<?xml version="1.0" encoding="UTF-8"?>
<!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=""><NAME>plus</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="2" unmarked="plus"><NAME>Marked_plus</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>and</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>x</NAME></SYMBOL><SYMBOL arity="2" unmarked="and"><NAME>Marked_and</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>tt</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="2" unmarked="x"><NAME>Marked_x</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>activate</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>M</VAR><VAR>N</VAR><VAR>X</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>and(tt,X)</LHS><RHS>activate(X)</RHS></RULE><RULE><LHS>plus(N,0)</LHS><RHS>N</RHS></RULE><RULE><LHS>plus(N,s(M))</LHS><RHS>s(plus(N,M))</RHS></RULE><RULE><LHS>x(N,0)</LHS><RHS>0</RHS></RULE><RULE><LHS>x(N,s(M))</LHS><RHS>plus(x(N,M),N)</RHS></RULE><RULE><LHS>activate(X)</LHS><RHS>X</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>and(tt,X)</LHS><RHS>activate(X)</RHS></RULE><RULE><LHS>plus(N,0)</LHS><RHS>N</RHS></RULE><RULE><LHS>plus(N,s(M))</LHS><RHS>s(plus(N,M))</RHS></RULE><RULE><LHS>x(N,0)</LHS><RHS>0</RHS></RULE><RULE><LHS>x(N,s(M))</LHS><RHS>plus(x(N,M),N)</RHS></RULE><RULE><LHS>activate(X)</LHS><RHS>X</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_and(tt,X)</LHS><RHS>Marked_activate(X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_plus(N,s(M))</LHS><RHS>Marked_plus(N,M)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_x(N,s(M))</LHS><RHS>Marked_plus(x(N,M),N)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_x(N,s(M))</LHS><RHS>Marked_x(N,M)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="3"/></SCC><NONSCC num="1"><NODE ref="2"/></NONSCC><SCC num="2"><NODE ref="1"/></SCC><NONSCC num="3"><NODE ref="0"/></NONSCC></CCLIST><EDGE end="1" start="0"/><EDGE end="2" start="1"/></DAG></CRITERION><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>and(tt,X)</LHS><RHS>activate(X)</RHS></RULE><RULE><LHS>plus(N,0)</LHS><RHS>N</RHS></RULE><RULE><LHS>plus(N,s(M))</LHS><RHS>s(plus(N,M))</RHS></RULE><RULE><LHS>x(N,0)</LHS><RHS>0</RHS></RULE><RULE><LHS>x(N,s(M))</LHS><RHS>plus(x(N,M),N)</RHS></RULE><RULE><LHS>activate(X)</LHS><RHS>X</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_x(N,s(M))</LHS><RHS>Marked_x(N,M)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="rpo"><STATUSLIST><STATUS status="lex"><SYMBOL><NAME>activate</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>x</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>and</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>s</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>plus</NAME></SYMBOL></STATUS></STATUSLIST><PRECEDENCE><PREC><OLHS><SYMBOL><NAME>activate</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>and</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>plus</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>x</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>s</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>plus</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>s</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>x</NAME></SYMBOL></ORHS></PREC></PRECEDENCE><EQUIVALENCE/></ORDERING></CRITERION></PROPERTY><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>and(tt,X)</LHS><RHS>activate(X)</RHS></RULE><RULE><LHS>plus(N,0)</LHS><RHS>N</RHS></RULE><RULE><LHS>plus(N,s(M))</LHS><RHS>s(plus(N,M))</RHS></RULE><RULE><LHS>x(N,0)</LHS><RHS>0</RHS></RULE><RULE><LHS>x(N,s(M))</LHS><RHS>plus(x(N,M),N)</RHS></RULE><RULE><LHS>activate(X)</LHS><RHS>X</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_plus(N,s(M))</LHS><RHS>Marked_plus(N,M)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="rpo"><STATUSLIST><STATUS status="lex"><SYMBOL><NAME>activate</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>x</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>and</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>s</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>plus</NAME></SYMBOL></STATUS></STATUSLIST><PRECEDENCE><PREC><OLHS><SYMBOL><NAME>activate</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>and</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>plus</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>x</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>s</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>plus</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>s</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>x</NAME></SYMBOL></ORHS></PREC></PRECEDENCE><EQUIVALENCE/></ORDERING></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROOF>
