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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="1" unmarked="top"><NAME>Marked_top</NAME></SYMBOL><SYMBOL arity="2" unmarked="f"><NAME>Marked_f</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>active</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>ok</NAME></SYMBOL><SYMBOL arity="1" unmarked="proper"><NAME>Marked_proper</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>f</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>top</NAME></SYMBOL><SYMBOL arity="1" unmarked="g"><NAME>Marked_g</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>g</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>proper</NAME></SYMBOL><SYMBOL arity="1" unmarked="active"><NAME>Marked_active</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>mark</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>X2</VAR><VAR>X1</VAR><VAR>Y</VAR><VAR>X</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>active(f(g(X),Y))</LHS><RHS>mark(f(X,f(g(X),Y)))</RHS></RULE><RULE><LHS>active(f(X1,X2))</LHS><RHS>f(active(X1),X2)</RHS></RULE><RULE><LHS>active(g(X))</LHS><RHS>g(active(X))</RHS></RULE><RULE><LHS>f(mark(X1),X2)</LHS><RHS>mark(f(X1,X2))</RHS></RULE><RULE><LHS>g(mark(X))</LHS><RHS>mark(g(X))</RHS></RULE><RULE><LHS>proper(f(X1,X2))</LHS><RHS>f(proper(X1),proper(X2))</RHS></RULE><RULE><LHS>proper(g(X))</LHS><RHS>g(proper(X))</RHS></RULE><RULE><LHS>f(ok(X1),ok(X2))</LHS><RHS>ok(f(X1,X2))</RHS></RULE><RULE><LHS>g(ok(X))</LHS><RHS>ok(g(X))</RHS></RULE><RULE><LHS>top(mark(X))</LHS><RHS>top(proper(X))</RHS></RULE><RULE><LHS>top(ok(X))</LHS><RHS>top(active(X))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>active(f(g(X),Y))</LHS><RHS>mark(f(X,f(g(X),Y)))</RHS></RULE><RULE><LHS>active(f(X1,X2))</LHS><RHS>f(active(X1),X2)</RHS></RULE><RULE><LHS>active(g(X))</LHS><RHS>g(active(X))</RHS></RULE><RULE><LHS>f(mark(X1),X2)</LHS><RHS>mark(f(X1,X2))</RHS></RULE><RULE><LHS>g(mark(X))</LHS><RHS>mark(g(X))</RHS></RULE><RULE><LHS>proper(f(X1,X2))</LHS><RHS>f(proper(X1),proper(X2))</RHS></RULE><RULE><LHS>proper(g(X))</LHS><RHS>g(proper(X))</RHS></RULE><RULE><LHS>f(ok(X1),ok(X2))</LHS><RHS>ok(f(X1,X2))</RHS></RULE><RULE><LHS>g(ok(X))</LHS><RHS>ok(g(X))</RHS></RULE><RULE><LHS>top(mark(X))</LHS><RHS>top(proper(X))</RHS></RULE><RULE><LHS>top(ok(X))</LHS><RHS>top(active(X))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_active(f(g(X),Y))</LHS><RHS>Marked_f(X,f(g(X),Y))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_active(f(X1,X2))</LHS><RHS>Marked_f(active(X1),X2)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_active(f(X1,X2))</LHS><RHS>Marked_active(X1)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_active(g(X))</LHS><RHS>Marked_g(active(X))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_active(g(X))</LHS><RHS>Marked_active(X)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_f(mark(X1),X2)</LHS><RHS>Marked_f(X1,X2)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_g(mark(X))</LHS><RHS>Marked_g(X)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_proper(f(X1,X2))</LHS><RHS>Marked_f(proper(X1),proper(X2))</RHS></DPRULE><DPRULE num="8"><LHS>Marked_proper(f(X1,X2))</LHS><RHS>Marked_proper(X1)</RHS></DPRULE><DPRULE num="9"><LHS>Marked_proper(f(X1,X2))</LHS><RHS>Marked_proper(X2)</RHS></DPRULE><DPRULE num="10"><LHS>Marked_proper(g(X))</LHS><RHS>Marked_g(proper(X))</RHS></DPRULE><DPRULE num="11"><LHS>Marked_proper(g(X))</LHS><RHS>Marked_proper(X)</RHS></DPRULE><DPRULE num="12"><LHS>Marked_f(ok(X1),ok(X2))</LHS><RHS>Marked_f(X1,X2)</RHS></DPRULE><DPRULE num="13"><LHS>Marked_g(ok(X))</LHS><RHS>Marked_g(X)</RHS></DPRULE><DPRULE num="14"><LHS>Marked_top(mark(X))</LHS><RHS>Marked_top(proper(X))</RHS></DPRULE><DPRULE num="15"><LHS>Marked_top(mark(X))</LHS><RHS>Marked_proper(X)</RHS></DPRULE><DPRULE num="16"><LHS>Marked_top(ok(X))</LHS><RHS>Marked_top(active(X))</RHS></DPRULE><DPRULE num="17"><LHS>Marked_top(ok(X))</LHS><RHS>Marked_active(X)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="17"/></NONSCC><SCC num="1"><NODE ref="2"/><NODE ref="4"/></SCC><NONSCC num="2"><NODE ref="3"/></NONSCC><NONSCC num="3"><NODE ref="1"/></NONSCC><NONSCC num="4"><NODE ref="0"/></NONSCC><SCC num="5"><NODE ref="14"/><NODE ref="16"/></SCC><NONSCC num="6"><NODE ref="15"/></NONSCC><SCC num="7"><NODE ref="8"/><NODE ref="9"/><NODE ref="11"/></SCC><NONSCC num="8"><NODE ref="10"/></NONSCC><NONSCC num="9"><NODE ref="7"/></NONSCC><SCC num="10"><NODE ref="6"/><NODE ref="13"/></SCC><SCC num="11"><NODE ref="5"/><NODE ref="12"/></SCC></CCLIST><EDGE end="4" start="0"/><EDGE end="3" start="0"/><EDGE end="2" start="0"/><EDGE end="1" start="0"/><EDGE end="4" start="1"/><EDGE end="3" start="1"/><EDGE end="2" start="1"/><EDGE end="10" start="2"/><EDGE end="11" start="3"/><EDGE end="11" start="4"/><EDGE end="6" start="5"/><EDGE end="0" start="5"/><EDGE end="9" start="6"/><EDGE end="8" start="6"/><EDGE end="7" start="6"/><EDGE end="9" start="7"/><EDGE end="8" start="7"/><EDGE end="10" start="8"/><EDGE end="11" start="9"/></DAG></CRITERION><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>active(f(g(X),Y))</LHS><RHS>mark(f(X,f(g(X),Y)))</RHS></RULE><RULE><LHS>active(f(X1,X2))</LHS><RHS>f(active(X1),X2)</RHS></RULE><RULE><LHS>active(g(X))</LHS><RHS>g(active(X))</RHS></RULE><RULE><LHS>f(mark(X1),X2)</LHS><RHS>mark(f(X1,X2))</RHS></RULE><RULE><LHS>g(mark(X))</LHS><RHS>mark(g(X))</RHS></RULE><RULE><LHS>proper(f(X1,X2))</LHS><RHS>f(proper(X1),proper(X2))</RHS></RULE><RULE><LHS>proper(g(X))</LHS><RHS>g(proper(X))</RHS></RULE><RULE><LHS>f(ok(X1),ok(X2))</LHS><RHS>ok(f(X1,X2))</RHS></RULE><RULE><LHS>g(ok(X))</LHS><RHS>ok(g(X))</RHS></RULE><RULE><LHS>top(mark(X))</LHS><RHS>top(proper(X))</RHS></RULE><RULE><LHS>top(ok(X))</LHS><RHS>top(active(X))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_top(mark(X))</LHS><RHS>Marked_top(proper(X))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_top(ok(X))</LHS><RHS>Marked_top(active(X))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>mark</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>proper</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>g</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>top</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>f</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>ok</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>active</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_top</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB></ORDERING></CRITERION></PROPERTY><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>active(f(g(X),Y))</LHS><RHS>mark(f(X,f(g(X),Y)))</RHS></RULE><RULE><LHS>active(f(X1,X2))</LHS><RHS>f(active(X1),X2)</RHS></RULE><RULE><LHS>active(g(X))</LHS><RHS>g(active(X))</RHS></RULE><RULE><LHS>f(mark(X1),X2)</LHS><RHS>mark(f(X1,X2))</RHS></RULE><RULE><LHS>g(mark(X))</LHS><RHS>mark(g(X))</RHS></RULE><RULE><LHS>proper(f(X1,X2))</LHS><RHS>f(proper(X1),proper(X2))</RHS></RULE><RULE><LHS>proper(g(X))</LHS><RHS>g(proper(X))</RHS></RULE><RULE><LHS>f(ok(X1),ok(X2))</LHS><RHS>ok(f(X1,X2))</RHS></RULE><RULE><LHS>g(ok(X))</LHS><RHS>ok(g(X))</RHS></RULE><RULE><LHS>top(mark(X))</LHS><RHS>top(proper(X))</RHS></RULE><RULE><LHS>top(ok(X))</LHS><RHS>top(active(X))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_proper(f(X1,X2))</LHS><RHS>Marked_proper(X1)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_proper(f(X1,X2))</LHS><RHS>Marked_proper(X2)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_proper(g(X))</LHS><RHS>Marked_proper(X)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_proper</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_proper(g(X))</LHS><RHS>Marked_proper(X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_proper(f(X1,X2))</LHS><RHS>Marked_proper(X2)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_proper(f(X1,X2))</LHS><RHS>Marked_proper(X1)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>active(f(g(X),Y))</LHS><RHS>mark(f(X,f(g(X),Y)))</RHS></RULE><RULE><LHS>active(f(X1,X2))</LHS><RHS>f(active(X1),X2)</RHS></RULE><RULE><LHS>active(g(X))</LHS><RHS>g(active(X))</RHS></RULE><RULE><LHS>f(mark(X1),X2)</LHS><RHS>mark(f(X1,X2))</RHS></RULE><RULE><LHS>g(mark(X))</LHS><RHS>mark(g(X))</RHS></RULE><RULE><LHS>proper(f(X1,X2))</LHS><RHS>f(proper(X1),proper(X2))</RHS></RULE><RULE><LHS>proper(g(X))</LHS><RHS>g(proper(X))</RHS></RULE><RULE><LHS>f(ok(X1),ok(X2))</LHS><RHS>ok(f(X1,X2))</RHS></RULE><RULE><LHS>g(ok(X))</LHS><RHS>ok(g(X))</RHS></RULE><RULE><LHS>top(mark(X))</LHS><RHS>top(proper(X))</RHS></RULE><RULE><LHS>top(ok(X))</LHS><RHS>top(active(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>active(f(g(X),Y))</LHS><RHS>mark(f(X,f(g(X),Y)))</RHS></RULE><RULE><LHS>active(f(X1,X2))</LHS><RHS>f(active(X1),X2)</RHS></RULE><RULE><LHS>active(g(X))</LHS><RHS>g(active(X))</RHS></RULE><RULE><LHS>f(mark(X1),X2)</LHS><RHS>mark(f(X1,X2))</RHS></RULE><RULE><LHS>g(mark(X))</LHS><RHS>mark(g(X))</RHS></RULE><RULE><LHS>proper(f(X1,X2))</LHS><RHS>f(proper(X1),proper(X2))</RHS></RULE><RULE><LHS>proper(g(X))</LHS><RHS>g(proper(X))</RHS></RULE><RULE><LHS>f(ok(X1),ok(X2))</LHS><RHS>ok(f(X1,X2))</RHS></RULE><RULE><LHS>g(ok(X))</LHS><RHS>ok(g(X))</RHS></RULE><RULE><LHS>top(mark(X))</LHS><RHS>top(proper(X))</RHS></RULE><RULE><LHS>top(ok(X))</LHS><RHS>top(active(X))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_active(f(X1,X2))</LHS><RHS>Marked_active(X1)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_active(g(X))</LHS><RHS>Marked_active(X)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_active</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_active(g(X))</LHS><RHS>Marked_active(X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_active(f(X1,X2))</LHS><RHS>Marked_active(X1)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>active(f(g(X),Y))</LHS><RHS>mark(f(X,f(g(X),Y)))</RHS></RULE><RULE><LHS>active(f(X1,X2))</LHS><RHS>f(active(X1),X2)</RHS></RULE><RULE><LHS>active(g(X))</LHS><RHS>g(active(X))</RHS></RULE><RULE><LHS>f(mark(X1),X2)</LHS><RHS>mark(f(X1,X2))</RHS></RULE><RULE><LHS>g(mark(X))</LHS><RHS>mark(g(X))</RHS></RULE><RULE><LHS>proper(f(X1,X2))</LHS><RHS>f(proper(X1),proper(X2))</RHS></RULE><RULE><LHS>proper(g(X))</LHS><RHS>g(proper(X))</RHS></RULE><RULE><LHS>f(ok(X1),ok(X2))</LHS><RHS>ok(f(X1,X2))</RHS></RULE><RULE><LHS>g(ok(X))</LHS><RHS>ok(g(X))</RHS></RULE><RULE><LHS>top(mark(X))</LHS><RHS>top(proper(X))</RHS></RULE><RULE><LHS>top(ok(X))</LHS><RHS>top(active(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>active(f(g(X),Y))</LHS><RHS>mark(f(X,f(g(X),Y)))</RHS></RULE><RULE><LHS>active(f(X1,X2))</LHS><RHS>f(active(X1),X2)</RHS></RULE><RULE><LHS>active(g(X))</LHS><RHS>g(active(X))</RHS></RULE><RULE><LHS>f(mark(X1),X2)</LHS><RHS>mark(f(X1,X2))</RHS></RULE><RULE><LHS>g(mark(X))</LHS><RHS>mark(g(X))</RHS></RULE><RULE><LHS>proper(f(X1,X2))</LHS><RHS>f(proper(X1),proper(X2))</RHS></RULE><RULE><LHS>proper(g(X))</LHS><RHS>g(proper(X))</RHS></RULE><RULE><LHS>f(ok(X1),ok(X2))</LHS><RHS>ok(f(X1,X2))</RHS></RULE><RULE><LHS>g(ok(X))</LHS><RHS>ok(g(X))</RHS></RULE><RULE><LHS>top(mark(X))</LHS><RHS>top(proper(X))</RHS></RULE><RULE><LHS>top(ok(X))</LHS><RHS>top(active(X))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_f(mark(X1),X2)</LHS><RHS>Marked_f(X1,X2)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_f(ok(X1),ok(X2))</LHS><RHS>Marked_f(X1,X2)</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(ok(X1),ok(X2))</LHS><RHS>Marked_f(X1,X2)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>active(f(g(X),Y))</LHS><RHS>mark(f(X,f(g(X),Y)))</RHS></RULE><RULE><LHS>active(f(X1,X2))</LHS><RHS>f(active(X1),X2)</RHS></RULE><RULE><LHS>active(g(X))</LHS><RHS>g(active(X))</RHS></RULE><RULE><LHS>f(mark(X1),X2)</LHS><RHS>mark(f(X1,X2))</RHS></RULE><RULE><LHS>g(mark(X))</LHS><RHS>mark(g(X))</RHS></RULE><RULE><LHS>proper(f(X1,X2))</LHS><RHS>f(proper(X1),proper(X2))</RHS></RULE><RULE><LHS>proper(g(X))</LHS><RHS>g(proper(X))</RHS></RULE><RULE><LHS>f(ok(X1),ok(X2))</LHS><RHS>ok(f(X1,X2))</RHS></RULE><RULE><LHS>g(ok(X))</LHS><RHS>ok(g(X))</RHS></RULE><RULE><LHS>top(mark(X))</LHS><RHS>top(proper(X))</RHS></RULE><RULE><LHS>top(ok(X))</LHS><RHS>top(active(X))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_f(mark(X1),X2)</LHS><RHS>Marked_f(X1,X2)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/></SCC></CCLIST></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>active(f(g(X),Y))</LHS><RHS>mark(f(X,f(g(X),Y)))</RHS></RULE><RULE><LHS>active(f(X1,X2))</LHS><RHS>f(active(X1),X2)</RHS></RULE><RULE><LHS>active(g(X))</LHS><RHS>g(active(X))</RHS></RULE><RULE><LHS>f(mark(X1),X2)</LHS><RHS>mark(f(X1,X2))</RHS></RULE><RULE><LHS>g(mark(X))</LHS><RHS>mark(g(X))</RHS></RULE><RULE><LHS>proper(f(X1,X2))</LHS><RHS>f(proper(X1),proper(X2))</RHS></RULE><RULE><LHS>proper(g(X))</LHS><RHS>g(proper(X))</RHS></RULE><RULE><LHS>f(ok(X1),ok(X2))</LHS><RHS>ok(f(X1,X2))</RHS></RULE><RULE><LHS>g(ok(X))</LHS><RHS>ok(g(X))</RHS></RULE><RULE><LHS>top(mark(X))</LHS><RHS>top(proper(X))</RHS></RULE><RULE><LHS>top(ok(X))</LHS><RHS>top(active(X))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_f(mark(X1),X2)</LHS><RHS>Marked_f(X1,X2)</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(mark(X1),X2)</LHS><RHS>Marked_f(X1,X2)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>active(f(g(X),Y))</LHS><RHS>mark(f(X,f(g(X),Y)))</RHS></RULE><RULE><LHS>active(f(X1,X2))</LHS><RHS>f(active(X1),X2)</RHS></RULE><RULE><LHS>active(g(X))</LHS><RHS>g(active(X))</RHS></RULE><RULE><LHS>f(mark(X1),X2)</LHS><RHS>mark(f(X1,X2))</RHS></RULE><RULE><LHS>g(mark(X))</LHS><RHS>mark(g(X))</RHS></RULE><RULE><LHS>proper(f(X1,X2))</LHS><RHS>f(proper(X1),proper(X2))</RHS></RULE><RULE><LHS>proper(g(X))</LHS><RHS>g(proper(X))</RHS></RULE><RULE><LHS>f(ok(X1),ok(X2))</LHS><RHS>ok(f(X1,X2))</RHS></RULE><RULE><LHS>g(ok(X))</LHS><RHS>ok(g(X))</RHS></RULE><RULE><LHS>top(mark(X))</LHS><RHS>top(proper(X))</RHS></RULE><RULE><LHS>top(ok(X))</LHS><RHS>top(active(X))</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>active(f(g(X),Y))</LHS><RHS>mark(f(X,f(g(X),Y)))</RHS></RULE><RULE><LHS>active(f(X1,X2))</LHS><RHS>f(active(X1),X2)</RHS></RULE><RULE><LHS>active(g(X))</LHS><RHS>g(active(X))</RHS></RULE><RULE><LHS>f(mark(X1),X2)</LHS><RHS>mark(f(X1,X2))</RHS></RULE><RULE><LHS>g(mark(X))</LHS><RHS>mark(g(X))</RHS></RULE><RULE><LHS>proper(f(X1,X2))</LHS><RHS>f(proper(X1),proper(X2))</RHS></RULE><RULE><LHS>proper(g(X))</LHS><RHS>g(proper(X))</RHS></RULE><RULE><LHS>f(ok(X1),ok(X2))</LHS><RHS>ok(f(X1,X2))</RHS></RULE><RULE><LHS>g(ok(X))</LHS><RHS>ok(g(X))</RHS></RULE><RULE><LHS>top(mark(X))</LHS><RHS>top(proper(X))</RHS></RULE><RULE><LHS>top(ok(X))</LHS><RHS>top(active(X))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_g(mark(X))</LHS><RHS>Marked_g(X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_g(ok(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(ok(X))</LHS><RHS>Marked_g(X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_g(mark(X))</LHS><RHS>Marked_g(X)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>active(f(g(X),Y))</LHS><RHS>mark(f(X,f(g(X),Y)))</RHS></RULE><RULE><LHS>active(f(X1,X2))</LHS><RHS>f(active(X1),X2)</RHS></RULE><RULE><LHS>active(g(X))</LHS><RHS>g(active(X))</RHS></RULE><RULE><LHS>f(mark(X1),X2)</LHS><RHS>mark(f(X1,X2))</RHS></RULE><RULE><LHS>g(mark(X))</LHS><RHS>mark(g(X))</RHS></RULE><RULE><LHS>proper(f(X1,X2))</LHS><RHS>f(proper(X1),proper(X2))</RHS></RULE><RULE><LHS>proper(g(X))</LHS><RHS>g(proper(X))</RHS></RULE><RULE><LHS>f(ok(X1),ok(X2))</LHS><RHS>ok(f(X1,X2))</RHS></RULE><RULE><LHS>g(ok(X))</LHS><RHS>ok(g(X))</RHS></RULE><RULE><LHS>top(mark(X))</LHS><RHS>top(proper(X))</RHS></RULE><RULE><LHS>top(ok(X))</LHS><RHS>top(active(X))</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
