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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="1" unmarked=""><NAME>f</NAME></SYMBOL><SYMBOL arity="2" unmarked="+"><NAME>Marked_+</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>+</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>z</VAR><VAR>y</VAR><VAR>x</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>+(+(x,y),z)</LHS><RHS>+(x,+(y,z))</RHS></RULE><RULE><LHS>+(f(x),f(y))</LHS><RHS>f(+(x,y))</RHS></RULE><RULE><LHS>+(f(x),+(f(y),z))</LHS><RHS>+(f(+(x,y)),z)</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(+(x,y),z)</LHS><RHS>+(x,+(y,z))</RHS></RULE><RULE><LHS>+(f(x),f(y))</LHS><RHS>f(+(x,y))</RHS></RULE><RULE><LHS>+(f(x),+(f(y),z))</LHS><RHS>+(f(+(x,y)),z)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_+(+(x,y),z)</LHS><RHS>Marked_+(x,+(y,z))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_+(+(x,y),z)</LHS><RHS>Marked_+(y,z)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_+(f(x),f(y))</LHS><RHS>Marked_+(x,y)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_+(f(x),+(f(y),z))</LHS><RHS>Marked_+(f(+(x,y)),z)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_+(f(x),+(f(y),z))</LHS><RHS>Marked_+(x,y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/><NODE ref="1"/><NODE ref="2"/><NODE ref="3"/><NODE ref="4"/></SCC></CCLIST></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(+(x,y),z)</LHS><RHS>+(x,+(y,z))</RHS></RULE><RULE><LHS>+(f(x),f(y))</LHS><RHS>f(+(x,y))</RHS></RULE><RULE><LHS>+(f(x),+(f(y),z))</LHS><RHS>+(f(+(x,y)),z)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_+(+(x,y),z)</LHS><RHS>Marked_+(x,+(y,z))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_+(+(x,y),z)</LHS><RHS>Marked_+(y,z)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_+(f(x),f(y))</LHS><RHS>Marked_+(x,y)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_+(f(x),+(f(y),z))</LHS><RHS>Marked_+(f(+(x,y)),z)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_+(f(x),+(f(y),z))</LHS><RHS>Marked_+(x,y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>+</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_+</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>f</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_+(f(x),f(y))</LHS><RHS>Marked_+(x,y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_+(f(x),+(f(y),z))</LHS><RHS>Marked_+(f(+(x,y)),z)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_+(f(x),+(f(y),z))</LHS><RHS>Marked_+(x,y)</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(+(x,y),z)</LHS><RHS>+(x,+(y,z))</RHS></RULE><RULE><LHS>+(f(x),f(y))</LHS><RHS>f(+(x,y))</RHS></RULE><RULE><LHS>+(f(x),+(f(y),z))</LHS><RHS>+(f(+(x,y)),z)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_+(+(x,y),z)</LHS><RHS>Marked_+(x,+(y,z))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_+(+(x,y),z)</LHS><RHS>Marked_+(y,z)</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="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(+(x,y),z)</LHS><RHS>+(x,+(y,z))</RHS></RULE><RULE><LHS>+(f(x),f(y))</LHS><RHS>f(+(x,y))</RHS></RULE><RULE><LHS>+(f(x),+(f(y),z))</LHS><RHS>+(f(+(x,y)),z)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_+(+(x,y),z)</LHS><RHS>Marked_+(x,+(y,z))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_+(+(x,y),z)</LHS><RHS>Marked_+(y,z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_+</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_+(+(x,y),z)</LHS><RHS>Marked_+(y,z)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_+(+(x,y),z)</LHS><RHS>Marked_+(x,+(y,z))</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(+(x,y),z)</LHS><RHS>+(x,+(y,z))</RHS></RULE><RULE><LHS>+(f(x),f(y))</LHS><RHS>f(+(x,y))</RHS></RULE><RULE><LHS>+(f(x),+(f(y),z))</LHS><RHS>+(f(+(x,y)),z)</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
