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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="2" unmarked=""><NAME>g</NAME></SYMBOL><SYMBOL arity="2" unmarked="f"><NAME>Marked_f</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="1" unmarked="s"><NAME>Marked_s</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>f</NAME></SYMBOL><SYMBOL arity="2" unmarked="g"><NAME>Marked_g</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>a</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>v</VAR><VAR>u</VAR><VAR>y</VAR><VAR>x</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>s(a)</LHS><RHS>a</RHS></RULE><RULE><LHS>s(s(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>s(f(x,y))</LHS><RHS>f(s(y),s(x))</RHS></RULE><RULE><LHS>s(g(x,y))</LHS><RHS>g(s(x),s(y))</RHS></RULE><RULE><LHS>f(x,a)</LHS><RHS>x</RHS></RULE><RULE><LHS>f(a,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>f(g(x,y),g(u,v))</LHS><RHS>g(f(x,u),f(y,v))</RHS></RULE><RULE><LHS>g(a,a)</LHS><RHS>a</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>s(a)</LHS><RHS>a</RHS></RULE><RULE><LHS>s(s(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>s(f(x,y))</LHS><RHS>f(s(y),s(x))</RHS></RULE><RULE><LHS>s(g(x,y))</LHS><RHS>g(s(x),s(y))</RHS></RULE><RULE><LHS>f(x,a)</LHS><RHS>x</RHS></RULE><RULE><LHS>f(a,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>f(g(x,y),g(u,v))</LHS><RHS>g(f(x,u),f(y,v))</RHS></RULE><RULE><LHS>g(a,a)</LHS><RHS>a</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_s(f(x,y))</LHS><RHS>Marked_f(s(y),s(x))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_s(f(x,y))</LHS><RHS>Marked_s(y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_s(f(x,y))</LHS><RHS>Marked_s(x)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_s(g(x,y))</LHS><RHS>Marked_g(s(x),s(y))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_s(g(x,y))</LHS><RHS>Marked_s(x)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_s(g(x,y))</LHS><RHS>Marked_s(y)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_f(g(x,y),g(u,v))</LHS><RHS>Marked_g(f(x,u),f(y,v))</RHS></DPRULE><DPRULE num="7"><LHS>Marked_f(g(x,y),g(u,v))</LHS><RHS>Marked_f(x,u)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_f(g(x,y),g(u,v))</LHS><RHS>Marked_f(y,v)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="7"/><NODE ref="8"/></SCC><NONSCC num="1"><NODE ref="6"/></NONSCC><SCC num="2"><NODE ref="1"/><NODE ref="2"/><NODE ref="4"/><NODE ref="5"/></SCC><NONSCC num="3"><NODE ref="3"/></NONSCC><NONSCC num="4"><NODE ref="0"/></NONSCC></CCLIST><EDGE end="1" start="0"/><EDGE end="4" start="2"/><EDGE end="3" start="2"/><EDGE end="1" start="4"/><EDGE end="0" start="4"/></DAG></CRITERION><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>s(a)</LHS><RHS>a</RHS></RULE><RULE><LHS>s(s(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>s(f(x,y))</LHS><RHS>f(s(y),s(x))</RHS></RULE><RULE><LHS>s(g(x,y))</LHS><RHS>g(s(x),s(y))</RHS></RULE><RULE><LHS>f(x,a)</LHS><RHS>x</RHS></RULE><RULE><LHS>f(a,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>f(g(x,y),g(u,v))</LHS><RHS>g(f(x,u),f(y,v))</RHS></RULE><RULE><LHS>g(a,a)</LHS><RHS>a</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_s(f(x,y))</LHS><RHS>Marked_s(y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_s(f(x,y))</LHS><RHS>Marked_s(x)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_s(g(x,y))</LHS><RHS>Marked_s(x)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_s(g(x,y))</LHS><RHS>Marked_s(y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>a</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>f</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</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><POLYSYMB><SYMBOL><NAME>Marked_s</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF><ARG degree="1" num="0"/></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><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>g</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</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></ORDERING></CRITERION></PROPERTY><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>s(a)</LHS><RHS>a</RHS></RULE><RULE><LHS>s(s(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>s(f(x,y))</LHS><RHS>f(s(y),s(x))</RHS></RULE><RULE><LHS>s(g(x,y))</LHS><RHS>g(s(x),s(y))</RHS></RULE><RULE><LHS>f(x,a)</LHS><RHS>x</RHS></RULE><RULE><LHS>f(a,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>f(g(x,y),g(u,v))</LHS><RHS>g(f(x,u),f(y,v))</RHS></RULE><RULE><LHS>g(a,a)</LHS><RHS>a</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_f(g(x,y),g(u,v))</LHS><RHS>Marked_f(x,u)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_f(g(x,y),g(u,v))</LHS><RHS>Marked_f(y,v)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>a</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>f</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>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>s</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>Marked_f</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>g</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</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></PROPERTY></PROOF>
