<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="4" unmarked="match_5"><NAME>Marked_match_5</NAME></SYMBOL><SYMBOL arity="3" unmarked="match_1"><NAME>Marked_match_1</NAME></SYMBOL><SYMBOL arity="3" unmarked=""><NAME>match_1</NAME></SYMBOL><SYMBOL arity="7" unmarked=""><NAME>match_3</NAME></SYMBOL><SYMBOL arity="2" unmarked="part"><NAME>Marked_part</NAME></SYMBOL><SYMBOL arity="3" unmarked=""><NAME>match_0</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>quick</NAME></SYMBOL><SYMBOL arity="5" unmarked="match_2"><NAME>Marked_match_2</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>Nil</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>Pair</NAME></SYMBOL><SYMBOL arity="1" unmarked="quick"><NAME>Marked_quick</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>test</NAME></SYMBOL><SYMBOL arity="4" unmarked=""><NAME>match_5</NAME></SYMBOL><SYMBOL arity="2" unmarked="test"><NAME>Marked_test</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>Cons</NAME></SYMBOL><SYMBOL arity="5" unmarked=""><NAME>match_2</NAME></SYMBOL><SYMBOL arity="2" unmarked="match_4"><NAME>Marked_match_4</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>False</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>match_4</NAME></SYMBOL><SYMBOL arity="7" unmarked="match_3"><NAME>Marked_match_3</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>append</NAME></SYMBOL><SYMBOL arity="3" unmarked="match_0"><NAME>Marked_match_0</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>part</NAME></SYMBOL><SYMBOL arity="2" unmarked="append"><NAME>Marked_append</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>True</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>x_0</VAR><VAR>y</VAR><VAR>l1_2</VAR><VAR>l2_1</VAR><VAR>l1</VAR><VAR>l2</VAR><VAR>l_3</VAR><VAR>a_4</VAR><VAR>l</VAR><VAR>l'</VAR><VAR>x</VAR><VAR>a</VAR><VAR>l_5</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>test(x_0,y)</LHS><RHS>True</RHS></RULE><RULE><LHS>test(x_0,y)</LHS><RHS>False</RHS></RULE><RULE><LHS>append(l1_2,l2_1)</LHS><RHS>match_0(l1_2,l2_1,l1_2)</RHS></RULE><RULE><LHS>match_0(l1_2,l2_1,Nil)</LHS><RHS>l2_1</RHS></RULE><RULE><LHS>match_0(l1_2,l2_1,Cons(x,l))</LHS><RHS>Cons(x,append(l,l2_1))</RHS></RULE><RULE><LHS>part(a_4,l_3)</LHS><RHS>match_1(a_4,l_3,l_3)</RHS></RULE><RULE><LHS>match_1(a_4,l_3,Nil)</LHS><RHS>Pair(Nil,Nil)</RHS></RULE><RULE><LHS>match_1(a_4,l_3,Cons(x,l'))</LHS><RHS>match_2(x,l',a_4,l_3,part(a_4,l'))</RHS></RULE><RULE><LHS>match_2(x,l',a_4,l_3,Pair(l1,l2))</LHS><RHS>match_3(l1,l2,x,l',a_4,l_3,test(a_4,x))</RHS></RULE><RULE><LHS>match_3(l1,l2,x,l',a_4,l_3,False)</LHS><RHS>Pair(Cons(x,l1),l2)</RHS></RULE><RULE><LHS>match_3(l1,l2,x,l',a_4,l_3,True)</LHS><RHS>Pair(l1,Cons(x,l2))</RHS></RULE><RULE><LHS>quick(l_5)</LHS><RHS>match_4(l_5,l_5)</RHS></RULE><RULE><LHS>match_4(l_5,Nil)</LHS><RHS>Nil</RHS></RULE><RULE><LHS>match_4(l_5,Cons(a,l'))</LHS><RHS>match_5(a,l',l_5,part(a,l'))</RHS></RULE><RULE><LHS>match_5(a,l',l_5,Pair(l1,l2))</LHS><RHS>append(quick(l1),Cons(a,quick(l2)))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>test(x_0,y)</LHS><RHS>True</RHS></RULE><RULE><LHS>test(x_0,y)</LHS><RHS>False</RHS></RULE><RULE><LHS>append(l1_2,l2_1)</LHS><RHS>match_0(l1_2,l2_1,l1_2)</RHS></RULE><RULE><LHS>match_0(l1_2,l2_1,Nil)</LHS><RHS>l2_1</RHS></RULE><RULE><LHS>match_0(l1_2,l2_1,Cons(x,l))</LHS><RHS>Cons(x,append(l,l2_1))</RHS></RULE><RULE><LHS>part(a_4,l_3)</LHS><RHS>match_1(a_4,l_3,l_3)</RHS></RULE><RULE><LHS>match_1(a_4,l_3,Nil)</LHS><RHS>Pair(Nil,Nil)</RHS></RULE><RULE><LHS>match_1(a_4,l_3,Cons(x,l'))</LHS><RHS>match_2(x,l',a_4,l_3,part(a_4,l'))</RHS></RULE><RULE><LHS>match_2(x,l',a_4,l_3,Pair(l1,l2))</LHS><RHS>match_3(l1,l2,x,l',a_4,l_3,test(a_4,x))</RHS></RULE><RULE><LHS>match_3(l1,l2,x,l',a_4,l_3,False)</LHS><RHS>Pair(Cons(x,l1),l2)</RHS></RULE><RULE><LHS>match_3(l1,l2,x,l',a_4,l_3,True)</LHS><RHS>Pair(l1,Cons(x,l2))</RHS></RULE><RULE><LHS>quick(l_5)</LHS><RHS>match_4(l_5,l_5)</RHS></RULE><RULE><LHS>match_4(l_5,Nil)</LHS><RHS>Nil</RHS></RULE><RULE><LHS>match_4(l_5,Cons(a,l'))</LHS><RHS>match_5(a,l',l_5,part(a,l'))</RHS></RULE><RULE><LHS>match_5(a,l',l_5,Pair(l1,l2))</LHS><RHS>append(quick(l1),Cons(a,quick(l2)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_append(l1_2,l2_1)</LHS><RHS>Marked_match_0(l1_2,l2_1,l1_2)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_match_0(l1_2,l2_1,Cons(x,l))</LHS><RHS>Marked_append(l,l2_1)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_part(a_4,l_3)</LHS><RHS>Marked_match_1(a_4,l_3,l_3)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_match_1(a_4,l_3,Cons(x,l'))</LHS><RHS>Marked_match_2(x,l',a_4,l_3,part(a_4,l'))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_match_1(a_4,l_3,Cons(x,l'))</LHS><RHS>Marked_part(a_4,l')</RHS></DPRULE><DPRULE num="5"><LHS>Marked_match_2(x,l',a_4,l_3,Pair(l1,l2))</LHS><RHS>Marked_match_3(l1,l2,x,l',a_4,l_3,test(a_4,x))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_match_2(x,l',a_4,l_3,Pair(l1,l2))</LHS><RHS>Marked_test(a_4,x)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_quick(l_5)</LHS><RHS>Marked_match_4(l_5,l_5)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_match_4(l_5,Cons(a,l'))</LHS><RHS>Marked_match_5(a,l',l_5,part(a,l'))</RHS></DPRULE><DPRULE num="9"><LHS>Marked_match_4(l_5,Cons(a,l'))</LHS><RHS>Marked_part(a,l')</RHS></DPRULE><DPRULE num="10"><LHS>Marked_match_5(a,l',l_5,Pair(l1,l2))</LHS><RHS>Marked_append(quick(l1),Cons(a,quick(l2)))</RHS></DPRULE><DPRULE num="11"><LHS>Marked_match_5(a,l',l_5,Pair(l1,l2))</LHS><RHS>Marked_quick(l1)</RHS></DPRULE><DPRULE num="12"><LHS>Marked_match_5(a,l',l_5,Pair(l1,l2))</LHS><RHS>Marked_quick(l2)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="7"/><NODE ref="8"/><NODE ref="11"/><NODE ref="12"/></SCC><NONSCC num="1"><NODE ref="10"/></NONSCC><SCC num="2"><NODE ref="0"/><NODE ref="1"/></SCC><NONSCC num="3"><NODE ref="9"/></NONSCC><SCC num="4"><NODE ref="2"/><NODE ref="4"/></SCC><NONSCC num="5"><NODE ref="6"/></NONSCC><NONSCC num="6"><NODE ref="5"/></NONSCC><NONSCC num="7"><NODE ref="3"/></NONSCC></CCLIST><EDGE end="3" start="0"/><EDGE end="1" start="0"/><EDGE end="2" start="1"/><EDGE end="4" start="3"/><EDGE end="7" start="4"/><EDGE end="6" start="7"/><EDGE end="5" start="7"/></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>test(x_0,y)</LHS><RHS>True</RHS></RULE><RULE><LHS>test(x_0,y)</LHS><RHS>False</RHS></RULE><RULE><LHS>append(l1_2,l2_1)</LHS><RHS>match_0(l1_2,l2_1,l1_2)</RHS></RULE><RULE><LHS>match_0(l1_2,l2_1,Nil)</LHS><RHS>l2_1</RHS></RULE><RULE><LHS>match_0(l1_2,l2_1,Cons(x,l))</LHS><RHS>Cons(x,append(l,l2_1))</RHS></RULE><RULE><LHS>part(a_4,l_3)</LHS><RHS>match_1(a_4,l_3,l_3)</RHS></RULE><RULE><LHS>match_1(a_4,l_3,Nil)</LHS><RHS>Pair(Nil,Nil)</RHS></RULE><RULE><LHS>match_1(a_4,l_3,Cons(x,l'))</LHS><RHS>match_2(x,l',a_4,l_3,part(a_4,l'))</RHS></RULE><RULE><LHS>match_2(x,l',a_4,l_3,Pair(l1,l2))</LHS><RHS>match_3(l1,l2,x,l',a_4,l_3,test(a_4,x))</RHS></RULE><RULE><LHS>match_3(l1,l2,x,l',a_4,l_3,False)</LHS><RHS>Pair(Cons(x,l1),l2)</RHS></RULE><RULE><LHS>match_3(l1,l2,x,l',a_4,l_3,True)</LHS><RHS>Pair(l1,Cons(x,l2))</RHS></RULE><RULE><LHS>quick(l_5)</LHS><RHS>match_4(l_5,l_5)</RHS></RULE><RULE><LHS>match_4(l_5,Nil)</LHS><RHS>Nil</RHS></RULE><RULE><LHS>match_4(l_5,Cons(a,l'))</LHS><RHS>match_5(a,l',l_5,part(a,l'))</RHS></RULE><RULE><LHS>match_5(a,l',l_5,Pair(l1,l2))</LHS><RHS>append(quick(l1),Cons(a,quick(l2)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_quick(l_5)</LHS><RHS>Marked_match_4(l_5,l_5)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_match_4(l_5,Cons(a,l'))</LHS><RHS>Marked_match_5(a,l',l_5,part(a,l'))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_match_5(a,l',l_5,Pair(l1,l2))</LHS><RHS>Marked_quick(l1)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_match_5(a,l',l_5,Pair(l1,l2))</LHS><RHS>Marked_quick(l2)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>True</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>part</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>append</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>match_4</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="1"/></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>Marked_match_4</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>match_2</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><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="4"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Cons</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><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>match_5</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><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="3"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>test</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_quick</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>Pair</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Nil</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>quick</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>match_0</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="2"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>match_3</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="2"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>match_1</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="2"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_match_5</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="3"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_match_4(l_5,Cons(a,l'))</LHS><RHS>Marked_match_5(a,l',l_5,part(a,l'))</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>test(x_0,y)</LHS><RHS>True</RHS></RULE><RULE><LHS>test(x_0,y)</LHS><RHS>False</RHS></RULE><RULE><LHS>append(l1_2,l2_1)</LHS><RHS>match_0(l1_2,l2_1,l1_2)</RHS></RULE><RULE><LHS>match_0(l1_2,l2_1,Nil)</LHS><RHS>l2_1</RHS></RULE><RULE><LHS>match_0(l1_2,l2_1,Cons(x,l))</LHS><RHS>Cons(x,append(l,l2_1))</RHS></RULE><RULE><LHS>part(a_4,l_3)</LHS><RHS>match_1(a_4,l_3,l_3)</RHS></RULE><RULE><LHS>match_1(a_4,l_3,Nil)</LHS><RHS>Pair(Nil,Nil)</RHS></RULE><RULE><LHS>match_1(a_4,l_3,Cons(x,l'))</LHS><RHS>match_2(x,l',a_4,l_3,part(a_4,l'))</RHS></RULE><RULE><LHS>match_2(x,l',a_4,l_3,Pair(l1,l2))</LHS><RHS>match_3(l1,l2,x,l',a_4,l_3,test(a_4,x))</RHS></RULE><RULE><LHS>match_3(l1,l2,x,l',a_4,l_3,False)</LHS><RHS>Pair(Cons(x,l1),l2)</RHS></RULE><RULE><LHS>match_3(l1,l2,x,l',a_4,l_3,True)</LHS><RHS>Pair(l1,Cons(x,l2))</RHS></RULE><RULE><LHS>quick(l_5)</LHS><RHS>match_4(l_5,l_5)</RHS></RULE><RULE><LHS>match_4(l_5,Nil)</LHS><RHS>Nil</RHS></RULE><RULE><LHS>match_4(l_5,Cons(a,l'))</LHS><RHS>match_5(a,l',l_5,part(a,l'))</RHS></RULE><RULE><LHS>match_5(a,l',l_5,Pair(l1,l2))</LHS><RHS>append(quick(l1),Cons(a,quick(l2)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_quick(l_5)</LHS><RHS>Marked_match_4(l_5,l_5)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_match_5(a,l',l_5,Pair(l1,l2))</LHS><RHS>Marked_quick(l1)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_match_5(a,l',l_5,Pair(l1,l2))</LHS><RHS>Marked_quick(l2)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="2"/></NONSCC><NONSCC num="1"><NODE ref="0"/></NONSCC><NONSCC num="2"><NODE ref="1"/></NONSCC></CCLIST><EDGE end="1" start="0"/><EDGE end="1" start="2"/></DAG></CRITERION></PROPERTY></PROPERTY><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>test(x_0,y)</LHS><RHS>True</RHS></RULE><RULE><LHS>test(x_0,y)</LHS><RHS>False</RHS></RULE><RULE><LHS>append(l1_2,l2_1)</LHS><RHS>match_0(l1_2,l2_1,l1_2)</RHS></RULE><RULE><LHS>match_0(l1_2,l2_1,Nil)</LHS><RHS>l2_1</RHS></RULE><RULE><LHS>match_0(l1_2,l2_1,Cons(x,l))</LHS><RHS>Cons(x,append(l,l2_1))</RHS></RULE><RULE><LHS>part(a_4,l_3)</LHS><RHS>match_1(a_4,l_3,l_3)</RHS></RULE><RULE><LHS>match_1(a_4,l_3,Nil)</LHS><RHS>Pair(Nil,Nil)</RHS></RULE><RULE><LHS>match_1(a_4,l_3,Cons(x,l'))</LHS><RHS>match_2(x,l',a_4,l_3,part(a_4,l'))</RHS></RULE><RULE><LHS>match_2(x,l',a_4,l_3,Pair(l1,l2))</LHS><RHS>match_3(l1,l2,x,l',a_4,l_3,test(a_4,x))</RHS></RULE><RULE><LHS>match_3(l1,l2,x,l',a_4,l_3,False)</LHS><RHS>Pair(Cons(x,l1),l2)</RHS></RULE><RULE><LHS>match_3(l1,l2,x,l',a_4,l_3,True)</LHS><RHS>Pair(l1,Cons(x,l2))</RHS></RULE><RULE><LHS>quick(l_5)</LHS><RHS>match_4(l_5,l_5)</RHS></RULE><RULE><LHS>match_4(l_5,Nil)</LHS><RHS>Nil</RHS></RULE><RULE><LHS>match_4(l_5,Cons(a,l'))</LHS><RHS>match_5(a,l',l_5,part(a,l'))</RHS></RULE><RULE><LHS>match_5(a,l',l_5,Pair(l1,l2))</LHS><RHS>append(quick(l1),Cons(a,quick(l2)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_append(l1_2,l2_1)</LHS><RHS>Marked_match_0(l1_2,l2_1,l1_2)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_match_0(l1_2,l2_1,Cons(x,l))</LHS><RHS>Marked_append(l,l2_1)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_match_0</NAME></SYMBOL><INT>3</INT></PROJ><PROJ><SYMBOL><NAME>Marked_append</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_match_0(l1_2,l2_1,Cons(x,l))</LHS><RHS>Marked_append(l,l2_1)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>test(x_0,y)</LHS><RHS>True</RHS></RULE><RULE><LHS>test(x_0,y)</LHS><RHS>False</RHS></RULE><RULE><LHS>append(l1_2,l2_1)</LHS><RHS>match_0(l1_2,l2_1,l1_2)</RHS></RULE><RULE><LHS>match_0(l1_2,l2_1,Nil)</LHS><RHS>l2_1</RHS></RULE><RULE><LHS>match_0(l1_2,l2_1,Cons(x,l))</LHS><RHS>Cons(x,append(l,l2_1))</RHS></RULE><RULE><LHS>part(a_4,l_3)</LHS><RHS>match_1(a_4,l_3,l_3)</RHS></RULE><RULE><LHS>match_1(a_4,l_3,Nil)</LHS><RHS>Pair(Nil,Nil)</RHS></RULE><RULE><LHS>match_1(a_4,l_3,Cons(x,l'))</LHS><RHS>match_2(x,l',a_4,l_3,part(a_4,l'))</RHS></RULE><RULE><LHS>match_2(x,l',a_4,l_3,Pair(l1,l2))</LHS><RHS>match_3(l1,l2,x,l',a_4,l_3,test(a_4,x))</RHS></RULE><RULE><LHS>match_3(l1,l2,x,l',a_4,l_3,False)</LHS><RHS>Pair(Cons(x,l1),l2)</RHS></RULE><RULE><LHS>match_3(l1,l2,x,l',a_4,l_3,True)</LHS><RHS>Pair(l1,Cons(x,l2))</RHS></RULE><RULE><LHS>quick(l_5)</LHS><RHS>match_4(l_5,l_5)</RHS></RULE><RULE><LHS>match_4(l_5,Nil)</LHS><RHS>Nil</RHS></RULE><RULE><LHS>match_4(l_5,Cons(a,l'))</LHS><RHS>match_5(a,l',l_5,part(a,l'))</RHS></RULE><RULE><LHS>match_5(a,l',l_5,Pair(l1,l2))</LHS><RHS>append(quick(l1),Cons(a,quick(l2)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_append(l1_2,l2_1)</LHS><RHS>Marked_match_0(l1_2,l2_1,l1_2)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="0"/></NONSCC></CCLIST></DAG></CRITERION></PROPERTY></PROPERTY><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>test(x_0,y)</LHS><RHS>True</RHS></RULE><RULE><LHS>test(x_0,y)</LHS><RHS>False</RHS></RULE><RULE><LHS>append(l1_2,l2_1)</LHS><RHS>match_0(l1_2,l2_1,l1_2)</RHS></RULE><RULE><LHS>match_0(l1_2,l2_1,Nil)</LHS><RHS>l2_1</RHS></RULE><RULE><LHS>match_0(l1_2,l2_1,Cons(x,l))</LHS><RHS>Cons(x,append(l,l2_1))</RHS></RULE><RULE><LHS>part(a_4,l_3)</LHS><RHS>match_1(a_4,l_3,l_3)</RHS></RULE><RULE><LHS>match_1(a_4,l_3,Nil)</LHS><RHS>Pair(Nil,Nil)</RHS></RULE><RULE><LHS>match_1(a_4,l_3,Cons(x,l'))</LHS><RHS>match_2(x,l',a_4,l_3,part(a_4,l'))</RHS></RULE><RULE><LHS>match_2(x,l',a_4,l_3,Pair(l1,l2))</LHS><RHS>match_3(l1,l2,x,l',a_4,l_3,test(a_4,x))</RHS></RULE><RULE><LHS>match_3(l1,l2,x,l',a_4,l_3,False)</LHS><RHS>Pair(Cons(x,l1),l2)</RHS></RULE><RULE><LHS>match_3(l1,l2,x,l',a_4,l_3,True)</LHS><RHS>Pair(l1,Cons(x,l2))</RHS></RULE><RULE><LHS>quick(l_5)</LHS><RHS>match_4(l_5,l_5)</RHS></RULE><RULE><LHS>match_4(l_5,Nil)</LHS><RHS>Nil</RHS></RULE><RULE><LHS>match_4(l_5,Cons(a,l'))</LHS><RHS>match_5(a,l',l_5,part(a,l'))</RHS></RULE><RULE><LHS>match_5(a,l',l_5,Pair(l1,l2))</LHS><RHS>append(quick(l1),Cons(a,quick(l2)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_part(a_4,l_3)</LHS><RHS>Marked_match_1(a_4,l_3,l_3)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_match_1(a_4,l_3,Cons(x,l'))</LHS><RHS>Marked_part(a_4,l')</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_match_1</NAME></SYMBOL><INT>3</INT></PROJ><PROJ><SYMBOL><NAME>Marked_part</NAME></SYMBOL><INT>2</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_match_1(a_4,l_3,Cons(x,l'))</LHS><RHS>Marked_part(a_4,l')</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>test(x_0,y)</LHS><RHS>True</RHS></RULE><RULE><LHS>test(x_0,y)</LHS><RHS>False</RHS></RULE><RULE><LHS>append(l1_2,l2_1)</LHS><RHS>match_0(l1_2,l2_1,l1_2)</RHS></RULE><RULE><LHS>match_0(l1_2,l2_1,Nil)</LHS><RHS>l2_1</RHS></RULE><RULE><LHS>match_0(l1_2,l2_1,Cons(x,l))</LHS><RHS>Cons(x,append(l,l2_1))</RHS></RULE><RULE><LHS>part(a_4,l_3)</LHS><RHS>match_1(a_4,l_3,l_3)</RHS></RULE><RULE><LHS>match_1(a_4,l_3,Nil)</LHS><RHS>Pair(Nil,Nil)</RHS></RULE><RULE><LHS>match_1(a_4,l_3,Cons(x,l'))</LHS><RHS>match_2(x,l',a_4,l_3,part(a_4,l'))</RHS></RULE><RULE><LHS>match_2(x,l',a_4,l_3,Pair(l1,l2))</LHS><RHS>match_3(l1,l2,x,l',a_4,l_3,test(a_4,x))</RHS></RULE><RULE><LHS>match_3(l1,l2,x,l',a_4,l_3,False)</LHS><RHS>Pair(Cons(x,l1),l2)</RHS></RULE><RULE><LHS>match_3(l1,l2,x,l',a_4,l_3,True)</LHS><RHS>Pair(l1,Cons(x,l2))</RHS></RULE><RULE><LHS>quick(l_5)</LHS><RHS>match_4(l_5,l_5)</RHS></RULE><RULE><LHS>match_4(l_5,Nil)</LHS><RHS>Nil</RHS></RULE><RULE><LHS>match_4(l_5,Cons(a,l'))</LHS><RHS>match_5(a,l',l_5,part(a,l'))</RHS></RULE><RULE><LHS>match_5(a,l',l_5,Pair(l1,l2))</LHS><RHS>append(quick(l1),Cons(a,quick(l2)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_part(a_4,l_3)</LHS><RHS>Marked_match_1(a_4,l_3,l_3)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="0"/></NONSCC></CCLIST></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
