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Runtime_Complexity_Innermost_Rewriting 2019-04-01 06.43 pair #433314468
details
property
value
status
complete
benchmark
ordered_better.xml
ran by
Akihisa Yamada
cpu timeout
1200 seconds
wallclock timeout
300 seconds
memory limit
137438953472 bytes
execution host
n037.star.cs.uiowa.edu
space
Frederiksen_Glenstrup
run statistics
property
value
solver
AProVE
configuration
certified
runtime (wallclock)
2.088 seconds
cpu usage
4.91362
user time
4.69977
system time
0.213858
max virtual memory
1.8657488E7
max residence set size
355836.0
stage attributes
key
value
certification-result
CERTIFIED
output-size
36131
starexec-result
WORST_CASE(?,O(n^1))
certification-time
0.1
output
WORST_CASE(?, O(n^1)) <?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" xsi:noNamespaceSchemaLocation="cpf.xsd"><input><complexityInput><trsInput><trs><rules><rule><lhs><funapp><name>ordered</name><arg><funapp><name>Cons</name><arg><var>x'</var></arg><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>ordered[Ite]</name><arg><funapp><name><</name><arg><var>x'</var></arg><arg><var>x</var></arg></funapp></arg><arg><funapp><name>Cons</name><arg><var>x'</var></arg><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>ordered</name><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><funapp><name>Nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>True</name></funapp></rhs></rule><rule><lhs><funapp><name>ordered</name><arg><funapp><name>Nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>True</name></funapp></rhs></rule><rule><lhs><funapp><name>notEmpty</name><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>True</name></funapp></rhs></rule><rule><lhs><funapp><name>notEmpty</name><arg><funapp><name>Nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>False</name></funapp></rhs></rule><rule><lhs><funapp><name>goal</name><arg><var>xs</var></arg></funapp></lhs><rhs><funapp><name>ordered</name><arg><var>xs</var></arg></funapp></rhs></rule></rules></trs><strategy><innermost/></strategy><relativeRules><rules><rule><lhs><funapp><name><</name><arg><funapp><name>S</name><arg><var>x</var></arg></funapp></arg><arg><funapp><name>S</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name><</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></rhs></rule><rule><lhs><funapp><name><</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>S</name><arg><var>y</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>True</name></funapp></rhs></rule><rule><lhs><funapp><name><</name><arg><var>x</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>False</name></funapp></rhs></rule><rule><lhs><funapp><name>ordered[Ite]</name><arg><funapp><name>True</name></funapp></arg><arg><funapp><name>Cons</name><arg><var>x</var></arg><arg><var>xs</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>ordered</name><arg><var>xs</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>ordered[Ite]</name><arg><funapp><name>False</name></funapp></arg><arg><var>xs</var></arg></funapp></lhs><rhs><funapp><name>False</name></funapp></rhs></rule></rules></relativeRules></trsInput><runtimeComplexity><signature><symbol><name>Cons</name><arity>2</arity></symbol><symbol><name>Nil</name><arity>0</arity></symbol><symbol><name>True</name><arity>0</arity></symbol><symbol><name>False</name><arity>0</arity></symbol><symbol><name>S</name><arity>1</arity></symbol><symbol><name>0</name><arity>0</arity></symbol></signature><signature><symbol><name>ordered</name><arity>1</arity></symbol><symbol><name>notEmpty</name><arity>1</arity></symbol><symbol><name>goal</name><arity>1</arity></symbol><symbol><name><</name><arity>2</arity></symbol><symbol><name>ordered[Ite]</name><arity>2</arity></symbol></signature></runtimeComplexity><polynomial>1</polynomial></complexityInput></input><cpfVersion>2.1</cpfVersion><proof><complexityProof><dtTransformation><strictDTs><ruleWithDT><rule><lhs><funapp><name>ordered</name><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>ordered[Ite]</name><arg><funapp><name><</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>ordered</name></sharp><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c5</name><arg><funapp><sharp><name>ordered[Ite]</name></sharp><arg><funapp><name><</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><sharp><name><</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></rhs></rule></ruleWithDT><ruleWithDT><rule><lhs><funapp><name>ordered</name><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>True</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>ordered</name></sharp><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c6</name></funapp></rhs></rule></ruleWithDT><ruleWithDT><rule><lhs><funapp><name>ordered</name><arg><funapp><name>Nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>True</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>ordered</name></sharp><arg><funapp><name>Nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>c7</name></funapp></rhs></rule></ruleWithDT><ruleWithDT><rule><lhs><funapp><name>notEmpty</name><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>True</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>notEmpty</name></sharp><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c8</name></funapp></rhs></rule></ruleWithDT><ruleWithDT><rule><lhs><funapp><name>notEmpty</name><arg><funapp><name>Nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>False</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>notEmpty</name></sharp><arg><funapp><name>Nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>c9</name></funapp></rhs></rule></ruleWithDT><ruleWithDT><rule><lhs><funapp><name>goal</name><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>ordered</name><arg><var>z0</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>goal</name></sharp><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>c10</name><arg><funapp><sharp><name>ordered</name></sharp><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule></ruleWithDT></strictDTs><weakDTs><ruleWithDT><rule><lhs><funapp><name><</name><arg><funapp><name>S</name><arg><var>z0</var></arg></funapp></arg><arg><funapp><name>S</name><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name><</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name><</name></sharp><arg><funapp><name>S</name><arg><var>z0</var></arg></funapp></arg><arg><funapp><name>S</name><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c</name><arg><funapp><sharp><name><</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></rhs></rule></ruleWithDT><ruleWithDT><rule><lhs><funapp><name><</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>S</name><arg><var>z0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>True</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name><</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>S</name><arg><var>z0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c1</name></funapp></rhs></rule></ruleWithDT><ruleWithDT><rule><lhs><funapp><name><</name><arg><var>z0</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>False</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name><</name></sharp><arg><var>z0</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>c2</name></funapp></rhs></rule></ruleWithDT><ruleWithDT><rule><lhs><funapp><name>ordered[Ite]</name><arg><funapp><name>True</name></funapp></arg><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>ordered</name><arg><var>z1</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>ordered[Ite]</name></sharp><arg><funapp><name>True</name></funapp></arg><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c3</name><arg><funapp><sharp><name>ordered</name></sharp><arg><var>z1</var></arg></funapp></arg></funapp></rhs></rule></ruleWithDT><ruleWithDT><rule><lhs><funapp><name>ordered[Ite]</name><arg><funapp><name>False</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>False</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>ordered[Ite]</name></sharp><arg><funapp><name>False</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>c4</name></funapp></rhs></rule></ruleWithDT></weakDTs><innermostLhss><funapp><sharp><name><</name></sharp><arg><funapp><name>S</name><arg><var>z0</var></arg></funapp></arg><arg><funapp><name>S</name><arg><var>z1</var></arg></funapp></arg></funapp><funapp><sharp><name><</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>S</name><arg><var>z0</var></arg></funapp></arg></funapp><funapp><sharp><name><</name></sharp><arg><var>z0</var></arg><arg><funapp><name>0</name></funapp></arg></funapp><funapp><sharp><name>ordered[Ite]</name></sharp><arg><funapp><name>True</name></funapp></arg><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp><funapp><sharp><name>ordered[Ite]</name></sharp><arg><funapp><name>False</name></funapp></arg><arg><var>z0</var></arg></funapp><funapp><sharp><name>ordered</name></sharp><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp><funapp><sharp><name>ordered</name></sharp><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Nil</name></funapp></arg></funapp></arg></funapp><funapp><sharp><name>ordered</name></sharp><arg><funapp><name>Nil</name></funapp></arg></funapp><funapp><sharp><name>notEmpty</name></sharp><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp><funapp><sharp><name>notEmpty</name></sharp><arg><funapp><name>Nil</name></funapp></arg></funapp><funapp><sharp><name>goal</name></sharp><arg><var>z0</var></arg></funapp></innermostLhss><complexityProof><usableRules><nonUsableRules><rules><rule><lhs><funapp><name>ordered[Ite]</name><arg><funapp><name>True</name></funapp></arg><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>ordered</name><arg><var>z1</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>ordered[Ite]</name><arg><funapp><name>False</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>False</name></funapp></rhs></rule><rule><lhs><funapp><name>ordered</name><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>ordered[Ite]</name><arg><funapp><name><</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name>ordered</name><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>True</name></funapp></rhs></rule><rule><lhs><funapp><name>ordered</name><arg><funapp><name>Nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>True</name></funapp></rhs></rule><rule><lhs><funapp><name>notEmpty</name><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>True</name></funapp></rhs></rule><rule><lhs><funapp><name>notEmpty</name><arg><funapp><name>Nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>False</name></funapp></rhs></rule><rule><lhs><funapp><name>goal</name><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>ordered</name><arg><var>z0</var></arg></funapp></rhs></rule></rules></nonUsableRules><complexityProof><ruleShifting><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>c</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c1</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c2</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c3</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c4</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c5</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c6</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c7</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c8</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c9</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c10</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name><</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name><</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>ordered[Ite]</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>ordered</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>notEmpty</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>goal</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>True</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>False</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>Cons</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>Nil</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><trs><rules><rule><lhs><funapp><sharp><name>ordered</name></sharp><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c6</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>ordered</name></sharp><arg><funapp><name>Nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>c7</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>notEmpty</name></sharp><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c8</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>notEmpty</name></sharp><arg><funapp><name>Nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>c9</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>goal</name></sharp><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>c10</name><arg><funapp><sharp><name>ordered</name></sharp><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><usableRules><rules><rule><lhs><funapp><sharp><name><</name></sharp><arg><funapp><name>S</name><arg><var>z0</var></arg></funapp></arg><arg><funapp><name>S</name><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c</name><arg><funapp><sharp><name><</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name><</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>S</name><arg><var>z0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c1</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name><</name></sharp><arg><var>z0</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>c2</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>ordered[Ite]</name></sharp><arg><funapp><name>True</name></funapp></arg><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c3</name><arg><funapp><sharp><name>ordered</name></sharp><arg><var>z1</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>ordered[Ite]</name></sharp><arg><funapp><name>False</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>c4</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>ordered</name></sharp><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c5</name><arg><funapp><sharp><name>ordered[Ite]</name></sharp><arg><funapp><name><</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><sharp><name><</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>ordered</name></sharp><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c6</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>ordered</name></sharp><arg><funapp><name>Nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>c7</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>notEmpty</name></sharp><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c8</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>notEmpty</name></sharp><arg><funapp><name>Nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>c9</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>goal</name></sharp><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>c10</name><arg><funapp><sharp><name>ordered</name></sharp><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><name><</name><arg><var>z0</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>False</name></funapp></rhs></rule><rule><lhs><funapp><name><</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>S</name><arg><var>z0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>True</name></funapp></rhs></rule><rule><lhs><funapp><name><</name><arg><funapp><name>S</name><arg><var>z0</var></arg></funapp></arg><arg><funapp><name>S</name><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name><</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></rhs></rule></rules></usableRules><complexityProof><ruleShifting><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><naturals/></domain><degree>1</degree></polynomial></type><interpret><name>c</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c1</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c2</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c3</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c4</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c5</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c6</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c7</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c8</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c9</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c10</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name><</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name><</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>ordered[Ite]</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>ordered</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>notEmpty</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>goal</name></sharp><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>2</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>S</name><arity>1</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>True</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>False</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>3</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>Cons</name><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>Nil</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><trs><rules><rule><lhs><funapp><sharp><name>ordered</name></sharp><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c5</name><arg><funapp><sharp><name>ordered[Ite]</name></sharp><arg><funapp><name><</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><sharp><name><</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><usableRules><rules><rule><lhs><funapp><sharp><name><</name></sharp><arg><funapp><name>S</name><arg><var>z0</var></arg></funapp></arg><arg><funapp><name>S</name><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c</name><arg><funapp><sharp><name><</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name><</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>S</name><arg><var>z0</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c1</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name><</name></sharp><arg><var>z0</var></arg><arg><funapp><name>0</name></funapp></arg></funapp></lhs><rhs><funapp><name>c2</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>ordered[Ite]</name></sharp><arg><funapp><name>True</name></funapp></arg><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c3</name><arg><funapp><sharp><name>ordered</name></sharp><arg><var>z1</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>ordered[Ite]</name></sharp><arg><funapp><name>False</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>c4</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>ordered</name></sharp><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c5</name><arg><funapp><sharp><name>ordered[Ite]</name></sharp><arg><funapp><name><</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Cons</name><arg><var>z1</var></arg><arg><var>z2</var></arg></funapp></arg></funapp></arg></funapp></arg><arg><funapp><sharp><name><</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>ordered</name></sharp><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><funapp><name>Nil</name></funapp></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c6</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>ordered</name></sharp><arg><funapp><name>Nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>c7</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>notEmpty</name></sharp><arg><funapp><name>Cons</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c8</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>notEmpty</name></sharp><arg><funapp><name>Nil</name></funapp></arg></funapp></lhs><rhs><funapp><name>c9</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>goal</name></sharp><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>c10</name><arg><funapp><sharp><name>ordered</name></sharp><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><complexityProof><rIsEmpty/></complexityProof></ruleShifting></complexityProof></ruleShifting></complexityProof></usableRules></complexityProof></dtTransformation></complexityProof></proof><origin><proofOrigin><tool><name>AProVE</name><version>AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty </version><strategy>Statistics for single proof: 100.00 % (5 real / 0 unknown / 0 assumptions / 5 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>
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return to Runtime_Complexity_Innermost_Rewriting 2019-04-01 06.43
