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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="1" unmarked="u22"><NAME>Marked_u22</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>u21</NAME></SYMBOL><SYMBOL arity="2" unmarked="ack_in"><NAME>Marked_ack_in</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>u22</NAME></SYMBOL><SYMBOL arity="1" unmarked="u11"><NAME>Marked_u11</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>ack_in</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>u11</NAME></SYMBOL><SYMBOL arity="2" unmarked="u21"><NAME>Marked_u21</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>ack_out</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>n</VAR><VAR>m</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>ack_in(0,n)</LHS><RHS>ack_out(s(n))</RHS></RULE><RULE><LHS>ack_in(s(m),0)</LHS><RHS>u11(ack_in(m,s(0)))</RHS></RULE><RULE><LHS>u11(ack_out(n))</LHS><RHS>ack_out(n)</RHS></RULE><RULE><LHS>ack_in(s(m),s(n))</LHS><RHS>u21(ack_in(s(m),n),m)</RHS></RULE><RULE><LHS>u21(ack_out(n),m)</LHS><RHS>u22(ack_in(m,n))</RHS></RULE><RULE><LHS>u22(ack_out(n))</LHS><RHS>ack_out(n)</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>ack_in(0,n)</LHS><RHS>ack_out(s(n))</RHS></RULE><RULE><LHS>ack_in(s(m),0)</LHS><RHS>u11(ack_in(m,s(0)))</RHS></RULE><RULE><LHS>u11(ack_out(n))</LHS><RHS>ack_out(n)</RHS></RULE><RULE><LHS>ack_in(s(m),s(n))</LHS><RHS>u21(ack_in(s(m),n),m)</RHS></RULE><RULE><LHS>u21(ack_out(n),m)</LHS><RHS>u22(ack_in(m,n))</RHS></RULE><RULE><LHS>u22(ack_out(n))</LHS><RHS>ack_out(n)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_ack_in(s(m),0)</LHS><RHS>Marked_u11(ack_in(m,s(0)))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_ack_in(s(m),0)</LHS><RHS>Marked_ack_in(m,s(0))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_ack_in(s(m),s(n))</LHS><RHS>Marked_u21(ack_in(s(m),n),m)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_ack_in(s(m),s(n))</LHS><RHS>Marked_ack_in(s(m),n)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_u21(ack_out(n),m)</LHS><RHS>Marked_u22(ack_in(m,n))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_u21(ack_out(n),m)</LHS><RHS>Marked_ack_in(m,n)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="1"/><NODE ref="2"/><NODE ref="3"/><NODE ref="5"/></SCC><NONSCC num="1"><NODE ref="0"/></NONSCC><NONSCC num="2"><NODE ref="4"/></NONSCC></CCLIST><EDGE end="2" start="0"/><EDGE end="1" start="0"/></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>ack_in(0,n)</LHS><RHS>ack_out(s(n))</RHS></RULE><RULE><LHS>ack_in(s(m),0)</LHS><RHS>u11(ack_in(m,s(0)))</RHS></RULE><RULE><LHS>u11(ack_out(n))</LHS><RHS>ack_out(n)</RHS></RULE><RULE><LHS>ack_in(s(m),s(n))</LHS><RHS>u21(ack_in(s(m),n),m)</RHS></RULE><RULE><LHS>u21(ack_out(n),m)</LHS><RHS>u22(ack_in(m,n))</RHS></RULE><RULE><LHS>u22(ack_out(n))</LHS><RHS>ack_out(n)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_ack_in(s(m),0)</LHS><RHS>Marked_ack_in(m,s(0))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_ack_in(s(m),s(n))</LHS><RHS>Marked_u21(ack_in(s(m),n),m)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_ack_in(s(m),s(n))</LHS><RHS>Marked_ack_in(s(m),n)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_u21(ack_out(n),m)</LHS><RHS>Marked_ack_in(m,n)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_u21</NAME></SYMBOL><INT>2</INT></PROJ><PROJ><SYMBOL><NAME>Marked_ack_in</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_ack_in(s(m),s(n))</LHS><RHS>Marked_u21(ack_in(s(m),n),m)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_ack_in(s(m),0)</LHS><RHS>Marked_ack_in(m,s(0))</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>ack_in(0,n)</LHS><RHS>ack_out(s(n))</RHS></RULE><RULE><LHS>ack_in(s(m),0)</LHS><RHS>u11(ack_in(m,s(0)))</RHS></RULE><RULE><LHS>u11(ack_out(n))</LHS><RHS>ack_out(n)</RHS></RULE><RULE><LHS>ack_in(s(m),s(n))</LHS><RHS>u21(ack_in(s(m),n),m)</RHS></RULE><RULE><LHS>u21(ack_out(n),m)</LHS><RHS>u22(ack_in(m,n))</RHS></RULE><RULE><LHS>u22(ack_out(n))</LHS><RHS>ack_out(n)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_u21(ack_out(n),m)</LHS><RHS>Marked_ack_in(m,n)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_ack_in(s(m),s(n))</LHS><RHS>Marked_ack_in(s(m),n)</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>ack_in(0,n)</LHS><RHS>ack_out(s(n))</RHS></RULE><RULE><LHS>ack_in(s(m),0)</LHS><RHS>u11(ack_in(m,s(0)))</RHS></RULE><RULE><LHS>u11(ack_out(n))</LHS><RHS>ack_out(n)</RHS></RULE><RULE><LHS>ack_in(s(m),s(n))</LHS><RHS>u21(ack_in(s(m),n),m)</RHS></RULE><RULE><LHS>u21(ack_out(n),m)</LHS><RHS>u22(ack_in(m,n))</RHS></RULE><RULE><LHS>u22(ack_out(n))</LHS><RHS>ack_out(n)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_ack_in(s(m),s(n))</LHS><RHS>Marked_ack_in(s(m),n)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_ack_in</NAME></SYMBOL><INT>2</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_ack_in(s(m),s(n))</LHS><RHS>Marked_ack_in(s(m),n)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>ack_in(0,n)</LHS><RHS>ack_out(s(n))</RHS></RULE><RULE><LHS>ack_in(s(m),0)</LHS><RHS>u11(ack_in(m,s(0)))</RHS></RULE><RULE><LHS>u11(ack_out(n))</LHS><RHS>ack_out(n)</RHS></RULE><RULE><LHS>ack_in(s(m),s(n))</LHS><RHS>u21(ack_in(s(m),n),m)</RHS></RULE><RULE><LHS>u21(ack_out(n),m)</LHS><RHS>u22(ack_in(m,n))</RHS></RULE><RULE><LHS>u22(ack_out(n))</LHS><RHS>ack_out(n)</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
