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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="2" unmarked="low"><NAME>Marked_low</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>cons</NAME></SYMBOL><SYMBOL arity="3" unmarked=""><NAME>ifhigh</NAME></SYMBOL><SYMBOL arity="3" unmarked="iflow"><NAME>Marked_iflow</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>false</NAME></SYMBOL><SYMBOL arity="1" unmarked="quicksort"><NAME>Marked_quicksort</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>app</NAME></SYMBOL><SYMBOL arity="3" unmarked=""><NAME>iflow</NAME></SYMBOL><SYMBOL arity="3" unmarked="ifhigh"><NAME>Marked_ifhigh</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>le</NAME></SYMBOL><SYMBOL arity="2" unmarked="app"><NAME>Marked_app</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>nil</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>high</NAME></SYMBOL><SYMBOL arity="2" unmarked="le"><NAME>Marked_le</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>quicksort</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>low</NAME></SYMBOL><SYMBOL arity="2" unmarked="high"><NAME>Marked_high</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>true</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>L</VAR><VAR>Y</VAR><VAR>X</VAR><VAR>M</VAR><VAR>N</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>le(0,Y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(X),s(Y))</LHS><RHS>le(X,Y)</RHS></RULE><RULE><LHS>app(nil,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>app(cons(N,L),Y)</LHS><RHS>cons(N,app(L,Y))</RHS></RULE><RULE><LHS>low(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>low(N,cons(M,L))</LHS><RHS>iflow(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>iflow(true,N,cons(M,L))</LHS><RHS>cons(M,low(N,L))</RHS></RULE><RULE><LHS>iflow(false,N,cons(M,L))</LHS><RHS>low(N,L)</RHS></RULE><RULE><LHS>high(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>high(N,cons(M,L))</LHS><RHS>ifhigh(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>ifhigh(true,N,cons(M,L))</LHS><RHS>high(N,L)</RHS></RULE><RULE><LHS>ifhigh(false,N,cons(M,L))</LHS><RHS>cons(M,high(N,L))</RHS></RULE><RULE><LHS>quicksort(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>quicksort(cons(N,L))</LHS><RHS>app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>le(0,Y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(X),s(Y))</LHS><RHS>le(X,Y)</RHS></RULE><RULE><LHS>app(nil,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>app(cons(N,L),Y)</LHS><RHS>cons(N,app(L,Y))</RHS></RULE><RULE><LHS>low(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>low(N,cons(M,L))</LHS><RHS>iflow(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>iflow(true,N,cons(M,L))</LHS><RHS>cons(M,low(N,L))</RHS></RULE><RULE><LHS>iflow(false,N,cons(M,L))</LHS><RHS>low(N,L)</RHS></RULE><RULE><LHS>high(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>high(N,cons(M,L))</LHS><RHS>ifhigh(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>ifhigh(true,N,cons(M,L))</LHS><RHS>high(N,L)</RHS></RULE><RULE><LHS>ifhigh(false,N,cons(M,L))</LHS><RHS>cons(M,high(N,L))</RHS></RULE><RULE><LHS>quicksort(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>quicksort(cons(N,L))</LHS><RHS>app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_le(s(X),s(Y))</LHS><RHS>Marked_le(X,Y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(cons(N,L),Y)</LHS><RHS>Marked_app(L,Y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_low(N,cons(M,L))</LHS><RHS>Marked_iflow(le(M,N),N,cons(M,L))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_low(N,cons(M,L))</LHS><RHS>Marked_le(M,N)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_iflow(true,N,cons(M,L))</LHS><RHS>Marked_low(N,L)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_iflow(false,N,cons(M,L))</LHS><RHS>Marked_low(N,L)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_high(N,cons(M,L))</LHS><RHS>Marked_ifhigh(le(M,N),N,cons(M,L))</RHS></DPRULE><DPRULE num="7"><LHS>Marked_high(N,cons(M,L))</LHS><RHS>Marked_le(M,N)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_ifhigh(true,N,cons(M,L))</LHS><RHS>Marked_high(N,L)</RHS></DPRULE><DPRULE num="9"><LHS>Marked_ifhigh(false,N,cons(M,L))</LHS><RHS>Marked_high(N,L)</RHS></DPRULE><DPRULE num="10"><LHS>Marked_quicksort(cons(N,L))</LHS><RHS>Marked_app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))</RHS></DPRULE><DPRULE num="11"><LHS>Marked_quicksort(cons(N,L))</LHS><RHS>Marked_quicksort(low(N,L))</RHS></DPRULE><DPRULE num="12"><LHS>Marked_quicksort(cons(N,L))</LHS><RHS>Marked_low(N,L)</RHS></DPRULE><DPRULE num="13"><LHS>Marked_quicksort(cons(N,L))</LHS><RHS>Marked_quicksort(high(N,L))</RHS></DPRULE><DPRULE num="14"><LHS>Marked_quicksort(cons(N,L))</LHS><RHS>Marked_high(N,L)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="14"/></NONSCC><NONSCC num="1"><NODE ref="7"/></NONSCC><SCC num="2"><NODE ref="6"/><NODE ref="8"/><NODE ref="9"/></SCC><SCC num="3"><NODE ref="11"/><NODE ref="13"/></SCC><NONSCC num="4"><NODE ref="12"/></NONSCC><NONSCC num="5"><NODE ref="10"/></NONSCC><NONSCC num="6"><NODE ref="3"/></NONSCC><SCC num="7"><NODE ref="2"/><NODE ref="4"/><NODE ref="5"/></SCC><SCC num="8"><NODE ref="1"/></SCC><SCC num="9"><NODE ref="0"/></SCC></CCLIST><EDGE end="2" start="0"/><EDGE end="1" start="0"/><EDGE end="9" start="1"/><EDGE end="1" start="2"/><EDGE end="5" start="3"/><EDGE end="4" start="3"/><EDGE end="0" start="3"/><EDGE end="7" start="4"/><EDGE end="6" start="4"/><EDGE end="8" start="5"/><EDGE end="9" start="6"/><EDGE end="6" start="7"/></DAG></CRITERION><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>le(0,Y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(X),s(Y))</LHS><RHS>le(X,Y)</RHS></RULE><RULE><LHS>app(nil,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>app(cons(N,L),Y)</LHS><RHS>cons(N,app(L,Y))</RHS></RULE><RULE><LHS>low(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>low(N,cons(M,L))</LHS><RHS>iflow(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>iflow(true,N,cons(M,L))</LHS><RHS>cons(M,low(N,L))</RHS></RULE><RULE><LHS>iflow(false,N,cons(M,L))</LHS><RHS>low(N,L)</RHS></RULE><RULE><LHS>high(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>high(N,cons(M,L))</LHS><RHS>ifhigh(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>ifhigh(true,N,cons(M,L))</LHS><RHS>high(N,L)</RHS></RULE><RULE><LHS>ifhigh(false,N,cons(M,L))</LHS><RHS>cons(M,high(N,L))</RHS></RULE><RULE><LHS>quicksort(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>quicksort(cons(N,L))</LHS><RHS>app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_quicksort(cons(N,L))</LHS><RHS>Marked_quicksort(low(N,L))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_quicksort(cons(N,L))</LHS><RHS>Marked_quicksort(high(N,L))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="rpo"><AFS><PROJARG num="2"><SYMBOL><NAME>ifhigh</NAME></SYMBOL></PROJARG><PROJARG num="2"><SYMBOL><NAME>iflow</NAME></SYMBOL></PROJARG><PROJARG num="1"><SYMBOL><NAME>high</NAME></SYMBOL></PROJARG><PROJARG num="1"><SYMBOL><NAME>low</NAME></SYMBOL></PROJARG></AFS><STATUSLIST><STATUS status="lex"><SYMBOL><NAME>true</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>quicksort</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>le</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>app</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>false</NAME></SYMBOL></STATUS><STATUS status="lex"><SYMBOL><NAME>cons</NAME></SYMBOL></STATUS></STATUSLIST><PRECEDENCE><PREC><OLHS><SYMBOL><NAME>true</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>le</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>false</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>le</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>app</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>quicksort</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>cons</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>app</NAME></SYMBOL></ORHS></PREC><PREC><OLHS><SYMBOL><NAME>cons</NAME></SYMBOL></OLHS><ORHS><SYMBOL><NAME>quicksort</NAME></SYMBOL></ORHS></PREC></PRECEDENCE><EQUIVALENCE/></ORDERING></CRITERION></PROPERTY><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>le(0,Y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(X),s(Y))</LHS><RHS>le(X,Y)</RHS></RULE><RULE><LHS>app(nil,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>app(cons(N,L),Y)</LHS><RHS>cons(N,app(L,Y))</RHS></RULE><RULE><LHS>low(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>low(N,cons(M,L))</LHS><RHS>iflow(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>iflow(true,N,cons(M,L))</LHS><RHS>cons(M,low(N,L))</RHS></RULE><RULE><LHS>iflow(false,N,cons(M,L))</LHS><RHS>low(N,L)</RHS></RULE><RULE><LHS>high(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>high(N,cons(M,L))</LHS><RHS>ifhigh(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>ifhigh(true,N,cons(M,L))</LHS><RHS>high(N,L)</RHS></RULE><RULE><LHS>ifhigh(false,N,cons(M,L))</LHS><RHS>cons(M,high(N,L))</RHS></RULE><RULE><LHS>quicksort(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>quicksort(cons(N,L))</LHS><RHS>app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(cons(N,L),Y)</LHS><RHS>Marked_app(L,Y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_app</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_app(cons(N,L),Y)</LHS><RHS>Marked_app(L,Y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>le(0,Y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(X),s(Y))</LHS><RHS>le(X,Y)</RHS></RULE><RULE><LHS>app(nil,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>app(cons(N,L),Y)</LHS><RHS>cons(N,app(L,Y))</RHS></RULE><RULE><LHS>low(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>low(N,cons(M,L))</LHS><RHS>iflow(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>iflow(true,N,cons(M,L))</LHS><RHS>cons(M,low(N,L))</RHS></RULE><RULE><LHS>iflow(false,N,cons(M,L))</LHS><RHS>low(N,L)</RHS></RULE><RULE><LHS>high(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>high(N,cons(M,L))</LHS><RHS>ifhigh(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>ifhigh(true,N,cons(M,L))</LHS><RHS>high(N,L)</RHS></RULE><RULE><LHS>ifhigh(false,N,cons(M,L))</LHS><RHS>cons(M,high(N,L))</RHS></RULE><RULE><LHS>quicksort(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>quicksort(cons(N,L))</LHS><RHS>app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))</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>le(0,Y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(X),s(Y))</LHS><RHS>le(X,Y)</RHS></RULE><RULE><LHS>app(nil,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>app(cons(N,L),Y)</LHS><RHS>cons(N,app(L,Y))</RHS></RULE><RULE><LHS>low(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>low(N,cons(M,L))</LHS><RHS>iflow(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>iflow(true,N,cons(M,L))</LHS><RHS>cons(M,low(N,L))</RHS></RULE><RULE><LHS>iflow(false,N,cons(M,L))</LHS><RHS>low(N,L)</RHS></RULE><RULE><LHS>high(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>high(N,cons(M,L))</LHS><RHS>ifhigh(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>ifhigh(true,N,cons(M,L))</LHS><RHS>high(N,L)</RHS></RULE><RULE><LHS>ifhigh(false,N,cons(M,L))</LHS><RHS>cons(M,high(N,L))</RHS></RULE><RULE><LHS>quicksort(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>quicksort(cons(N,L))</LHS><RHS>app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_low(N,cons(M,L))</LHS><RHS>Marked_iflow(le(M,N),N,cons(M,L))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_iflow(true,N,cons(M,L))</LHS><RHS>Marked_low(N,L)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_iflow(false,N,cons(M,L))</LHS><RHS>Marked_low(N,L)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_iflow</NAME></SYMBOL><INT>3</INT></PROJ><PROJ><SYMBOL><NAME>Marked_low</NAME></SYMBOL><INT>2</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_iflow(false,N,cons(M,L))</LHS><RHS>Marked_low(N,L)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_iflow(true,N,cons(M,L))</LHS><RHS>Marked_low(N,L)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>le(0,Y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(X),s(Y))</LHS><RHS>le(X,Y)</RHS></RULE><RULE><LHS>app(nil,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>app(cons(N,L),Y)</LHS><RHS>cons(N,app(L,Y))</RHS></RULE><RULE><LHS>low(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>low(N,cons(M,L))</LHS><RHS>iflow(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>iflow(true,N,cons(M,L))</LHS><RHS>cons(M,low(N,L))</RHS></RULE><RULE><LHS>iflow(false,N,cons(M,L))</LHS><RHS>low(N,L)</RHS></RULE><RULE><LHS>high(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>high(N,cons(M,L))</LHS><RHS>ifhigh(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>ifhigh(true,N,cons(M,L))</LHS><RHS>high(N,L)</RHS></RULE><RULE><LHS>ifhigh(false,N,cons(M,L))</LHS><RHS>cons(M,high(N,L))</RHS></RULE><RULE><LHS>quicksort(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>quicksort(cons(N,L))</LHS><RHS>app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_low(N,cons(M,L))</LHS><RHS>Marked_iflow(le(M,N),N,cons(M,L))</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>le(0,Y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(X),s(Y))</LHS><RHS>le(X,Y)</RHS></RULE><RULE><LHS>app(nil,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>app(cons(N,L),Y)</LHS><RHS>cons(N,app(L,Y))</RHS></RULE><RULE><LHS>low(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>low(N,cons(M,L))</LHS><RHS>iflow(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>iflow(true,N,cons(M,L))</LHS><RHS>cons(M,low(N,L))</RHS></RULE><RULE><LHS>iflow(false,N,cons(M,L))</LHS><RHS>low(N,L)</RHS></RULE><RULE><LHS>high(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>high(N,cons(M,L))</LHS><RHS>ifhigh(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>ifhigh(true,N,cons(M,L))</LHS><RHS>high(N,L)</RHS></RULE><RULE><LHS>ifhigh(false,N,cons(M,L))</LHS><RHS>cons(M,high(N,L))</RHS></RULE><RULE><LHS>quicksort(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>quicksort(cons(N,L))</LHS><RHS>app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_high(N,cons(M,L))</LHS><RHS>Marked_ifhigh(le(M,N),N,cons(M,L))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_ifhigh(true,N,cons(M,L))</LHS><RHS>Marked_high(N,L)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_ifhigh(false,N,cons(M,L))</LHS><RHS>Marked_high(N,L)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_ifhigh</NAME></SYMBOL><INT>3</INT></PROJ><PROJ><SYMBOL><NAME>Marked_high</NAME></SYMBOL><INT>2</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_ifhigh(false,N,cons(M,L))</LHS><RHS>Marked_high(N,L)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_ifhigh(true,N,cons(M,L))</LHS><RHS>Marked_high(N,L)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>le(0,Y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(X),s(Y))</LHS><RHS>le(X,Y)</RHS></RULE><RULE><LHS>app(nil,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>app(cons(N,L),Y)</LHS><RHS>cons(N,app(L,Y))</RHS></RULE><RULE><LHS>low(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>low(N,cons(M,L))</LHS><RHS>iflow(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>iflow(true,N,cons(M,L))</LHS><RHS>cons(M,low(N,L))</RHS></RULE><RULE><LHS>iflow(false,N,cons(M,L))</LHS><RHS>low(N,L)</RHS></RULE><RULE><LHS>high(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>high(N,cons(M,L))</LHS><RHS>ifhigh(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>ifhigh(true,N,cons(M,L))</LHS><RHS>high(N,L)</RHS></RULE><RULE><LHS>ifhigh(false,N,cons(M,L))</LHS><RHS>cons(M,high(N,L))</RHS></RULE><RULE><LHS>quicksort(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>quicksort(cons(N,L))</LHS><RHS>app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_high(N,cons(M,L))</LHS><RHS>Marked_ifhigh(le(M,N),N,cons(M,L))</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>le(0,Y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(X),s(Y))</LHS><RHS>le(X,Y)</RHS></RULE><RULE><LHS>app(nil,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>app(cons(N,L),Y)</LHS><RHS>cons(N,app(L,Y))</RHS></RULE><RULE><LHS>low(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>low(N,cons(M,L))</LHS><RHS>iflow(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>iflow(true,N,cons(M,L))</LHS><RHS>cons(M,low(N,L))</RHS></RULE><RULE><LHS>iflow(false,N,cons(M,L))</LHS><RHS>low(N,L)</RHS></RULE><RULE><LHS>high(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>high(N,cons(M,L))</LHS><RHS>ifhigh(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>ifhigh(true,N,cons(M,L))</LHS><RHS>high(N,L)</RHS></RULE><RULE><LHS>ifhigh(false,N,cons(M,L))</LHS><RHS>cons(M,high(N,L))</RHS></RULE><RULE><LHS>quicksort(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>quicksort(cons(N,L))</LHS><RHS>app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_le(s(X),s(Y))</LHS><RHS>Marked_le(X,Y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_le</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_le(s(X),s(Y))</LHS><RHS>Marked_le(X,Y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>le(0,Y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(X),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(X),s(Y))</LHS><RHS>le(X,Y)</RHS></RULE><RULE><LHS>app(nil,Y)</LHS><RHS>Y</RHS></RULE><RULE><LHS>app(cons(N,L),Y)</LHS><RHS>cons(N,app(L,Y))</RHS></RULE><RULE><LHS>low(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>low(N,cons(M,L))</LHS><RHS>iflow(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>iflow(true,N,cons(M,L))</LHS><RHS>cons(M,low(N,L))</RHS></RULE><RULE><LHS>iflow(false,N,cons(M,L))</LHS><RHS>low(N,L)</RHS></RULE><RULE><LHS>high(N,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>high(N,cons(M,L))</LHS><RHS>ifhigh(le(M,N),N,cons(M,L))</RHS></RULE><RULE><LHS>ifhigh(true,N,cons(M,L))</LHS><RHS>high(N,L)</RHS></RULE><RULE><LHS>ifhigh(false,N,cons(M,L))</LHS><RHS>cons(M,high(N,L))</RHS></RULE><RULE><LHS>quicksort(nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>quicksort(cons(N,L))</LHS><RHS>app(quicksort(low(N,L)),cons(N,quicksort(high(N,L))))</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
