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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="2" unmarked="div"><NAME>Marked_div</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>div</NAME></SYMBOL><SYMBOL arity="3" unmarked=""><NAME>quot</NAME></SYMBOL><SYMBOL arity="3" unmarked="quot"><NAME>Marked_quot</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>z</VAR><VAR>y</VAR><VAR>x</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>div(0,y)</LHS><RHS>0</RHS></RULE><RULE><LHS>div(x,y)</LHS><RHS>quot(x,y,y)</RHS></RULE><RULE><LHS>quot(0,s(y),z)</LHS><RHS>0</RHS></RULE><RULE><LHS>quot(s(x),s(y),z)</LHS><RHS>quot(x,y,z)</RHS></RULE><RULE><LHS>quot(x,0,s(z))</LHS><RHS>s(div(x,s(z)))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>div(0,y)</LHS><RHS>0</RHS></RULE><RULE><LHS>div(x,y)</LHS><RHS>quot(x,y,y)</RHS></RULE><RULE><LHS>quot(0,s(y),z)</LHS><RHS>0</RHS></RULE><RULE><LHS>quot(s(x),s(y),z)</LHS><RHS>quot(x,y,z)</RHS></RULE><RULE><LHS>quot(x,0,s(z))</LHS><RHS>s(div(x,s(z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_div(x,y)</LHS><RHS>Marked_quot(x,y,y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_quot(s(x),s(y),z)</LHS><RHS>Marked_quot(x,y,z)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_quot(x,0,s(z))</LHS><RHS>Marked_div(x,s(z))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/><NODE ref="1"/><NODE ref="2"/></SCC></CCLIST></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>div(0,y)</LHS><RHS>0</RHS></RULE><RULE><LHS>div(x,y)</LHS><RHS>quot(x,y,y)</RHS></RULE><RULE><LHS>quot(0,s(y),z)</LHS><RHS>0</RHS></RULE><RULE><LHS>quot(s(x),s(y),z)</LHS><RHS>quot(x,y,z)</RHS></RULE><RULE><LHS>quot(x,0,s(z))</LHS><RHS>s(div(x,s(z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_div(x,y)</LHS><RHS>Marked_quot(x,y,y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_quot(s(x),s(y),z)</LHS><RHS>Marked_quot(x,y,z)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_quot(x,0,s(z))</LHS><RHS>Marked_div(x,s(z))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_quot</NAME></SYMBOL><INT>1</INT></PROJ><PROJ><SYMBOL><NAME>Marked_div</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_quot(s(x),s(y),z)</LHS><RHS>Marked_quot(x,y,z)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>div(0,y)</LHS><RHS>0</RHS></RULE><RULE><LHS>div(x,y)</LHS><RHS>quot(x,y,y)</RHS></RULE><RULE><LHS>quot(0,s(y),z)</LHS><RHS>0</RHS></RULE><RULE><LHS>quot(s(x),s(y),z)</LHS><RHS>quot(x,y,z)</RHS></RULE><RULE><LHS>quot(x,0,s(z))</LHS><RHS>s(div(x,s(z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_quot(x,0,s(z))</LHS><RHS>Marked_div(x,s(z))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_div(x,y)</LHS><RHS>Marked_quot(x,y,y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/><NODE ref="1"/></SCC></CCLIST></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>div(0,y)</LHS><RHS>0</RHS></RULE><RULE><LHS>div(x,y)</LHS><RHS>quot(x,y,y)</RHS></RULE><RULE><LHS>quot(0,s(y),z)</LHS><RHS>0</RHS></RULE><RULE><LHS>quot(s(x),s(y),z)</LHS><RHS>quot(x,y,z)</RHS></RULE><RULE><LHS>quot(x,0,s(z))</LHS><RHS>s(div(x,s(z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_quot(x,0,s(z))</LHS><RHS>Marked_div(x,s(z))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_div(x,y)</LHS><RHS>Marked_quot(x,y,y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>0</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_quot</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>quot</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>div</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_div</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>s</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_quot(x,0,s(z))</LHS><RHS>Marked_div(x,s(z))</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>div(0,y)</LHS><RHS>0</RHS></RULE><RULE><LHS>div(x,y)</LHS><RHS>quot(x,y,y)</RHS></RULE><RULE><LHS>quot(0,s(y),z)</LHS><RHS>0</RHS></RULE><RULE><LHS>quot(s(x),s(y),z)</LHS><RHS>quot(x,y,z)</RHS></RULE><RULE><LHS>quot(x,0,s(z))</LHS><RHS>s(div(x,s(z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_div(x,y)</LHS><RHS>Marked_quot(x,y,y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="0"/></NONSCC></CCLIST></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
