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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="0" unmarked=""><NAME>node</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>nil</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>leaf</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>cons</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>mapt</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>maptlist</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>app</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>xs</VAR><VAR>x</VAR><VAR>f</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>app(app(mapt,f),app(leaf,x))</LHS><RHS>app(leaf,app(f,x))</RHS></RULE><RULE><LHS>app(app(mapt,f),app(node,xs))</LHS><RHS>app(node,app(app(maptlist,f),xs))</RHS></RULE><RULE><LHS>app(app(maptlist,f),nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>app(app(cons,app(app(mapt,f),x)),app(app(maptlist,f),xs))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(mapt,f),app(leaf,x))</LHS><RHS>app(leaf,app(f,x))</RHS></RULE><RULE><LHS>app(app(mapt,f),app(node,xs))</LHS><RHS>app(node,app(app(maptlist,f),xs))</RHS></RULE><RULE><LHS>app(app(maptlist,f),nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>app(app(cons,app(app(mapt,f),x)),app(app(maptlist,f),xs))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(mapt,f),app(leaf,x))</LHS><RHS>Marked_app(leaf,app(f,x))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(mapt,f),app(leaf,x))</LHS><RHS>Marked_app(f,x)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(app(mapt,f),app(node,xs))</LHS><RHS>Marked_app(node,app(app(maptlist,f),xs))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_app(app(mapt,f),app(node,xs))</LHS><RHS>Marked_app(app(maptlist,f),xs)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_app(app(mapt,f),app(node,xs))</LHS><RHS>Marked_app(maptlist,f)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>Marked_app(app(cons,app(app(mapt,f),x)),app(app(maptlist,f),xs))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>Marked_app(cons,app(app(mapt,f),x))</RHS></DPRULE><DPRULE num="7"><LHS>Marked_app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>Marked_app(app(mapt,f),x)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>Marked_app(mapt,f)</RHS></DPRULE><DPRULE num="9"><LHS>Marked_app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>Marked_app(app(maptlist,f),xs)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="1"/><NODE ref="3"/><NODE ref="5"/><NODE ref="7"/><NODE ref="9"/></SCC><NONSCC num="1"><NODE ref="8"/></NONSCC><NONSCC num="2"><NODE ref="6"/></NONSCC><NONSCC num="3"><NODE ref="4"/></NONSCC><NONSCC num="4"><NODE ref="2"/></NONSCC><NONSCC num="5"><NODE ref="0"/></NONSCC></CCLIST><EDGE end="5" start="0"/><EDGE end="4" start="0"/><EDGE end="3" start="0"/><EDGE end="2" start="0"/><EDGE end="1" start="0"/></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(mapt,f),app(leaf,x))</LHS><RHS>app(leaf,app(f,x))</RHS></RULE><RULE><LHS>app(app(mapt,f),app(node,xs))</LHS><RHS>app(node,app(app(maptlist,f),xs))</RHS></RULE><RULE><LHS>app(app(maptlist,f),nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>app(app(cons,app(app(mapt,f),x)),app(app(maptlist,f),xs))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(mapt,f),app(leaf,x))</LHS><RHS>Marked_app(f,x)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(mapt,f),app(node,xs))</LHS><RHS>Marked_app(app(maptlist,f),xs)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>Marked_app(app(cons,app(app(mapt,f),x)),app(app(maptlist,f),xs))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>Marked_app(app(mapt,f),x)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>Marked_app(app(maptlist,f),xs)</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>2</INT></COEF><ARG degree="1" num="0"/><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>maptlist</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>mapt</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>cons</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>leaf</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>nil</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>node</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_app</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></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(mapt,f),app(leaf,x))</LHS><RHS>Marked_app(f,x)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>Marked_app(app(cons,app(app(mapt,f),x)),app(app(maptlist,f),xs))</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(mapt,f),app(leaf,x))</LHS><RHS>app(leaf,app(f,x))</RHS></RULE><RULE><LHS>app(app(mapt,f),app(node,xs))</LHS><RHS>app(node,app(app(maptlist,f),xs))</RHS></RULE><RULE><LHS>app(app(maptlist,f),nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>app(app(cons,app(app(mapt,f),x)),app(app(maptlist,f),xs))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(mapt,f),app(node,xs))</LHS><RHS>Marked_app(app(maptlist,f),xs)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>Marked_app(app(mapt,f),x)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>Marked_app(app(maptlist,f),xs)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/><NODE ref="1"/><NODE ref="2"/></SCC></CCLIST></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(mapt,f),app(leaf,x))</LHS><RHS>app(leaf,app(f,x))</RHS></RULE><RULE><LHS>app(app(mapt,f),app(node,xs))</LHS><RHS>app(node,app(app(maptlist,f),xs))</RHS></RULE><RULE><LHS>app(app(maptlist,f),nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>app(app(cons,app(app(mapt,f),x)),app(app(maptlist,f),xs))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(mapt,f),app(node,xs))</LHS><RHS>Marked_app(app(maptlist,f),xs)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>Marked_app(app(mapt,f),x)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>Marked_app(app(maptlist,f),xs)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_app</NAME></SYMBOL><INT>2</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>Marked_app(app(maptlist,f),xs)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>Marked_app(app(mapt,f),x)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(app(mapt,f),app(node,xs))</LHS><RHS>Marked_app(app(maptlist,f),xs)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(mapt,f),app(leaf,x))</LHS><RHS>app(leaf,app(f,x))</RHS></RULE><RULE><LHS>app(app(mapt,f),app(node,xs))</LHS><RHS>app(node,app(app(maptlist,f),xs))</RHS></RULE><RULE><LHS>app(app(maptlist,f),nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>app(app(maptlist,f),app(app(cons,x),xs))</LHS><RHS>app(app(cons,app(app(mapt,f),x)),app(app(maptlist,f),xs))</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
