details

property	value
status	complete
benchmark	sumList.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n146.star.cs.uiowa.edu
space	Secret_06_TRS

run statistics

property	value
solver	AProVE
configuration	certified
runtime (wallclock)	3.46813 seconds
cpu usage	7.81775
user time	7.53295
system time	0.284795
max virtual memory	1.8598092E7
max residence set size	424116.0

stage attributes

key	value
certification-result	CERTIFIED
starexec-result	CERTIFIED YES
certification-time	0.49
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><trsInput><trs><rules><rule><lhs><funapp><name>isEmpty</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule><rule><lhs><funapp><name>isEmpty</name><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>isZero</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>isZero</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule><rule><lhs><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><var>xs</var></rhs></rule><rule><lhs><funapp><name>tail</name><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>nil</name></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>inc</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>inc</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>inc</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>sumList</name><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if</name><arg><funapp><name>isEmpty</name><arg><var>xs</var></arg></funapp></arg><arg><funapp><name>isZero</name><arg><funapp><name>head</name><arg><var>xs</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg><arg><funapp><name>tail</name><arg><var>xs</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>head</name><arg><var>xs</var></arg></funapp></arg></funapp></arg><arg><funapp><name>tail</name><arg><var>xs</var></arg></funapp></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if</name><arg><funapp><name>true</name></funapp></arg><arg><var>b</var></arg><arg><var>y</var></arg><arg><var>xs</var></arg><arg><var>ys</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>if</name><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>true</name></funapp></arg><arg><var>y</var></arg><arg><var>xs</var></arg><arg><var>ys</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>sumList</name><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>if</name><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><var>y</var></arg><arg><var>xs</var></arg><arg><var>ys</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>sumList</name><arg><var>ys</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>sum</name><arg><var>xs</var></arg></funapp></lhs><rhs><funapp><name>sumList</name><arg><var>xs</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule></rules></trs></trsInput></input><cpfVersion>2.1</cpfVersion><proof><trsTerminationProof><switchInnermost><wcrProof><joinableCriticalPairsBFS>10</joinableCriticalPairsBFS></wcrProof><trsTerminationProof><dpTrans><dps><rules><rule><lhs><funapp><sharp><name>p</name></sharp><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>p</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>inc</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>inc</name></sharp><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>isEmpty</name><arg><var>xs</var></arg></funapp></arg><arg><funapp><name>isZero</name><arg><funapp><name>head</name><arg><var>xs</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg><arg><funapp><name>tail</name><arg><var>xs</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>head</name><arg><var>xs</var></arg></funapp></arg></funapp></arg><arg><funapp><name>tail</name><arg><var>xs</var></arg></funapp></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>isEmpty</name></sharp><arg><var>xs</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>isZero</name></sharp><arg><funapp><name>head</name><arg><var>xs</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>head</name></sharp><arg><var>xs</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>tail</name></sharp><arg><var>xs</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>p</name></sharp><arg><funapp><name>head</name><arg><var>xs</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>inc</name></sharp><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>true</name></funapp></arg><arg><var>y</var></arg><arg><var>xs</var></arg><arg><var>ys</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><var>y</var></arg><arg><var>xs</var></arg><arg><var>ys</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>sumList</name></sharp><arg><var>ys</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>sum</name></sharp><arg><var>xs</var></arg></funapp></lhs><rhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule></rules></dps><markedSymbols>true</markedSymbols><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>sum</name></sharp><arg><var>xs</var></arg></funapp></lhs><rhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>isEmpty</name><arg><var>xs</var></arg></funapp></arg><arg><funapp><name>isZero</name><arg><funapp><name>head</name><arg><var>xs</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg><arg><funapp><name>tail</name><arg><var>xs</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>head</name><arg><var>xs</var></arg></funapp></arg></funapp></arg><arg><funapp><name>tail</name><arg><var>xs</var></arg></funapp></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>true</name></funapp></arg><arg><var>y</var></arg><arg><var>xs</var></arg><arg><var>ys</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><var>y</var></arg><arg><var>xs</var></arg><arg><var>ys</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>sumList</name></sharp><arg><var>ys</var></arg><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>isEmpty</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule><rule><lhs><funapp><name>isEmpty</name><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>isZero</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>isZero</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule><rule><lhs><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><var>xs</var></rhs></rule><rule><lhs><funapp><name>tail</name><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>nil</name></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>inc</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>inc</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>inc</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>isEmpty</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp><funapp><name>isEmpty</name><arg><funapp><name>nil</name></funapp></arg></funapp><funapp><name>isZero</name><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>isZero</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp><funapp><name>tail</name><arg><funapp><name>nil</name></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>inc</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp><funapp><name>inc</name><arg><funapp><name>0</name></funapp></arg></funapp></innermostLhss><dpProof><narrowingProc><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>isEmpty</name><arg><var>xs</var></arg></funapp></arg><arg><funapp><name>isZero</name><arg><funapp><name>head</name><arg><var>xs</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg><arg><funapp><name>tail</name><arg><var>xs</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>head</name><arg><var>xs</var></arg></funapp></arg></funapp></arg><arg><funapp><name>tail</name><arg><var>xs</var></arg></funapp></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><positionInTerm><position>1</position></positionInTerm><narrowings><rules><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>isZero</name><arg><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>y1</var></arg><arg><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>nil</name></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>isZero</name><arg><funapp><name>head</name><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></arg><arg><var>y1</var></arg><arg><funapp><name>tail</name><arg><funapp><name>nil</name></funapp></arg></funapp></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>head</name><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>tail</name><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule></rules></narrowings><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>isZero</name><arg><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>y1</var></arg><arg><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>true</name></funapp></arg><arg><var>y</var></arg><arg><var>xs</var></arg><arg><var>ys</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><var>y</var></arg><arg><var>xs</var></arg><arg><var>ys</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>sumList</name></sharp><arg><var>ys</var></arg><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>isZero</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>isZero</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule><rule><lhs><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><var>xs</var></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>inc</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>inc</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>inc</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>isZero</name><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>isZero</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp><funapp><name>tail</name><arg><funapp><name>nil</name></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>inc</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp><funapp><name>inc</name><arg><funapp><name>0</name></funapp></arg></funapp></innermostLhss><dpProof><rewritingProc><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>isZero</name><arg><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>y1</var></arg><arg><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule><rewriteStep><positionInTerm><position>2</position><position>1</position></positionInTerm><rule><lhs><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg><arg><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y1</var></arg></funapp></arg></funapp></rewriteStep><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg><arg><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule><usableRules><rules><rule><lhs><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule></rules></usableRules><dpProof><rewritingProc><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg><arg><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule><rewriteStep><positionInTerm><position>4</position></positionInTerm><rule><lhs><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><var>xs</var></rhs></rule><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg><arg><var>x1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y1</var></arg></funapp></arg></funapp></rewriteStep><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg><arg><var>x1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule><usableRules><rules><rule><lhs><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><var>xs</var></rhs></rule></rules></usableRules><dpProof><rewritingProc><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg><arg><var>x1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule><rewriteStep><positionInTerm><position>5</position><position>1</position><position>1</position></positionInTerm><rule><lhs><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg><arg><var>x1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><var>x0</var></arg></funapp></arg><arg><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y1</var></arg></funapp></arg></funapp></rewriteStep><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg><arg><var>x1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><var>x0</var></arg></funapp></arg><arg><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule><usableRules><rules><rule><lhs><funapp><name>head</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule></rules></usableRules><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>isZero</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>isZero</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><var>xs</var></rhs></rule><rule><lhs><funapp><name>inc</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>inc</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>inc</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>isZero</name><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>isZero</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp><funapp><name>tail</name><arg><funapp><name>nil</name></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>inc</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp><funapp><name>inc</name><arg><funapp><name>0</name></funapp></arg></funapp></innermostLhss><dpProof><rewritingProc><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg><arg><var>x1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><var>x0</var></arg></funapp></arg><arg><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule><rewriteStep><positionInTerm><position>5</position><position>2</position></positionInTerm><rule><lhs><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><var>xs</var></rhs></rule><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg><arg><var>x1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><var>x0</var></arg></funapp></arg><arg><var>x1</var></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y1</var></arg></funapp></arg></funapp></rewriteStep><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg><arg><var>x1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><var>x0</var></arg></funapp></arg><arg><var>x1</var></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule><usableRules><rules><rule><lhs><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><var>xs</var></rhs></rule></rules></usableRules><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>isZero</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>isZero</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>inc</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>inc</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>inc</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>isZero</name><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>isZero</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>inc</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp><funapp><name>inc</name><arg><funapp><name>0</name></funapp></arg></funapp></innermostLhss><dpProof><narrowingProc><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg><arg><var>x1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><var>x0</var></arg></funapp></arg><arg><var>x1</var></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule><positionInTerm><position>2</position></positionInTerm><narrowings><rules><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>0</name></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><var>y2</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>true</name></funapp></arg><arg><var>y2</var></arg><arg><var>y1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><var>y2</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><var>y2</var></arg><arg><var>y1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y2</var></arg></funapp></arg></funapp></rhs></rule></rules></narrowings><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>inc</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>inc</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>inc</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>inc</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp><funapp><name>inc</name><arg><funapp><name>0</name></funapp></arg></funapp></innermostLhss><dpProof><rewritingProc><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>0</name></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><var>y2</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>true</name></funapp></arg><arg><var>y2</var></arg><arg><var>y1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y2</var></arg></funapp></arg></funapp></rhs></rule><rewriteStep><positionInTerm><position>5</position><position>1</position></positionInTerm><rule><lhs><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>true</name></funapp></arg><arg><var>y2</var></arg><arg><var>y1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>0</name></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y2</var></arg></funapp></arg></funapp></rewriteStep><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>0</name></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><var>y2</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>true</name></funapp></arg><arg><var>y2</var></arg><arg><var>y1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>0</name></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y2</var></arg></funapp></arg></funapp></rhs></rule><usableRules><rules><rule><lhs><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule></rules></usableRules><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>inc</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>inc</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>inc</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule></rules></usableRules><dpProof><instantiationProc><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>true</name></funapp></arg><arg><var>y</var></arg><arg><var>xs</var></arg><arg><var>ys</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></rhs></rule><instantiations><rules><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>true</name></funapp></arg><arg><var>z1</var></arg><arg><var>z0</var></arg><arg><funapp><name>cons</name><arg><funapp><name>0</name></funapp></arg><arg><var>z0</var></arg></funapp></arg><arg><var>y_0</var></arg></funapp></lhs><rhs><funapp><sharp><name>sumList</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></rhs></rule></rules></instantiations><dpProof><redPairUrProc><orderingConstraintProof><redPair><knuthBendixOrder><w0>1</w0><precedenceWeight><precedenceWeightEntry><name>cons</name><arity>2</arity><precedence>1</precedence><weight>1</weight></precedenceWeightEntry></precedenceWeight><argumentFilter><argumentFilterEntry><sharp><name>if</name></sharp><arity>6</arity><collapsing>5</collapsing></argumentFilterEntry><argumentFilterEntry><sharp><name>sumList</name></sharp><arity>2</arity><collapsing>1</collapsing></argumentFilterEntry><argumentFilterEntry><name>cons</name><arity>2</arity><nonCollapsing><position>2</position></nonCollapsing></argumentFilterEntry></argumentFilter></knuthBendixOrder></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><var>y</var></arg><arg><var>xs</var></arg><arg><var>ys</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>sumList</name></sharp><arg><var>ys</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><var>y2</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><var>y2</var></arg><arg><var>y1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>0</name></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><var>y2</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>true</name></funapp></arg><arg><var>y2</var></arg><arg><var>y1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>0</name></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y2</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><usableRules><rules/></usableRules><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><var>y2</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><var>y2</var></arg><arg><var>y1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><var>y</var></arg><arg><var>xs</var></arg><arg><var>ys</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>sumList</name></sharp><arg><var>ys</var></arg><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><redPairUrProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>if</name></sharp><arity>6</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><polynomial><variable>2</variable></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>5</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>cons</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>-2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>p</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>-2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>inc</name><arity>1</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret><interpret><sharp><name>sumList</name></sharp><arity>2</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret><interpret><name>false</name><arity>0</arity><polynomial><coefficient><integer>1</integer></coefficient></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><var>y2</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><var>y2</var></arg><arg><var>y1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y2</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><usableRules><rules><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule></rules></usableRules><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><var>y2</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><var>y2</var></arg><arg><var>y1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y2</var></arg></funapp></arg></funapp></rhs></rule><edge><position>2</position><strict>false</strict><position>3</position></edge><edge><position>1</position><strict>true</strict><position>4</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></redPairUrProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>0</name></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><var>y2</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>true</name></funapp></arg><arg><var>y2</var></arg><arg><var>y1</var></arg><arg><funapp><name>cons</name><arg><funapp><name>0</name></funapp></arg><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y2</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component></depGraphProc></dpProof></redPairUrProc></dpProof></instantiationProc></dpProof></usableRulesProc></dpProof></rewritingProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></narrowingProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></rewritingProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></rewritingProc></dpProof></rewritingProc></dpProof></rewritingProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><funapp><name>nil</name></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>isZero</name><arg><funapp><name>head</name><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></arg><arg><var>y1</var></arg><arg><funapp><name>tail</name><arg><funapp><name>nil</name></funapp></arg></funapp></arg><arg><funapp><name>cons</name><arg><funapp><name>p</name><arg><funapp><name>head</name><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>tail</name><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>inc</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component></depGraphProc></dpProof></narrowingProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>p</name></sharp><arg><funapp><name>head</name><arg><var>xs</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>inc</name></sharp><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>p</name></sharp><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>p</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules/></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss/><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>p</name></sharp><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>p</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><edge><position>1</position><strict>true</strict><position>1</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>inc</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>inc</name></sharp><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules/></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss/><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>inc</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>inc</name></sharp><arg><var>x</var></arg></funapp></rhs></rule><edge><position>1</position><strict>true</strict><position>1</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>isEmpty</name></sharp><arg><var>xs</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>isZero</name></sharp><arg><funapp><name>head</name><arg><var>xs</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>head</name></sharp><arg><var>xs</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>sumList</name></sharp><arg><var>xs</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>tail</name></sharp><arg><var>xs</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component></depGraphProc></dpProof></dpTrans></trsTerminationProof></switchInnermost></trsTerminationProof></proof><origin><proofOrigin><tool><name>AProVE</name><version>AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty </version><strategy>Statistics for single proof: 100.00 % (33 real / 0 unknown / 0 assumptions / 33 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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all output return to TRS Standard Certified