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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="2" unmarked=""><NAME>cons</NAME></SYMBOL><SYMBOL arity="2" unmarked="f"><NAME>Marked_f</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>empty</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>f</NAME></SYMBOL><SYMBOL arity="2" unmarked="g"><NAME>Marked_g</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>g</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>k</VAR><VAR>x</VAR><VAR>d</VAR><VAR>a</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>f(a,empty)</LHS><RHS>g(a,empty)</RHS></RULE><RULE><LHS>f(a,cons(x,k))</LHS><RHS>f(cons(x,a),k)</RHS></RULE><RULE><LHS>g(empty,d)</LHS><RHS>d</RHS></RULE><RULE><LHS>g(cons(x,k),d)</LHS><RHS>g(k,cons(x,d))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>f(a,empty)</LHS><RHS>g(a,empty)</RHS></RULE><RULE><LHS>f(a,cons(x,k))</LHS><RHS>f(cons(x,a),k)</RHS></RULE><RULE><LHS>g(empty,d)</LHS><RHS>d</RHS></RULE><RULE><LHS>g(cons(x,k),d)</LHS><RHS>g(k,cons(x,d))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_f(a,empty)</LHS><RHS>Marked_g(a,empty)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_f(a,cons(x,k))</LHS><RHS>Marked_f(cons(x,a),k)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_g(cons(x,k),d)</LHS><RHS>Marked_g(k,cons(x,d))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="2"/></SCC><SCC num="1"><NODE ref="1"/></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>f(a,empty)</LHS><RHS>g(a,empty)</RHS></RULE><RULE><LHS>f(a,cons(x,k))</LHS><RHS>f(cons(x,a),k)</RHS></RULE><RULE><LHS>g(empty,d)</LHS><RHS>d</RHS></RULE><RULE><LHS>g(cons(x,k),d)</LHS><RHS>g(k,cons(x,d))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_f(a,cons(x,k))</LHS><RHS>Marked_f(cons(x,a),k)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_f</NAME></SYMBOL><INT>2</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_f(a,cons(x,k))</LHS><RHS>Marked_f(cons(x,a),k)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>f(a,empty)</LHS><RHS>g(a,empty)</RHS></RULE><RULE><LHS>f(a,cons(x,k))</LHS><RHS>f(cons(x,a),k)</RHS></RULE><RULE><LHS>g(empty,d)</LHS><RHS>d</RHS></RULE><RULE><LHS>g(cons(x,k),d)</LHS><RHS>g(k,cons(x,d))</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>f(a,empty)</LHS><RHS>g(a,empty)</RHS></RULE><RULE><LHS>f(a,cons(x,k))</LHS><RHS>f(cons(x,a),k)</RHS></RULE><RULE><LHS>g(empty,d)</LHS><RHS>d</RHS></RULE><RULE><LHS>g(cons(x,k),d)</LHS><RHS>g(k,cons(x,d))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_g(cons(x,k),d)</LHS><RHS>Marked_g(k,cons(x,d))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_g</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_g(cons(x,k),d)</LHS><RHS>Marked_g(k,cons(x,d))</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>f(a,empty)</LHS><RHS>g(a,empty)</RHS></RULE><RULE><LHS>f(a,cons(x,k))</LHS><RHS>f(cons(x,a),k)</RHS></RULE><RULE><LHS>g(empty,d)</LHS><RHS>d</RHS></RULE><RULE><LHS>g(cons(x,k),d)</LHS><RHS>g(k,cons(x,d))</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
