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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="2" unmarked="ack"><NAME>Marked_ack</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>ack</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>z</VAR><VAR>y</VAR><VAR>x</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>ack(0,y)</LHS><RHS>s(y)</RHS></RULE><RULE><LHS>ack(s(x),0)</LHS><RHS>ack(x,s(0))</RHS></RULE><RULE><LHS>ack(s(x),s(y))</LHS><RHS>ack(x,ack(s(x),y))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>ack(0,y)</LHS><RHS>s(y)</RHS></RULE><RULE><LHS>ack(s(x),0)</LHS><RHS>ack(x,s(0))</RHS></RULE><RULE><LHS>ack(s(x),s(y))</LHS><RHS>ack(x,ack(s(x),y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_ack(s(x),0)</LHS><RHS>Marked_ack(x,s(0))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_ack(s(x),s(y))</LHS><RHS>Marked_ack(x,ack(s(x),y))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_ack(s(x),s(y))</LHS><RHS>Marked_ack(s(x),y)</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="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>ack(0,y)</LHS><RHS>s(y)</RHS></RULE><RULE><LHS>ack(s(x),0)</LHS><RHS>ack(x,s(0))</RHS></RULE><RULE><LHS>ack(s(x),s(y))</LHS><RHS>ack(x,ack(s(x),y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_ack(s(x),0)</LHS><RHS>Marked_ack(x,s(0))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_ack(s(x),s(y))</LHS><RHS>Marked_ack(x,ack(s(x),y))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_ack(s(x),s(y))</LHS><RHS>Marked_ack(s(x),y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="rpo"><STATUSLIST><STATUS status="lex"><SYMBOL><NAME>s</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>ack</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>Marked_ack</NAME></SYMBOL></STATUS></STATUSLIST><PRECEDENCE><PREC><OLHS><SYMBOL><NAME>s</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>ack</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>s</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>Marked_ack</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>ack</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>Marked_ack</NAME></SYMBOL></ORHS></PREC></PRECEDENCE><EQUIVALENCE/></ORDERING></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROOF>
