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<!DOCTYPE PROOF SYSTEM "http://a3pat.ensiie.fr/pub/a3pat.dtd">
<PROOF><SIGNATURE><SYMBOLLIST><SYMBOL arity="3" unmarked="if_minsort"><NAME>Marked_if_minsort</NAME></SYMBOL><SYMBOL arity="2" unmarked="le"><NAME>Marked_le</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>nil</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>rm</NAME></SYMBOL><SYMBOL arity="2" unmarked="eq"><NAME>Marked_eq</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>false</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>minsort</NAME></SYMBOL><SYMBOL arity="3" unmarked="if_rm"><NAME>Marked_if_rm</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>le</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>min</NAME></SYMBOL><SYMBOL arity="2" unmarked="app"><NAME>Marked_app</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>eq</NAME></SYMBOL><SYMBOL arity="3" unmarked=""><NAME>if_minsort</NAME></SYMBOL><SYMBOL arity="2" unmarked="if_min"><NAME>Marked_if_min</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>app</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>if_min</NAME></SYMBOL><SYMBOL arity="2" unmarked="rm"><NAME>Marked_rm</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>0</NAME></SYMBOL><SYMBOL arity="3" unmarked=""><NAME>if_rm</NAME></SYMBOL><SYMBOL arity="1" unmarked="min"><NAME>Marked_min</NAME></SYMBOL><SYMBOL arity="1" unmarked=""><NAME>s</NAME></SYMBOL><SYMBOL arity="2" unmarked=""><NAME>add</NAME></SYMBOL><SYMBOL arity="2" unmarked="minsort"><NAME>Marked_minsort</NAME></SYMBOL><SYMBOL arity="0" unmarked=""><NAME>true</NAME></SYMBOL></SYMBOLLIST><VARLIST><VAR>n</VAR><VAR>m</VAR><VAR>y</VAR><VAR>x</VAR></VARLIST></SIGNATURE><PROPERTY criterion="dp" prop="sntrs"><SYSTEM><REWSYS><RULE><LHS>eq(0,0)</LHS><RHS>true</RHS></RULE><RULE><LHS>eq(0,s(x))</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),s(y))</LHS><RHS>eq(x,y)</RHS></RULE><RULE><LHS>le(0,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(x),s(y))</LHS><RHS>le(x,y)</RHS></RULE><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>min(add(n,nil))</LHS><RHS>n</RHS></RULE><RULE><LHS>min(add(n,add(m,x)))</LHS><RHS>if_min(le(n,m),add(n,add(m,x)))</RHS></RULE><RULE><LHS>if_min(true,add(n,add(m,x)))</LHS><RHS>min(add(n,x))</RHS></RULE><RULE><LHS>if_min(false,add(n,add(m,x)))</LHS><RHS>min(add(m,x))</RHS></RULE><RULE><LHS>rm(n,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rm(n,add(m,x))</LHS><RHS>if_rm(eq(n,m),n,add(m,x))</RHS></RULE><RULE><LHS>if_rm(true,n,add(m,x))</LHS><RHS>rm(n,x)</RHS></RULE><RULE><LHS>if_rm(false,n,add(m,x))</LHS><RHS>add(m,rm(n,x))</RHS></RULE><RULE><LHS>minsort(nil,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>minsort(add(n,x),y)</LHS><RHS>if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></RULE><RULE><LHS>if_minsort(true,add(n,x),y)</LHS><RHS>add(n,minsort(app(rm(n,x),y),nil))</RHS></RULE><RULE><LHS>if_minsort(false,add(n,x),y)</LHS><RHS>minsort(x,add(n,y))</RHS></RULE></REWSYS></SYSTEM><CRITERION/><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>eq(0,0)</LHS><RHS>true</RHS></RULE><RULE><LHS>eq(0,s(x))</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),s(y))</LHS><RHS>eq(x,y)</RHS></RULE><RULE><LHS>le(0,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(x),s(y))</LHS><RHS>le(x,y)</RHS></RULE><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>min(add(n,nil))</LHS><RHS>n</RHS></RULE><RULE><LHS>min(add(n,add(m,x)))</LHS><RHS>if_min(le(n,m),add(n,add(m,x)))</RHS></RULE><RULE><LHS>if_min(true,add(n,add(m,x)))</LHS><RHS>min(add(n,x))</RHS></RULE><RULE><LHS>if_min(false,add(n,add(m,x)))</LHS><RHS>min(add(m,x))</RHS></RULE><RULE><LHS>rm(n,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rm(n,add(m,x))</LHS><RHS>if_rm(eq(n,m),n,add(m,x))</RHS></RULE><RULE><LHS>if_rm(true,n,add(m,x))</LHS><RHS>rm(n,x)</RHS></RULE><RULE><LHS>if_rm(false,n,add(m,x))</LHS><RHS>add(m,rm(n,x))</RHS></RULE><RULE><LHS>minsort(nil,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>minsort(add(n,x),y)</LHS><RHS>if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></RULE><RULE><LHS>if_minsort(true,add(n,x),y)</LHS><RHS>add(n,minsort(app(rm(n,x),y),nil))</RHS></RULE><RULE><LHS>if_minsort(false,add(n,x),y)</LHS><RHS>minsort(x,add(n,y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_eq(s(x),s(y))</LHS><RHS>Marked_eq(x,y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_le(s(x),s(y))</LHS><RHS>Marked_le(x,y)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_app(add(n,x),y)</LHS><RHS>Marked_app(x,y)</RHS></DPRULE><DPRULE num="3"><LHS>Marked_min(add(n,add(m,x)))</LHS><RHS>Marked_if_min(le(n,m),add(n,add(m,x)))</RHS></DPRULE><DPRULE num="4"><LHS>Marked_min(add(n,add(m,x)))</LHS><RHS>Marked_le(n,m)</RHS></DPRULE><DPRULE num="5"><LHS>Marked_if_min(true,add(n,add(m,x)))</LHS><RHS>Marked_min(add(n,x))</RHS></DPRULE><DPRULE num="6"><LHS>Marked_if_min(false,add(n,add(m,x)))</LHS><RHS>Marked_min(add(m,x))</RHS></DPRULE><DPRULE num="7"><LHS>Marked_rm(n,add(m,x))</LHS><RHS>Marked_if_rm(eq(n,m),n,add(m,x))</RHS></DPRULE><DPRULE num="8"><LHS>Marked_rm(n,add(m,x))</LHS><RHS>Marked_eq(n,m)</RHS></DPRULE><DPRULE num="9"><LHS>Marked_if_rm(true,n,add(m,x))</LHS><RHS>Marked_rm(n,x)</RHS></DPRULE><DPRULE num="10"><LHS>Marked_if_rm(false,n,add(m,x))</LHS><RHS>Marked_rm(n,x)</RHS></DPRULE><DPRULE num="11"><LHS>Marked_minsort(add(n,x),y)</LHS><RHS>Marked_if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></DPRULE><DPRULE num="12"><LHS>Marked_minsort(add(n,x),y)</LHS><RHS>Marked_eq(n,min(add(n,x)))</RHS></DPRULE><DPRULE num="13"><LHS>Marked_minsort(add(n,x),y)</LHS><RHS>Marked_min(add(n,x))</RHS></DPRULE><DPRULE num="14"><LHS>Marked_if_minsort(true,add(n,x),y)</LHS><RHS>Marked_minsort(app(rm(n,x),y),nil)</RHS></DPRULE><DPRULE num="15"><LHS>Marked_if_minsort(true,add(n,x),y)</LHS><RHS>Marked_app(rm(n,x),y)</RHS></DPRULE><DPRULE num="16"><LHS>Marked_if_minsort(true,add(n,x),y)</LHS><RHS>Marked_rm(n,x)</RHS></DPRULE><DPRULE num="17"><LHS>Marked_if_minsort(false,add(n,x),y)</LHS><RHS>Marked_minsort(x,add(n,y))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><SCC num="0"><NODE ref="11"/><NODE ref="14"/><NODE ref="17"/></SCC><NONSCC num="1"><NODE ref="13"/></NONSCC><NONSCC num="2"><NODE ref="12"/></NONSCC><NONSCC num="3"><NODE ref="16"/></NONSCC><NONSCC num="4"><NODE ref="8"/></NONSCC><SCC num="5"><NODE ref="7"/><NODE ref="9"/><NODE ref="10"/></SCC><NONSCC num="6"><NODE ref="15"/></NONSCC><SCC num="7"><NODE ref="2"/></SCC><NONSCC num="8"><NODE ref="4"/></NONSCC><SCC num="9"><NODE ref="3"/><NODE ref="5"/><NODE ref="6"/></SCC><SCC num="10"><NODE ref="0"/></SCC><SCC num="11"><NODE ref="1"/></SCC></CCLIST><EDGE end="6" start="0"/><EDGE end="3" start="0"/><EDGE end="2" start="0"/><EDGE end="1" start="0"/><EDGE end="9" start="1"/><EDGE end="8" start="1"/><EDGE end="10" start="2"/><EDGE end="5" start="3"/><EDGE end="4" start="3"/><EDGE end="10" start="4"/><EDGE end="4" start="5"/><EDGE end="7" start="6"/><EDGE end="11" start="8"/><EDGE end="8" start="9"/></DAG></CRITERION><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>eq(0,0)</LHS><RHS>true</RHS></RULE><RULE><LHS>eq(0,s(x))</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),s(y))</LHS><RHS>eq(x,y)</RHS></RULE><RULE><LHS>le(0,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(x),s(y))</LHS><RHS>le(x,y)</RHS></RULE><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>min(add(n,nil))</LHS><RHS>n</RHS></RULE><RULE><LHS>min(add(n,add(m,x)))</LHS><RHS>if_min(le(n,m),add(n,add(m,x)))</RHS></RULE><RULE><LHS>if_min(true,add(n,add(m,x)))</LHS><RHS>min(add(n,x))</RHS></RULE><RULE><LHS>if_min(false,add(n,add(m,x)))</LHS><RHS>min(add(m,x))</RHS></RULE><RULE><LHS>rm(n,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rm(n,add(m,x))</LHS><RHS>if_rm(eq(n,m),n,add(m,x))</RHS></RULE><RULE><LHS>if_rm(true,n,add(m,x))</LHS><RHS>rm(n,x)</RHS></RULE><RULE><LHS>if_rm(false,n,add(m,x))</LHS><RHS>add(m,rm(n,x))</RHS></RULE><RULE><LHS>minsort(nil,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>minsort(add(n,x),y)</LHS><RHS>if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></RULE><RULE><LHS>if_minsort(true,add(n,x),y)</LHS><RHS>add(n,minsort(app(rm(n,x),y),nil))</RHS></RULE><RULE><LHS>if_minsort(false,add(n,x),y)</LHS><RHS>minsort(x,add(n,y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_minsort(add(n,x),y)</LHS><RHS>Marked_if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_if_minsort(true,add(n,x),y)</LHS><RHS>Marked_minsort(app(rm(n,x),y),nil)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_if_minsort(false,add(n,x),y)</LHS><RHS>Marked_minsort(x,add(n,y))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>true</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_minsort</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>add</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><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>s</NAME></SYMBOL><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></POLYSYMB><POLYSYMB><SYMBOL><NAME>if_rm</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</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>0</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>if_min</NAME></SYMBOL><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></POLYSYMB><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>if_minsort</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</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>eq</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>min</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>le</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>minsort</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>false</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>rm</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>nil</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_if_minsort</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="1"/></MONOME></POLYNOMIAL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>2</INT></COEF><ARG degree="1" num="2"/></MONOME></POLYNOMIAL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></SUMPOLY></POLYNOMIAL></SUMPOLY></POLYNOMIAL></POLYSYMB></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_if_minsort(true,add(n,x),y)</LHS><RHS>Marked_minsort(app(rm(n,x),y),nil)</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>eq(0,0)</LHS><RHS>true</RHS></RULE><RULE><LHS>eq(0,s(x))</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),s(y))</LHS><RHS>eq(x,y)</RHS></RULE><RULE><LHS>le(0,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(x),s(y))</LHS><RHS>le(x,y)</RHS></RULE><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>min(add(n,nil))</LHS><RHS>n</RHS></RULE><RULE><LHS>min(add(n,add(m,x)))</LHS><RHS>if_min(le(n,m),add(n,add(m,x)))</RHS></RULE><RULE><LHS>if_min(true,add(n,add(m,x)))</LHS><RHS>min(add(n,x))</RHS></RULE><RULE><LHS>if_min(false,add(n,add(m,x)))</LHS><RHS>min(add(m,x))</RHS></RULE><RULE><LHS>rm(n,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rm(n,add(m,x))</LHS><RHS>if_rm(eq(n,m),n,add(m,x))</RHS></RULE><RULE><LHS>if_rm(true,n,add(m,x))</LHS><RHS>rm(n,x)</RHS></RULE><RULE><LHS>if_rm(false,n,add(m,x))</LHS><RHS>add(m,rm(n,x))</RHS></RULE><RULE><LHS>minsort(nil,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>minsort(add(n,x),y)</LHS><RHS>if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></RULE><RULE><LHS>if_minsort(true,add(n,x),y)</LHS><RHS>add(n,minsort(app(rm(n,x),y),nil))</RHS></RULE><RULE><LHS>if_minsort(false,add(n,x),y)</LHS><RHS>minsort(x,add(n,y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_minsort(add(n,x),y)</LHS><RHS>Marked_if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_if_minsort(false,add(n,x),y)</LHS><RHS>Marked_minsort(x,add(n,y))</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="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>eq(0,0)</LHS><RHS>true</RHS></RULE><RULE><LHS>eq(0,s(x))</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),s(y))</LHS><RHS>eq(x,y)</RHS></RULE><RULE><LHS>le(0,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(x),s(y))</LHS><RHS>le(x,y)</RHS></RULE><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>min(add(n,nil))</LHS><RHS>n</RHS></RULE><RULE><LHS>min(add(n,add(m,x)))</LHS><RHS>if_min(le(n,m),add(n,add(m,x)))</RHS></RULE><RULE><LHS>if_min(true,add(n,add(m,x)))</LHS><RHS>min(add(n,x))</RHS></RULE><RULE><LHS>if_min(false,add(n,add(m,x)))</LHS><RHS>min(add(m,x))</RHS></RULE><RULE><LHS>rm(n,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rm(n,add(m,x))</LHS><RHS>if_rm(eq(n,m),n,add(m,x))</RHS></RULE><RULE><LHS>if_rm(true,n,add(m,x))</LHS><RHS>rm(n,x)</RHS></RULE><RULE><LHS>if_rm(false,n,add(m,x))</LHS><RHS>add(m,rm(n,x))</RHS></RULE><RULE><LHS>minsort(nil,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>minsort(add(n,x),y)</LHS><RHS>if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></RULE><RULE><LHS>if_minsort(true,add(n,x),y)</LHS><RHS>add(n,minsort(app(rm(n,x),y),nil))</RHS></RULE><RULE><LHS>if_minsort(false,add(n,x),y)</LHS><RHS>minsort(x,add(n,y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_minsort(add(n,x),y)</LHS><RHS>Marked_if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_if_minsort(false,add(n,x),y)</LHS><RHS>Marked_minsort(x,add(n,y))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_if_minsort</NAME></SYMBOL><INT>2</INT></PROJ><PROJ><SYMBOL><NAME>Marked_minsort</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_if_minsort(false,add(n,x),y)</LHS><RHS>Marked_minsort(x,add(n,y))</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>eq(0,0)</LHS><RHS>true</RHS></RULE><RULE><LHS>eq(0,s(x))</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),s(y))</LHS><RHS>eq(x,y)</RHS></RULE><RULE><LHS>le(0,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(x),s(y))</LHS><RHS>le(x,y)</RHS></RULE><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>min(add(n,nil))</LHS><RHS>n</RHS></RULE><RULE><LHS>min(add(n,add(m,x)))</LHS><RHS>if_min(le(n,m),add(n,add(m,x)))</RHS></RULE><RULE><LHS>if_min(true,add(n,add(m,x)))</LHS><RHS>min(add(n,x))</RHS></RULE><RULE><LHS>if_min(false,add(n,add(m,x)))</LHS><RHS>min(add(m,x))</RHS></RULE><RULE><LHS>rm(n,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rm(n,add(m,x))</LHS><RHS>if_rm(eq(n,m),n,add(m,x))</RHS></RULE><RULE><LHS>if_rm(true,n,add(m,x))</LHS><RHS>rm(n,x)</RHS></RULE><RULE><LHS>if_rm(false,n,add(m,x))</LHS><RHS>add(m,rm(n,x))</RHS></RULE><RULE><LHS>minsort(nil,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>minsort(add(n,x),y)</LHS><RHS>if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></RULE><RULE><LHS>if_minsort(true,add(n,x),y)</LHS><RHS>add(n,minsort(app(rm(n,x),y),nil))</RHS></RULE><RULE><LHS>if_minsort(false,add(n,x),y)</LHS><RHS>minsort(x,add(n,y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_minsort(add(n,x),y)</LHS><RHS>Marked_if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="0"/></NONSCC></CCLIST></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>eq(0,0)</LHS><RHS>true</RHS></RULE><RULE><LHS>eq(0,s(x))</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),s(y))</LHS><RHS>eq(x,y)</RHS></RULE><RULE><LHS>le(0,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(x),s(y))</LHS><RHS>le(x,y)</RHS></RULE><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>min(add(n,nil))</LHS><RHS>n</RHS></RULE><RULE><LHS>min(add(n,add(m,x)))</LHS><RHS>if_min(le(n,m),add(n,add(m,x)))</RHS></RULE><RULE><LHS>if_min(true,add(n,add(m,x)))</LHS><RHS>min(add(n,x))</RHS></RULE><RULE><LHS>if_min(false,add(n,add(m,x)))</LHS><RHS>min(add(m,x))</RHS></RULE><RULE><LHS>rm(n,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rm(n,add(m,x))</LHS><RHS>if_rm(eq(n,m),n,add(m,x))</RHS></RULE><RULE><LHS>if_rm(true,n,add(m,x))</LHS><RHS>rm(n,x)</RHS></RULE><RULE><LHS>if_rm(false,n,add(m,x))</LHS><RHS>add(m,rm(n,x))</RHS></RULE><RULE><LHS>minsort(nil,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>minsort(add(n,x),y)</LHS><RHS>if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></RULE><RULE><LHS>if_minsort(true,add(n,x),y)</LHS><RHS>add(n,minsort(app(rm(n,x),y),nil))</RHS></RULE><RULE><LHS>if_minsort(false,add(n,x),y)</LHS><RHS>minsort(x,add(n,y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_app(add(n,x),y)</LHS><RHS>Marked_app(x,y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_app</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_app(add(n,x),y)</LHS><RHS>Marked_app(x,y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>eq(0,0)</LHS><RHS>true</RHS></RULE><RULE><LHS>eq(0,s(x))</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),s(y))</LHS><RHS>eq(x,y)</RHS></RULE><RULE><LHS>le(0,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(x),s(y))</LHS><RHS>le(x,y)</RHS></RULE><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>min(add(n,nil))</LHS><RHS>n</RHS></RULE><RULE><LHS>min(add(n,add(m,x)))</LHS><RHS>if_min(le(n,m),add(n,add(m,x)))</RHS></RULE><RULE><LHS>if_min(true,add(n,add(m,x)))</LHS><RHS>min(add(n,x))</RHS></RULE><RULE><LHS>if_min(false,add(n,add(m,x)))</LHS><RHS>min(add(m,x))</RHS></RULE><RULE><LHS>rm(n,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rm(n,add(m,x))</LHS><RHS>if_rm(eq(n,m),n,add(m,x))</RHS></RULE><RULE><LHS>if_rm(true,n,add(m,x))</LHS><RHS>rm(n,x)</RHS></RULE><RULE><LHS>if_rm(false,n,add(m,x))</LHS><RHS>add(m,rm(n,x))</RHS></RULE><RULE><LHS>minsort(nil,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>minsort(add(n,x),y)</LHS><RHS>if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></RULE><RULE><LHS>if_minsort(true,add(n,x),y)</LHS><RHS>add(n,minsort(app(rm(n,x),y),nil))</RHS></RULE><RULE><LHS>if_minsort(false,add(n,x),y)</LHS><RHS>minsort(x,add(n,y))</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>eq(0,0)</LHS><RHS>true</RHS></RULE><RULE><LHS>eq(0,s(x))</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),s(y))</LHS><RHS>eq(x,y)</RHS></RULE><RULE><LHS>le(0,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(x),s(y))</LHS><RHS>le(x,y)</RHS></RULE><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>min(add(n,nil))</LHS><RHS>n</RHS></RULE><RULE><LHS>min(add(n,add(m,x)))</LHS><RHS>if_min(le(n,m),add(n,add(m,x)))</RHS></RULE><RULE><LHS>if_min(true,add(n,add(m,x)))</LHS><RHS>min(add(n,x))</RHS></RULE><RULE><LHS>if_min(false,add(n,add(m,x)))</LHS><RHS>min(add(m,x))</RHS></RULE><RULE><LHS>rm(n,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rm(n,add(m,x))</LHS><RHS>if_rm(eq(n,m),n,add(m,x))</RHS></RULE><RULE><LHS>if_rm(true,n,add(m,x))</LHS><RHS>rm(n,x)</RHS></RULE><RULE><LHS>if_rm(false,n,add(m,x))</LHS><RHS>add(m,rm(n,x))</RHS></RULE><RULE><LHS>minsort(nil,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>minsort(add(n,x),y)</LHS><RHS>if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></RULE><RULE><LHS>if_minsort(true,add(n,x),y)</LHS><RHS>add(n,minsort(app(rm(n,x),y),nil))</RHS></RULE><RULE><LHS>if_minsort(false,add(n,x),y)</LHS><RHS>minsort(x,add(n,y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_rm(n,add(m,x))</LHS><RHS>Marked_if_rm(eq(n,m),n,add(m,x))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_if_rm(true,n,add(m,x))</LHS><RHS>Marked_rm(n,x)</RHS></DPRULE><DPRULE num="2"><LHS>Marked_if_rm(false,n,add(m,x))</LHS><RHS>Marked_rm(n,x)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_if_rm</NAME></SYMBOL><INT>3</INT></PROJ><PROJ><SYMBOL><NAME>Marked_rm</NAME></SYMBOL><INT>2</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_if_rm(false,n,add(m,x))</LHS><RHS>Marked_rm(n,x)</RHS></DPRULE><DPRULE num="1"><LHS>Marked_if_rm(true,n,add(m,x))</LHS><RHS>Marked_rm(n,x)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>eq(0,0)</LHS><RHS>true</RHS></RULE><RULE><LHS>eq(0,s(x))</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),s(y))</LHS><RHS>eq(x,y)</RHS></RULE><RULE><LHS>le(0,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(x),s(y))</LHS><RHS>le(x,y)</RHS></RULE><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>min(add(n,nil))</LHS><RHS>n</RHS></RULE><RULE><LHS>min(add(n,add(m,x)))</LHS><RHS>if_min(le(n,m),add(n,add(m,x)))</RHS></RULE><RULE><LHS>if_min(true,add(n,add(m,x)))</LHS><RHS>min(add(n,x))</RHS></RULE><RULE><LHS>if_min(false,add(n,add(m,x)))</LHS><RHS>min(add(m,x))</RHS></RULE><RULE><LHS>rm(n,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rm(n,add(m,x))</LHS><RHS>if_rm(eq(n,m),n,add(m,x))</RHS></RULE><RULE><LHS>if_rm(true,n,add(m,x))</LHS><RHS>rm(n,x)</RHS></RULE><RULE><LHS>if_rm(false,n,add(m,x))</LHS><RHS>add(m,rm(n,x))</RHS></RULE><RULE><LHS>minsort(nil,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>minsort(add(n,x),y)</LHS><RHS>if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></RULE><RULE><LHS>if_minsort(true,add(n,x),y)</LHS><RHS>add(n,minsort(app(rm(n,x),y),nil))</RHS></RULE><RULE><LHS>if_minsort(false,add(n,x),y)</LHS><RHS>minsort(x,add(n,y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_rm(n,add(m,x))</LHS><RHS>Marked_if_rm(eq(n,m),n,add(m,x))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="0"/></NONSCC></CCLIST></DAG></CRITERION></PROPERTY></PROPERTY><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>eq(0,0)</LHS><RHS>true</RHS></RULE><RULE><LHS>eq(0,s(x))</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),s(y))</LHS><RHS>eq(x,y)</RHS></RULE><RULE><LHS>le(0,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(x),s(y))</LHS><RHS>le(x,y)</RHS></RULE><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>min(add(n,nil))</LHS><RHS>n</RHS></RULE><RULE><LHS>min(add(n,add(m,x)))</LHS><RHS>if_min(le(n,m),add(n,add(m,x)))</RHS></RULE><RULE><LHS>if_min(true,add(n,add(m,x)))</LHS><RHS>min(add(n,x))</RHS></RULE><RULE><LHS>if_min(false,add(n,add(m,x)))</LHS><RHS>min(add(m,x))</RHS></RULE><RULE><LHS>rm(n,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rm(n,add(m,x))</LHS><RHS>if_rm(eq(n,m),n,add(m,x))</RHS></RULE><RULE><LHS>if_rm(true,n,add(m,x))</LHS><RHS>rm(n,x)</RHS></RULE><RULE><LHS>if_rm(false,n,add(m,x))</LHS><RHS>add(m,rm(n,x))</RHS></RULE><RULE><LHS>minsort(nil,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>minsort(add(n,x),y)</LHS><RHS>if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></RULE><RULE><LHS>if_minsort(true,add(n,x),y)</LHS><RHS>add(n,minsort(app(rm(n,x),y),nil))</RHS></RULE><RULE><LHS>if_minsort(false,add(n,x),y)</LHS><RHS>minsort(x,add(n,y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_eq(s(x),s(y))</LHS><RHS>Marked_eq(x,y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_eq</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_eq(s(x),s(y))</LHS><RHS>Marked_eq(x,y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>eq(0,0)</LHS><RHS>true</RHS></RULE><RULE><LHS>eq(0,s(x))</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),s(y))</LHS><RHS>eq(x,y)</RHS></RULE><RULE><LHS>le(0,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(x),s(y))</LHS><RHS>le(x,y)</RHS></RULE><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>min(add(n,nil))</LHS><RHS>n</RHS></RULE><RULE><LHS>min(add(n,add(m,x)))</LHS><RHS>if_min(le(n,m),add(n,add(m,x)))</RHS></RULE><RULE><LHS>if_min(true,add(n,add(m,x)))</LHS><RHS>min(add(n,x))</RHS></RULE><RULE><LHS>if_min(false,add(n,add(m,x)))</LHS><RHS>min(add(m,x))</RHS></RULE><RULE><LHS>rm(n,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rm(n,add(m,x))</LHS><RHS>if_rm(eq(n,m),n,add(m,x))</RHS></RULE><RULE><LHS>if_rm(true,n,add(m,x))</LHS><RHS>rm(n,x)</RHS></RULE><RULE><LHS>if_rm(false,n,add(m,x))</LHS><RHS>add(m,rm(n,x))</RHS></RULE><RULE><LHS>minsort(nil,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>minsort(add(n,x),y)</LHS><RHS>if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></RULE><RULE><LHS>if_minsort(true,add(n,x),y)</LHS><RHS>add(n,minsort(app(rm(n,x),y),nil))</RHS></RULE><RULE><LHS>if_minsort(false,add(n,x),y)</LHS><RHS>minsort(x,add(n,y))</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY><PROPERTY criterion="stronggraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>eq(0,0)</LHS><RHS>true</RHS></RULE><RULE><LHS>eq(0,s(x))</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),s(y))</LHS><RHS>eq(x,y)</RHS></RULE><RULE><LHS>le(0,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(x),s(y))</LHS><RHS>le(x,y)</RHS></RULE><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>min(add(n,nil))</LHS><RHS>n</RHS></RULE><RULE><LHS>min(add(n,add(m,x)))</LHS><RHS>if_min(le(n,m),add(n,add(m,x)))</RHS></RULE><RULE><LHS>if_min(true,add(n,add(m,x)))</LHS><RHS>min(add(n,x))</RHS></RULE><RULE><LHS>if_min(false,add(n,add(m,x)))</LHS><RHS>min(add(m,x))</RHS></RULE><RULE><LHS>rm(n,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rm(n,add(m,x))</LHS><RHS>if_rm(eq(n,m),n,add(m,x))</RHS></RULE><RULE><LHS>if_rm(true,n,add(m,x))</LHS><RHS>rm(n,x)</RHS></RULE><RULE><LHS>if_rm(false,n,add(m,x))</LHS><RHS>add(m,rm(n,x))</RHS></RULE><RULE><LHS>minsort(nil,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>minsort(add(n,x),y)</LHS><RHS>if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></RULE><RULE><LHS>if_minsort(true,add(n,x),y)</LHS><RHS>add(n,minsort(app(rm(n,x),y),nil))</RHS></RULE><RULE><LHS>if_minsort(false,add(n,x),y)</LHS><RHS>minsort(x,add(n,y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_min(add(n,add(m,x)))</LHS><RHS>Marked_if_min(le(n,m),add(n,add(m,x)))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_if_min(true,add(n,add(m,x)))</LHS><RHS>Marked_min(add(n,x))</RHS></DPRULE><DPRULE num="2"><LHS>Marked_if_min(false,add(n,add(m,x)))</LHS><RHS>Marked_min(add(m,x))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><ORDERING type="poly"><POLYSYMB><SYMBOL><NAME>true</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>add</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>s</NAME></SYMBOL><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></POLYSYMB><POLYSYMB><SYMBOL><NAME>Marked_min</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>if_rm</NAME></SYMBOL><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></POLYSYMB><POLYSYMB><SYMBOL><NAME>0</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>if_min</NAME></SYMBOL><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></POLYSYMB><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>Marked_if_min</NAME></SYMBOL><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></POLYSYMB><POLYSYMB><SYMBOL><NAME>if_minsort</NAME></SYMBOL><POLYNOMIAL><SUMPOLY><POLYNOMIAL><MONOME><COEF><INT>1</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>eq</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>min</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>le</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>minsort</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>false</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB><POLYSYMB><SYMBOL><NAME>rm</NAME></SYMBOL><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></POLYSYMB><POLYSYMB><SYMBOL><NAME>nil</NAME></SYMBOL><POLYNOMIAL><MONOME><COEF><INT>0</INT></COEF></MONOME></POLYNOMIAL></POLYSYMB></ORDERING><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_if_min(true,add(n,add(m,x)))</LHS><RHS>Marked_min(add(n,x))</RHS></DPRULE><DPRULE num="1"><LHS>Marked_if_min(false,add(n,add(m,x)))</LHS><RHS>Marked_min(add(m,x))</RHS></DPRULE></DPLIST></STRICTPAIRS></CRITERION><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>eq(0,0)</LHS><RHS>true</RHS></RULE><RULE><LHS>eq(0,s(x))</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),s(y))</LHS><RHS>eq(x,y)</RHS></RULE><RULE><LHS>le(0,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(x),s(y))</LHS><RHS>le(x,y)</RHS></RULE><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>min(add(n,nil))</LHS><RHS>n</RHS></RULE><RULE><LHS>min(add(n,add(m,x)))</LHS><RHS>if_min(le(n,m),add(n,add(m,x)))</RHS></RULE><RULE><LHS>if_min(true,add(n,add(m,x)))</LHS><RHS>min(add(n,x))</RHS></RULE><RULE><LHS>if_min(false,add(n,add(m,x)))</LHS><RHS>min(add(m,x))</RHS></RULE><RULE><LHS>rm(n,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rm(n,add(m,x))</LHS><RHS>if_rm(eq(n,m),n,add(m,x))</RHS></RULE><RULE><LHS>if_rm(true,n,add(m,x))</LHS><RHS>rm(n,x)</RHS></RULE><RULE><LHS>if_rm(false,n,add(m,x))</LHS><RHS>add(m,rm(n,x))</RHS></RULE><RULE><LHS>minsort(nil,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>minsort(add(n,x),y)</LHS><RHS>if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></RULE><RULE><LHS>if_minsort(true,add(n,x),y)</LHS><RHS>add(n,minsort(app(rm(n,x),y),nil))</RHS></RULE><RULE><LHS>if_minsort(false,add(n,x),y)</LHS><RHS>minsort(x,add(n,y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_min(add(n,add(m,x)))</LHS><RHS>Marked_if_min(le(n,m),add(n,add(m,x)))</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST><NONSCC num="0"><NODE ref="0"/></NONSCC></CCLIST></DAG></CRITERION></PROPERTY></PROPERTY><PROPERTY criterion="sterm" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>eq(0,0)</LHS><RHS>true</RHS></RULE><RULE><LHS>eq(0,s(x))</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),s(y))</LHS><RHS>eq(x,y)</RHS></RULE><RULE><LHS>le(0,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(x),s(y))</LHS><RHS>le(x,y)</RHS></RULE><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>min(add(n,nil))</LHS><RHS>n</RHS></RULE><RULE><LHS>min(add(n,add(m,x)))</LHS><RHS>if_min(le(n,m),add(n,add(m,x)))</RHS></RULE><RULE><LHS>if_min(true,add(n,add(m,x)))</LHS><RHS>min(add(n,x))</RHS></RULE><RULE><LHS>if_min(false,add(n,add(m,x)))</LHS><RHS>min(add(m,x))</RHS></RULE><RULE><LHS>rm(n,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rm(n,add(m,x))</LHS><RHS>if_rm(eq(n,m),n,add(m,x))</RHS></RULE><RULE><LHS>if_rm(true,n,add(m,x))</LHS><RHS>rm(n,x)</RHS></RULE><RULE><LHS>if_rm(false,n,add(m,x))</LHS><RHS>add(m,rm(n,x))</RHS></RULE><RULE><LHS>minsort(nil,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>minsort(add(n,x),y)</LHS><RHS>if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></RULE><RULE><LHS>if_minsort(true,add(n,x),y)</LHS><RHS>add(n,minsort(app(rm(n,x),y),nil))</RHS></RULE><RULE><LHS>if_minsort(false,add(n,x),y)</LHS><RHS>minsort(x,add(n,y))</RHS></RULE></REWSYS><DPLIST><DPRULE num="0"><LHS>Marked_le(s(x),s(y))</LHS><RHS>Marked_le(x,y)</RHS></DPRULE></DPLIST></DPSYS></SYSTEM><PROJECTION><PROJ><SYMBOL><NAME>Marked_le</NAME></SYMBOL><INT>1</INT></PROJ></PROJECTION><STRICTPAIRS><DPLIST><DPRULE num="0"><LHS>Marked_le(s(x),s(y))</LHS><RHS>Marked_le(x,y)</RHS></DPRULE></DPLIST></STRICTPAIRS><PROPERTY criterion="weakgraph" prop="sndp"><SYSTEM><DPSYS><REWSYS><RULE><LHS>eq(0,0)</LHS><RHS>true</RHS></RULE><RULE><LHS>eq(0,s(x))</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>eq(s(x),s(y))</LHS><RHS>eq(x,y)</RHS></RULE><RULE><LHS>le(0,y)</LHS><RHS>true</RHS></RULE><RULE><LHS>le(s(x),0)</LHS><RHS>false</RHS></RULE><RULE><LHS>le(s(x),s(y))</LHS><RHS>le(x,y)</RHS></RULE><RULE><LHS>app(nil,y)</LHS><RHS>y</RHS></RULE><RULE><LHS>app(add(n,x),y)</LHS><RHS>add(n,app(x,y))</RHS></RULE><RULE><LHS>min(add(n,nil))</LHS><RHS>n</RHS></RULE><RULE><LHS>min(add(n,add(m,x)))</LHS><RHS>if_min(le(n,m),add(n,add(m,x)))</RHS></RULE><RULE><LHS>if_min(true,add(n,add(m,x)))</LHS><RHS>min(add(n,x))</RHS></RULE><RULE><LHS>if_min(false,add(n,add(m,x)))</LHS><RHS>min(add(m,x))</RHS></RULE><RULE><LHS>rm(n,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>rm(n,add(m,x))</LHS><RHS>if_rm(eq(n,m),n,add(m,x))</RHS></RULE><RULE><LHS>if_rm(true,n,add(m,x))</LHS><RHS>rm(n,x)</RHS></RULE><RULE><LHS>if_rm(false,n,add(m,x))</LHS><RHS>add(m,rm(n,x))</RHS></RULE><RULE><LHS>minsort(nil,nil)</LHS><RHS>nil</RHS></RULE><RULE><LHS>minsort(add(n,x),y)</LHS><RHS>if_minsort(eq(n,min(add(n,x))),add(n,x),y)</RHS></RULE><RULE><LHS>if_minsort(true,add(n,x),y)</LHS><RHS>add(n,minsort(app(rm(n,x),y),nil))</RHS></RULE><RULE><LHS>if_minsort(false,add(n,x),y)</LHS><RHS>minsort(x,add(n,y))</RHS></RULE></REWSYS><DPLIST/></DPSYS></SYSTEM><CRITERION><DAG approx="simpl"><CCLIST/></DAG></CRITERION></PROPERTY></PROPERTY></PROPERTY></PROPERTY></PROOF>