details

property	value
status	complete
benchmark	AC21.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n005.star.cs.uiowa.edu
space	AProVE_AC_04

run statistics

property	value
solver	AProVE
configuration	certified
runtime (wallclock)	5.55525588989 seconds
cpu usage	15.086092228
max memory	1.241739264E9

stage attributes

key	value
certification-result	CERTIFIED
output-size	298804
starexec-result	YES
certification-time	0.1

output

/export/starexec/sandbox2/solver/bin/starexec_run_certified /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES <?xml version="1.0" encoding="UTF-8" standalone="no"?><?xml-stylesheet type="text/xsl" href="cpfHTML.xsl"?><certificationProblem xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" 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g><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>0</name></sharp><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>opp</name></sharp><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>0</name></sharp><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>opp</name></sharp><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>opp</name></sharp><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>opp</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>opp</name></sharp><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>opp</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>opp</name></sharp><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>minus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>opp</name><arg><var>y</var></arg></funapp></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>minus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>opp</name></sharp><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>0</name></sharp><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>0</name></sharp><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>minus</name></sharp><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>0</name></sharp><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>sign</name></sharp><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_sign</name></sharp><arg><var>x</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_sign</name></sharp><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_sign</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_sign</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_sign</name></sharp><arg><var>x</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_sign</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_sign</name></sharp><arg><var>x</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>abs</name></sharp><arg><var>x</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_abs</name></sharp><arg><var>x</var></arg><arg><var>x</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_abs</name></sharp><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_abs</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_abs</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_abs</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_abs</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_abs</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_abs</name></sharp><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>opp</name></sharp><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>min</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>minus</name><arg><funapp><name>abs</name><arg><var>y</var></arg></funapp></arg><arg><funapp><name>abs</name><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>min</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>minus</name></sharp><arg><funapp><name>abs</name><arg><var>y</var></arg></funapp></arg><arg><funapp><name>abs</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>min</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>abs</name></sharp><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>min</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>abs</name></sharp><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>min'</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>minus</name><arg><funapp><name>abs</name><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><funapp><name>abs</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>min'</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>minus</name></sharp><arg><funapp><name>abs</name><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><funapp><name>abs</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>min'</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>abs</name></sharp><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>min'</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>abs</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>min''</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>minus</name><arg><funapp><name>abs</name><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><funapp><name>abs</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>min''</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>minus</name></sharp><arg><funapp><name>abs</name><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><funapp><name>abs</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>min''</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>abs</name></sharp><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>min''</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>abs</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_min</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_min</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_min</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>0</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>0</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>0</name></sharp><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>0</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>0</name></sharp><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>S</name></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>0</name></sharp><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>0</name></sharp><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>minus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>minus</name></sharp><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>0</name></sharp><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><extensions><rules><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>0</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>minus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule></rules></extensions><acDPTerminationProof><acDepGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>S</name></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>minus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>y</var></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><funapp><name>times</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><acDPTerminationProof><acRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>2</degree></polynomial></type><interpret><sharp><name>times</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>times</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>j</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>minus</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>1</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>plus</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</i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pp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>1</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>minus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>opp</name><arg><var>y</var></arg></funapp></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>0</name><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>minus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>minus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>1</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><acDPTerminationProof><acRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>2</degree></polynomial></type><interpret><sharp><name>times</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>times</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>j</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>minus</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>plus</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>opp</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>1</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>minus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule></rules></dps><usableRules><rules><rule><lhs><funapp><name>minus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>opp</name><arg><var>y</var></arg></funapp></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>0</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>0</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>1</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>1</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>1</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>minus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>minus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>0</name><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><acDPTerminationProof><acRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>2</degree></polynomial></type><interpret><sharp><name>times</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>times</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>1</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>plus</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>j</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>minus</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>opp</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><usableRules><rules><rule><lhs><funapp><name>0</name><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>minus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>minus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>0</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>0</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>1</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>1</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>minus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>opp</name><arg><var>y</var></arg></funapp></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>1</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule></rules></usableRules><acDPTerminationProof><acRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>times</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>times</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>3</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>minus</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>plus</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>opp</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>1</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>j</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><var>y</var></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>S</name></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule></rules></dps><usableRules><rules><rule><lhs><funapp><name>minus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>opp</name><arg><var>y</var></arg></funapp></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>1</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>0</name><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp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p><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>minus</name></sharp><arg><funapp><name>abs</name><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><funapp><name>abs</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>minus</name></sharp><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>minus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>opp</name><arg><var>y</var></arg></funapp></arg><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>0</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>0</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>0</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>y</var></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><acDPTerminationProof><acMonoRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>plus</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>plus</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>j</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>1</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules/></dps><trs><rules><rule><lhs><funapp><name>opp</name><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>1</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>minus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>opp</name><arg><var>y</var></arg></funapp></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>minus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>sign</name><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>if_sign</name><arg><var>x</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_sign</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>if_sign</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_sign</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_sign</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_sign</name><arg><var>x</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_sign</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_sign</name><arg><var>x</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>abs</name><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>if_abs</name><arg><var>x</var></arg><arg><var>x</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>if_abs</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>if_abs</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>if_abs</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></lh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app></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule></rules></usableRules><acDPTerminationProof><acRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>plus</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>plus</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>j</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>1</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule></rules></dps><usableRules><rules><rule><lhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>1</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></a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/var></arg></funapp></lhs><rhs><funapp><name>if_sign</name><arg><var>x</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_sign</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_sign</name><arg><var>x</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>abs</name><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>if_abs</name><arg><var>x</var></arg><arg><var>x</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>if_abs</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>if_abs</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>if_abs</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>opp</name><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>min</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_min</name><arg><funapp><name>minus</name><arg><funapp><name>abs</name><arg><var>y</var></arg></funapp></arg><arg><funapp><name>abs</name><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>min'</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_min</name><arg><funapp><name>minus</name><arg><funapp><name>abs</name><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><funapp><name>abs</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>min''</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_min</name><arg><funapp><name>minus</name><arg><funapp><name>abs</name><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><funapp><name>abs</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_min</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></lhs><rhs><funapp><name>if_min</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_min</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></lhs><rhs><funapp><name>if_min</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_min</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></lhs><rhs><funapp><name>if_min</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_min</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>if_min</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>if_min</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>0</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>minus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule></rules></trs><usableRules><rules/></usableRules><acDPTerminationProof><acMonoRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>plus</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>times</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>if_abs</name></sharp><arity>3</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>1</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>j</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>if_abs</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_abs</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><trs><rules/></trs><usableRules><rules><rule><lhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><acDPTerminationProof><acMonoRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>plus</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>times</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>if_abs</name></sharp><arity>3</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>1</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>if_abs</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_abs</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><trs><rules/></trs><usableRules><rules><rule><lhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><acDPTerminationProof><acTrivialProc/></acDPTerminationProof></acMonoRedPairProc></acDPTerminationProof></acMonoRedPairProc></acDPTerminationProof></acMonoRedPairProc></acDPTerminationProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>if_abs</name></sharp><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>opp</name></sharp><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>opp</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>opp</name></sharp><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>opp</name></sharp><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>opp</name></sharp><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>opp</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>opp</name></sharp><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><acDPTerminationProof><acMonoRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>opp</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>j</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>1</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>opp</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>opp</name></sharp><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>opp</name></sharp><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>opp</name></sharp><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>opp</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>opp</name></sharp><arg><var>x</var></arg></funapp></rhs></rule></rules></dps><trs><rules><rule><lhs><funapp><name>0</name><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>0</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>1</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>1</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>1</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>minus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funa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><funapp><name>opp</name><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>1</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>minus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>opp</name><arg><var>y</var></arg></funapp></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>minus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>sign</name><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>if_sign</name><arg><var>x</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_sign</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>if_sign</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_sign</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_sign</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_sign</name><arg><var>x</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_sign</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_sign</name><arg><var>x</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>abs</name><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>if_abs</name><arg><var>x</var></arg><arg><var>x</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>if_abs</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>if_abs</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>if_abs</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>opp</name><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>min</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_min</name><arg><funapp><name>minus</name><arg><funapp><name>abs</name><arg><var>y</var></arg></funapp></arg><arg><funapp><name>abs</name><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>min'</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_min</name><arg><funapp><name>minus</name><arg><funapp><name>abs</name><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><funapp><name>abs</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>min''</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_min</name><arg><funapp><name>minus</name><arg><funapp><name>abs</name><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><funapp><name>abs</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_min</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></lhs><rhs><funapp><name>if_min</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_min</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></lhs><rhs><funapp><name>if_min</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_min</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></lhs><rhs><funapp><name>if_min</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_min</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>if_min</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>if_min</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>0</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>minus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule></rules></trs><usableRules><rules/></usableRules><acDPTerminationProof><acMonoRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>plus</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>times</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>if_sign</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>1</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>j</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>if_sign</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_sign</name></sharp><arg><var>x</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><trs><rules/></trs><usableRules><rules><rule><lhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><acDPTerminationProof><acMonoRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>plus</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>times</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>if_sign</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>1</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>if_sign</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_sign</name></sharp><arg><var>x</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><trs><rules/></trs><usableRules><rules><rule><lhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><acDPTerminationProof><acTrivialProc/></acDPTerminationProof></acMonoRedPairProc></acDPTerminationProof></acMonoRedPairProc></acDPTerminationProof></acMonoRedPairProc></acDPTerminationProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>min</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>minus</name><arg><funapp><name>abs</name><arg><var>y</var></arg></funapp></arg><arg><funapp><name>abs</name><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>min'</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>minus</name><arg><funapp><name>abs</name><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><funapp><name>abs</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>min''</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>minus</name><arg><funapp><name>abs</name><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><funapp><name>abs</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_min</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_min</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_min</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><acDPTerminationProof><acMonoRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>if_min</name></sharp><arity>4</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>4</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>1</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>j</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_min</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></rhs></rule></rules></dps><trs><rules><rule><lhs><funapp><name>0</name><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>0</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>1</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>1</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>j</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>opp</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>1</name><arg><funapp><name>opp</name><arg><var>x</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>minus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>opp</name><arg><var>y</var></arg></funapp></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>minus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>sign</name><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>if_sign</name><arg><var>x</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_sign</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>if_sign</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_sign</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_sign</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_sign</name><arg><var>x</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_sign</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_sign</name><arg><var>x</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>abs</name><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>if_abs</name><arg><var>x</var></arg><arg><var>x</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>if_abs</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>if_abs</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>if_abs</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><funapp><name>S</name></funapp></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>if_abs</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>opp</name><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>min</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_min</name><arg><funapp><name>minus</name><arg><funapp><name>abs</name><arg><var>y</var></arg></funapp></arg><arg><funapp><name>abs</name><arg><var>x</var></arg></funapp></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>min'</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_min</name><arg><funapp><name>minus</name><arg><funapp><name>abs</name><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><funapp><name>abs</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>min''</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>if_min</name><arg><funapp><name>minus</name><arg><funapp><name>abs</name><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><funapp><name>abs</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_min</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></lhs><rhs><funapp><name>if_min</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_min</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></lhs><rhs><funapp><name>if_min</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_min</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></lhs><rhs><funapp><name>if_min</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>if_min</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>S</name></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>if_min</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>if_min</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>0</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>1</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>1</name><arg><funapp><name>plus</name><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>x</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>S</name></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><funapp><name>minus</name><arg><funapp><name>0</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></rhs></rule></rules></trs><usableRules><rules/></usableRules><acDPTerminationProof><acMonoRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>plus</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>times</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>if_min</name></sharp><arity>4</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>4</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>1</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>0</integer></coefficient></polynomial></sum></polynomial></interpret><interpret><name>j</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_min</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><funapp><name>j</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><trs><rules/></trs><usableRules><rules><rule><lhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><acDPTerminationProof><acMonoRedPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>plus</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>times</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>if_min</name></sharp><arity>4</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>4</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>1</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>3</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>0</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>if_min</name></sharp><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><var>u</var></arg></funapp></lhs><rhs><funapp><sharp><name>if_min</name></sharp><arg><var>x</var></arg><arg><var>y</var></arg><arg><var>z</var></arg><arg><funapp><name>1</name><arg><funapp><name>S</name></funapp></arg></funapp></arg></funapp></rhs></rule></rules></dps><trs><rules/></trs><usableRules><rules><rule><lhs><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>y</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>plus</name><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>plus</name><arg><var>x</var></arg><arg><funapp><name>plus</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>times</name><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg><arg><var>z</var></arg></funapp></lhs><rhs><funapp><name>times</name><arg><var>x</var></arg><arg><funapp><name>times</name><arg><var>y</var></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><acDPTerminationProof><acTrivialProc/></acDPTerminationProof></acMonoRedPairProc></acDPTerminationProof></acMonoRedPairProc></acDPTerminationProof></acMonoRedPairProc></acDPTerminationProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>0</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>0</name></sharp><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>plus</name></sharp><arg><funapp><name>plus</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>j</name><arg><var>y</var></arg></funapp></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>0</name></sharp><arg><funapp><name>plus</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>0</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>0</name></sharp><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>1</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>0</name></sharp><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>times</name></sharp><arg><funapp><name>times</name><arg><funapp><name>j</name><arg><var>x</var></arg></funapp></arg><arg><var>y</var></arg></funapp></arg><arg><var>ext</var></arg></funapp></lhs><rhs><funapp><sharp><name>0</name></sharp><arg><funapp><name>times</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component></acDepGraphProc></acDPTerminationProof></acDependencyPairs></acTerminationProof></proof><origin><proofOrigin><tool><name>AProVE</name><version>AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty </version><strategy>Statistics for single proof: 100.00 % (28 real / 0 unknown / 0 assumptions / 28 total proof steps)</strategy><url>http://aprove.informatik.rwth-aachen.de</url></tool><toolUser><firstName>John</firstName><lastName>Doe</lastName></toolUser></proofOrigin><inputOrigin/></origin></certificationProblem> popout

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all output return to TRS Equat certi 47471