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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="2" unmarked=""><NAME>union</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>mem</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>=</NAME></SYMBOL><SYMBOL arity="2" unmarked="or"><NAME>Marked_or</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>or</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>set</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>false</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>nil</NAME></SYMBOL><SYMBOL arity="2" unmarked="mem"><NAME>Marked_mem</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>true</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>z</VAR><VAR>y</VAR><VAR>x</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>or(true,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>or(x,true)</LHS><RHS>true</RHS></RULE><RULE><LHS>or(false,false)</LHS><RHS>false</RHS></RULE><RULE><LHS>mem(x,nil)</LHS><RHS>false</RHS></RULE><RULE><LHS>mem(x,set(y))</LHS><RHS>=(x,y)</RHS></RULE><RULE><LHS>mem(x,union(y,z))</LHS><RHS>or(mem(x,y),mem(x,z))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>or(true,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>or(x,true)</LHS><RHS>true</RHS></RULE><RULE><LHS>or(false,false)</LHS><RHS>false</RHS></RULE><RULE><LHS>mem(x,nil)</LHS><RHS>false</RHS></RULE><RULE><LHS>mem(x,set(y))</LHS><RHS>=(x,y)</RHS></RULE><RULE><LHS>mem(x,union(y,z))</LHS><RHS>or(mem(x,y),mem(x,z))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_mem(x,union(y,z))</LHS><RHS>Marked_or(mem(x,y),mem(x,z))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_mem(x,union(y,z))</LHS><RHS>Marked_mem(x,y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_mem(x,union(y,z))</LHS><RHS>Marked_mem(x,z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="1"/><NODE ref="2"/></SCC><NONSCC num="1"><NODE ref="0"/></NONSCC></CCLIST><EDGE end="1" start="0"/></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>or(true,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>or(x,true)</LHS><RHS>true</RHS></RULE><RULE><LHS>or(false,false)</LHS><RHS>false</RHS></RULE><RULE><LHS>mem(x,nil)</LHS><RHS>false</RHS></RULE><RULE><LHS>mem(x,set(y))</LHS><RHS>=(x,y)</RHS></RULE><RULE><LHS>mem(x,union(y,z))</LHS><RHS>or(mem(x,y),mem(x,z))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_mem(x,union(y,z))</LHS><RHS>Marked_mem(x,y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_mem(x,union(y,z))</LHS><RHS>Marked_mem(x,z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_mem</NAME></SYMBOL><INT>2</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_mem(x,union(y,z))</LHS><RHS>Marked_mem(x,z)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_mem(x,union(y,z))</LHS><RHS>Marked_mem(x,y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>or(true,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>or(x,true)</LHS><RHS>true</RHS></RULE><RULE><LHS>or(false,false)</LHS><RHS>false</RHS></RULE><RULE><LHS>mem(x,nil)</LHS><RHS>false</RHS></RULE><RULE><LHS>mem(x,set(y))</LHS><RHS>=(x,y)</RHS></RULE><RULE><LHS>mem(x,union(y,z))</LHS><RHS>or(mem(x,y),mem(x,z))</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
