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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="2" unmarked=":"><NAME>Marked_:</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>:</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>i</NAME></SYMBOL><SYMBOL arity="1" unmarked="i"><NAME>Marked_i</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>e</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,x)</LHS><RHS>e</RHS></RULE><RULE><LHS>:(x,e)</LHS><RHS>x</RHS></RULE><RULE><LHS>i(:(x,y))</LHS><RHS>:(y,x)</RHS></RULE><RULE><LHS>:(:(x,y),z)</LHS><RHS>:(x,:(z,i(y)))</RHS></RULE><RULE><LHS>:(e,x)</LHS><RHS>i(x)</RHS></RULE><RULE><LHS>i(i(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>i(e)</LHS><RHS>e</RHS></RULE><RULE><LHS>:(x,:(y,i(x)))</LHS><RHS>i(y)</RHS></RULE><RULE><LHS>:(x,:(y,:(i(x),z)))</LHS><RHS>:(i(z),y)</RHS></RULE><RULE><LHS>:(i(x),:(y,x))</LHS><RHS>i(y)</RHS></RULE><RULE><LHS>:(i(x),:(y,:(x,z)))</LHS><RHS>:(i(z),y)</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>:(x,x)</LHS><RHS>e</RHS></RULE><RULE><LHS>:(x,e)</LHS><RHS>x</RHS></RULE><RULE><LHS>i(:(x,y))</LHS><RHS>:(y,x)</RHS></RULE><RULE><LHS>:(:(x,y),z)</LHS><RHS>:(x,:(z,i(y)))</RHS></RULE><RULE><LHS>:(e,x)</LHS><RHS>i(x)</RHS></RULE><RULE><LHS>i(i(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>i(e)</LHS><RHS>e</RHS></RULE><RULE><LHS>:(x,:(y,i(x)))</LHS><RHS>i(y)</RHS></RULE><RULE><LHS>:(x,:(y,:(i(x),z)))</LHS><RHS>:(i(z),y)</RHS></RULE><RULE><LHS>:(i(x),:(y,x))</LHS><RHS>i(y)</RHS></RULE><RULE><LHS>:(i(x),:(y,:(x,z)))</LHS><RHS>:(i(z),y)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_i(:(x,y))</LHS><RHS>Marked_:(y,x)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_:(:(x,y),z)</LHS><RHS>Marked_:(x,:(z,i(y)))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_:(:(x,y),z)</LHS><RHS>Marked_:(z,i(y))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_:(:(x,y),z)</LHS><RHS>Marked_i(y)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_:(e,x)</LHS><RHS>Marked_i(x)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_:(x,:(y,i(x)))</LHS><RHS>Marked_i(y)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_:(x,:(y,:(i(x),z)))</LHS><RHS>Marked_:(i(z),y)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_:(x,:(y,:(i(x),z)))</LHS><RHS>Marked_i(z)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_:(i(x),:(y,x))</LHS><RHS>Marked_i(y)</RHS></DPRULE><DPRULE num="9"><LHS>Marked_:(i(x),:(y,:(x,z)))</LHS><RHS>Marked_:(i(z),y)</RHS></DPRULE><DPRULE num="10"><LHS>Marked_:(i(x),:(y,:(x,z)))</LHS><RHS>Marked_i(z)</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"/><NODE ref="5"/><NODE ref="6"/><NODE ref="7"/><NODE ref="8"/><NODE ref="9"/><NODE ref="10"/></SCC></CCLIST></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>:(x,x)</LHS><RHS>e</RHS></RULE><RULE><LHS>:(x,e)</LHS><RHS>x</RHS></RULE><RULE><LHS>i(:(x,y))</LHS><RHS>:(y,x)</RHS></RULE><RULE><LHS>:(:(x,y),z)</LHS><RHS>:(x,:(z,i(y)))</RHS></RULE><RULE><LHS>:(e,x)</LHS><RHS>i(x)</RHS></RULE><RULE><LHS>i(i(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>i(e)</LHS><RHS>e</RHS></RULE><RULE><LHS>:(x,:(y,i(x)))</LHS><RHS>i(y)</RHS></RULE><RULE><LHS>:(x,:(y,:(i(x),z)))</LHS><RHS>:(i(z),y)</RHS></RULE><RULE><LHS>:(i(x),:(y,x))</LHS><RHS>i(y)</RHS></RULE><RULE><LHS>:(i(x),:(y,:(x,z)))</LHS><RHS>:(i(z),y)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_i(:(x,y))</LHS><RHS>Marked_:(y,x)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_:(:(x,y),z)</LHS><RHS>Marked_:(x,:(z,i(y)))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_:(:(x,y),z)</LHS><RHS>Marked_:(z,i(y))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_:(:(x,y),z)</LHS><RHS>Marked_i(y)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_:(e,x)</LHS><RHS>Marked_i(x)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_:(x,:(y,i(x)))</LHS><RHS>Marked_i(y)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_:(x,:(y,:(i(x),z)))</LHS><RHS>Marked_:(i(z),y)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_:(x,:(y,:(i(x),z)))</LHS><RHS>Marked_i(z)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_:(i(x),:(y,x))</LHS><RHS>Marked_i(y)</RHS></DPRULE><DPRULE num="9"><LHS>Marked_:(i(x),:(y,:(x,z)))</LHS><RHS>Marked_:(i(z),y)</RHS></DPRULE><DPRULE num="10"><LHS>Marked_:(i(x),:(y,:(x,z)))</LHS><RHS>Marked_i(z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>e</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_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>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>:</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_:</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><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_:(:(x,y),z)</LHS><RHS>Marked_:(z,i(y))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_:(:(x,y),z)</LHS><RHS>Marked_i(y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_:(e,x)</LHS><RHS>Marked_i(x)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_:(x,:(y,i(x)))</LHS><RHS>Marked_i(y)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_:(x,:(y,:(i(x),z)))</LHS><RHS>Marked_:(i(z),y)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_:(x,:(y,:(i(x),z)))</LHS><RHS>Marked_i(z)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_:(i(x),:(y,x))</LHS><RHS>Marked_i(y)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_:(i(x),:(y,:(x,z)))</LHS><RHS>Marked_:(i(z),y)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_:(i(x),:(y,:(x,z)))</LHS><RHS>Marked_i(z)</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>:(x,x)</LHS><RHS>e</RHS></RULE><RULE><LHS>:(x,e)</LHS><RHS>x</RHS></RULE><RULE><LHS>i(:(x,y))</LHS><RHS>:(y,x)</RHS></RULE><RULE><LHS>:(:(x,y),z)</LHS><RHS>:(x,:(z,i(y)))</RHS></RULE><RULE><LHS>:(e,x)</LHS><RHS>i(x)</RHS></RULE><RULE><LHS>i(i(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>i(e)</LHS><RHS>e</RHS></RULE><RULE><LHS>:(x,:(y,i(x)))</LHS><RHS>i(y)</RHS></RULE><RULE><LHS>:(x,:(y,:(i(x),z)))</LHS><RHS>:(i(z),y)</RHS></RULE><RULE><LHS>:(i(x),:(y,x))</LHS><RHS>i(y)</RHS></RULE><RULE><LHS>:(i(x),:(y,:(x,z)))</LHS><RHS>:(i(z),y)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_i(:(x,y))</LHS><RHS>Marked_:(y,x)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_:(:(x,y),z)</LHS><RHS>Marked_:(x,:(z,i(y)))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="1"/></SCC><NONSCC num="1"><NODE ref="0"/></NONSCC></CCLIST><EDGE end="0" start="1"/></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>:(x,x)</LHS><RHS>e</RHS></RULE><RULE><LHS>:(x,e)</LHS><RHS>x</RHS></RULE><RULE><LHS>i(:(x,y))</LHS><RHS>:(y,x)</RHS></RULE><RULE><LHS>:(:(x,y),z)</LHS><RHS>:(x,:(z,i(y)))</RHS></RULE><RULE><LHS>:(e,x)</LHS><RHS>i(x)</RHS></RULE><RULE><LHS>i(i(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>i(e)</LHS><RHS>e</RHS></RULE><RULE><LHS>:(x,:(y,i(x)))</LHS><RHS>i(y)</RHS></RULE><RULE><LHS>:(x,:(y,:(i(x),z)))</LHS><RHS>:(i(z),y)</RHS></RULE><RULE><LHS>:(i(x),:(y,x))</LHS><RHS>i(y)</RHS></RULE><RULE><LHS>:(i(x),:(y,:(x,z)))</LHS><RHS>:(i(z),y)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_:(:(x,y),z)</LHS><RHS>Marked_:(x,:(z,i(y)))</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_:(x,:(z,i(y)))</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>:(x,x)</LHS><RHS>e</RHS></RULE><RULE><LHS>:(x,e)</LHS><RHS>x</RHS></RULE><RULE><LHS>i(:(x,y))</LHS><RHS>:(y,x)</RHS></RULE><RULE><LHS>:(:(x,y),z)</LHS><RHS>:(x,:(z,i(y)))</RHS></RULE><RULE><LHS>:(e,x)</LHS><RHS>i(x)</RHS></RULE><RULE><LHS>i(i(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>i(e)</LHS><RHS>e</RHS></RULE><RULE><LHS>:(x,:(y,i(x)))</LHS><RHS>i(y)</RHS></RULE><RULE><LHS>:(x,:(y,:(i(x),z)))</LHS><RHS>:(i(z),y)</RHS></RULE><RULE><LHS>:(i(x),:(y,x))</LHS><RHS>i(y)</RHS></RULE><RULE><LHS>:(i(x),:(y,:(x,z)))</LHS><RHS>:(i(z),y)</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
