details

property	value
status	complete
benchmark	#3.17a_mset.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n126.star.cs.uiowa.edu
space	INVY_15

run statistics

property	value
solver	AProVE
configuration	certified
runtime (wallclock)	3.40218 seconds
cpu usage	9.81935
user time	9.34483
system time	0.474521
max virtual memory	1.8794752E7
max residence set size	1179176.0

stage attributes

key	value
certification-result	CERTIFIED
output-size	44956
starexec-result	YES
certification-time	0.1

output

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><trsInput><trs><rules><rule><lhs><funapp><name>app</name><arg><funapp><name>nil</name></funapp></arg><arg><var>k</var></arg></funapp></lhs><rhs><var>k</var></rhs></rule><rule><lhs><funapp><name>app</name><arg><var>l</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><var>l</var></rhs></rule><rule><lhs><funapp><name>app</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg><arg><var>k</var></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>app</name><arg><var>l</var></arg><arg><var>k</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>sum</name><arg><funapp><name>cons</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>l</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>app</name><arg><var>l</var></arg><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>k</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>sum</name><arg><funapp><name>app</name><arg><var>l</var></arg><arg><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>k</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>0</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</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>sum</name><arg><funapp><name>plus</name><arg><funapp><name>cons</name><arg><funapp><name>0</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>pred</name><arg><funapp><name>sum</name><arg><funapp><name>cons</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>pred</name><arg><funapp><name>cons</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></rhs></rule></rules></trs><relativeRules><rules><rule><lhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>y</var></arg><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg></funapp></rhs></rule></rules></relativeRules></trsInput></input><cpfVersion>2.1</cpfVersion><proof><relativeTerminationProof><ruleRemoval><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>app</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>nil</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>cons</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>sum</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>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>0</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></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><interpret><name>pred</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>app</name><arg><funapp><name>nil</name></funapp></arg><arg><var>k</var></arg></funapp></lhs><rhs><var>k</var></rhs></rule><rule><lhs><funapp><name>app</name><arg><var>l</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><var>l</var></rhs></rule><rule><lhs><funapp><name>app</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg><arg><var>k</var></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>app</name><arg><var>l</var></arg><arg><var>k</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>sum</name><arg><funapp><name>cons</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>l</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>app</name><arg><var>l</var></arg><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>k</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>sum</name><arg><funapp><name>app</name><arg><var>l</var></arg><arg><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>k</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</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>sum</name><arg><funapp><name>plus</name><arg><funapp><name>cons</name><arg><funapp><name>0</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>pred</name><arg><funapp><name>sum</name><arg><funapp><name>cons</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></trs><trs><rules><rule><lhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>y</var></arg><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><relativeTerminationProof><ruleRemoval><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>app</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>nil</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>cons</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>sum</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>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>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><interpret><name>0</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>pred</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>app</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg><arg><var>k</var></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>app</name><arg><var>l</var></arg><arg><var>k</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>sum</name><arg><funapp><name>cons</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>l</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>app</name><arg><var>l</var></arg><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>k</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>sum</name><arg><funapp><name>app</name><arg><var>l</var></arg><arg><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>k</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</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>sum</name><arg><funapp><name>plus</name><arg><funapp><name>cons</name><arg><funapp><name>0</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>pred</name><arg><funapp><name>sum</name><arg><funapp><name>cons</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></trs><trs><rules><rule><lhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>y</var></arg><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><relativeTerminationProof><ruleRemoval><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>app</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>cons</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>sum</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>nil</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></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>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><interpret><name>0</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>pred</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>app</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg><arg><var>k</var></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>app</name><arg><var>l</var></arg><arg><var>k</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>sum</name><arg><funapp><name>cons</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>l</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>app</name><arg><var>l</var></arg><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>k</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>sum</name><arg><funapp><name>app</name><arg><var>l</var></arg><arg><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>k</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><trs><rules><rule><lhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>y</var></arg><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><relativeTerminationProof><ruleRemoval><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>app</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>cons</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>sum</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>nil</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></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>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>app</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg><arg><var>k</var></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>app</name><arg><var>l</var></arg><arg><var>k</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>app</name><arg><var>l</var></arg><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>k</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>sum</name><arg><funapp><name>app</name><arg><var>l</var></arg><arg><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>k</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><trs><rules><rule><lhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>y</var></arg><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><relativeTerminationProof><ruleRemoval><orderingConstraintProof><redPair><interpretation><type><matrixInterpretation><domain><integers/></domain><dimension>2</dimension><strictDimension>1</strictDimension></matrixInterpretation></type><interpret><name>app</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>cons</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>sum</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>plus</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>1</integer></coefficient></vector><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>1</integer></coefficient></vector><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><trs><rules><rule><lhs><funapp><name>app</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg><arg><var>k</var></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>app</name><arg><var>l</var></arg><arg><var>k</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>app</name><arg><var>l</var></arg><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>k</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>sum</name><arg><funapp><name>app</name><arg><var>l</var></arg><arg><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>k</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></trs><trs><rules><rule><lhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>y</var></arg><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><relativeTerminationProof><ruleRemoval><orderingConstraintProof><redPair><interpretation><type><matrixInterpretation><domain><integers/></domain><dimension>2</dimension><strictDimension>1</strictDimension></matrixInterpretation></type><interpret><name>app</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>cons</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>sum</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><trs><rules><rule><lhs><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><funapp><name>app</name><arg><var>l</var></arg><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>k</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>sum</name><arg><funapp><name>app</name><arg><var>l</var></arg><arg><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>k</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></trs><trs><rules><rule><lhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>y</var></arg><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><relativeTerminationProof><ruleRemoval><orderingConstraintProof><redPair><interpretation><type><matrixInterpretation><domain><integers/></domain><dimension>2</dimension><strictDimension>1</strictDimension></matrixInterpretation></type><interpret><name>sum</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>cons</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>1</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>app</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><trs><rules><rule><lhs><funapp><name>sum</name><arg><funapp><name>app</name><arg><var>l</var></arg><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>k</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>sum</name><arg><funapp><name>app</name><arg><var>l</var></arg><arg><funapp><name>sum</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>k</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></trs><trs><rules><rule><lhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>y</var></arg><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><relativeTerminationProof><ruleRemoval><orderingConstraintProof><redPair><interpretation><type><matrixInterpretation><domain><integers/></domain><dimension>2</dimension><strictDimension>1</strictDimension></matrixInterpretation></type><interpret><name>sum</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>app</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>cons</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>1</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><trs><rules/></trs><trs><rules><rule><lhs><funapp><name>cons</name><arg><var>x</var></arg><arg><funapp><name>cons</name><arg><var>y</var></arg><arg><var>l</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>cons</name><arg><var>y</var></arg><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><relativeTerminationProof><rIsEmpty/></relativeTerminationProof></ruleRemoval></relativeTerminationProof></ruleRemoval></relativeTerminationProof></ruleRemoval></relativeTerminationProof></ruleRemoval></relativeTerminationProof></ruleRemoval></relativeTerminationProof></ruleRemoval></relativeTerminationProof></ruleRemoval></relativeTerminationProof></ruleRemoval></relativeTerminationProof></proof><origin><proofOrigin><tool><name>AProVE</name><version>AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 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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