<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="2" unmarked="app"><NAME>Marked_app</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>add</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>plus</NAME></SYMBOL><SYMBOL arity="0" unmarked="add"><NAME>Marked_add</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>curry</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>app</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>f</VAR><VAR>x</VAR><VAR>y</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>app(app(plus,0),y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(app(plus,app(s,x)),y)</LHS><RHS>app(s,app(app(plus,x),y))</RHS></RULE><RULE><LHS>app(app(app(curry,f),x),y)</LHS><RHS>app(app(f,x),y)</RHS></RULE><RULE><LHS>add</LHS><RHS>app(curry,plus)</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(plus,0),y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(app(plus,app(s,x)),y)</LHS><RHS>app(s,app(app(plus,x),y))</RHS></RULE><RULE><LHS>app(app(app(curry,f),x),y)</LHS><RHS>app(app(f,x),y)</RHS></RULE><RULE><LHS>add</LHS><RHS>app(curry,plus)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(plus,app(s,x)),y)</LHS><RHS>Marked_app(s,app(app(plus,x),y))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(plus,app(s,x)),y)</LHS><RHS>Marked_app(app(plus,x),y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(app(plus,app(s,x)),y)</LHS><RHS>Marked_app(plus,x)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_app(app(app(curry,f),x),y)</LHS><RHS>Marked_app(app(f,x),y)</RHS></DPRULE><DPRULE num="4"><LHS>Marked_app(app(app(curry,f),x),y)</LHS><RHS>Marked_app(f,x)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_add</LHS><RHS>Marked_app(curry,plus)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="5"/></NONSCC><SCC num="1"><NODE ref="1"/><NODE ref="3"/><NODE ref="4"/></SCC><NONSCC num="2"><NODE ref="2"/></NONSCC><NONSCC num="3"><NODE ref="0"/></NONSCC></CCLIST><EDGE end="3" start="1"/><EDGE end="2" start="1"/></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(plus,0),y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(app(plus,app(s,x)),y)</LHS><RHS>app(s,app(app(plus,x),y))</RHS></RULE><RULE><LHS>app(app(app(curry,f),x),y)</LHS><RHS>app(app(f,x),y)</RHS></RULE><RULE><LHS>add</LHS><RHS>app(curry,plus)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(plus,app(s,x)),y)</LHS><RHS>Marked_app(app(plus,x),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(app(curry,f),x),y)</LHS><RHS>Marked_app(app(f,x),y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(app(app(curry,f),x),y)</LHS><RHS>Marked_app(f,x)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>app</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>curry</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>0</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>add</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>s</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>Marked_app</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></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(plus,app(s,x)),y)</LHS><RHS>Marked_app(app(plus,x),y)</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(plus,0),y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(app(plus,app(s,x)),y)</LHS><RHS>app(s,app(app(plus,x),y))</RHS></RULE><RULE><LHS>app(app(app(curry,f),x),y)</LHS><RHS>app(app(f,x),y)</RHS></RULE><RULE><LHS>add</LHS><RHS>app(curry,plus)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(app(curry,f),x),y)</LHS><RHS>Marked_app(app(f,x),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(app(curry,f),x),y)</LHS><RHS>Marked_app(f,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="ordering" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>app(app(plus,0),y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(app(plus,app(s,x)),y)</LHS><RHS>app(s,app(app(plus,x),y))</RHS></RULE><RULE><LHS>app(app(app(curry,f),x),y)</LHS><RHS>app(app(f,x),y)</RHS></RULE><RULE><LHS>add</LHS><RHS>app(curry,plus)</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(app(app(curry,f),x),y)</LHS><RHS>Marked_app(app(f,x),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_app(app(app(curry,f),x),y)</LHS><RHS>Marked_app(f,x)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>app</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><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>curry</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>0</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>add</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>s</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_app</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></ORDERING></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>
