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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="1" unmarked="activate"><NAME>Marked_activate</NAME></SYMBOL><SYMBOL arity="2" unmarked="U11"><NAME>Marked_U11</NAME></SYMBOL><SYMBOL arity="3" unmarked=""><NAME>U42</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>n__0</NAME></SYMBOL><SYMBOL arity="3" unmarked="U41"><NAME>Marked_U41</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>U11</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>n__s</NAME></SYMBOL><SYMBOL arity="2" unmarked="U31"><NAME>Marked_U31</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>U21</NAME></SYMBOL><SYMBOL arity="1" unmarked="U12"><NAME>Marked_U12</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="2" unmarked="plus"><NAME>Marked_plus</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>isNat</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="1" unmarked="U21"><NAME>Marked_U21</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>U31</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>plus</NAME></SYMBOL><SYMBOL arity="0" unmarked="0"><NAME>Marked_0</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>activate</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>n__plus</NAME></SYMBOL><SYMBOL arity="1" unmarked="isNat"><NAME>Marked_isNat</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>tt</NAME></SYMBOL><SYMBOL arity="3" unmarked="U42"><NAME>Marked_U42</NAME></SYMBOL><SYMBOL arity="3" unmarked=""><NAME>U41</NAME></SYMBOL><SYMBOL arity="1" unmarked="s"><NAME>Marked_s</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>U12</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>X</VAR><VAR>X2</VAR><VAR>X1</VAR><VAR>V1</VAR><VAR>M</VAR><VAR>N</VAR><VAR>V2</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>U11(tt,V2)</LHS><RHS>U12(isNat(activate(V2)))</RHS></RULE><RULE><LHS>U12(tt)</LHS><RHS>tt</RHS></RULE><RULE><LHS>U21(tt)</LHS><RHS>tt</RHS></RULE><RULE><LHS>U31(tt,N)</LHS><RHS>activate(N)</RHS></RULE><RULE><LHS>U41(tt,M,N)</LHS><RHS>U42(isNat(activate(N)),activate(M),activate(N))</RHS></RULE><RULE><LHS>U42(tt,M,N)</LHS><RHS>s(plus(activate(N),activate(M)))</RHS></RULE><RULE><LHS>isNat(n__0)</LHS><RHS>tt</RHS></RULE><RULE><LHS>isNat(n__plus(V1,V2))</LHS><RHS>U11(isNat(activate(V1)),activate(V2))</RHS></RULE><RULE><LHS>isNat(n__s(V1))</LHS><RHS>U21(isNat(activate(V1)))</RHS></RULE><RULE><LHS>plus(N,0)</LHS><RHS>U31(isNat(N),N)</RHS></RULE><RULE><LHS>plus(N,s(M))</LHS><RHS>U41(isNat(M),M,N)</RHS></RULE><RULE><LHS>0</LHS><RHS>n__0</RHS></RULE><RULE><LHS>plus(X1,X2)</LHS><RHS>n__plus(X1,X2)</RHS></RULE><RULE><LHS>s(X)</LHS><RHS>n__s(X)</RHS></RULE><RULE><LHS>activate(n__0)</LHS><RHS>0</RHS></RULE><RULE><LHS>activate(n__plus(X1,X2))</LHS><RHS>plus(X1,X2)</RHS></RULE><RULE><LHS>activate(n__s(X))</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>activate(X)</LHS><RHS>X</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>U11(tt,V2)</LHS><RHS>U12(isNat(activate(V2)))</RHS></RULE><RULE><LHS>U12(tt)</LHS><RHS>tt</RHS></RULE><RULE><LHS>U21(tt)</LHS><RHS>tt</RHS></RULE><RULE><LHS>U31(tt,N)</LHS><RHS>activate(N)</RHS></RULE><RULE><LHS>U41(tt,M,N)</LHS><RHS>U42(isNat(activate(N)),activate(M),activate(N))</RHS></RULE><RULE><LHS>U42(tt,M,N)</LHS><RHS>s(plus(activate(N),activate(M)))</RHS></RULE><RULE><LHS>isNat(n__0)</LHS><RHS>tt</RHS></RULE><RULE><LHS>isNat(n__plus(V1,V2))</LHS><RHS>U11(isNat(activate(V1)),activate(V2))</RHS></RULE><RULE><LHS>isNat(n__s(V1))</LHS><RHS>U21(isNat(activate(V1)))</RHS></RULE><RULE><LHS>plus(N,0)</LHS><RHS>U31(isNat(N),N)</RHS></RULE><RULE><LHS>plus(N,s(M))</LHS><RHS>U41(isNat(M),M,N)</RHS></RULE><RULE><LHS>0</LHS><RHS>n__0</RHS></RULE><RULE><LHS>plus(X1,X2)</LHS><RHS>n__plus(X1,X2)</RHS></RULE><RULE><LHS>s(X)</LHS><RHS>n__s(X)</RHS></RULE><RULE><LHS>activate(n__0)</LHS><RHS>0</RHS></RULE><RULE><LHS>activate(n__plus(X1,X2))</LHS><RHS>plus(X1,X2)</RHS></RULE><RULE><LHS>activate(n__s(X))</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>activate(X)</LHS><RHS>X</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_U11(tt,V2)</LHS><RHS>Marked_U12(isNat(activate(V2)))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_U11(tt,V2)</LHS><RHS>Marked_isNat(activate(V2))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_U11(tt,V2)</LHS><RHS>Marked_activate(V2)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_U31(tt,N)</LHS><RHS>Marked_activate(N)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_U41(tt,M,N)</LHS><RHS>Marked_U42(isNat(activate(N)),activate(M),activate(N))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_U41(tt,M,N)</LHS><RHS>Marked_isNat(activate(N))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_U41(tt,M,N)</LHS><RHS>Marked_activate(N)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_U41(tt,M,N)</LHS><RHS>Marked_activate(M)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_U41(tt,M,N)</LHS><RHS>Marked_activate(N)</RHS></DPRULE><DPRULE num="9"><LHS>Marked_U42(tt,M,N)</LHS><RHS>Marked_s(plus(activate(N),activate(M)))</RHS></DPRULE><DPRULE num="10"><LHS>Marked_U42(tt,M,N)</LHS><RHS>Marked_plus(activate(N),activate(M))</RHS></DPRULE><DPRULE num="11"><LHS>Marked_U42(tt,M,N)</LHS><RHS>Marked_activate(N)</RHS></DPRULE><DPRULE num="12"><LHS>Marked_U42(tt,M,N)</LHS><RHS>Marked_activate(M)</RHS></DPRULE><DPRULE num="13"><LHS>Marked_isNat(n__plus(V1,V2))</LHS><RHS>Marked_U11(isNat(activate(V1)),activate(V2))</RHS></DPRULE><DPRULE num="14"><LHS>Marked_isNat(n__plus(V1,V2))</LHS><RHS>Marked_isNat(activate(V1))</RHS></DPRULE><DPRULE num="15"><LHS>Marked_isNat(n__plus(V1,V2))</LHS><RHS>Marked_activate(V1)</RHS></DPRULE><DPRULE num="16"><LHS>Marked_isNat(n__plus(V1,V2))</LHS><RHS>Marked_activate(V2)</RHS></DPRULE><DPRULE num="17"><LHS>Marked_isNat(n__s(V1))</LHS><RHS>Marked_U21(isNat(activate(V1)))</RHS></DPRULE><DPRULE num="18"><LHS>Marked_isNat(n__s(V1))</LHS><RHS>Marked_isNat(activate(V1))</RHS></DPRULE><DPRULE num="19"><LHS>Marked_isNat(n__s(V1))</LHS><RHS>Marked_activate(V1)</RHS></DPRULE><DPRULE num="20"><LHS>Marked_plus(N,0)</LHS><RHS>Marked_U31(isNat(N),N)</RHS></DPRULE><DPRULE num="21"><LHS>Marked_plus(N,0)</LHS><RHS>Marked_isNat(N)</RHS></DPRULE><DPRULE num="22"><LHS>Marked_plus(N,s(M))</LHS><RHS>Marked_U41(isNat(M),M,N)</RHS></DPRULE><DPRULE num="23"><LHS>Marked_plus(N,s(M))</LHS><RHS>Marked_isNat(M)</RHS></DPRULE><DPRULE num="24"><LHS>Marked_activate(n__0)</LHS><RHS>Marked_0</RHS></DPRULE><DPRULE num="25"><LHS>Marked_activate(n__plus(X1,X2))</LHS><RHS>Marked_plus(X1,X2)</RHS></DPRULE><DPRULE num="26"><LHS>Marked_activate(n__s(X))</LHS><RHS>Marked_s(X)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="26"/></NONSCC><SCC num="1"><NODE ref="1"/><NODE ref="2"/><NODE ref="3"/><NODE ref="4"/><NODE ref="5"/><NODE ref="6"/><NODE ref="7"/><NODE ref="6"/><NODE ref="10"/><NODE ref="11"/><NODE ref="12"/><NODE ref="13"/><NODE ref="14"/><NODE ref="15"/><NODE ref="16"/><NODE ref="18"/><NODE ref="19"/><NODE ref="20"/><NODE ref="21"/><NODE ref="22"/><NODE ref="23"/><NODE ref="25"/></SCC><NONSCC num="2"><NODE ref="24"/></NONSCC><NONSCC num="3"><NODE ref="17"/></NONSCC><NONSCC num="4"><NODE ref="0"/></NONSCC><NONSCC num="5"><NODE ref="9"/></NONSCC></CCLIST><EDGE end="5" start="1"/><EDGE end="4" start="1"/><EDGE end="3" start="1"/><EDGE end="2" start="1"/><EDGE end="0" start="1"/></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>U11(tt,V2)</LHS><RHS>U12(isNat(activate(V2)))</RHS></RULE><RULE><LHS>U12(tt)</LHS><RHS>tt</RHS></RULE><RULE><LHS>U21(tt)</LHS><RHS>tt</RHS></RULE><RULE><LHS>U31(tt,N)</LHS><RHS>activate(N)</RHS></RULE><RULE><LHS>U41(tt,M,N)</LHS><RHS>U42(isNat(activate(N)),activate(M),activate(N))</RHS></RULE><RULE><LHS>U42(tt,M,N)</LHS><RHS>s(plus(activate(N),activate(M)))</RHS></RULE><RULE><LHS>isNat(n__0)</LHS><RHS>tt</RHS></RULE><RULE><LHS>isNat(n__plus(V1,V2))</LHS><RHS>U11(isNat(activate(V1)),activate(V2))</RHS></RULE><RULE><LHS>isNat(n__s(V1))</LHS><RHS>U21(isNat(activate(V1)))</RHS></RULE><RULE><LHS>plus(N,0)</LHS><RHS>U31(isNat(N),N)</RHS></RULE><RULE><LHS>plus(N,s(M))</LHS><RHS>U41(isNat(M),M,N)</RHS></RULE><RULE><LHS>0</LHS><RHS>n__0</RHS></RULE><RULE><LHS>plus(X1,X2)</LHS><RHS>n__plus(X1,X2)</RHS></RULE><RULE><LHS>s(X)</LHS><RHS>n__s(X)</RHS></RULE><RULE><LHS>activate(n__0)</LHS><RHS>0</RHS></RULE><RULE><LHS>activate(n__plus(X1,X2))</LHS><RHS>plus(X1,X2)</RHS></RULE><RULE><LHS>activate(n__s(X))</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>activate(X)</LHS><RHS>X</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_U11(tt,V2)</LHS><RHS>Marked_isNat(activate(V2))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_U11(tt,V2)</LHS><RHS>Marked_activate(V2)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_U31(tt,N)</LHS><RHS>Marked_activate(N)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_U41(tt,M,N)</LHS><RHS>Marked_U42(isNat(activate(N)),activate(M),activate(N))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_U41(tt,M,N)</LHS><RHS>Marked_isNat(activate(N))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_U41(tt,M,N)</LHS><RHS>Marked_activate(N)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_U41(tt,M,N)</LHS><RHS>Marked_activate(M)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_U41(tt,M,N)</LHS><RHS>Marked_activate(N)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_U42(tt,M,N)</LHS><RHS>Marked_plus(activate(N),activate(M))</RHS></DPRULE><DPRULE num="9"><LHS>Marked_U42(tt,M,N)</LHS><RHS>Marked_activate(N)</RHS></DPRULE><DPRULE num="10"><LHS>Marked_U42(tt,M,N)</LHS><RHS>Marked_activate(M)</RHS></DPRULE><DPRULE num="11"><LHS>Marked_isNat(n__plus(V1,V2))</LHS><RHS>Marked_U11(isNat(activate(V1)),activate(V2))</RHS></DPRULE><DPRULE num="12"><LHS>Marked_isNat(n__plus(V1,V2))</LHS><RHS>Marked_isNat(activate(V1))</RHS></DPRULE><DPRULE num="13"><LHS>Marked_isNat(n__plus(V1,V2))</LHS><RHS>Marked_activate(V1)</RHS></DPRULE><DPRULE num="14"><LHS>Marked_isNat(n__plus(V1,V2))</LHS><RHS>Marked_activate(V2)</RHS></DPRULE><DPRULE num="15"><LHS>Marked_isNat(n__s(V1))</LHS><RHS>Marked_isNat(activate(V1))</RHS></DPRULE><DPRULE num="16"><LHS>Marked_isNat(n__s(V1))</LHS><RHS>Marked_activate(V1)</RHS></DPRULE><DPRULE num="17"><LHS>Marked_plus(N,0)</LHS><RHS>Marked_U31(isNat(N),N)</RHS></DPRULE><DPRULE num="18"><LHS>Marked_plus(N,0)</LHS><RHS>Marked_isNat(N)</RHS></DPRULE><DPRULE num="19"><LHS>Marked_plus(N,s(M))</LHS><RHS>Marked_U41(isNat(M),M,N)</RHS></DPRULE><DPRULE num="20"><LHS>Marked_plus(N,s(M))</LHS><RHS>Marked_isNat(M)</RHS></DPRULE><DPRULE num="21"><LHS>Marked_activate(n__plus(X1,X2))</LHS><RHS>Marked_plus(X1,X2)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>U12</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>U41</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="2"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_U42</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="2"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>tt</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_isNat</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>n__plus</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>activate</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>plus</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>U31</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>0</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>isNat</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_plus</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>s</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>U21</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_U31</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>n__s</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>U11</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_U41</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="2"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>n__0</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>U42</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="2"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_U11</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_activate</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_U11(tt,V2)</LHS><RHS>Marked_isNat(activate(V2))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_U11(tt,V2)</LHS><RHS>Marked_activate(V2)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_U31(tt,N)</LHS><RHS>Marked_activate(N)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_U41(tt,M,N)</LHS><RHS>Marked_isNat(activate(N))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_U41(tt,M,N)</LHS><RHS>Marked_activate(N)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_U41(tt,M,N)</LHS><RHS>Marked_activate(M)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_U41(tt,M,N)</LHS><RHS>Marked_activate(N)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_U42(tt,M,N)</LHS><RHS>Marked_plus(activate(N),activate(M))</RHS></DPRULE><DPRULE num="8"><LHS>Marked_U42(tt,M,N)</LHS><RHS>Marked_activate(N)</RHS></DPRULE><DPRULE num="9"><LHS>Marked_U42(tt,M,N)</LHS><RHS>Marked_activate(M)</RHS></DPRULE><DPRULE num="10"><LHS>Marked_isNat(n__plus(V1,V2))</LHS><RHS>Marked_isNat(activate(V1))</RHS></DPRULE><DPRULE num="11"><LHS>Marked_isNat(n__plus(V1,V2))</LHS><RHS>Marked_activate(V1)</RHS></DPRULE><DPRULE num="12"><LHS>Marked_isNat(n__plus(V1,V2))</LHS><RHS>Marked_activate(V2)</RHS></DPRULE><DPRULE num="13"><LHS>Marked_isNat(n__s(V1))</LHS><RHS>Marked_isNat(activate(V1))</RHS></DPRULE><DPRULE num="14"><LHS>Marked_isNat(n__s(V1))</LHS><RHS>Marked_activate(V1)</RHS></DPRULE><DPRULE num="15"><LHS>Marked_plus(N,0)</LHS><RHS>Marked_isNat(N)</RHS></DPRULE><DPRULE num="16"><LHS>Marked_plus(N,s(M))</LHS><RHS>Marked_U41(isNat(M),M,N)</RHS></DPRULE><DPRULE num="17"><LHS>Marked_plus(N,s(M))</LHS><RHS>Marked_isNat(M)</RHS></DPRULE><DPRULE num="18"><LHS>Marked_activate(n__plus(X1,X2))</LHS><RHS>Marked_plus(X1,X2)</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>U11(tt,V2)</LHS><RHS>U12(isNat(activate(V2)))</RHS></RULE><RULE><LHS>U12(tt)</LHS><RHS>tt</RHS></RULE><RULE><LHS>U21(tt)</LHS><RHS>tt</RHS></RULE><RULE><LHS>U31(tt,N)</LHS><RHS>activate(N)</RHS></RULE><RULE><LHS>U41(tt,M,N)</LHS><RHS>U42(isNat(activate(N)),activate(M),activate(N))</RHS></RULE><RULE><LHS>U42(tt,M,N)</LHS><RHS>s(plus(activate(N),activate(M)))</RHS></RULE><RULE><LHS>isNat(n__0)</LHS><RHS>tt</RHS></RULE><RULE><LHS>isNat(n__plus(V1,V2))</LHS><RHS>U11(isNat(activate(V1)),activate(V2))</RHS></RULE><RULE><LHS>isNat(n__s(V1))</LHS><RHS>U21(isNat(activate(V1)))</RHS></RULE><RULE><LHS>plus(N,0)</LHS><RHS>U31(isNat(N),N)</RHS></RULE><RULE><LHS>plus(N,s(M))</LHS><RHS>U41(isNat(M),M,N)</RHS></RULE><RULE><LHS>0</LHS><RHS>n__0</RHS></RULE><RULE><LHS>plus(X1,X2)</LHS><RHS>n__plus(X1,X2)</RHS></RULE><RULE><LHS>s(X)</LHS><RHS>n__s(X)</RHS></RULE><RULE><LHS>activate(n__0)</LHS><RHS>0</RHS></RULE><RULE><LHS>activate(n__plus(X1,X2))</LHS><RHS>plus(X1,X2)</RHS></RULE><RULE><LHS>activate(n__s(X))</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>activate(X)</LHS><RHS>X</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_U41(tt,M,N)</LHS><RHS>Marked_U42(isNat(activate(N)),activate(M),activate(N))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_isNat(n__plus(V1,V2))</LHS><RHS>Marked_U11(isNat(activate(V1)),activate(V2))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_plus(N,0)</LHS><RHS>Marked_U31(isNat(N),N)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="2"/></NONSCC><NONSCC num="1"><NODE ref="1"/></NONSCC><NONSCC num="2"><NODE ref="0"/></NONSCC></CCLIST></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
