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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>true</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>false</NAME></SYMBOL><SYMBOL arity="4" unmarked="g"><NAME>Marked_g</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>&lt;=</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="3" unmarked="f"><NAME>Marked_f</NAME></SYMBOL><SYMBOL arity="3" unmarked=""><NAME>f</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>p</NAME></SYMBOL><SYMBOL arity="1" unmarked="p"><NAME>Marked_p</NAME></SYMBOL><SYMBOL arity="4" unmarked=""><NAME>g</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>z</VAR><VAR>y</VAR><VAR>x</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>f(x,y,z)</LHS><RHS>g(&lt;=(x,y),x,y,z)</RHS></RULE><RULE><LHS>g(true,x,y,z)</LHS><RHS>z</RHS></RULE><RULE><LHS>g(false,x,y,z)</LHS><RHS>f(f(p(x),y,z),f(p(y),z,x),f(p(z),x,y))</RHS></RULE><RULE><LHS>p(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>p(s(x))</LHS><RHS>x</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>f(x,y,z)</LHS><RHS>g(&lt;=(x,y),x,y,z)</RHS></RULE><RULE><LHS>g(true,x,y,z)</LHS><RHS>z</RHS></RULE><RULE><LHS>g(false,x,y,z)</LHS><RHS>f(f(p(x),y,z),f(p(y),z,x),f(p(z),x,y))</RHS></RULE><RULE><LHS>p(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>p(s(x))</LHS><RHS>x</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_f(x,y,z)</LHS><RHS>Marked_g(&lt;=(x,y),x,y,z)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_g(false,x,y,z)</LHS><RHS>Marked_f(f(p(x),y,z),f(p(y),z,x),f(p(z),x,y))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_g(false,x,y,z)</LHS><RHS>Marked_f(p(x),y,z)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_g(false,x,y,z)</LHS><RHS>Marked_p(x)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_g(false,x,y,z)</LHS><RHS>Marked_f(p(y),z,x)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_g(false,x,y,z)</LHS><RHS>Marked_p(y)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_g(false,x,y,z)</LHS><RHS>Marked_f(p(z),x,y)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_g(false,x,y,z)</LHS><RHS>Marked_p(z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="7"/></NONSCC><NONSCC num="1"><NODE ref="6"/></NONSCC><NONSCC num="2"><NODE ref="0"/></NONSCC><NONSCC num="3"><NODE ref="5"/></NONSCC><NONSCC num="4"><NODE ref="4"/></NONSCC><NONSCC num="5"><NODE ref="3"/></NONSCC><NONSCC num="6"><NODE ref="2"/></NONSCC><NONSCC num="7"><NODE ref="1"/></NONSCC></CCLIST><EDGE end="2" start="1"/><EDGE end="2" start="4"/><EDGE end="2" start="6"/><EDGE end="2" start="7"/></DAG></CRITERION></PROPERTY></PROPERTY></PROOF>
