details

property	value
status	complete
benchmark	Ex4_DLMMU04_GM.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	Transformed_CSR_innermost_04

run statistics

property	value
solver	AProVE
configuration	certified
runtime (wallclock)	11.4819 seconds
cpu usage	40.6163
user time	39.6814
system time	0.934928
max virtual memory	3.7765908E7
max residence set size	1834888.0

stage attributes

key	value
certification-result	CERTIFIED
output-size	173610
starexec-result	YES
certification-time	0.4

output

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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app></rhs></rule><rule><lhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>T</var></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>T</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X2</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatIList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatList</name><arg><var>L</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNat</name></sharp><arg><funapp><name>s</name><arg><var>N</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNat</name></sharp><arg><funapp><name>length</name><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>take</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>take</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>take</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNat</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>length</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__length</name></sharp><arg><funapp><name>mark</name><arg><var>X</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__length</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__uLength</name></sharp><arg><funapp><name>a__and</name><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatList</name><arg><var>L</var></arg></funapp></arg></funapp></arg><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__uLength</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>L</var></arg></funapp></lhs><rhs><funapp><sharp><name>a__length</name></sharp><arg><funapp><name>mark</name><arg><var>L</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__length</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatList</name><arg><var>L</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__length</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__length</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__uLength</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>L</var></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>length</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>take</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__take</name></sharp><arg><funapp><name>mark</name><arg><var>X1</var></arg></funapp></arg><arg><funapp><name>mark</name><arg><var>X2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__take</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><var>IL</var></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__take</name></sharp><arg><funapp><name>s</name><arg><var>M</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__uTake2</name></sharp><arg><funapp><name>a__and</name><arg><funapp><name>a__isNat</name><arg><var>M</var></arg></funapp></arg><arg><funapp><name>a__and</name><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><var>M</var></arg><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__uTake2</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>M</var></arg><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>take</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>take</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X2</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>uTake1</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>uTake2</name><arg><var>X1</var></arg><arg><var>X2</var></arg><arg><var>X3</var></arg><arg><var>X4</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__uTake2</name></sharp><arg><funapp><name>mark</name><arg><var>X1</var></arg></funapp></arg><arg><var>X2</var></arg><arg><var>X3</var></arg><arg><var>X4</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>uTake2</name><arg><var>X1</var></arg><arg><var>X2</var></arg><arg><var>X3</var></arg><arg><var>X4</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>uLength</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__uLength</name></sharp><arg><funapp><name>mark</name><arg><var>X1</var></arg></funapp></arg><arg><var>X2</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>uLength</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>s</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>cons</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__take</name></sharp><arg><funapp><name>s</name><arg><var>M</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>M</var></arg></funapp></arg><arg><funapp><name>a__and</name><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__take</name></sharp><arg><funapp><name>s</name><arg><var>M</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>M</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__take</name></sharp><arg><funapp><name>s</name><arg><var>M</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__take</name></sharp><arg><funapp><name>s</name><arg><var>M</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__take</name></sharp><arg><funapp><name>s</name><arg><var>M</var></arg></funapp></arg><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><redPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>a__and</name></sharp><arity>2</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></interpret><interpret><sharp><name>a__length</name></sharp><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret><interpret><sharp><name>a__take</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>a__uLength</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><sharp><name>a__uTake2</name></sharp><arity>4</arity><polynomial><variable>3</variable></polynomial></interpret><interpret><name>mark</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>and</name><arity>2</arity><polynomial><sum><polynomial><variable>1</variable></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__and</name><arity>2</arity><polynomial><sum><polynomial><variable>1</variable></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>tt</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>isNatIList</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__isNatIList</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__isNatList</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>cons</name><arity>2</arity><polynomial><sum><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>a__isNat</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>take</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>isNatList</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>isNat</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>length</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret><interpret><name>a__length</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret><interpret><name>zeros</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__zeros</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__take</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>uTake1</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>a__uTake1</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>uTake2</name><arity>4</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial><polynomial><variable>3</variable></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>4</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__uTake2</name><arity>4</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial><polynomial><variable>3</variable></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>4</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>uLength</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>a__uLength</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><sharp><name>mark</name></sharp><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><sharp><name>a__isNatIList</name></sharp><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><sharp><name>a__isNatList</name></sharp><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><sharp><name>a__isNat</name></sharp><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>mark</name><arg><var>X1</var></arg></funapp></arg><arg><funapp><name>mark</name><arg><var>X2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>T</var></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>T</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X2</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatIList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatList</name><arg><var>L</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNat</name></sharp><arg><funapp><name>s</name><arg><var>N</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNat</name></sharp><arg><funapp><name>length</name><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>take</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>take</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>take</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNat</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>length</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__length</name></sharp><arg><funapp><name>mark</name><arg><var>X</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__length</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__uLength</name></sharp><arg><funapp><name>a__and</name><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatList</name><arg><var>L</var></arg></funapp></arg></funapp></arg><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__uLength</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>L</var></arg></funapp></lhs><rhs><funapp><sharp><name>a__length</name></sharp><arg><funapp><name>mark</name><arg><var>L</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__uTake2</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>M</var></arg><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>uTake1</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>uLength</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__uLength</name></sharp><arg><funapp><name>mark</name><arg><var>X1</var></arg></funapp></arg><arg><var>X2</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>s</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>cons</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X1</var></arg></funapp></rhs></rule></rules></dps><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>a__uTake2</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>M</var></arg><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>N</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>T</var></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>T</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>mark</name><arg><var>X1</var></arg></funapp></arg><arg><funapp><name>mark</name><arg><var>X2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X2</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatIList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatList</name><arg><var>L</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNat</name></sharp><arg><funapp><name>s</name><arg><var>N</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNat</name></sharp><arg><funapp><name>length</name><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>take</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>take</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>take</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNat</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>uTake1</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>s</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>cons</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X1</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><redPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>a__and</name></sharp><arity>2</arity><polynomial><variable>2</variable></polynomial></interpret><interpret><name>mark</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>and</name><arity>2</arity><polynomial><sum><polynomial><variable>1</variable></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__and</name><arity>2</arity><polynomial><sum><polynomial><variable>1</variable></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>tt</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>isNatIList</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__isNatIList</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__isNatList</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>cons</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__isNat</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>take</name><arity>2</arity><polynomial><variable>2</variable></polynomial></interpret><interpret><name>isNatList</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>isNat</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>length</name><arity>1</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret><interpret><name>a__length</name><arity>1</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret><interpret><name>zeros</name><arity>0</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret><interpret><name>a__zeros</name><arity>0</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret><interpret><name>a__take</name><arity>2</arity><polynomial><variable>2</variable></polynomial></interpret><interpret><name>uTake1</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>a__uTake1</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>uTake2</name><arity>4</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__uTake2</name><arity>4</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>uLength</name><arity>2</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret><interpret><name>a__uLength</name><arity>2</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><sharp><name>mark</name></sharp><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><sharp><name>a__isNatIList</name></sharp><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><sharp><name>a__isNatList</name></sharp><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><sharp><name>a__isNat</name></sharp><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>T</var></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>T</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>mark</name><arg><var>X1</var></arg></funapp></arg><arg><funapp><name>mark</name><arg><var>X2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X2</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatIList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatList</name><arg><var>L</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNat</name></sharp><arg><funapp><name>s</name><arg><var>N</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNat</name></sharp><arg><funapp><name>length</name><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>take</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>take</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>take</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNat</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>uTake1</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>s</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule></rules></dps><dpProof><redPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>a__and</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>mark</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>and</name><arity>2</arity><polynomial><sum><polynomial><variable>1</variable></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__and</name><arity>2</arity><polynomial><sum><polynomial><variable>1</variable></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>tt</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>isNatIList</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__isNatIList</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__isNatList</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>cons</name><arity>2</arity><polynomial><coefficient><integer>-2</integer></coefficient></polynomial></interpret><interpret><name>a__isNat</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>take</name><arity>2</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret><interpret><name>isNatList</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>isNat</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>length</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__length</name><arity>1</arity><polynomial><coefficient><integer>-2</integer></coefficient></polynomial></interpret><interpret><name>zeros</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__zeros</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__take</name><arity>2</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret><interpret><name>uTake1</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__uTake1</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>uTake2</name><arity>4</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__uTake2</name><arity>4</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>uLength</name><arity>2</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__uLength</name><arity>2</arity><polynomial><coefficient><integer>-2</integer></coefficient></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><sharp><name>mark</name></sharp><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret><interpret><sharp><name>a__isNatIList</name></sharp><arity>1</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret><interpret><sharp><name>a__isNatList</name></sharp><arity>1</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret><interpret><sharp><name>a__isNat</name></sharp><arity>1</arity><polynomial><coefficient><integer>2</integer></coefficient></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>T</var></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>T</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>mark</name><arg><var>X1</var></arg></funapp></arg><arg><funapp><name>mark</name><arg><var>X2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X2</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatIList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatList</name><arg><var>L</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNat</name></sharp><arg><funapp><name>s</name><arg><var>N</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNat</name></sharp><arg><funapp><name>length</name><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>take</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>take</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>take</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNat</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>s</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule></rules></dps><dpProof><redPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>a__and</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>mark</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>and</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>a__and</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>tt</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>isNatIList</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>a__isNatIList</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>a__isNatList</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>cons</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>a__isNat</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>take</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><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>isNatList</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>isNat</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>length</name><arity>1</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret><interpret><name>a__length</name><arity>1</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret><interpret><name>zeros</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__zeros</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__take</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><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>uTake1</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__uTake1</name><arity>1</arity><polynomial><coefficient><integer>-2</integer></coefficient></polynomial></interpret><interpret><name>uTake2</name><arity>4</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</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><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>4</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__uTake2</name><arity>4</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</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><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>4</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>uLength</name><arity>2</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></interpret><interpret><name>a__uLength</name><arity>2</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><sharp><name>mark</name></sharp><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>a__isNatIList</name></sharp><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>a__isNatList</name></sharp><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>a__isNat</name></sharp><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</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>a__and</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>T</var></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>T</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>mark</name><arg><var>X1</var></arg></funapp></arg><arg><funapp><name>mark</name><arg><var>X2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X2</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatIList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatList</name><arg><var>L</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNat</name></sharp><arg><funapp><name>s</name><arg><var>N</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNat</name></sharp><arg><funapp><name>length</name><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNat</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>s</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule></rules></dps><dpProof><redPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>a__and</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>mark</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>and</name><arity>2</arity><polynomial><sum><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>a__and</name><arity>2</arity><polynomial><sum><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>tt</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>isNatIList</name><arity>1</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret><interpret><name>a__isNatIList</name><arity>1</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret><interpret><name>a__isNatList</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>cons</name><arity>2</arity><polynomial><sum><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>a__isNat</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>take</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><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>isNatList</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>isNat</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>length</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret><interpret><name>a__length</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret><interpret><name>zeros</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__zeros</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__take</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><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>uTake1</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__uTake1</name><arity>1</arity><polynomial><coefficient><integer>-2</integer></coefficient></polynomial></interpret><interpret><name>uTake2</name><arity>4</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</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><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>4</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__uTake2</name><arity>4</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</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><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>4</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>uLength</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>a__uLength</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><sharp><name>mark</name></sharp><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret><interpret><sharp><name>a__isNatIList</name></sharp><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>a__isNatList</name></sharp><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret><interpret><sharp><name>a__isNat</name></sharp><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>T</var></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>T</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>mark</name><arg><var>X1</var></arg></funapp></arg><arg><funapp><name>mark</name><arg><var>X2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X2</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatIList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatList</name><arg><var>L</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNat</name></sharp><arg><funapp><name>s</name><arg><var>N</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNat</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>s</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule></rules></dps><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>mark</name><arg><var>X1</var></arg></funapp></arg><arg><funapp><name>mark</name><arg><var>X2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>T</var></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>T</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X2</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatIList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatList</name><arg><var>L</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>s</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><redPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>a__and</name></sharp><arity>2</arity><polynomial><variable>2</variable></polynomial></interpret><interpret><name>mark</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>and</name><arity>2</arity><polynomial><sum><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>a__and</name><arity>2</arity><polynomial><sum><polynomial><variable>1</variable></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>tt</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>isNatIList</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__isNatIList</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__isNatList</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>cons</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>a__isNat</name><arity>1</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret><interpret><name>take</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><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>isNatList</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>isNat</name><arity>1</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>length</name><arity>1</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret><interpret><name>a__length</name><arity>1</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret><interpret><name>zeros</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__zeros</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__take</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><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>uTake1</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__uTake1</name><arity>1</arity><polynomial><coefficient><integer>-2</integer></coefficient></polynomial></interpret><interpret><name>uTake2</name><arity>4</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</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><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>4</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__uTake2</name><arity>4</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</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><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>4</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>uLength</name><arity>2</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></interpret><interpret><name>a__uLength</name><arity>2</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><sharp><name>mark</name></sharp><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><sharp><name>a__isNatIList</name></sharp><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>a__isNatList</name></sharp><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>mark</name><arg><var>X1</var></arg></funapp></arg><arg><funapp><name>mark</name><arg><var>X2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>T</var></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>T</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X2</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatIList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatList</name><arg><var>L</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>s</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule></rules></dps><dpProof><redPairProc><orderingConstraintProof><redPair><interpretation><type><matrixInterpretation><domain><arctic><domain><naturals/></domain></arctic></domain><dimension>1</dimension><strictDimension>1</strictDimension></matrixInterpretation></type><interpret><sharp><name>mark</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>and</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>a__and</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>mark</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>tt</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>isNatIList</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>a__isNatIList</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>a__isNatList</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>cons</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__isNat</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__isNatList</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__isNatIList</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>isNatList</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__and</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>take</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>isNat</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>length</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__length</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>zeros</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__zeros</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__take</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>uTake1</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__uTake1</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>uTake2</name><arity>4</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>4</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__uTake2</name><arity>4</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>4</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>uLength</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__uLength</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>mark</name><arg><var>X1</var></arg></funapp></arg><arg><funapp><name>mark</name><arg><var>X2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>T</var></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>T</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X2</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatIList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatList</name><arg><var>L</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>L</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>IL</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>s</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule></rules></dps><dpProof><redPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>a__and</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>mark</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret><interpret><name>and</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>a__and</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>tt</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>isNatIList</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>a__isNatIList</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>a__isNatList</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>cons</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>a__isNat</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>take</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>isNatList</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>isNat</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>length</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>a__length</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>zeros</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__zeros</name><arity>0</arity><polynomial><coefficient><integer>1</integer></coefficient></polynomial></interpret><interpret><name>a__take</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>uTake1</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__uTake1</name><arity>1</arity><polynomial><coefficient><integer>-2</integer></coefficient></polynomial></interpret><interpret><name>uTake2</name><arity>4</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>4</variable></polynomial></sum></polynomial></interpret><interpret><name>a__uTake2</name><arity>4</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>4</variable></polynomial></sum></polynomial></interpret><interpret><name>uLength</name><arity>2</arity><polynomial><variable>2</variable></polynomial></interpret><interpret><name>a__uLength</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><sharp><name>mark</name></sharp><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret><interpret><sharp><name>a__isNatIList</name></sharp><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret><interpret><sharp><name>a__isNatList</name></sharp><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>mark</name><arg><var>X1</var></arg></funapp></arg><arg><funapp><name>mark</name><arg><var>X2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>T</var></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>T</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatIList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>s</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule></rules></dps><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>T</var></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>T</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>and</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>mark</name><arg><var>X1</var></arg></funapp></arg><arg><funapp><name>mark</name><arg><var>X2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>s</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><redPairProc><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><sharp><name>a__and</name></sharp><arity>2</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></interpret><interpret><name>mark</name><arity>1</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></sum></polynomial></interpret><interpret><name>and</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>a__and</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>tt</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>isNatIList</name><arity>1</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret><interpret><name>a__isNatIList</name><arity>1</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret><interpret><name>a__isNatList</name><arity>1</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret><interpret><name>cons</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>a__isNat</name><arity>1</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret><interpret><name>take</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>isNatList</name><arity>1</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret><interpret><name>isNat</name><arity>1</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>length</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>a__length</name><arity>1</arity><polynomial><variable>1</variable></polynomial></interpret><interpret><name>zeros</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__zeros</name><arity>0</arity><polynomial><coefficient><integer>1</integer></coefficient></polynomial></interpret><interpret><name>a__take</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>uTake1</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>a__uTake1</name><arity>1</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>uTake2</name><arity>4</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>4</variable></polynomial></sum></polynomial></interpret><interpret><name>a__uTake2</name><arity>4</arity><polynomial><sum><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>4</variable></polynomial></sum></polynomial></interpret><interpret><name>uLength</name><arity>2</arity><polynomial><variable>2</variable></polynomial></interpret><interpret><name>a__uLength</name><arity>2</arity><polynomial><sum><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><coefficient><integer>0</integer></coefficient></polynomial></interpret><interpret><sharp><name>mark</name></sharp><arity>1</arity><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>T</var></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>T</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>s</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule></rules></dps><dpProof><depGraphProc><component><dps><rules><rule><lhs><funapp><sharp><name>a__and</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>T</var></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>T</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>s</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules/></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss/><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>s</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>mark</name></sharp><arg><var>X</var></arg></funapp></rhs></rule><edge><position>1</position><strict>true</strict><position>1</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component></depGraphProc></dpProof></redPairProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatIList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNatList</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNatList</name></sharp><arg><var>X</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component></depGraphProc></dpProof></redPairProc></dpProof></redPairProc></dpProof></redPairProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>a__isNatList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>a__isNatIList</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>IL</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>isNat</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>X</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>a__isNat</name></sharp><arg><funapp><name>s</name><arg><var>N</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><usableRulesProc><usableRules><rules/></usableRules><dpProof><innermostLhssRemovalProc><innermostLhss/><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>a__isNat</name></sharp><arg><funapp><name>s</name><arg><var>N</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__isNat</name></sharp><arg><var>N</var></arg></funapp></rhs></rule><edge><position>1</position><strict>true</strict><position>1</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></innermostLhssRemovalProc></dpProof></usableRulesProc></dpProof></component></depGraphProc></dpProof></redPairProc></dpProof></redPairProc></dpProof></redPairProc></dpProof></redPairProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>length</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__length</name></sharp><arg><funapp><name>mark</name><arg><var>X</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>uLength</name><arg><var>X1</var></arg><arg><var>X2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__uLength</name></sharp><arg><funapp><name>mark</name><arg><var>X1</var></arg></funapp></arg><arg><var>X2</var></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>a__uLength</name></sharp><arg><funapp><name>tt</name></funapp></arg><arg><var>L</var></arg></funapp></lhs><rhs><funapp><sharp><name>a__length</name></sharp><arg><funapp><name>mark</name><arg><var>L</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>a__length</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__uLength</name></sharp><arg><funapp><name>a__and</name><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatList</name><arg><var>L</var></arg></funapp></arg></funapp></arg><arg><var>L</var></arg></funapp></rhs></rule></rules></dps><realScc>true</realScc><dpProof><redPairProc><orderingConstraintProof><redPair><interpretation><type><matrixInterpretation><domain><arctic><domain><naturals/></domain></arctic></domain><dimension>1</dimension><strictDimension>1</strictDimension></matrixInterpretation></type><interpret><sharp><name>a__uLength</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>tt</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>5</integer></coefficient></vector></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>a__length</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>4</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>mark</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>cons</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__and</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__isNat</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>2</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__isNatList</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>4</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>and</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>isNatIList</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>5</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__isNatIList</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>5</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>take</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>4</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>3</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>isNatList</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>4</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>isNat</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>2</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>2</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>length</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>4</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__length</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>4</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>zeros</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__zeros</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__take</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>4</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>3</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>uTake1</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>4</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__uTake1</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>4</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>uTake2</name><arity>4</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>5</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>4</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>4</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__uTake2</name><arity>4</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>5</integer></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>4</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>4</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>uLength</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>a__uLength</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><minusInfinity/></coefficient></vector></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>0</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><vector><coefficient><integer>1</integer></coefficient></vector></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>3</integer></coefficient></vector></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>nil</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><vector><coefficient><integer>4</integer></coefficient></vector></coefficient></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><dps><rules><rule><lhs><funapp><sharp><name>a__length</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__uLength</name></sharp><arg><funapp><name>a__and</name><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatList</name><arg><var>L</var></arg></funapp></arg></funapp></arg><arg><var>L</var></arg></funapp></rhs></rule></rules></dps><dpProof><sizeChangeProc><subtermCriterion/><sizeChangeGraph><rule><lhs><funapp><sharp><name>a__length</name></sharp><arg><funapp><name>cons</name><arg><var>N</var></arg><arg><var>L</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__uLength</name></sharp><arg><funapp><name>a__and</name><arg><funapp><name>a__isNat</name><arg><var>N</var></arg></funapp></arg><arg><funapp><name>a__isNatList</name><arg><var>L</var></arg></funapp></arg></funapp></arg><arg><var>L</var></arg></funapp></rhs></rule><edge><position>1</position><strict>true</strict><position>2</position></edge></sizeChangeGraph></sizeChangeProc></dpProof></redPairProc></dpProof></component></depGraphProc></dpProof></redPairProc></dpProof></component><component><dps><rules><rule><lhs><funapp><sharp><name>a__take</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><var>IL</var></arg></funapp></lhs><rhs><funapp><sharp><name>a__uTake1</name></sharp><arg><funapp><name>a__isNatIList</name><arg><var>IL</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>zeros</name></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__zeros</name></sharp></funapp></rhs></rule></rules></dps><realScc>false</realScc></component><component><dps><rules><rule><lhs><funapp><sharp><name>mark</name></sharp><arg><funapp><name>uTake1</name><arg><var>X</var></arg></funapp></arg></funapp></lhs><rhs><funapp><sharp><name>a__uTake1</name></sharp><arg><funapp><name>mark</name><arg><var>X</var></arg></funapp></arg></funapp></rhs></rule></rules></dps><realScc>false</realScc></component></depGraphProc></dpProof></dpTrans></trsTerminationProof></proof><origin><proofOrigin><tool><name>AProVE</name><version>AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty </version><strategy>Statistics for single proof: 100.00 % (23 real / 0 unknown / 0 assumptions / 23 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_Innermost Certified 2019-04-01 06.48