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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="2" unmarked=""><NAME>gcd</NAME></SYMBOL><SYMBOL arity="2" unmarked="minus"><NAME>Marked_minus</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="1" unmarked="pred"><NAME>Marked_pred</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>false</NAME></SYMBOL><SYMBOL arity="3" unmarked="if"><NAME>Marked_if</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>minus</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>true</NAME></SYMBOL><SYMBOL arity="2" unmarked="gcd"><NAME>Marked_gcd</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="2" unmarked="le"><NAME>Marked_le</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>le</NAME></SYMBOL><SYMBOL arity="3" unmarked=""><NAME>if</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>pred</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>Y</VAR><VAR>X</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>minus(X,s(Y))</LHS><RHS>pred(minus(X,Y))</RHS></RULE><RULE><LHS>minus(X,0)</LHS><RHS>X</RHS></RULE><RULE><LHS>pred(s(X))</LHS><RHS>X</RHS></RULE><RULE><LHS>le(s(X),s(Y))</LHS><RHS>le(X,Y)</RHS></RULE><RULE><LHS>le(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(0,Y)</LHS><RHS>true</RHS></RULE><RULE><LHS>gcd(0,Y)</LHS><RHS>0</RHS></RULE><RULE><LHS>gcd(s(X),0)</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>gcd(s(X),s(Y))</LHS><RHS>if(le(Y,X),s(X),s(Y))</RHS></RULE><RULE><LHS>if(true,s(X),s(Y))</LHS><RHS>gcd(minus(X,Y),s(Y))</RHS></RULE><RULE><LHS>if(false,s(X),s(Y))</LHS><RHS>gcd(minus(Y,X),s(X))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>minus(X,s(Y))</LHS><RHS>pred(minus(X,Y))</RHS></RULE><RULE><LHS>minus(X,0)</LHS><RHS>X</RHS></RULE><RULE><LHS>pred(s(X))</LHS><RHS>X</RHS></RULE><RULE><LHS>le(s(X),s(Y))</LHS><RHS>le(X,Y)</RHS></RULE><RULE><LHS>le(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(0,Y)</LHS><RHS>true</RHS></RULE><RULE><LHS>gcd(0,Y)</LHS><RHS>0</RHS></RULE><RULE><LHS>gcd(s(X),0)</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>gcd(s(X),s(Y))</LHS><RHS>if(le(Y,X),s(X),s(Y))</RHS></RULE><RULE><LHS>if(true,s(X),s(Y))</LHS><RHS>gcd(minus(X,Y),s(Y))</RHS></RULE><RULE><LHS>if(false,s(X),s(Y))</LHS><RHS>gcd(minus(Y,X),s(X))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_minus(X,s(Y))</LHS><RHS>Marked_pred(minus(X,Y))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_minus(X,s(Y))</LHS><RHS>Marked_minus(X,Y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_le(s(X),s(Y))</LHS><RHS>Marked_le(X,Y)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_gcd(s(X),s(Y))</LHS><RHS>Marked_if(le(Y,X),s(X),s(Y))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_gcd(s(X),s(Y))</LHS><RHS>Marked_le(Y,X)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_if(true,s(X),s(Y))</LHS><RHS>Marked_gcd(minus(X,Y),s(Y))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_if(true,s(X),s(Y))</LHS><RHS>Marked_minus(X,Y)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_if(false,s(X),s(Y))</LHS><RHS>Marked_gcd(minus(Y,X),s(X))</RHS></DPRULE><DPRULE num="8"><LHS>Marked_if(false,s(X),s(Y))</LHS><RHS>Marked_minus(Y,X)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="8"/></NONSCC><SCC num="1"><NODE ref="1"/></SCC><NONSCC num="2"><NODE ref="0"/></NONSCC><SCC num="3"><NODE ref="3"/><NODE ref="5"/><NODE ref="7"/></SCC><NONSCC num="4"><NODE ref="4"/></NONSCC><NONSCC num="5"><NODE ref="6"/></NONSCC><SCC num="6"><NODE ref="2"/></SCC></CCLIST><EDGE end="2" start="0"/><EDGE end="1" start="0"/><EDGE end="2" start="1"/><EDGE end="5" start="3"/><EDGE end="4" start="3"/><EDGE end="0" start="3"/><EDGE end="6" start="4"/><EDGE end="2" start="5"/><EDGE end="1" start="5"/></DAG></CRITERION><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>minus(X,s(Y))</LHS><RHS>pred(minus(X,Y))</RHS></RULE><RULE><LHS>minus(X,0)</LHS><RHS>X</RHS></RULE><RULE><LHS>pred(s(X))</LHS><RHS>X</RHS></RULE><RULE><LHS>le(s(X),s(Y))</LHS><RHS>le(X,Y)</RHS></RULE><RULE><LHS>le(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(0,Y)</LHS><RHS>true</RHS></RULE><RULE><LHS>gcd(0,Y)</LHS><RHS>0</RHS></RULE><RULE><LHS>gcd(s(X),0)</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>gcd(s(X),s(Y))</LHS><RHS>if(le(Y,X),s(X),s(Y))</RHS></RULE><RULE><LHS>if(true,s(X),s(Y))</LHS><RHS>gcd(minus(X,Y),s(Y))</RHS></RULE><RULE><LHS>if(false,s(X),s(Y))</LHS><RHS>gcd(minus(Y,X),s(X))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_gcd(s(X),s(Y))</LHS><RHS>Marked_if(le(Y,X),s(X),s(Y))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_if(true,s(X),s(Y))</LHS><RHS>Marked_gcd(minus(X,Y),s(Y))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_if(false,s(X),s(Y))</LHS><RHS>Marked_gcd(minus(Y,X),s(X))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>pred</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>if</NAME></SYMBOL><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="2"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>le</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>s</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_gcd</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</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><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>true</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>minus</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>Marked_if</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="2"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>false</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>0</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>gcd</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>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 criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>minus(X,s(Y))</LHS><RHS>pred(minus(X,Y))</RHS></RULE><RULE><LHS>minus(X,0)</LHS><RHS>X</RHS></RULE><RULE><LHS>pred(s(X))</LHS><RHS>X</RHS></RULE><RULE><LHS>le(s(X),s(Y))</LHS><RHS>le(X,Y)</RHS></RULE><RULE><LHS>le(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(0,Y)</LHS><RHS>true</RHS></RULE><RULE><LHS>gcd(0,Y)</LHS><RHS>0</RHS></RULE><RULE><LHS>gcd(s(X),0)</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>gcd(s(X),s(Y))</LHS><RHS>if(le(Y,X),s(X),s(Y))</RHS></RULE><RULE><LHS>if(true,s(X),s(Y))</LHS><RHS>gcd(minus(X,Y),s(Y))</RHS></RULE><RULE><LHS>if(false,s(X),s(Y))</LHS><RHS>gcd(minus(Y,X),s(X))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_le(s(X),s(Y))</LHS><RHS>Marked_le(X,Y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_le</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_le(s(X),s(Y))</LHS><RHS>Marked_le(X,Y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>minus(X,s(Y))</LHS><RHS>pred(minus(X,Y))</RHS></RULE><RULE><LHS>minus(X,0)</LHS><RHS>X</RHS></RULE><RULE><LHS>pred(s(X))</LHS><RHS>X</RHS></RULE><RULE><LHS>le(s(X),s(Y))</LHS><RHS>le(X,Y)</RHS></RULE><RULE><LHS>le(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(0,Y)</LHS><RHS>true</RHS></RULE><RULE><LHS>gcd(0,Y)</LHS><RHS>0</RHS></RULE><RULE><LHS>gcd(s(X),0)</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>gcd(s(X),s(Y))</LHS><RHS>if(le(Y,X),s(X),s(Y))</RHS></RULE><RULE><LHS>if(true,s(X),s(Y))</LHS><RHS>gcd(minus(X,Y),s(Y))</RHS></RULE><RULE><LHS>if(false,s(X),s(Y))</LHS><RHS>gcd(minus(Y,X),s(X))</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,s(Y))</LHS><RHS>pred(minus(X,Y))</RHS></RULE><RULE><LHS>minus(X,0)</LHS><RHS>X</RHS></RULE><RULE><LHS>pred(s(X))</LHS><RHS>X</RHS></RULE><RULE><LHS>le(s(X),s(Y))</LHS><RHS>le(X,Y)</RHS></RULE><RULE><LHS>le(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(0,Y)</LHS><RHS>true</RHS></RULE><RULE><LHS>gcd(0,Y)</LHS><RHS>0</RHS></RULE><RULE><LHS>gcd(s(X),0)</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>gcd(s(X),s(Y))</LHS><RHS>if(le(Y,X),s(X),s(Y))</RHS></RULE><RULE><LHS>if(true,s(X),s(Y))</LHS><RHS>gcd(minus(X,Y),s(Y))</RHS></RULE><RULE><LHS>if(false,s(X),s(Y))</LHS><RHS>gcd(minus(Y,X),s(X))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_minus(X,s(Y))</LHS><RHS>Marked_minus(X,Y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_minus</NAME></SYMBOL><INT>2</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_minus(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,s(Y))</LHS><RHS>pred(minus(X,Y))</RHS></RULE><RULE><LHS>minus(X,0)</LHS><RHS>X</RHS></RULE><RULE><LHS>pred(s(X))</LHS><RHS>X</RHS></RULE><RULE><LHS>le(s(X),s(Y))</LHS><RHS>le(X,Y)</RHS></RULE><RULE><LHS>le(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(0,Y)</LHS><RHS>true</RHS></RULE><RULE><LHS>gcd(0,Y)</LHS><RHS>0</RHS></RULE><RULE><LHS>gcd(s(X),0)</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>gcd(s(X),s(Y))</LHS><RHS>if(le(Y,X),s(X),s(Y))</RHS></RULE><RULE><LHS>if(true,s(X),s(Y))</LHS><RHS>gcd(minus(X,Y),s(Y))</RHS></RULE><RULE><LHS>if(false,s(X),s(Y))</LHS><RHS>gcd(minus(Y,X),s(X))</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
