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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="2" unmarked="app"><NAME>Marked_app</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>reverse</NAME></SYMBOL><SYMBOL arity="1" unmarked="shuffle"><NAME>Marked_shuffle</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>nil</NAME></SYMBOL><SYMBOL arity="1" unmarked="reverse"><NAME>Marked_reverse</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>add</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>shuffle</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>app</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>n</VAR><VAR>y</VAR><VAR>x</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>reverse(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>reverse(add(n,x))</LHS><RHS>app(reverse(x),add(n,nil))</RHS></RULE><RULE><LHS>shuffle(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>shuffle(add(n,x))</LHS><RHS>add(n,shuffle(reverse(x)))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>reverse(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>reverse(add(n,x))</LHS><RHS>app(reverse(x),add(n,nil))</RHS></RULE><RULE><LHS>shuffle(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>shuffle(add(n,x))</LHS><RHS>add(n,shuffle(reverse(x)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(add(n,x),y)</LHS><RHS>Marked_app(x,y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_reverse(add(n,x))</LHS><RHS>Marked_app(reverse(x),add(n,nil))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_reverse(add(n,x))</LHS><RHS>Marked_reverse(x)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_shuffle(add(n,x))</LHS><RHS>Marked_shuffle(reverse(x))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_shuffle(add(n,x))</LHS><RHS>Marked_reverse(x)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="4"/></NONSCC><SCC num="1"><NODE ref="2"/></SCC><NONSCC num="2"><NODE ref="1"/></NONSCC><SCC num="3"><NODE ref="3"/></SCC><SCC num="4"><NODE ref="0"/></SCC></CCLIST><EDGE end="2" start="0"/><EDGE end="1" start="0"/><EDGE end="2" start="1"/><EDGE end="4" start="2"/><EDGE end="0" start="3"/></DAG></CRITERION><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>reverse(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>reverse(add(n,x))</LHS><RHS>app(reverse(x),add(n,nil))</RHS></RULE><RULE><LHS>shuffle(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>shuffle(add(n,x))</LHS><RHS>add(n,shuffle(reverse(x)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_shuffle(add(n,x))</LHS><RHS>Marked_shuffle(reverse(x))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>app</NAME></SYMBOL><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></POLYSYMB><POLYSYMB><SYMBOL><NAME>shuffle</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>add</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="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>nil</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_shuffle</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>reverse</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>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>reverse(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>reverse(add(n,x))</LHS><RHS>app(reverse(x),add(n,nil))</RHS></RULE><RULE><LHS>shuffle(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>shuffle(add(n,x))</LHS><RHS>add(n,shuffle(reverse(x)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_reverse(add(n,x))</LHS><RHS>Marked_reverse(x)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>app</NAME></SYMBOL><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></POLYSYMB><POLYSYMB><SYMBOL><NAME>shuffle</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>add</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="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_reverse</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>nil</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>reverse</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>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>reverse(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>reverse(add(n,x))</LHS><RHS>app(reverse(x),add(n,nil))</RHS></RULE><RULE><LHS>shuffle(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>shuffle(add(n,x))</LHS><RHS>add(n,shuffle(reverse(x)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(add(n,x),y)</LHS><RHS>Marked_app(x,y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>app</NAME></SYMBOL><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></POLYSYMB><POLYSYMB><SYMBOL><NAME>shuffle</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>add</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="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>nil</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>reverse</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_app</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></ORDERING></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROOF>
