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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="1" unmarked="rev"><NAME>Marked_rev</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>.</NAME></SYMBOL><SYMBOL arity="1" unmarked="make"><NAME>Marked_make</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>rev</NAME></SYMBOL><SYMBOL arity="2" unmarked="++"><NAME>Marked_++</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>++</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>make</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>nil</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>z</VAR><VAR>y</VAR><VAR>x</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>rev(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rev(rev(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>rev(++(x,y))</LHS><RHS>++(rev(y),rev(x))</RHS></RULE><RULE><LHS>++(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>++(x,nil)</LHS><RHS>x</RHS></RULE><RULE><LHS>++(.(x,y),z)</LHS><RHS>.(x,++(y,z))</RHS></RULE><RULE><LHS>++(x,++(y,z))</LHS><RHS>++(++(x,y),z)</RHS></RULE><RULE><LHS>make(x)</LHS><RHS>.(x,nil)</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>rev(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rev(rev(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>rev(++(x,y))</LHS><RHS>++(rev(y),rev(x))</RHS></RULE><RULE><LHS>++(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>++(x,nil)</LHS><RHS>x</RHS></RULE><RULE><LHS>++(.(x,y),z)</LHS><RHS>.(x,++(y,z))</RHS></RULE><RULE><LHS>++(x,++(y,z))</LHS><RHS>++(++(x,y),z)</RHS></RULE><RULE><LHS>make(x)</LHS><RHS>.(x,nil)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_rev(++(x,y))</LHS><RHS>Marked_++(rev(y),rev(x))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_rev(++(x,y))</LHS><RHS>Marked_rev(y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_rev(++(x,y))</LHS><RHS>Marked_rev(x)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_++(.(x,y),z)</LHS><RHS>Marked_++(y,z)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_++(x,++(y,z))</LHS><RHS>Marked_++(++(x,y),z)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_++(x,++(y,z))</LHS><RHS>Marked_++(x,y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="3"/><NODE ref="4"/><NODE ref="5"/></SCC><SCC num="1"><NODE ref="1"/><NODE ref="2"/></SCC><NONSCC num="2"><NODE ref="0"/></NONSCC></CCLIST><EDGE end="2" start="1"/><EDGE end="0" start="2"/></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>rev(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rev(rev(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>rev(++(x,y))</LHS><RHS>++(rev(y),rev(x))</RHS></RULE><RULE><LHS>++(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>++(x,nil)</LHS><RHS>x</RHS></RULE><RULE><LHS>++(.(x,y),z)</LHS><RHS>.(x,++(y,z))</RHS></RULE><RULE><LHS>++(x,++(y,z))</LHS><RHS>++(++(x,y),z)</RHS></RULE><RULE><LHS>make(x)</LHS><RHS>.(x,nil)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_rev(++(x,y))</LHS><RHS>Marked_rev(y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_rev(++(x,y))</LHS><RHS>Marked_rev(x)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_rev</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_rev(++(x,y))</LHS><RHS>Marked_rev(x)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_rev(++(x,y))</LHS><RHS>Marked_rev(y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>rev(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rev(rev(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>rev(++(x,y))</LHS><RHS>++(rev(y),rev(x))</RHS></RULE><RULE><LHS>++(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>++(x,nil)</LHS><RHS>x</RHS></RULE><RULE><LHS>++(.(x,y),z)</LHS><RHS>.(x,++(y,z))</RHS></RULE><RULE><LHS>++(x,++(y,z))</LHS><RHS>++(++(x,y),z)</RHS></RULE><RULE><LHS>make(x)</LHS><RHS>.(x,nil)</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>rev(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rev(rev(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>rev(++(x,y))</LHS><RHS>++(rev(y),rev(x))</RHS></RULE><RULE><LHS>++(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>++(x,nil)</LHS><RHS>x</RHS></RULE><RULE><LHS>++(.(x,y),z)</LHS><RHS>.(x,++(y,z))</RHS></RULE><RULE><LHS>++(x,++(y,z))</LHS><RHS>++(++(x,y),z)</RHS></RULE><RULE><LHS>make(x)</LHS><RHS>.(x,nil)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_++(.(x,y),z)</LHS><RHS>Marked_++(y,z)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_++(x,++(y,z))</LHS><RHS>Marked_++(++(x,y),z)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_++(x,++(y,z))</LHS><RHS>Marked_++(x,y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_++</NAME></SYMBOL><INT>2</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_++(x,++(y,z))</LHS><RHS>Marked_++(x,y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_++(x,++(y,z))</LHS><RHS>Marked_++(++(x,y),z)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>rev(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rev(rev(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>rev(++(x,y))</LHS><RHS>++(rev(y),rev(x))</RHS></RULE><RULE><LHS>++(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>++(x,nil)</LHS><RHS>x</RHS></RULE><RULE><LHS>++(.(x,y),z)</LHS><RHS>.(x,++(y,z))</RHS></RULE><RULE><LHS>++(x,++(y,z))</LHS><RHS>++(++(x,y),z)</RHS></RULE><RULE><LHS>make(x)</LHS><RHS>.(x,nil)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_++(.(x,y),z)</LHS><RHS>Marked_++(y,z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/></SCC></CCLIST></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>rev(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rev(rev(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>rev(++(x,y))</LHS><RHS>++(rev(y),rev(x))</RHS></RULE><RULE><LHS>++(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>++(x,nil)</LHS><RHS>x</RHS></RULE><RULE><LHS>++(.(x,y),z)</LHS><RHS>.(x,++(y,z))</RHS></RULE><RULE><LHS>++(x,++(y,z))</LHS><RHS>++(++(x,y),z)</RHS></RULE><RULE><LHS>make(x)</LHS><RHS>.(x,nil)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_++(.(x,y),z)</LHS><RHS>Marked_++(y,z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_++</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_++(.(x,y),z)</LHS><RHS>Marked_++(y,z)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>rev(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rev(rev(x))</LHS><RHS>x</RHS></RULE><RULE><LHS>rev(++(x,y))</LHS><RHS>++(rev(y),rev(x))</RHS></RULE><RULE><LHS>++(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>++(x,nil)</LHS><RHS>x</RHS></RULE><RULE><LHS>++(.(x,y),z)</LHS><RHS>.(x,++(y,z))</RHS></RULE><RULE><LHS>++(x,++(y,z))</LHS><RHS>++(++(x,y),z)</RHS></RULE><RULE><LHS>make(x)</LHS><RHS>.(x,nil)</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
