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


--------------------------------------------------------------------------------


YES
<?xml version="1.0" encoding="UTF-8" standalone="no"?><?xml-stylesheet type="text/xsl" href="cpfHTML.xsl"?><certificationProblem xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="cpf.xsd"><input><trsInput><trs><rules><rule><lhs><funapp><name>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><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>plus</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>plus</name><arg><funapp><name>0</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>plus</name><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>0</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>div</name><arg><funapp><name>0</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>div</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>quot</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>quot</name><arg><funapp><name>zero</name><arg><var>y</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>quot</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><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>quot</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>quot</name><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>div</name><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>z</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>div</name><arg><funapp><name>div</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>div</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><funapp><name>zero</name><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>eq</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>eq</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>eq</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule><rule><lhs><funapp><name>eq</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>eq</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>divides</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>eq</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><funapp><name>div</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>prime</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>pr</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>pr</name><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>pr</name><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>if</name><arg><funapp><name>divides</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if</name><arg><funapp><name>true</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule><rule><lhs><funapp><name>if</name><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>pr</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>zero</name><arg><funapp><name>div</name><arg><var>x</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>zero</name><arg><funapp><name>divides</name><arg><var>x</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>zero</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>zero</name><arg><funapp><name>quot</name><arg><var>x</var></arg><arg><var>x</var></arg><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></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>if</name><arg><funapp><name>eq</name><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><funapp><name>zero</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>plus</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>zero</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></trs></trsInput></input><cpfVersion>2.1</cpfVersion><proof><trsTerminationProof><dpTrans><dps><rules><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>p</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>p</name></sharp><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>y</var></arg><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>div</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>quot</name></sharp><arg><var>x</var></arg><arg><var>y</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><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>quot</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>quot</name></sharp><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>div</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>div</name></sharp><arg><funapp><name>div</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>div</name></sharp><arg><var>x</var></arg><arg><funapp><name>times</name><arg><funapp><name>zero</name><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>div</name></sharp><arg><funapp><name>div</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>zero</name><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>div</name></sharp><arg><funapp><name>div</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>zero</name></sharp><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>eq</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>eq</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>divides</name></sharp><arg><var>y</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>eq</name></sharp><arg><var>x</var></arg><arg><funapp><name>times</name><arg><funapp><name>div</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>divides</name></sharp><arg><var>y</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>div</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>divides</name></sharp><arg><var>y</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>div</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>prime</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>pr</name></sharp><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>pr</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>divides</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>pr</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>divides</name></sharp><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if</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>pr</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>zero</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>eq</name><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><funapp><name>zero</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>plus</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>zero</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>zero</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>eq</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>zero</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>zero</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>zero</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>zero</name></sharp><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>zero</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>zero</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><markedSymbols>true</markedSymbols><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>prime</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>pr</name></sharp><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>div</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>quot</name></sharp><arg><var>x</var></arg><arg><var>y</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><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>quot</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>quot</name></sharp><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>div</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>div</name></sharp><arg><funapp><name>div</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>div</name></sharp><arg><var>x</var></arg><arg><funapp><name>times</name><arg><funapp><name>zero</name><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>div</name></sharp><arg><funapp><name>div</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>zero</name></sharp><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>zero</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>eq</name><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><funapp><name>zero</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>plus</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>zero</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if</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>pr</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>pr</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>divides</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>pr</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>divides</name></sharp><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>divides</name></sharp><arg><var>y</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>div</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><redPairUrProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>div</name></sharp><arity>2</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>times</name><arity>2</arity><polynomial><coefficient><integer>-2</integer></coefficient></polynomial></interpret><interpret><sharp><name>if</name></sharp><arity>3</arity><polynomial><sum><polynomial><coefficient><integer>-1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>div</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret><interpret><name>zero</name><arity>1</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret><interpret><name>divides</name><arity>2</arity><polynomial><coefficient><integer>1</integer></coefficient></polynomial></interpret><interpret><name>quot</name><arity>3</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>if</name><arity>3</arity><polynomial><sum><polynomial><coefficient><integer>-1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>eq</name><arity>2</arity><polynomial><coefficient><integer>1</integer></coefficient></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>plus</name><arity>2</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>false</name><arity>0</arity><polynomial><coefficient><integer>1</integer></coefficient></polynomial></interpret><interpret><name>true</name><arity>0</arity><polynomial><coefficient><integer>1</integer></coefficient></polynomial></interpret><interpret><name>pr</name><arity>2</arity><polynomial><coefficient><integer>1</integer></coefficient></polynomial></interpret><interpret><name>p</name><arity>1</arity><polynomial><coefficient><integer>-2</integer></coefficient></polynomial></interpret><interpret><sharp><name>quot</name></sharp><arity>3</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><sharp><name>zero</name></sharp><arity>1</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret><interpret><sharp><name>pr</name></sharp><arity>2</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><sharp><name>divides</name></sharp><arity>2</arity><polynomial><variable>2</variable></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>div</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>quot</name></sharp><arg><var>x</var></arg><arg><var>y</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><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>quot</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>quot</name></sharp><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>div</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>div</name></sharp><arg><funapp><name>div</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>zero</name></sharp><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>zero</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>eq</name><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><funapp><name>zero</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>plus</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>zero</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if</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>pr</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>pr</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>if</name></sharp><arg><funapp><name>divides</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>pr</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>divides</name></sharp><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>divides</name></sharp><arg><var>y</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>div</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><usableRules><rules><rule><lhs><funapp><name>zero</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>if</name><arg><funapp><name>eq</name><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><funapp><name>zero</name><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></arg><arg><funapp><name>s</name><arg><funapp><name>plus</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>zero</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>eq</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule><rule><lhs><funapp><name>eq</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>eq</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>divides</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>eq</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><funapp><name>div</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if</name><arg><funapp><name>true</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>false</name></funapp></rhs></rule><rule><lhs><funapp><name>pr</name><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>if</name><arg><funapp><name>divides</name><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if</name><arg><funapp><name>false</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>pr</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>pr</name><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>div</name><arg><funapp><name>0</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>eq</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>true</name></funapp></rhs></rule><rule><lhs><funapp><name>eq</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>div</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>quot</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>quot</name><arg><funapp><name>zero</name><arg><var>y</var></arg></funapp></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>0</name></funapp></rhs></rule><rule><lhs><funapp><name>quot</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><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>quot</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>quot</name><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>div</name><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>z</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>div</name><arg><funapp><name>div</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>div</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><funapp><name>zero</name><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><dpProof><redPairUrProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>if</name></sharp><arity>3</arity><polynomial><sum><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>eq</name><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>0</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></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>false</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>div</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>if</name><arity>3</arity><polynomial><variable>3</variable></polynomial></interpret><interpret><name>plus</name><arity>2</arity><polynomial><sum><polynomial><variable>1</variable></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>divides</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>-2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>times</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>zero</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>true</name><arity>0</arity><polynomial><coefficient><integer>1</integer></coefficient></polynomial></interpret><interpret><name>pr</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>quot</name><arity>3</arity><polynomial><sum><polynomial><variable>2</variable></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>p</name><arity>1</arity><polynomial><coefficient><integer>-2</integer></coefficient></polynomial></interpret><interpret><sharp><name>div</name></sharp><arity>2</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret><interpret><sharp><name>quot</name></sharp><arity>3</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret><interpret><sharp><name>zero</name></sharp><arity>1</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret><interpret><sharp><name>pr</name></sharp><arity>2</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><interpret><sharp><name>divides</name></sharp><arity>2</arity><polynomial><sum><polynomial><variable>1</variable></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>div</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>quot</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>quot</name></sharp><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>div</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if</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>pr</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>pr</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>divides</name></sharp><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>divides</name></sharp><arg><var>y</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>div</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><usableRules><rules><rule><lhs><funapp><name>plus</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>plus</name><arg><funapp><name>0</name></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule></rules></usableRules><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>if</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>pr</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>pr</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>divides</name></sharp><arg><funapp><name>s</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>divides</name></sharp><arg><var>y</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>div</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>quot</name></sharp><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>s</name><arg><var>z</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>div</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>div</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>quot</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><redPairUrProc><orderingConstraintProof><redPair><knuthBendixOrder><w0>1</w0><precedenceWeight><precedenceWeightEntry><name>0</name><arity>0</arity><precedence>1</precedence><weight>2</weight></precedenceWeightEntry><precedenceWeightEntry><name>s</name><arity>1</arity><precedence>0</precedence><weight>1</weight></precedenceWeightEntry></precedenceWeight><argumentFilter><argumentFilterEntry><sharp><name>quot</name></sharp><arity>3</arity><collapsing>2</collapsing></argumentFilterEntry><argumentFilterEntry><name>0</name><arity>0</arity><nonCollapsing/></argumentFilterEntry><argumentFilterEntry><sharp><name>div</name></sharp><arity>2</arity><collapsing>2</collapsing></argumentFilterEntry><argumentFilterEntry><name>s</name><arity>1</arity><nonCollapsing/></argumentFilterEntry></argumentFilter></knuthBendixOrder></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>div</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>quot</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><usableRules><rules/></usableRules><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>div</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>quot</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>y</var></arg></funapp></rhs></rule><edge><position>1</position><strict>false</strict><position>1</position></edge><edge><position>2</position><strict>false</strict><position>2</position></edge><edge><position>2</position><strict>false</strict><position>3</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></redPairUrProc></dpProof></component></depGraphProc></dpProof></redPairUrProc></dpProof></redPairUrProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>div</name></sharp><arg><funapp><name>div</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>zero</name><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>divides</name></sharp><arg><var>y</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>div</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><monoRedPairUrProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>times</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></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><trs><rules/></trs><usableRules><rules/></usableRules><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</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></monoRedPairUrProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>zero</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>zero</name></sharp><arg><funapp><name>0</name></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>y</var></arg><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>zero</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>zero</name><arg><funapp><name>0</name></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>zero</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>zero</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><monoRedPairUrProc><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>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>plus</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></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><trs><rules><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule></rules></trs><usableRules><rules><rule><lhs><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule></rules></usableRules><dpProof><switchInnermostProc><wcrProof><joinableCriticalPairsBFS>20</joinableCriticalPairsBFS></wcrProof><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><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>plus</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></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><trs><rules/></trs><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>p</name><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component></depGraphProc></dpProof></monoRedPairProc></dpProof></switchInnermostProc></dpProof></monoRedPairUrProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>divides</name></sharp><arg><var>y</var></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>eq</name></sharp><arg><var>x</var></arg><arg><funapp><name>times</name><arg><funapp><name>div</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>zero</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>eq</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><funapp><name>0</name></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></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>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>p</name></sharp><arg><funapp><name>s</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>eq</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>eq</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><monoRedPairUrProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>eq</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></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>eq</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>eq</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><trs><rules/></trs><usableRules><rules/></usableRules><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>eq</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>eq</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></monoRedPairUrProc></dpProof></component></depGraphProc></dpProof></dpTrans></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 % (15 real / 0 unknown / 0 assumptions / 15 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>