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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="1" unmarked="i"><NAME>Marked_i</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>j</NAME></SYMBOL><SYMBOL arity="1" unmarked="g"><NAME>Marked_g</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>h</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>f</NAME></SYMBOL><SYMBOL arity="2" unmarked="f"><NAME>Marked_f</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>h1</NAME></SYMBOL><SYMBOL arity="1" unmarked="k"><NAME>Marked_k</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>k</NAME></SYMBOL><SYMBOL arity="3" unmarked="h2"><NAME>Marked_h2</NAME></SYMBOL><SYMBOL arity="3" unmarked=""><NAME>h2</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>i</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>g</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>u</VAR><VAR>z</VAR><VAR>y</VAR><VAR>x</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>f(j(x,y),y)</LHS><RHS>g(f(x,k(y)))</RHS></RULE><RULE><LHS>f(x,h1(y,z))</LHS><RHS>h2(0,x,h1(y,z))</RHS></RULE><RULE><LHS>g(h2(x,y,h1(z,u)))</LHS><RHS>h2(s(x),y,h1(z,u))</RHS></RULE><RULE><LHS>h2(x,j(y,h1(z,u)),h1(z,u))</LHS><RHS>h2(s(x),y,h1(s(z),u))</RHS></RULE><RULE><LHS>i(f(x,h(y)))</LHS><RHS>y</RHS></RULE><RULE><LHS>i(h2(s(x),y,h1(x,z)))</LHS><RHS>z</RHS></RULE><RULE><LHS>k(h(x))</LHS><RHS>h1(0,x)</RHS></RULE><RULE><LHS>k(h1(x,y))</LHS><RHS>h1(s(x),y)</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>f(j(x,y),y)</LHS><RHS>g(f(x,k(y)))</RHS></RULE><RULE><LHS>f(x,h1(y,z))</LHS><RHS>h2(0,x,h1(y,z))</RHS></RULE><RULE><LHS>g(h2(x,y,h1(z,u)))</LHS><RHS>h2(s(x),y,h1(z,u))</RHS></RULE><RULE><LHS>h2(x,j(y,h1(z,u)),h1(z,u))</LHS><RHS>h2(s(x),y,h1(s(z),u))</RHS></RULE><RULE><LHS>i(f(x,h(y)))</LHS><RHS>y</RHS></RULE><RULE><LHS>i(h2(s(x),y,h1(x,z)))</LHS><RHS>z</RHS></RULE><RULE><LHS>k(h(x))</LHS><RHS>h1(0,x)</RHS></RULE><RULE><LHS>k(h1(x,y))</LHS><RHS>h1(s(x),y)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_f(j(x,y),y)</LHS><RHS>Marked_g(f(x,k(y)))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_f(j(x,y),y)</LHS><RHS>Marked_f(x,k(y))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_f(j(x,y),y)</LHS><RHS>Marked_k(y)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_f(x,h1(y,z))</LHS><RHS>Marked_h2(0,x,h1(y,z))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_g(h2(x,y,h1(z,u)))</LHS><RHS>Marked_h2(s(x),y,h1(z,u))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_h2(x,j(y,h1(z,u)),h1(z,u))</LHS><RHS>Marked_h2(s(x),y,h1(s(z),u))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="5"/></SCC><NONSCC num="1"><NODE ref="4"/></NONSCC><NONSCC num="2"><NODE ref="3"/></NONSCC><NONSCC num="3"><NODE ref="2"/></NONSCC><SCC num="4"><NODE ref="1"/></SCC><NONSCC num="5"><NODE ref="0"/></NONSCC></CCLIST><EDGE end="0" start="1"/><EDGE end="0" start="2"/><EDGE end="5" start="4"/><EDGE end="3" start="4"/><EDGE end="2" start="4"/><EDGE end="1" start="5"/></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>f(j(x,y),y)</LHS><RHS>g(f(x,k(y)))</RHS></RULE><RULE><LHS>f(x,h1(y,z))</LHS><RHS>h2(0,x,h1(y,z))</RHS></RULE><RULE><LHS>g(h2(x,y,h1(z,u)))</LHS><RHS>h2(s(x),y,h1(z,u))</RHS></RULE><RULE><LHS>h2(x,j(y,h1(z,u)),h1(z,u))</LHS><RHS>h2(s(x),y,h1(s(z),u))</RHS></RULE><RULE><LHS>i(f(x,h(y)))</LHS><RHS>y</RHS></RULE><RULE><LHS>i(h2(s(x),y,h1(x,z)))</LHS><RHS>z</RHS></RULE><RULE><LHS>k(h(x))</LHS><RHS>h1(0,x)</RHS></RULE><RULE><LHS>k(h1(x,y))</LHS><RHS>h1(s(x),y)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_f(j(x,y),y)</LHS><RHS>Marked_f(x,k(y))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_f</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_f(j(x,y),y)</LHS><RHS>Marked_f(x,k(y))</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>f(j(x,y),y)</LHS><RHS>g(f(x,k(y)))</RHS></RULE><RULE><LHS>f(x,h1(y,z))</LHS><RHS>h2(0,x,h1(y,z))</RHS></RULE><RULE><LHS>g(h2(x,y,h1(z,u)))</LHS><RHS>h2(s(x),y,h1(z,u))</RHS></RULE><RULE><LHS>h2(x,j(y,h1(z,u)),h1(z,u))</LHS><RHS>h2(s(x),y,h1(s(z),u))</RHS></RULE><RULE><LHS>i(f(x,h(y)))</LHS><RHS>y</RHS></RULE><RULE><LHS>i(h2(s(x),y,h1(x,z)))</LHS><RHS>z</RHS></RULE><RULE><LHS>k(h(x))</LHS><RHS>h1(0,x)</RHS></RULE><RULE><LHS>k(h1(x,y))</LHS><RHS>h1(s(x),y)</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(j(x,y),y)</LHS><RHS>g(f(x,k(y)))</RHS></RULE><RULE><LHS>f(x,h1(y,z))</LHS><RHS>h2(0,x,h1(y,z))</RHS></RULE><RULE><LHS>g(h2(x,y,h1(z,u)))</LHS><RHS>h2(s(x),y,h1(z,u))</RHS></RULE><RULE><LHS>h2(x,j(y,h1(z,u)),h1(z,u))</LHS><RHS>h2(s(x),y,h1(s(z),u))</RHS></RULE><RULE><LHS>i(f(x,h(y)))</LHS><RHS>y</RHS></RULE><RULE><LHS>i(h2(s(x),y,h1(x,z)))</LHS><RHS>z</RHS></RULE><RULE><LHS>k(h(x))</LHS><RHS>h1(0,x)</RHS></RULE><RULE><LHS>k(h1(x,y))</LHS><RHS>h1(s(x),y)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_h2(x,j(y,h1(z,u)),h1(z,u))</LHS><RHS>Marked_h2(s(x),y,h1(s(z),u))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_h2</NAME></SYMBOL><INT>2</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_h2(x,j(y,h1(z,u)),h1(z,u))</LHS><RHS>Marked_h2(s(x),y,h1(s(z),u))</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>f(j(x,y),y)</LHS><RHS>g(f(x,k(y)))</RHS></RULE><RULE><LHS>f(x,h1(y,z))</LHS><RHS>h2(0,x,h1(y,z))</RHS></RULE><RULE><LHS>g(h2(x,y,h1(z,u)))</LHS><RHS>h2(s(x),y,h1(z,u))</RHS></RULE><RULE><LHS>h2(x,j(y,h1(z,u)),h1(z,u))</LHS><RHS>h2(s(x),y,h1(s(z),u))</RHS></RULE><RULE><LHS>i(f(x,h(y)))</LHS><RHS>y</RHS></RULE><RULE><LHS>i(h2(s(x),y,h1(x,z)))</LHS><RHS>z</RHS></RULE><RULE><LHS>k(h(x))</LHS><RHS>h1(0,x)</RHS></RULE><RULE><LHS>k(h1(x,y))</LHS><RHS>h1(s(x),y)</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
