details

property	value
status	complete
benchmark	thiemann07.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n148.star.cs.uiowa.edu
space	AProVE_07

run statistics

property	value
solver	AProVE
configuration	certified
runtime (wallclock)	9.04486 seconds
cpu usage	27.88
user time	26.9003
system time	0.979658
max virtual memory	3.8164512E7
max residence set size	2599120.0

stage attributes

key	value
certification-result	CERTIFIED
starexec-result	CERTIFIED YES
certification-time	0.99
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" 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napp><name>car</name><arg><var>l</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>if2</name><arg><funapp><name>false</name></funapp></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><name>prod</name><arg><funapp><name>cons</name><arg><funapp><name>times</name><arg><funapp><name>car</name><arg><var>l</var></arg></funapp></arg><arg><funapp><name>cadr</name><arg><var>l</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cddr</name><arg><var>l</var></arg></funapp></arg></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>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>ifplus</name></sharp><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>isZero</name></sharp><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>ifplus</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>ifplus</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>p</name></sharp><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>iftimes</name></sharp><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>isZero</name></sharp><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>iftimes</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>y</var></arg><arg><funapp><name>times</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>iftimes</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>iftimes</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>p</name></sharp><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>shorter</name></sharp><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>shorter</name></sharp><arg><var>l</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>shorter</name><arg><var>l</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>shorter</name><arg><var>l</var></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg><arg><var>l</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>shorter</name></sharp><arg><var>l</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>shorter</name></sharp><arg><var>l</var></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>b</var></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><var>b</var></arg><arg><var>l</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>car</name></sharp><arg><var>l</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>times</name><arg><funapp><name>car</name><arg><var>l</var></arg></funapp></arg><arg><funapp><name>cadr</name><arg><var>l</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cddr</name><arg><var>l</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>car</name><arg><var>l</var></arg></funapp></arg><arg><funapp><name>cadr</name><arg><var>l</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>car</name></sharp><arg><var>l</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>cadr</name></sharp><arg><var>l</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>cddr</name></sharp><arg><var>l</var></arg></funapp></rhs></rule></rules></dps><markedSymbols>true</markedSymbols><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>times</name><arg><funapp><name>car</name><arg><var>l</var></arg></funapp></arg><arg><funapp><name>cadr</name><arg><var>l</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cddr</name><arg><var>l</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>shorter</name><arg><var>l</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>shorter</name><arg><var>l</var></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg><arg><var>l</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>b</var></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><var>b</var></arg><arg><var>l</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>shorter</name><arg><funapp><name>nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>shorter</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule><rule><lhs><funapp><name>shorter</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>shorter</name><arg><var>l</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>car</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>cadr</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><var>y</var></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>cddr</name><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>nil</name></funapp></rhs></rule><rule><lhs><funapp><name>cddr</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>nil</name></funapp></rhs></rule><rule><lhs><funapp><name>cddr</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><var>l</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>iftimes</name><arg><funapp><name>true</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>iftimes</name><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>times</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></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>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>ifplus</name><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>ifplus</name><arg><funapp><name>true</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>ifplus</name><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>plus</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>car</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp><funapp><name>cddr</name><arg><funapp><name>nil</name></funapp></arg></funapp><funapp><name>cddr</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp><funapp><name>cddr</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></arg></funapp><funapp><name>cadr</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></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>plus</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>ifplus</name><arg><funapp><name>true</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>ifplus</name><arg><funapp><name>false</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>times</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>iftimes</name><arg><funapp><name>true</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>iftimes</name><arg><funapp><name>false</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>shorter</name><arg><funapp><name>nil</name></funapp></arg><arg><var>x0</var></arg></funapp><funapp><name>shorter</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>shorter</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x2</var></arg></funapp></arg></funapp></innermostLhss><dpProof><rewritingProc><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>times</name><arg><funapp><name>car</name><arg><var>l</var></arg></funapp></arg><arg><funapp><name>cadr</name><arg><var>l</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cddr</name><arg><var>l</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rewriteStep><positionInTerm><position>1</position><position>1</position></positionInTerm><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><funapp><name>car</name><arg><var>l</var></arg></funapp></arg></funapp></arg><arg><funapp><name>car</name><arg><var>l</var></arg></funapp></arg><arg><funapp><name>cadr</name><arg><var>l</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cddr</name><arg><var>l</var></arg></funapp></arg></funapp></arg></funapp></rewriteStep><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><funapp><name>car</name><arg><var>l</var></arg></funapp></arg></funapp></arg><arg><funapp><name>car</name><arg><var>l</var></arg></funapp></arg><arg><funapp><name>cadr</name><arg><var>l</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cddr</name><arg><var>l</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><usableRules><rules><rule><lhs><funapp><name>car</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>cadr</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><var>y</var></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></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>iftimes</name><arg><funapp><name>true</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>iftimes</name><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>times</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></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>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>ifplus</name><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>ifplus</name><arg><funapp><name>true</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>ifplus</name><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>plus</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><narrowingProc><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>shorter</name><arg><var>l</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>shorter</name><arg><var>l</var></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg><arg><var>l</var></arg></funapp></rhs></rule><positionInTerm><position>1</position></positionInTerm><narrowings><rules><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>shorter</name><arg><funapp><name>nil</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>nil</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>shorter</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></rhs></rule></rules></narrowings><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><funapp><name>car</name><arg><var>l</var></arg></funapp></arg></funapp></arg><arg><funapp><name>car</name><arg><var>l</var></arg></funapp></arg><arg><funapp><name>cadr</name><arg><var>l</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cddr</name><arg><var>l</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>shorter</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>b</var></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><var>b</var></arg><arg><var>l</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><rewritingProc><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>shorter</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></rhs></rule><rewriteStep><positionInTerm><position>2</position></positionInTerm><rule><lhs><funapp><name>shorter</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>shorter</name><arg><var>l</var></arg><arg><var>y</var></arg></funapp></rhs></rule><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>shorter</name><arg><var>x1</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></rewriteStep><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>shorter</name><arg><var>x1</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></rhs></rule><usableRules><rules><rule><lhs><funapp><name>shorter</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>shorter</name><arg><var>l</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>shorter</name><arg><funapp><name>nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>shorter</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule></rules></usableRules><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>shorter</name><arg><funapp><name>nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>shorter</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule><rule><lhs><funapp><name>car</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</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>cadr</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><var>y</var></rhs></rule><rule><lhs><funapp><name>iftimes</name><arg><funapp><name>true</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>iftimes</name><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>times</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>cddr</name><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>nil</name></funapp></rhs></rule><rule><lhs><funapp><name>cddr</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>nil</name></funapp></rhs></rule><rule><lhs><funapp><name>cddr</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><var>l</var></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></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>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>ifplus</name><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>ifplus</name><arg><funapp><name>true</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>ifplus</name><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>plus</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><instantiationProc><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>b</var></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><var>b</var></arg><arg><var>l</var></arg></funapp></rhs></rule><instantiations><rules><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>y_0</var></arg><arg><funapp><name>cons</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><var>y_0</var></arg><arg><funapp><name>cons</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></rhs></rule></rules></instantiations><dpProof><instantiationProc><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><funapp><name>car</name><arg><var>l</var></arg></funapp></arg></funapp></arg><arg><funapp><name>car</name><arg><var>l</var></arg></funapp></arg><arg><funapp><name>cadr</name><arg><var>l</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cddr</name><arg><var>l</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><instantiations><rules><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><funapp><name>car</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>car</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cadr</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>cddr</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></instantiations><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>car</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</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>cadr</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><var>y</var></rhs></rule><rule><lhs><funapp><name>iftimes</name><arg><funapp><name>true</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>iftimes</name><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>times</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>cddr</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>nil</name></funapp></rhs></rule><rule><lhs><funapp><name>cddr</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><var>l</var></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></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>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>ifplus</name><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>ifplus</name><arg><funapp><name>true</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>ifplus</name><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>plus</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>shorter</name><arg><funapp><name>nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>shorter</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule></rules></usableRules><dpProof><rewritingProc><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><funapp><name>car</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>car</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cadr</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>cddr</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rewriteStep><positionInTerm><position>1</position><position>1</position><position>1</position><position>1</position></positionInTerm><rule><lhs><funapp><name>car</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>z1</var></arg></funapp></arg><arg><funapp><name>car</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cadr</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>cddr</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rewriteStep><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>z1</var></arg></funapp></arg><arg><funapp><name>car</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cadr</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>cddr</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><usableRules><rules><rule><lhs><funapp><name>car</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule></rules></usableRules><dpProof><rewritingProc><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>z1</var></arg></funapp></arg><arg><funapp><name>car</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg><arg><funapp><name>cadr</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>cddr</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rewriteStep><positionInTerm><position>1</position><position>1</position><position>2</position></positionInTerm><rule><lhs><funapp><name>car</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>z1</var></arg></funapp></arg><arg><var>z1</var></arg><arg><funapp><name>cadr</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>cddr</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rewriteStep><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>z1</var></arg></funapp></arg><arg><var>z1</var></arg><arg><funapp><name>cadr</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>cddr</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><usableRules><rules><rule><lhs><funapp><name>car</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</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>cadr</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><var>y</var></rhs></rule><rule><lhs><funapp><name>iftimes</name><arg><funapp><name>true</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>iftimes</name><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>times</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>cddr</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>nil</name></funapp></rhs></rule><rule><lhs><funapp><name>cddr</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><var>l</var></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></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>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>ifplus</name><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>ifplus</name><arg><funapp><name>true</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>ifplus</name><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>plus</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>shorter</name><arg><funapp><name>nil</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>shorter</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>cddr</name><arg><funapp><name>nil</name></funapp></arg></funapp><funapp><name>cddr</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp><funapp><name>cddr</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></arg></funapp><funapp><name>cadr</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></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>plus</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>ifplus</name><arg><funapp><name>true</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>ifplus</name><arg><funapp><name>false</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>times</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>iftimes</name><arg><funapp><name>true</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>iftimes</name><arg><funapp><name>false</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>shorter</name><arg><funapp><name>nil</name></funapp></arg><arg><var>x0</var></arg></funapp><funapp><name>shorter</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>shorter</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x2</var></arg></funapp></arg></funapp></innermostLhss><dpProof><forwardInstantiationProc><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>y_0</var></arg><arg><funapp><name>cons</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><var>y_0</var></arg><arg><funapp><name>cons</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></rhs></rule><instantiations><rules><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></rhs></rule></rules></instantiations><usableRules><rules/></usableRules><dpProof><narrowingProc><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>shorter</name><arg><var>x1</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></rhs></rule><positionInTerm><position>2</position></positionInTerm><narrowings><rules><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><var>y0</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></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><funapp><name>cons</name><arg><var>y0</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><var>y0</var></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></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><funapp><name>cons</name><arg><var>y0</var></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></narrowings><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><var>y0</var></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></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><funapp><name>cons</name><arg><var>y0</var></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></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><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>z1</var></arg></funapp></arg><arg><var>z1</var></arg><arg><funapp><name>cadr</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>cddr</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><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>cadr</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><var>y</var></rhs></rule><rule><lhs><funapp><name>iftimes</name><arg><funapp><name>true</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>iftimes</name><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>times</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>cddr</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>nil</name></funapp></rhs></rule><rule><lhs><funapp><name>cddr</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><var>l</var></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></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>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>ifplus</name><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>ifplus</name><arg><funapp><name>true</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>ifplus</name><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>plus</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>cddr</name><arg><funapp><name>nil</name></funapp></arg></funapp><funapp><name>cddr</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp><funapp><name>cddr</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></arg></funapp><funapp><name>cadr</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></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>plus</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>ifplus</name><arg><funapp><name>true</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>ifplus</name><arg><funapp><name>false</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>times</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>iftimes</name><arg><funapp><name>true</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>iftimes</name><arg><funapp><name>false</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp></innermostLhss><dpProof><narrowingProc><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>z1</var></arg></funapp></arg><arg><var>z1</var></arg><arg><funapp><name>cadr</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>cddr</name><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><positionInTerm><position>1</position><position>2</position></positionInTerm><narrowings><rules><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>x0</var></arg><arg><funapp><name>cadr</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>x0</var></arg><arg><funapp><name>cadr</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>x2</var></arg></funapp></arg></funapp></rhs></rule></rules></narrowings><dpProof><depGraphProc><component><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><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>x0</var></arg><arg><funapp><name>cadr</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>x2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><var>y0</var></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></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><funapp><name>cons</name><arg><var>y0</var></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><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>cadr</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><var>y</var></rhs></rule><rule><lhs><funapp><name>iftimes</name><arg><funapp><name>true</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>iftimes</name><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>times</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></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>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>ifplus</name><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>ifplus</name><arg><funapp><name>true</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>ifplus</name><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>plus</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>cadr</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></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>plus</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>ifplus</name><arg><funapp><name>true</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>ifplus</name><arg><funapp><name>false</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>times</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>iftimes</name><arg><funapp><name>true</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>iftimes</name><arg><funapp><name>false</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp></innermostLhss><dpProof><rewritingProc><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>x0</var></arg><arg><funapp><name>cadr</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>x2</var></arg></funapp></arg></funapp></rhs></rule><rewriteStep><positionInTerm><position>1</position><position>1</position><position>3</position></positionInTerm><rule><lhs><funapp><name>cadr</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><var>y</var></rhs></rule><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><var>x2</var></arg></funapp></arg></funapp></rewriteStep><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg><arg><var>x2</var></arg></funapp></arg></funapp></rhs></rule><usableRules><rules><rule><lhs><funapp><name>cadr</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><var>y</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>iftimes</name><arg><funapp><name>true</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>iftimes</name><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>times</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></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>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>ifplus</name><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>ifplus</name><arg><funapp><name>true</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>ifplus</name><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>plus</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></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>plus</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>ifplus</name><arg><funapp><name>true</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>ifplus</name><arg><funapp><name>false</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>times</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>iftimes</name><arg><funapp><name>true</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>iftimes</name><arg><funapp><name>false</name></funapp></arg><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp></innermostLhss><dpProof><instantiationProc><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></rhs></rule><instantiations><rules><rule><lhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>z0</var></arg><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>z0</var></arg><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></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><precedenceWeightEntry><name>false</name><arity>0</arity><precedence>0</precedence><weight>2</weight></precedenceWeightEntry></precedenceWeight><argumentFilter><argumentFilterEntry><sharp><name>prod</name></sharp><arity>1</arity><collapsing>1</collapsing></argumentFilterEntry><argumentFilterEntry><name>cons</name><arity>2</arity><nonCollapsing><position>2</position></nonCollapsing></argumentFilterEntry><argumentFilterEntry><sharp><name>if</name></sharp><arity>3</arity><collapsing>3</collapsing></argumentFilterEntry><argumentFilterEntry><sharp><name>if2</name></sharp><arity>2</arity><collapsing>2</collapsing></argumentFilterEntry><argumentFilterEntry><name>false</name><arity>0</arity><nonCollapsing/></argumentFilterEntry></argumentFilter></knuthBendixOrder></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><var>y0</var></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></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><funapp><name>cons</name><arg><var>y0</var></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></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><funapp><name>cons</name><arg><var>z0</var></arg><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>z0</var></arg><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><usableRules><rules/></usableRules><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><var>y0</var></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></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><funapp><name>cons</name><arg><var>y0</var></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><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><funapp><name>cons</name><arg><var>z0</var></arg><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>z0</var></arg><arg><funapp><name>cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component></depGraphProc></dpProof></redPairUrProc></dpProof></instantiationProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></rewritingProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>cons</name><arg><funapp><name>iftimes</name><arg><funapp><name>isZero</name><arg><var>x0</var></arg></funapp></arg><arg><var>x0</var></arg><arg><funapp><name>cadr</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>nil</name></funapp></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>prod</name></sharp><arg><funapp><name>cons</name><arg><var>y0</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></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><funapp><name>cons</name><arg><var>y0</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component></depGraphProc></dpProof></narrowingProc></dpProof></forwardInstantiationProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></rewritingProc></dpProof></rewritingProc></dpProof></usableRulesProc></dpProof></instantiationProc></dpProof></instantiationProc></dpProof></usableRulesProc></dpProof></rewritingProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>shorter</name><arg><funapp><name>nil</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>nil</name></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component></depGraphProc></dpProof></narrowingProc></dpProof></rewritingProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>car</name><arg><var>l</var></arg></funapp></arg><arg><funapp><name>cadr</name><arg><var>l</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>iftimes</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>iftimes</name></sharp><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><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><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></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>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><var>x0</var></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp></innermostLhss><dpProof><redPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><rationals><delta><rational><numerator>1</numerator><denominator>16</denominator></rational></delta></rationals></domain><degree>1</degree></polynomial></type><interpret><sharp><name>iftimes</name></sharp><arity>3</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>0</numerator><denominator>1</denominator></rational></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>1</numerator><denominator>4</denominator></rational></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>1</numerator><denominator>4</denominator></rational></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>0</numerator><denominator>1</denominator></rational></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>false</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>1</numerator><denominator>4</denominator></rational></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>times</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>0</numerator><denominator>1</denominator></rational></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>1</numerator><denominator>1</denominator></rational></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>0</numerator><denominator>1</denominator></rational></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>p</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>0</numerator><denominator>1</denominator></rational></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>1</numerator><denominator>4</denominator></rational></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>isZero</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>0</numerator><denominator>1</denominator></rational></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>1</numerator><denominator>2</denominator></rational></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>2</numerator><denominator>1</denominator></rational></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>4</numerator><denominator>1</denominator></rational></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>0</numerator><denominator>1</denominator></rational></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>true</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>0</numerator><denominator>1</denominator></rational></coefficient></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>iftimes</name></sharp><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><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></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></innermostLhss><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>iftimes</name></sharp><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><edge><position>1</position><strict>false</strict><position>2</position></edge><edge><position>2</position><strict>false</strict><position>3</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></redPairProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>iftimes</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>y</var></arg><arg><funapp><name>times</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>ifplus</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>ifplus</name></sharp><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><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><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></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>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><var>x0</var></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp></innermostLhss><dpProof><redPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><rationals><delta><rational><numerator>1</numerator><denominator>16</denominator></rational></delta></rationals></domain><degree>1</degree></polynomial></type><interpret><sharp><name>ifplus</name></sharp><arity>3</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>0</numerator><denominator>1</denominator></rational></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>1</numerator><denominator>4</denominator></rational></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>1</numerator><denominator>4</denominator></rational></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>0</numerator><denominator>1</denominator></rational></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>false</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>1</numerator><denominator>4</denominator></rational></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>plus</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>0</numerator><denominator>1</denominator></rational></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>1</numerator><denominator>1</denominator></rational></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>0</numerator><denominator>1</denominator></rational></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>p</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>0</numerator><denominator>1</denominator></rational></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>1</numerator><denominator>4</denominator></rational></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>isZero</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>0</numerator><denominator>1</denominator></rational></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>1</numerator><denominator>2</denominator></rational></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>2</numerator><denominator>1</denominator></rational></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>4</numerator><denominator>1</denominator></rational></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>0</numerator><denominator>1</denominator></rational></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>true</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>0</numerator><denominator>1</denominator></rational></coefficient></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>ifplus</name></sharp><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><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></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></innermostLhss><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>ifplus</name></sharp><arg><funapp><name>isZero</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><edge><position>1</position><strict>false</strict><position>2</position></edge><edge><position>2</position><strict>false</strict><position>3</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></redPairProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>prod</name></sharp><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>shorter</name></sharp><arg><var>l</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>prod</name></sharp><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>shorter</name></sharp><arg><var>l</var></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>isZero</name></sharp><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>ifplus</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>p</name></sharp><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>isZero</name></sharp><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>iftimes</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>p</name></sharp><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>shorter</name></sharp><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>shorter</name></sharp><arg><var>l</var></arg><arg><var>y</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>shorter</name></sharp><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>l</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>shorter</name></sharp><arg><var>l</var></arg><arg><var>y</var></arg></funapp></rhs></rule><edge><position>1</position><strict>true</strict><position>1</position></edge><edge><position>2</position><strict>true</strict><position>2</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>car</name></sharp><arg><var>l</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>car</name></sharp><arg><var>l</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>cadr</name></sharp><arg><var>l</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>l</var></arg></funapp></lhs><rhs><funapp><sharp><name>cddr</name></sharp><arg><var>l</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 % (47 real / 0 unknown / 0 assumptions / 47 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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