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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="1" unmarked="double"><NAME>Marked_double</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>double</NAME></SYMBOL><SYMBOL arity="2" unmarked="plus"><NAME>Marked_plus</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="2" unmarked="minus"><NAME>Marked_minus</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>plus</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>minus</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>z</VAR><VAR>y</VAR><VAR>x</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>minus(x,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>minus(s(x),s(y))</LHS><RHS>minus(x,y)</RHS></RULE><RULE><LHS>double(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>double(s(x))</LHS><RHS>s(s(double(x)))</RHS></RULE><RULE><LHS>plus(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(x,y))</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>plus(x,s(y))</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(minus(x,y),double(y)))</RHS></RULE><RULE><LHS>plus(s(plus(x,y)),z)</LHS><RHS>s(plus(plus(x,y),z))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>minus(x,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>minus(s(x),s(y))</LHS><RHS>minus(x,y)</RHS></RULE><RULE><LHS>double(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>double(s(x))</LHS><RHS>s(s(double(x)))</RHS></RULE><RULE><LHS>plus(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(x,y))</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>plus(x,s(y))</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(minus(x,y),double(y)))</RHS></RULE><RULE><LHS>plus(s(plus(x,y)),z)</LHS><RHS>s(plus(plus(x,y),z))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_minus(s(x),s(y))</LHS><RHS>Marked_minus(x,y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_double(s(x))</LHS><RHS>Marked_double(x)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_plus(s(x),y)</LHS><RHS>Marked_plus(x,y)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_plus(s(x),y)</LHS><RHS>Marked_plus(x,s(y))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_plus(s(x),y)</LHS><RHS>Marked_plus(minus(x,y),double(y))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_plus(s(x),y)</LHS><RHS>Marked_minus(x,y)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_plus(s(x),y)</LHS><RHS>Marked_double(y)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_plus(s(plus(x,y)),z)</LHS><RHS>Marked_plus(plus(x,y),z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="2"/><NODE ref="3"/><NODE ref="4"/><NODE ref="7"/></SCC><NONSCC num="1"><NODE ref="6"/></NONSCC><NONSCC num="2"><NODE ref="5"/></NONSCC><SCC num="3"><NODE ref="1"/></SCC><SCC num="4"><NODE ref="0"/></SCC></CCLIST><EDGE end="2" start="0"/><EDGE end="1" start="0"/><EDGE end="3" start="1"/><EDGE end="4" start="2"/></DAG></CRITERION><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>minus(x,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>minus(s(x),s(y))</LHS><RHS>minus(x,y)</RHS></RULE><RULE><LHS>double(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>double(s(x))</LHS><RHS>s(s(double(x)))</RHS></RULE><RULE><LHS>plus(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(x,y))</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>plus(x,s(y))</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(minus(x,y),double(y)))</RHS></RULE><RULE><LHS>plus(s(plus(x,y)),z)</LHS><RHS>s(plus(plus(x,y),z))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_plus(s(x),y)</LHS><RHS>Marked_plus(x,y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_plus(s(x),y)</LHS><RHS>Marked_plus(x,s(y))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_plus(s(x),y)</LHS><RHS>Marked_plus(minus(x,y),double(y))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_plus(s(plus(x,y)),z)</LHS><RHS>Marked_plus(plus(x,y),z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="rpo"><AFS><PROJARG num="0"><SYMBOL><NAME>minus</NAME></SYMBOL></PROJARG></AFS><STATUSLIST><STATUS status="lex"><SYMBOL><NAME>plus</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>s</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>Marked_plus</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>double</NAME></SYMBOL></STATUS></STATUSLIST><PRECEDENCE><PREC><OLHS><SYMBOL><NAME>s</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>double</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>s</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>plus</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>s</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>Marked_plus</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>double</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>plus</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>double</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>Marked_plus</NAME></SYMBOL></ORHS></PREC></PRECEDENCE><EQUIVALENCE/></ORDERING></CRITERION></PROPERTY><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>minus(x,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>minus(s(x),s(y))</LHS><RHS>minus(x,y)</RHS></RULE><RULE><LHS>double(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>double(s(x))</LHS><RHS>s(s(double(x)))</RHS></RULE><RULE><LHS>plus(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(x,y))</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>plus(x,s(y))</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(minus(x,y),double(y)))</RHS></RULE><RULE><LHS>plus(s(plus(x,y)),z)</LHS><RHS>s(plus(plus(x,y),z))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_minus(s(x),s(y))</LHS><RHS>Marked_minus(x,y)</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(s(x),s(y))</LHS><RHS>Marked_minus(x,y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>minus(x,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>minus(s(x),s(y))</LHS><RHS>minus(x,y)</RHS></RULE><RULE><LHS>double(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>double(s(x))</LHS><RHS>s(s(double(x)))</RHS></RULE><RULE><LHS>plus(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(x,y))</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>plus(x,s(y))</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(minus(x,y),double(y)))</RHS></RULE><RULE><LHS>plus(s(plus(x,y)),z)</LHS><RHS>s(plus(plus(x,y),z))</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>minus(x,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>minus(s(x),s(y))</LHS><RHS>minus(x,y)</RHS></RULE><RULE><LHS>double(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>double(s(x))</LHS><RHS>s(s(double(x)))</RHS></RULE><RULE><LHS>plus(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(x,y))</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>plus(x,s(y))</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(minus(x,y),double(y)))</RHS></RULE><RULE><LHS>plus(s(plus(x,y)),z)</LHS><RHS>s(plus(plus(x,y),z))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_double(s(x))</LHS><RHS>Marked_double(x)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_double</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_double(s(x))</LHS><RHS>Marked_double(x)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>minus(x,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>minus(s(x),s(y))</LHS><RHS>minus(x,y)</RHS></RULE><RULE><LHS>double(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>double(s(x))</LHS><RHS>s(s(double(x)))</RHS></RULE><RULE><LHS>plus(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(x,y))</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>plus(x,s(y))</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(minus(x,y),double(y)))</RHS></RULE><RULE><LHS>plus(s(plus(x,y)),z)</LHS><RHS>s(plus(plus(x,y),z))</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>