details

property	value
status	complete
benchmark	AC22.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n145.star.cs.uiowa.edu
space	AProVE_AC_04

run statistics

property	value
solver	AProVE
configuration	certified
runtime (wallclock)	4.09744 seconds
cpu usage	9.20343
user time	8.87953
system time	0.323898
max virtual memory	1.8439456E7
max residence set size	539228.0

stage attributes

key	value
certification-result	CERTIFIED
starexec-result	CERTIFIED YES
certification-time	0.45
bare-result	YES

output

<?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>zero</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>plus</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>plus</name><arg><funapp><name>zero</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>zero</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>zero</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><funapp><name>zero</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>un</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><funapp><name>zero</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</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><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>zero</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><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>un</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>minus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>neg</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>neg</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>neg</name><arg><funapp><name>zero</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>zero</name><arg><funapp><name>neg</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>neg</name><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>neg</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>neg</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>un</name><arg><funapp><name>neg</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><funapp><name>0</name></funapp></arg><arg><var>z</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>0</name></funapp></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>zero</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>zero</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><funapp><name>zero</name><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>zero</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>zero</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>zero</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>zero</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><funapp><name>neg</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>plus</name><arg><funapp><name>zero</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><funapp><name>neg</name><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><var>z</var></arg></funapp></rhs></rule></rules></trs><Asymbols><name>plus</name><name>times</name></Asymbols><Csymbols><name>plus</name><name>times</name></Csymbols></acRewriteSystem></input><cpfVersion>2.1</cpfVersion><proof><acTerminationProof><acRuleRemoval><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>2</degree></polynomial></type><interpret><name>plus</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>times</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><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>zero</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>un</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><interpret><name>j</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><interpret><name>minus</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>neg</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>plus</name><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>zero</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>times</name><arg><var>x</var></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>zero</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>zero</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>zero</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><funapp><name>neg</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>plus</name><arg><funapp><name>zero</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><funapp><name>neg</name><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><var>z</var></arg></funapp></rhs></rule></rules></trs><acTerminationProof><acRuleRemoval><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>2</degree></polynomial></type><interpret><name>plus</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><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>times</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>zero</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>un</name><arity>1</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>j</name><arity>1</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><sum><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>3</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><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>neg</name><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><trs><rules><rule><lhs><funapp><name>minus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>neg</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>neg</name><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>neg</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>neg</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>un</name><arg><funapp><name>neg</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><acTerminationProof><acRuleRemoval><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>2</degree></polynomial></type><interpret><name>plus</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>times</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>zero</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><interpret><name>0</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>un</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>j</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>neg</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><trs><rules><rule><lhs><funapp><name>zero</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>zero</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>zero</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>zero</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><funapp><name>zero</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>un</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><funapp><name>zero</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</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>neg</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>neg</name><arg><funapp><name>zero</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>zero</name><arg><funapp><name>neg</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><funapp><name>0</name></funapp></arg><arg><var>z</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>0</name></funapp></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>zero</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>zero</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><funapp><name>zero</name><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>zero</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>z</var></arg></funapp></rhs></rule></rules></trs><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>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><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></lhs><rhs><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></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></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><funapp><name>plus</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>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>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></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>y</var></arg><arg><var>z</var></arg></funapp></rhs></rule></rules></dpEquations><dps><rules><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>un</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><extensions><rules><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>un</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule></rules></extensions><acDPTerminationProof><acMonoRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>plus</name></sharp><arity>2</arity><polynomial><sum><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>plus</name><arity>2</arity><polynomial><sum><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>times</name><arity>2</arity><polynomial><sum><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>0</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>un</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><interpret><name>j</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><dps><rules><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><trs><rules/></trs><usableRules><rules><rule><lhs><funapp><name>plus</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>plus</name><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>un</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>un</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></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>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><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></lhs><rhs><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></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</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>plus</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>plus</name><arity>2</arity><polynomial><sum><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>0</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>un</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>j</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>plus</name></sharp><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>ext</var></arg></funapp></rhs></rule></rules></dps><usableRules><rules><rule><lhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>un</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</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>plus</name><arg><funapp><name>plus</name><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>un</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><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></lhs><rhs><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></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></rules></usableRules><acDPTerminationProof><acRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>plus</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>plus</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>3</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>un</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>3</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>j</name><arity>1</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></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><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></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>y</var></arg><arg><var>z</var></arg></funapp></rhs></rule></rules></dps><usableRules><rules><rule><lhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>un</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</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>plus</name><arg><funapp><name>plus</name><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>un</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><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></lhs><rhs><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></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></rules></usableRules><acDPTerminationProof><acRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>plus</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>plus</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>j</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>un</name><arity>1</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></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>un</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule></rules></dps><usableRules><rules><rule><lhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>un</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</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>plus</name><arg><funapp><name>plus</name><arg><funapp><name>un</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>un</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>un</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>un</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><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></lhs><rhs><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></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></rules></usableRules><acDPTerminationProof><acTrivialProc/></acDPTerminationProof></acRedPairProc></acDPTerminationProof></acRedPairProc></acDPTerminationProof></acRedPairProc></acDPTerminationProof></acMonoRedPairProc></acDPTerminationProof></acDependencyPairs></acTerminationProof></acRuleRemoval></acTerminationProof></acRuleRemoval></acTerminationProof></acRuleRemoval></acTerminationProof></proof><origin><proofOrigin><tool><name>AProVE</name><version>AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty </version><strategy>Statistics for single proof: 100.00 % (9 real / 0 unknown / 0 assumptions / 9 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> popout

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