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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="2" unmarked=""><NAME>=</NAME></SYMBOL><SYMBOL arity="2" unmarked="implies"><NAME>Marked_implies</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>not</NAME></SYMBOL><SYMBOL arity="2" unmarked="and"><NAME>Marked_and</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>or</NAME></SYMBOL><SYMBOL arity="3" unmarked="if"><NAME>Marked_if</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>false</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>implies</NAME></SYMBOL><SYMBOL arity="1" unmarked="not"><NAME>Marked_not</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>true</NAME></SYMBOL><SYMBOL arity="2" unmarked="or"><NAME>Marked_or</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>and</NAME></SYMBOL><SYMBOL arity="2" unmarked="="><NAME>Marked_=</NAME></SYMBOL><SYMBOL arity="3" unmarked=""><NAME>if</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>z</VAR><VAR>y</VAR><VAR>x</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>not(x)</LHS><RHS>if(x,false,true)</RHS></RULE><RULE><LHS>and(x,y)</LHS><RHS>if(x,y,false)</RHS></RULE><RULE><LHS>or(x,y)</LHS><RHS>if(x,true,y)</RHS></RULE><RULE><LHS>implies(x,y)</LHS><RHS>if(x,y,true)</RHS></RULE><RULE><LHS>=(x,x)</LHS><RHS>true</RHS></RULE><RULE><LHS>=(x,y)</LHS><RHS>if(x,y,not(y))</RHS></RULE><RULE><LHS>if(true,x,y)</LHS><RHS>x</RHS></RULE><RULE><LHS>if(false,x,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>if(x,x,if(x,false,true))</LHS><RHS>true</RHS></RULE><RULE><LHS>=(x,y)</LHS><RHS>if(x,y,if(y,false,true))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>not(x)</LHS><RHS>if(x,false,true)</RHS></RULE><RULE><LHS>and(x,y)</LHS><RHS>if(x,y,false)</RHS></RULE><RULE><LHS>or(x,y)</LHS><RHS>if(x,true,y)</RHS></RULE><RULE><LHS>implies(x,y)</LHS><RHS>if(x,y,true)</RHS></RULE><RULE><LHS>=(x,x)</LHS><RHS>true</RHS></RULE><RULE><LHS>=(x,y)</LHS><RHS>if(x,y,not(y))</RHS></RULE><RULE><LHS>if(true,x,y)</LHS><RHS>x</RHS></RULE><RULE><LHS>if(false,x,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>if(x,x,if(x,false,true))</LHS><RHS>true</RHS></RULE><RULE><LHS>=(x,y)</LHS><RHS>if(x,y,if(y,false,true))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_not(x)</LHS><RHS>Marked_if(x,false,true)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_and(x,y)</LHS><RHS>Marked_if(x,y,false)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_or(x,y)</LHS><RHS>Marked_if(x,true,y)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_implies(x,y)</LHS><RHS>Marked_if(x,y,true)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_=(x,y)</LHS><RHS>Marked_if(x,y,not(y))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_=(x,y)</LHS><RHS>Marked_not(y)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_=(x,y)</LHS><RHS>Marked_if(x,y,if(y,false,true))</RHS></DPRULE><DPRULE num="7"><LHS>Marked_=(x,y)</LHS><RHS>Marked_if(y,false,true)</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="5"/></NONSCC><NONSCC num="3"><NODE ref="0"/></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="3" start="2"/></DAG></CRITERION></PROPERTY></PROPERTY></PROOF>
