/export/starexec/sandbox2/solver/bin/starexec_run_certified /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES
<?xml version="1.0" encoding="UTF-8" standalone="no"?><?xml-stylesheet type="text/xsl" href="cpfHTML.xsl"?><certificationProblem xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" 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var></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>zero</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>zero</name><arg><var>y</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><var>b</var></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if1</name></sharp><arg><var>b</var></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if1</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>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>b</var></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><var>b</var></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>zero</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>zero</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>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>zero</name><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>zero</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>p</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></innermostLhss><dpProof><narrowingProc><rule><lhs><funapp><sharp><name>int</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>zero</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>zero</name><arg><var>y</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><positionInTerm><position>1</position></positionInTerm><narrowings><rules><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>zero</name><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><var>y1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>zero</name><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></rhs></rule></rules></narrowings><dpProof><instantiationProc><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><var>b</var></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if1</name></sharp><arg><var>b</var></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><instantiations><rules><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><var>y_0</var></arg><arg><funapp><name>0</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><sharp><name>if1</name></sharp><arg><var>y_0</var></arg><arg><funapp><name>0</name></funapp></arg><arg><var>z0</var></arg></funapp></rhs></rule></rules></instantiations><dpProof><instantiationProc><rule><lhs><funapp><sharp><name>if1</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>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><instantiations><rules><rule><lhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>z1</var></arg></funapp></rhs></rule></rules></instantiations><dpProof><narrowingProc><rule><lhs><funapp><sharp><name>if2</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>int</name></sharp><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>p</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><positionInTerm><position>1</position></positionInTerm><narrowings><rules><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>p</name><arg><var>y1</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>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>p</name><arg><var>y1</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>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>p</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule></rules></narrowings><dpProof><instantiationProc><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>b</var></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><var>b</var></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><instantiations><rules><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>y_0</var></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><var>y_0</var></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg><arg><var>z1</var></arg></funapp></rhs></rule></rules></instantiations><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>p</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><var>y_0</var></arg><arg><funapp><name>0</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><sharp><name>if1</name></sharp><arg><var>y_0</var></arg><arg><funapp><name>0</name></funapp></arg><arg><var>z0</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>z1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>zero</name><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>y_0</var></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><var>y_0</var></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg><arg><var>z1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>p</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>zero</name><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><var>y1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>p</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><forwardInstantiationProc><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><var>y_0</var></arg><arg><funapp><name>0</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><sharp><name>if1</name></sharp><arg><var>y_0</var></arg><arg><funapp><name>0</name></funapp></arg><arg><var>z0</var></arg></funapp></rhs></rule><instantiations><rules><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><var>x1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><var>x1</var></arg></funapp></rhs></rule></rules></instantiations><usableRules><rules/></usableRules><dpProof><narrowingProc><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>zero</name><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><var>y1</var></arg></funapp></rhs></rule><positionInTerm><position>2</position></positionInTerm><narrowings><rules><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></rhs></rule></rules></narrowings><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>zero</name><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>y_0</var></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><var>y_0</var></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg><arg><var>z1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>p</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><var>x1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><var>x1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>z1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>p</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><narrowingProc><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>p</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule><positionInTerm><position>2</position></positionInTerm><narrowings><rules><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></narrowings><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>y_0</var></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><var>y_0</var></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg><arg><var>z1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>p</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>zero</name><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><var>x1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><var>x1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>z1</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><instantiationProc><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><var>x1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><var>x1</var></arg></funapp></rhs></rule><instantiations><rules><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule></rules></instantiations><dpProof><instantiationProc><rule><lhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>z1</var></arg></funapp></rhs></rule><instantiations><rules><rule><lhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule></rules></instantiations><dpProof><forwardInstantiationProc><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>y_0</var></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><var>y_0</var></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg><arg><var>z1</var></arg></funapp></rhs></rule><instantiations><rules><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg><arg><var>x2</var></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg><arg><var>x2</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</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>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></instantiations><usableRules><rules/></usableRules><dpProof><narrowingProc><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>zero</name><arg><var>y1</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg><arg><var>y1</var></arg></funapp></rhs></rule><positionInTerm><position>2</position></positionInTerm><narrowings><rules><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></rhs></rule></rules></narrowings><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg><arg><var>x2</var></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg><arg><var>x2</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>p</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</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>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</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><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>p</name><arg><funapp><name>0</name></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></innermostLhss><dpProof><narrowingProc><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><var>y1</var></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>p</name><arg><var>y1</var></arg></funapp></arg></funapp></rhs></rule><positionInTerm><position>2</position></positionInTerm><narrowings><rules><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></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>s</name><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule></rules></narrowings><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg><arg><var>x2</var></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg><arg><var>x2</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</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>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</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><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><instantiationProc><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg><arg><var>x2</var></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg><arg><var>x2</var></arg></funapp></rhs></rule><instantiations><rules><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>z1</var></arg></funapp></arg></funapp></rhs></rule></rules></instantiations><dpProof><forwardInstantiationProc><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></rhs></rule><instantiations><rules><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x1</var></arg></funapp></arg></funapp></rhs></rule></rules></instantiations><usableRules><rules/></usableRules><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x1</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>z1</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>s</name><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><monoRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>p</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><sharp><name>int</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>if_int</name></sharp><arity>4</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>4</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>false</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><sharp><name>if2</name></sharp><arity>3</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x1</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>z1</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>s</name><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><trs><rules><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></trs><dpProof><redPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>int</name></sharp><arity>2</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>p</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>-1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret><interpret><sharp><name>if_int</name></sharp><arity>4</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><variable>3</variable></polynomial></sum></polynomial></interpret><interpret><name>false</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><sharp><name>if2</name></sharp><arity>3</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x1</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>z1</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x1</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>z1</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component></depGraphProc></dpProof></redPairProc></dpProof></monoRedPairProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</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>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><forwardInstantiationProc><rule><lhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule><instantiations><rules><rule><lhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></instantiations><usableRules><rules/></usableRules><dpProof><redPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>int</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>-2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>p</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>-1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><coefficient><integer>1</integer></coefficient></polynomial></interpret><interpret><sharp><name>if2</name></sharp><arity>3</arity><polynomial><sum><polynomial><coefficient><integer>-2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>false</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><sharp><name>if_int</name></sharp><arity>4</arity><polynomial><sum><polynomial><coefficient><integer>-2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>4</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>true</name><arity>0</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret><interpret><sharp><name>if1</name></sharp><arity>3</arity><polynomial><sum><polynomial><coefficient><integer>-2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</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>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>x0</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>if1</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</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>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y_0</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component></depGraphProc></dpProof></redPairProc></dpProof></forwardInstantiationProc></dpProof></component></depGraphProc></dpProof></forwardInstantiationProc></dpProof></instantiationProc></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>s</name><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg></funapp></arg></funapp></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>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>s</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>0</name></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>int</name></sharp><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>s</name><arg><var>y0</var></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component></depGraphProc></dpProof></narrowingProc></dpProof></forwardInstantiationProc></dpProof></instantiationProc></dpProof></instantiationProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></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>if2</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component></depGraphProc></dpProof></narrowingProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>int</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if_int</name></sharp><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>true</name></funapp></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component></depGraphProc></dpProof></narrowingProc></dpProof></forwardInstantiationProc></dpProof></component></depGraphProc></dpProof></instantiationProc></dpProof></narrowingProc></dpProof></instantiationProc></dpProof></instantiationProc></dpProof></narrowingProc></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>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>intlist</name></sharp><arg><funapp><name>int</name><arg><funapp><name>p</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>p</name><arg><var>y</var></arg></funapp></arg></funapp></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>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>if2</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>y</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>if_intlist</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>intlist</name></sharp><arg><funapp><name>tail</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>intlist</name></sharp><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_intlist</name></sharp><arg><funapp><name>empty</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>empty</name><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>empty</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule><rule><lhs><funapp><name>tail</name><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>nil</name></funapp></rhs></rule><rule><lhs><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><var>y</var></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>empty</name><arg><funapp><name>nil</name></funapp></arg></funapp><funapp><name>empty</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp><funapp><name>tail</name><arg><funapp><name>nil</name></funapp></arg></funapp><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></innermostLhss><dpProof><redPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><rationals><delta><rational><numerator>3</numerator><denominator>16</denominator></rational></delta></rationals></domain><degree>1</degree></polynomial></type><interpret><sharp><name>if_intlist</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>4</denominator></rational></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>1</numerator><denominator>1</denominator></rational></coefficient></polynomial><polynomial><variable>2</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>1</denominator></rational></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>intlist</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>1</numerator><denominator>4</denominator></rational></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>2</numerator><denominator>1</denominator></rational></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>tail</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>empty</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>1</numerator><denominator>4</denominator></rational></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>2</numerator><denominator>1</denominator></rational></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>0</numerator><denominator>1</denominator></rational></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>cons</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><rational><numerator>1</numerator><denominator>1</denominator></rational></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>0</numerator><denominator>1</denominator></rational></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><rational><numerator>4</numerator><denominator>1</denominator></rational></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>true</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></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>if_intlist</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>intlist</name></sharp><arg><funapp><name>tail</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><dpProof><usableRulesProc><usableRules><rules><rule><lhs><funapp><name>tail</name><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>nil</name></funapp></rhs></rule><rule><lhs><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><var>y</var></rhs></rule></rules></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss><funapp><name>tail</name><arg><funapp><name>nil</name></funapp></arg></funapp><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x0</var></arg><arg><var>x1</var></arg></funapp></arg></funapp></innermostLhss><dpProof><monoRedPairUrProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>tail</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>cons</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>if_intlist</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>false</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><sharp><name>intlist</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules/></dps><trs><rules><rule><lhs><funapp><name>tail</name><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>nil</name></funapp></rhs></rule></rules></trs><usableRules><rules><rule><lhs><funapp><name>tail</name><arg><funapp><name>nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>nil</name></funapp></rhs></rule><rule><lhs><funapp><name>tail</name><arg><funapp><name>cons</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><var>y</var></rhs></rule></rules></usableRules><dpProof><pIsEmpty/></dpProof></monoRedPairUrProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></redPairProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>p</name></sharp><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>p</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules/></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss/><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>p</name></sharp><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>p</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><edge><position>1</position><strict>true</strict><position>1</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>intlist</name></sharp><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>empty</name></sharp><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>if_intlist</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>head</name></sharp><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>if_intlist</name></sharp><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>tail</name></sharp><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>int</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>zero</name></sharp><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>int</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>zero</name></sharp><arg><var>y</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 % (45 real / 0 unknown / 0 assumptions / 45 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>