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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="0" unmarked=""><NAME>member</NAME></SYMBOL><SYMBOL arity="2" unmarked="app"><NAME>Marked_app</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>true</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>false</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>if</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>lt</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>eq</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>fork</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>null</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>app</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>z</VAR><VAR>w</VAR><VAR>y</VAR><VAR>x</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>app(app(lt,app(s,x)),app(s,y))</LHS><RHS>app(app(lt,x),y)</RHS></RULE><RULE><LHS>app(app(lt,0),app(s,y))</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(lt,y),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,x),x)</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(eq,app(s,x)),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,0),app(s,x))</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),null)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(lt,app(s,x)),app(s,y))</LHS><RHS>app(app(lt,x),y)</RHS></RULE><RULE><LHS>app(app(lt,0),app(s,y))</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(lt,y),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,x),x)</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(eq,app(s,x)),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,0),app(s,x))</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),null)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(lt,app(s,x)),app(s,y))</LHS><RHS>Marked_app(app(lt,x),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(lt,app(s,x)),app(s,y))</LHS><RHS>Marked_app(lt,x)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(if,app(app(lt,w),y)),app(app(member,w),x))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(if,app(app(lt,w),y))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(lt,w),y)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(lt,w)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),x)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z))</RHS></DPRULE><DPRULE num="9"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(if,app(app(eq,w),y)),true)</RHS></DPRULE><DPRULE num="10"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(if,app(app(eq,w),y))</RHS></DPRULE><DPRULE num="11"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(eq,w),y)</RHS></DPRULE><DPRULE num="12"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(eq,w)</RHS></DPRULE><DPRULE num="13"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/><NODE ref="2"/><NODE ref="3"/><NODE ref="5"/><NODE ref="7"/><NODE ref="8"/><NODE ref="11"/><NODE ref="13"/></SCC><NONSCC num="1"><NODE ref="12"/></NONSCC><NONSCC num="2"><NODE ref="10"/></NONSCC><NONSCC num="3"><NODE ref="9"/></NONSCC><NONSCC num="4"><NODE ref="6"/></NONSCC><NONSCC num="5"><NODE ref="4"/></NONSCC><NONSCC num="6"><NODE ref="1"/></NONSCC></CCLIST><EDGE end="6" start="0"/><EDGE end="5" start="0"/><EDGE end="4" start="0"/><EDGE end="3" start="0"/><EDGE end="2" start="0"/><EDGE end="1" start="0"/></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(lt,app(s,x)),app(s,y))</LHS><RHS>app(app(lt,x),y)</RHS></RULE><RULE><LHS>app(app(lt,0),app(s,y))</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(lt,y),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,x),x)</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(eq,app(s,x)),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,0),app(s,x))</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),null)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(lt,app(s,x)),app(s,y))</LHS><RHS>Marked_app(app(lt,x),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(if,app(app(lt,w),y)),app(app(member,w),x))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(lt,w),y)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),x)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(eq,w),y)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>app</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>null</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>0</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>s</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>fork</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>eq</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>lt</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>if</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>false</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>true</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_app</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>member</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(if,app(app(lt,w),y)),app(app(member,w),x))</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(lt,app(s,x)),app(s,y))</LHS><RHS>app(app(lt,x),y)</RHS></RULE><RULE><LHS>app(app(lt,0),app(s,y))</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(lt,y),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,x),x)</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(eq,app(s,x)),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,0),app(s,x))</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),null)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(lt,app(s,x)),app(s,y))</LHS><RHS>Marked_app(app(lt,x),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(lt,w),y)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),x)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(eq,w),y)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/><NODE ref="1"/><NODE ref="2"/><NODE ref="3"/><NODE ref="4"/><NODE ref="5"/><NODE ref="6"/></SCC></CCLIST></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(lt,app(s,x)),app(s,y))</LHS><RHS>app(app(lt,x),y)</RHS></RULE><RULE><LHS>app(app(lt,0),app(s,y))</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(lt,y),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,x),x)</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(eq,app(s,x)),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,0),app(s,x))</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),null)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(lt,app(s,x)),app(s,y))</LHS><RHS>Marked_app(app(lt,x),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(lt,w),y)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),x)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(eq,w),y)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>app</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><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></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>null</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>0</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>s</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>fork</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>eq</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>lt</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>if</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>false</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>true</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_app</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>member</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(eq,w),y)</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(lt,app(s,x)),app(s,y))</LHS><RHS>app(app(lt,x),y)</RHS></RULE><RULE><LHS>app(app(lt,0),app(s,y))</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(lt,y),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,x),x)</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(eq,app(s,x)),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,0),app(s,x))</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),null)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(lt,app(s,x)),app(s,y))</LHS><RHS>Marked_app(app(lt,x),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(lt,w),y)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),x)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/><NODE ref="1"/><NODE ref="2"/><NODE ref="3"/><NODE ref="4"/><NODE ref="5"/></SCC></CCLIST></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(lt,app(s,x)),app(s,y))</LHS><RHS>app(app(lt,x),y)</RHS></RULE><RULE><LHS>app(app(lt,0),app(s,y))</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(lt,y),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,x),x)</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(eq,app(s,x)),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,0),app(s,x))</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),null)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(lt,app(s,x)),app(s,y))</LHS><RHS>Marked_app(app(lt,x),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(lt,w),y)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),x)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>app</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>null</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>0</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>s</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>fork</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>eq</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>lt</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>if</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>false</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>true</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_app</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>member</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(lt,w),y)</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(lt,app(s,x)),app(s,y))</LHS><RHS>app(app(lt,x),y)</RHS></RULE><RULE><LHS>app(app(lt,0),app(s,y))</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(lt,y),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,x),x)</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(eq,app(s,x)),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,0),app(s,x))</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),null)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(lt,app(s,x)),app(s,y))</LHS><RHS>Marked_app(app(lt,x),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),x)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/><NODE ref="1"/><NODE ref="2"/><NODE ref="3"/><NODE ref="4"/></SCC></CCLIST></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(lt,app(s,x)),app(s,y))</LHS><RHS>app(app(lt,x),y)</RHS></RULE><RULE><LHS>app(app(lt,0),app(s,y))</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(lt,y),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,x),x)</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(eq,app(s,x)),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,0),app(s,x))</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),null)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(lt,app(s,x)),app(s,y))</LHS><RHS>Marked_app(app(lt,x),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),x)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>app</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>null</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>0</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>s</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>fork</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>eq</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>lt</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>if</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>false</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>true</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_app</NAME></SYMBOL><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></POLYSYMB><POLYSYMB><SYMBOL><NAME>member</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z))</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(lt,app(s,x)),app(s,y))</LHS><RHS>app(app(lt,x),y)</RHS></RULE><RULE><LHS>app(app(lt,0),app(s,y))</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(lt,y),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,x),x)</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(eq,app(s,x)),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,0),app(s,x))</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),null)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(lt,app(s,x)),app(s,y))</LHS><RHS>Marked_app(app(lt,x),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),x)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/><NODE ref="1"/><NODE ref="2"/></SCC></CCLIST></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(lt,app(s,x)),app(s,y))</LHS><RHS>app(app(lt,x),y)</RHS></RULE><RULE><LHS>app(app(lt,0),app(s,y))</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(lt,y),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,x),x)</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(eq,app(s,x)),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,0),app(s,x))</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),null)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(lt,app(s,x)),app(s,y))</LHS><RHS>Marked_app(app(lt,x),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),x)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_app</NAME></SYMBOL><INT>2</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),z)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>Marked_app(app(member,w),x)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(app(lt,app(s,x)),app(s,y))</LHS><RHS>Marked_app(app(lt,x),y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(lt,app(s,x)),app(s,y))</LHS><RHS>app(app(lt,x),y)</RHS></RULE><RULE><LHS>app(app(lt,0),app(s,y))</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(lt,y),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,x),x)</LHS><RHS>true</RHS></RULE><RULE><LHS>app(app(eq,app(s,x)),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(eq,0),app(s,x))</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),null)</LHS><RHS>false</RHS></RULE><RULE><LHS>app(app(member,w),app(app(app(fork,x),y),z))</LHS><RHS>app(app(app(if,app(app(lt,w),y)),app(app(member,w),x)),
 app(app(app(if,app(app(eq,w),y)),true),app(app(member,w),z)))</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
