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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="2" unmarked="*"><NAME>Marked_*</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="1" unmarked="minus"><NAME>Marked_minus</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>minus</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>1</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,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>+(minus(x),x)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(minus(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>minus(+(x,y))</LHS><RHS>+(minus(y),minus(x))</RHS></RULE><RULE><LHS> *(x,1)</LHS><RHS>x</RHS></RULE><RULE><LHS> *(x,0)</LHS><RHS>0</RHS></RULE><RULE><LHS> *(x,+(y,z))</LHS><RHS>+( *(x,y), *(x,z))</RHS></RULE><RULE><LHS> *(x,minus(y))</LHS><RHS>minus( *(x,y))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(x,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>+(minus(x),x)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(minus(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>minus(+(x,y))</LHS><RHS>+(minus(y),minus(x))</RHS></RULE><RULE><LHS> *(x,1)</LHS><RHS>x</RHS></RULE><RULE><LHS> *(x,0)</LHS><RHS>0</RHS></RULE><RULE><LHS> *(x,+(y,z))</LHS><RHS>+( *(x,y), *(x,z))</RHS></RULE><RULE><LHS> *(x,minus(y))</LHS><RHS>minus( *(x,y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_minus(+(x,y))</LHS><RHS>Marked_+(minus(y),minus(x))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_minus(+(x,y))</LHS><RHS>Marked_minus(y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_minus(+(x,y))</LHS><RHS>Marked_minus(x)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_*(x,+(y,z))</LHS><RHS>Marked_+( *(x,y), *(x,z))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_*(x,+(y,z))</LHS><RHS>Marked_*(x,y)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_*(x,+(y,z))</LHS><RHS>Marked_*(x,z)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_*(x,minus(y))</LHS><RHS>Marked_minus( *(x,y))</RHS></DPRULE><DPRULE num="7"><LHS>Marked_*(x,minus(y))</LHS><RHS>Marked_*(x,y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="4"/><NODE ref="5"/><NODE ref="7"/></SCC><NONSCC num="1"><NODE ref="6"/></NONSCC><NONSCC num="2"><NODE ref="3"/></NONSCC><SCC num="3"><NODE ref="1"/><NODE ref="2"/></SCC><NONSCC num="4"><NODE ref="0"/></NONSCC></CCLIST><EDGE end="2" start="0"/><EDGE end="1" start="0"/><EDGE end="4" start="1"/><EDGE end="3" start="1"/><EDGE end="4" start="3"/></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(x,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>+(minus(x),x)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(minus(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>minus(+(x,y))</LHS><RHS>+(minus(y),minus(x))</RHS></RULE><RULE><LHS> *(x,1)</LHS><RHS>x</RHS></RULE><RULE><LHS> *(x,0)</LHS><RHS>0</RHS></RULE><RULE><LHS> *(x,+(y,z))</LHS><RHS>+( *(x,y), *(x,z))</RHS></RULE><RULE><LHS> *(x,minus(y))</LHS><RHS>minus( *(x,y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_*(x,+(y,z))</LHS><RHS>Marked_*(x,y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_*(x,+(y,z))</LHS><RHS>Marked_*(x,z)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_*(x,minus(y))</LHS><RHS>Marked_*(x,y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_*</NAME></SYMBOL><INT>2</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_*(x,minus(y))</LHS><RHS>Marked_*(x,y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_*(x,+(y,z))</LHS><RHS>Marked_*(x,z)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_*(x,+(y,z))</LHS><RHS>Marked_*(x,y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(x,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>+(minus(x),x)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(minus(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>minus(+(x,y))</LHS><RHS>+(minus(y),minus(x))</RHS></RULE><RULE><LHS> *(x,1)</LHS><RHS>x</RHS></RULE><RULE><LHS> *(x,0)</LHS><RHS>0</RHS></RULE><RULE><LHS> *(x,+(y,z))</LHS><RHS>+( *(x,y), *(x,z))</RHS></RULE><RULE><LHS> *(x,minus(y))</LHS><RHS>minus( *(x,y))</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(x,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>+(minus(x),x)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(minus(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>minus(+(x,y))</LHS><RHS>+(minus(y),minus(x))</RHS></RULE><RULE><LHS> *(x,1)</LHS><RHS>x</RHS></RULE><RULE><LHS> *(x,0)</LHS><RHS>0</RHS></RULE><RULE><LHS> *(x,+(y,z))</LHS><RHS>+( *(x,y), *(x,z))</RHS></RULE><RULE><LHS> *(x,minus(y))</LHS><RHS>minus( *(x,y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_minus(+(x,y))</LHS><RHS>Marked_minus(y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_minus(+(x,y))</LHS><RHS>Marked_minus(x)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_minus</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_minus(+(x,y))</LHS><RHS>Marked_minus(x)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_minus(+(x,y))</LHS><RHS>Marked_minus(y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(x,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>+(minus(x),x)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(minus(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>minus(+(x,y))</LHS><RHS>+(minus(y),minus(x))</RHS></RULE><RULE><LHS> *(x,1)</LHS><RHS>x</RHS></RULE><RULE><LHS> *(x,0)</LHS><RHS>0</RHS></RULE><RULE><LHS> *(x,+(y,z))</LHS><RHS>+( *(x,y), *(x,z))</RHS></RULE><RULE><LHS> *(x,minus(y))</LHS><RHS>minus( *(x,y))</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
