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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="1" unmarked=""><NAME>p</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME> *</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="2" unmarked="*"><NAME>Marked_*</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>minus</NAME></SYMBOL><SYMBOL arity="1" unmarked="minus"><NAME>Marked_minus</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>+</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>y</VAR><VAR>x</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>+(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>+(s(x),y)</LHS><RHS>s(+(x,y))</RHS></RULE><RULE><LHS>+(p(x),y)</LHS><RHS>p(+(x,y))</RHS></RULE><RULE><LHS>minus(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(s(x))</LHS><RHS>p(minus(x))</RHS></RULE><RULE><LHS>minus(p(x))</LHS><RHS>s(minus(x))</RHS></RULE><RULE><LHS> *(0,y)</LHS><RHS>0</RHS></RULE><RULE><LHS> *(s(x),y)</LHS><RHS>+( *(x,y),y)</RHS></RULE><RULE><LHS> *(p(x),y)</LHS><RHS>+( *(x,y),minus(y))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>+(s(x),y)</LHS><RHS>s(+(x,y))</RHS></RULE><RULE><LHS>+(p(x),y)</LHS><RHS>p(+(x,y))</RHS></RULE><RULE><LHS>minus(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(s(x))</LHS><RHS>p(minus(x))</RHS></RULE><RULE><LHS>minus(p(x))</LHS><RHS>s(minus(x))</RHS></RULE><RULE><LHS> *(0,y)</LHS><RHS>0</RHS></RULE><RULE><LHS> *(s(x),y)</LHS><RHS>+( *(x,y),y)</RHS></RULE><RULE><LHS> *(p(x),y)</LHS><RHS>+( *(x,y),minus(y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_+(s(x),y)</LHS><RHS>Marked_+(x,y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_+(p(x),y)</LHS><RHS>Marked_+(x,y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_minus(s(x))</LHS><RHS>Marked_minus(x)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_minus(p(x))</LHS><RHS>Marked_minus(x)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_*(s(x),y)</LHS><RHS>Marked_+( *(x,y),y)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_*(s(x),y)</LHS><RHS>Marked_*(x,y)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_*(p(x),y)</LHS><RHS>Marked_+( *(x,y),minus(y))</RHS></DPRULE><DPRULE num="7"><LHS>Marked_*(p(x),y)</LHS><RHS>Marked_*(x,y)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_*(p(x),y)</LHS><RHS>Marked_minus(y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="8"/></NONSCC><SCC num="1"><NODE ref="2"/><NODE ref="3"/></SCC><SCC num="2"><NODE ref="5"/><NODE ref="7"/></SCC><NONSCC num="3"><NODE ref="6"/></NONSCC><NONSCC num="4"><NODE ref="4"/></NONSCC><SCC num="5"><NODE ref="0"/><NODE ref="1"/></SCC></CCLIST><EDGE end="1" start="0"/><EDGE end="4" start="2"/><EDGE end="3" start="2"/><EDGE end="0" start="2"/><EDGE end="5" start="3"/><EDGE end="5" start="4"/></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>+(s(x),y)</LHS><RHS>s(+(x,y))</RHS></RULE><RULE><LHS>+(p(x),y)</LHS><RHS>p(+(x,y))</RHS></RULE><RULE><LHS>minus(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(s(x))</LHS><RHS>p(minus(x))</RHS></RULE><RULE><LHS>minus(p(x))</LHS><RHS>s(minus(x))</RHS></RULE><RULE><LHS> *(0,y)</LHS><RHS>0</RHS></RULE><RULE><LHS> *(s(x),y)</LHS><RHS>+( *(x,y),y)</RHS></RULE><RULE><LHS> *(p(x),y)</LHS><RHS>+( *(x,y),minus(y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_*(s(x),y)</LHS><RHS>Marked_*(x,y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_*(p(x),y)</LHS><RHS>Marked_*(x,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_*(p(x),y)</LHS><RHS>Marked_*(x,y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_*(s(x),y)</LHS><RHS>Marked_*(x,y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>+(s(x),y)</LHS><RHS>s(+(x,y))</RHS></RULE><RULE><LHS>+(p(x),y)</LHS><RHS>p(+(x,y))</RHS></RULE><RULE><LHS>minus(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(s(x))</LHS><RHS>p(minus(x))</RHS></RULE><RULE><LHS>minus(p(x))</LHS><RHS>s(minus(x))</RHS></RULE><RULE><LHS> *(0,y)</LHS><RHS>0</RHS></RULE><RULE><LHS> *(s(x),y)</LHS><RHS>+( *(x,y),y)</RHS></RULE><RULE><LHS> *(p(x),y)</LHS><RHS>+( *(x,y),minus(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>+(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>+(s(x),y)</LHS><RHS>s(+(x,y))</RHS></RULE><RULE><LHS>+(p(x),y)</LHS><RHS>p(+(x,y))</RHS></RULE><RULE><LHS>minus(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(s(x))</LHS><RHS>p(minus(x))</RHS></RULE><RULE><LHS>minus(p(x))</LHS><RHS>s(minus(x))</RHS></RULE><RULE><LHS> *(0,y)</LHS><RHS>0</RHS></RULE><RULE><LHS> *(s(x),y)</LHS><RHS>+( *(x,y),y)</RHS></RULE><RULE><LHS> *(p(x),y)</LHS><RHS>+( *(x,y),minus(y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_+(s(x),y)</LHS><RHS>Marked_+(x,y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_+(p(x),y)</LHS><RHS>Marked_+(x,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_+(p(x),y)</LHS><RHS>Marked_+(x,y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_+(s(x),y)</LHS><RHS>Marked_+(x,y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>+(s(x),y)</LHS><RHS>s(+(x,y))</RHS></RULE><RULE><LHS>+(p(x),y)</LHS><RHS>p(+(x,y))</RHS></RULE><RULE><LHS>minus(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(s(x))</LHS><RHS>p(minus(x))</RHS></RULE><RULE><LHS>minus(p(x))</LHS><RHS>s(minus(x))</RHS></RULE><RULE><LHS> *(0,y)</LHS><RHS>0</RHS></RULE><RULE><LHS> *(s(x),y)</LHS><RHS>+( *(x,y),y)</RHS></RULE><RULE><LHS> *(p(x),y)</LHS><RHS>+( *(x,y),minus(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>+(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>+(s(x),y)</LHS><RHS>s(+(x,y))</RHS></RULE><RULE><LHS>+(p(x),y)</LHS><RHS>p(+(x,y))</RHS></RULE><RULE><LHS>minus(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(s(x))</LHS><RHS>p(minus(x))</RHS></RULE><RULE><LHS>minus(p(x))</LHS><RHS>s(minus(x))</RHS></RULE><RULE><LHS> *(0,y)</LHS><RHS>0</RHS></RULE><RULE><LHS> *(s(x),y)</LHS><RHS>+( *(x,y),y)</RHS></RULE><RULE><LHS> *(p(x),y)</LHS><RHS>+( *(x,y),minus(y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_minus(s(x))</LHS><RHS>Marked_minus(x)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_minus(p(x))</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(p(x))</LHS><RHS>Marked_minus(x)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_minus(s(x))</LHS><RHS>Marked_minus(x)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>+(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>+(s(x),y)</LHS><RHS>s(+(x,y))</RHS></RULE><RULE><LHS>+(p(x),y)</LHS><RHS>p(+(x,y))</RHS></RULE><RULE><LHS>minus(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>minus(s(x))</LHS><RHS>p(minus(x))</RHS></RULE><RULE><LHS>minus(p(x))</LHS><RHS>s(minus(x))</RHS></RULE><RULE><LHS> *(0,y)</LHS><RHS>0</RHS></RULE><RULE><LHS> *(s(x),y)</LHS><RHS>+( *(x,y),y)</RHS></RULE><RULE><LHS> *(p(x),y)</LHS><RHS>+( *(x,y),minus(y))</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
