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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="1" unmarked=""><NAME>tl</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>incr</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>a__hd</NAME></SYMBOL><SYMBOL arity="0" unmarked="a__zeros"><NAME>Marked_a__zeros</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>cons</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>nats</NAME></SYMBOL><SYMBOL arity="0" unmarked="a__nats"><NAME>Marked_a__nats</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>zeros</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>adx</NAME></SYMBOL><SYMBOL arity="1" unmarked="a__hd"><NAME>Marked_a__hd</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>a__zeros</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>hd</NAME></SYMBOL><SYMBOL arity="1" unmarked="mark"><NAME>Marked_mark</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>mark</NAME></SYMBOL><SYMBOL arity="1" unmarked="a__incr"><NAME>Marked_a__incr</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>a__nats</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>a__tl</NAME></SYMBOL><SYMBOL arity="1" unmarked="a__adx"><NAME>Marked_a__adx</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>a__incr</NAME></SYMBOL><SYMBOL arity="1" unmarked="a__tl"><NAME>Marked_a__tl</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>a__adx</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>X2</VAR><VAR>X1</VAR><VAR>Y</VAR><VAR>X</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>a__nats</LHS><RHS>a__adx(a__zeros)</RHS></RULE><RULE><LHS>a__zeros</LHS><RHS>cons(0,zeros)</RHS></RULE><RULE><LHS>a__incr(cons(X,Y))</LHS><RHS>cons(s(X),incr(Y))</RHS></RULE><RULE><LHS>a__adx(cons(X,Y))</LHS><RHS>a__incr(cons(X,adx(Y)))</RHS></RULE><RULE><LHS>a__hd(cons(X,Y))</LHS><RHS>mark(X)</RHS></RULE><RULE><LHS>a__tl(cons(X,Y))</LHS><RHS>mark(Y)</RHS></RULE><RULE><LHS>mark(nats)</LHS><RHS>a__nats</RHS></RULE><RULE><LHS>mark(adx(X))</LHS><RHS>a__adx(mark(X))</RHS></RULE><RULE><LHS>mark(zeros)</LHS><RHS>a__zeros</RHS></RULE><RULE><LHS>mark(incr(X))</LHS><RHS>a__incr(mark(X))</RHS></RULE><RULE><LHS>mark(hd(X))</LHS><RHS>a__hd(mark(X))</RHS></RULE><RULE><LHS>mark(tl(X))</LHS><RHS>a__tl(mark(X))</RHS></RULE><RULE><LHS>mark(cons(X1,X2))</LHS><RHS>cons(X1,X2)</RHS></RULE><RULE><LHS>mark(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>mark(s(X))</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>a__nats</LHS><RHS>nats</RHS></RULE><RULE><LHS>a__adx(X)</LHS><RHS>adx(X)</RHS></RULE><RULE><LHS>a__zeros</LHS><RHS>zeros</RHS></RULE><RULE><LHS>a__incr(X)</LHS><RHS>incr(X)</RHS></RULE><RULE><LHS>a__hd(X)</LHS><RHS>hd(X)</RHS></RULE><RULE><LHS>a__tl(X)</LHS><RHS>tl(X)</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>a__nats</LHS><RHS>a__adx(a__zeros)</RHS></RULE><RULE><LHS>a__zeros</LHS><RHS>cons(0,zeros)</RHS></RULE><RULE><LHS>a__incr(cons(X,Y))</LHS><RHS>cons(s(X),incr(Y))</RHS></RULE><RULE><LHS>a__adx(cons(X,Y))</LHS><RHS>a__incr(cons(X,adx(Y)))</RHS></RULE><RULE><LHS>a__hd(cons(X,Y))</LHS><RHS>mark(X)</RHS></RULE><RULE><LHS>a__tl(cons(X,Y))</LHS><RHS>mark(Y)</RHS></RULE><RULE><LHS>mark(nats)</LHS><RHS>a__nats</RHS></RULE><RULE><LHS>mark(adx(X))</LHS><RHS>a__adx(mark(X))</RHS></RULE><RULE><LHS>mark(zeros)</LHS><RHS>a__zeros</RHS></RULE><RULE><LHS>mark(incr(X))</LHS><RHS>a__incr(mark(X))</RHS></RULE><RULE><LHS>mark(hd(X))</LHS><RHS>a__hd(mark(X))</RHS></RULE><RULE><LHS>mark(tl(X))</LHS><RHS>a__tl(mark(X))</RHS></RULE><RULE><LHS>mark(cons(X1,X2))</LHS><RHS>cons(X1,X2)</RHS></RULE><RULE><LHS>mark(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>mark(s(X))</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>a__nats</LHS><RHS>nats</RHS></RULE><RULE><LHS>a__adx(X)</LHS><RHS>adx(X)</RHS></RULE><RULE><LHS>a__zeros</LHS><RHS>zeros</RHS></RULE><RULE><LHS>a__incr(X)</LHS><RHS>incr(X)</RHS></RULE><RULE><LHS>a__hd(X)</LHS><RHS>hd(X)</RHS></RULE><RULE><LHS>a__tl(X)</LHS><RHS>tl(X)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_a__nats</LHS><RHS>Marked_a__adx(a__zeros)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_a__nats</LHS><RHS>Marked_a__zeros</RHS></DPRULE><DPRULE num="2"><LHS>Marked_a__adx(cons(X,Y))</LHS><RHS>Marked_a__incr(cons(X,adx(Y)))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_a__hd(cons(X,Y))</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_a__tl(cons(X,Y))</LHS><RHS>Marked_mark(Y)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_mark(nats)</LHS><RHS>Marked_a__nats</RHS></DPRULE><DPRULE num="6"><LHS>Marked_mark(adx(X))</LHS><RHS>Marked_a__adx(mark(X))</RHS></DPRULE><DPRULE num="7"><LHS>Marked_mark(adx(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="8"><LHS>Marked_mark(zeros)</LHS><RHS>Marked_a__zeros</RHS></DPRULE><DPRULE num="9"><LHS>Marked_mark(incr(X))</LHS><RHS>Marked_a__incr(mark(X))</RHS></DPRULE><DPRULE num="10"><LHS>Marked_mark(incr(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="11"><LHS>Marked_mark(hd(X))</LHS><RHS>Marked_a__hd(mark(X))</RHS></DPRULE><DPRULE num="12"><LHS>Marked_mark(hd(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="13"><LHS>Marked_mark(tl(X))</LHS><RHS>Marked_a__tl(mark(X))</RHS></DPRULE><DPRULE num="14"><LHS>Marked_mark(tl(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="3"/><NODE ref="4"/><NODE ref="7"/><NODE ref="10"/><NODE ref="11"/><NODE ref="12"/><NODE ref="13"/><NODE ref="14"/></SCC><NONSCC num="1"><NODE ref="9"/></NONSCC><NONSCC num="2"><NODE ref="8"/></NONSCC><NONSCC num="3"><NODE ref="6"/></NONSCC><NONSCC num="4"><NODE ref="5"/></NONSCC><NONSCC num="5"><NODE ref="2"/></NONSCC><NONSCC num="6"><NODE ref="1"/></NONSCC><NONSCC num="7"><NODE ref="0"/></NONSCC></CCLIST><EDGE end="4" start="0"/><EDGE end="3" start="0"/><EDGE end="2" start="0"/><EDGE end="1" start="0"/><EDGE end="5" start="3"/><EDGE end="7" start="4"/><EDGE end="6" start="4"/><EDGE end="5" start="7"/></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>a__nats</LHS><RHS>a__adx(a__zeros)</RHS></RULE><RULE><LHS>a__zeros</LHS><RHS>cons(0,zeros)</RHS></RULE><RULE><LHS>a__incr(cons(X,Y))</LHS><RHS>cons(s(X),incr(Y))</RHS></RULE><RULE><LHS>a__adx(cons(X,Y))</LHS><RHS>a__incr(cons(X,adx(Y)))</RHS></RULE><RULE><LHS>a__hd(cons(X,Y))</LHS><RHS>mark(X)</RHS></RULE><RULE><LHS>a__tl(cons(X,Y))</LHS><RHS>mark(Y)</RHS></RULE><RULE><LHS>mark(nats)</LHS><RHS>a__nats</RHS></RULE><RULE><LHS>mark(adx(X))</LHS><RHS>a__adx(mark(X))</RHS></RULE><RULE><LHS>mark(zeros)</LHS><RHS>a__zeros</RHS></RULE><RULE><LHS>mark(incr(X))</LHS><RHS>a__incr(mark(X))</RHS></RULE><RULE><LHS>mark(hd(X))</LHS><RHS>a__hd(mark(X))</RHS></RULE><RULE><LHS>mark(tl(X))</LHS><RHS>a__tl(mark(X))</RHS></RULE><RULE><LHS>mark(cons(X1,X2))</LHS><RHS>cons(X1,X2)</RHS></RULE><RULE><LHS>mark(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>mark(s(X))</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>a__nats</LHS><RHS>nats</RHS></RULE><RULE><LHS>a__adx(X)</LHS><RHS>adx(X)</RHS></RULE><RULE><LHS>a__zeros</LHS><RHS>zeros</RHS></RULE><RULE><LHS>a__incr(X)</LHS><RHS>incr(X)</RHS></RULE><RULE><LHS>a__hd(X)</LHS><RHS>hd(X)</RHS></RULE><RULE><LHS>a__tl(X)</LHS><RHS>tl(X)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_a__hd(cons(X,Y))</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_a__tl(cons(X,Y))</LHS><RHS>Marked_mark(Y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_mark(adx(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_mark(incr(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_mark(hd(X))</LHS><RHS>Marked_a__hd(mark(X))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_mark(hd(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_mark(tl(X))</LHS><RHS>Marked_a__tl(mark(X))</RHS></DPRULE><DPRULE num="7"><LHS>Marked_mark(tl(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>a__adx</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>Marked_a__tl</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>a__incr</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>a__tl</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>a__nats</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>mark</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>s</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_mark</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>hd</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>a__zeros</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_a__hd</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>adx</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>zeros</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>nats</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>cons</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="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>a__hd</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>incr</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>tl</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></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_mark(tl(X))</LHS><RHS>Marked_a__tl(mark(X))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_mark(tl(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>a__nats</LHS><RHS>a__adx(a__zeros)</RHS></RULE><RULE><LHS>a__zeros</LHS><RHS>cons(0,zeros)</RHS></RULE><RULE><LHS>a__incr(cons(X,Y))</LHS><RHS>cons(s(X),incr(Y))</RHS></RULE><RULE><LHS>a__adx(cons(X,Y))</LHS><RHS>a__incr(cons(X,adx(Y)))</RHS></RULE><RULE><LHS>a__hd(cons(X,Y))</LHS><RHS>mark(X)</RHS></RULE><RULE><LHS>a__tl(cons(X,Y))</LHS><RHS>mark(Y)</RHS></RULE><RULE><LHS>mark(nats)</LHS><RHS>a__nats</RHS></RULE><RULE><LHS>mark(adx(X))</LHS><RHS>a__adx(mark(X))</RHS></RULE><RULE><LHS>mark(zeros)</LHS><RHS>a__zeros</RHS></RULE><RULE><LHS>mark(incr(X))</LHS><RHS>a__incr(mark(X))</RHS></RULE><RULE><LHS>mark(hd(X))</LHS><RHS>a__hd(mark(X))</RHS></RULE><RULE><LHS>mark(tl(X))</LHS><RHS>a__tl(mark(X))</RHS></RULE><RULE><LHS>mark(cons(X1,X2))</LHS><RHS>cons(X1,X2)</RHS></RULE><RULE><LHS>mark(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>mark(s(X))</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>a__nats</LHS><RHS>nats</RHS></RULE><RULE><LHS>a__adx(X)</LHS><RHS>adx(X)</RHS></RULE><RULE><LHS>a__zeros</LHS><RHS>zeros</RHS></RULE><RULE><LHS>a__incr(X)</LHS><RHS>incr(X)</RHS></RULE><RULE><LHS>a__hd(X)</LHS><RHS>hd(X)</RHS></RULE><RULE><LHS>a__tl(X)</LHS><RHS>tl(X)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_a__hd(cons(X,Y))</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_a__tl(cons(X,Y))</LHS><RHS>Marked_mark(Y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_mark(adx(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_mark(incr(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_mark(hd(X))</LHS><RHS>Marked_a__hd(mark(X))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_mark(hd(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/><NODE ref="2"/><NODE ref="3"/><NODE ref="4"/><NODE ref="5"/></SCC><NONSCC num="1"><NODE ref="1"/></NONSCC></CCLIST><EDGE end="0" start="1"/></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>a__nats</LHS><RHS>a__adx(a__zeros)</RHS></RULE><RULE><LHS>a__zeros</LHS><RHS>cons(0,zeros)</RHS></RULE><RULE><LHS>a__incr(cons(X,Y))</LHS><RHS>cons(s(X),incr(Y))</RHS></RULE><RULE><LHS>a__adx(cons(X,Y))</LHS><RHS>a__incr(cons(X,adx(Y)))</RHS></RULE><RULE><LHS>a__hd(cons(X,Y))</LHS><RHS>mark(X)</RHS></RULE><RULE><LHS>a__tl(cons(X,Y))</LHS><RHS>mark(Y)</RHS></RULE><RULE><LHS>mark(nats)</LHS><RHS>a__nats</RHS></RULE><RULE><LHS>mark(adx(X))</LHS><RHS>a__adx(mark(X))</RHS></RULE><RULE><LHS>mark(zeros)</LHS><RHS>a__zeros</RHS></RULE><RULE><LHS>mark(incr(X))</LHS><RHS>a__incr(mark(X))</RHS></RULE><RULE><LHS>mark(hd(X))</LHS><RHS>a__hd(mark(X))</RHS></RULE><RULE><LHS>mark(tl(X))</LHS><RHS>a__tl(mark(X))</RHS></RULE><RULE><LHS>mark(cons(X1,X2))</LHS><RHS>cons(X1,X2)</RHS></RULE><RULE><LHS>mark(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>mark(s(X))</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>a__nats</LHS><RHS>nats</RHS></RULE><RULE><LHS>a__adx(X)</LHS><RHS>adx(X)</RHS></RULE><RULE><LHS>a__zeros</LHS><RHS>zeros</RHS></RULE><RULE><LHS>a__incr(X)</LHS><RHS>incr(X)</RHS></RULE><RULE><LHS>a__hd(X)</LHS><RHS>hd(X)</RHS></RULE><RULE><LHS>a__tl(X)</LHS><RHS>tl(X)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_a__hd(cons(X,Y))</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_mark(adx(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_mark(incr(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_mark(hd(X))</LHS><RHS>Marked_a__hd(mark(X))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_mark(hd(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>a__adx</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>a__incr</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>a__tl</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>a__nats</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>mark</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>s</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_mark</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>hd</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</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>a__zeros</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_a__hd</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</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>adx</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>zeros</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>nats</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>cons</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>a__hd</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</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>incr</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>tl</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></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_a__hd(cons(X,Y))</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_mark(hd(X))</LHS><RHS>Marked_a__hd(mark(X))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_mark(hd(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>a__nats</LHS><RHS>a__adx(a__zeros)</RHS></RULE><RULE><LHS>a__zeros</LHS><RHS>cons(0,zeros)</RHS></RULE><RULE><LHS>a__incr(cons(X,Y))</LHS><RHS>cons(s(X),incr(Y))</RHS></RULE><RULE><LHS>a__adx(cons(X,Y))</LHS><RHS>a__incr(cons(X,adx(Y)))</RHS></RULE><RULE><LHS>a__hd(cons(X,Y))</LHS><RHS>mark(X)</RHS></RULE><RULE><LHS>a__tl(cons(X,Y))</LHS><RHS>mark(Y)</RHS></RULE><RULE><LHS>mark(nats)</LHS><RHS>a__nats</RHS></RULE><RULE><LHS>mark(adx(X))</LHS><RHS>a__adx(mark(X))</RHS></RULE><RULE><LHS>mark(zeros)</LHS><RHS>a__zeros</RHS></RULE><RULE><LHS>mark(incr(X))</LHS><RHS>a__incr(mark(X))</RHS></RULE><RULE><LHS>mark(hd(X))</LHS><RHS>a__hd(mark(X))</RHS></RULE><RULE><LHS>mark(tl(X))</LHS><RHS>a__tl(mark(X))</RHS></RULE><RULE><LHS>mark(cons(X1,X2))</LHS><RHS>cons(X1,X2)</RHS></RULE><RULE><LHS>mark(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>mark(s(X))</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>a__nats</LHS><RHS>nats</RHS></RULE><RULE><LHS>a__adx(X)</LHS><RHS>adx(X)</RHS></RULE><RULE><LHS>a__zeros</LHS><RHS>zeros</RHS></RULE><RULE><LHS>a__incr(X)</LHS><RHS>incr(X)</RHS></RULE><RULE><LHS>a__hd(X)</LHS><RHS>hd(X)</RHS></RULE><RULE><LHS>a__tl(X)</LHS><RHS>tl(X)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_mark(adx(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_mark(incr(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/><NODE ref="1"/></SCC></CCLIST></DAG></CRITERION><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>a__nats</LHS><RHS>a__adx(a__zeros)</RHS></RULE><RULE><LHS>a__zeros</LHS><RHS>cons(0,zeros)</RHS></RULE><RULE><LHS>a__incr(cons(X,Y))</LHS><RHS>cons(s(X),incr(Y))</RHS></RULE><RULE><LHS>a__adx(cons(X,Y))</LHS><RHS>a__incr(cons(X,adx(Y)))</RHS></RULE><RULE><LHS>a__hd(cons(X,Y))</LHS><RHS>mark(X)</RHS></RULE><RULE><LHS>a__tl(cons(X,Y))</LHS><RHS>mark(Y)</RHS></RULE><RULE><LHS>mark(nats)</LHS><RHS>a__nats</RHS></RULE><RULE><LHS>mark(adx(X))</LHS><RHS>a__adx(mark(X))</RHS></RULE><RULE><LHS>mark(zeros)</LHS><RHS>a__zeros</RHS></RULE><RULE><LHS>mark(incr(X))</LHS><RHS>a__incr(mark(X))</RHS></RULE><RULE><LHS>mark(hd(X))</LHS><RHS>a__hd(mark(X))</RHS></RULE><RULE><LHS>mark(tl(X))</LHS><RHS>a__tl(mark(X))</RHS></RULE><RULE><LHS>mark(cons(X1,X2))</LHS><RHS>cons(X1,X2)</RHS></RULE><RULE><LHS>mark(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>mark(s(X))</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>a__nats</LHS><RHS>nats</RHS></RULE><RULE><LHS>a__adx(X)</LHS><RHS>adx(X)</RHS></RULE><RULE><LHS>a__zeros</LHS><RHS>zeros</RHS></RULE><RULE><LHS>a__incr(X)</LHS><RHS>incr(X)</RHS></RULE><RULE><LHS>a__hd(X)</LHS><RHS>hd(X)</RHS></RULE><RULE><LHS>a__tl(X)</LHS><RHS>tl(X)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_mark(adx(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_mark(incr(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_mark</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_mark(incr(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_mark(adx(X))</LHS><RHS>Marked_mark(X)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>a__nats</LHS><RHS>a__adx(a__zeros)</RHS></RULE><RULE><LHS>a__zeros</LHS><RHS>cons(0,zeros)</RHS></RULE><RULE><LHS>a__incr(cons(X,Y))</LHS><RHS>cons(s(X),incr(Y))</RHS></RULE><RULE><LHS>a__adx(cons(X,Y))</LHS><RHS>a__incr(cons(X,adx(Y)))</RHS></RULE><RULE><LHS>a__hd(cons(X,Y))</LHS><RHS>mark(X)</RHS></RULE><RULE><LHS>a__tl(cons(X,Y))</LHS><RHS>mark(Y)</RHS></RULE><RULE><LHS>mark(nats)</LHS><RHS>a__nats</RHS></RULE><RULE><LHS>mark(adx(X))</LHS><RHS>a__adx(mark(X))</RHS></RULE><RULE><LHS>mark(zeros)</LHS><RHS>a__zeros</RHS></RULE><RULE><LHS>mark(incr(X))</LHS><RHS>a__incr(mark(X))</RHS></RULE><RULE><LHS>mark(hd(X))</LHS><RHS>a__hd(mark(X))</RHS></RULE><RULE><LHS>mark(tl(X))</LHS><RHS>a__tl(mark(X))</RHS></RULE><RULE><LHS>mark(cons(X1,X2))</LHS><RHS>cons(X1,X2)</RHS></RULE><RULE><LHS>mark(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>mark(s(X))</LHS><RHS>s(X)</RHS></RULE><RULE><LHS>a__nats</LHS><RHS>nats</RHS></RULE><RULE><LHS>a__adx(X)</LHS><RHS>adx(X)</RHS></RULE><RULE><LHS>a__zeros</LHS><RHS>zeros</RHS></RULE><RULE><LHS>a__incr(X)</LHS><RHS>incr(X)</RHS></RULE><RULE><LHS>a__hd(X)</LHS><RHS>hd(X)</RHS></RULE><RULE><LHS>a__tl(X)</LHS><RHS>tl(X)</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
