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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="0" unmarked=""><NAME>id</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>msubst</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>sop</NAME></SYMBOL><SYMBOL arity="1" unmarked="te"><NAME>Marked_te</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>cons</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>lift</NAME></SYMBOL><SYMBOL arity="1" unmarked="sortSu"><NAME>Marked_sortSu</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>te</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>circ</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>subst</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>sortSu</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>u</VAR><VAR>t</VAR><VAR>s</VAR><VAR>a</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>sortSu(circ(sortSu(cons(te(a),sortSu(s))),sortSu(t)))</LHS><RHS>sortSu(cons(te(msubst(te(a),sortSu(t))),sortSu(circ(sortSu(s),sortSu(t)))))</RHS></RULE><RULE><LHS>sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),sortSu(cons(te(a),sortSu(t)))))</LHS><RHS>sortSu(cons(te(a),sortSu(circ(sortSu(s),sortSu(t)))))</RHS></RULE><RULE><LHS>sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),
        sortSu(cons(sop(lift),sortSu(t)))))</LHS><RHS>sortSu(cons(sop(lift),sortSu(circ(sortSu(s),sortSu(t)))))</RHS></RULE><RULE><LHS>sortSu(circ(sortSu(circ(sortSu(s),sortSu(t))),sortSu(u)))</LHS><RHS>sortSu(circ(sortSu(s),sortSu(circ(sortSu(t),sortSu(u)))))</RHS></RULE><RULE><LHS>sortSu(circ(sortSu(s),sortSu(id)))</LHS><RHS>sortSu(s)</RHS></RULE><RULE><LHS>sortSu(circ(sortSu(id),sortSu(s)))</LHS><RHS>sortSu(s)</RHS></RULE><RULE><LHS>sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),
        sortSu(circ(sortSu(cons(sop(lift),sortSu(t))),sortSu(u)))))</LHS><RHS>sortSu(circ(sortSu(cons(sop(lift),sortSu(circ(sortSu(s),sortSu(t))))),
        sortSu(u)))</RHS></RULE><RULE><LHS>te(subst(te(a),sortSu(id)))</LHS><RHS>te(a)</RHS></RULE><RULE><LHS>te(msubst(te(a),sortSu(id)))</LHS><RHS>te(a)</RHS></RULE><RULE><LHS>te(msubst(te(msubst(te(a),sortSu(s))),sortSu(t)))</LHS><RHS>te(msubst(te(a),sortSu(circ(sortSu(s),sortSu(t)))))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>sortSu(circ(sortSu(cons(te(a),sortSu(s))),sortSu(t)))</LHS><RHS>sortSu(cons(te(msubst(te(a),sortSu(t))),sortSu(circ(sortSu(s),sortSu(t)))))</RHS></RULE><RULE><LHS>sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),sortSu(cons(te(a),sortSu(t)))))</LHS><RHS>sortSu(cons(te(a),sortSu(circ(sortSu(s),sortSu(t)))))</RHS></RULE><RULE><LHS>sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),
        sortSu(cons(sop(lift),sortSu(t)))))</LHS><RHS>sortSu(cons(sop(lift),sortSu(circ(sortSu(s),sortSu(t)))))</RHS></RULE><RULE><LHS>sortSu(circ(sortSu(circ(sortSu(s),sortSu(t))),sortSu(u)))</LHS><RHS>sortSu(circ(sortSu(s),sortSu(circ(sortSu(t),sortSu(u)))))</RHS></RULE><RULE><LHS>sortSu(circ(sortSu(s),sortSu(id)))</LHS><RHS>sortSu(s)</RHS></RULE><RULE><LHS>sortSu(circ(sortSu(id),sortSu(s)))</LHS><RHS>sortSu(s)</RHS></RULE><RULE><LHS>sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),
        sortSu(circ(sortSu(cons(sop(lift),sortSu(t))),sortSu(u)))))</LHS><RHS>sortSu(circ(sortSu(cons(sop(lift),sortSu(circ(sortSu(s),sortSu(t))))),
        sortSu(u)))</RHS></RULE><RULE><LHS>te(subst(te(a),sortSu(id)))</LHS><RHS>te(a)</RHS></RULE><RULE><LHS>te(msubst(te(a),sortSu(id)))</LHS><RHS>te(a)</RHS></RULE><RULE><LHS>te(msubst(te(msubst(te(a),sortSu(s))),sortSu(t)))</LHS><RHS>te(msubst(te(a),sortSu(circ(sortSu(s),sortSu(t)))))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_sortSu(circ(sortSu(cons(te(a),sortSu(s))),sortSu(t)))</LHS><RHS>Marked_sortSu(cons(te(msubst(te(a),sortSu(t))),
               sortSu(circ(sortSu(s),sortSu(t)))))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_sortSu(circ(sortSu(cons(te(a),sortSu(s))),sortSu(t)))</LHS><RHS>Marked_te(msubst(te(a),sortSu(t)))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_sortSu(circ(sortSu(cons(te(a),sortSu(s))),sortSu(t)))</LHS><RHS>Marked_sortSu(circ(sortSu(s),sortSu(t)))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),
               sortSu(cons(te(a),sortSu(t)))))</LHS><RHS>Marked_sortSu(cons(te(a),sortSu(circ(sortSu(s),sortSu(t)))))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),
               sortSu(cons(te(a),sortSu(t)))))</LHS><RHS>Marked_sortSu(circ(sortSu(s),sortSu(t)))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),
               sortSu(cons(sop(lift),sortSu(t)))))</LHS><RHS>Marked_sortSu(cons(sop(lift),sortSu(circ(sortSu(s),sortSu(t)))))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),
               sortSu(cons(sop(lift),sortSu(t)))))</LHS><RHS>Marked_sortSu(circ(sortSu(s),sortSu(t)))</RHS></DPRULE><DPRULE num="7"><LHS>Marked_sortSu(circ(sortSu(circ(sortSu(s),sortSu(t))),sortSu(u)))</LHS><RHS>Marked_sortSu(circ(sortSu(s),sortSu(circ(sortSu(t),sortSu(u)))))</RHS></DPRULE><DPRULE num="8"><LHS>Marked_sortSu(circ(sortSu(circ(sortSu(s),sortSu(t))),sortSu(u)))</LHS><RHS>Marked_sortSu(circ(sortSu(t),sortSu(u)))</RHS></DPRULE><DPRULE num="9"><LHS>Marked_sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),
               sortSu(circ(sortSu(cons(sop(lift),sortSu(t))),sortSu(u)))))</LHS><RHS>Marked_sortSu(circ(sortSu(cons(sop(lift),sortSu(circ(sortSu(s),sortSu(t))))),
               sortSu(u)))</RHS></DPRULE><DPRULE num="10"><LHS>Marked_sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),
               sortSu(circ(sortSu(cons(sop(lift),sortSu(t))),sortSu(u)))))</LHS><RHS>Marked_sortSu(cons(sop(lift),sortSu(circ(sortSu(s),sortSu(t)))))</RHS></DPRULE><DPRULE num="11"><LHS>Marked_sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),
               sortSu(circ(sortSu(cons(sop(lift),sortSu(t))),sortSu(u)))))</LHS><RHS>Marked_sortSu(circ(sortSu(s),sortSu(t)))</RHS></DPRULE><DPRULE num="12"><LHS>Marked_te(msubst(te(msubst(te(a),sortSu(s))),sortSu(t)))</LHS><RHS>Marked_te(msubst(te(a),sortSu(circ(sortSu(s),sortSu(t)))))</RHS></DPRULE><DPRULE num="13"><LHS>Marked_te(msubst(te(msubst(te(a),sortSu(s))),sortSu(t)))</LHS><RHS>Marked_sortSu(circ(sortSu(s),sortSu(t)))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="1"/><NODE ref="2"/><NODE ref="4"/><NODE ref="6"/><NODE ref="7"/><NODE ref="8"/><NODE ref="9"/><NODE ref="11"/><NODE ref="12"/><NODE ref="13"/></SCC><NONSCC num="1"><NODE ref="10"/></NONSCC><NONSCC num="2"><NODE ref="5"/></NONSCC><NONSCC num="3"><NODE ref="3"/></NONSCC><NONSCC num="4"><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"/></DAG></CRITERION><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>sortSu(circ(sortSu(cons(te(a),sortSu(s))),sortSu(t)))</LHS><RHS>sortSu(cons(te(msubst(te(a),sortSu(t))),sortSu(circ(sortSu(s),sortSu(t)))))</RHS></RULE><RULE><LHS>sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),sortSu(cons(te(a),sortSu(t)))))</LHS><RHS>sortSu(cons(te(a),sortSu(circ(sortSu(s),sortSu(t)))))</RHS></RULE><RULE><LHS>sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),
        sortSu(cons(sop(lift),sortSu(t)))))</LHS><RHS>sortSu(cons(sop(lift),sortSu(circ(sortSu(s),sortSu(t)))))</RHS></RULE><RULE><LHS>sortSu(circ(sortSu(circ(sortSu(s),sortSu(t))),sortSu(u)))</LHS><RHS>sortSu(circ(sortSu(s),sortSu(circ(sortSu(t),sortSu(u)))))</RHS></RULE><RULE><LHS>sortSu(circ(sortSu(s),sortSu(id)))</LHS><RHS>sortSu(s)</RHS></RULE><RULE><LHS>sortSu(circ(sortSu(id),sortSu(s)))</LHS><RHS>sortSu(s)</RHS></RULE><RULE><LHS>sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),
        sortSu(circ(sortSu(cons(sop(lift),sortSu(t))),sortSu(u)))))</LHS><RHS>sortSu(circ(sortSu(cons(sop(lift),sortSu(circ(sortSu(s),sortSu(t))))),
        sortSu(u)))</RHS></RULE><RULE><LHS>te(subst(te(a),sortSu(id)))</LHS><RHS>te(a)</RHS></RULE><RULE><LHS>te(msubst(te(a),sortSu(id)))</LHS><RHS>te(a)</RHS></RULE><RULE><LHS>te(msubst(te(msubst(te(a),sortSu(s))),sortSu(t)))</LHS><RHS>te(msubst(te(a),sortSu(circ(sortSu(s),sortSu(t)))))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_sortSu(circ(sortSu(cons(te(a),sortSu(s))),sortSu(t)))</LHS><RHS>Marked_te(msubst(te(a),sortSu(t)))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_sortSu(circ(sortSu(cons(te(a),sortSu(s))),sortSu(t)))</LHS><RHS>Marked_sortSu(circ(sortSu(s),sortSu(t)))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),
               sortSu(cons(te(a),sortSu(t)))))</LHS><RHS>Marked_sortSu(circ(sortSu(s),sortSu(t)))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),
               sortSu(cons(sop(lift),sortSu(t)))))</LHS><RHS>Marked_sortSu(circ(sortSu(s),sortSu(t)))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_sortSu(circ(sortSu(circ(sortSu(s),sortSu(t))),sortSu(u)))</LHS><RHS>Marked_sortSu(circ(sortSu(s),sortSu(circ(sortSu(t),sortSu(u)))))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_sortSu(circ(sortSu(circ(sortSu(s),sortSu(t))),sortSu(u)))</LHS><RHS>Marked_sortSu(circ(sortSu(t),sortSu(u)))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),
               sortSu(circ(sortSu(cons(sop(lift),sortSu(t))),sortSu(u)))))</LHS><RHS>Marked_sortSu(circ(sortSu(cons(sop(lift),sortSu(circ(sortSu(s),sortSu(t))))),
               sortSu(u)))</RHS></DPRULE><DPRULE num="7"><LHS>Marked_sortSu(circ(sortSu(cons(sop(lift),sortSu(s))),
               sortSu(circ(sortSu(cons(sop(lift),sortSu(t))),sortSu(u)))))</LHS><RHS>Marked_sortSu(circ(sortSu(s),sortSu(t)))</RHS></DPRULE><DPRULE num="8"><LHS>Marked_te(msubst(te(msubst(te(a),sortSu(s))),sortSu(t)))</LHS><RHS>Marked_te(msubst(te(a),sortSu(circ(sortSu(s),sortSu(t)))))</RHS></DPRULE><DPRULE num="9"><LHS>Marked_te(msubst(te(msubst(te(a),sortSu(s))),sortSu(t)))</LHS><RHS>Marked_sortSu(circ(sortSu(s),sortSu(t)))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>sortSu</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>subst</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>circ</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><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="0"/><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>te</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_sortSu</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>lift</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>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>Marked_te</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>sop</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>msubst</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><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="0"/><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>id</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB></ORDERING></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROOF>
