<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="2" unmarked="rem"><NAME>Marked_rem</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>f</NAME></SYMBOL><SYMBOL arity="1" unmarked="norm"><NAME>Marked_norm</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>norm</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>rem</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>nil</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>g</NAME></SYMBOL><SYMBOL arity="2" unmarked="f"><NAME>Marked_f</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>norm(nil)</LHS><RHS>0</RHS></RULE><RULE><LHS>norm(g(x,y))</LHS><RHS>s(norm(x))</RHS></RULE><RULE><LHS>f(x,nil)</LHS><RHS>g(nil,x)</RHS></RULE><RULE><LHS>f(x,g(y,z))</LHS><RHS>g(f(x,y),z)</RHS></RULE><RULE><LHS>rem(nil,y)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rem(g(x,y),0)</LHS><RHS>g(x,y)</RHS></RULE><RULE><LHS>rem(g(x,y),s(z))</LHS><RHS>rem(x,z)</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>norm(nil)</LHS><RHS>0</RHS></RULE><RULE><LHS>norm(g(x,y))</LHS><RHS>s(norm(x))</RHS></RULE><RULE><LHS>f(x,nil)</LHS><RHS>g(nil,x)</RHS></RULE><RULE><LHS>f(x,g(y,z))</LHS><RHS>g(f(x,y),z)</RHS></RULE><RULE><LHS>rem(nil,y)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rem(g(x,y),0)</LHS><RHS>g(x,y)</RHS></RULE><RULE><LHS>rem(g(x,y),s(z))</LHS><RHS>rem(x,z)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_norm(g(x,y))</LHS><RHS>Marked_norm(x)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_f(x,g(y,z))</LHS><RHS>Marked_f(x,y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_rem(g(x,y),s(z))</LHS><RHS>Marked_rem(x,z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="2"/></SCC><SCC num="1"><NODE ref="1"/></SCC><SCC num="2"><NODE ref="0"/></SCC></CCLIST></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>norm(nil)</LHS><RHS>0</RHS></RULE><RULE><LHS>norm(g(x,y))</LHS><RHS>s(norm(x))</RHS></RULE><RULE><LHS>f(x,nil)</LHS><RHS>g(nil,x)</RHS></RULE><RULE><LHS>f(x,g(y,z))</LHS><RHS>g(f(x,y),z)</RHS></RULE><RULE><LHS>rem(nil,y)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rem(g(x,y),0)</LHS><RHS>g(x,y)</RHS></RULE><RULE><LHS>rem(g(x,y),s(z))</LHS><RHS>rem(x,z)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_norm(g(x,y))</LHS><RHS>Marked_norm(x)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_norm</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_norm(g(x,y))</LHS><RHS>Marked_norm(x)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>norm(nil)</LHS><RHS>0</RHS></RULE><RULE><LHS>norm(g(x,y))</LHS><RHS>s(norm(x))</RHS></RULE><RULE><LHS>f(x,nil)</LHS><RHS>g(nil,x)</RHS></RULE><RULE><LHS>f(x,g(y,z))</LHS><RHS>g(f(x,y),z)</RHS></RULE><RULE><LHS>rem(nil,y)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rem(g(x,y),0)</LHS><RHS>g(x,y)</RHS></RULE><RULE><LHS>rem(g(x,y),s(z))</LHS><RHS>rem(x,z)</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>norm(nil)</LHS><RHS>0</RHS></RULE><RULE><LHS>norm(g(x,y))</LHS><RHS>s(norm(x))</RHS></RULE><RULE><LHS>f(x,nil)</LHS><RHS>g(nil,x)</RHS></RULE><RULE><LHS>f(x,g(y,z))</LHS><RHS>g(f(x,y),z)</RHS></RULE><RULE><LHS>rem(nil,y)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rem(g(x,y),0)</LHS><RHS>g(x,y)</RHS></RULE><RULE><LHS>rem(g(x,y),s(z))</LHS><RHS>rem(x,z)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_f(x,g(y,z))</LHS><RHS>Marked_f(x,y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_f</NAME></SYMBOL><INT>2</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_f(x,g(y,z))</LHS><RHS>Marked_f(x,y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>norm(nil)</LHS><RHS>0</RHS></RULE><RULE><LHS>norm(g(x,y))</LHS><RHS>s(norm(x))</RHS></RULE><RULE><LHS>f(x,nil)</LHS><RHS>g(nil,x)</RHS></RULE><RULE><LHS>f(x,g(y,z))</LHS><RHS>g(f(x,y),z)</RHS></RULE><RULE><LHS>rem(nil,y)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rem(g(x,y),0)</LHS><RHS>g(x,y)</RHS></RULE><RULE><LHS>rem(g(x,y),s(z))</LHS><RHS>rem(x,z)</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>norm(nil)</LHS><RHS>0</RHS></RULE><RULE><LHS>norm(g(x,y))</LHS><RHS>s(norm(x))</RHS></RULE><RULE><LHS>f(x,nil)</LHS><RHS>g(nil,x)</RHS></RULE><RULE><LHS>f(x,g(y,z))</LHS><RHS>g(f(x,y),z)</RHS></RULE><RULE><LHS>rem(nil,y)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rem(g(x,y),0)</LHS><RHS>g(x,y)</RHS></RULE><RULE><LHS>rem(g(x,y),s(z))</LHS><RHS>rem(x,z)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_rem(g(x,y),s(z))</LHS><RHS>Marked_rem(x,z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_rem</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_rem(g(x,y),s(z))</LHS><RHS>Marked_rem(x,z)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>norm(nil)</LHS><RHS>0</RHS></RULE><RULE><LHS>norm(g(x,y))</LHS><RHS>s(norm(x))</RHS></RULE><RULE><LHS>f(x,nil)</LHS><RHS>g(nil,x)</RHS></RULE><RULE><LHS>f(x,g(y,z))</LHS><RHS>g(f(x,y),z)</RHS></RULE><RULE><LHS>rem(nil,y)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rem(g(x,y),0)</LHS><RHS>g(x,y)</RHS></RULE><RULE><LHS>rem(g(x,y),s(z))</LHS><RHS>rem(x,z)</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
