<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="3" unmarked="u31"><NAME>Marked_u31</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>times</NAME></SYMBOL><SYMBOL arity="3" unmarked="u42"><NAME>Marked_u42</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>plus</NAME></SYMBOL><SYMBOL arity="1" unmarked="din"><NAME>Marked_din</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>dout</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>u41</NAME></SYMBOL><SYMBOL arity="3" unmarked="u21"><NAME>Marked_u21</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>din</NAME></SYMBOL><SYMBOL arity="4" unmarked="u32"><NAME>Marked_u32</NAME></SYMBOL><SYMBOL arity="3" unmarked=""><NAME>u31</NAME></SYMBOL><SYMBOL arity="3" unmarked=""><NAME>u42</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>der</NAME></SYMBOL><SYMBOL arity="2" unmarked="u41"><NAME>Marked_u41</NAME></SYMBOL><SYMBOL arity="4" unmarked=""><NAME>u22</NAME></SYMBOL><SYMBOL arity="4" unmarked=""><NAME>u32</NAME></SYMBOL><SYMBOL arity="4" unmarked="u22"><NAME>Marked_u22</NAME></SYMBOL><SYMBOL arity="3" unmarked=""><NAME>u21</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>DDX</VAR><VAR>DY</VAR><VAR>DX</VAR><VAR>Y</VAR><VAR>X</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>din(der(plus(X,Y)))</LHS><RHS>u21(din(der(X)),X,Y)</RHS></RULE><RULE><LHS>u21(dout(DX),X,Y)</LHS><RHS>u22(din(der(Y)),X,Y,DX)</RHS></RULE><RULE><LHS>u22(dout(DY),X,Y,DX)</LHS><RHS>dout(plus(DX,DY))</RHS></RULE><RULE><LHS>din(der(times(X,Y)))</LHS><RHS>u31(din(der(X)),X,Y)</RHS></RULE><RULE><LHS>u31(dout(DX),X,Y)</LHS><RHS>u32(din(der(Y)),X,Y,DX)</RHS></RULE><RULE><LHS>u32(dout(DY),X,Y,DX)</LHS><RHS>dout(plus(times(X,DY),times(Y,DX)))</RHS></RULE><RULE><LHS>din(der(der(X)))</LHS><RHS>u41(din(der(X)),X)</RHS></RULE><RULE><LHS>u41(dout(DX),X)</LHS><RHS>u42(din(der(DX)),X,DX)</RHS></RULE><RULE><LHS>u42(dout(DDX),X,DX)</LHS><RHS>dout(DDX)</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>din(der(plus(X,Y)))</LHS><RHS>u21(din(der(X)),X,Y)</RHS></RULE><RULE><LHS>u21(dout(DX),X,Y)</LHS><RHS>u22(din(der(Y)),X,Y,DX)</RHS></RULE><RULE><LHS>u22(dout(DY),X,Y,DX)</LHS><RHS>dout(plus(DX,DY))</RHS></RULE><RULE><LHS>din(der(times(X,Y)))</LHS><RHS>u31(din(der(X)),X,Y)</RHS></RULE><RULE><LHS>u31(dout(DX),X,Y)</LHS><RHS>u32(din(der(Y)),X,Y,DX)</RHS></RULE><RULE><LHS>u32(dout(DY),X,Y,DX)</LHS><RHS>dout(plus(times(X,DY),times(Y,DX)))</RHS></RULE><RULE><LHS>din(der(der(X)))</LHS><RHS>u41(din(der(X)),X)</RHS></RULE><RULE><LHS>u41(dout(DX),X)</LHS><RHS>u42(din(der(DX)),X,DX)</RHS></RULE><RULE><LHS>u42(dout(DDX),X,DX)</LHS><RHS>dout(DDX)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_din(der(plus(X,Y)))</LHS><RHS>Marked_u21(din(der(X)),X,Y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_din(der(plus(X,Y)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_u21(dout(DX),X,Y)</LHS><RHS>Marked_u22(din(der(Y)),X,Y,DX)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_u21(dout(DX),X,Y)</LHS><RHS>Marked_din(der(Y))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_din(der(times(X,Y)))</LHS><RHS>Marked_u31(din(der(X)),X,Y)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_din(der(times(X,Y)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_u31(dout(DX),X,Y)</LHS><RHS>Marked_u32(din(der(Y)),X,Y,DX)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_u31(dout(DX),X,Y)</LHS><RHS>Marked_din(der(Y))</RHS></DPRULE><DPRULE num="8"><LHS>Marked_din(der(der(X)))</LHS><RHS>Marked_u41(din(der(X)),X)</RHS></DPRULE><DPRULE num="9"><LHS>Marked_din(der(der(X)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE><DPRULE num="10"><LHS>Marked_u41(dout(DX),X)</LHS><RHS>Marked_u42(din(der(DX)),X,DX)</RHS></DPRULE><DPRULE num="11"><LHS>Marked_u41(dout(DX),X)</LHS><RHS>Marked_din(der(DX))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/><NODE ref="1"/><NODE ref="3"/><NODE ref="4"/><NODE ref="5"/><NODE ref="7"/><NODE ref="8"/><NODE ref="9"/><NODE ref="11"/></SCC><NONSCC num="1"><NODE ref="10"/></NONSCC><NONSCC num="2"><NODE ref="6"/></NONSCC><NONSCC num="3"><NODE ref="2"/></NONSCC></CCLIST><EDGE end="3" start="0"/><EDGE end="2" start="0"/><EDGE end="1" start="0"/></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>din(der(plus(X,Y)))</LHS><RHS>u21(din(der(X)),X,Y)</RHS></RULE><RULE><LHS>u21(dout(DX),X,Y)</LHS><RHS>u22(din(der(Y)),X,Y,DX)</RHS></RULE><RULE><LHS>u22(dout(DY),X,Y,DX)</LHS><RHS>dout(plus(DX,DY))</RHS></RULE><RULE><LHS>din(der(times(X,Y)))</LHS><RHS>u31(din(der(X)),X,Y)</RHS></RULE><RULE><LHS>u31(dout(DX),X,Y)</LHS><RHS>u32(din(der(Y)),X,Y,DX)</RHS></RULE><RULE><LHS>u32(dout(DY),X,Y,DX)</LHS><RHS>dout(plus(times(X,DY),times(Y,DX)))</RHS></RULE><RULE><LHS>din(der(der(X)))</LHS><RHS>u41(din(der(X)),X)</RHS></RULE><RULE><LHS>u41(dout(DX),X)</LHS><RHS>u42(din(der(DX)),X,DX)</RHS></RULE><RULE><LHS>u42(dout(DDX),X,DX)</LHS><RHS>dout(DDX)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_din(der(plus(X,Y)))</LHS><RHS>Marked_u21(din(der(X)),X,Y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_din(der(plus(X,Y)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_u21(dout(DX),X,Y)</LHS><RHS>Marked_din(der(Y))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_din(der(times(X,Y)))</LHS><RHS>Marked_u31(din(der(X)),X,Y)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_din(der(times(X,Y)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_u31(dout(DX),X,Y)</LHS><RHS>Marked_din(der(Y))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_din(der(der(X)))</LHS><RHS>Marked_u41(din(der(X)),X)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_din(der(der(X)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE><DPRULE num="8"><LHS>Marked_u41(dout(DX),X)</LHS><RHS>Marked_din(der(DX))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>u21</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>u32</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>u22</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_u41</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>der</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>u42</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>u31</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>din</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_u21</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>u41</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>dout</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_din</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>plus</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>times</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_u31</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></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_u21(dout(DX),X,Y)</LHS><RHS>Marked_din(der(Y))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_u31(dout(DX),X,Y)</LHS><RHS>Marked_din(der(Y))</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>din(der(plus(X,Y)))</LHS><RHS>u21(din(der(X)),X,Y)</RHS></RULE><RULE><LHS>u21(dout(DX),X,Y)</LHS><RHS>u22(din(der(Y)),X,Y,DX)</RHS></RULE><RULE><LHS>u22(dout(DY),X,Y,DX)</LHS><RHS>dout(plus(DX,DY))</RHS></RULE><RULE><LHS>din(der(times(X,Y)))</LHS><RHS>u31(din(der(X)),X,Y)</RHS></RULE><RULE><LHS>u31(dout(DX),X,Y)</LHS><RHS>u32(din(der(Y)),X,Y,DX)</RHS></RULE><RULE><LHS>u32(dout(DY),X,Y,DX)</LHS><RHS>dout(plus(times(X,DY),times(Y,DX)))</RHS></RULE><RULE><LHS>din(der(der(X)))</LHS><RHS>u41(din(der(X)),X)</RHS></RULE><RULE><LHS>u41(dout(DX),X)</LHS><RHS>u42(din(der(DX)),X,DX)</RHS></RULE><RULE><LHS>u42(dout(DDX),X,DX)</LHS><RHS>dout(DDX)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_din(der(plus(X,Y)))</LHS><RHS>Marked_u21(din(der(X)),X,Y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_din(der(plus(X,Y)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_din(der(times(X,Y)))</LHS><RHS>Marked_u31(din(der(X)),X,Y)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_din(der(times(X,Y)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_din(der(der(X)))</LHS><RHS>Marked_u41(din(der(X)),X)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_din(der(der(X)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_u41(dout(DX),X)</LHS><RHS>Marked_din(der(DX))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="1"/><NODE ref="3"/><NODE ref="4"/><NODE ref="5"/><NODE ref="6"/></SCC><NONSCC num="1"><NODE ref="2"/></NONSCC><NONSCC num="2"><NODE ref="0"/></NONSCC></CCLIST><EDGE end="2" start="0"/><EDGE end="1" start="0"/></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>din(der(plus(X,Y)))</LHS><RHS>u21(din(der(X)),X,Y)</RHS></RULE><RULE><LHS>u21(dout(DX),X,Y)</LHS><RHS>u22(din(der(Y)),X,Y,DX)</RHS></RULE><RULE><LHS>u22(dout(DY),X,Y,DX)</LHS><RHS>dout(plus(DX,DY))</RHS></RULE><RULE><LHS>din(der(times(X,Y)))</LHS><RHS>u31(din(der(X)),X,Y)</RHS></RULE><RULE><LHS>u31(dout(DX),X,Y)</LHS><RHS>u32(din(der(Y)),X,Y,DX)</RHS></RULE><RULE><LHS>u32(dout(DY),X,Y,DX)</LHS><RHS>dout(plus(times(X,DY),times(Y,DX)))</RHS></RULE><RULE><LHS>din(der(der(X)))</LHS><RHS>u41(din(der(X)),X)</RHS></RULE><RULE><LHS>u41(dout(DX),X)</LHS><RHS>u42(din(der(DX)),X,DX)</RHS></RULE><RULE><LHS>u42(dout(DDX),X,DX)</LHS><RHS>dout(DDX)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_din(der(plus(X,Y)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_din(der(times(X,Y)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_din(der(der(X)))</LHS><RHS>Marked_u41(din(der(X)),X)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_din(der(der(X)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_u41(dout(DX),X)</LHS><RHS>Marked_din(der(DX))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>u21</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>u32</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="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>u22</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="3"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_u41</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</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>der</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>u42</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>u31</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="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>din</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>u41</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>dout</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>Marked_din</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>plus</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>times</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_din(der(der(X)))</LHS><RHS>Marked_u41(din(der(X)),X)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_din(der(der(X)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_u41(dout(DX),X)</LHS><RHS>Marked_din(der(DX))</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>din(der(plus(X,Y)))</LHS><RHS>u21(din(der(X)),X,Y)</RHS></RULE><RULE><LHS>u21(dout(DX),X,Y)</LHS><RHS>u22(din(der(Y)),X,Y,DX)</RHS></RULE><RULE><LHS>u22(dout(DY),X,Y,DX)</LHS><RHS>dout(plus(DX,DY))</RHS></RULE><RULE><LHS>din(der(times(X,Y)))</LHS><RHS>u31(din(der(X)),X,Y)</RHS></RULE><RULE><LHS>u31(dout(DX),X,Y)</LHS><RHS>u32(din(der(Y)),X,Y,DX)</RHS></RULE><RULE><LHS>u32(dout(DY),X,Y,DX)</LHS><RHS>dout(plus(times(X,DY),times(Y,DX)))</RHS></RULE><RULE><LHS>din(der(der(X)))</LHS><RHS>u41(din(der(X)),X)</RHS></RULE><RULE><LHS>u41(dout(DX),X)</LHS><RHS>u42(din(der(DX)),X,DX)</RHS></RULE><RULE><LHS>u42(dout(DDX),X,DX)</LHS><RHS>dout(DDX)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_din(der(plus(X,Y)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_din(der(times(X,Y)))</LHS><RHS>Marked_din(der(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="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>din(der(plus(X,Y)))</LHS><RHS>u21(din(der(X)),X,Y)</RHS></RULE><RULE><LHS>u21(dout(DX),X,Y)</LHS><RHS>u22(din(der(Y)),X,Y,DX)</RHS></RULE><RULE><LHS>u22(dout(DY),X,Y,DX)</LHS><RHS>dout(plus(DX,DY))</RHS></RULE><RULE><LHS>din(der(times(X,Y)))</LHS><RHS>u31(din(der(X)),X,Y)</RHS></RULE><RULE><LHS>u31(dout(DX),X,Y)</LHS><RHS>u32(din(der(Y)),X,Y,DX)</RHS></RULE><RULE><LHS>u32(dout(DY),X,Y,DX)</LHS><RHS>dout(plus(times(X,DY),times(Y,DX)))</RHS></RULE><RULE><LHS>din(der(der(X)))</LHS><RHS>u41(din(der(X)),X)</RHS></RULE><RULE><LHS>u41(dout(DX),X)</LHS><RHS>u42(din(der(DX)),X,DX)</RHS></RULE><RULE><LHS>u42(dout(DDX),X,DX)</LHS><RHS>dout(DDX)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_din(der(plus(X,Y)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_din(der(times(X,Y)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>u21</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>u32</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>u22</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>der</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>u42</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>u31</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>din</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>u41</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>dout</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_din</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>plus</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>times</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></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_din(der(plus(X,Y)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>din(der(plus(X,Y)))</LHS><RHS>u21(din(der(X)),X,Y)</RHS></RULE><RULE><LHS>u21(dout(DX),X,Y)</LHS><RHS>u22(din(der(Y)),X,Y,DX)</RHS></RULE><RULE><LHS>u22(dout(DY),X,Y,DX)</LHS><RHS>dout(plus(DX,DY))</RHS></RULE><RULE><LHS>din(der(times(X,Y)))</LHS><RHS>u31(din(der(X)),X,Y)</RHS></RULE><RULE><LHS>u31(dout(DX),X,Y)</LHS><RHS>u32(din(der(Y)),X,Y,DX)</RHS></RULE><RULE><LHS>u32(dout(DY),X,Y,DX)</LHS><RHS>dout(plus(times(X,DY),times(Y,DX)))</RHS></RULE><RULE><LHS>din(der(der(X)))</LHS><RHS>u41(din(der(X)),X)</RHS></RULE><RULE><LHS>u41(dout(DX),X)</LHS><RHS>u42(din(der(DX)),X,DX)</RHS></RULE><RULE><LHS>u42(dout(DDX),X,DX)</LHS><RHS>dout(DDX)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_din(der(times(X,Y)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/></SCC></CCLIST></DAG></CRITERION><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>din(der(plus(X,Y)))</LHS><RHS>u21(din(der(X)),X,Y)</RHS></RULE><RULE><LHS>u21(dout(DX),X,Y)</LHS><RHS>u22(din(der(Y)),X,Y,DX)</RHS></RULE><RULE><LHS>u22(dout(DY),X,Y,DX)</LHS><RHS>dout(plus(DX,DY))</RHS></RULE><RULE><LHS>din(der(times(X,Y)))</LHS><RHS>u31(din(der(X)),X,Y)</RHS></RULE><RULE><LHS>u31(dout(DX),X,Y)</LHS><RHS>u32(din(der(Y)),X,Y,DX)</RHS></RULE><RULE><LHS>u32(dout(DY),X,Y,DX)</LHS><RHS>dout(plus(times(X,DY),times(Y,DX)))</RHS></RULE><RULE><LHS>din(der(der(X)))</LHS><RHS>u41(din(der(X)),X)</RHS></RULE><RULE><LHS>u41(dout(DX),X)</LHS><RHS>u42(din(der(DX)),X,DX)</RHS></RULE><RULE><LHS>u42(dout(DDX),X,DX)</LHS><RHS>dout(DDX)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_din(der(times(X,Y)))</LHS><RHS>Marked_din(der(X))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>u21</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>u32</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>u22</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>der</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>u42</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>u31</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>din</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>u41</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>dout</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>3</INT></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_din</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>plus</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>times</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></ORDERING></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
