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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="1" unmarked=""><NAME>bits</NAME></SYMBOL><SYMBOL arity="1" unmarked="half"><NAME>Marked_half</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>half</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="1" unmarked="bits"><NAME>Marked_bits</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>x</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>half(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>half(s(0))</LHS><RHS>0</RHS></RULE><RULE><LHS>half(s(s(x)))</LHS><RHS>s(half(x))</RHS></RULE><RULE><LHS>bits(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>bits(s(x))</LHS><RHS>s(bits(half(s(x))))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>half(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>half(s(0))</LHS><RHS>0</RHS></RULE><RULE><LHS>half(s(s(x)))</LHS><RHS>s(half(x))</RHS></RULE><RULE><LHS>bits(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>bits(s(x))</LHS><RHS>s(bits(half(s(x))))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_half(s(s(x)))</LHS><RHS>Marked_half(x)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_bits(s(x))</LHS><RHS>Marked_bits(half(s(x)))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_bits(s(x))</LHS><RHS>Marked_half(s(x))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="2"/></NONSCC><SCC num="1"><NODE ref="0"/></SCC><SCC num="2"><NODE ref="1"/></SCC></CCLIST><EDGE end="1" start="0"/><EDGE end="0" start="2"/></DAG></CRITERION><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>half(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>half(s(0))</LHS><RHS>0</RHS></RULE><RULE><LHS>half(s(s(x)))</LHS><RHS>s(half(x))</RHS></RULE><RULE><LHS>bits(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>bits(s(x))</LHS><RHS>s(bits(half(s(x))))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_bits(s(x))</LHS><RHS>Marked_bits(half(s(x)))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING numstrictcol="1" size="3" type="matrix"><POLYSYMB><SYMBOL><NAME>0</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_bits</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>1</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>s</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>1</INT><INT>0</INT><INT>1</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>1</INT><INT>1</INT></COLUMN><COLUMN><INT>1</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>half</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>1</INT><INT>1</INT><INT>1</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>bits</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>1</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>1</INT></COLUMN><COLUMN><INT>1</INT><INT>1</INT><INT>0</INT></COLUMN></MATRIX></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB></ORDERING></CRITERION></PROPERTY><PROPERTY criterion="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>half(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>half(s(0))</LHS><RHS>0</RHS></RULE><RULE><LHS>half(s(s(x)))</LHS><RHS>s(half(x))</RHS></RULE><RULE><LHS>bits(0)</LHS><RHS>0</RHS></RULE><RULE><LHS>bits(s(x))</LHS><RHS>s(bits(half(s(x))))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_half(s(s(x)))</LHS><RHS>Marked_half(x)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING numstrictcol="1" size="3" type="matrix"><POLYSYMB><SYMBOL><NAME>0</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>s</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>1</INT><INT>1</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>1</INT><INT>1</INT><INT>1</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>half</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>1</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>1</INT><INT>0</INT><INT>1</INT></COLUMN><COLUMN><INT>0</INT><INT>1</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_half</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>1</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>bits</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>1</INT><INT>1</INT><INT>1</INT></COLUMN></MATRIX></COEF><ARG degree="1" num="0"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><MATRIX><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN><COLUMN><INT>0</INT><INT>0</INT><INT>0</INT></COLUMN></MATRIX></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB></ORDERING></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROOF>
