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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="2" unmarked=""><NAME>f</NAME></SYMBOL><SYMBOL arity="2" unmarked="+"><NAME>Marked_+</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>b</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>a</NAME></SYMBOL><SYMBOL arity="2" unmarked="f"><NAME>Marked_f</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>+(a,b)</LHS><RHS>+(b,a)</RHS></RULE><RULE><LHS>+(a,+(b,z))</LHS><RHS>+(b,+(a,z))</RHS></RULE><RULE><LHS>+(+(x,y),z)</LHS><RHS>+(x,+(y,z))</RHS></RULE><RULE><LHS>f(a,y)</LHS><RHS>a</RHS></RULE><RULE><LHS>f(b,y)</LHS><RHS>b</RHS></RULE><RULE><LHS>f(+(x,y),z)</LHS><RHS>+(f(x,z),f(y,z))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(a,b)</LHS><RHS>+(b,a)</RHS></RULE><RULE><LHS>+(a,+(b,z))</LHS><RHS>+(b,+(a,z))</RHS></RULE><RULE><LHS>+(+(x,y),z)</LHS><RHS>+(x,+(y,z))</RHS></RULE><RULE><LHS>f(a,y)</LHS><RHS>a</RHS></RULE><RULE><LHS>f(b,y)</LHS><RHS>b</RHS></RULE><RULE><LHS>f(+(x,y),z)</LHS><RHS>+(f(x,z),f(y,z))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_+(a,b)</LHS><RHS>Marked_+(b,a)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_+(a,+(b,z))</LHS><RHS>Marked_+(b,+(a,z))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_+(a,+(b,z))</LHS><RHS>Marked_+(a,z)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_+(+(x,y),z)</LHS><RHS>Marked_+(x,+(y,z))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_+(+(x,y),z)</LHS><RHS>Marked_+(y,z)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_f(+(x,y),z)</LHS><RHS>Marked_+(f(x,z),f(y,z))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_f(+(x,y),z)</LHS><RHS>Marked_f(x,z)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_f(+(x,y),z)</LHS><RHS>Marked_f(y,z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="6"/><NODE ref="7"/></SCC><NONSCC num="1"><NODE ref="5"/></NONSCC><SCC num="2"><NODE ref="3"/><NODE ref="4"/></SCC><SCC num="3"><NODE ref="2"/></SCC><NONSCC num="4"><NODE ref="1"/></NONSCC><NONSCC num="5"><NODE ref="0"/></NONSCC></CCLIST><EDGE end="1" start="0"/><EDGE end="5" start="1"/><EDGE end="4" start="1"/><EDGE end="3" start="1"/><EDGE end="2" start="1"/><EDGE end="5" start="2"/><EDGE end="4" start="2"/><EDGE end="3" start="2"/><EDGE end="5" start="3"/><EDGE end="4" start="3"/></DAG></CRITERION><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(a,b)</LHS><RHS>+(b,a)</RHS></RULE><RULE><LHS>+(a,+(b,z))</LHS><RHS>+(b,+(a,z))</RHS></RULE><RULE><LHS>+(+(x,y),z)</LHS><RHS>+(x,+(y,z))</RHS></RULE><RULE><LHS>f(a,y)</LHS><RHS>a</RHS></RULE><RULE><LHS>f(b,y)</LHS><RHS>b</RHS></RULE><RULE><LHS>f(+(x,y),z)</LHS><RHS>+(f(x,z),f(y,z))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_f(+(x,y),z)</LHS><RHS>Marked_f(x,z)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_f(+(x,y),z)</LHS><RHS>Marked_f(y,z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>+</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><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></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_f</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>a</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>b</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>f</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB></ORDERING></CRITERION></PROPERTY><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(a,b)</LHS><RHS>+(b,a)</RHS></RULE><RULE><LHS>+(a,+(b,z))</LHS><RHS>+(b,+(a,z))</RHS></RULE><RULE><LHS>+(+(x,y),z)</LHS><RHS>+(x,+(y,z))</RHS></RULE><RULE><LHS>f(a,y)</LHS><RHS>a</RHS></RULE><RULE><LHS>f(b,y)</LHS><RHS>b</RHS></RULE><RULE><LHS>f(+(x,y),z)</LHS><RHS>+(f(x,z),f(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><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>+</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><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></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>a</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>b</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_+</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>f</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB></ORDERING></CRITERION></PROPERTY><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(a,b)</LHS><RHS>+(b,a)</RHS></RULE><RULE><LHS>+(a,+(b,z))</LHS><RHS>+(b,+(a,z))</RHS></RULE><RULE><LHS>+(+(x,y),z)</LHS><RHS>+(x,+(y,z))</RHS></RULE><RULE><LHS>f(a,y)</LHS><RHS>a</RHS></RULE><RULE><LHS>f(b,y)</LHS><RHS>b</RHS></RULE><RULE><LHS>f(+(x,y),z)</LHS><RHS>+(f(x,z),f(y,z))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_+(a,+(b,z))</LHS><RHS>Marked_+(a,z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>+</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</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></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>a</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>b</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_+</NAME></SYMBOL><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></POLYSYMB><POLYSYMB><SYMBOL><NAME>f</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB></ORDERING></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROOF>
