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