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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="3" unmarked=""><NAME>if</NAME></SYMBOL><SYMBOL arity="4" unmarked=""><NAME>f</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>true</NAME></SYMBOL><SYMBOL arity="4" unmarked="f"><NAME>Marked_f</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>perfectp</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>minus</NAME></SYMBOL><SYMBOL arity="1" unmarked="perfectp"><NAME>Marked_perfectp</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>le</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>false</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>z</VAR><VAR>y</VAR><VAR>x</VAR><VAR>u</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>perfectp(0)</LHS><RHS>false</RHS></RULE><RULE><LHS>perfectp(s(x))</LHS><RHS>f(x,s(0),s(x),s(x))</RHS></RULE><RULE><LHS>f(0,y,0,u)</LHS><RHS>true</RHS></RULE><RULE><LHS>f(0,y,s(z),u)</LHS><RHS>false</RHS></RULE><RULE><LHS>f(s(x),0,z,u)</LHS><RHS>f(x,u,minus(z,s(x)),u)</RHS></RULE><RULE><LHS>f(s(x),s(y),z,u)</LHS><RHS>if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>perfectp(0)</LHS><RHS>false</RHS></RULE><RULE><LHS>perfectp(s(x))</LHS><RHS>f(x,s(0),s(x),s(x))</RHS></RULE><RULE><LHS>f(0,y,0,u)</LHS><RHS>true</RHS></RULE><RULE><LHS>f(0,y,s(z),u)</LHS><RHS>false</RHS></RULE><RULE><LHS>f(s(x),0,z,u)</LHS><RHS>f(x,u,minus(z,s(x)),u)</RHS></RULE><RULE><LHS>f(s(x),s(y),z,u)</LHS><RHS>if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_perfectp(s(x))</LHS><RHS>Marked_f(x,s(0),s(x),s(x))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_f(s(x),0,z,u)</LHS><RHS>Marked_f(x,u,minus(z,s(x)),u)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_f(s(x),s(y),z,u)</LHS><RHS>Marked_f(s(x),minus(y,x),z,u)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_f(s(x),s(y),z,u)</LHS><RHS>Marked_f(x,u,z,u)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="1"/><NODE ref="3"/></SCC><NONSCC num="1"><NODE ref="2"/></NONSCC><NONSCC num="2"><NODE ref="0"/></NONSCC></CCLIST><EDGE end="1" start="0"/><EDGE end="1" start="2"/><EDGE end="0" start="2"/></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>perfectp(0)</LHS><RHS>false</RHS></RULE><RULE><LHS>perfectp(s(x))</LHS><RHS>f(x,s(0),s(x),s(x))</RHS></RULE><RULE><LHS>f(0,y,0,u)</LHS><RHS>true</RHS></RULE><RULE><LHS>f(0,y,s(z),u)</LHS><RHS>false</RHS></RULE><RULE><LHS>f(s(x),0,z,u)</LHS><RHS>f(x,u,minus(z,s(x)),u)</RHS></RULE><RULE><LHS>f(s(x),s(y),z,u)</LHS><RHS>if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_f(s(x),0,z,u)</LHS><RHS>Marked_f(x,u,minus(z,s(x)),u)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_f(s(x),s(y),z,u)</LHS><RHS>Marked_f(x,u,z,u)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_f</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_f(s(x),s(y),z,u)</LHS><RHS>Marked_f(x,u,z,u)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_f(s(x),0,z,u)</LHS><RHS>Marked_f(x,u,minus(z,s(x)),u)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>perfectp(0)</LHS><RHS>false</RHS></RULE><RULE><LHS>perfectp(s(x))</LHS><RHS>f(x,s(0),s(x),s(x))</RHS></RULE><RULE><LHS>f(0,y,0,u)</LHS><RHS>true</RHS></RULE><RULE><LHS>f(0,y,s(z),u)</LHS><RHS>false</RHS></RULE><RULE><LHS>f(s(x),0,z,u)</LHS><RHS>f(x,u,minus(z,s(x)),u)</RHS></RULE><RULE><LHS>f(s(x),s(y),z,u)</LHS><RHS>if(le(x,y),f(s(x),minus(y,x),z,u),f(x,u,z,u))</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
