Runtime_Complexity_Innermost_Rewriting 2019-04-01 06.43 pair #433314418

details
property	value
status	complete
benchmark	#3.53b.xml
ran by	Akihisa Yamada
cpu timeout	1200 seconds
wallclock timeout	300 seconds
memory limit	137438953472 bytes
execution host	n018.star.cs.uiowa.edu
space	AG01
run statistics
property	value
solver	AProVE
configuration	certified
runtime (wallclock)	2.02597 seconds
cpu usage	4.77257
user time	4.55888
system time	0.213691
max virtual memory	1.865566E7
max residence set size	330400.0
stage attributes
key	value
certification-result	CERTIFIED
output-size	20139
starexec-result	WORST_CASE(?,O(n^2))
certification-time	0.1
output
WORST_CASE(?, O(n^2))
<?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>g</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>x</var></rhs></rule><rule><lhs><funapp><name>g</name><arg><var>x</var></arg><arg><var>y</var></arg></funapp></lhs><rhs><var>y</var></rhs></rule><rule><lhs><funapp><name>f</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>1</name></funapp></arg><arg><var>x</var></arg></funapp></lhs><rhs><funapp><name>f</name><arg><funapp><name>s</name><arg><var>x</var></arg></funapp></arg><arg><var>x</var></arg><arg><var>x</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>f</name><arg><var>x</var></arg><arg><var>y</var></arg><arg><funapp><name>s</name><arg><var>z</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>f</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>1</name></funapp></arg><arg><var>z</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><strategy><innermost/></strategy></trsInput><runtimeComplexity><signature><symbol><name>0</name><arity>0</arity></symbol><symbol><name>1</name><arity>0</arity></symbol><symbol><name>s</name><arity>1</arity></symbol></signature><signature><symbol><name>g</name><arity>2</arity></symbol><symbol><name>f</name><arity>3</arity></symbol></signature></runtimeComplexity><polynomial>2</polynomial></complexityInput></input><cpfVersion>2.1</cpfVersion><proof><complexityProof><dtTransformation><strictDTs><ruleWithDT><rule><lhs><funapp><name>g</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></lhs><rhs><var>z0</var></rhs></rule><rule><lhs><funapp><sharp><name>g</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><name>c</name></funapp></rhs></rule></ruleWithDT><ruleWithDT><rule><lhs><funapp><name>g</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></lhs><rhs><var>z1</var></rhs></rule><rule><lhs><funapp><sharp><name>g</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><name>c1</name></funapp></rhs></rule></ruleWithDT><ruleWithDT><rule><lhs><funapp><name>f</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>1</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>f</name><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg><arg><var>z0</var></arg><arg><var>z0</var></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>f</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>1</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>c2</name><arg><funapp><sharp><name>f</name></sharp><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg><arg><var>z0</var></arg><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule></ruleWithDT><ruleWithDT><rule><lhs><funapp><name>f</name><arg><var>z0</var></arg><arg><var>z1</var></arg><arg><funapp><name>s</name><arg><var>z2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>f</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>1</name></funapp></arg><arg><var>z2</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>f</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg><arg><funapp><name>s</name><arg><var>z2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c3</name><arg><funapp><sharp><name>f</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>1</name></funapp></arg><arg><var>z2</var></arg></funapp></arg></funapp></rhs></rule></ruleWithDT></strictDTs><weakDTs/><innermostLhss><funapp><sharp><name>g</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp><funapp><sharp><name>g</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp><funapp><sharp><name>f</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>1</name></funapp></arg><arg><var>z0</var></arg></funapp><funapp><sharp><name>f</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg><arg><funapp><name>s</name><arg><var>z2</var></arg></funapp></arg></funapp></innermostLhss><complexityProof><usableRules><nonUsableRules><rules><rule><lhs><funapp><name>g</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></lhs><rhs><var>z0</var></rhs></rule><rule><lhs><funapp><name>g</name><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></lhs><rhs><var>z1</var></rhs></rule><rule><lhs><funapp><name>f</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>1</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>f</name><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg><arg><var>z0</var></arg><arg><var>z0</var></arg></funapp></rhs></rule><rule><lhs><funapp><name>f</name><arg><var>z0</var></arg><arg><var>z1</var></arg><arg><funapp><name>s</name><arg><var>z2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>s</name><arg><funapp><name>f</name><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>1</name></funapp></arg><arg><var>z2</var></arg></funapp></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>0</arity><polynomial><sum><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>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>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><sharp><name>g</name></sharp><arity>2</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>f</name></sharp><arity>3</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>1</integer></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></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>1</name><arity>0</arity><polynomial><sum><polynomial><product><polynomial><coefficient><integer>0</integer></coefficient></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></interpretation></redPair></orderingConstraintProof><trs><rules><rule><lhs><funapp><sharp><name>g</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><name>c</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>g</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><name>c1</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>f</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg><arg><funapp><name>s</name><arg><var>z2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c3</name><arg><funapp><sharp><name>f</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>1</name></funapp></arg><arg><var>z2</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><usableRules><rules><rule><lhs><funapp><sharp><name>g</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><name>c</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>g</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><name>c1</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>f</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>1</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>c2</name><arg><funapp><sharp><name>f</name></sharp><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg><arg><var>z0</var></arg><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>f</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg><arg><funapp><name>s</name><arg><var>z2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c3</name><arg><funapp><sharp><name>f</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>1</name></funapp></arg><arg><var>z2</var></arg></funapp></arg></funapp></rhs></rule></rules></usableRules><complexityProof><ruleShifting><orderingConstraintProof><redPair><interpretation><type><polynomial><domain><matrices><dimension>2</dimension><strictDimension>1</strictDimension><domain><rationals><delta><rational><numerator>1</numerator><denominator>1</denominator></rational></delta></rationals></domain></matrices></domain><degree>1</degree></polynomial></type><interpret><name>c</name><arity>0</arity><polynomial><sum><polynomial><coefficient><matrix><vector><coefficient><rational><numerator>4</numerator><denominator>1</denominator></rational></coefficient><coefficient><rational><numerator>4</numerator><denominator>1</denominator></rational></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial></sum></polynomial></interpret><interpret><sharp><name>g</name></sharp><arity>2</arity><polynomial><sum><polynomial><coefficient><matrix><vector><coefficient><rational><numerator>4</numerator><denominator>1</denominator></rational></coefficient><coefficient><rational><numerator>4</numerator><denominator>1</denominator></rational></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><rational><numerator>1</numerator><denominator>1</denominator></rational></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><rational><numerator>1</numerator><denominator>1</denominator></rational></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c1</name><arity>0</arity><polynomial><sum><polynomial><coefficient><matrix><vector><coefficient><rational><numerator>4</numerator><denominator>1</denominator></rational></coefficient><coefficient><rational><numerator>4</numerator><denominator>1</denominator></rational></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial></sum></polynomial></interpret><interpret><name>c3</name><arity>1</arity><polynomial><sum><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><rational><numerator>1</numerator><denominator>1</denominator></rational></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><sharp><name>f</name></sharp><arity>3</arity><polynomial><sum><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><rational><numerator>2</numerator><denominator>1</denominator></rational></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>2</variable></polynomial></product></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><rational><numerator>1</numerator><denominator>1</denominator></rational></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>3</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>c2</name><arity>1</arity><polynomial><sum><polynomial><coefficient><matrix><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><rational><numerator>1</numerator><denominator>1</denominator></rational></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>s</name><arity>1</arity><polynomial><sum><polynomial><coefficient><matrix><vector><coefficient><rational><numerator>4</numerator><denominator>1</denominator></rational></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><product><polynomial><coefficient><matrix><vector><coefficient><rational><numerator>1</numerator><denominator>1</denominator></rational></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><rational><numerator>2</numerator><denominator>1</denominator></rational></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial><polynomial><variable>1</variable></polynomial></product></polynomial></sum></polynomial></interpret><interpret><name>0</name><arity>0</arity><polynomial><sum><polynomial><coefficient><matrix><vector><coefficient><rational><numerator>4</numerator><denominator>1</denominator></rational></coefficient><coefficient><rational><numerator>2</numerator><denominator>1</denominator></rational></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial></sum></polynomial></interpret><interpret><name>1</name><arity>0</arity><polynomial><sum><polynomial><coefficient><matrix><vector><coefficient><rational><numerator>3</numerator><denominator>1</denominator></rational></coefficient><coefficient><integer>0</integer></coefficient></vector><vector><coefficient><integer>0</integer></coefficient><coefficient><integer>0</integer></coefficient></vector></matrix></coefficient></polynomial></sum></polynomial></interpret></interpretation></redPair></orderingConstraintProof><trs><rules><rule><lhs><funapp><sharp><name>f</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>1</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>c2</name><arg><funapp><sharp><name>f</name></sharp><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg><arg><var>z0</var></arg><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule></rules></trs><usableRules><rules><rule><lhs><funapp><sharp><name>g</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><name>c</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>g</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg></funapp></lhs><rhs><funapp><name>c1</name></funapp></rhs></rule><rule><lhs><funapp><sharp><name>f</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>1</name></funapp></arg><arg><var>z0</var></arg></funapp></lhs><rhs><funapp><name>c2</name><arg><funapp><sharp><name>f</name></sharp><arg><funapp><name>s</name><arg><var>z0</var></arg></funapp></arg><arg><var>z0</var></arg><arg><var>z0</var></arg></funapp></arg></funapp></rhs></rule><rule><lhs><funapp><sharp><name>f</name></sharp><arg><var>z0</var></arg><arg><var>z1</var></arg><arg><funapp><name>s</name><arg><var>z2</var></arg></funapp></arg></funapp></lhs><rhs><funapp><name>c3</name><arg><funapp><sharp><name>f</name></sharp><arg><funapp><name>0</name></funapp></arg><arg><funapp><name>1</name></funapp></arg><arg><var>z2</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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actions all output return to Runtime_Complexity_Innermost_Rewriting 2019-04-01 06.43