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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="2" unmarked=""><NAME>k</NAME></SYMBOL><SYMBOL arity="2" unmarked="*"><NAME>Marked_*</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>1</NAME></SYMBOL><SYMBOL arity="1" unmarked="i"><NAME>Marked_i</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>i</NAME></SYMBOL><SYMBOL arity="2" unmarked="k"><NAME>Marked_k</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,1)</LHS><RHS>x</RHS></RULE><RULE><LHS> *(1,y)</LHS><RHS>y</RHS></RULE><RULE><LHS> *(i(x),x)</LHS><RHS>1</RHS></RULE><RULE><LHS> *(x,i(x))</LHS><RHS>1</RHS></RULE><RULE><LHS> *(x, *(y,z))</LHS><RHS> *( *(x,y),z)</RHS></RULE><RULE><LHS>i(1)</LHS><RHS>1</RHS></RULE><RULE><LHS> *( *(x,y),i(y))</LHS><RHS>x</RHS></RULE><RULE><LHS> *( *(x,i(y)),y)</LHS><RHS>x</RHS></RULE><RULE><LHS>i(i(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>i( *(x,y))</LHS><RHS> *(i(y),i(x))</RHS></RULE><RULE><LHS>k(x,1)</LHS><RHS>1</RHS></RULE><RULE><LHS>k(x,x)</LHS><RHS>1</RHS></RULE><RULE><LHS> *(k(x,y),k(y,x))</LHS><RHS>1</RHS></RULE><RULE><LHS> *( *(i(x),k(y,z)),x)</LHS><RHS>k( *( *(i(x),y),x), *( *(i(x),z),x))</RHS></RULE><RULE><LHS>k( *(x,i(y)), *(y,i(x)))</LHS><RHS>1</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS> *(x,1)</LHS><RHS>x</RHS></RULE><RULE><LHS> *(1,y)</LHS><RHS>y</RHS></RULE><RULE><LHS> *(i(x),x)</LHS><RHS>1</RHS></RULE><RULE><LHS> *(x,i(x))</LHS><RHS>1</RHS></RULE><RULE><LHS> *(x, *(y,z))</LHS><RHS> *( *(x,y),z)</RHS></RULE><RULE><LHS>i(1)</LHS><RHS>1</RHS></RULE><RULE><LHS> *( *(x,y),i(y))</LHS><RHS>x</RHS></RULE><RULE><LHS> *( *(x,i(y)),y)</LHS><RHS>x</RHS></RULE><RULE><LHS>i(i(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>i( *(x,y))</LHS><RHS> *(i(y),i(x))</RHS></RULE><RULE><LHS>k(x,1)</LHS><RHS>1</RHS></RULE><RULE><LHS>k(x,x)</LHS><RHS>1</RHS></RULE><RULE><LHS> *(k(x,y),k(y,x))</LHS><RHS>1</RHS></RULE><RULE><LHS> *( *(i(x),k(y,z)),x)</LHS><RHS>k( *( *(i(x),y),x), *( *(i(x),z),x))</RHS></RULE><RULE><LHS>k( *(x,i(y)), *(y,i(x)))</LHS><RHS>1</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_*(x,y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_i( *(x,y))</LHS><RHS>Marked_*(i(y),i(x))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_i( *(x,y))</LHS><RHS>Marked_i(y)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_i( *(x,y))</LHS><RHS>Marked_i(x)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_*( *(i(x),k(y,z)),x)</LHS><RHS>Marked_k( *( *(i(x),y),x), *( *(i(x),z),x))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_*( *(i(x),k(y,z)),x)</LHS><RHS>Marked_*( *(i(x),y),x)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_*( *(i(x),k(y,z)),x)</LHS><RHS>Marked_*(i(x),y)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_*( *(i(x),k(y,z)),x)</LHS><RHS>Marked_*( *(i(x),z),x)</RHS></DPRULE><DPRULE num="9"><LHS>Marked_*( *(i(x),k(y,z)),x)</LHS><RHS>Marked_*(i(x),z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/><NODE ref="1"/><NODE ref="6"/><NODE ref="7"/><NODE ref="8"/><NODE ref="9"/></SCC><NONSCC num="1"><NODE ref="5"/></NONSCC><SCC num="2"><NODE ref="3"/><NODE ref="4"/></SCC><NONSCC num="3"><NODE ref="2"/></NONSCC></CCLIST><EDGE end="1" start="0"/><EDGE end="3" start="2"/><EDGE end="1" start="3"/><EDGE end="0" start="3"/></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS> *(x,1)</LHS><RHS>x</RHS></RULE><RULE><LHS> *(1,y)</LHS><RHS>y</RHS></RULE><RULE><LHS> *(i(x),x)</LHS><RHS>1</RHS></RULE><RULE><LHS> *(x,i(x))</LHS><RHS>1</RHS></RULE><RULE><LHS> *(x, *(y,z))</LHS><RHS> *( *(x,y),z)</RHS></RULE><RULE><LHS>i(1)</LHS><RHS>1</RHS></RULE><RULE><LHS> *( *(x,y),i(y))</LHS><RHS>x</RHS></RULE><RULE><LHS> *( *(x,i(y)),y)</LHS><RHS>x</RHS></RULE><RULE><LHS>i(i(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>i( *(x,y))</LHS><RHS> *(i(y),i(x))</RHS></RULE><RULE><LHS>k(x,1)</LHS><RHS>1</RHS></RULE><RULE><LHS>k(x,x)</LHS><RHS>1</RHS></RULE><RULE><LHS> *(k(x,y),k(y,x))</LHS><RHS>1</RHS></RULE><RULE><LHS> *( *(i(x),k(y,z)),x)</LHS><RHS>k( *( *(i(x),y),x), *( *(i(x),z),x))</RHS></RULE><RULE><LHS>k( *(x,i(y)), *(y,i(x)))</LHS><RHS>1</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_i( *(x,y))</LHS><RHS>Marked_i(y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_i( *(x,y))</LHS><RHS>Marked_i(x)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_i</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_i( *(x,y))</LHS><RHS>Marked_i(x)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_i( *(x,y))</LHS><RHS>Marked_i(y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS> *(x,1)</LHS><RHS>x</RHS></RULE><RULE><LHS> *(1,y)</LHS><RHS>y</RHS></RULE><RULE><LHS> *(i(x),x)</LHS><RHS>1</RHS></RULE><RULE><LHS> *(x,i(x))</LHS><RHS>1</RHS></RULE><RULE><LHS> *(x, *(y,z))</LHS><RHS> *( *(x,y),z)</RHS></RULE><RULE><LHS>i(1)</LHS><RHS>1</RHS></RULE><RULE><LHS> *( *(x,y),i(y))</LHS><RHS>x</RHS></RULE><RULE><LHS> *( *(x,i(y)),y)</LHS><RHS>x</RHS></RULE><RULE><LHS>i(i(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>i( *(x,y))</LHS><RHS> *(i(y),i(x))</RHS></RULE><RULE><LHS>k(x,1)</LHS><RHS>1</RHS></RULE><RULE><LHS>k(x,x)</LHS><RHS>1</RHS></RULE><RULE><LHS> *(k(x,y),k(y,x))</LHS><RHS>1</RHS></RULE><RULE><LHS> *( *(i(x),k(y,z)),x)</LHS><RHS>k( *( *(i(x),y),x), *( *(i(x),z),x))</RHS></RULE><RULE><LHS>k( *(x,i(y)), *(y,i(x)))</LHS><RHS>1</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS> *(x,1)</LHS><RHS>x</RHS></RULE><RULE><LHS> *(1,y)</LHS><RHS>y</RHS></RULE><RULE><LHS> *(i(x),x)</LHS><RHS>1</RHS></RULE><RULE><LHS> *(x,i(x))</LHS><RHS>1</RHS></RULE><RULE><LHS> *(x, *(y,z))</LHS><RHS> *( *(x,y),z)</RHS></RULE><RULE><LHS>i(1)</LHS><RHS>1</RHS></RULE><RULE><LHS> *( *(x,y),i(y))</LHS><RHS>x</RHS></RULE><RULE><LHS> *( *(x,i(y)),y)</LHS><RHS>x</RHS></RULE><RULE><LHS>i(i(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>i( *(x,y))</LHS><RHS> *(i(y),i(x))</RHS></RULE><RULE><LHS>k(x,1)</LHS><RHS>1</RHS></RULE><RULE><LHS>k(x,x)</LHS><RHS>1</RHS></RULE><RULE><LHS> *(k(x,y),k(y,x))</LHS><RHS>1</RHS></RULE><RULE><LHS> *( *(i(x),k(y,z)),x)</LHS><RHS>k( *( *(i(x),y),x), *( *(i(x),z),x))</RHS></RULE><RULE><LHS>k( *(x,i(y)), *(y,i(x)))</LHS><RHS>1</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_*(x,y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_*( *(i(x),k(y,z)),x)</LHS><RHS>Marked_*( *(i(x),y),x)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_*( *(i(x),k(y,z)),x)</LHS><RHS>Marked_*(i(x),y)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_*( *(i(x),k(y,z)),x)</LHS><RHS>Marked_*( *(i(x),z),x)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_*( *(i(x),k(y,z)),x)</LHS><RHS>Marked_*(i(x),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>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="0"/><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>i</NAME></SYMBOL><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></POLYSYMB><POLYSYMB><SYMBOL><NAME>1</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="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="0"/><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>k</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></ORDERING></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROOF>
