<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="2" unmarked="append"><NAME>Marked_append</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>add</NAME></SYMBOL><SYMBOL arity="6" unmarked=""><NAME>f_2</NAME></SYMBOL><SYMBOL arity="4" unmarked="f_1"><NAME>Marked_f_1</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="3" unmarked=""><NAME>f_3</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>append</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>split</NAME></SYMBOL><SYMBOL arity="2" unmarked="split"><NAME>Marked_split</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>lt</NAME></SYMBOL><SYMBOL arity="3" unmarked="f_3"><NAME>Marked_f_3</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>nil</NAME></SYMBOL><SYMBOL arity="4" unmarked=""><NAME>f_1</NAME></SYMBOL><SYMBOL arity="6" unmarked="f_2"><NAME>Marked_f_2</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>qsort</NAME></SYMBOL><SYMBOL arity="2" unmarked="lt"><NAME>Marked_lt</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>false</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>pair</NAME></SYMBOL><SYMBOL arity="1" unmarked="qsort"><NAME>Marked_qsort</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>true</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>N</VAR><VAR>M</VAR><VAR>Z</VAR><VAR>Y</VAR><VAR>X</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>lt(0,s(X))</LHS><RHS>true</RHS></RULE><RULE><LHS>lt(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>lt(s(X),s(Y))</LHS><RHS>lt(X,Y)</RHS></RULE><RULE><LHS>append(nil,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>append(add(N,X),Y)</LHS><RHS>add(N,append(X,Y))</RHS></RULE><RULE><LHS>split(N,nil)</LHS><RHS>pair(nil,nil)</RHS></RULE><RULE><LHS>split(N,add(M,Y))</LHS><RHS>f_1(split(N,Y),N,M,Y)</RHS></RULE><RULE><LHS>f_1(pair(X,Z),N,M,Y)</LHS><RHS>f_2(lt(N,M),N,M,Y,X,Z)</RHS></RULE><RULE><LHS>f_2(true,N,M,Y,X,Z)</LHS><RHS>pair(X,add(M,Z))</RHS></RULE><RULE><LHS>f_2(false,N,M,Y,X,Z)</LHS><RHS>pair(add(M,X),Z)</RHS></RULE><RULE><LHS>qsort(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>qsort(add(N,X))</LHS><RHS>f_3(split(N,X),N,X)</RHS></RULE><RULE><LHS>f_3(pair(Y,Z),N,X)</LHS><RHS>append(qsort(Y),add(X,qsort(Z)))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>lt(0,s(X))</LHS><RHS>true</RHS></RULE><RULE><LHS>lt(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>lt(s(X),s(Y))</LHS><RHS>lt(X,Y)</RHS></RULE><RULE><LHS>append(nil,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>append(add(N,X),Y)</LHS><RHS>add(N,append(X,Y))</RHS></RULE><RULE><LHS>split(N,nil)</LHS><RHS>pair(nil,nil)</RHS></RULE><RULE><LHS>split(N,add(M,Y))</LHS><RHS>f_1(split(N,Y),N,M,Y)</RHS></RULE><RULE><LHS>f_1(pair(X,Z),N,M,Y)</LHS><RHS>f_2(lt(N,M),N,M,Y,X,Z)</RHS></RULE><RULE><LHS>f_2(true,N,M,Y,X,Z)</LHS><RHS>pair(X,add(M,Z))</RHS></RULE><RULE><LHS>f_2(false,N,M,Y,X,Z)</LHS><RHS>pair(add(M,X),Z)</RHS></RULE><RULE><LHS>qsort(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>qsort(add(N,X))</LHS><RHS>f_3(split(N,X),N,X)</RHS></RULE><RULE><LHS>f_3(pair(Y,Z),N,X)</LHS><RHS>append(qsort(Y),add(X,qsort(Z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_lt(s(X),s(Y))</LHS><RHS>Marked_lt(X,Y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_append(add(N,X),Y)</LHS><RHS>Marked_append(X,Y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_split(N,add(M,Y))</LHS><RHS>Marked_f_1(split(N,Y),N,M,Y)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_split(N,add(M,Y))</LHS><RHS>Marked_split(N,Y)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_f_1(pair(X,Z),N,M,Y)</LHS><RHS>Marked_f_2(lt(N,M),N,M,Y,X,Z)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_f_1(pair(X,Z),N,M,Y)</LHS><RHS>Marked_lt(N,M)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_qsort(add(N,X))</LHS><RHS>Marked_f_3(split(N,X),N,X)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_qsort(add(N,X))</LHS><RHS>Marked_split(N,X)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_f_3(pair(Y,Z),N,X)</LHS><RHS>Marked_append(qsort(Y),add(X,qsort(Z)))</RHS></DPRULE><DPRULE num="9"><LHS>Marked_f_3(pair(Y,Z),N,X)</LHS><RHS>Marked_qsort(Y)</RHS></DPRULE><DPRULE num="10"><LHS>Marked_f_3(pair(Y,Z),N,X)</LHS><RHS>Marked_qsort(Z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="6"/><NODE ref="9"/><NODE ref="10"/></SCC><NONSCC num="1"><NODE ref="7"/></NONSCC><NONSCC num="2"><NODE ref="8"/></NONSCC><SCC num="3"><NODE ref="1"/></SCC><SCC num="4"><NODE ref="3"/></SCC><NONSCC num="5"><NODE ref="2"/></NONSCC><NONSCC num="6"><NODE ref="5"/></NONSCC><SCC num="7"><NODE ref="0"/></SCC><NONSCC num="8"><NODE ref="4"/></NONSCC></CCLIST><EDGE end="2" start="0"/><EDGE end="1" start="0"/><EDGE end="5" start="1"/><EDGE end="4" start="1"/><EDGE end="3" start="2"/><EDGE end="5" start="4"/><EDGE end="8" start="5"/><EDGE end="6" start="5"/><EDGE end="7" start="6"/></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>lt(0,s(X))</LHS><RHS>true</RHS></RULE><RULE><LHS>lt(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>lt(s(X),s(Y))</LHS><RHS>lt(X,Y)</RHS></RULE><RULE><LHS>append(nil,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>append(add(N,X),Y)</LHS><RHS>add(N,append(X,Y))</RHS></RULE><RULE><LHS>split(N,nil)</LHS><RHS>pair(nil,nil)</RHS></RULE><RULE><LHS>split(N,add(M,Y))</LHS><RHS>f_1(split(N,Y),N,M,Y)</RHS></RULE><RULE><LHS>f_1(pair(X,Z),N,M,Y)</LHS><RHS>f_2(lt(N,M),N,M,Y,X,Z)</RHS></RULE><RULE><LHS>f_2(true,N,M,Y,X,Z)</LHS><RHS>pair(X,add(M,Z))</RHS></RULE><RULE><LHS>f_2(false,N,M,Y,X,Z)</LHS><RHS>pair(add(M,X),Z)</RHS></RULE><RULE><LHS>qsort(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>qsort(add(N,X))</LHS><RHS>f_3(split(N,X),N,X)</RHS></RULE><RULE><LHS>f_3(pair(Y,Z),N,X)</LHS><RHS>append(qsort(Y),add(X,qsort(Z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_qsort(add(N,X))</LHS><RHS>Marked_f_3(split(N,X),N,X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_f_3(pair(Y,Z),N,X)</LHS><RHS>Marked_qsort(Y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_f_3(pair(Y,Z),N,X)</LHS><RHS>Marked_qsort(Z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>true</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>Marked_qsort</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>false</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>qsort</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>0</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>f_1</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><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>Marked_f_3</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>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>split</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>f_3</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>s</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>f_2</NAME></SYMBOL><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><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="5"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>add</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="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_qsort(add(N,X))</LHS><RHS>Marked_f_3(split(N,X),N,X)</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>lt(0,s(X))</LHS><RHS>true</RHS></RULE><RULE><LHS>lt(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>lt(s(X),s(Y))</LHS><RHS>lt(X,Y)</RHS></RULE><RULE><LHS>append(nil,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>append(add(N,X),Y)</LHS><RHS>add(N,append(X,Y))</RHS></RULE><RULE><LHS>split(N,nil)</LHS><RHS>pair(nil,nil)</RHS></RULE><RULE><LHS>split(N,add(M,Y))</LHS><RHS>f_1(split(N,Y),N,M,Y)</RHS></RULE><RULE><LHS>f_1(pair(X,Z),N,M,Y)</LHS><RHS>f_2(lt(N,M),N,M,Y,X,Z)</RHS></RULE><RULE><LHS>f_2(true,N,M,Y,X,Z)</LHS><RHS>pair(X,add(M,Z))</RHS></RULE><RULE><LHS>f_2(false,N,M,Y,X,Z)</LHS><RHS>pair(add(M,X),Z)</RHS></RULE><RULE><LHS>qsort(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>qsort(add(N,X))</LHS><RHS>f_3(split(N,X),N,X)</RHS></RULE><RULE><LHS>f_3(pair(Y,Z),N,X)</LHS><RHS>append(qsort(Y),add(X,qsort(Z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_f_3(pair(Y,Z),N,X)</LHS><RHS>Marked_qsort(Y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_f_3(pair(Y,Z),N,X)</LHS><RHS>Marked_qsort(Z)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="1"/></NONSCC><NONSCC num="1"><NODE ref="0"/></NONSCC></CCLIST></DAG></CRITERION></PROPERTY></PROPERTY><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>lt(0,s(X))</LHS><RHS>true</RHS></RULE><RULE><LHS>lt(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>lt(s(X),s(Y))</LHS><RHS>lt(X,Y)</RHS></RULE><RULE><LHS>append(nil,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>append(add(N,X),Y)</LHS><RHS>add(N,append(X,Y))</RHS></RULE><RULE><LHS>split(N,nil)</LHS><RHS>pair(nil,nil)</RHS></RULE><RULE><LHS>split(N,add(M,Y))</LHS><RHS>f_1(split(N,Y),N,M,Y)</RHS></RULE><RULE><LHS>f_1(pair(X,Z),N,M,Y)</LHS><RHS>f_2(lt(N,M),N,M,Y,X,Z)</RHS></RULE><RULE><LHS>f_2(true,N,M,Y,X,Z)</LHS><RHS>pair(X,add(M,Z))</RHS></RULE><RULE><LHS>f_2(false,N,M,Y,X,Z)</LHS><RHS>pair(add(M,X),Z)</RHS></RULE><RULE><LHS>qsort(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>qsort(add(N,X))</LHS><RHS>f_3(split(N,X),N,X)</RHS></RULE><RULE><LHS>f_3(pair(Y,Z),N,X)</LHS><RHS>append(qsort(Y),add(X,qsort(Z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_append(add(N,X),Y)</LHS><RHS>Marked_append(X,Y)</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>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>false</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>qsort</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>0</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>f_1</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><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>lt</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>split</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>f_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><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>s</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>f_2</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="4"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="5"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>add</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="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_append</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></ORDERING></CRITERION></PROPERTY><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>lt(0,s(X))</LHS><RHS>true</RHS></RULE><RULE><LHS>lt(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>lt(s(X),s(Y))</LHS><RHS>lt(X,Y)</RHS></RULE><RULE><LHS>append(nil,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>append(add(N,X),Y)</LHS><RHS>add(N,append(X,Y))</RHS></RULE><RULE><LHS>split(N,nil)</LHS><RHS>pair(nil,nil)</RHS></RULE><RULE><LHS>split(N,add(M,Y))</LHS><RHS>f_1(split(N,Y),N,M,Y)</RHS></RULE><RULE><LHS>f_1(pair(X,Z),N,M,Y)</LHS><RHS>f_2(lt(N,M),N,M,Y,X,Z)</RHS></RULE><RULE><LHS>f_2(true,N,M,Y,X,Z)</LHS><RHS>pair(X,add(M,Z))</RHS></RULE><RULE><LHS>f_2(false,N,M,Y,X,Z)</LHS><RHS>pair(add(M,X),Z)</RHS></RULE><RULE><LHS>qsort(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>qsort(add(N,X))</LHS><RHS>f_3(split(N,X),N,X)</RHS></RULE><RULE><LHS>f_3(pair(Y,Z),N,X)</LHS><RHS>append(qsort(Y),add(X,qsort(Z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_split(N,add(M,Y))</LHS><RHS>Marked_split(N,Y)</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>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>false</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>qsort</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>0</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>f_1</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><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>lt</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_split</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>split</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="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></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>f_3</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>s</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>f_2</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="4"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="5"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>add</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="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB></ORDERING></CRITERION></PROPERTY><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>lt(0,s(X))</LHS><RHS>true</RHS></RULE><RULE><LHS>lt(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>lt(s(X),s(Y))</LHS><RHS>lt(X,Y)</RHS></RULE><RULE><LHS>append(nil,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>append(add(N,X),Y)</LHS><RHS>add(N,append(X,Y))</RHS></RULE><RULE><LHS>split(N,nil)</LHS><RHS>pair(nil,nil)</RHS></RULE><RULE><LHS>split(N,add(M,Y))</LHS><RHS>f_1(split(N,Y),N,M,Y)</RHS></RULE><RULE><LHS>f_1(pair(X,Z),N,M,Y)</LHS><RHS>f_2(lt(N,M),N,M,Y,X,Z)</RHS></RULE><RULE><LHS>f_2(true,N,M,Y,X,Z)</LHS><RHS>pair(X,add(M,Z))</RHS></RULE><RULE><LHS>f_2(false,N,M,Y,X,Z)</LHS><RHS>pair(add(M,X),Z)</RHS></RULE><RULE><LHS>qsort(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>qsort(add(N,X))</LHS><RHS>f_3(split(N,X),N,X)</RHS></RULE><RULE><LHS>f_3(pair(Y,Z),N,X)</LHS><RHS>append(qsort(Y),add(X,qsort(Z)))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_lt(s(X),s(Y))</LHS><RHS>Marked_lt(X,Y)</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>pair</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>false</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_lt</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>qsort</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>0</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>f_1</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>nil</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>split</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>append</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>f_3</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>s</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</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>f_2</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>add</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB></ORDERING></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROOF>
