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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="2" unmarked=""><NAME>plus</NAME></SYMBOL><SYMBOL arity="2" unmarked="plus"><NAME>Marked_plus</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>posrecip</NAME></SYMBOL><SYMBOL arity="1" unmarked="s"><NAME>Marked_s</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>from</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>negrecip</NAME></SYMBOL><SYMBOL arity="1" unmarked="square"><NAME>Marked_square</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>2ndspos</NAME></SYMBOL><SYMBOL arity="2" unmarked="2ndspos"><NAME>Marked_2ndspos</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>activate</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>square</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>n__from</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>pi</NAME></SYMBOL><SYMBOL arity="2" unmarked="times"><NAME>Marked_times</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="2" unmarked="2ndsneg"><NAME>Marked_2ndsneg</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>rcons</NAME></SYMBOL><SYMBOL arity="1" unmarked="activate"><NAME>Marked_activate</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>n__s</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>2ndsneg</NAME></SYMBOL><SYMBOL arity="1" unmarked="from"><NAME>Marked_from</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>rnil</NAME></SYMBOL><SYMBOL arity="1" unmarked="pi"><NAME>Marked_pi</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>cons2</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>times</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>cons</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>Z</VAR><VAR>Y</VAR><VAR>X</VAR><VAR>N</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>from(X)</LHS><RHS>cons(X,n__from(n__s(X)))</RHS></RULE><RULE><LHS>2ndspos(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndspos(s(N),cons(X,Z))</LHS><RHS>2ndspos(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndspos(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(posrecip(Y),2ndsneg(N,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons(X,Z))</LHS><RHS>2ndsneg(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(negrecip(Y),2ndspos(N,activate(Z)))</RHS></RULE><RULE><LHS>pi(X)</LHS><RHS>2ndspos(X,from(0))</RHS></RULE><RULE><LHS>plus(0,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>plus(s(X),Y)</LHS><RHS>s(plus(X,Y))</RHS></RULE><RULE><LHS>times(0,Y)</LHS><RHS>0</RHS></RULE><RULE><LHS>times(s(X),Y)</LHS><RHS>plus(Y,times(X,Y))</RHS></RULE><RULE><LHS>square(X)</LHS><RHS>times(X,X)</RHS></RULE><RULE><LHS>from(X)</LHS><RHS>n__from(X)</RHS></RULE><RULE><LHS>s(X)</LHS><RHS>n__s(X)</RHS></RULE><RULE><LHS>activate(n__from(X))</LHS><RHS>from(activate(X))</RHS></RULE><RULE><LHS>activate(n__s(X))</LHS><RHS>s(activate(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>from(X)</LHS><RHS>cons(X,n__from(n__s(X)))</RHS></RULE><RULE><LHS>2ndspos(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndspos(s(N),cons(X,Z))</LHS><RHS>2ndspos(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndspos(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(posrecip(Y),2ndsneg(N,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons(X,Z))</LHS><RHS>2ndsneg(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(negrecip(Y),2ndspos(N,activate(Z)))</RHS></RULE><RULE><LHS>pi(X)</LHS><RHS>2ndspos(X,from(0))</RHS></RULE><RULE><LHS>plus(0,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>plus(s(X),Y)</LHS><RHS>s(plus(X,Y))</RHS></RULE><RULE><LHS>times(0,Y)</LHS><RHS>0</RHS></RULE><RULE><LHS>times(s(X),Y)</LHS><RHS>plus(Y,times(X,Y))</RHS></RULE><RULE><LHS>square(X)</LHS><RHS>times(X,X)</RHS></RULE><RULE><LHS>from(X)</LHS><RHS>n__from(X)</RHS></RULE><RULE><LHS>s(X)</LHS><RHS>n__s(X)</RHS></RULE><RULE><LHS>activate(n__from(X))</LHS><RHS>from(activate(X))</RHS></RULE><RULE><LHS>activate(n__s(X))</LHS><RHS>s(activate(X))</RHS></RULE><RULE><LHS>activate(X)</LHS><RHS>X</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_2ndspos(s(N),cons(X,Z))</LHS><RHS>Marked_2ndspos(s(N),cons2(X,activate(Z)))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_2ndspos(s(N),cons(X,Z))</LHS><RHS>Marked_activate(Z)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_2ndspos(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>Marked_2ndsneg(N,activate(Z))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_2ndspos(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>Marked_activate(Z)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_2ndsneg(s(N),cons(X,Z))</LHS><RHS>Marked_2ndsneg(s(N),cons2(X,activate(Z)))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_2ndsneg(s(N),cons(X,Z))</LHS><RHS>Marked_activate(Z)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_2ndsneg(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>Marked_2ndspos(N,activate(Z))</RHS></DPRULE><DPRULE num="7"><LHS>Marked_2ndsneg(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>Marked_activate(Z)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_pi(X)</LHS><RHS>Marked_2ndspos(X,from(0))</RHS></DPRULE><DPRULE num="9"><LHS>Marked_pi(X)</LHS><RHS>Marked_from(0)</RHS></DPRULE><DPRULE num="10"><LHS>Marked_plus(s(X),Y)</LHS><RHS>Marked_s(plus(X,Y))</RHS></DPRULE><DPRULE num="11"><LHS>Marked_plus(s(X),Y)</LHS><RHS>Marked_plus(X,Y)</RHS></DPRULE><DPRULE num="12"><LHS>Marked_times(s(X),Y)</LHS><RHS>Marked_plus(Y,times(X,Y))</RHS></DPRULE><DPRULE num="13"><LHS>Marked_times(s(X),Y)</LHS><RHS>Marked_times(X,Y)</RHS></DPRULE><DPRULE num="14"><LHS>Marked_square(X)</LHS><RHS>Marked_times(X,X)</RHS></DPRULE><DPRULE num="15"><LHS>Marked_activate(n__from(X))</LHS><RHS>Marked_from(activate(X))</RHS></DPRULE><DPRULE num="16"><LHS>Marked_activate(n__from(X))</LHS><RHS>Marked_activate(X)</RHS></DPRULE><DPRULE num="17"><LHS>Marked_activate(n__s(X))</LHS><RHS>Marked_s(activate(X))</RHS></DPRULE><DPRULE num="18"><LHS>Marked_activate(n__s(X))</LHS><RHS>Marked_activate(X)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="16"/><NODE ref="18"/></SCC><NONSCC num="1"><NODE ref="17"/></NONSCC><NONSCC num="2"><NODE ref="15"/></NONSCC><NONSCC num="3"><NODE ref="14"/></NONSCC><SCC num="4"><NODE ref="13"/></SCC><NONSCC num="5"><NODE ref="12"/></NONSCC><SCC num="6"><NODE ref="11"/></SCC><NONSCC num="7"><NODE ref="10"/></NONSCC><NONSCC num="8"><NODE ref="9"/></NONSCC><NONSCC num="9"><NODE ref="8"/></NONSCC><NONSCC num="10"><NODE ref="3"/></NONSCC><SCC num="11"><NODE ref="0"/><NODE ref="2"/><NODE ref="4"/><NODE ref="6"/></SCC><NONSCC num="12"><NODE ref="1"/></NONSCC><NONSCC num="13"><NODE ref="7"/></NONSCC><NONSCC num="14"><NODE ref="5"/></NONSCC></CCLIST><EDGE end="2" start="0"/><EDGE end="1" start="0"/><EDGE end="5" start="3"/><EDGE end="4" start="3"/><EDGE end="5" start="4"/><EDGE end="7" start="5"/><EDGE end="6" start="5"/><EDGE end="7" start="6"/><EDGE end="12" start="9"/><EDGE end="11" start="9"/><EDGE end="10" start="9"/><EDGE end="2" start="10"/><EDGE end="1" start="10"/><EDGE end="0" start="10"/><EDGE end="14" start="11"/><EDGE end="13" start="11"/><EDGE end="12" start="11"/><EDGE end="10" start="11"/><EDGE end="2" start="12"/><EDGE end="1" start="12"/><EDGE end="0" start="12"/><EDGE end="2" start="13"/><EDGE end="1" start="13"/><EDGE end="0" start="13"/><EDGE end="2" start="14"/><EDGE end="1" start="14"/><EDGE end="0" start="14"/></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>from(X)</LHS><RHS>cons(X,n__from(n__s(X)))</RHS></RULE><RULE><LHS>2ndspos(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndspos(s(N),cons(X,Z))</LHS><RHS>2ndspos(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndspos(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(posrecip(Y),2ndsneg(N,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons(X,Z))</LHS><RHS>2ndsneg(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(negrecip(Y),2ndspos(N,activate(Z)))</RHS></RULE><RULE><LHS>pi(X)</LHS><RHS>2ndspos(X,from(0))</RHS></RULE><RULE><LHS>plus(0,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>plus(s(X),Y)</LHS><RHS>s(plus(X,Y))</RHS></RULE><RULE><LHS>times(0,Y)</LHS><RHS>0</RHS></RULE><RULE><LHS>times(s(X),Y)</LHS><RHS>plus(Y,times(X,Y))</RHS></RULE><RULE><LHS>square(X)</LHS><RHS>times(X,X)</RHS></RULE><RULE><LHS>from(X)</LHS><RHS>n__from(X)</RHS></RULE><RULE><LHS>s(X)</LHS><RHS>n__s(X)</RHS></RULE><RULE><LHS>activate(n__from(X))</LHS><RHS>from(activate(X))</RHS></RULE><RULE><LHS>activate(n__s(X))</LHS><RHS>s(activate(X))</RHS></RULE><RULE><LHS>activate(X)</LHS><RHS>X</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_2ndspos(s(N),cons(X,Z))</LHS><RHS>Marked_2ndspos(s(N),cons2(X,activate(Z)))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_2ndspos(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>Marked_2ndsneg(N,activate(Z))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_2ndsneg(s(N),cons(X,Z))</LHS><RHS>Marked_2ndsneg(s(N),cons2(X,activate(Z)))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_2ndsneg(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>Marked_2ndspos(N,activate(Z))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_2ndsneg</NAME></SYMBOL><INT>1</INT></PROJ><PROJ><SYMBOL><NAME>Marked_2ndspos</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_2ndsneg(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>Marked_2ndspos(N,activate(Z))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_2ndspos(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>Marked_2ndsneg(N,activate(Z))</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>from(X)</LHS><RHS>cons(X,n__from(n__s(X)))</RHS></RULE><RULE><LHS>2ndspos(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndspos(s(N),cons(X,Z))</LHS><RHS>2ndspos(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndspos(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(posrecip(Y),2ndsneg(N,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons(X,Z))</LHS><RHS>2ndsneg(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(negrecip(Y),2ndspos(N,activate(Z)))</RHS></RULE><RULE><LHS>pi(X)</LHS><RHS>2ndspos(X,from(0))</RHS></RULE><RULE><LHS>plus(0,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>plus(s(X),Y)</LHS><RHS>s(plus(X,Y))</RHS></RULE><RULE><LHS>times(0,Y)</LHS><RHS>0</RHS></RULE><RULE><LHS>times(s(X),Y)</LHS><RHS>plus(Y,times(X,Y))</RHS></RULE><RULE><LHS>square(X)</LHS><RHS>times(X,X)</RHS></RULE><RULE><LHS>from(X)</LHS><RHS>n__from(X)</RHS></RULE><RULE><LHS>s(X)</LHS><RHS>n__s(X)</RHS></RULE><RULE><LHS>activate(n__from(X))</LHS><RHS>from(activate(X))</RHS></RULE><RULE><LHS>activate(n__s(X))</LHS><RHS>s(activate(X))</RHS></RULE><RULE><LHS>activate(X)</LHS><RHS>X</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_2ndsneg(s(N),cons(X,Z))</LHS><RHS>Marked_2ndsneg(s(N),cons2(X,activate(Z)))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_2ndspos(s(N),cons(X,Z))</LHS><RHS>Marked_2ndspos(s(N),cons2(X,activate(Z)))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="1"/></NONSCC><NONSCC num="1"><NODE ref="0"/></NONSCC></CCLIST></DAG></CRITERION></PROPERTY></PROPERTY><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>from(X)</LHS><RHS>cons(X,n__from(n__s(X)))</RHS></RULE><RULE><LHS>2ndspos(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndspos(s(N),cons(X,Z))</LHS><RHS>2ndspos(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndspos(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(posrecip(Y),2ndsneg(N,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons(X,Z))</LHS><RHS>2ndsneg(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(negrecip(Y),2ndspos(N,activate(Z)))</RHS></RULE><RULE><LHS>pi(X)</LHS><RHS>2ndspos(X,from(0))</RHS></RULE><RULE><LHS>plus(0,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>plus(s(X),Y)</LHS><RHS>s(plus(X,Y))</RHS></RULE><RULE><LHS>times(0,Y)</LHS><RHS>0</RHS></RULE><RULE><LHS>times(s(X),Y)</LHS><RHS>plus(Y,times(X,Y))</RHS></RULE><RULE><LHS>square(X)</LHS><RHS>times(X,X)</RHS></RULE><RULE><LHS>from(X)</LHS><RHS>n__from(X)</RHS></RULE><RULE><LHS>s(X)</LHS><RHS>n__s(X)</RHS></RULE><RULE><LHS>activate(n__from(X))</LHS><RHS>from(activate(X))</RHS></RULE><RULE><LHS>activate(n__s(X))</LHS><RHS>s(activate(X))</RHS></RULE><RULE><LHS>activate(X)</LHS><RHS>X</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_times(s(X),Y)</LHS><RHS>Marked_times(X,Y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_times</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_times(s(X),Y)</LHS><RHS>Marked_times(X,Y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>from(X)</LHS><RHS>cons(X,n__from(n__s(X)))</RHS></RULE><RULE><LHS>2ndspos(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndspos(s(N),cons(X,Z))</LHS><RHS>2ndspos(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndspos(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(posrecip(Y),2ndsneg(N,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons(X,Z))</LHS><RHS>2ndsneg(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(negrecip(Y),2ndspos(N,activate(Z)))</RHS></RULE><RULE><LHS>pi(X)</LHS><RHS>2ndspos(X,from(0))</RHS></RULE><RULE><LHS>plus(0,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>plus(s(X),Y)</LHS><RHS>s(plus(X,Y))</RHS></RULE><RULE><LHS>times(0,Y)</LHS><RHS>0</RHS></RULE><RULE><LHS>times(s(X),Y)</LHS><RHS>plus(Y,times(X,Y))</RHS></RULE><RULE><LHS>square(X)</LHS><RHS>times(X,X)</RHS></RULE><RULE><LHS>from(X)</LHS><RHS>n__from(X)</RHS></RULE><RULE><LHS>s(X)</LHS><RHS>n__s(X)</RHS></RULE><RULE><LHS>activate(n__from(X))</LHS><RHS>from(activate(X))</RHS></RULE><RULE><LHS>activate(n__s(X))</LHS><RHS>s(activate(X))</RHS></RULE><RULE><LHS>activate(X)</LHS><RHS>X</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>from(X)</LHS><RHS>cons(X,n__from(n__s(X)))</RHS></RULE><RULE><LHS>2ndspos(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndspos(s(N),cons(X,Z))</LHS><RHS>2ndspos(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndspos(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(posrecip(Y),2ndsneg(N,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons(X,Z))</LHS><RHS>2ndsneg(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(negrecip(Y),2ndspos(N,activate(Z)))</RHS></RULE><RULE><LHS>pi(X)</LHS><RHS>2ndspos(X,from(0))</RHS></RULE><RULE><LHS>plus(0,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>plus(s(X),Y)</LHS><RHS>s(plus(X,Y))</RHS></RULE><RULE><LHS>times(0,Y)</LHS><RHS>0</RHS></RULE><RULE><LHS>times(s(X),Y)</LHS><RHS>plus(Y,times(X,Y))</RHS></RULE><RULE><LHS>square(X)</LHS><RHS>times(X,X)</RHS></RULE><RULE><LHS>from(X)</LHS><RHS>n__from(X)</RHS></RULE><RULE><LHS>s(X)</LHS><RHS>n__s(X)</RHS></RULE><RULE><LHS>activate(n__from(X))</LHS><RHS>from(activate(X))</RHS></RULE><RULE><LHS>activate(n__s(X))</LHS><RHS>s(activate(X))</RHS></RULE><RULE><LHS>activate(X)</LHS><RHS>X</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_plus(s(X),Y)</LHS><RHS>Marked_plus(X,Y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_plus</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_plus(s(X),Y)</LHS><RHS>Marked_plus(X,Y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>from(X)</LHS><RHS>cons(X,n__from(n__s(X)))</RHS></RULE><RULE><LHS>2ndspos(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndspos(s(N),cons(X,Z))</LHS><RHS>2ndspos(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndspos(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(posrecip(Y),2ndsneg(N,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons(X,Z))</LHS><RHS>2ndsneg(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(negrecip(Y),2ndspos(N,activate(Z)))</RHS></RULE><RULE><LHS>pi(X)</LHS><RHS>2ndspos(X,from(0))</RHS></RULE><RULE><LHS>plus(0,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>plus(s(X),Y)</LHS><RHS>s(plus(X,Y))</RHS></RULE><RULE><LHS>times(0,Y)</LHS><RHS>0</RHS></RULE><RULE><LHS>times(s(X),Y)</LHS><RHS>plus(Y,times(X,Y))</RHS></RULE><RULE><LHS>square(X)</LHS><RHS>times(X,X)</RHS></RULE><RULE><LHS>from(X)</LHS><RHS>n__from(X)</RHS></RULE><RULE><LHS>s(X)</LHS><RHS>n__s(X)</RHS></RULE><RULE><LHS>activate(n__from(X))</LHS><RHS>from(activate(X))</RHS></RULE><RULE><LHS>activate(n__s(X))</LHS><RHS>s(activate(X))</RHS></RULE><RULE><LHS>activate(X)</LHS><RHS>X</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>from(X)</LHS><RHS>cons(X,n__from(n__s(X)))</RHS></RULE><RULE><LHS>2ndspos(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndspos(s(N),cons(X,Z))</LHS><RHS>2ndspos(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndspos(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(posrecip(Y),2ndsneg(N,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons(X,Z))</LHS><RHS>2ndsneg(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(negrecip(Y),2ndspos(N,activate(Z)))</RHS></RULE><RULE><LHS>pi(X)</LHS><RHS>2ndspos(X,from(0))</RHS></RULE><RULE><LHS>plus(0,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>plus(s(X),Y)</LHS><RHS>s(plus(X,Y))</RHS></RULE><RULE><LHS>times(0,Y)</LHS><RHS>0</RHS></RULE><RULE><LHS>times(s(X),Y)</LHS><RHS>plus(Y,times(X,Y))</RHS></RULE><RULE><LHS>square(X)</LHS><RHS>times(X,X)</RHS></RULE><RULE><LHS>from(X)</LHS><RHS>n__from(X)</RHS></RULE><RULE><LHS>s(X)</LHS><RHS>n__s(X)</RHS></RULE><RULE><LHS>activate(n__from(X))</LHS><RHS>from(activate(X))</RHS></RULE><RULE><LHS>activate(n__s(X))</LHS><RHS>s(activate(X))</RHS></RULE><RULE><LHS>activate(X)</LHS><RHS>X</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_activate(n__from(X))</LHS><RHS>Marked_activate(X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_activate(n__s(X))</LHS><RHS>Marked_activate(X)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_activate</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_activate(n__s(X))</LHS><RHS>Marked_activate(X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_activate(n__from(X))</LHS><RHS>Marked_activate(X)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>from(X)</LHS><RHS>cons(X,n__from(n__s(X)))</RHS></RULE><RULE><LHS>2ndspos(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndspos(s(N),cons(X,Z))</LHS><RHS>2ndspos(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndspos(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(posrecip(Y),2ndsneg(N,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(0,Z)</LHS><RHS>rnil</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons(X,Z))</LHS><RHS>2ndsneg(s(N),cons2(X,activate(Z)))</RHS></RULE><RULE><LHS>2ndsneg(s(N),cons2(X,cons(Y,Z)))</LHS><RHS>rcons(negrecip(Y),2ndspos(N,activate(Z)))</RHS></RULE><RULE><LHS>pi(X)</LHS><RHS>2ndspos(X,from(0))</RHS></RULE><RULE><LHS>plus(0,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>plus(s(X),Y)</LHS><RHS>s(plus(X,Y))</RHS></RULE><RULE><LHS>times(0,Y)</LHS><RHS>0</RHS></RULE><RULE><LHS>times(s(X),Y)</LHS><RHS>plus(Y,times(X,Y))</RHS></RULE><RULE><LHS>square(X)</LHS><RHS>times(X,X)</RHS></RULE><RULE><LHS>from(X)</LHS><RHS>n__from(X)</RHS></RULE><RULE><LHS>s(X)</LHS><RHS>n__s(X)</RHS></RULE><RULE><LHS>activate(n__from(X))</LHS><RHS>from(activate(X))</RHS></RULE><RULE><LHS>activate(n__s(X))</LHS><RHS>s(activate(X))</RHS></RULE><RULE><LHS>activate(X)</LHS><RHS>X</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
