name><arg><funapp><name>singl</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>singl</name><arg><funapp><name>max</name><arg><funapp><name>union</name><arg><var>Y</var></arg><arg><var>Z</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><usableRules><rules><rule><lhs><funapp><name>union</name><arg><funapp><name>union</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>union</name><arg><var>x</var></arg><arg><funapp><name>union</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>union</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>union</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule></rules></usableRules><acDPTerminationProof><acTrivialProc/></acDPTerminationProof></acRedPairProc></acDPTerminationProof></acMonoRedPairProc></acDPTerminationProof></acDependencyPairs></acTerminationProof></acRuleRemoval></acTerminationProof></acRuleRemoval></acTerminationProof></acRuleRemoval></acTerminationProof></proof><origin><proofOrigin><tool><name>AProVE</name><version>AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
</version><strategy>Statistics for single proof: 100.00 % (7 real / 0 unknown / 0 assumptions / 7 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>