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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="2" unmarked="concat"><NAME>Marked_concat</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>false</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>true</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>leaf</NAME></SYMBOL><SYMBOL arity="2" unmarked="less_leaves"><NAME>Marked_less_leaves</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>cons</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>less_leaves</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>concat</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>z</VAR><VAR>y</VAR><VAR>x</VAR><VAR>w</VAR><VAR>v</VAR><VAR>u</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>concat(leaf,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>concat(cons(u,v),y)</LHS><RHS>cons(u,concat(v,y))</RHS></RULE><RULE><LHS>less_leaves(x,leaf)</LHS><RHS>false</RHS></RULE><RULE><LHS>less_leaves(leaf,cons(w,z))</LHS><RHS>true</RHS></RULE><RULE><LHS>less_leaves(cons(u,v),cons(w,z))</LHS><RHS>less_leaves(concat(u,v),concat(w,z))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>concat(leaf,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>concat(cons(u,v),y)</LHS><RHS>cons(u,concat(v,y))</RHS></RULE><RULE><LHS>less_leaves(x,leaf)</LHS><RHS>false</RHS></RULE><RULE><LHS>less_leaves(leaf,cons(w,z))</LHS><RHS>true</RHS></RULE><RULE><LHS>less_leaves(cons(u,v),cons(w,z))</LHS><RHS>less_leaves(concat(u,v),concat(w,z))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_concat(cons(u,v),y)</LHS><RHS>Marked_concat(v,y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_less_leaves(cons(u,v),cons(w,z))</LHS><RHS>Marked_less_leaves(concat(u,v),concat(w,z))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_less_leaves(cons(u,v),cons(w,z))</LHS><RHS>Marked_concat(u,v)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_less_leaves(cons(u,v),cons(w,z))</LHS><RHS>Marked_concat(w,z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="3"/></NONSCC><SCC num="1"><NODE ref="0"/></SCC><NONSCC num="2"><NODE ref="2"/></NONSCC><SCC num="3"><NODE ref="1"/></SCC></CCLIST><EDGE end="1" start="0"/><EDGE end="1" start="2"/><EDGE end="2" start="3"/><EDGE end="0" start="3"/></DAG></CRITERION><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>concat(leaf,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>concat(cons(u,v),y)</LHS><RHS>cons(u,concat(v,y))</RHS></RULE><RULE><LHS>less_leaves(x,leaf)</LHS><RHS>false</RHS></RULE><RULE><LHS>less_leaves(leaf,cons(w,z))</LHS><RHS>true</RHS></RULE><RULE><LHS>less_leaves(cons(u,v),cons(w,z))</LHS><RHS>less_leaves(concat(u,v),concat(w,z))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_less_leaves(cons(u,v),cons(w,z))</LHS><RHS>Marked_less_leaves(concat(u,v),concat(w,z))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>concat</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>less_leaves</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</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>cons</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><POLYSYMB><SYMBOL><NAME>Marked_less_leaves</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>leaf</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>true</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>false</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></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>concat(leaf,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>concat(cons(u,v),y)</LHS><RHS>cons(u,concat(v,y))</RHS></RULE><RULE><LHS>less_leaves(x,leaf)</LHS><RHS>false</RHS></RULE><RULE><LHS>less_leaves(leaf,cons(w,z))</LHS><RHS>true</RHS></RULE><RULE><LHS>less_leaves(cons(u,v),cons(w,z))</LHS><RHS>less_leaves(concat(u,v),concat(w,z))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_concat(cons(u,v),y)</LHS><RHS>Marked_concat(v,y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>concat</NAME></SYMBOL><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></POLYSYMB><POLYSYMB><SYMBOL><NAME>less_leaves</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>cons</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>leaf</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>true</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>false</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_concat</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>
