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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="1" unmarked=""><NAME>u22</NAME></SYMBOL><SYMBOL arity="2" unmarked="u21"><NAME>Marked_u21</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>ackin</NAME></SYMBOL><SYMBOL arity="2" unmarked="ackin"><NAME>Marked_ackin</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>ackout</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>u21</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>Y</VAR><VAR>X</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>ackin(s(X),s(Y))</LHS><RHS>u21(ackin(s(X),Y),X)</RHS></RULE><RULE><LHS>u21(ackout(X),Y)</LHS><RHS>u22(ackin(Y,X))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>ackin(s(X),s(Y))</LHS><RHS>u21(ackin(s(X),Y),X)</RHS></RULE><RULE><LHS>u21(ackout(X),Y)</LHS><RHS>u22(ackin(Y,X))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_ackin(s(X),s(Y))</LHS><RHS>Marked_u21(ackin(s(X),Y),X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_ackin(s(X),s(Y))</LHS><RHS>Marked_ackin(s(X),Y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_u21(ackout(X),Y)</LHS><RHS>Marked_ackin(Y,X)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/><NODE ref="1"/><NODE ref="2"/></SCC></CCLIST></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>ackin(s(X),s(Y))</LHS><RHS>u21(ackin(s(X),Y),X)</RHS></RULE><RULE><LHS>u21(ackout(X),Y)</LHS><RHS>u22(ackin(Y,X))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_ackin(s(X),s(Y))</LHS><RHS>Marked_u21(ackin(s(X),Y),X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_ackin(s(X),s(Y))</LHS><RHS>Marked_ackin(s(X),Y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_u21(ackout(X),Y)</LHS><RHS>Marked_ackin(Y,X)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_u21</NAME></SYMBOL><INT>2</INT></PROJ><PROJ><SYMBOL><NAME>Marked_ackin</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_ackin(s(X),s(Y))</LHS><RHS>Marked_u21(ackin(s(X),Y),X)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>ackin(s(X),s(Y))</LHS><RHS>u21(ackin(s(X),Y),X)</RHS></RULE><RULE><LHS>u21(ackout(X),Y)</LHS><RHS>u22(ackin(Y,X))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_u21(ackout(X),Y)</LHS><RHS>Marked_ackin(Y,X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_ackin(s(X),s(Y))</LHS><RHS>Marked_ackin(s(X),Y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="1"/></SCC><NONSCC num="1"><NODE ref="0"/></NONSCC></CCLIST><EDGE end="0" start="1"/></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>ackin(s(X),s(Y))</LHS><RHS>u21(ackin(s(X),Y),X)</RHS></RULE><RULE><LHS>u21(ackout(X),Y)</LHS><RHS>u22(ackin(Y,X))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_ackin(s(X),s(Y))</LHS><RHS>Marked_ackin(s(X),Y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_ackin</NAME></SYMBOL><INT>2</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_ackin(s(X),s(Y))</LHS><RHS>Marked_ackin(s(X),Y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>ackin(s(X),s(Y))</LHS><RHS>u21(ackin(s(X),Y),X)</RHS></RULE><RULE><LHS>u21(ackout(X),Y)</LHS><RHS>u22(ackin(Y,X))</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
