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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="1" unmarked="g"><NAME>Marked_g</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>cons</NAME></SYMBOL><SYMBOL arity="1" unmarked="f''"><NAME>Marked_f''</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>C</NAME></SYMBOL><SYMBOL arity="2" unmarked="f'"><NAME>Marked_f'</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>nil</NAME></SYMBOL><SYMBOL arity="3" unmarked=""><NAME>triple</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>g</NAME></SYMBOL><SYMBOL arity="2" unmarked="f"><NAME>Marked_f</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>f</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>f''</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>B</NAME></SYMBOL><SYMBOL arity="2" unmarked="foldf"><NAME>Marked_foldf</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>foldf</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>f'</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>A</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>x</VAR><VAR>y</VAR><VAR>z</VAR><VAR>a</VAR><VAR>b</VAR><VAR>c</VAR><VAR>t</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>g(A)</LHS><RHS>A</RHS></RULE><RULE><LHS>g(B)</LHS><RHS>A</RHS></RULE><RULE><LHS>g(B)</LHS><RHS>B</RHS></RULE><RULE><LHS>g(C)</LHS><RHS>A</RHS></RULE><RULE><LHS>g(C)</LHS><RHS>B</RHS></RULE><RULE><LHS>g(C)</LHS><RHS>C</RHS></RULE><RULE><LHS>foldf(x,nil)</LHS><RHS>x</RHS></RULE><RULE><LHS>foldf(x,cons(y,z))</LHS><RHS>f(foldf(x,z),y)</RHS></RULE><RULE><LHS>f(t,x)</LHS><RHS>f'(t,g(x))</RHS></RULE><RULE><LHS>f'(triple(a,b,c),C)</LHS><RHS>triple(a,b,cons(C,c))</RHS></RULE><RULE><LHS>f'(triple(a,b,c),B)</LHS><RHS>f(triple(a,b,c),A)</RHS></RULE><RULE><LHS>f'(triple(a,b,c),A)</LHS><RHS>f''(foldf(triple(cons(A,a),nil,c),b))</RHS></RULE><RULE><LHS>f''(triple(a,b,c))</LHS><RHS>foldf(triple(a,b,nil),c)</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>g(A)</LHS><RHS>A</RHS></RULE><RULE><LHS>g(B)</LHS><RHS>A</RHS></RULE><RULE><LHS>g(B)</LHS><RHS>B</RHS></RULE><RULE><LHS>g(C)</LHS><RHS>A</RHS></RULE><RULE><LHS>g(C)</LHS><RHS>B</RHS></RULE><RULE><LHS>g(C)</LHS><RHS>C</RHS></RULE><RULE><LHS>foldf(x,nil)</LHS><RHS>x</RHS></RULE><RULE><LHS>foldf(x,cons(y,z))</LHS><RHS>f(foldf(x,z),y)</RHS></RULE><RULE><LHS>f(t,x)</LHS><RHS>f'(t,g(x))</RHS></RULE><RULE><LHS>f'(triple(a,b,c),C)</LHS><RHS>triple(a,b,cons(C,c))</RHS></RULE><RULE><LHS>f'(triple(a,b,c),B)</LHS><RHS>f(triple(a,b,c),A)</RHS></RULE><RULE><LHS>f'(triple(a,b,c),A)</LHS><RHS>f''(foldf(triple(cons(A,a),nil,c),b))</RHS></RULE><RULE><LHS>f''(triple(a,b,c))</LHS><RHS>foldf(triple(a,b,nil),c)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_foldf(x,cons(y,z))</LHS><RHS>Marked_f(foldf(x,z),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_foldf(x,cons(y,z))</LHS><RHS>Marked_foldf(x,z)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_f(t,x)</LHS><RHS>Marked_f'(t,g(x))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_f(t,x)</LHS><RHS>Marked_g(x)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_f'(triple(a,b,c),B)</LHS><RHS>Marked_f(triple(a,b,c),A)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_f'(triple(a,b,c),A)</LHS><RHS>Marked_f''(foldf(triple(cons(A,a),nil,c),b))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_f'(triple(a,b,c),A)</LHS><RHS>Marked_foldf(triple(cons(A,a),nil,c),b)</RHS></DPRULE><DPRULE num="7"><LHS>Marked_f''(triple(a,b,c))</LHS><RHS>Marked_foldf(triple(a,b,nil),c)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/><NODE ref="1"/><NODE ref="2"/><NODE ref="4"/><NODE ref="5"/><NODE ref="6"/><NODE ref="7"/></SCC><NONSCC num="1"><NODE ref="3"/></NONSCC></CCLIST><EDGE end="1" start="0"/></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>g(A)</LHS><RHS>A</RHS></RULE><RULE><LHS>g(B)</LHS><RHS>A</RHS></RULE><RULE><LHS>g(B)</LHS><RHS>B</RHS></RULE><RULE><LHS>g(C)</LHS><RHS>A</RHS></RULE><RULE><LHS>g(C)</LHS><RHS>B</RHS></RULE><RULE><LHS>g(C)</LHS><RHS>C</RHS></RULE><RULE><LHS>foldf(x,nil)</LHS><RHS>x</RHS></RULE><RULE><LHS>foldf(x,cons(y,z))</LHS><RHS>f(foldf(x,z),y)</RHS></RULE><RULE><LHS>f(t,x)</LHS><RHS>f'(t,g(x))</RHS></RULE><RULE><LHS>f'(triple(a,b,c),C)</LHS><RHS>triple(a,b,cons(C,c))</RHS></RULE><RULE><LHS>f'(triple(a,b,c),B)</LHS><RHS>f(triple(a,b,c),A)</RHS></RULE><RULE><LHS>f'(triple(a,b,c),A)</LHS><RHS>f''(foldf(triple(cons(A,a),nil,c),b))</RHS></RULE><RULE><LHS>f''(triple(a,b,c))</LHS><RHS>foldf(triple(a,b,nil),c)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_foldf(x,cons(y,z))</LHS><RHS>Marked_f(foldf(x,z),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_foldf(x,cons(y,z))</LHS><RHS>Marked_foldf(x,z)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_f(t,x)</LHS><RHS>Marked_f'(t,g(x))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_f'(triple(a,b,c),B)</LHS><RHS>Marked_f(triple(a,b,c),A)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_f'(triple(a,b,c),A)</LHS><RHS>Marked_f''(foldf(triple(cons(A,a),nil,c),b))</RHS></DPRULE><DPRULE num="5"><LHS>Marked_f'(triple(a,b,c),A)</LHS><RHS>Marked_foldf(triple(cons(A,a),nil,c),b)</RHS></DPRULE><DPRULE num="6"><LHS>Marked_f''(triple(a,b,c))</LHS><RHS>Marked_foldf(triple(a,b,nil),c)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>A</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>f'</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><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></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>foldf</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>Marked_foldf</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><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></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>B</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>f''</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>f</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><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></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_f</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>g</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>triple</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="2"/></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'</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>C</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_f''</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>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></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_f'(triple(a,b,c),B)</LHS><RHS>Marked_f(triple(a,b,c),A)</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>g(A)</LHS><RHS>A</RHS></RULE><RULE><LHS>g(B)</LHS><RHS>A</RHS></RULE><RULE><LHS>g(B)</LHS><RHS>B</RHS></RULE><RULE><LHS>g(C)</LHS><RHS>A</RHS></RULE><RULE><LHS>g(C)</LHS><RHS>B</RHS></RULE><RULE><LHS>g(C)</LHS><RHS>C</RHS></RULE><RULE><LHS>foldf(x,nil)</LHS><RHS>x</RHS></RULE><RULE><LHS>foldf(x,cons(y,z))</LHS><RHS>f(foldf(x,z),y)</RHS></RULE><RULE><LHS>f(t,x)</LHS><RHS>f'(t,g(x))</RHS></RULE><RULE><LHS>f'(triple(a,b,c),C)</LHS><RHS>triple(a,b,cons(C,c))</RHS></RULE><RULE><LHS>f'(triple(a,b,c),B)</LHS><RHS>f(triple(a,b,c),A)</RHS></RULE><RULE><LHS>f'(triple(a,b,c),A)</LHS><RHS>f''(foldf(triple(cons(A,a),nil,c),b))</RHS></RULE><RULE><LHS>f''(triple(a,b,c))</LHS><RHS>foldf(triple(a,b,nil),c)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_foldf(x,cons(y,z))</LHS><RHS>Marked_f(foldf(x,z),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_foldf(x,cons(y,z))</LHS><RHS>Marked_foldf(x,z)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_f(t,x)</LHS><RHS>Marked_f'(t,g(x))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_f'(triple(a,b,c),A)</LHS><RHS>Marked_f''(foldf(triple(cons(A,a),nil,c),b))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_f'(triple(a,b,c),A)</LHS><RHS>Marked_foldf(triple(cons(A,a),nil,c),b)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_f''(triple(a,b,c))</LHS><RHS>Marked_foldf(triple(a,b,nil),c)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="0"/><NODE ref="1"/><NODE ref="2"/><NODE ref="3"/><NODE ref="4"/><NODE ref="5"/></SCC></CCLIST></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>g(A)</LHS><RHS>A</RHS></RULE><RULE><LHS>g(B)</LHS><RHS>A</RHS></RULE><RULE><LHS>g(B)</LHS><RHS>B</RHS></RULE><RULE><LHS>g(C)</LHS><RHS>A</RHS></RULE><RULE><LHS>g(C)</LHS><RHS>B</RHS></RULE><RULE><LHS>g(C)</LHS><RHS>C</RHS></RULE><RULE><LHS>foldf(x,nil)</LHS><RHS>x</RHS></RULE><RULE><LHS>foldf(x,cons(y,z))</LHS><RHS>f(foldf(x,z),y)</RHS></RULE><RULE><LHS>f(t,x)</LHS><RHS>f'(t,g(x))</RHS></RULE><RULE><LHS>f'(triple(a,b,c),C)</LHS><RHS>triple(a,b,cons(C,c))</RHS></RULE><RULE><LHS>f'(triple(a,b,c),B)</LHS><RHS>f(triple(a,b,c),A)</RHS></RULE><RULE><LHS>f'(triple(a,b,c),A)</LHS><RHS>f''(foldf(triple(cons(A,a),nil,c),b))</RHS></RULE><RULE><LHS>f''(triple(a,b,c))</LHS><RHS>foldf(triple(a,b,nil),c)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_foldf(x,cons(y,z))</LHS><RHS>Marked_f(foldf(x,z),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_foldf(x,cons(y,z))</LHS><RHS>Marked_foldf(x,z)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_f(t,x)</LHS><RHS>Marked_f'(t,g(x))</RHS></DPRULE><DPRULE num="3"><LHS>Marked_f'(triple(a,b,c),A)</LHS><RHS>Marked_f''(foldf(triple(cons(A,a),nil,c),b))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_f'(triple(a,b,c),A)</LHS><RHS>Marked_foldf(triple(cons(A,a),nil,c),b)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_f''(triple(a,b,c))</LHS><RHS>Marked_foldf(triple(a,b,nil),c)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>A</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>f'</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>foldf</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>Marked_foldf</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><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></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>B</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>f''</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>f</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>Marked_f</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>g</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>triple</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</INT></COEF><ARG degree="1" num="2"/></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'</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>C</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_f''</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>cons</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_foldf(x,cons(y,z))</LHS><RHS>Marked_f(foldf(x,z),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_foldf(x,cons(y,z))</LHS><RHS>Marked_foldf(x,z)</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>g(A)</LHS><RHS>A</RHS></RULE><RULE><LHS>g(B)</LHS><RHS>A</RHS></RULE><RULE><LHS>g(B)</LHS><RHS>B</RHS></RULE><RULE><LHS>g(C)</LHS><RHS>A</RHS></RULE><RULE><LHS>g(C)</LHS><RHS>B</RHS></RULE><RULE><LHS>g(C)</LHS><RHS>C</RHS></RULE><RULE><LHS>foldf(x,nil)</LHS><RHS>x</RHS></RULE><RULE><LHS>foldf(x,cons(y,z))</LHS><RHS>f(foldf(x,z),y)</RHS></RULE><RULE><LHS>f(t,x)</LHS><RHS>f'(t,g(x))</RHS></RULE><RULE><LHS>f'(triple(a,b,c),C)</LHS><RHS>triple(a,b,cons(C,c))</RHS></RULE><RULE><LHS>f'(triple(a,b,c),B)</LHS><RHS>f(triple(a,b,c),A)</RHS></RULE><RULE><LHS>f'(triple(a,b,c),A)</LHS><RHS>f''(foldf(triple(cons(A,a),nil,c),b))</RHS></RULE><RULE><LHS>f''(triple(a,b,c))</LHS><RHS>foldf(triple(a,b,nil),c)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_f(t,x)</LHS><RHS>Marked_f'(t,g(x))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_f'(triple(a,b,c),A)</LHS><RHS>Marked_f''(foldf(triple(cons(A,a),nil,c),b))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_f'(triple(a,b,c),A)</LHS><RHS>Marked_foldf(triple(cons(A,a),nil,c),b)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_f''(triple(a,b,c))</LHS><RHS>Marked_foldf(triple(a,b,nil),c)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="3"/></NONSCC><NONSCC num="1"><NODE ref="2"/></NONSCC><NONSCC num="2"><NODE ref="1"/></NONSCC><NONSCC num="3"><NODE ref="0"/></NONSCC></CCLIST><EDGE end="0" start="2"/><EDGE end="2" start="3"/><EDGE end="1" start="3"/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
