details

property	value
status	complete
benchmark	#3.55.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n140.star.cs.uiowa.edu
space	AG01

run statistics

property	value
solver	AProVE
configuration	certified
runtime (wallclock)	2.29742 seconds
cpu usage	5.67519
user time	5.45365
system time	0.221549
max virtual memory	1.9856456E7
max residence set size	399764.0

stage attributes

key	value
certification-result	CERTIFIED
starexec-result	CERTIFIED YES
certification-time	0.36
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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><arg><funapp><name>add</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>app</name><arg><funapp><name>quicksort</name><arg><funapp><name>low</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><funapp><name>add</name><arg><var>n</var></arg><arg><funapp><name>quicksort</name><arg><funapp><name>high</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></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>minus</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>minus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>quot</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>quot</name></sharp><arg><funapp><name>minus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>quot</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>minus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>le</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>le</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>app</name></sharp><arg><funapp><name>add</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>app</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>low</name></sharp><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_low</name></sharp><arg><funapp><name>le</name><arg><var>m</var></arg><arg><var>n</var></arg></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>low</name></sharp><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>le</name></sharp><arg><var>m</var></arg><arg><var>n</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_low</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>low</name></sharp><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_low</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>low</name></sharp><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>high</name></sharp><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_high</name></sharp><arg><funapp><name>le</name><arg><var>m</var></arg><arg><var>n</var></arg></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>high</name></sharp><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>le</name></sharp><arg><var>m</var></arg><arg><var>n</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_high</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>high</name></sharp><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_high</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>high</name></sharp><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>quicksort</name></sharp><arg><funapp><name>add</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>app</name></sharp><arg><funapp><name>quicksort</name><arg><funapp><name>low</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><funapp><name>add</name><arg><var>n</var></arg><arg><funapp><name>quicksort</name><arg><funapp><name>high</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>quicksort</name></sharp><arg><funapp><name>add</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>quicksort</name></sharp><arg><funapp><name>low</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>quicksort</name></sharp><arg><funapp><name>add</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>low</name></sharp><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>quicksort</name></sharp><arg><funapp><name>add</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>quicksort</name></sharp><arg><funapp><name>high</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>quicksort</name></sharp><arg><funapp><name>add</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>high</name></sharp><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><markedSymbols>true</markedSymbols><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>quicksort</name></sharp><arg><funapp><name>add</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>quicksort</name></sharp><arg><funapp><name>high</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>quicksort</name></sharp><arg><funapp><name>add</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>quicksort</name></sharp><arg><funapp><name>low</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>low</name><arg><var>n</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>nil</name></funapp></rhs></rule><rule><lhs><funapp><name>low</name><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>if_low</name><arg><funapp><name>le</name><arg><var>m</var></arg><arg><var>n</var></arg></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_low</name><arg><funapp><name>false</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>low</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>le</name><arg><funapp><name>0</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>le</name><arg><funapp><name>s</name><arg><var>x</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>le</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>le</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_low</name><arg><funapp><name>true</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>add</name><arg><var>m</var></arg><arg><funapp><name>low</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>high</name><arg><var>n</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>nil</name></funapp></rhs></rule><rule><lhs><funapp><name>high</name><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>if_high</name><arg><funapp><name>le</name><arg><var>m</var></arg><arg><var>n</var></arg></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_high</name><arg><funapp><name>true</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>high</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_high</name><arg><funapp><name>false</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>add</name><arg><var>m</var></arg><arg><funapp><name>high</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>le</name><arg><funapp><name>0</name></funapp></arg><arg><var>x0</var></arg></funapp><funapp><name>le</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>le</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x1</var></arg></funapp></arg></funapp><funapp><name>low</name><arg><var>x0</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp><funapp><name>low</name><arg><var>x0</var></arg><arg><funapp><name>add</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp><funapp><name>if_low</name><arg><funapp><name>true</name></funapp></arg><arg><var>x0</var></arg><arg><funapp><name>add</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp><funapp><name>if_low</name><arg><funapp><name>false</name></funapp></arg><arg><var>x0</var></arg><arg><funapp><name>add</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp><funapp><name>high</name><arg><var>x0</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp><funapp><name>high</name><arg><var>x0</var></arg><arg><funapp><name>add</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp><funapp><name>if_high</name><arg><funapp><name>true</name></funapp></arg><arg><var>x0</var></arg><arg><funapp><name>add</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp><funapp><name>if_high</name><arg><funapp><name>false</name></funapp></arg><arg><var>x0</var></arg><arg><funapp><name>add</name><arg><var>x1</var></arg><arg><var>x2</var></arg></funapp></arg></funapp></innermostLhss><dpProof><redPairUrProc><orderingConstraintProof><redPair><pathOrder><statusPrecedence><statusPrecedenceEntry><sharp><name>quicksort</name></sharp><arity>1</arity><precedence>2</precedence><mul/></statusPrecedenceEntry><statusPrecedenceEntry><name>add</name><arity>2</arity><precedence>1</precedence><mul/></statusPrecedenceEntry><statusPrecedenceEntry><name>low</name><arity>2</arity><precedence>1</precedence><mul/></statusPrecedenceEntry><statusPrecedenceEntry><name>nil</name><arity>0</arity><precedence>0</precedence><mul/></statusPrecedenceEntry><statusPrecedenceEntry><name>le</name><arity>2</arity><precedence>1</precedence><mul/></statusPrecedenceEntry><statusPrecedenceEntry><name>true</name><arity>0</arity><precedence>3</precedence><mul/></statusPrecedenceEntry><statusPrecedenceEntry><name>if_low</name><arity>3</arity><precedence>1</precedence><mul/></statusPrecedenceEntry><statusPrecedenceEntry><name>false</name><arity>0</arity><precedence>4</precedence><mul/></statusPrecedenceEntry><statusPrecedenceEntry><name>0</name><arity>0</arity><precedence>5</precedence><mul/></statusPrecedenceEntry><statusPrecedenceEntry><name>s</name><arity>1</arity><precedence>6</precedence><mul/></statusPrecedenceEntry></statusPrecedence><argumentFilter><argumentFilterEntry><sharp><name>quicksort</name></sharp><arity>1</arity><nonCollapsing><position>1</position></nonCollapsing></argumentFilterEntry><argumentFilterEntry><name>add</name><arity>2</arity><nonCollapsing><position>1</position><position>2</position></nonCollapsing></argumentFilterEntry><argumentFilterEntry><name>high</name><arity>2</arity><collapsing>2</collapsing></argumentFilterEntry><argumentFilterEntry><name>low</name><arity>2</arity><nonCollapsing><position>2</position></nonCollapsing></argumentFilterEntry><argumentFilterEntry><name>nil</name><arity>0</arity><nonCollapsing/></argumentFilterEntry><argumentFilterEntry><name>if_high</name><arity>3</arity><collapsing>3</collapsing></argumentFilterEntry><argumentFilterEntry><name>le</name><arity>2</arity><nonCollapsing/></argumentFilterEntry><argumentFilterEntry><name>true</name><arity>0</arity><nonCollapsing/></argumentFilterEntry><argumentFilterEntry><name>if_low</name><arity>3</arity><nonCollapsing><position>3</position></nonCollapsing></argumentFilterEntry><argumentFilterEntry><name>false</name><arity>0</arity><nonCollapsing/></argumentFilterEntry><argumentFilterEntry><name>0</name><arity>0</arity><nonCollapsing/></argumentFilterEntry><argumentFilterEntry><name>s</name><arity>1</arity><nonCollapsing><position>1</position></nonCollapsing></argumentFilterEntry></argumentFilter></pathOrder></redPair></orderingConstraintProof><dps><rules/></dps><usableRules><rules><rule><lhs><funapp><name>high</name><arg><var>n</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>nil</name></funapp></rhs></rule><rule><lhs><funapp><name>high</name><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>if_high</name><arg><funapp><name>le</name><arg><var>m</var></arg><arg><var>n</var></arg></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_high</name><arg><funapp><name>true</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>high</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>low</name><arg><var>n</var></arg><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>nil</name></funapp></rhs></rule><rule><lhs><funapp><name>low</name><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>if_low</name><arg><funapp><name>le</name><arg><var>m</var></arg><arg><var>n</var></arg></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_low</name><arg><funapp><name>false</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>low</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_low</name><arg><funapp><name>true</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>add</name><arg><var>m</var></arg><arg><funapp><name>low</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_high</name><arg><funapp><name>false</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>add</name><arg><var>m</var></arg><arg><funapp><name>high</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><pIsEmpty/></dpProof></redPairUrProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>quicksort</name></sharp><arg><funapp><name>add</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>low</name></sharp><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>quicksort</name></sharp><arg><funapp><name>add</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>high</name></sharp><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>quot</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>quot</name></sharp><arg><funapp><name>minus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>minus</name><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>minus</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>minus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>minus</name><arg><var>x0</var></arg><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>minus</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x1</var></arg></funapp></arg></funapp></innermostLhss><dpProof><redPairProc><orderingConstraintProof><redPair><knuthBendixOrder><w0>1</w0><precedenceWeight><precedenceWeightEntry><name>s</name><arity>1</arity><precedence>0</precedence><weight>1</weight></precedenceWeightEntry></precedenceWeight><argumentFilter><argumentFilterEntry><sharp><name>quot</name></sharp><arity>2</arity><collapsing>1</collapsing></argumentFilterEntry><argumentFilterEntry><name>s</name><arity>1</arity><nonCollapsing><position>1</position></nonCollapsing></argumentFilterEntry><argumentFilterEntry><name>minus</name><arity>2</arity><collapsing>1</collapsing></argumentFilterEntry></argumentFilter></knuthBendixOrder></redPair></orderingConstraintProof><dps><rules/></dps><dpProof><pIsEmpty/></dpProof></redPairProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>low</name></sharp><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_low</name></sharp><arg><funapp><name>le</name><arg><var>m</var></arg><arg><var>n</var></arg></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_low</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>low</name></sharp><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_low</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>low</name></sharp><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>le</name><arg><funapp><name>0</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>le</name><arg><funapp><name>s</name><arg><var>x</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>le</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>le</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>le</name><arg><funapp><name>0</name></funapp></arg><arg><var>x0</var></arg></funapp><funapp><name>le</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>le</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x1</var></arg></funapp></arg></funapp></innermostLhss><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>low</name></sharp><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_low</name></sharp><arg><funapp><name>le</name><arg><var>m</var></arg><arg><var>n</var></arg></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></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><sizeChangeGraph><rule><lhs><funapp><sharp><name>if_low</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>low</name></sharp><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule><edge><position>2</position><strict>false</strict><position>1</position></edge><edge><position>3</position><strict>true</strict><position>2</position></edge></sizeChangeGraph><sizeChangeGraph><rule><lhs><funapp><sharp><name>if_low</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>low</name></sharp><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule><edge><position>2</position><strict>false</strict><position>1</position></edge><edge><position>3</position><strict>true</strict><position>2</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>high</name></sharp><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_high</name></sharp><arg><funapp><name>le</name><arg><var>m</var></arg><arg><var>n</var></arg></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_high</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>high</name></sharp><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_high</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>high</name></sharp><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>le</name><arg><funapp><name>0</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>le</name><arg><funapp><name>s</name><arg><var>x</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>le</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>le</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>le</name><arg><funapp><name>0</name></funapp></arg><arg><var>x0</var></arg></funapp><funapp><name>le</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>le</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x1</var></arg></funapp></arg></funapp></innermostLhss><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>high</name></sharp><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_high</name></sharp><arg><funapp><name>le</name><arg><var>m</var></arg><arg><var>n</var></arg></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></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><sizeChangeGraph><rule><lhs><funapp><sharp><name>if_high</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>high</name></sharp><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule><edge><position>2</position><strict>false</strict><position>1</position></edge><edge><position>3</position><strict>true</strict><position>2</position></edge></sizeChangeGraph><sizeChangeGraph><rule><lhs><funapp><sharp><name>if_high</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>high</name></sharp><arg><var>n</var></arg><arg><var>x</var></arg></funapp></rhs></rule><edge><position>2</position><strict>false</strict><position>1</position></edge><edge><position>3</position><strict>true</strict><position>2</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>quot</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>minus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>low</name></sharp><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>le</name></sharp><arg><var>m</var></arg><arg><var>n</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>high</name></sharp><arg><var>n</var></arg><arg><funapp><name>add</name><arg><var>m</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>le</name></sharp><arg><var>m</var></arg><arg><var>n</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>quicksort</name></sharp><arg><funapp><name>add</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>app</name></sharp><arg><funapp><name>quicksort</name><arg><funapp><name>low</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><funapp><name>add</name><arg><var>n</var></arg><arg><funapp><name>quicksort</name><arg><funapp><name>high</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>minus</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>minus</name></sharp><arg><var>x</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>minus</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>minus</name></sharp><arg><var>x</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>le</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>le</name></sharp><arg><var>x</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>le</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>le</name></sharp><arg><var>x</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>app</name></sharp><arg><funapp><name>add</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>app</name></sharp><arg><var>x</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>app</name></sharp><arg><funapp><name>add</name><arg><var>n</var></arg><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>app</name></sharp><arg><var>x</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>false</strict><position>2</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></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 % (26 real / 0 unknown / 0 assumptions / 26 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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