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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="2" unmarked=""><NAME>+</NAME></SYMBOL><SYMBOL arity="1" unmarked="p"><NAME>Marked_p</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>p</NAME></SYMBOL><SYMBOL arity="1" unmarked="f"><NAME>Marked_f</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>g</NAME></SYMBOL><SYMBOL arity="1" unmarked="g"><NAME>Marked_g</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>f</NAME></SYMBOL><SYMBOL arity="1" unmarked="h"><NAME>Marked_h</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>pair</NAME></SYMBOL><SYMBOL arity="2" unmarked="+"><NAME>Marked_+</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="1" unmarked="q"><NAME>Marked_q</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>h</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>q</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>z</VAR><VAR>y</VAR><VAR>x</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>f(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>f(s(0))</LHS><RHS>s(0)</RHS></RULE><RULE><LHS>f(s(s(x)))</LHS><RHS>p(h(g(x)))</RHS></RULE><RULE><LHS>g(0)</LHS><RHS>pair(s(0),s(0))</RHS></RULE><RULE><LHS>g(s(x))</LHS><RHS>h(g(x))</RHS></RULE><RULE><LHS>h(x)</LHS><RHS>pair(+(p(x),q(x)),p(x))</RHS></RULE><RULE><LHS>p(pair(x,y))</LHS><RHS>x</RHS></RULE><RULE><LHS>q(pair(x,y))</LHS><RHS>y</RHS></RULE><RULE><LHS>+(x,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>+(x,s(y))</LHS><RHS>s(+(x,y))</RHS></RULE><RULE><LHS>f(s(s(x)))</LHS><RHS>+(p(g(x)),q(g(x)))</RHS></RULE><RULE><LHS>g(s(x))</LHS><RHS>pair(+(p(g(x)),q(g(x))),p(g(x)))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>f(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>f(s(0))</LHS><RHS>s(0)</RHS></RULE><RULE><LHS>f(s(s(x)))</LHS><RHS>p(h(g(x)))</RHS></RULE><RULE><LHS>g(0)</LHS><RHS>pair(s(0),s(0))</RHS></RULE><RULE><LHS>g(s(x))</LHS><RHS>h(g(x))</RHS></RULE><RULE><LHS>h(x)</LHS><RHS>pair(+(p(x),q(x)),p(x))</RHS></RULE><RULE><LHS>p(pair(x,y))</LHS><RHS>x</RHS></RULE><RULE><LHS>q(pair(x,y))</LHS><RHS>y</RHS></RULE><RULE><LHS>+(x,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>+(x,s(y))</LHS><RHS>s(+(x,y))</RHS></RULE><RULE><LHS>f(s(s(x)))</LHS><RHS>+(p(g(x)),q(g(x)))</RHS></RULE><RULE><LHS>g(s(x))</LHS><RHS>pair(+(p(g(x)),q(g(x))),p(g(x)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_f(s(s(x)))</LHS><RHS>Marked_p(h(g(x)))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_f(s(s(x)))</LHS><RHS>Marked_h(g(x))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_f(s(s(x)))</LHS><RHS>Marked_g(x)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_g(s(x))</LHS><RHS>Marked_h(g(x))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_g(s(x))</LHS><RHS>Marked_g(x)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_h(x)</LHS><RHS>Marked_+(p(x),q(x))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_h(x)</LHS><RHS>Marked_p(x)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_h(x)</LHS><RHS>Marked_q(x)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_h(x)</LHS><RHS>Marked_p(x)</RHS></DPRULE><DPRULE num="9"><LHS>Marked_+(x,s(y))</LHS><RHS>Marked_+(x,y)</RHS></DPRULE><DPRULE num="10"><LHS>Marked_f(s(s(x)))</LHS><RHS>Marked_+(p(g(x)),q(g(x)))</RHS></DPRULE><DPRULE num="11"><LHS>Marked_f(s(s(x)))</LHS><RHS>Marked_p(g(x))</RHS></DPRULE><DPRULE num="12"><LHS>Marked_f(s(s(x)))</LHS><RHS>Marked_g(x)</RHS></DPRULE><DPRULE num="13"><LHS>Marked_f(s(s(x)))</LHS><RHS>Marked_q(g(x))</RHS></DPRULE><DPRULE num="14"><LHS>Marked_f(s(s(x)))</LHS><RHS>Marked_g(x)</RHS></DPRULE><DPRULE num="15"><LHS>Marked_g(s(x))</LHS><RHS>Marked_+(p(g(x)),q(g(x)))</RHS></DPRULE><DPRULE num="16"><LHS>Marked_g(s(x))</LHS><RHS>Marked_p(g(x))</RHS></DPRULE><DPRULE num="17"><LHS>Marked_g(s(x))</LHS><RHS>Marked_g(x)</RHS></DPRULE><DPRULE num="18"><LHS>Marked_g(s(x))</LHS><RHS>Marked_q(g(x))</RHS></DPRULE><DPRULE num="19"><LHS>Marked_g(s(x))</LHS><RHS>Marked_g(x)</RHS></DPRULE><DPRULE num="20"><LHS>Marked_g(s(x))</LHS><RHS>Marked_p(g(x))</RHS></DPRULE><DPRULE num="21"><LHS>Marked_g(s(x))</LHS><RHS>Marked_g(x)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="4"/><NODE ref="4"/><NODE ref="4"/><NODE ref="4"/></SCC><NONSCC num="1"><NODE ref="16"/></NONSCC><NONSCC num="2"><NODE ref="18"/></NONSCC><NONSCC num="3"><NODE ref="15"/></NONSCC><NONSCC num="4"><NODE ref="3"/></NONSCC><SCC num="5"><NODE ref="9"/></SCC><NONSCC num="6"><NODE ref="2"/></NONSCC><NONSCC num="7"><NODE ref="13"/></NONSCC><NONSCC num="8"><NODE ref="11"/></NONSCC><NONSCC num="9"><NODE ref="10"/></NONSCC><NONSCC num="10"><NODE ref="6"/></NONSCC><NONSCC num="11"><NODE ref="7"/></NONSCC><NONSCC num="12"><NODE ref="5"/></NONSCC><NONSCC num="13"><NODE ref="1"/></NONSCC><NONSCC num="14"><NODE ref="0"/></NONSCC></CCLIST><EDGE end="4" start="0"/><EDGE end="3" start="0"/><EDGE end="2" start="0"/><EDGE end="1" start="0"/><EDGE end="5" start="3"/><EDGE end="12" start="4"/><EDGE end="11" start="4"/><EDGE end="10" start="4"/><EDGE end="4" start="6"/><EDGE end="3" start="6"/><EDGE end="2" start="6"/><EDGE end="1" start="6"/><EDGE end="0" start="6"/><EDGE end="5" start="9"/><EDGE end="5" start="12"/><EDGE end="12" start="13"/><EDGE end="11" start="13"/><EDGE end="10" start="13"/></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>f(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>f(s(0))</LHS><RHS>s(0)</RHS></RULE><RULE><LHS>f(s(s(x)))</LHS><RHS>p(h(g(x)))</RHS></RULE><RULE><LHS>g(0)</LHS><RHS>pair(s(0),s(0))</RHS></RULE><RULE><LHS>g(s(x))</LHS><RHS>h(g(x))</RHS></RULE><RULE><LHS>h(x)</LHS><RHS>pair(+(p(x),q(x)),p(x))</RHS></RULE><RULE><LHS>p(pair(x,y))</LHS><RHS>x</RHS></RULE><RULE><LHS>q(pair(x,y))</LHS><RHS>y</RHS></RULE><RULE><LHS>+(x,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>+(x,s(y))</LHS><RHS>s(+(x,y))</RHS></RULE><RULE><LHS>f(s(s(x)))</LHS><RHS>+(p(g(x)),q(g(x)))</RHS></RULE><RULE><LHS>g(s(x))</LHS><RHS>pair(+(p(g(x)),q(g(x))),p(g(x)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_g(s(x))</LHS><RHS>Marked_g(x)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_g(s(x))</LHS><RHS>Marked_g(x)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_g(s(x))</LHS><RHS>Marked_g(x)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_g(s(x))</LHS><RHS>Marked_g(x)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_g</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_g(s(x))</LHS><RHS>Marked_g(x)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_g(s(x))</LHS><RHS>Marked_g(x)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_g(s(x))</LHS><RHS>Marked_g(x)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_g(s(x))</LHS><RHS>Marked_g(x)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>f(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>f(s(0))</LHS><RHS>s(0)</RHS></RULE><RULE><LHS>f(s(s(x)))</LHS><RHS>p(h(g(x)))</RHS></RULE><RULE><LHS>g(0)</LHS><RHS>pair(s(0),s(0))</RHS></RULE><RULE><LHS>g(s(x))</LHS><RHS>h(g(x))</RHS></RULE><RULE><LHS>h(x)</LHS><RHS>pair(+(p(x),q(x)),p(x))</RHS></RULE><RULE><LHS>p(pair(x,y))</LHS><RHS>x</RHS></RULE><RULE><LHS>q(pair(x,y))</LHS><RHS>y</RHS></RULE><RULE><LHS>+(x,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>+(x,s(y))</LHS><RHS>s(+(x,y))</RHS></RULE><RULE><LHS>f(s(s(x)))</LHS><RHS>+(p(g(x)),q(g(x)))</RHS></RULE><RULE><LHS>g(s(x))</LHS><RHS>pair(+(p(g(x)),q(g(x))),p(g(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>f(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>f(s(0))</LHS><RHS>s(0)</RHS></RULE><RULE><LHS>f(s(s(x)))</LHS><RHS>p(h(g(x)))</RHS></RULE><RULE><LHS>g(0)</LHS><RHS>pair(s(0),s(0))</RHS></RULE><RULE><LHS>g(s(x))</LHS><RHS>h(g(x))</RHS></RULE><RULE><LHS>h(x)</LHS><RHS>pair(+(p(x),q(x)),p(x))</RHS></RULE><RULE><LHS>p(pair(x,y))</LHS><RHS>x</RHS></RULE><RULE><LHS>q(pair(x,y))</LHS><RHS>y</RHS></RULE><RULE><LHS>+(x,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>+(x,s(y))</LHS><RHS>s(+(x,y))</RHS></RULE><RULE><LHS>f(s(s(x)))</LHS><RHS>+(p(g(x)),q(g(x)))</RHS></RULE><RULE><LHS>g(s(x))</LHS><RHS>pair(+(p(g(x)),q(g(x))),p(g(x)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_+(x,s(y))</LHS><RHS>Marked_+(x,y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_+</NAME></SYMBOL><INT>2</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_+(x,s(y))</LHS><RHS>Marked_+(x,y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>f(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>f(s(0))</LHS><RHS>s(0)</RHS></RULE><RULE><LHS>f(s(s(x)))</LHS><RHS>p(h(g(x)))</RHS></RULE><RULE><LHS>g(0)</LHS><RHS>pair(s(0),s(0))</RHS></RULE><RULE><LHS>g(s(x))</LHS><RHS>h(g(x))</RHS></RULE><RULE><LHS>h(x)</LHS><RHS>pair(+(p(x),q(x)),p(x))</RHS></RULE><RULE><LHS>p(pair(x,y))</LHS><RHS>x</RHS></RULE><RULE><LHS>q(pair(x,y))</LHS><RHS>y</RHS></RULE><RULE><LHS>+(x,0)</LHS><RHS>x</RHS></RULE><RULE><LHS>+(x,s(y))</LHS><RHS>s(+(x,y))</RHS></RULE><RULE><LHS>f(s(s(x)))</LHS><RHS>+(p(g(x)),q(g(x)))</RHS></RULE><RULE><LHS>g(s(x))</LHS><RHS>pair(+(p(g(x)),q(g(x))),p(g(x)))</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
