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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="1" unmarked=""><NAME>pred</NAME></SYMBOL><SYMBOL arity="2" unmarked="app"><NAME>Marked_app</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>sum</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="1" unmarked="sum"><NAME>Marked_sum</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>nil</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="2" unmarked="plus"><NAME>Marked_plus</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>cons</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>plus</NAME></SYMBOL><SYMBOL arity="1" unmarked="pred"><NAME>Marked_pred</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>app</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>y</VAR><VAR>x</VAR><VAR>l</VAR><VAR>k</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>app(nil,k)</LHS><RHS>k</RHS></RULE><RULE><LHS>app(l,nil)</LHS><RHS>l</RHS></RULE><RULE><LHS>app(cons(x,l),k)</LHS><RHS>cons(x,app(l,k))</RHS></RULE><RULE><LHS>sum(cons(x,nil))</LHS><RHS>cons(x,nil)</RHS></RULE><RULE><LHS>sum(cons(x,cons(y,l)))</LHS><RHS>sum(cons(plus(x,y),l))</RHS></RULE><RULE><LHS>sum(app(l,cons(x,cons(y,k))))</LHS><RHS>sum(app(l,sum(cons(x,cons(y,k)))))</RHS></RULE><RULE><LHS>plus(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(x,y))</RHS></RULE><RULE><LHS>sum(plus(cons(0,x),cons(y,l)))</LHS><RHS>pred(sum(cons(s(x),cons(y,l))))</RHS></RULE><RULE><LHS>pred(cons(s(x),nil))</LHS><RHS>cons(x,nil)</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(nil,k)</LHS><RHS>k</RHS></RULE><RULE><LHS>app(l,nil)</LHS><RHS>l</RHS></RULE><RULE><LHS>app(cons(x,l),k)</LHS><RHS>cons(x,app(l,k))</RHS></RULE><RULE><LHS>sum(cons(x,nil))</LHS><RHS>cons(x,nil)</RHS></RULE><RULE><LHS>sum(cons(x,cons(y,l)))</LHS><RHS>sum(cons(plus(x,y),l))</RHS></RULE><RULE><LHS>sum(app(l,cons(x,cons(y,k))))</LHS><RHS>sum(app(l,sum(cons(x,cons(y,k)))))</RHS></RULE><RULE><LHS>plus(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(x,y))</RHS></RULE><RULE><LHS>sum(plus(cons(0,x),cons(y,l)))</LHS><RHS>pred(sum(cons(s(x),cons(y,l))))</RHS></RULE><RULE><LHS>pred(cons(s(x),nil))</LHS><RHS>cons(x,nil)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(cons(x,l),k)</LHS><RHS>Marked_app(l,k)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_sum(cons(x,cons(y,l)))</LHS><RHS>Marked_sum(cons(plus(x,y),l))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_sum(cons(x,cons(y,l)))</LHS><RHS>Marked_plus(x,y)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_sum(app(l,cons(x,cons(y,k))))</LHS><RHS>Marked_sum(app(l,sum(cons(x,cons(y,k)))))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_sum(app(l,cons(x,cons(y,k))))</LHS><RHS>Marked_app(l,sum(cons(x,cons(y,k))))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_sum(app(l,cons(x,cons(y,k))))</LHS><RHS>Marked_sum(cons(x,cons(y,k)))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_plus(s(x),y)</LHS><RHS>Marked_plus(x,y)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_sum(plus(cons(0,x),cons(y,l)))</LHS><RHS>Marked_pred(sum(cons(s(x),cons(y,l))))</RHS></DPRULE><DPRULE num="8"><LHS>Marked_sum(plus(cons(0,x),cons(y,l)))</LHS><RHS>Marked_sum(cons(s(x),cons(y,l)))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="8"/></NONSCC><NONSCC num="1"><NODE ref="2"/></NONSCC><SCC num="2"><NODE ref="1"/></SCC><NONSCC num="3"><NODE ref="7"/></NONSCC><SCC num="4"><NODE ref="6"/></SCC><NONSCC num="5"><NODE ref="5"/></NONSCC><NONSCC num="6"><NODE ref="4"/></NONSCC><SCC num="7"><NODE ref="0"/></SCC><SCC num="8"><NODE ref="3"/></SCC></CCLIST><EDGE end="2" start="0"/><EDGE end="1" start="0"/><EDGE end="4" start="1"/><EDGE end="1" start="2"/><EDGE end="2" start="5"/><EDGE end="1" start="5"/><EDGE end="7" start="6"/><EDGE end="6" start="8"/><EDGE end="5" start="8"/><EDGE end="3" start="8"/><EDGE end="2" start="8"/><EDGE end="1" start="8"/><EDGE end="0" start="8"/></DAG></CRITERION><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(nil,k)</LHS><RHS>k</RHS></RULE><RULE><LHS>app(l,nil)</LHS><RHS>l</RHS></RULE><RULE><LHS>app(cons(x,l),k)</LHS><RHS>cons(x,app(l,k))</RHS></RULE><RULE><LHS>sum(cons(x,nil))</LHS><RHS>cons(x,nil)</RHS></RULE><RULE><LHS>sum(cons(x,cons(y,l)))</LHS><RHS>sum(cons(plus(x,y),l))</RHS></RULE><RULE><LHS>sum(app(l,cons(x,cons(y,k))))</LHS><RHS>sum(app(l,sum(cons(x,cons(y,k)))))</RHS></RULE><RULE><LHS>plus(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(x,y))</RHS></RULE><RULE><LHS>sum(plus(cons(0,x),cons(y,l)))</LHS><RHS>pred(sum(cons(s(x),cons(y,l))))</RHS></RULE><RULE><LHS>pred(cons(s(x),nil))</LHS><RHS>cons(x,nil)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_sum(app(l,cons(x,cons(y,k))))</LHS><RHS>Marked_sum(app(l,sum(cons(x,cons(y,k)))))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>app</NAME></SYMBOL><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></POLYSYMB><POLYSYMB><SYMBOL><NAME>plus</NAME></SYMBOL><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></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>s</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>nil</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_sum</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>sum</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>pred</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="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(nil,k)</LHS><RHS>k</RHS></RULE><RULE><LHS>app(l,nil)</LHS><RHS>l</RHS></RULE><RULE><LHS>app(cons(x,l),k)</LHS><RHS>cons(x,app(l,k))</RHS></RULE><RULE><LHS>sum(cons(x,nil))</LHS><RHS>cons(x,nil)</RHS></RULE><RULE><LHS>sum(cons(x,cons(y,l)))</LHS><RHS>sum(cons(plus(x,y),l))</RHS></RULE><RULE><LHS>sum(app(l,cons(x,cons(y,k))))</LHS><RHS>sum(app(l,sum(cons(x,cons(y,k)))))</RHS></RULE><RULE><LHS>plus(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(x,y))</RHS></RULE><RULE><LHS>sum(plus(cons(0,x),cons(y,l)))</LHS><RHS>pred(sum(cons(s(x),cons(y,l))))</RHS></RULE><RULE><LHS>pred(cons(s(x),nil))</LHS><RHS>cons(x,nil)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(cons(x,l),k)</LHS><RHS>Marked_app(l,k)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_app</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_app(cons(x,l),k)</LHS><RHS>Marked_app(l,k)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(nil,k)</LHS><RHS>k</RHS></RULE><RULE><LHS>app(l,nil)</LHS><RHS>l</RHS></RULE><RULE><LHS>app(cons(x,l),k)</LHS><RHS>cons(x,app(l,k))</RHS></RULE><RULE><LHS>sum(cons(x,nil))</LHS><RHS>cons(x,nil)</RHS></RULE><RULE><LHS>sum(cons(x,cons(y,l)))</LHS><RHS>sum(cons(plus(x,y),l))</RHS></RULE><RULE><LHS>sum(app(l,cons(x,cons(y,k))))</LHS><RHS>sum(app(l,sum(cons(x,cons(y,k)))))</RHS></RULE><RULE><LHS>plus(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(x,y))</RHS></RULE><RULE><LHS>sum(plus(cons(0,x),cons(y,l)))</LHS><RHS>pred(sum(cons(s(x),cons(y,l))))</RHS></RULE><RULE><LHS>pred(cons(s(x),nil))</LHS><RHS>cons(x,nil)</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(nil,k)</LHS><RHS>k</RHS></RULE><RULE><LHS>app(l,nil)</LHS><RHS>l</RHS></RULE><RULE><LHS>app(cons(x,l),k)</LHS><RHS>cons(x,app(l,k))</RHS></RULE><RULE><LHS>sum(cons(x,nil))</LHS><RHS>cons(x,nil)</RHS></RULE><RULE><LHS>sum(cons(x,cons(y,l)))</LHS><RHS>sum(cons(plus(x,y),l))</RHS></RULE><RULE><LHS>sum(app(l,cons(x,cons(y,k))))</LHS><RHS>sum(app(l,sum(cons(x,cons(y,k)))))</RHS></RULE><RULE><LHS>plus(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(x,y))</RHS></RULE><RULE><LHS>sum(plus(cons(0,x),cons(y,l)))</LHS><RHS>pred(sum(cons(s(x),cons(y,l))))</RHS></RULE><RULE><LHS>pred(cons(s(x),nil))</LHS><RHS>cons(x,nil)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_sum(cons(x,cons(y,l)))</LHS><RHS>Marked_sum(cons(plus(x,y),l))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>app</NAME></SYMBOL><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></POLYSYMB><POLYSYMB><SYMBOL><NAME>plus</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>cons</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>s</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>nil</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_sum</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>0</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>sum</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><POLYSYMB><SYMBOL><NAME>pred</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="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(nil,k)</LHS><RHS>k</RHS></RULE><RULE><LHS>app(l,nil)</LHS><RHS>l</RHS></RULE><RULE><LHS>app(cons(x,l),k)</LHS><RHS>cons(x,app(l,k))</RHS></RULE><RULE><LHS>sum(cons(x,nil))</LHS><RHS>cons(x,nil)</RHS></RULE><RULE><LHS>sum(cons(x,cons(y,l)))</LHS><RHS>sum(cons(plus(x,y),l))</RHS></RULE><RULE><LHS>sum(app(l,cons(x,cons(y,k))))</LHS><RHS>sum(app(l,sum(cons(x,cons(y,k)))))</RHS></RULE><RULE><LHS>plus(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(x,y))</RHS></RULE><RULE><LHS>sum(plus(cons(0,x),cons(y,l)))</LHS><RHS>pred(sum(cons(s(x),cons(y,l))))</RHS></RULE><RULE><LHS>pred(cons(s(x),nil))</LHS><RHS>cons(x,nil)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_plus(s(x),y)</LHS><RHS>Marked_plus(x,y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_plus</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_plus(s(x),y)</LHS><RHS>Marked_plus(x,y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(nil,k)</LHS><RHS>k</RHS></RULE><RULE><LHS>app(l,nil)</LHS><RHS>l</RHS></RULE><RULE><LHS>app(cons(x,l),k)</LHS><RHS>cons(x,app(l,k))</RHS></RULE><RULE><LHS>sum(cons(x,nil))</LHS><RHS>cons(x,nil)</RHS></RULE><RULE><LHS>sum(cons(x,cons(y,l)))</LHS><RHS>sum(cons(plus(x,y),l))</RHS></RULE><RULE><LHS>sum(app(l,cons(x,cons(y,k))))</LHS><RHS>sum(app(l,sum(cons(x,cons(y,k)))))</RHS></RULE><RULE><LHS>plus(0,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>plus(s(x),y)</LHS><RHS>s(plus(x,y))</RHS></RULE><RULE><LHS>sum(plus(cons(0,x),cons(y,l)))</LHS><RHS>pred(sum(cons(s(x),cons(y,l))))</RHS></RULE><RULE><LHS>pred(cons(s(x),nil))</LHS><RHS>cons(x,nil)</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
